seventh international conference on...

EUROPEAN POLYTECHNICAL INSTITUTE, KUNOVICE

P R O C E E D I N G S

SEVENTH INTERNATIONAL CONFERENCE ON SOFT COMPUTING APPLIED IN COMPUTER AND

ECONOMIC ENVIRONMENTS

ICSC 2009

January 23, Hodonín, Czech Republic

Edited by:

Prof. Ing. Imrich Rukovanský, CSc., Doc. Ing. Pavel Ošmera, CSc.

Prepared for print by:

Ing. Andrea Šimonová, DiS.

Martin Tuček

Printed by:

© European Polytechnical Institute Kunovice, 2009

ISBN : 978-80-7314-163-9

SEVENTH INTERNATIONAL CONFERENCE ON SOFT COMPUTING

APPLIED IN COMPUTER AND ECONOMIC ENVIRONMENTS

ICSC 2009

Organized by

THE EUROPEAN POLYTECHNICAL INSTITUTE, KUNOVICE THE CZECH REPUBLIC

Conference Chairman

H. prof. Ing. Oldřich Kratochvíl, Dr.h.c., MBA

rector

Conference Co-Chairmen

Prof. Ing. Imrich Rukovanský, CSc.

Assoc. Prof. Ing. Pavel Ošmera, CSc.

INTERNATIONAL PROGRAMME COMMITEE

O. Kratochvíl – Chairman (CZ) W. M. Baraňski (Poland) J. Baštinec (Czech Republic) J. Brzobohatý (Czech Republic) J. Ďaďo (Slovak Republic) J. Diblík (Czech Republic) P. Dostál (Czech Republic) T. Dostál (Czech Republic) U. K. Chakraborthy (USA) M. Kubát (USA) B. Kulcsár (Hungary)

K. Matiaško (Slovak Republic) P. Ošmera (Czech Republic) J. Petrucha (Czech Republic) K. Rais (Czech Republic) I. Rukovanský (Czech Republic) G. N. Smirnov (Russian) J. Strišš (Slovak Republic) G. Vértesy (Hungary) W. Zamojski (Poland) J. Zapletal (Czech Republic) T. Walkowiak (Poland)

ORGANIZING COMMITEE

I. Rukovanský (Chairman) P. Ošmera J. Šáchová A. Šimonová I. Matušíková J. Kavka Z. Omelková

Z. Pospíšilová M. Zálešák T. Chmela J. Míšek M. Balus P. Trnečka

Session 1: ICSC – Soft Computing a jeho uplatnění v managementu, marketingu a v moderních finančních systémech

Chairman: Doc. Ing. Petr Dostál, CSc.

Session 2: ICSC – Soft Computing – tvorba moderních počítačových nástrojů pro optimalizaci procesů

Chairman: Doc. Ing. Pavel Ošmera, CSc.; Ing. Jindřich Petrucha, Ph.D.

Oponentní rada

Doc. RNDr. Jaromír Baštinec, CSc. – Vysoké učení technické v Brně

Doc. Wlodzimierz M. Baraňski – University of Technology, Wroclaw

Doc. Ing. Petr Dostál, CSc. – Vysoké učení technické v Brně

„ICSC– SEVENTH INTERNATIONAL CONFERENCE ON SOFT COMPUTING APPLIED IN COMPUTER AND ECONOMIC

ENVIRONMENTS” EPI Kunovice, Czech Republic. January 23, 2009. 5

OBSAH

A MESSAGE FROM THE GENERAL CHAIRMAN OF THE CONFERENCE............................................................ 7

THE DECISION MAKING UNDER LAW UNCERTAINTY BY MEANS OF FUZZY LOGIC .............. 11 PETR DOSTÁL ........................................................................................................................................................ 11 DISTRIBUTION PROBLEM SOLVED BY GENETIC ALGORITHM ...................................................... 19 PETR DOSTÁL, OLDŘICH KRATOCHVÍL.................................................................................................................. 19 METHOD OF USING SEASONALITY FOR DETERMINATION OF TOURIST-ECREATION SYSTEM DEVELOPMENT STRATEGY ....................................................................................................... 25 SERHIY LYAKHOV.................................................................................................................................................. 25 GRANULAR NEURAL NETWORK IN MANAGERIAL FORECASTING SYSTEMS............................ 29 ZUZANA MEČIAROVÁ, JÁN BÁBEL, LUCIA PANČÍKOVÁ......................................................................................... 29 SOFT COMPUTING AND ITS APPLICABILITY FOR ESTIMATING THE EFFICIENCY OF FORMING AND USING THE CAPITAL OF FOOD RETAIL INDUSTRY ENTERPRISES.................. 35 JULIYA VERBITSKAYA ........................................................................................................................................... 35 MODELING METHODOLOGY OF SOCIO-ECONOMIC OBJECTS FLUCTUATION GROWS ........ 39 YAROSLAV VYKLYUK ........................................................................................................................................... 39 THE USE OF FUZZY LOGIC FOR SEASONAL RECREATION ATTRACTIVENESS OF TERRITORY CARTOGRAPHIC DESIGN .................................................................................................... 45 OLGA ARTEMENKO................................................................................................................................................ 45 FORECASTING OF ECONOMIC QUANTITIES USING FUZZY AUTOREGRESSIVE - MODEL AND SOFT RBF NEURAL NETWORK ................................................................................................................... 51 DUŠAN MARČEK.................................................................................................................................................... 51 APPROXIMATION AND FORECASTIG ABILITY OF VARIOUS RBF AND GRANULAR NNW: APPLICATION TO SALES PROCESS MODELLING................................................................................. 57 MILAN MARČEK .................................................................................................................................................... 57 USING DATAMINING AND OLAP TECHNOLOGY AT INFORMATION SYSTEM ............................ 63 JINDŘICH PETRUCHA, DAN SLOVÁČEK .................................................................................................................. 63 IMAGE PROCESSING FOR DETERMINATION OF SURFACE PARAMETERS.................................. 69 JIŘÍ ŠŤASTNÝ, PETR LUDÍK, MILAN ŠTENCL.......................................................................................................... 69 VARIABLE LATERAL SILICON CONTROLLED RECTIFIER FOR IC’S ESD PROTECTION DESIGN ............................................................................................................................................................... 75 PETR BĚŤÁK, JAROMÍR BRZOBOHATÝ, VLADISLAV MUSIL ................................................................................... 75 THE METHODOLOGY OF VALUE DETERMINATION OF CLIENT FOR INSURANCE BUSINESS.............................................................................................................................................................................. 83 RADEK DOSKOČIL.................................................................................................................................................. 83 USING NE URELA NETWORKS AT FINANCIAL ARREA....................................................................... 91 JINDŘICH PETRUCHA.............................................................................................................................................. 91 PARALLEL GRAMMATICAL EVOLUTION FOR OPTIMIZATION OF COMPUTER NETWORK STRUCTURE...................................................................................................................................................... 95 PAVEL OŠMERA, IMRICH RUKOVANSKÝ ................................................................................................................ 95 INDUCTION IN MULTI-LABEL TEXT CATEGORIZATION DOMAINS ............................................ 101 SAREEWAN DENDAMRONGVIT, MIROSLAV KUBAT, ZEYNEL SENDUR................................................................. 101 STATISTICAL AND SOFT COMPUTING METHODS IN CROSS RATES MODELING .................... 107 JÁN BÁBEL, ZUZANA MEČIAROVÁ, LUCIA PANČÍKOVÁ....................................................................................... 107



CELULÁRNÍ PROCESOR LOGICKÝCH FUNKCÍ PRO POUŽITÍ PŘI GENETICKÉM PROGRAMOVÁNÍ A EVOLUČNÍM VÝVOJI ALGORITMŮ.................................................................. 113 PETR SKORKOVSKÝ ............................................................................................................................................. 113 INSPECTION OF STEEL DEGRADATION BY MAGNETIC ADAPTIVE TESTING .......................... 119 GÁBOR VÉRTESY, IVAN TOMÁŠ........................................................................................................................... 119 ON AN EXAMPLE OF NON-LINEAR PROGRAMMING......................................................................... 123 MARIE TOMŠOVÁ ................................................................................................................................................ 123 MODELING OF HIGH ORDER TRANSFER FUNCTION IN CURRENT MODE ................................ 129 TOMÁŠ DOSTÁL................................................................................................................................................... 129 ANALOG BEHAVIORAL MODELING OF THE MULTIFUNCTIONAL FILTER WITH CURRENT CONVEYORS ................................................................................................................................................... 135 JOSEF SLEZÁK, ROMAN ŠOTNER .......................................................................................................................... 135 NONLINEAR CIRCUIT SYNTHESIS USING MATHCAD ....................................................................... 139 JIŘÍ PETRŽELA...................................................................................................................................................... 139 STRUCTURE OF GRAVITATION................................................................................................................ 145 PAVEL OŠMERA ................................................................................................................................................... 145 IMAGE PROCESSING BY MEANS OF LABVIEW ENVIRONMENT.................................................... 153 JIŘÍ LIŠKA, JIŘÍ ŠŤASTNÝ, MIROSLAV CEPL......................................................................................................... 153 PRIORITY SWITCHING BY MEANS OF NEURAL NETWORK ........................................................... 159 JIŘÍ STASTNY, P. POKORNY, O. POPELKA ............................................................................................................ 159

EXISTENCE OF POSITIVE SOILUTIONS OF DISCRETE LINEAR DELAYED EQUATIONS .............. 163

JAROMÍR BAŠTINEC, JOSEF DIBLÍK, ZDENĚK ŠMARDA .........................................................................................163

GENERATED FUZZY IMPLICATORS ..........................................................................................................175

VLADISLAV BIBA..................................................................................................................................................175

EXISTENCE AND UNIQUENESS OF SOLUTIONS OF FRACTIONAL DELAY DIFFERENTIAL

EQUATIONS ......................................................................................................................................................181

BŘETISLAV FAJMON .............................................................................................................................................181

ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF CERTAIN CLASSES OF NONLINEAR SINGULAR

INTEGRODIFFERENTIAL EQUATIONS.......................................................................................................185

OLGA FILIPOVÁ, ZDENĚK ŠMARDA.......................................................................................................................185

P-SEMIHYPERGROUPS OF PREFERENCE RELATIONS AND ASSOCIATED BITOPOLOGIES.........189

JAN CHVALINA, MICHAL NOVÁK..........................................................................................................................189

ASYMPTOTIC BEHAVIOR OF ONE AUXILIARY SYSTÉM .....................................................................195

ZDENĚK SVOBODA................................................................................................................................................195

DECOMPOSITION METHOD OF SOLVING OF INTEGRODIFFERENTIAL EQUATIONS...................201

ZDENĚK ŠMARDA .................................................................................................................................................201

ON THE TOLERANCE RELATION Τ ON (β) LATTICES ...........................................................................207

JOSEF ZAPLETAL...................................................................................................................................................207

INVESTIGATION ON OPTIMIZATION OF CHEMICAL PROCESS.........................................................221

TRAN TRONG DAO, IVAN ZELINKA.......................................................................................................................221



Prof. Ing. Imrich Rukovanský, CSc.

c.

Rukovanský, CSc.

c.

A MESSAGE FROM THE GENERAL CHAIRMAN OF THE CONFERENCE A MESSAGE FROM THE GENERAL CHAIRMAN OF THE CONFERENCE Dear guests and participants at this conference. Dear guests and participants at this conference.

Let me welcome you at the 7th International Conference on Soft Computing Applied in Computer and Economic Environment ICSC 2009. During last seven years the every year conference has appeared to become an important meeting for introduction the latest knowledge and results of collaborating universities and work places involved in modern optimizing methods and tools of soft computing such as fuzzy control, evolutional algorithms, usage of neuron web etc. We witness that the

papers from this conference are cited at many international conferences abroad which make our school penetrate into the subconscious of broader professional

Let me welcome you at the 7th International Conference on Soft Computing Applied in Computer and Economic Environment ICSC 2009. During last seven years the every year conference has appeared to become an important meeting for introduction the latest knowledge and results of collaborating universities and work places involved in modern optimizing methods and tools of soft computing such as fuzzy control, evolutional algorithms, usage of neuron web etc. We witness that the

papers from this conference are cited at many international conferences abroad which make our school penetrate into the subconscious of broader professional publi

publi

Due to this reality the writers not only from the Czech Republic, but also from Russia, the Slovak republic, Poland, Ukraine, Hungary, USA, read papers at our conference.

Due to this reality the writers not only from the Czech Republic, but also from Russia, the Slovak republic, Poland, Ukraine, Hungary, USA, read papers at our conference.

As in every year the papers are divided into two groups. In the first one the focus is on the soft computing and its application in marketing management and in the modern financial systems the other one concentrates on modern computer tools for the process optimizing.

As in every year the papers are divided into two groups. In the first one the focus is on the soft computing and its application in marketing management and in the modern financial systems the other one concentrates on modern computer tools for the process optimizing. Dear guests, I believe that this anniversary seventh ICSC 2009 will support the further depth of contacts and information exchange among the collaborating universities and other institutions both at home and abroad in the area of the development of the modern optimizing methods and application opportunities of soft computing.

Dear guests, I believe that this anniversary seventh ICSC 2009 will support the further depth of contacts and information exchange among the collaborating universities and other institutions both at home and abroad in the area of the development of the modern optimizing methods and application opportunities of soft computing. Kunovice, January 23, 2009 Oldřich Kratochvíl Kunovice, January 23, 2009 Oldřich Kratochvíl Honorary professor, Ing., Dr.h.c., MBA Honorary professor, Ing., Dr.h.c., MBA rector rector

H. prof., Ing. Oldřich Kratochvíl, Dr.h.c., MBA



SESSION 1



THE DECISION MAKING UNDER LAW UNCERTAINTY BY MEANS OF FUZZY LOGIC

Petr Dostál

Brno University of Technology

Abstract: The article deals with the use of fuzzy logic at support of manager decision making. The brief description of fuzzy logic and the process of calculation are mentioned. The use is demonstrated on the example such as the decision making whether to accept, entertain or reject the client of law office. The scheme of models, rule blocks, attributes and their membership functions are mentioned. The use of fuzzy logic is the advantage especially at decision making processes at law uncertainty. Keywords: Decision making, law, uncertainty, fuzzy logic, Matlab

1. INTRODUCTION The use of fuzzy logic is the advantage especially at decision making processes where the description by algorithms is very difficult and criteria are multiplied. The advantage is that the linguistic variables are used. The fuzzy logic measures the certainty or uncertainty of membership of element of the set. Analogously the man makes decision during the mental and physical activities. The solution of certain case is found on the principle of rules that were defined by fuzzy logics for similar cases. The fuzzy logics belong among methods that are used in the area of decision making of firms and offices. 2. FUZZY LOGIC The calculation of fuzzy logics consists of three steps: fuzzification, fuzzy inference and defuzzification. The fuzzification means that the real variables are transferred on linguistic variables. The definition of

linguistic variable goes out from basic linguistic variables, for example, at the variable profit is set up the following attributes: none, very low, low, medium, high, very high profit. Usually there are used from three to seven attributes of variable. The attributes are defined by the so called membership function, such as , Z, S and some others. The membership function is set up for input and output variables.

The fuzzy inference defines the behavior of system by means of rules of type <When>, <Then> on linguistic level. The conditional clauses evaluate the state of input variables by the rules. The conditional clauses are in the form

<When> Inputa <And> Inputb ….. Inputx <Or> Inputy …….. <Then> Output1 , <With> x

it means, when (the state occurs) Inputa and Inputb, ….., Inputx or Inputy, ……, then (the situation is) Output1 with x. The fuzzy logic represents the expert systems. Each combination of attributes of variables, incoming into the system and occurring in condition <When>, <Then>, presents one rule. Every condition behind <When> has a corresponding result behind <Then>. It is necessary to determine every rule and its degree of supports (the weight of rule in the system). The rules are created by the expert himself.

The defuzzification transfers the results of fuzzy inference on the output variables, that describe the results

verbally (for example, whether to accept or reject the client). The system with fuzzy logics can work as an automatic system with entering input data. The input data can be represented by many variables. 3. CASE STUDY Let us mention an example, when it will be evaluated whether the client of law office will be accepted – (Accept), entertained the acceptance – (Entertain) or rejected (Reject) on the basis of inputs. The input variables and their attributes are: Profit - low, medium and high profit from case, Time - low, medium and high of time costing ness, Professional - low, medium, high professional requirements, Value - small, medium high value of client, Justification - low, medium, high justification of claim. The task will be solved by means of MATLAB



program using Fuzzy Logic Toolbox. It is suitable to set up an executable program to realize the mentioned task. The sequence of orders is as follows. See prog.1.

b2 = readfis('LO.fis'); Data = input('Set up the values in an order Profit; Time; Professional; Value; Justification [Profit; Time; Professional; Value; Justification]: '); result = evalfis(Data, b2); fuzzy(b2) mfedit(b2) ruleedit(b2) surfview(b2) ruleview(b2) result if result<0.20 'Accept' elseif result<0.50 'Entertain' else 'Reject' end

Prog. 1 File LO.m The first row reads the parameters of fuzzy model into variable b2 by order readfis from file LO.fis. The second row reads the five input variables such as Profit, Time, Professional, Value and Justification. The fourth row contains the order evalfis, that evaluates the input variables Data and b2. The output is in variable result. The following order fuzzy(b2) enables the set up of fuzzy model, the order mfedit(b2) enables the set up of membership functions, the order ruleedit(b2) enables the set up fuzzy rules, the order surfview(b2) enables the graphical search of dependence of input and output variables and order ruleview(b2) enables to evaluate the output on the basis of inputs. The order result writes off the final value. The following four rows make a vague result. If the output value result is in the range R(0,20>, the result is to Accept the client. If the output value result is in the range R(20,50>, the result is to Entertain the client. If the output value result is in the range R (50,100), the result is to Reject the client. The order fuzzy(b2) enables the set up of fuzzy model. See fig.1. The order mfedit(b2) enables the set up the membership functions of variables Profit, Time, Professional, Value, Justification and Result.

Fig. 1 Order mfedit(b2) – AK

The order mfedit(b2) enables the set up of membership functions. Fig. 2 defines the attributes and membership functions for Profit - low (LP), medium (MP) and high (HP) profit from case. Fig. 3 defines the attributes and



membership functions for Time - low (LT), medium (MT) and (HT) high of time costing ness. Fig. 4 defines the attributes and membership functions for Professional - low (LR), medium (MR), high (HR) Professional requirements. Fig. 5 defines the attributes and membership functions for Value low (LV), medium (MV) high (HV) value of a client. Fig. 6 defines the attributes and membership functions for Justification - low (LJ), medium (MJ) and high (HJ) justification of claim.

Fig. 2 Order mfedit(b2) – Profit

Fig. 3 Order mfedit(b2) – Time



Fig. 4 Order mfedit(b2) – Professional

Fig. 5 Order mfedit(b2) – Value

Fig. 6 Order mfedit(b2) – Justification

Fig. 7 defines the attributes and membership functions for Result Accept (RA), Entertain (RE) or Rejected (RR) the client



Fig. 7 Order mfedit(b2) – Result

The way of set up of the first rule is on the fig. 8 and its verbal interpretation is as follows: If the profit is low (Profit = LP) and together the time costing ness is high (Time = HT) and together professional requirements is high (Professional = HR) and together the value of an client is low (Value = LV) and together the justification of a claim is low (Justification = LJ), then the result of an client is its rejection (Result = RR) with weight 1. The other rules are set up accordingly. The created rules are possible to check in the table of rules at upper part of display. It is necessary to set up such number of rules that describe the solved task. The order surfview(b2) enables the graphical search of dependence of input and output variables, for example the dependence of Result on Profit and Justification. See fig. 9.

Fig. 8 Order ruleedit(b2)



Fig. 9 Order surfview(b2) – Result = f(Profit, Justification)

The order ruleview(b2) enables to evaluate output Result on the basis of inputs Profit, Time, Professional, Value and Justification. The following cases represent two set up rules. The ranges of variables are normalized in range <0,1>. Fig.10 presents the input variables with values Profit =1.0, Time = 0.0, Professional = 0.0, Value = 1.0, Justification = 1.0, that results in this case to value Result = 0.13, it means to reject the client. Fig. 11 presents the input variables with values Profit =0.0, Time = 1.0, Professional = 1.0, Value = 0.0, Justification = 0.0, that results in this case to value Result = 0.87, it means to accept the client. The ranges and abbreviation of attributes of variables is as follows: Profit <0-1> <LP,MP,HP>; Time <0-1> <LT,MT,HT>; Professional <0-1> <LR,MR,HR>; Value <0-1> <LV,MV,HV>; Justification <0-1> <LJ,MJ, HJ> and Result <0-1> <RA,RE,RR>.

Fig. 10 Order ruleview(b2 ) – Input [1;0;0;1;1]



Fig. 11 Order ruleview(b2 ) - Input [0;1;1;0;0]

Fig. 12 Order ruleview(b2 ) - Input [0.74;0.24;0.83;0.22;0.47]

It is possible to use the model in practice, when the build up model is verified and its result corresponds to the reality. In this case the values of input variables are set up on the basis of real client. The set up is done by movement of vertical lines. See fig. 12 for values Profit =0.74, Time = 0.24, Professional = 0.83, Value = 0.22 and Justification = 0.47. When the program LO.m is started, the request for entering the input data is displayed in the form Profit, Time, Professional, Value, Justification. When the input values are written [0;1;1;0;0], the result is to Reject the client. When the input values are written [1;0;0;1;1], the result is to Accept the client. When the input values are written [0.74; 0.24; 0.83; 0.22; 0.47] the result is to Entertain the acceptance of client. See fig. 13.

Set up the values in an order Profit; Time; Professional; Value; Justification [Profit; Time; Professional; Value; Justification]: [0;1;1;0;0] ans = Reject Set up the values in an order Profit; Time; Professional; Value; Justification [Profit; Time; Professional; Value; Justification]: [1;0;0;1;1] ans = Accept Set up the values in an order Profit; Time; Professional; Value; Justification [Profit; Time; Professional; Value; Justification]: [0.74;0.24;0.83;0.22;0.47] ans = Entertain

Fig. 13 Result of a client



The results correspond to reality therefore it is possible to consider the build up model to be functional. All graph generated by orders fuzzy(b), mfedit(b), ruleedit(b), surfview(b) and ruleview(b) are at disposal after calculation. The parameters of the build up model are saved in a LO.fis file. 5. CONCLUSION The fuzzy logic is used for the decision making process whether the client of law office will be accepted, entertained or rejected. The use of fuzzy logic is suitable in the cases when vague notes are used. The low profession belongs among these cases. LITERATURE [1] ALLIEV, A.–ALLIEV, R. Soft Computing and Its Applications, UK : World Scientific Publishing Co,

2002, ISBN 981-02-4700-1. [2] DOSTÁL, P. Pokročilé metody analýz a modelování v podnikatelství a veřejné správě, (The Advanced

Methods of Analyses and Simulation in Business and Public Service - in Czech), Brno : CERM 2008, ISBN 978-80-7204-605-8.

[3] DOSTÁL, P.; RAIS, K.; SOJKA, Z. Pokročilé metody manažerského rozhodování, Grada 2005. [4] KAZABOV, K. Neuro-Fuzzy – Techniques for Intelligent Information Systems, Physica-Verlag 1998,

ISBN 3-7908-1187-4. [5] KLIR,G. J., YUAN, B. Fuzzy Sets and Fuzzy Logic, Theory and Applicatio Applications, New Jersy :

Prentice Hall, USA, 1995, ISBN 0-13-101171-5. [6] THE MATHWORKS. MATLAB – Fuzzy Logic Toolbox - User’s Guide, The MathWorks, Inc., 2008. ADDRESS Ass. Prof. Petr Dostál, MSc, Ph.D. Faculty of Business and Management Department of Informatics Kolejní 4 612 00 Brno Tel. +420 541 143714 Fax. +420 541 142 692 E-mail: [email protected]



DISTRIBUTION PROBLEM SOLVED BY GENETIC ALGORITHM

1Petr Dostál, 2Oldřich Kratochvíl

1Brno University of Technology 2European Polytechnic Institute, Ltd., Kunovice

Abstract: This paper deals with the use of genetic algorithms that can solved many problems in entrepreneurial and economic field. This method can solve many problems from practice. The distribution problems belong among them. The correct solution of such problems enables us to minimize the cost, save time and nature. The methods mentioned in the article can be used for many similar cases and various places.

Keywords: Distribution problems, genetic algorithms, Matlab program

1. INTRODUCTION They are problems that are necessary to solve in practice. The distribution problems belong among them. The correct solution of such problems enables us to minimize the cost, save time and nature. The genetic algorithms can help us with these problems. 2. GENETIC ALGORITHMS AS AN OPTIMIZATION TASK The genetic algorithms simulate the evolution of human population. During the calculation by means of genetic algorithms we use such operators as selection, crossover and mutation. The selection means the choice of the best individuals. The crossover represents the exchange of so-called chromosomes among single individuals of the population. The mutation means the modification of a part of a particular chromosome if a random change happens. Genetic algorithms operate such that the initial population of chromosomes is created first; this population is changed by means of genetic operators until the process is finished. One cycle of the reproduction process is called the epoch of evaluation of a population (generation). The aim of genetic algorithm as an optimization task is to divide the set of N existing objects into M groups. Each object is characterized by the values of K variables of a K-dimensional vector. The aim is to divide the objects into groups so that the variability inside groups is minimized.

Let be a set of N objects. Let x Nii ,,2,1; x

or i ,2,1

and ights

lues, i.e.

il denote the value of l-th variable for i-th object. Let us

define f N, M, the wej ,2,1

.0

,1

otherwise

groupth -j ofpart a isobject th -i theifijw

The matrix W = [w ij] has the following properties

1;0ijw and .11

M

jijw

Let centroid of j-th group c j = [cj1, cj2, …, cjK] be calculated in such a way, that each of its elements is a weighted arithmetic mean of relevant va



.

1

1

N

iij

N

iilij

jl

w

xwc

The inner stability of j-th group is defined as

2

11

)( )()(

K

ljlil

N

iij

j cxwWS

and its total inner group variance as

.)()(1 1 1

2

1

)(

M

j

N

i

K

ljlilij

M

j

j cxwSWS

The distances between an object and a centroid can be calculated in this case by means of common Euclidean distances

qp

K

lqlplqpE xxD xxxx

2

1

)(),( .

The aim is to find such matrix W* = [w*ij], that minimizes the sum of squares of distances in groups from their centroids (over all M centroids), i.e. )(min* WSWS W

ula

3. CASE STUDY The case study represents the situation where the coordinates of towns are known and the places of distribution are searched. See tab.1. The places can be searched by calculation of centroids. The centroids have the minimal distances to allocated places. This task can by solved by genetic algorithms. It is used software MATLAB R2008a and its Genetic Algorithm Toolbox to prepare software applications that can be used to solve these types of problems. The input data are represented by coordinates x1, x2, …, xK that characterize the objects. It is possible to define any number of groups. The fitness function represents the sum of squares of distances between the objects and centroids. The coordinates of centroids cj1, cj2, …, cjK (j=1,2,…,M) are changed. The calculation assigns the objects to their centroids. The whole process is repeated until the condition of optimum (minimum) of fitness function is reached. The process of optimization ensures that the defined coordinates xi1, xi2, …, xiK (i=1,2,…,N) of objects and assigned coordinates cj1, cj2, …, cjK of groups have the minimum distances. The fitness function is expressed by following form

),)((min1 1

2min

),...,2,1(

N

i

K

ljlil cxf

Mj

where N is the number of objects, M the number of groups and K dimension. Input data are represented by 14 objects with x1 and x2 coordinates.

Town X Y Town X Y

1 Kyjov 0 35 8 Otrokovice 117 131 2 Veselí n. Mor. 76 0 9 Uh. Brod 149 48 3 Uherský Ostroh 87 34 10 Luhačovice 183 81 4 St. Město 79 65 11 Zlín - a 143 126 5 Kunovice 112 47 12 Zlín - b 191 132 6 Uh. Hradiště 104 63 13 Vizovice 210 130 7 Napajedla 114 109 14 Slavičín 225 74

Tab. 1 Coordinates of places



It is convenient to program the task. See the prog. 1 called DP.m. The input data are in an MS Excel format file DPD.xls and it corresponds to tab.1. The prog. 2 Group.m is used for calculation of Euclidean distances and prog. 3 Draw.m is used for drawing the graph.

function DP global LOCATION; num=input( umber of groups:'); 'Nnum=3*num; PopSize=input('Population size:'); FitnessFcn = @Group; numberOfVariables = num; LOCATION=(xlsread('DPD','Coordinates')) my_plot = @(Options,state,flag) Draw(Options,state,flag,LOCATION,num); Options = gaoptimset('PlotFcns',my_plot,'PopInitRange',[0;300],'PopulationSize',PopSize); [x,fval] = ga(FitnessFcn,numberOfVariables,Options); assign=zeros(1,size(LOCATION,1)); for i=1:size(LOCATION,1) distances=zeros(num/3,1); for j=1:(size(x,2)/3) distances(j)=sqrt((LOCATION(i,1)-x(j))^2+(LOCATION(i,2)-x(size(x,2)/3+j))^2+(LOCATION(i,3)-x(2*size(x,2)/3+j))^2); end [min_distance,assign(i)]=min(distances); end assign fval xy=zeros(num/3,3); for i=1:(num/3) xy(i,1)=x(1,i); xy(i,2)=x(1,num/3+i); xy(i,3)=x(1,2*num/3+i); end xy

Prog. 1 DP.m

function z=Group(x) global LOCATION z=0; for i=1:size(LOCATION,1) for j=1:(size(x,2)/3) distances(j)=sqrt((LOCATION(i,1)-x(j))^2+(LOCATION(i,2)-x(size(x,2)/3+j))^2+(LOCATION(i,3)-x(2*size(x,2)/3+j))^2); end min_distance=min(distances); z=z+min_distance; end

Prog. 2 Group.m

function state = Draw(Options,state,flag,LOCATION,num) [unused,i] = min(state.Score); x=state.Population(i,:); for i=1:size(LOCATION,1) for j=1:(size(x,2)/3) distances(j)=sqrt((LOCATION(i,1)-x(j))^2+(LOCATION(i,2)-x(size(x,2)/3+j))^2+(LOCATION(i,3)-x(2*size(x,2)/3+j))^2); end [min_distance,assign(i)]=min(distances); end for i=1:size(LOCATION,1)



plot3(LOCATION(i,1),LOCATION(i,2),LOCATION(i,3),'sr','MarkerFaceColor',[3*(assign(i))/num,3*(assign(i))/num,3*(assign(i))/num],'MarkerSize',10); xlabel('x');ylabel('y');zlabel('z'); grid on; hold on; end plot3(x(1:size(x,2)/3),x((size(x,2)/3+1):2*size(x,2)/3),x(2*size(x,2)/3+1:size(x,2)) sr','MarkerFaceColor','b','MarkerSize',10); ,'hold off;

Prog. 3 Draw.m The program enables us to set up the number of required groups and the population size. The higher number of individuals the more precise solution but the higher duration of the calculation. Futher, the program sets up the options for optimization and the optimization command ga is called. The program involves the calculation of fitness function and it fills the variables with data that inform us about the coordinates of centroids and the assignment of objects to groups and displays them. The two (z-axes is zero) and three dimensional tasks can be solved. The program is started by command DP in MATLAB. Then it is necessary to set up the requested number of groups, e.g. Number of groups to be 3 and Population size to be 1000. When the calculation is terminated, the input parameters and results of calculation are displayed on the screen. The results are presented by coordinates of centroids and assignment of places to groups. See Res. 1. The fig.1 presents the graph.

Number of groups: 3 Population size:1000 assign = 3 3 3 3 3 3 2 2 3 1 2 1 1 1 fval = 447.1 xyz =

200 114 120 126

92 44 Res. 1 Coordinates of centroids and location of places



Fig. 1 Places and their centroids

5. CONCLUSION The genetic algorithms and neural network enable us to solve complicated distribution problems. The correct optimization and application of results in practice enables us to minimize the costs, increase the profit and save our environment. The problem that was solved in practice results in construction of three places of distributions A, B and C, that uses places x1, x2, …, x15. See fig. 3. The distribution problems have a wide range of use in various branches. One of the branches is economy and business. We can mention for example the search of best location of a market, bank or firm. The advantage of the use of genetic algorithms is their applicability in various types of optimization problems with a high speed of calculation and found solution very close to the optimal one. it is presented in real application in the article. The real application is presented on fig. 2. Genetic algorithms can be successfully applied in many traditional areas of the operations research where only deterministic models are used. This method can be used for other similar cases and various places.



Fig. 2 The distribution places and their places

LITERATURE [7] DAVIS, L. Handbook of Genetic Algorithms, USA : Int. Thomson Com. Press 1991, 385p.,

ISBN 1-850-32825-0. [8] DOSTÁL P.; RAIS, K. Risk Management and Artificial Neural Network, In Word of Informations

Systems, Zlín : 2005, s. 292-297, ISBN 80-7318-276-9, ISSN 1214-9489. [9] DOSTÁL, P. Avanced Economic Analyses, Brno : CERM 2008, 80s., ISBN 978-80-214-356-3. [10] DOSTÁL, P.; POKORNÝ, P. Group analysis and genetic algorithms. In Management, Economics

and Business Development in the New European Conditions, Brno University of Technology, Faculty of Business and Management, 2008. s. (9 s.) ISBN: 978-80-7204-582-2.

[11] DOSTÁL, P. Pokročilé metody analýz a modelování v podnikatelství a veřejné správě, (The Advanced Methods of Analyses and Simulation in Business and Public Service – in Czech), Brno : CERM, 2008,342s, ISBN 978-80-7204-605-8

[12] OŠMERA, P. Evolution of Complexity, In the book Zhong Li, Halang W.A., Chen G.: Integration of Fuzzy Logic and Chaos Theory, 51 pages (527-578), New York : Springer 2006, 625p., ISBN 3-540-26899-5.

ADDRESS: Ass. Prof. Petr Dostál, MSc, Ph.D. Brno University of Technology Faculty of Business and Management Department of Informatics Kolejní 4 612 00 Brno Tel. +420 541 143714 Fax. +420 541 142 692 E-mail: [email protected] Hon. Prof. Ing. Oldřich Kratochvíl, Dr.h.c., MBA European Polytechnic Institute, Ltd. Osvobození 699 686 04 Kunovice E-mail: [email protected]

12

B

C



METHOD OF USING SEASONALITY FOR DETERMINATION OF TOURIST-ECREATION SYSTEM DEVELOPMENT STRATEGY

Serhiy Lyakhov

European University Ukraine

Abstract: In this paper territory seasonal attractiveness index in the improved «gravitation» model was used first time. Using new model were conducted the calculation of relative amount of holiday-makers in the tourist recreation systems. Comparative description of calculations is conducted after a «gravitational» model which used the seasonality attractiveness and general territory attractiveness index. Recommendations are given in relation to the use of «gravitation» model with different indexes for determining the most attractive location for building tourist complex.

Territory attractiveness for resting could be estimated using different attractiveness indexes. Main factor for placing tourist complex is income that why using only attractiveness index is not enough. We should take into account possibility to rest there in all seasons. For example there are a lot of hotels placed in mountains that are fully packed in winter but are almost empty 3 other seasons. This situation is caused by area climate. That’s why beginning of such business could be unprofitable even if territory is very attractive. In this paper we try to compare efficiency of using gravitational model with attractiveness index and seasonal index that show probability of resting on some territory in different seasons. Taking into account possibility to rest in some area in different seasons we should also estimate people’s possibility and wish to rest in these seasons. For calculation of possible people quantity who wants to visit some tourism complex we have used improved gravitational model [5], where main multiplayer is common attractiveness index. This multiplayer is a function of many territory and tourist complex attractiveness factors. Upon this model visitors quantity is calculated as followed:

price

catТРС

ТРСcatrij

nj

miicat

Att

n

l

catl

ТРСl

catij n

BBTPTP

r

nmD

n

AttAttkTK

22

1 11 (1)

where catijK – the number of the guests in a j-th TRS who came from an i-th demand locality;

mim – population number in the i-th demand locality;

njn – maximum capacity of the j-th TRS;

rijr – distance between the j-th TRS and the i-th demand locality;

k – empirical “gravity” (attractiveness) factor; m, n, r – empirical factors [2];

icatD – the proportion of the “cat” category people in the i-th demand locality;

TPcat – probability that the “cat” people will have their vacation in the time T;

TPТРS – probability that the given TRS will work in the time T;

ТРSB – TRS price category;

catB – the desired TRS category for the guests of the “cat” category;

pricen – the normalizing factor which is equal to the degree of the та rating scale; ТРSB catBl – the “attractiveness” type;

ТРSlAtt – the rating grade of the l-th TRS “attractiveness”;



catlAtt – the rating grade defining the importance of the l-th “attractiveness” for the “cat” category guest;

Attn – the maximum permissible value of . ТРSlAtt

All the factors in (1) except for k, m, n, r have statistical nature and were defined by means of expert assessments performed by the leading experts in this field. Common attractiveness index is calculated as:

Att

n

l

catl

ТРСl

n

AttAtt

cattrsAtr 11),( (2)

We replaced index (2) with territory recreation potential. Mathematical model of recreation potential and all necessary data for its calculation were taken from paper [9]. Methods of fuzzy logic were applied for attractiveness and recreation potential indexes calculation. Aggregated territory attractiveness index for tourists consist of several independent attractiveness indexes that based on defined recreation types. For Chernivtsi region all rest types and recreations could be combined into 4 groups:

1p – winter recreation;

2p – summer water-based recreation;

3p – spring and autumn nature-based recreation;

4p – excursions and sightseeing.

All this types depend on several factors and were calculated using Fuzzy Logic methods in MathLab [1, 3, 9]

Season recreation potential of territory is defined as:

tptpftP 41 ,..., . (3)

Gravitational model with multiplier (3) has following structure:

price

catТРС

ТРСcatrij

nj

miicat

jcatij n

BBTPTP

r

nmDTPkTK

22

1)( (4)

COMPUTER EXPERIMENT For model approbation and making calculation were chosen tourists systems which are situated in Ukrainian Carpathians in Chernivtsi refion: “Nimchych”, “Mygovo”, “Lekeche”. We assumed that all tourist systems open whole year and could serve the same quantity of visitors. Received results are shown on figures 1, 2 and 3. These results are well correlated with statistical data if the tourist systems. As you can see from figure 1 graph is look like two saddles. Maximums are in the points of the 1st, 8th and 12th months. So, maximum quantity of visitors is in winter and summer seasons. In tourist complex “Mygovo” the biggest amount of visitors is in winter season. It depends on its specific location.



Main feature of the “Nimchych” complex is that it is most attractive in summer season. It is stipulated by comfort climate conditions.

From all figures you can see that most attractive is complex of the 3d class, because it is mostly suit for middle class, which is 76.7 % [4] of whole population of Ukraine. Also you could see from figures that people do not like to rest in spring and autumn. This is caused by the rest specific during these seasons and by the recreation complexes location. Also people mostly like to take vacation on summer or winter. On figures 4 and 5 are shown results with models (4) and (1) accordingly. These two models have different results for the same complexes. So, we couldn’t get adequate results using models (1) and (4) separately. Better approach for evaluating most attractive location for rest is the following:

Define most attractive territory using model (4) taking into consideration seasonal prevalence, for most attractive territory defined in a) find more attractive place using model (1).

Fig. 2. Relative amount of visitors in TRS «Mygovo»

Fig. 1. Relative amount of visitors in TRS

«Lekeche»

Рис 3. Relative amount of visitors in TRS

«Nimchych»



0

1000

2000

3000

4000

5000

6000

7000

8000

9000

1 2 3 4 5

Клас ТРС

Кількість

відпочиваю

чих,

ум

.од

.

Мигово Німчич Лекече Fig 4. Relative amount of visitors on TRS classes

(Seasonality index)

LITERATURE: [1] ZADEH, L. A. Fuzzy Sets - Information and Control, 1965. – #8 [2] БОЄВ Ю. Аналіз рекреаційного потенціалу території національного парку «Подільські Товтри».

"Социально - экономические, экологические и гуманитарные проблемы развития туристического бизнеса":Сб.науч.тр./Донецкий ин-т туристического бизнеса, 2003. С. 177-182.

[3] П.ДЬЯКОНОВ, В. П.КРУГЛОВ В. MATLAB 6.5 SP1/7/7 SP1/7 SP2 Simulink 5/6 Инструменты исскуственного интеллекта и биоинформатики // Серис «Библиотека проффесионала». – М.:СОЛОН-ПРЕСС, 2006.–456с.

[4] Дослідження середнього класу в Україні. Міжнародний центр перспективних досліджень. http://www.icps.kiev.ua/project.html?pid=10

[5] ЄВДОКИМЕНКО, В. К.; ВИКЛЮК Я. І.; ЛЯХОВ С. О. Застосування нечіткої логіки для вдосконалення визначення потоків рекреантів за допомогою модифікованої «гравітаційної» моделі. Регіональна Економіка. – 2008. 2(48). – с. 198-212.

[6] Карпатский рекреационный комплекс / Под ред. М. И. Долишнего, М. С. Нудельмана, К. К. Ткаченка и др. – Киев: Наук. думка, 1984. – 145 с.

[7] КРАВЦІВ В.; КАТОЛИЧ Б.; ГУЛИЧ О.; ПОЛЮГА, В. Рекреаційний потенціал Львівської області та стратегія його освоєння. Регіональна економіка. – 2002. - 2. – с. 134-143.

[8] Розвиток туристичного бізнесу регіону: Монографія / За ред. доктора економічних наук, професора Школи І.М. – Чернівці: Книги – ХХІ, 2007. – 292 с.

[9] ВИКЛЮК, Я. І., АРТЕМЕНКО, О. І., Розрахунок рекреаційної привабливості територій з використанням нечіткої логіки Матеріали Третьої Міжнародної конференції "Комп'ютерні науки та інформаційні технології" CSIT'2008, 2008, ст. 347- 350.

[10] ЧОРНЕНЬКА, П. Принципи виділення перспективних рекреаційних територій. Вісник Львівського Університету, серія географічна, 2004. Вип. 30. С. 312-316.

[11] ЧУЛАНОВА, Л. І. Екологічна оцінка рекреаційного потенціалу територій природно-заповідного фонду Донецької області. Магістерська робота, 2004.

ADDRESS Serhiy Lyakhov post graduate student, European University Ukraine, 02192, Kyiv, Generala Zhmachenka 12, app. 321 tel. +38-063-599-777-9 E-mail: [email protected]

0

1000

2000

3000

4000

5000

6000

7000

8000

1 2 3 4 5

Клас ТРС

Кількість

відпочиваючих

, ум

.од

.

Мигово Німчич Лекече Fig 5. Relative amount of visitors on TRS classes

(Attractiveness index)



GRANULAR NEURAL NETWORK IN MANAGERIAL FORECASTING SYSTEMS

Zuzana Mečiarová, Ján Bábel, Lucia Pančíková

University of Žilina

Abstract: Forecasting forms the foundation for all decision processes in many business organizations. Forecasting helps to reduce uncertainty measurement of the future, eliminates the risk associated with decision-making and so improves the quality of decision process. This article discusses granular neural network as a new forecasting tool and a part of managerial forecasting system in companies. After a brief introduction and theoretical background of granular neural network, the article describes granular neural network learning algorithm. The case study is described in the fourth section. The section five deals with the application part of our case study and with reviewing experimental results. Finally, the conclusion is introduced.

Keywords: ARIMA methodology, cloud activation function, forecasting, granular neural network

1. INTRODUCTION One of the most important managerial activities is to make the right decisions. The manager as decision-maker uses forecasting models to assist him or her in decision-making process. Almost all managerial decisions are based on forecasts. Forecasting is the process of estimation unknown situation. Forecasting helps to reduce uncertainty measurement of the feature, eliminates the risk associated with decision-making and so improves the quality of decision process. Various methods have been developed and applied to forecasting problem. One of the most attractive approaches is the neural network. The reasons to use neural networks for forecasting are many. First, neural networks are nonparametric and nonlinear in nature. They do not require any specific assumptions about the underlying model form and are powerful and flexible in modeling real-world phenomena which have more or less nonlinearities. Second, neural networks are universal functional approximator and they can capture any type of complex relationship. Third, neural networks are data-driven and self-adaptive. They have the capability to learn from experience. All these features of neural networks make them a very useful tool for forecasting tasks [1]. This article is organized as follow. The theoretical background of granular network is given firstly. Next the neural learning algorithm is described. The section 4 describes the real dataset and the case study. In the section 5, we deal with the application of granular neural network for forecasting this real dataset and evaluate the network performance by comparing to ARIMA methodology. Then the conclusion is introduced. 2. THEORETICAL BACKGROUND OF GRANULAR NEURAL NETWORK Granular neural network is a feed forward neural network consists of one hidden layer. The activation function of hidden neurons is based on cloud activation function given by the form

2'

2

2 )(2

)(exp)(

En

cxou jtj

j , (1)

where are the outputs from the hidden neurons, is an input vector, is the mean of

cloud activation function and ' is the mean of random numbers with mean

jo ),,,( 21 kt xxxx jc

En and standard deviation . Figure 1 shows the cloud activation function (CAF) of hidden neurons. As shown in Figure 1, the cloud activation function is represented by the three numerical characters: the expected value c, the entropy σ and the hyperentropy He. The expected value is the most representative sample of the dataset, the entropy represents the uncertainty measurement of the dataset and the hyperentropy is the uncertainty measurement of the entropy [2]. The three numerical characters can be calculated using backward normal cloud generator without certainty degree [3].

He



Figure 1 CAF – Cloud activation function

The outputs from hidden neurons are normalized using the following form

s

j

jo1

, (3)

and then the normalized outputs from hidden neurons are given by the form

s

j

j

jNj

o

oo

1

, (4)

where is normalized output value of jth hidden neuron, is the output value of jth hidden neuron and s is

the number of hidden neurons.

Njo jo

The activation function of output neuron is given by form

j

s

j

Nj voy

1

ˆ , (5)

where are the weights between the output neuron and neurons of hidden layer. jvGranular neural network has typically three layers – an input layer, a hidden layer and an output layer (Figure 2). The input layer is composed of input neurons. Input neurons standardize the range of input values by substracting the mean and dividing by the standard deviation. Then the input neurons feed the input values to each of the neurons in hidden layer. The hidden layer has a variable number of hidden neurons with non-linear cloud activation function (CAF). The output layer consists of one output neuron. The architecture of granular neural network is shown in Figure 2.

s

jj

Nj voy

1

ˆ

Figure 2 Architecture of granular neural network



3. LEARNING ALGORITHM OF GRANULAR NEURAL NETWORK Learning of granular neural network with cloud activation function is generally divided into two phases. The first one is an unsupervised learning phase in which appropriate locations for the centers of the cloud functions in the hidden layer and the standard deviations are estimated. We can use K-means clustering algorithm or method based on competitive learning for finding the centers of cloud activation function. Competitive learning is a class of unsupervised learning algorithms based on the idea of adjusting a weight matrix in such a way that the weights represent cluster centers. This method is based on Kohonen’s adaptive rule [4]. The detail description of this algorithm is described in the experimental section of our study. The second phase is a supervised learning in which the weights between hidden and output layer are calculated using backpropagation algorithm [5] or linear regression technique. 4. THE CASE STUDY AND DATASET The aim of this paper is to judge the forecast accuracy of granular neural network in comparison with ARIMA methodology. In this regard the available real dataset (the daily product purchasebility of one Slovak spa portal) have been collected and divided into two sets for model training and verification. The training set is data from 1st March 2007 up to 31st December 2008 and the rest of the data, i.e. from 1st January 2008 up to 31st May 2008 is used to verify the model. The training set has been statistically analyzed using the statistical software Statgraphics Centurion XV. The model obtained through the analysis is given by the form

1765418,0ˆ tt yy . (6)

The equation (6) represents an autoregressive process of order one which means that only the immediately

previous value has a direct effect on the current value . 1ty tyThe equation (7) was used to transform input and output data of model (6) into the normalized dataset due to computing simplicity

z

tt

N

s

Zzz

)( , (7)

where is normalized data, is original input or output data, tN z tz Z is the mean and the standard deviation

of original data. zs

5. EXPERIMENT DESIGN AND RESULTS The proposed architecture of granular neural network is 1-s-1, i.e. one input neuron – s hidden neurons – one output neuron. The number of lagged observations used in the equation (6) determines the number of input nodes of granular neural network. One input neuron in input layer is selected by reason that the time series has one lagged observation. The number of hidden nodes is another important factor for the neural network model. Hidden nodes play a major role for the nonlinear modeling of neural networks [1]. Since there is no systematic method to determine this parameter, it is varied with 5 levels ranging from 5 to 20. The transfer function of hidden nodes is the cloud activation function given by the equation (1) and the activation function given by the equation (5) is employed for the output node. In this study, we use a two-phase learning method to train the granular neural network. In the first phase the centers and the standard deviations of cloud activation function are calculated by means of Kohonen’s adaptive algorithm [4]. A Kohonen network is composed of a grid of output units and N input units. The input pattern is fed to each output unit. The input lines to each output unit are weighted. At the beginning of learning process the weights are initialized to small random numbers. The winning output unit is simply the unit with the weight vector that has the smallest Euclidian distance to the input pattern. The weights of every unit in the neighbourhood of the winning unit (including the winning unit itself) are updated using the form

),.(*jtjj wxww (8)

where are the weights or centers after update, wjw j are the weights before update, is the learning rate and xt

are input patterns. Equation (8) will move each unit in the neighbourhood closer to the input pattern. As time progresses the learning rate and the neighbourhood size are reduced. The final network captures clusters with their centers in the data. The standard deviations of cloud activation function are calculated by the form

M

tmjj xc

M 1

21 , (9)



where xm is the m-th input vector with input patterns which belongs to the center cj. Then, in the second phase the weights of output layer are adapted by back-propagation algorithm. Modeling of granular neural network with cloud activation function is implemented by our own software tool developed in Borland Delphi 7.0.. The results of the simulation for different learning coefficient η are shown in the table 1, 2, 3 and 4. The performance of neural network is compared to ARIMA model given by the equation (6). Error measure of the root mean square error (RMSE) is used as the major performance measure. It is defined as

N

yyRMSE

N

ttt

1

2)ˆ(, (10)

where is the actual observation at time t, is the predicted value and N is the number of predictions. Less

the RMSE value the better prediction accuracy. ty ty

Table 1 Simulation results of granular neural network, η = 0,001

Hidden neurons RMSEVAL Epoch number 3 0,6329 64 5 0,6389 115 10 0,6363 151 15 0,6318 123 20 0,6314 96







The table 2 shows, the minimum RMSE of validation period is obtained for 20 hidden neurons and after 28 training epochs. The table 5 presents the result comparison of the neural network simulation with ARIMA methodology. It clearly shows the superiority of neural network with cloud activation function over ARIMA methodology in forecasting the AR(1) time series.

Table 5 Result comparison RMSEVAL ARIMA methodology 0,6971 Granular neural network 0,6187



6. CONCLUSION In this article we have applied the granular neural network to forecast the daily product purchasebility of one Slovak spa portal. The perfomance of neural network is compared to ARIMA model by means of root means square error. The experimental results show that the granular neural network with cloud activation function outperforms the ARIMA methodology that is widely used by managers as forecasting tool. ACKNOWLEDGEMENTS This work was supported by Slovak grant foundation under the grant VEGA No. 1/0024/08. REFERENCES [1] ZHANG, G. P. 2001. An investigation of neural networks for linear time-series forecasting. In Computer

and Operations Research, Vol. 28, No. 12. Elsevier, October 2001, pp. 1183-1202. [2] LI, D.; DU, Y. 2008. Artificial intelligence with uncertainty. Boca Raton: Chapman&Hall/CRC, Taylor

and Francis Group, 2008. [3] LU, H.; WANG, Y.; LI, D.; LIU, C. 2003. The Application of Backward Cloud in Quantitative

Evaluation. In Chinese Journal of Computers, Vol. 26, No. 8. Beijing: China Computer Federation, 2003, pp. 1009-1014.

[4] KOHONEN, T. 1997. Self-Organizing Maps. In Series in Information Sciences, Vol. 30. Heidelberg: Springer Verlag, Second edition, 1997.

[5] RUMELHART, R. E.; McCLELLAND, J. L. and the PDP Research Group. 1980. Parallel distributed processing explorations in the microstructure of cognition. Cambridge: MIT Press, 1980.

ADDRESS Ing. Zuzana Mečiarová University of Žilina Faculty of Management Science and Informatics Department of Macro and Microeconomics Univerzitná 8215/1 010 26 Žilina [email protected] Ing. Ján Bábel University of Žilina Faculty of Management Science and Informatics Department of Macro and Microeconomics Univerzitná 8215/1 010 26 Žilina [email protected] Ing. Lucia Pančíková, PhD. University of Žilina Faculty of Management Science and Informatics Department of Macro and Microeconomics Univerzitná 8215/1 010 26 Žilina [email protected]



SOFT COMPUTING AND ITS APPLICABILITY FOR ESTIMATING THE EFFICIENCY

OF FORMING AND USING THE CAPITAL OF FOOD RETAIL INDUSTRY

ENTERPRISES

Juliya Verbitskaya

Donetsk National University of Economics and Trade named after Michail Tugan-Baranovsky

Abstract: The article considers and analyses forming and using the capital of food retail industry enterprises. Methodology of financial analysis of forming and using the capital of food retail industry enterprises of Ukraine is offered. It studies how capital structure influences the efficiency of food retail industry enterprises and their value on the stock market.

Key words: Capital structure, forming of capital, use of capital structure, efficiency of activity of food retail industry enterprises, cost of enterprises, financial solvency (firmness), liquidity, business activity, profitability, property state.

Among the production-economic problems of enterprises, presumably, there is no other, which was so often examined by research workers and practical workers and was seemed so simple to them, as analysis and estimation of financial job performance of enterprise. From our point of view financial results on a line depend on efficiency of forming and using of capital by the enterprise. In our opinion, such a considerable attention is paid to the analysis of forming and using the capital of enterprise and its estimation because, on the one hand, it interprets the results of its activity, testifies the level of implementation of the planned tasks and is the base of comparison with particular branch leading enterprises, and on the other hand, - reveals ways, backlogs and prospects of development. A wide circle of the questions, related to the financial analysis of enterprises, its estimation and management, is explored in the work of foreign and domestic economists. Methodical bases of the quantitative measuring and estimation of forming and using the capital of enterprise and interpretation of its essence are fixed also in the normative acts of the corresponding ministries (departments). Among scientists which pay considerable attention to the research of the financial state of enterprises, we should mention such scientists as Andriychuk V. G., Aptekar S. S., Belyk M. D., Breghem J., Gapencke L., Кresanov M. F., Laxteonova L. A., Оmelyanovich L. A., Panchenco А. E., Podderegin А. M., Saveckaya G. V., Savchyk V. P., Suchevskiy М. P., Yolsh К., Хуngenberg Х., Chumachenko М. G., Sheremet А. D. Studying methods of analysis of forming and using the capital of enterprises of the USA, experience of Russia, Ukraine and other countries of the CIS gave possibility to choose and form the groups of indexes for carrying out analysis of forming and using the capital of enterprises of food retail industry of Ukraine (in the cut of analysis of forming and using the capital of enterprises in producing of cacao, chocolate and saccharine pastry wares (КVЕD - 15.84). It should be noted that in scientific literature a concept «estimation of forming the capital of the enterprise» was absent for a long time and, consequently, there was no method, by which it was possible to conduct a complex estimation of forming capital of the enterprise and its counterparts. As there is a process of forming market infrastructure and transformation of economy in Ukraine and other countries of the CIS, the clever forming and using of capital of enterprises is one of major factors of its vital functions, especially in the conditions of economic crisis. [1 p. 44] Every enterprise must necessarily develop, foremost, the method of conducting of analysis, which gave a possibility to have a complete estimation of forming and using the capital of enterprise taking into account external (stages of development of economy of Ukraine, economic and financial position of the state, financial policy and other) and internal factors (features of production-economic and financial activity and their indexes). The main for every subject of the household is to estimate how capital is formed and used, to reveal drawbacks in the work of enterprises and develop measures on improving financial results of its activity. Judging from this, methodical approaches of conducting financial analysis of forming and using the capital of enterprises in Ukraine on the base of experience of foreign and domestic economists-financiers on this problem were



considered. Taking into account the above mentioned experience and recommendations of banks of Ukraine for carrying out financial analysis of forming and using the capital of food retail industry enterprises (in the cut of analysis of forming and using capital of enterprises in producing cacao, chocolate and saccharine pastry wares (КVЕD - 15.84) of Ukraine the method of estimating the efficiency of forming and using the capital of food retail industry enterprises adapted to Ukrainian realities is offered. The above-mentioned problems are topical because of the absence of statistic information, first of all, at a particular branch and at its divisions, due to commercial secret. The information, which is published at national level, has: firstly, contradictory enough character; secondly, most industries are combined larger into units. (Such industries, as food retail industry and

processing of agricultural raw materials, incorporated in one of the same name industry, one of the representatives of which is the enterprise-producer of tobacco wares, the presence of which in this group has quite a contradictory character. As a result of researches in relation to food retail industry to a certain extent it is possible to consider unreliable);

thirdly, the information that is accessible, has address direction, to the list of which most fundamental scientific establishments do not enter.

The noted situation negatively influenced publicizing the state (in particular financial) of the whole economy of Ukraine, in the cut of industries and subindustries. Agrarian scientific school which systematically publishes the basic job performances of АIC of Ukraine due to the ramified network of collection is the exception, treatments and revealing of statistical information, what can not be said both about food retail industry of Ukraine on the whole and about its subindustries. Taking into account a multivector and many-sided nature of the problem of estimating the forming and using the capital of enterprises of Ukraine, it is suggested to select the row of conditional groups, which are the constituents of a complex analysis, which integrates such elements as «numerical material > analytical material». The task consists in determination of the modern financial and economic state of food retail industry of Ukraine, finding and systematising of backlogs and ways both a subsequent growth and strengthening of this industry. Scientifically the substantial providing of population of some country with a high quality food is the global problem of development of international concord. A special role in solving food problem on regional, national and global levels belongs to food retail industry, that is not only the finishing functional link of production of food goods but also as a real organizer and integrator of the effective, rational and balanced functioning of food subcomplex of every country. Judging from the above mentioned, providing dynamic, stable and at the same time effective enough development of food retail industry, is one of topical and exigent tasks of economic policy of the young Ukrainian state, realization of which is the important condition of not only satisfaction of necessities of internal market of Ukraine in food stuffs but also in forming mighty export potential. [2, p. 44-45] From data of enterprises [3] that is explored, the author conducted the analysis of expedience of forming and using capital. In table 1 basic results of the conducted analysis of the system of indexes of estimation are represented: financial firmness, liquidity (solvency), business activity, profitability, property state of pastry factories of Ukraine after 2005 - 2007. Characterizing basic tendencies it is possible to say in the changes of the system of indexes, that at enterprises, at which there is a slump of profitability, negative consequences for financial solvency (firmness) are forecast, because possibilities of enterprises to settle accounts in time, to get and grow incomes, to remain financially proof and stable grow short, and it, in the turn, will entail a slump of both liquidity and business activity of enterprises.



Table 1 - Tendency in the changes of the system of indexes of estimation… System of the indexes of estimation*… pastry factories of Ukraine 2005 - 2007 р.р.

Enterprises

fina

ncia

l so

lven

cy

(fir

mne

ss)

liqu

idit

y

busi

ness

ac

tivi

ty

prof

itab

ilit

y

prop

erty

sta

te

1 «Winnitca pastry factory» 2 «Zaporozhian pastry factory» 3 «АVК» 4 «Production amalgamation «Кonti» 5 «Kiev pastry factory of the name of C. Маrcs”s» 6 «Кremencheg pastry factory» 7 «Lvov pastry firm «Svitoch» 8 «Mariupol pastry factory» 9 «Sumskaya pastry factory» __ 10 «Kherson pastry factory» *Conditional notes - it is growth; - it is slump; - it is the transitional state. Analysing the information of table 1 enterprises can take the following places according to their progress: the first place is the Joint-stock company «АVK», second, is the Joint-stock COMPANY «Kherson pastry factory», you grind is the Joint-stock company «Production amalgamation «Konti». By the financial reporting a capital structure is analysed, to define intercommunication between efficiency of activity of enterprise and capital structure that is by correlation of property and loan assets (results are represented in table 2).

Table 2 - Analysis of own and loan capital structures of pastry factories in Ukraine 2005 - 2007. 2. Loan capital Indexes 1.

Property asset

Long-term debt

Current liabilities

Only:

2005 % 27,70 0,00 72,30 100,00 2006 % 16,41 25,12 58,47 100,00 1

«Winnitca pastry factory»

2007 % 14,87 24,83 60,30 100,00 2005 % 61,84 0,00 38,16 100,00 2006 % 68,79 0,00 31,21 100,00 2

«Zaporozhian pastry factory»

2007 % 64,75 0,00 35,25 100,00 2005 % 89,73 0,00 10,27 100,00 2006 % 91,77 0,01 8,23 100,00 3 «АVК» 2007 % 85,16 0,00 14,84 100,00 2005 % 26,78 44,18 29,05 100,00 2006 % 24,17 45,86 29,97 100,00 4

«Production amalgamation «Коnti»

2007 % 19,07 37,33 43,60 100,00 2005 % 36,90 0,00 63,10 100,00 2006 % 22,69 0,00 77,31 100,00 5

«Kiev pastry factory of the name of C. Маrcs”s» 2007 % 15,17 0,00 84,83 100,00

2005 % 9,52 0,00 90,48 100,00 2006 % 8,81 0,00 91,19 100,00 6

«Кremenchug pastry factory»

2007 % 7,05 0,16 92,79 100,00 2005 % 51,57 0,00 48,43 100,00 2006 % 79,37 0,00 20,63 100,00 7

«Lvov pastry firm «Svitoch»

2007 % 78,42 0,00 21,58 100,00

2005 % 59,58 0,00 40,42 100,00 2006 % 52,30 0,01 47,68 100,00 8

«Mariupol pastry factory»

2007 % 51,26 0,02 48,72 100,00 2005 % 34,74 0,73 64,53 100,00 2006 % 31,00 0,77 68,23 100,00 9

«Sumskaya pastry factory»

2007 % 12,30 0,78 86,92 100,00



2005 % 74,28 0,00 25,72 100,00 2006 % 72,54 0,00 27,46 100,00 10

«Kherson pastry factory»

2007 % 67,31 9,85 22,83 100,00 Analysing table 2, it is possible to draw a conclusion, that efficiency of activity of the enterprise and capital structure are not interdependent. The Моdelyane and Міller theory testifies also about it: the cost of the company, that uses a loan capital, equals the cost of the company, that does not use a loan capital, enlarged by the size of tax shield, that equals the increase of amount of debt multiplied by the rate of taxation. [4] Consequently, the given material does not reveal the common picture of efficiency forming and using the capital, and, moreover, about functioning of food retail industry of Ukraine, as it touches only a part of this many-sided problem that is estimation of efficiency of forming and using capital of food retail industry enterprises. The prospects of subsequent researches consist in determining factors that influence profitability of food retail industry enterprises, determining backlogs of increasing efficiency of functioning and strengthening its financial state. LITERATURE [1] PETRUXA, С. Methodology complex analysis of innovative activity of domestic food industry / Petruxa

С., Коlotysha // Economist. - 2007. - 4. - С. 50-56. [2] PETRUXA, С. Financially-economic indexes of work of food retail industry of Ukraine: analysis of

financial results / Petruxa С., Коlotysha // Economist. - 2006. - 11. - С. 44-50. [3] State establishment "Agency from development of infrastructure of fund market of Ukraine" is

incorporated in 26.10.2006 - in 2008 - www.smida.gov.ua. [4] MODIGLIANI F.; MILLER M. H. The Cost of Capital, Corporation Finance and the Theory of

Investment // Amer. Econ. Rev. 1958. June. P. 261-297; Modigliani F„ Miller M. H. Taxes and the Cost of Capital : And Correction // Ibid. 1963. June. P. 433-443.

ADDRESS Juliya Verbitskaya Donetsk National University of Economics and Trade named after Michail Tugan-Baranovsky 83017 Ukraine, city Donetsk, boulevard Schevchenko, 30 Tel.: +380505307771 E-Mail: [email protected]



MODELING METHODOLOGY OF SOCIO-ECONOMIC OBJECTS FLUCTUATION

GROWS

Yaroslav Vyklyuk

NU Lvvska Politechnika

Abstract: Methodology of crystals fractal growth methods appliqué in the unclear potential attractiveness field for prognostication of socio-economic processes is offered. The method of the unclear potential attractiveness field construction is considered. The method of modification and integration of fractal growth classic methods is offered: diffusely limited aggregation and «casual rain» with the theory of molecular dynamics.The segmentation method for prognostication of socio-economic processes pursuant to methodology of crystals fractal growth methods appliqué in the unclear potential attractiveness field is offered. Influence of model empiric parameters on appearance of fractal structure fluctuations as formation of additional aggregation centers was explored. Mechanism of stagnation phenomenon appearance and system self organization in the evolution process was explored.

Key words: Soft Computing, econophysics, attractor, fractal.

INTRODUCTION As a rule territory attraction and prognostication of prognostication of the poorly guided processes in society determined or by expert estimations, or after marketing’s researches which also are not deprived lacks of expert estimations. In addition, the adopted methods can give an objective picture only for ideal cases., consequently they must be complemented exact quantitative models [1]. Development of region would take place more dynamically if the program of development was based on scientific results got by quantitative methods. Application of mathematical vehicle in economic researches enables to solve special tasks with the construction of prognostic scenarios and possibility to foresee forming and development of difficult socio-economic processes. PURPOSE AND RESEARCH ACTUALITY A research purpose is creation and approbation of appliqué methods methodology of fractal crystals growth for the design of the socio-economic systems of different levels. Research actuality is development of structure prognostication conception of the poorly controlled social processes, such as development of cities and settlements which are related to active development of green tourism, formation of concomitant infrastructure and etc on a base well known in solid physics methods of crystals fractal growth and molecular dynamics in combination with the theory of fuzzy logic. Prognostication of growth geometry of the socio-economic systems and their underlying structure will let to plan development of the proper infrastructure and communications with a maximal economic value. It will let to optimize strategies of regions development, define specialization of separate segments and foresee the monies streams of such system [2]. POTENTIAL FIELD MODEL It is possible to describe territory attractiveness potential field for building by the mathematical vehicle of fuzzy logic. In general the potential U has following view:

naaaFU ...,,, 21 (1)

where –input parameters, ia F – function which determines the type of potential.

The type of function and choice of algorithm (Mamdani, Sugeno, Cucamoto and others [3]) of unclear conclusion depends on the mechanism of unclear productional rules construction, which are used in consulting



and leading models, and for basis have a base of knowledge, formed the specialists-experts of subject domain or got as a result of neuronet teaching, the educational great number of which, in same queue, is based on experimental information as an aggregate of unclear productional rules. The vehicle of fuzzy logic well recommend itself in researches of economic and social processes, in particular at the calculations of the efficiency integrated indexes, multicriteria tasks decision [5], determination of economy growing competition between regions in China [6]. In paper [7] we proved possibility using of Mamdani and Sugeno algorithms for determination of recreational potential. It is shown that results, got after these methods, are in good correlation with the experts estimations. Therefore in further calculations we used one of these algorithms, namely Sugeno algorithm with the Gaussian belonging functions [8]. The choice of this algorithm is argued that at presence of knowledge experimental bases the expedient will be become by the use of neurons hybrid networks of ANFIS (Adaptive Neuro-Fuzzy Inference System) where Sugeno method lies in basis. As input parameters of unclear attractiveness potential we suggest to choose distance to the nearest road and distance from road to the nearest center of crystallization. The calculation of the integral spatial distributing of attraction potential field closeness use methods of solid physics [11]. For modeling of settlements central part growth we took advantage of the «Casual rain (CR)» modified method [12].

If exist clusterization centers rationed weight is determined for every center. If settlement has several

centers of attractiveness gravimetric multipliers could be calculated as a relative amount of people which visited such objects for certain period of time:

n iw

nii

ii

S

Sw

,1

(2)

where – visitors quantity who visit the i-th object. iS

In obedience to the algorithm CR a particle moves on a casual chord to one of clusterization centers. Clusterization center for every particle gets out random appearance depending on the size of the rationed weight

[13]. To avoid appearance of empty areas [14] after particle aggregation, its copy («transparent particle»)

which continues the motion to the center is created irresponsive on the crystallized particles. As soon as she meets in an area, where are not the aggregated particles (a particle got in empty area) in a small radius, status of “ordinary particle” is appropriated a «transparent particle» and the accretion algorithm proceeds by classic rules.

iw

Influencing of the potential field is taken into account in such cases: the source potential field of the explored region is rationed; aggregation probability is determined, as probability of two independent events offensive, namely: presence in a number of the aggregated particle and «possibility» of aggregation in the set point from the side of the rationed potential field (1). In our calculations reliability yxU , yxPa , of stay next to the

moved particle of the aggregated cluster was adopted equal 1, if aggregated atom is located nearby on a verge with a particle; 0,5 – if the aggregated atom is located alongside bias; and 0,01 – in other case. So probability of particles aggregation defined as:

yxPyxUyxP a ,,, . (3)

An unzero aggregation probability in an area where the aggregated particles are absent in the nearest

neighbouring strengthens influence of the potential field on the form of created cluster, however results in appearance of the separated aggregation centers.

yxPa ,

Evaporation was designed as follows: if «transparent particle»during attractor growth gets in an area surrounded

by aggregated particles it evaporates with probability : dN dp

yxUdpd ,1 (4)

where – empiric evaporate index, – competitiveness level. d dN

For settlement periphery modeling the matrix of created structure is divided into the matrix of settlement central part which consists of particles that have next-door neighbours and free particles matrix. The first matrix is examined as single attractor and particles set by second matrix continue motion in the potential field in obedience to the model of diffusely limited aggregation (DLA). The classic DLA model is very simple: particles which accomplish the casual moving form a cluster as an



aggregation result. I.e. a particle beginning motion from the randomly chosen point joins to the clustering center point or to the before aggregated particles. Computer researches showed that the result of this process is creation of complicated ramified fractals [12, 15] which have a spherical form. In our case a particle must move in the potential field which must influence on the form of fractal. Modelling of this motion could be made using methods of molecular dynamics [16-20].

Particle aggregation takes place in the case when during motion it runs into the clustering center or before aggregated particles. In case if the input parameters of unclear potential hinder aggregation (coast, bog, and reservoir) a particle is withdrawn from a calculation. Particle mass and environment resistance index are empiric parameters of this theory. At prognostication of complicated social structures such as settlements, mass can be interpreted as a measure of investment ability into settlement certain object (sanatorium, hotel, office, cottage, summer residence, etc.) or infrastructure (supermarket, shop, booth, etc.). The closeness of environment can be interpreted as a measure of region investment assistance.

COMPUTER EXPERIMENT Adequacy of this methodology was approved for the group of socio-economic objects on mesonic level for different countries of the world. For example, modeling of region periphery in Sudak - Novyy Mir (Ukraine) got fractal consisted approximately of 74 000 aggregated particles. The general structure of the got fractal shows good correlation with the present region structure (fig. 1, fig. 2). Fractal growth at the modeling of this region reminds the projection of physical crystal growth on the flatness. Main attractiveness objects are located along a coast and near a road. As is obvious from fig. 2 limitations put obstacle in fractal growth in sea area. Fig. 2 evidently represents infrastructure segmentation of the explored region. You can see on a figure that the most expensive infrastructure is aggregated from heavy particles disposed along a coastline and surrounds the basic historic-cultural centers of attractiveness in Sudak i.e. engulfs most attractive area. The infrastructure of middle class is disposed close to the expensive elements and makes a small layer by comparison to the expensive elements in spite of the initial amount of these particles exceeds the amount of heavy twice. So, middle business is absorbed by large one. Penetration of middle class elements into the area of expensive infrastructure is insignificant what can not be said about small business. The elements of small business which correspond easy particles are disposed on considerable distance from main attractiveness centers and gravitate to the road. From fig. 2 (b) evidently that this segment engulfs both the outskirts of the explored region and «leaks» between the expensive infrastructure elements. It is possible to get to the coast only by one motor-car way. From fig. 2 (b) evidently that before researched region entrance the aggregation center is formed by easy particles. As is obvious from fig. 1 and fig. 2 (b) the aggregation center got in theory after distance to the coast and size well correlates with a present settlement. The differences between forma can be explained by using territorial limitations.

Fig. 1. Sudak – Novyj Svit

Fig. 2. Segments of fractal structure of Sudak – Novyj Svit

а) б)

в)



Modelling of attractor evolution depending on algorithm’s iterations confirmed features which are inherent the real systems. You could observe stagnation point on the graph of elements systems dependence which are characterized the high level of competition (fig. 3). Consequently, in the process of evolution the elements of the system multiply the level of the specialization from one side, from other – the level of monopolization grows constantly. So, the system itself – monopolized. Research of change symmetry showed (tab. 1) that at the beginning of fractal growth (100 particles) most closeness of aggregated particles is observed in the attractor center. Plenty of the separated aggregated particles predefined an unzero probability of aggragation appears on a perimeter. As is obvious from fig.3 the stagnation point appears when the amount of initiating particles achieves 10 000 – 20 000. To this moment the aggregation processes prevail above evaporation. As is obvious from a table 1 the underlying attractor structure on this stage has a chaotic structure. Under reaching the stagnation point the processes of evaporation are counterbalanced by the aggregation processes. At 10 000 initiating particles there is stratification of particles on two areas. First – the attractor district is carried by chaotic character, second is periphery which begins structured. Small empty areas which have a linear structure appear, which in future are multiplied and can be interpreted as streets. At growth of initiating particles amount the border of section between a chaotic structures becomes more expressive and its radius diminishes gradually.

Table 2. Dependence of attractor structure on the amount of algorithm iterations 100inN

0001inN

0005inN

00010inN

00020inN

00050inN

000100inN

000500inN

An alike structure is indeed observed as in the socio-economic systems (fig. 4-5) so in physical fractals (fig 6). It can serve as confirmation of adequacy of the offered model.

0 100 200 300 400 500 600 700 800 900 10000

2000

4000

6000

8000

10000

12000

Fig. 3. Dynamics of running the number of particles with the high level of competition depend on algorithm iterations



Accordance of theoretical structures and got segments with experimental confirms adequacy of the offered prognostication method and segmentation and can serve as foundation for further theoretical and practical researches. SUMMARY The method of construction of the unclear potential attractiveness field is resulted. The algorithm of input parameters calculation of unclear model is offered. The algorithm of calculation of fractal growth in the unclear potential field by the methods of «casual rain» and DLA is resulted. It is shown that taking into account elements of molecular dynamics vehicle, forces of viscid friction and limitations in the model of DLA lets adequately to describe motion of particle in the unclear potential field. The method of underlying structure segmentation of growing cluster is developed. Influence of model empiric parameters on the form of growing cluster is explored. The values of these empiric parameters at which observed fluctuation in growth of cluster which result in appearance of new aggregation centers are defined. Explained and grounded mechanism of such phenomenon. During computer calculations the structures of fractals which well correlate with present experimental information are got. It confirms supposition that deciding part in forming of settlements is acted by a present infrastructure, namely: connection ways and present attractiveness centers. The got segments are confirmed by the basic economic features of present infrastructure. It is shown that the aggregation centers formed due to fluctuations repeated basic features of the real settlements of the explored regions by from and location. Analogy of basic processes which are observed in the process of crystal growth of (evaporation, diffusion and other) in the socio-economic systems is grounded. It is proved that the phenomenon of evaporation is an inalienable process in the dynamics of society frames development. Therefore ignoring this phenomenon lead to incorrect results. It is shown that in the process of evolution there is a stagnation point after which the level of general competition of the open system diminishes at the unchanging common amount of elements. Consequently in course of time there is multiplying the level of specialization and growth of system constituent’s

monopoly. I.e. system left on itself – monopolized. From other side it is confirmation that the system is self organized. In paper is shown that an initial structure of base infrastructure like lift, hotel, complex, factory and others like that, in the process of development are almost not changed. Around the new socio-economic system middle and small business grows chaotically by rapid rates. Afterwards, in the process of decline and origin of business elements around attractor appear streets, quarters, dear and other infrastructures elements which put in order this system, minimizing chaos and competition. These processes are observed at any levels of competition and competitiveness. It is proved that exactly a competition leads to self organization of the system. It is shown that amount of aggregated particles and accordingly closeness to the crystal linearly depends on the level of competitiveness. It is possible to establish in general: loosening the competitiveness of objects of the system results in diminishing of fractal closeness. It predetermines diminishing of monies streams of the system. In other side absence of competition (monopolization of the system) from considerable slows development of the system and results in the phenomenon of «stagnation». The middle level of competitiveness speeds up the dynamics of system development. Good correlation of experimental and calculations information proves adequacy of the offered methodology and allows using it for further prognostication of both geometrical form and underlying structure of settlements.

Fig. 4. Central part of St. Petersburg

Fig. 5. Central part of New York

Fig .6. Microstructure of alloy of bromine silver with bromine potassium



Research and analysis of new aggregation centers appearance can serve as a scientific ground for planning of regions development strategy. LITERATURE: [1] ТКАЧЕНКО, Т. И. Постоянное развитие туризма: теория, методология, реалии бизнеса. -

К.:КНТЕУ, 2006. - 537с. [2] Маркетинг в туризме: Учеб. пособие / А.П. Дурович. 3-- е изд., стереотип. - Мн.: Новое знание,

2003. - 496 с. [3] ЛЕОНЕНКОВ, А. В. Нечеткое моделирование в среде MATLAB и fuzzy, TECH Спб.: Бхв-петербург,

2005. - 736 с. [4] ПЕТРЕНКО, В. Р, КАШУБА, С. В. Нечеткая модель анализа эффективности бизнесов-процессов

предприятия, Сложные системы и процессы : 2006. - 2, - с.18-26. [5] TSUNG-YU C., MEI-CHYI C., CHIA-LUN H., A fuzzy multi-criteria decision model for international

tourist hotels location selection, International Journal of Hospitality Management. In Press. [6] MA, S.; FENG, J.; CAO, H. Fuzzy model of regional economic competitiveness in GIS spatial analysis,

Case study of Gansu Western China, Fuzzy Optim Decis Making, 2006. - #5, p.99-111. [7] ВИКЛЮК Я.И. Картографическое моделирование рекреационного потенциала єврорегіону

„верхній Прут” на основе нечеткой логики, Отбор и обработка информации : 2008. - 28(104), в печати

[8] ДЬЯКОНОВ, В. П. ; КРУГЛОВ, В. В. MATLAB 6.5 SP1/1 SP2 + Simulink 5/6. Инструменты искусственного интеллекта и биоинформатики. Серия «Библиотека профессионала». - Г.:СОЛОН-ПРЕСС, 2006. - 456с.

[9] ШТОВБА, С. Д. Проектирование нечетких систем средствами Matlab. Г.: Горячая линия - Телеком, 2007. - 288с.

[10] ВИКЛЮК, Я. И. Прогнозирование геометрической структуры населенных пунктов методом модифицированной диффузно-ограниченной агрегации в нечетком потенциальном поле, Информационные технологии и компьютерная инженерия : 2008, в печати

[11] МАДЕЛУНГ, О. Теория твердого тела. М.:Наука, 1980. – 416с. [12] Фракталы в физике. Труды VI международного симпозиума по фракталам в физике. Под.ред

Л.Пьетронеро., Г. Мир, 1988г. - 670с. [13] ТОМАШЕВСЬКИЙ, В. М. Моделирование систем., К.:Издательская группа BHV, 2005. - 352с. [14] ВИКЛЮК, Я. И. Методология прогнозирования социально-экономических процессов методами

фрактального роста кристаллов в нечетком потенциальном поле, Вестник ТДТУ : 2008. - 2, в печати.

[15] КРОНОВЕР, Г. Фракталы и хаос в динамических системах, М.:Техносфера, 2006. - 488с. [16] MARI, A. ; PEREZ-MARTIN, C. ; JIMENEZ-RODRIGUEZ, J. J.; J. JIMENEZ-SAEZ, C. Shallow

boron dopant on silicon An MD study, Applied Surface Science : 2004.– #234, – p. 228–233. [17] SIBONA, G. J. ; SCHREIBER, S. ; HOPPE, R. H. W.; STRITZKER, B. ; REVNIC, A.; Numerical

simulation of the production processes of layered materials, Materials Science in Semiconductor Processing : 2003. - #6, - p.71-76.

[18] MOON, W. H.; HWANG, H. J.; Atomistic study of elastic constants and thermodynamic properties of cubic boron nitride, Materials Science and Engineering : 2003. - # B103, - p.253-257.

[19] ГУЛД, Х.; ТОБОЧНИК, Я. Компьютерное моделирование в физике, Пер. с англ.: В 2-х ч. - Г.:Мир,1990. - Ч.1. - 349 с.

[20] КАПЛАН, И. Г. Введение в теорию межмолекулярных взаимодействий, Г.:Наука, 1982. - 311с. ADDRESS Yaroslav Vyklyuk NU «Lvvska Politechnika» Ukraine, 58000,Chernivtsi, Simovycha str.21 tel.: +380974065842 e-mail: [email protected]



THE USE OF FUZZY LOGIC FOR SEASONAL RECREATION ATTRACTIVENESS OF

TERRITORY CARTOGRAPHIC DESIGN

Olga Artemenko

Bucovinian University

Abstract: This paper describes the application method of fuzzy complex recreation potential for drawing a map. The calculation method for territory’s recreation attractiveness index is considered on the base of fuzzy logic. The tourist attractiveness seasonal maps of the Chernivtsi region (Ukraine) are created. The developed maps represent the territory tourist attractiveness level and change during a year. The attractiveness of territory is examined as for the separate types of rest so in an integral form.

Keywords: fuzzy logic, potential, recreation attractiveness map.

INTRODUCTION Nowadays tourism not only purchased world popularity as the cultural phenomenon but also grew into one of the most perspective economics industries. Profits got in tourism are one of sources for filling the budgets of whole countries and separate regions. In addition, tourist activity inflicts considerably less harm ecology and peoples health, than, for example, industrial enterprises. Various small and middle tourist businesses quickly develop in Ukraine in all regions and in the Chernivtsi region in particular. However, often tourist complexes, firms, tourist centers, hotels and others like that, which are built and organized without proper scientific basis, do not get sufficient number of orders, and consequently appear unprofitable. The enterprises of tourist industry would function considerably more effective, if it was possible to determine potentially attractive for tourists and holiday-makers territories, and also to estimate the level of their attractiveness and develop specialization on the proper types of rest. PURPOSE AND RESEARCH ACTUALITY A research purpose was a construction of territory recreation attractiveness maps on the basis of fuzzy complex recreation potential. Research actuality was to determine the level of territory attractiveness for tourists and holiday-makers during a year with the purpose to form a strategy for tourist and recreation industries enterprises. The practical value of the paper is to give concrete recommendations to the institutions of local governing and investors about the optimum placing, specialization and development of the tourist-recreation systems (TRS) on territory of a region. Produced maps must serve as a scientific basis for a region economic development strategies. METHODOLOGY The recreation attractiveness of territory is determined by the types of rest and recreation, that can be organized and carry out on this territory. Rest and recreation, in the turn, depend on climatic, geographical, historic and cultural terms and peoples activity. In this research the potential of territory recreation attractiveness is estimated on the base of fuzzy logic mathematical instruments [1]. The universal form of the aggregated index for territory attractiveness P can be presented as:

nppfP ,...,1 (1)

where – are input values, – is a function which defines the form of potential. ip f



For finding places potentially attractive for tourism and recreation it is used a recreation potential maps

construction method [2]. The map of territory T is covered by a rectangle dcbaП ,, . Obviously, that

the rectangleП contains the set (territory) T ( ПT ). The rectangle П is divided with a net yx ,

where:

; (2); N

kkx x

0

; (3); M

lly y

0

Nkkhxx xok ,0, ; (4);

Mllhyy yl ,0,0 ; (5);

N

abhx

; (6);

M

cdhy

. (7).

For every centre of net the input parameters values are determined. The received matrices serve as the input values in fuzzy model for calculation an aggregated recreation attractiveness potential (1). The calculation result is a matrix which determines the potential form of territory T. The aggregated index of territory tourist attractiveness consists of a few separate attractiveness indexes, that are based on the certain types of rest. For territories of the Chernivtsi region the currently important types of rest and recreation can be united in four groups:

1p – winter rest;

2p – rest in a summer period at waters;

3p – rest in spring and autumn at the countryside;

4p – excursions and review of historic and cultural sights.

The seasonal recreation potential of territory is determined as:

tptpftP 41 ,..., . (8)

The linear convolution product is used for the calculation of the aggregated recreation attractiveness index. It allows getting the integral index in those cases, when the input variables are independent and equivalent sizes [3]:

4

1iii ttptP , (9)

where ti – are the parameters of groups attractiveness indexes normalized values.

The normalized value of coefficient i estimates with a formula:

n

ii

ii

t

tt

1

*

*

, (10)

where n is the common amount of parameters in given attractiveness potential, and is determined as: *

i



tHCt iii * , (11)

where – a percent of people that want to have the indicated type of rest, iC tHi – is seasonal possibility of

having a rest. The rest management and providing in a summer period of the year depends on 7 basic parameters which the followings linguistic variables are defined for: х1 – swimming; х2 – rafting, kayaking and etc.; х3 – fishing; х4 – boating, going for a drive on catamarans and others like that; х5 – a waters type (river, lake, pond, etc); х6 – approach roads condition; х7 – preparedness of the territory for arranging a rest. It is suggested by us to create two subsystems for the calculation of summer rest recreation potential. The first of them unites the kinds of rest on water and determines the potential amount of types of rest, accessible for these waters:

4111 ,..., xxfp . (12)

This indicator is one of the incoming parameters in the second subsystem which defines overall summer rest potential on this territory:

765111 ,,, xxxpfp . (13)

Winter rest is mainly related to the kinds of skiing and mountain-skiing. Particularly, it is important for the Chernivtsi region which relief is mainly mountainous. For a few last years businessmen opened about 10 bases for mountain-skiers in the different districts of region. The expert’s opinion such factors influence the favourableness of conditions for organization and running a tourist business in mountain-skiing rest field: x8 – slopes height; x9 – slopes length; x10 – slopes position; x11 – slopes gradient; x12 – approach roads condition. The territory attractiveness potential for winter rest is determined:

1282 ,..., xxfp . (14)

Rest in spring and autumn mainly consists in spending of weekends on countryside. As a rule, tourists do not move away on far distances from their home. The basic factors influencing on the territory attractiveness potential for tourists in spring and autumn is an ability to get tourist services. Most popular kinds of rest in considered period of the year are: х13 – picnics; х14 – gathering berries, mushrooms and other; х15 – other entertainments on countryside (for example, riding a horse, bicycle and others like that). The group index of territory attractiveness for a spring and autumn rest estimates as:

1514133 ,, xxxfp . (15)

The last group index – potential of historic and cultural attractiveness was estimated in work [4]. This index depends on two parameters: geographical coordinates of the Chernivtsi region historic and cultural sights and places interesting for

tourists; foregoing objects significance rating estimations, which are determined by experts.



The index of territory historic and cultural attractiveness is determined as:

m

i

rN

i

ikli

ep1

)(

4

2

2,

, (16)

where: – is distance between territory, for which potential is calculated, and a tourist object that has an

historic and\or cultural significance; – a standard deviate, it determines the form of function (the order quantile ½ defines “optimum distance” at which potential becomes two times less);

iklr ,

– is a rating estimation of the historic and/or cultural object recreation potential; N – is a rating maximal value (when m=N all tourists will visit this object). The coefficient of territory historic and cultural attractiveness shows, how much optimal this tourist object is located in relation to Chernivtsi region basic historic and\or cultural sights, those places which are interesting for a tourists sightseeing. The value of coefficient of historic and cultural attractiveness (16) for various territories can strongly differ in the numbers order, which negatively influences on a basic result. That is why, it is executed a normalization for historic and cultural attractiveness potential, using a fuzzy algorithm where this index is presented as a linguistic variable. As a result of normalization the values of coefficient are in a range between 0 and 1. On the whole, the complex seasonal attractiveness potential of territory for holiday-makers and tourists depends on 17 basic parameters, 15 of them are presented as fuzzy linguistic variables.

Results

To approbate the offered method it is decided to take the Chernivtsi region. On pict.1 it is represented potentially attractive for tourists and holiday-makers zones of the Chernivtsi region. An image on the left demonstrates the attractiveness of territory in January. Right part of picture contains the map of recreation potential for March.

a) January b) March

Picture 1. The integral recreation attractiveness maps of the Chernivtsi region. On the picture it is shown, that in winter time a south-west part of region is attractive for tourists. In particular, it is a result of its mountainous relief that furthers in organization of mountain-skiing rest. North and east parts of a



region are flat and almost unattractive for arranging a rest in winter. In March most of region’s territory is attractive for tourists. It means that in different districts of a region it is advisable to create and develop TRS for the different kinds of rest. On pict.2 the maps of recreation attractiveness are shown for June and September.

a) June b) September

Picture. 2. The Chernivtsi region seasonal maps of complex tourist attractiveness. The picture shows territories potentially attractive for rest in June and September. In summer attractiveness of territory is mainly related to the rest on waters. In this characteristic the Chernivtsi region does not belong to the popular areas of rest, as he does not have large and comfortable for swimming and etc. waters. However fishing and rafting are present and perspective enough. In September the territory of the region is very attractive for tourists. It is caused by numerous historic and\or cultural objects, and also forests, full of mushrooms, berries and others like that. In general, the seasonal maps of recreation potentials are developed for each month of a year. They represent the change of territory’s attractiveness that is influenced by the seasonality factor. CONCLUSIONS This paper describes the calculation method for the aggregated recreation attractiveness index of territory. The seasonal maps of tourist attractiveness of the Chernivtsi region are built with the recreation potential maps construction method. The obtained results enable to define perspective places for developing tourist infrastructure and TRS. The maps, produced with computer calculations, allow monitoring the change of territory’s attractiveness during a year. First of all, the offered method will allow the tourist industry investors to elect more effectively direction and dimensions of capital investments in planning their strategy, arranging PR-actions. Secondly, the regional government gets a scientific base which allows to make effective strategy for regional development and to optimize activity of tourist industry in the region. Extremum’s of recreation potential in most cases coincide with the places where tourism develops actively. In addition, the maps of recreation potentials show perspective for creation new TRS territories. The offered method allows building the maps of recreation attractiveness for any territory. The used method for calculation the potential attractiveness of territory takes into account various parameters; therefore the received results give a complex representation of territory’s perspectives for tourist business. Mathematical instruments applied in calculations can be easily integrated with the different informational systems.



REFERENCES [1] LEONENKOV, A. Fuzzy modelling in MATLAB and fuzzyTECH. St. Petersburg.: БХВ-Петербург 2005. [2] KYFJAK, V.; VYKLYUK, Y. A.; KYFJAK, O. (). Determination of optimal recreation-tourist zones in

the conditions of international collaboration. Forming of marketing relations in Ukraine, 2007, 1 (68), 132-136.

[3] GNATIENKO, G.; SNITYUK, V. (). Decision-making expert technologies: Monograph. Kyiv : 2008 “Маклаут” Ltd.

[4] YAKIN V.; RUDENKO V.; KOROLJ O.; KRACHILO M.; GOSTYUK M. and others (). Problems of geography and tourism management. Chernivtsi: Chernivtsi national university, 2006.

ADDRESS: Olga Artemenko Bucovinian University Simovich str. 21 Chernictsi, 58000 Ukraine E-mail: [email protected]



FORECASTING OF ECONOMIC QUANTITIES USING FUZZY AUTOREGRESSIVE -

MODEL AND SOFT RBF NEURAL NETWORK

Dušan Marček1, 2

1University of Žilina 2The Silesian University Opava

Abstract: Most models for the time series of stock prices have centered on autoregressive (AR) processes. Traditionally, fundamental Box-Jenkins analysis have been the mainstream methodology used to develop time series models. Next, we briefly describe the develop a classical AR model for stock price forecasting. Then a fuzzy regression model is introduced. Following this description, an artificial soft RBF neural network is presented as an alternative to the stock prediction method based on AR models. Finally, we present our preliminary results and some further experiments that we performed.

Keywords: Time series analysis and forecasting, fuzzy autoregressive model, soft RBF neural network.

1 INTRODUCTION In [4] the stock price autoregressive (AR) models based on the Box-Jenkins methodology [2] were described. Although an AR model can reflect well the reality, these models are not suitable for situations where the quantities are not functionally related. In economics, finance and so on, there are however many situations where we must deal with uncertainties in a manner like humans, one may incorporate the concept of fuzzy sets into the statistical models. The fuzzy regression is another efficient approach for computing the parameter of the structure for an uncertain situation and for predicting of uncertain events following the decision. The fuzzy regression models have been in use in analyses for many years. Lots of issues of journal Fuzzy Sets and Systems as well as many others have been articles whose analyses are based on the fuzzy regression models. From reviewing of these papers, it become clear that in economic applications the use of method is not on the same level as analyses using classical linear regression. Computers play an important role in fuzzy regression analyses and forecasting systems. The widespread use of the method is influenced by inclusion of fuzzy regression routines in major computer software packages and selection of appropriate forecasting procedure. The primary objective of this paper is a focused introduction to the fuzzy regression model and its application to the analyses and forecasting from classical regression model of view. In Section 2, we briefly describe some basic notions of AR modelling and a model based on signal processing. In Section 3 the basic principles of fuzzy regression modeling approaches are given. In Section 4 the forecasting model based on the soft RBF neural network is described. Section 5gives the summary statistical accuracy statistics for the AR(2) model and model based on soft RBF neural network. 2 AR MODELING AND MODELS BASED ON SIGNAL PROCESSING We give an example that illustrates one kind of possible results. We will regard these results as the referential values for the approach of fuzzy autoregressive and ANN modelling. To illustrate the Box-Jenkins methodology, consider the stock price time readings of a typical company (say VAHOSTAV company). We would like to develop a time series model for this process so that a predictor for the process output can be developed. The data was collected for the period January 2, 1997 to December 31, 1997 which provided a total of 163 observations (see Fig. 1). To build a forecast model the sample period for analysis y1, ..., y128 was defined, i.e. the period over which the forecasting model was developed and the ex post forecast period (validation data set), y129, ..., y163 as the time period from the first observation after the end of the sample period to the most recent observation. By using only the actual and forecast values within the ex post forecasting period only, the accuracy of the model can be calculated.



Fig. 1 The data for VAHOSTAV stock prices (January 1997 - August 1997) and the values of the AR(7) model

for VAHOSTAV stock prices estimated by GL algorithm After some experimentation, we have identified two models for this series (see [2]): the first one (1) based on Box-Jenkins methodology and the second one (2) based on signal processing.

y a y a yt t t t 1 1 2 2 (1) t N 1 2 2, , ... ,

y a yt k t kk

1

7

t (2)

t 1 2, , ... , N - 7 The final estimates of model parameters (1), (2) are obtained using OLS (Ordinary Last Square) and two adaptive filtering algorithms in signal processing [1]. The Gradient Lattice (GL) adaptive algorithm and Last Squares Latice (LSL) algorithm representing the parameter estimates of the predictors (1), (2) were used. In Tab. 1 the parameter estimates for model (2) and corresponding RMSE’s are given. 3 FUZZY AUTOREGRESSIVE (FAR) MODELLING Next, we examine the application of fuzzy linear regression model [6] to the stock price time readings used in (1) and (2). Recall that the models in (1) and (2) fit to the stock prices were the AR(2) and AR(7) processes. In the fuzzy regression model proposed by Tanaka et al. [6], the parameters are the fuzzy numbers. The regression function of such fuzzy parameters can be modeled by the following equation

Tab. 1 OLS, GL and LSL estimates of AR models

Model Order Est.proc a11.113

a2-0.127

a3 a4 a5 a6

a7 RMSE* (1) 2 OLS 26.639 67.758 (2) 7 GL -0.7513 -0.1701 -0.0230 -0.0128 -0.0028 -0.0472 0.0084 68.540 (2) 7 LSL -0.8941 -0.6672 0.7346 -0.2383 0.1805 -0.5692 0.4470 94.570

*ex post forecast period

Y A x A x A xt t t k k kt 0 0 0 1 1 1 ( ) ( ) ,..., ( ) = (3) A x t

where are fuzzy numbers, A A Ak0 1, ,..., and are fuzzy addition and fuzzy multiplication operators

respectively, Y is fuzzy subset of . This kind of fuzzy modelling is known as fuzzy parameter extension. t yt

The problem to find out fuzzy parameters gives the following linear programming solution [6, 7]

min s c c ck 0 1 ...

subject to c j 0

and (4) ( ) ( )h yt 1 c x x 0

( ) ( )1 h ytc x x 0

for t = 1, 2, ..., N

where , j = 0, 1, ..., k is the width or spread around the center of the fuzzy number,

denotes vector of center of the fuzzy numbers for model parameters,

denotes vector of regressor variables in (3), h is an inposed threshold h [0, 1] (see [5]). A

c j

( ,,x x0 1

, )0 1 ... , k

x' ( , ) ' xk...,



choice of the h value influences the widths c of the fuzzy parameters. The h value expresses a measure of the

fitting of the estimated fruzzy model (3) to the given data. The fuzzines of j

c ( , , )c c ck0 1 ... , of the

parameters ~A0 ,

~A1 , ... ,

~Ak for the models (1) and (2) are given in Tab. 2.

Tab. 2

h=0.5 ~Ak k: 0 1 2 3 4 5 6 7

Model AR(2) Modal values( )

26.639 1.113 -0.127

Spread (c) 0 0 0.229008 Model AR(7) Modal values( )

45.930 1.085 0.0861 -0.2531 0.0836 -0.0057 0.2081 -0.2281

Spread (c) 0 0 0 0 0.209587 0 0 0 The forecast for future observation is generated sucessively throught the Eq. (3) by replacing the functions of the

independent variables ( j jx( t ), j = 0, 1, ... , k by observations . Then the forecasting function of the fuzzy

AR process is

yt j

Y T A A y A y A yT T T k T k 1 0 1 2 1( ) ,..., 1

(5)

whereYT1 is the forecast for period T+1 made at origin T. We observe that the forecasting procedure (5)

produces forecast for one period ahead. As a new observation becomes available, we may set the new current period T+1 equal to T and compute the next forecast again according to (5).

T( )

), vc

4 SOFT RBF NEURAL NETWORK APPROACH The concept of soft RBF neural network can be approached from several different avenues. The one that we have used for stock price forecasts is shown in Fig. 2. The output layer neuron is linear and has a scalar output given by [3]

ty ,(x tG= =

s

tj

tj

o

o

1,

,

s

jv

1

j

tj , =

s

jjt

jt

cx

cx

,(

,s

jtjv

12

2

1,

)

)(

, t = 1, 2, ..., N.

(6)

where N is the size of data samples, s denotes the number of the hidden layer neurons, are the trainable

weights connecting the component of the output vector o . The hidden layer neurons receive the Euclidian distances

jv

)j( cx and compute the scalar values of the Gaussian function that form the hidden

layer output vector , where is a k-dimensional neural input vector, represents the hidden layer weights, tjo ,

),(2 jt cx

to tx jw

2 are radial basis (Gaussian) activation functions. Note that for an RBF network, the hidden layer weights

represent the centres of activation functions jw

jc2

Fig. 2 Fuzzy logic (soft) (b) RBF neural network architecture



A serious problem is how to determine the number of hidden layer (RBF) neurons. The most used selection method is to preprocess training (input) data by some clustering algorithm. After choosing the cluster centres, the shape parameters j must be determined. These parameters express an overlapping measure of basis functions.

For Gaussians, the standard deviations j can be selected, i.e. j ~ c , where c denotes the average distance

among the centres. The frequently used learning technique uses clustering to find a set of centres which more accurately reflect the distribution of the data points. For example by using K-means clustering algorithm, the member of K centres must be decided in advances. After choosing the centres w, the standard deviations σ j can be selected as σ j ~ Δc j where cj denotes the average distance among the centres w j. To train the weights v j, the first-order gradient procedure is used. These weights can be adapted by the error back-propagation algorithm. In this case, the weight update is particularly simple. If the estimated output for the single output neuron is

t, and the correct

output should be , then the error is given by = - and the learning rule has the form y

ty te te ty ty

tjv , ← + tjv , tjo , te , j = 1, 2, ..., s; t = 1, 2, N (7)

where the term is a constant called the learning rate, is the normalised output signal from the hidden

layer. Typically, the updating process is divided into epochs. Each epoch involves updating all the weights for all the examples.

tjo ,

5 EMPIRICAL RESULTS The network described in Section 4 was trained in software at the Faculty of Management Science and Informatics Zilina. Our soft RBF neural network was trained on the training data set. Periodically, during the training period, the RMSE of the soft RBF neural network were measured not only on the training set but also on the validation set. The RMSE’s of our predictor models are shown in Tab. 2. The initial results of the soft RBF neural network forecasting model are slightly better. Tab. 2

Model RMSE* AR(2) 67.7 Soft RBF neural network 66.9

* Validation set ACKNOWLEDGEMENT This work was supported by Slovak grant foundation under the grant No. VEGA 1/0024/08 and from the Grant Agency of the Czech Republic under the grant No. GAČR 402/08/0022. REFERENCES [1] BAYHAN, G. M.: Sales Forecasting Using Adaptive Signal Processing Algorithms. Neural Network

World 4-5/1997, Vol. 7, pp. 579-589 [2] BOX, G. E.; JENKINS, G. M.: Time Series Analysis, Forecasting and Control. Holden-Day, San

Francisco, CA 1976 [3] KECMANOSKO, B. Neural networks and fuzzy systems - a dynamical systems approach to machine

intelligence. Prentice-Hall International, Inc. 1992 [4] MARČEK, D. Stock Price Prediction Using Autoregressive Models and Signal Processing Procedures.

Proceedings of the 16th Conference MME’98, Cheb 8.-10.9.1998 [5] SAVIC, D. A.; PEDRICZ, W. Evaluation of fuzzy regression models. Fuzzy Sets and Systems. 39 (1991)

North-Holland (51-53) [6] TAKAGI T.; SUGENO M. Fuzzy identification of systems and its applications to modelling and control.

IEEE trans. System Man. Cybernet, 16 yr 1985, (116-132) [7] TANAKA, H.; UEJIMA, S. and ASAI, K. Linear Regression Analysis with Fuzzy Model. IEEE

Transaction on Systems, Man and Cybernetics, Vol. SMC - 12. No. 6, November/December 1982, (903-906)



ADDRESS Prof. Ing. Dušan Marček, CSc. The Faculty of Management Science and Informatics University of Žilina Univerzitná 8215/1 010 26 Žilina Tel.: +421-41-513 4061 Fax: +421-41-513 4055 Prof. Ing. Dušan Marček, CSc. Institute of Computer Science Faculty of Philosophy and Science The Silesian University Opava Bezručovo náměstí 13 746 01 Opava Tel.: +420 553 684 200 Fax: +420 553 716 948



APPROXIMATION AND FORECASTIG ABILITY OF VARIOUS RBF AND

GRANULAR NNW: APPLICATION TO SALES PROCESS MODELLING

Milan Marček1,2,3

1The Silesian University Opava 2University of Žilina

3MEDIS - Medical Innovations

Abstract: At first, we discuss the basic structure of the fuzzy system as a simple yet powerful fuzzy modeling technique. Neural networks and fuzzy logic models are based on very similar underlying mathematics. The similarity between RBF networks and fuzzy models is noted in detail. Then, we propose the extension of RBF neural networks by the cloud model. Time series approximation and prediction by applying RBF neural networks or fuzzy models and comparisons between the various types of RBF networks and statistical models are discussed

Key words: Probabilistic time-series models, fuzzy system, classic and soft RBF network, cloud models, granular computing.

INTRODUCTION In this paper, we consider the approximation ability of ARMA models and models based on RBF neural ntworksfuzzy systems to “explain” the behaviour of time-series variables. In addition, we explore some of the more important specifications associated with approximation of time-series variables using RBF networks. A MODEL OF FUZZY SYSTEMS This section concentrates on the basic principles of identifying input-output functions of systems using fuzzy systems. Fuzzy systems theory have been recently consolidated and presented by B. Kosko [5].

Fig. 1: Fuzzy system architecture.

The basic fuzzy system architecture is shown in Fig. 1. In this architecture the fuzzy system maps input fuzzy sets A to output fuzzy sets B. The fuzzy inference computes the output fuzzy sets , weights them with the

weights wiB

i, and sums to produce the output fuzzy set B, i.e. ;

iii BwB (1)

The fuzzy system is distributed and consists of a series of a separate fuzzy rules (relations) of the type of if Ai then Bi. Centroidal output converts fuzzy sets vector B to a scalar. The most popular centroidal defuzzification technique uses all the information in the fuzzy distribution B to compute the crisp y value as the centroid y~ or

centre of mass of B, i. e.

dyydyyyy BB )(/)(~ (2)



where B represents the union of all clipped output fuzzy sets. When the output membership functions are

singletons, then, in the case of an → function, Eq. (2) becomes k

n

jj

n

jjj xxyy

11

)(/)(~ (3)

where stands for the centre of gravidity of the jth output singleton, the notation jy is used for a membership

function and n denotes the number of rules. As mentioned earlier the output fuzzy sets can be calculated if all the separated fuzzy rules are known and the weights are determined. As in fuzzy logic systems all operations involve sets, the amount of calculation per inference rises dramatically. In a fuzzy system, powerful tools for generating fuzzy rules purely from data are neural networks. In next section we show, how to obtain fuzzy rules and how to determine the weights wi for fuzzy system using RBF networks.¨ RBF NEURAL NETWORK IMPLEMENTATION OF FUZZY LOGIC Fuzzy systems offer methodologies for managing uncertainty in a rule-based structure. In this section, RBF neural network structures are used (see Fig. 2) as tools of performing fuzzy logic inference for fuzzy system depicted in Fig. 1. We propose the neural architecture according to the Fig. 2 whereby the a priori knowledge of each rule is embedded directly into the weights of the network. The structure of a neural network is defined by its processing units and their interconnections, activation functions, methods of learning and so on. In Fig. 2, each circle or node represents the neuron. This neural network consists an input layer with input vector and an output layer with the output value . The layer

between the input and output layers is normally referred to as the hidden layer. Here, the input layer is not treated as a layer of neural processing units. One important feature of RBF networks is the way how output signals are calculated in computational neurons. The output signals of the hidden layer are

x ty

)(2 jjo wx (4)

where is a k-dimensional neural input vector, represents the hidden layer weights, x jw 2 are radial basis

(Gaussian) activation functions. Note that for an RBF network, the hidden layer weights represent the

centers of activation functions jw

jc 2 .

Fig. 2: RBF neural network architecture.

The output layer neuron is linear and has a scalar output given by = where are the trainable

weights connecting the component of the output vector . Then, the output of the hidden layer neurons are the radial basic functions of a proximity of weights and input values. A serious problem is how to determine the number of hidden layer (RBF) neurons. The most used selection method is to preprocess training (input) data by some clustering algorithm. After choosing the cluster centres, the shape parameters

y

s

jjjov

1jv

o

j must be determined.

These parameters express an overlapping measure of basis functions. For Gaussians, the standard deviations j

can be selected, i. e. j ~ c where denotes the average distance among the centres. c



To show the similarity of the RBF neural network and the fuzzy system, consider again the scalar output . The

RBF network computes the output data set as

y

ty = = = , t = 1, 2, ..., N ),,( vcx tG

s

jjttjv

12, ),( cx

s

jtjjov

1,

(5)

where N is the size of data samples, s denotes the number of the hidden layer neurons. The hidden layer neurons

receive the Euclidian distances )( jcx and compute the scalar values of the Gaussian function

that form the hidden layer output vector . Finally, the single linear output layer neuron computes

the weighted sum of the Gaussian functions that form the output value of .

tjo ,

),(2 jt cx to

ty

If the scalar output values from the hidden layer will be normalised, where the normalisation means that the

sum of the outputs from the hidden layer is equal to 1, then the RBF network will compute the “normalised” output data set as follows

tjo ,

ty

ty ),,( vcx tG = =

s

jtj

tjs

jtj

o

ov

1,

,

1, =

s

jjt

jts

jtj

cx

cxv

12

2

1,

),(

),(

, t = 1, 2, ..., N.

(6)

The similarity of approximation schemes (6) and (3) is obvious. From these schemes is shown that the weights

in Eq. (6) to be learned correspond to in Eq. (1), and to tjv , iw .)/(. 2 )(xj in Eq. (3). Thus, the adaptive

fuzzy system depicted in Fig. 1 uses neural techniques to abstract fuzzy principles and to choose the weights ,

and gradually refine those principles as the system samples new cases. These properties were firstly recognised by V. Kecman [3]. In Fig. 2, the network with one hidden layer and normalised output values is the fuzzy

logic model or the soft RBF network.

iw

tjo ,

Next, to improve the abstraction ability of soft RBF neural networks with architecture depicted in Fig. 2, we replaced the standard Gaussian activation (membership) function of RBF neurons with functions based on the normal cloud concept. Cloud models are described by three numerical characteristics [2]: Expectation (Ex) as most typical sample which represents a qualitative concept, Entropy (En) as the uncertainty measurement of the qualitative concept and Hyper Entropy (He) which represents the uncertain degree of entropy. En and He represent the granularity of the concept, because both the En and He not only represent fuzziness of the concept, but also randomness and their relations. This is very important, because in economics there are processes where the inherent uncertainty and randomness are associated with different time. Then, in the case of soft RBF

network, the Gaussian membership function in Eq. (6) has the form .)/(. 2

),( 2 jt cx = = 2)(2/)((exp nEE jt xx 2)(2/)(exp nEjt cx (7)

where is a normally distributed random number with mean and standard deviation , E is the expectation operator.

nE En He

AN APPLICATION We illustrate the classic, fuzzy logic (soft) and cloud (granular) RBF neural networks on the input – output function estimation of a sales process. The time plot of the data set used in this application (the 724 daily sales for Hansa Flex company, 2004-2005) is shown in Fig. 3.



0

0,2

0,4

0,6

0,8

1

1,2

1 36 71 106 141 176 211 246 281 316 351 386 421 456 491 526 561 596 631 666 701

Time (2004 - 2005)

Sal

es (

100

tho

usa

nd

of

euro

s)

Fig. 3: Daily sales from January 2004 to December 2005

Statistical models chosen after some experimentation using the Statgraphics procedures were

ttt yy 71 or (8)

tttt yy 717 . (9)

Both statistical models have typical seasonal behavior with the seventh lag. Fitted models have the following forms: or 71248.0ˆ tt yy 77 93868.0 ttt yy respectively. The usual diagnostic checking procedures

according to Box & Jenkins [1] do not reveal any inadequacies in these models. The Box-Jenkins theory was also used to specify the neural input variables. As shown from Eq. (8) and (9), these variables are here and 7ty

7t respectively.

In the RBF neural network framework, the non-linear function f(x) was estimated according to the expressions in Eq. (5). In the case of RBF fuzzy logic network, the non-linear input – output approximation function was estimated according to the formula (6). Next, the fuzzy logic RBF neural network was extended towards estimation with (a priori known) noise levels of the entropy. Noise levels are indicated by hyper entropy. It is assumed that the noise level is constant over time. We select, for practical reasons, that the noise level is a multiple, say 0.015, of entropy. In Table 1, we give the achieved results of approximation ability in dependence on various number of RBF neurons. The mean square error (MSE) was used to measure the approximation ability.

Table 1. The MSE´s measures of approximation accuracy of various RBF networks related to the different number of clusters (RBF neurons).

Numb. of RBF Neurons

NNW Archite-cture:

Gausian Classic RBF

Soft RBF

Classic with Normal Cloud Concept

Soft with Normal Cloud Concept

RBF network representations for model (8): 3 1.439 0.698 1.503 0.729 5 0.729 0.693 0.817 0.716 10 0.687 0.675 0.671 0.678 15 0.697 0.681 0.681 0.678 RBF network representations for model (9): 3 0.783 0.646 0.786 0.647 5 0.810 0.632 0.803 0.630 10 0.607 0.571 0.607 0.571 15 0.582 0.563 0.582 0.563

The mean (centre), standard deviation of the clusters (RBF neurons) are computed using K-means algorithm. The data used are the same as used in the previous statistical models. As shown in Table 1, models that generate the “best” MSE´s are soft RBF networks. Comparing both approaches, i. e. the models based on the Box-Jenkins methodology (the MSE for model expressed by Eq. (8) is 0.7793 and by Eq. (9) is 0.74606 respectively), and the models based on RBF networks approaches, we clearly see that models based on RBF networks are better approximation models because the estimated values are close to the actual values.



Next, a forecast model was produced. Forecasts are provided during the ex post forecast period ( , …, ,

i. e. the sample period ends with observation ). Table 2 presents the MSE´s measures of ex post forecast

accuracy. As can be seen from Table 2, the soft RBF networks have indeed a forecasting power: if anything, it seems that they manage to forecast better than other RBF network architectures.

525y 724y

524y

CONCLUSION In this article, we have extended RBF neural network methodology to approximate the non-linear time series data using normal cloud models in the role of standard Gaussian activation (membership) function for RBF neurons. This was done by formulating a hyper entropy of standard deviation (entropy) of the Gaussian cloud model.

Table 2. The MSE´s measures of ex post forecast accuracy of various RBF networks related to the different number of clusters (RBF neurons).

Numb. of RBF Neurons

NNW Archite-cture:

Gausian Classic RBF

Soft RBF

Classic with Normal Cloud Concept

Soft with Normal Cloud Concept

RBF network representations for model (8): 3 1.6634 0.8602 1.6092 0.8488 5 0.8509 0.8377 0.8489 0.8338 10 0.8055 0.8359 0.8051 0.8346 15 0.8433 0.8480 0.8391 0.8026 RBF network representations for model (9): 3 0.8452 0.6869 0.8451 0.6879 5 0.8806 0.6548 0.8801 0.6549 10 0.6600 0.6241 0.6649 0.6245 15 0.6307 0.6248 0.8795 0.6052

To approximate the input-output function of a business process, the RBF neural network approach was applied on the daily sales data of the Hansa Flex company and compared with an approach based on statistical procedures. For the sake of approximation abilities we evaluated 34 models. Two models are based on the Box-Jenkins time series analysis approach, and 32 models are based on the neural (fuzzy logic) methodology. Using the disposable data a very appropriate model is the soft RBF network with activation functions based on the granular concept. It is also interesting to note that the most computationally intensive models, the model based on the Box-Jenkins methodology, is newer considered “best”. ACKNOWLEDGEMENT This work was supported by Slovak grant foundation under the grant No. VEGA 1/0024/08 and from the Grant Agency of the Czech Republic under the grant No. GAČR 402/08/0022. REFERENCES [1] BOX, G. E. P. and JENKINS, G. M. Time Series Analysis, Forecasting and Control. San Francisco, CA :

Holden-Day, 1970. [2] CHANGYU, L.; DEYI L.; YI D.; XU H. Normal Cloud Models and Their Interpretation. The 11th World

Congress of International Fuzzy Systems Association (IFSA 2005). Beijing China : July 28-31, 2005, Springer, Volume III, 1540–1543.

[3] KECMAN, V. Learning and Soft Computing: Support Vector Machines, Neural Networks, and Fuzzy logic Models. Massachusetts Institute of Technology : The MIT Press, 2001.

[4] KELLER, J. M.; YAGER, R. R., TAHANI, H. Neural network implementation of fuzzy logic. Fuzzy Sets and Systems 45 (1992), 1–12.

[5] KOSKO, B. Neural networks and fuzzy systems a dynamic approach to machine intelligence. Prentice Hall, Inc., 1992.

[6] MARČEK, D. Determination of fuzzy relations for economic fuzzy time series models by SCL techniques. The 11th World Congress of International Fuzzy Systems Association (IFSA 2005). Beijing China, July 28-31, 2005, Tsinghua University Press, Springer, Volume III, 1419–1424.



[7] YOSHINARI, Z.; PEDRITZ, W.; HIROTA, K. Construction of fuzzy models through clustering techniques. Fuzzy Sets and Systems 54 (1993), 157–165.

ADDRESS Ing. Milan Marček Institute of Computer Science Faculty of Philosophy and Science The Silesian University Opava Bezručovo náměstí 13 746 01 Opava Tel.: +420 553 684 200 Fax: +420 553 716 948 The Faculty of Management Science and Informatics University of Žilina Univerzitná 8215/1 010 26 Žilina Tel.: +421-41-513 4061 Fax: +421-41-513 4055 MEDIS - Medical Innovations Pri Dobrotke 659/81 Nitra-Dražovce Slovakia



USING DATAMINING AND OLAP TECHNOLOGY AT INFORMATION SYSTEM

Jindřich Petrucha, Dan Slováček

Evropský polytechnický institut, s.r.o. Kunovice

Abstract: The paper deals about possibility of using data mining technology at information system and explain some processing in visual technique. There are some examples of using data cube with data in XML cube. The paper describes main window of OLAP program that can create dimension by drag and drop technology on the screen.

Keywords: Datamining, OLAP technology, visualisation of data, management mining project, Pivot Cube

1. INTRODUCTION In today's highly competitive environment for the organization, regardless of their size, very good ability to meet important needs, expectations and in particular the behavior of their customers or revealing weaknesses of business processes. To achieve this knowledge is now increasingly using predictive analysis and data mining in particular. Data mining is seen as a set of prediction methods for the transformation of data sources to support the organization of information management, decision making and implementation of business goals. This is a tool designed for large companies with extensive technology infrastructure, business information systems, data warehouses or databases with hundreds of thousands of records. And in the middle and small organizations can use these procedures with relatively the same benefits as large companies. The success of data mining role rather than the quantity of data available depends on correctly defined targets, the quality of the data, an appropriate model, etc. It is not true that this is only a matter of technical gurus and experts on the database. To successfully manage the simpler data mining project in a small organization, a fundamental understanding of processes and techniques, such as basic training, supported by business analyst. In conjunction with high-quality software tool may be technical solutions to the issue of visual programming without the need for registration of any programming code. The purpose of a major contribution data mining problems is the enrichment of existing processes, new knowledge and swift action recovery. By joining the algorithms may not be the only database data, but also the date of research, monitoring, caught manual data entry, etc. special inputs such as records of activity on the website (weblogs) or unstructured text documents. Combination and analysis of the above data are obtained valuable information for better management of business processes. 2. USING OF DATA MINING Regardless of the size of the organization in solving problems according to the standard methodology for data management mining projects. Commonly used methodology CRISPDM, independently developed in 1999 a consortium of organizations SPSS, NCR, Daimler-Chrysler and OHRA (www.CRISPDM.org). The methodology describes the steps in general terms the basic phase (understanding the role from a business perspective, understanding of data, data preparation, modeling, evaluation and transfer them to real use) to split the role that can be used as a guide in dealing with specific projects. A typical data mining roles include classification and direct prediction of future behavior of individual entities (such as an estimate of the likelihood of purchase, leaving customers to the competition or reply to e-mail), segmentation (formation of groups with similar characteristics, the determination of unusual or suspect cases), detection of relations between (an analysis of items in shopping cart) or the genome sequences (typical passages website). Wide application is also in quality control (analysis of the causes of error detection and quality) or the prediction of failure in the monitoring systems. On-line monitoring of processes, but also look at the historical, often caught the data automatically, for example, can predict the failure of the production unit, control systems and other important components of the decommissioning works considerable financial costs. In modeling the interaction may also include higher order (machine - operator), non-linear dependence, etc.



Especially for smaller organizations is not the demand for all types of the above problems and find it appropriate to use the modular data mining instruments. The modules contain a majority of groups of similar algorithms needed to solve specific tasks required. In other parts of the indicated examples of typical tasks, which can be found in small organizations. 2.1 EXAMPLES OF USING DATA MINING IN SMALL ORGANIZATION Small business organizations serve a limited number of customers, among which, however, there are some differences such as the corresponding average size of orders, the type of purchased goods and services, seasonality in the demand or payment history. Organizations are faced with the problem of leaving their customers to the competition. Finding answers to questions like: How to strengthen customer loyalty? There are groups of more or less loyal customers? What is typical for these groups? It can positively affect customer loyalty? A comprehensive analysis of the data stored in the database, including the preparation, modeling and evaluation phase will become the basis for the use of action. There will be mainly on the segmentation of customer bases, getting pricing policies and other marketing activities such as special offers or greater concentration in the segment. Companies will benefit from maintaining an overall satisfactory relationships with our customers, strengthening loyalty and generating additional revenue. 2.2 EXAMPLES OF USING DATA MINING IN REGIONAL INSTITUTIONS Regional institutions serving thousand customers looking for how to promote the sale of additional products. So far was the selection of potential candidates almost accidentally as a very low response. Institutions, therefore, looking for answers to key questions: By the customer with an offer to address? What is the probability of interest in the product offered? How much money you can save a well-targeted campaigns? The basic step is to understand their customers and identify those who have the additional product will be interested. The combination of database and research data and analysis of patterns of conduct for customers who have already bought a similar product can detect segments in which customers are more inclined to buy. Model is obtained as a result formula or rule summary predictor model to the end - here estimate the likelihood of purchase. Deeper look improve the interactions with customers and more efficient in achieving the desired objectives. 2.3 EXAMPLES OF USING DATA MINING IN COMPANY OPERATING THE WEBSITE The company operating the website, including on-line shop needs to better understand the behavior of their visitors. So far, only monitor the number of a number of on-line orders. The objective data mining the project is to increase efficiency in the sales channel, measured on-line share of the orders and also drive traffic to the Web. Data entry are detailed records of customer behavior in the past (weblogs), which are available for most providers of web connection. A comprehensive analysis to answer questions such as Who visited the company Web site? How to behave typical visitor? It is when you visit any problems? What are typical passages web presentation? Analysis may reveal a number of shortcomings, which exacerbate the orientation on the Web and ultimately reduce the use of its customers - poor menu arrangement, non-transparent structure, malfunctioning search improperly functioning personalization for fair visitors and others. 3 EXAMPLES OF USING OLAP TECHNOLOGY OLAP has evolved as users' needs for data analysis have grown. It provides executives, analysts and managers with valuable information via a " slice, dice and rotate" method of end user data access, augmenting or replacing the more complicated relational query. This slice and dice method gives the user consistently fast access to a wide variety of views of data organized by key selection criteria that match the real dimensions of the modern enterprise. OLAP performs multidimensional analysis of enterprise data including complex calculations, trend



analysis and modeling. Derived from end-user requirements, OLAP enables end-users to perform ad hoc analysis of data in multiple dimensions, thereby giving them the insight and understanding they need for better decision making. Typical OLAP Applications financial modeling (budgeting, planning) sales forecasting customer and product profitability exception reporting resource allocation and capacity planning variance analysis promotion planning market share analysis OLAP technology can lower IS costs and help end-users work more independently, saving time and costly resources.

Figure 1. Main window of Pivot Cube program with dimension Employer and Month On the figure 1 we can see The PivotCube user interface components. The are: dimension toolbar, which is used to manipulate the dimension items of the PivotCube. They are sequential

order Ware, Client, Store, Year, Day, Seasons, Terms,, Quarter, Country and ShipDate. The dimension column toolbar is used to manipulate the dimension items of the PivotCube, the most

important property is the Map property (in our particular case is Employer), which determines the source of all dimension items which can be displayed on this toolbar.

The dimension row toolbar determines the source of all dimension items which can be displayed on this toolbar (in our particular case are Month and Payment).

The measures toolbar is used to manipulate the measures defined in the PivotCube. The pivot grid is the component used to display the data source's data using the layouts defined in the

underlying TPivotMap.



Figure 2. visualization of data in bar graph This will display a multiple bar chart for the selected dimension item. Each measure's view values will be used to generate the chart. If you select any of the dimension's values from the treeview on the left, the chart will be redrawn to display only the values for the selected item.

Figure 3. Display graph for measure view item The figure 3 describes display a multiple bar chart for the selected dimension item and view. If you select any of the dimension's values from the treeview on the left, the chart will be redrawn to display only the values for the selected item.



Figure 4. Map Builder Main principle of Map Builder we can see on the figure 4. It used to organize the dimensions of the PivotCube quickly. Here, you can drag and drop dimension items between the map's rows and columns. You can set items to unused and disabled states. The advantage of using a MapBuilder is that you can quickly view all the available dimension items and their states, especially if you have a lot of items. In these situations, the dimension toolbar would have many items and you would need to repeatedly scroll the toolbar to select them. You can also activate the dimension editor for the individual dimensions by clicking on the button on the lower left corner. 4. CONCLUSION Predictive analysis is now becoming an essential part of a well-functioning organization. This applies to companies large and small. Wrongly established processes become a direct threat to the continued existence of the enterprise and therefore need to pay close attention. In particular the need for immediate (on-line) and the correct response to the increasingly rapid initiatives surroundings. It is worth also assess the financial return on investment in data mining (ROI). Financial benefits (such as measured generated additional income, a reduction in costs associated with financial coverage inefficient processes) usually cover the costs already invested after a few months of operation - depending on the scope of solutions. Development of data mining is far from completed. The aim is to use as much data as possible and get the maximum possible information. In support of software development is an emphasis on the openness of the system and its wide adaptation to existing information systems. For users is indisputable advantage simplicity of operation and a high degree of automation of all steps of the process of gathering data for the presentation of results. LITERATURE: [1] LACKO, L. Datové sklady analýza OLAP a dolování dat s příklady v Microsoft SQL Serveru a Oracle.

1. vyd. Brno: Computer Press, 2003. s. 486. ISBN 80-7226-969-0. [2] PETRUCHA, J. Technologie analýzy dat – OLAP systémy v prostředí DBPROVE. ACTA

UNIVERSITATIS AGRICULTURAE ET SILVICULTURAE MENDELIANAE BRUNENSIS, 2000, ročník XLVIII, číslo 2, s. 149-155. ISSN 1211-8516.



[3] NAOUALI, S.; QUAFAFOU, M.; NACHOUKI, G. Mining OLAP cubes: semantic links based on frequent itemsets, Publication Date: 19-23 April 2004, ISBN: 0-7803-8482-2

[4] WREMBEL, R.; KONCILIA, CH. Data Warehouses and OLAP: Concepts, Architectures and Solutions, Published on 2006-10-30, IRM Press, ISBN 1-59904-364-5

ADDRESS: Ing. Jindřich Petrucha, Ph.D. Evropský polytechnický institute Osvobození 699 686 04 Kunovice Tel.: +420 572 549 018 E-mail: [email protected] Mgr. Dan Slováček Evropský polytechnický institute Osvobození 699 686 04 Kunovice Tel.: +420 572 549 018 E-mail: slovacek@ edukomplex.cz

http://www.amazon.com/gp/redirect.html?ie=UTF8&location=http%3A%2F%2Fwww.amazon.com%2FData-Warehouses-Olap-Architectures-Solutions%2Fdp%2F1599043645&tag=abriefsqltutori&linkCode=ur2&camp=1789&creative=9325



IMAGE PROCESSING FOR DETERMINATION OF SURFACE PARAMETERS

Jiří Šťastný, Petr Ludík, Milan Štencl


Abstract: The paper describes the application of methods and algorithms of image processing in the field of determination of topographic surface parameters. In this paper the method of shearing interferometry for evaluating the basic statistical quantities of randomly flat and randomly curved surfaces of solids is described. This method can be utilized for characterizing the randomly rough surfaces of both transparent and opaque solids. The result of the statistical analysis obtained by this method were used in an application that was developed at Brno University of Technology to serve as comparison with stylus-type instruments.

Key-Words: Image processing, thresholding, shearing interferometry, topographic surface parameters, interferogram.

1 INTRODUCTION A lot of methods for characterizing randomly rough surfaces have already been published in [2,3]. One of the most important interferometric methods which has been used for testing purposes is called shearing intereferometry. This method can be realized by means of Zeiss Epival Interphako (ZEI) and also Zeiss Peraval Interphako (ZPI interference microscopes). Monochromatic light from Tungsten lamp is transmitted through the sample under investigation and splitted into two waves. These waves intefere and intereference fringes can be observed and recorded with lens and CCD camera. Between these mutually translated waves originates shear (see fig. 1).

Fig 1. Translation of waves a and b is „shear” . [1]

Math principles describing wave characteristics were described in [1]. From the CCD camera was obtained real interferogram stored as 8 bit image (see fig. 3). As the sample was used surface of the glass diffuser. 2 ANALYSIS OF INTERFEROGRAM Interferogram consists of two fringes, dark and bright. Dark fringes (minimum) can be defined as:

Δe=(2m-1)π, where m=0,±1,±2,…

Bright fringes (maximum) are defined as: Δe=2mπ, where m=0,±1,±2,…

To obtain parameters of randomly rough surface there must be performed image processing of interferogram. For the next processing it is necessary to transform interferogram into contours representing the centres of dark and bright fringes. For image processing is better to use dark fringes due to risk of overexposed image.



For recognizing centers of fringes were used these steps to obtain the best possible results: 1. median filter with neighbourhood of 3x3 pixels filtration to filter out small local errors and noise 2. seeking the minimal values of dark fringes-calculate average brightness of the whole image function 3. thresholding according to average brightness value 4. floodfill area of object 5. find minimal values of this marked object row by row 6. connect all found minimal values 7. erase floodfilled object 8. perform step 4 until end of entry image 9. draw matrix of minimal values

This algorithm was used during software implementation. After this algorithm is possible to perform statistical analysis of sample under investigation. For statistical analysis of interferogram is necessary to calculate random quantity h for ZPI microcope is defined as:

hn n

p

q

0

(1)

or for ZEI microscope:

q

p

nh

02

(2)

where wave length of monochromatic light 578 nm

])/([ 112 Rq e (3)

)]/([ 112 Rhp ee (4)

t n n 0 , r n 2 0

and ),(),( yxyxh (5)

Where refractive index of immersion liquid no= 1.469 is close to external environment n=1.51. The quantities p and q are shown in schematic diagram of the dark fringes of a randomly rough surface (see fig. 2).

Fig. 2 Schematic diagram of the dark fringes of a randomly rough surface [1]

In this figure the lines representing the centres of the dark fringes are plotted schematically. We can see that randomly rough surfaces which are curved exhibit the same interferograms as the randomly rough flat surfaces(1/R1=1/R2=0, where R1 and R2 are principal radii of curvature) from the qualitative point of view. Equation (1) and (2) show that values of random quantity h can be easily determined using the values of

distances p and q measured in the interferograms produced by the ZPI and ZEI microscopes( values of , no, n are known).

The standard deviation hof random quantity h is given as follows [4]:

)](1[2 2 ch (6)

where )(c is autocorrelation coefficient of the function ),( yx .

Owing to the mathematical properties of )(c [5] its apparent that :



2)(0 h and 0)0( h

From the foregoing one can see that:

2)( h (7)

For sufficiently large values of shear (i.e. values where denotes a certain crucial value for * * for the

surface. If the values of h corresponding to the values lying between 0 and are used following equation

for determining

*)(c

)2/()(1)( 22 hc (8)

When the dependence )( h can be measured only for its necessary to employ the dependence of * )( h in its entirety for determining the values of and autocorrelation length T of surface roughness

described in [1]. The values of h can using equations (1) and (2) can be calculated for each dark fringes of the inteferogram

recorded for a certain value by using ZPI or ZEI microscopes. The value of h corresponding to this

interferogram is expressed as:

M

i

ih M

h

1

2

1 (9)

where M is number of all points chosen at the centres of all dark fringes appearing in this interferogram. In equation (9) symbol hi represents one of the calculated values of h for the chosen point in the interferogram. 3 EXPERIMENT RESULTS To verify if image processing is working correctly were compared results given by stylus type instrument receieved in laboratory. In figure 3 intereferogram of a chosen flat glass diffuser recorded by CCD camera of the ZPI microscope for m 8.36 is shown.

Fig. 3 The interferogram of the chosen sample of the glass diffuser recored by CCD camera

The centers of intereference fringes corresponding to the interferogram obtained by methods built in the developed application (see fig. 4).



Fig. 4 Centers of dark fringes found by developed application

For the glass diffusers the values of were in range 0-90μm.

From the [1] was found that the best fit of experimental data was the Gaussian form of )(c

)/exp()( 22 Tc (10)

In this case with using manual image processing methods obtained earlier in [1] the values σ and T were determined as follows:

mT

and

m

)661(

)06.084.1(

this value has been determined equation (7). By using styles type instrument was found that :

mT

and

m

)65.60(

)06.090.1(

In [1] was found out that glass diffusers are generated by Gaussian stochastic processes. Because contrast of intereferogram strongly decreases for different values of for the glass diffusers this fact is not explanated in this paper. For the shear )8.36( m with new algorithm was received (μm)= m5.4 which strongly depends on

used methods of image processing.

Tab. 1 comparison of methods 4 EXPERIMENT RESULTS We can see there is not optimal agreement between values of image processing and stylus-type instrument so the other values were not necessary to calculate until there are image preprocessing methods improved. Results of image processing were very sensitive on methods of image preprocessing and processing. Algorithm has shown that its important to find centers of fringes carefully otherwise obtained results were not sufficient.

)8.36( mshear Stylus instrument Image processing

(μm) m)06.090.1( m)5.4(



5 CONCLUSION In this paper interferometry method used for determining the basic statistical quantities of the flat surface whose roughness is represented by the normal stochastic process was mentioned. Determination of surface parameters is based on statistical analysis of the centres of dark interference fringes recorded by CCD camera serving as the detector of shearing interference microscopes utilized for producing the interferograms of rough surfaces. The main advantages of the presented method can be summarized as follows: this method is nondesctructive this method can be used for analysing the basic statistical quantities of both rough flat and curved surfaces. it can be used for studies of various rough surfaces software application can be compared with the interefometric method based on measuring and interpeting

the contrast of fringes(coherence properties of the light sources used in the microscopes). To obtain the best results there has to be tested and developed new methods for image processing which can deal with the high values of shear parameter and low values of contrast.

ACKNOWLEDGEMENT This research was supported by the grants: MSM 0021630529 Intelligent Systems in Automation (Research design of Brno University of Technology)

No 102/07/1503 Advanced Optimisation of Communications Systems Design by Means of Neural Networks. The Grant Agency of the Czech Republic (GACR)

MSM 6215648904/03 Development of relationships in the business sphere as connected with changes in the

life style of purchasing behaviour of the Czech population and in the business environment in the course of processes of integration and globalization (Research design of Mendel University in Brno)

6 REFERENCES [1] OHLIDAL, M.; OHLIDAL, I.; DRUCKMULLER, M. and FRANTA, D. Pure appl. Opt 4, 599-616,

1995. [2] BENNET, J. M.: Appl Opt. 26, 2690-5, 1976. [3] OHLIDAL, M.; NAVRATIL, K.; DRUCKMULLER, M. Appl.Opt 33, 7838-45, 1994. [4] VELZEL C. H. F.: Optical instruments and techniques, 1970. [5] LEVIN, B. R. The theory of random processes from statistically rough surfaces, Oxfod:Pergamon, 1979.

ADDRESS Doc. RNDr. Ing. Jiří Šťastný, CSc. Department of Automation and Computer Science Brno University of Technology Technicka 2 616 69 Brno Czech Republic E-mail: [email protected] Ing. Petr Ludík Department of Automation and Computer Science Brno University of Technology Technicka 2 616 69 Brno Czech Republic E-mail: [email protected] Ing. Milan Štencl Department of Automation and Computer Science Brno University of Technology Technicka 2 616 69 Brno Czech Republic



VARIABLE LATERAL SILICON CONTROLLED RECTIFIER FOR IC’S ESD

PROTECTION DESIGN

Petr Běťák, Jaromír Brzobohatý, Vladislav Musil


Abstract: The Variable lateral Silicon Controlled Rectifier (VLSCR) is a SCR based structure with the possibility to tune I-V snapback characteristics. This effect is important for an ESD (electrostatic discharge) protection design. The ESD protection structures act as a protection of integrated circuits against parasitic electrostatic discharge. Among often used structures belong structures having snapback type of I-V characteristic. Typical is a gate-grounded NMOS transistor [3] or a SCR [3]. This text is dealing with the VLSCR structure which enables I-V snap-back characteristics tuning according to the application demand. Simulated technology was 0.5µm CMOS very high voltage (VHV Integrated Circuits). Measurement was done in 1.5 µm BiCMOS process.

Keywords: ESD, SCR, LVTSCR,VLSCR

1 INTRODUCTION Fig.1.1 illustrates a principal connection of ESD (Electro Static Discharge) protection cells and required I-V behaviour. A random ESD stress goes through the ESD clamps to ground and a core circuit is not endangered. The protections must be active only during the ESD event.

Fig. 1.1 Connection of ESD protection clamps

The typical snap-back I-V characteristic is shown in Fig. 1.2. The required snap-back protection device should fulfil subsequent conditions: value of a leakage current is low, the ESD current in the area of the snapback is high, robustness of the ESD cell is sufficient. An important feature of the ESD structure is a triggering voltage adjustability and a holding voltage adjustability. For the ESD protection design the holding voltage adjustability and the trigger voltage adjustability is the key to form required A/V characteristic. The voltage triggered ESD clamps are based mainly on diodes and they are often used as power clamps [3]. The power clamps need to have a voltage reference upon VDD level otherwise a power supply could short out.

Fig. 1.2 Snap-back I-V characteristic of common ESD device.

Nevertheless, the diode string for the supply clamps has several disadvantages like high resistances and slow performance. The ideal ESD protection structure is Silicon Controlled Rectifier (SCR) for the sake of its quick



performance, but the problem is in the high trigger condition (approximately 40-50V) and low holding voltage (~1V)[2]. This limits the application of SCR as a power supply protection. However, there is a way to design the SCR having better triggering and higher holding voltage. The first approach was introduced as a Modified Lateral SCR [2]. The trigger voltage is determined by N-well to a P-substrate breakdown. In MLSCR is inserted the N+ diffusion in N-well to decrease the trigger voltage. But the holding voltage in the MLSCR is still too low. The first power–supply–adjustable–SCR-based structure was introduced by Chatterjee [1]. The structure was termed as “Low Voltage Triggering SCR (LVTSCR)”. 2 CONVENTIONAL LVTSCR (LOW VOLTAGE-TRIGGER SILICON CONTROLLED RECTIFIER) The LVTSCR structure uses a MOS transistor in parallel with a SCR. Therefore, the triggering condition is set on the MOS breakdown level (in the submicron CMOS process approximately 10-15V) and the holding voltage level is given by X and L dimension.

Fig. 2.1: Cross section of LVTSCR structure. The Fig.2.1 shows the LVTSCR structure of conventional primary ESD protection. Fig. 2.2 illustrates a circuit scheme of the structure.

Fig. 2.2: Circuit scheme and connection of LVTSCR structure.

The dimension L (Fig. 2.1) represents a length of NPN base region in the SCR structure. By changing the L, the current gain of parasitic bipolar transistors is changed. This yields different I-V characteristics. The dimension L-X represents a size of N+ diffusion. This diffusion changes the concentration in N-well region. Usually the higher X+L is, the higher holding voltage is. Formula (2.1) determines a latch-up condition [2].

1 pnpnpn (2.1)

βnpn and βpnp are the current gains of the npn and pnp transistors. However, the collector of one is the base of the other. Therefore, it is not possible to use the discrete β when computing Equation (2.1). The illustration in Fig. 2.3 introduces a typical behaviour of LVTSCR as the function of sizes L and X. It is evident that the higher size L is, the higher holding voltage and “ON” resistance are. The trigger voltage is almost constant. The 0.5µm CMOS technology provides a trigger condition at 14.9V.



N-type LVTSCR I-V characteristics

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X=22um,L=36um

X=14um,L=48um

X=14um,L=68um

Fig. 2.3: I-V results for 0.5µm CMOS process (TCAD device simulation) .

The LVTSCR was simulated in a TCAD device simulator to verify declared performance [2]. The LVTSCR structure is not adjustable for higher holding voltages (for example to protect 10V, 15V, 30V as a supply clamp). The reason is in a grounded MOS gate which sets the trigger and the holding voltage level close to the MOS breakdown. 3 VARIABLE LATERAL SILICON CONTROLLED RECTIFIER To achieve a better holding and also trigger voltage adjustability a variable lateral SCR (VLSCR) was formed in N-well. This structure utilizes N++ diffusion between anode and cathode to change the N-region concentration for the trigger voltage tuning as was discussed at the beginning in the MLSCR [2]. To change the holding voltage we must manipulate with the P-region and the anode-to-cathode spacing as it is shown in Fig. 3.1.

Fig. 3.1: Cross section of VLSCR The Cross section of VLSCR is described by dimensions X, Y and L, where L represents the anode-to-cathode spacing and it affects the base length of PNP transistor. The size Y is the P+ region extension and it changes the base length of NPN transistor. Finally, X is the size of N++ diffusion to change the trigger voltage. The principal circuit scheme of this structure is presented in Fig. 3.2. The dimensions of diffusion regions also influence a value of the P-region and the N-region resistances which are illustrated in the scheme. The principal circuit scheme of this structure is presented in Fig. 3.2. The dimensions of diffusion regions also influence a value of the P-region and the N-region resistances which are illustrated in the scheme.



Fig. 3.2: Circuit scheme of SCR-based structure termed as VLSCR

Manipulating with the sizes X,Y the current gains of the NPN and PNP transistor are changed in the way that was discussed earlier. The Fig 3.3 presents the simulated I-V characteristics of the structure.

Simulation results, VLSCR [P+ anode],L=13,X=10um

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Fig. 3.3: Simulated I-V characteristics in 0.5µm CMOS process (TCAD device simulation). According to the graph, the holding voltage adjustability is 17V-31V and the trigger voltage adjustability is 25V-57V. This behaviour is applicable for the high voltage supply clamps. But the behaviour depends on the P+ anode resistance. If we use a different P+ anode concentration, we get a different I-V characteristics. The same VLSCR structure but with a different P+ anode concentration is illustrated in Fig. 3.4.

Fig. 3.4: Cross section of VLSCR with P++ anode This structure setting yields different voltage levels in I-V characteristics as seen in Fig. 3.5.



Simulation results, VLSCR, P++ anode

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Fig. 3.5: Simulated I-V characteristics in 0.5µm CMOS process (TCAD device simulation).

In this case, the used voltage at HVASCR or LVTSCR structures is similar. Therefore, the convenient P+ anode concentration has to be chosen. Fig. 3.6 shows measurement results of a VLSCR structure which was manufactured in 1.5µm BiCMOS technology. We can see the holding voltage adjustability by the dimensions X and L. The dimension Y was not used. It can be noticed that the trigger voltage is almost constant. It is given by the difference L-X, which is retained constant. The only change is the cathode-to-anode distance which results in different holding voltages.

VLSCR, BiCMOS 1,5um

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Fig. 3.6: Measurement results of VLSCR manufactured in BiCMOS 1.5 µm process.

Fig.3.7: Cross section of VLSCR structure manufactured in BiCMOS

A cross section of a VLSCR structure manufactured in BiCMOS is shown in Fig. 3.7. For a wider range of I-V characteristics is convenient to use the Y dimension which enables the setting of holding voltages on lower levels. BiCMOS technology provides a low doped epitaxial layer and therefore the voltage levels of I-V characteristics are relatively high. In a common CMOS process, the voltage levels are relatively low and the “ON” resistance is better. Fig. 3.8 illustrates a voltage variability of the VLSCR in the 0.5µm CMOS process



from the TCAD device simulator.

Comparison of different P+ anode concentrations,TCAD simulation results

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Fig.3.8: Comparison of different P+ anode concentration, TCAD device simulation.

4 CONCLUSION The VLSCR structure is useful in the high voltage integrated circuits but even in low voltage applications. Simulated and measured I-V characteristics of the new VLSCR structure have been commented. The VLSCR structure is convenient for low voltage applications in CMOS process but even for higher holding voltages in BiCMOS process. In the ESD protection design the VLSCR can form a supply clamp or a primary ESD protection. ACKNOWLEDGMENT This research has been supported by the Czech Ministry of Education in the frame of Research Plan MSM 0021630503 MIKROSYN New Trends in Microelectronic Systems and Nanotechnologies. REFERENCES [1] CHATTERJEE, A.; POLGREEN, T. „A low-voltage triggering SCR for on-chip protection at output and

input pads“ In Elec.Dev.Lett., EDL-12, 21-22, 1991. [2] AMERASEKERA, A.; DUVVURY, CH. ESD in Silicon Integrated Circuits. New York : John Wiley and

Sons, 2002. [3] VOLDMAN, S. H. ESD Circuits and Devices. New York : John Wiley and Sons, 2006. ADRESS: Ing. Petr Běťák Brno University of Technology Faculty of Electrical Engineering and Communication Udolní 53 602 00 Brno +420 54114-6160 prof. Ing. Jaromír Brzobohatý, CSc. Brno University of Technology Faculty of Electrical Engineering and Communication Udolní 53 602 00 Brno +420 54114-6160 E-mail: [email protected]



Prof. Ing. Vladislav Musil, CSc. Brno University of Technology Faculty of Electrical Engineering and Communication Udolní 53 602 00 Brno +420 54114-6160



THE METHODOLOGY OF VALUE DETERMINATION OF CLIENT FOR INSURANCE

BUSINESS

Radek Doskočil


Abstract: The article deals with the use of soft computing as a support of manager decision making. For this purpose the fuzzy logic is used. The brief description of fuzzy logic and the process of calculation are mentioned. The use is demonstrated on the problem of value of client for insurance business. The scheme of models, rule blocks, attributes and their membership functions are mentioned. The use of fuzzy logic is the advantage especially at decision making processes where the description by algorithms is very difficult and criteria are multiplied.

Keywords: Soft computing, fuzzy logic, decision making, insurance business.

INTRODUCTION The use of fuzzy logic is the advantage especially at decision making processes where the description by algorithms is very difficult and criteria are multiplied. The advantage is that the linguistic variables are used. The fuzzy logic measures the certainty or uncertainty of membership of element of the set. Analogously the man makes decision during the mental and physical activities. The solution of certain case is found on the principle of rules that were defined by fuzzy logics for similar cases. The fuzzy logics belong among methods that are used in the area of decision making of firms and offices. THE FUZZY PROCESS The calculation of fuzzy logics consists of three steps: fuzzification, fuzzy inference and defuzzification. The fuzzification means that the real variables are transferred on linguistic variables. The definition of linguistic variable goes out from basic linguistic variables, for example, at the variable risk it is set up the following attributes: none, very low, low, medium, high, very high risk. Usually there are used from three to seven attributes of variable. The attributes are defined by the so called membership function, such as , , Z, S and some others. The membership function is set up for input and output variables. The fuzzy inference defines the behavior of system by means of rules of type <When>, <Then> on linguistic level. The conditional clauses evaluate the state of input variables by the rules. The conditional clauses are in the form

<When> Inputa <And> Inputb ….. Inputx <Or> Inputy …….. <Then> Output1,

it means, when (the state occurs) Inputa and Inputb, ….., Inputx or Inputy, …… , then (the situation is) Output1. The fuzzy logic represents the expert systems. Each combination of attributes of variables, incoming into the system and occurring in condition <When>, <Then>, presents one rule. Every condition behind <When> has a corresponding result behind <Then>. It is necessary to determine every rule and its degree of supports (the weight of rule in the system). The rules are created by the expert himself. The defuzzification transfers the results of fuzzy inference on the output variables, that describes the results verbally (for example, whether the risk exists or not). The system with fuzzy logics can work as an automatic system with entering input data. The input data can be represented by many variables. EXAMPLE – VALUE OF CLIENT FOR INSURANCE BUSINESS The example presents the use of fuzzy logic at the decision making of value of client by the insurance company. The MATLAB program was used for the calculation. The application uses six input variables with five or ten



attributes, one rule box and one output variable with five attributes. See the model on the fig.1. It is necessary to design the variables, their attributes and their membership functions. The inputs are represented by variables Loss Ration (LR), Location (L), Length of Insurance (LI), Change of Liquidity (Liq), Change of Profit/Loss (PR) and Insured Risk (IR). The output from the rule box and the output variable is Value of Client (VC).

Fig. 1. Build up model

The variables Loss Ration, Length of Insurance, Change of Liquidity, Change of Profit/Loss and Insured Risk have five attributes and variable Location has ten attributes. The output variable Value of Client has five attributes. The membership functions are of the shape , S and Z. The following pictures show the attributes and membership functions some of the variables Loss Ration (fig. 2), Location (fig. 3) and Value of Client (fig. 4).



Fig. 2. The attributes and membership functions of variable Loss Ration

Fig. 3. The attributes and membership functions of variable Location



Fig. 4. The attributes and membership functions of variable Value of Client

The fig. 5 shows rule box with their rules and degree of support that set up the relation between input and output variables.

Fig. 5. Rule box

The fig. 6 shows the testing of fuzzy model where the inputs are set up (LR=0, L=0, LI=60, Liq=15, PL=0 and



IR=15) and it leads to result VC=0.0729 which means that the value of client is Excellent. The output vague value can be: Excellent, Very good, Good, Failed and Definitely Failed.

Fig. 6. The process of testing of fuzzy processing

The fig. 7 shows the testing of fuzzy model where the inputs are set up (LR=200, L=1, LI=0, Liq=30, PL=20 and IR=0) and it leads to result VC=0.927 which means that the value of client is Definitely Failed.

Fig. 7. The process of testing of fuzzy processing



The fig. 8 shows the evaluation of concrete client where the inputs are set up (LR=120, L=0,5, LI=9, Liq=30, PL=13 and IR=3) and it leads to result VC=0.25 which means that the value of client is Very good.

Fig. 8. The evaluation of concrete client

When the model is made, it is necessary to tuned it (to set up the inputs on known values, evaluate the results and to change the rules or weights, if necessary). If the system is tuned, it is possible to use it in practice. The fig.8 shows the result, where the value of client is evaluated to Very good. CONCLUSION The mentioned example is only the fractions of possible variants of the use of fuzzy logic in various areas of decision making. The methodology of value determination of client for insurance belongs among them. The theory of fuzzy logic contributes to the quality of decision making. The decision making process is an important activities of the firms. It is possible to say, that the successful decision making make the firm successful. It is necessary to emphasize, that these methods support the decision making and that the responsibility of optimal variant or variants are on those, who makes the decision.

LITERATURE [1] ALTROCK, C. Fuzzy Logic & Neurofuzzy – Applications in Business & Finance, Prentice Hall, USA,

1996, 375 p., ISBN 0-13-591512-0. [2] DOSTÁL, P. Pokročilé metody analýz a modelování v podnikatelství a veřejné správě, (The Advanced

Methods of Analyses and Simulation in Business and Public Service – in Czech), CERM, Brno, 2008, 432p, ISBN 978-80-7204-605-8.

[3] KLIR, G. J.; YUAN, B. Fuzzy Sets and Fuzzy Logic, Theory and Applications, Prentice Hall, New Jersy, USA, 1995, 279 p., ISBN 0-13-101171-5.

[4] RAIS, K., SMEJKAL, V. Řízení rizik ve firmách a jiných organizacích, Grada, Praha, 2006, 296 p., ISBN 80-247-1667-4.



ADDRESS: Radek Doskočil, MSc. Brno University of Technology Faculty of Business and Management Department of Informatics Kolejní 2906/4 612 00 Brno Tel. +420 541 143 722 Fax. +420 541 142 692 E-mail: [email protected] www.fbm.vutbr.cz



USING NE URELA NETWORKS AT FINANCIAL ARREA

Jindřich Petrucha

Evropský polytechnický institut, s.r.o. Kunovice

Abstract: The paper deals about possibility of using JOONE simulator at financial area. The specific example using the time series is describes and some parameters are explain. In the first part is explaining principle of learning mode by using special block yahoo financial and its properties during the learning mode. In the second part is explaining testing mode and saving results to the excel file. Using GUI components is most important part of JOONE system.

Keywords: Neural network, JOONE, time series, financial forecasting, delayed input, neural simulator, GUI components

1. INTRODUCTION During working with simulators of artificial neural networks is very important for user create user-friendly interface and easy approach to configure architecture of neural networks. Very offen we can meet with system SNNS or MATLAB that are very sophistical, but user has knowledge about many details to be able implement three level architecture. The same situation is when we need implemented input and output from neural networks, because we need external files very offen. In the simulator we have to create three level architecture and prepare parameters for teaching and we need see this process on the graph or another system. This situation is more complicated when we use time series for financial prediction. The simulator must be able create delayed inputs and working with them as temporary window. Now we will go step by step in the system JOONE that has prepared examples for this forecast time series at financial area. 2. EXAMPLE OF USING JOONE AT FINANCIAL AREA What type of components we will need? We will use typical level architecture of neural network – input layer with delayed inputs, two hidden layers and one output. The other components we will use for connection with real word. It is very important for this architecture that we can find special components for connection to market stock. These components can work as a data pump to the input level of neural architecture and are able to find real data on the yahoo server and create time series. The name of the components is yahoo because components create connection with server finace.yahoo where we can find historical prices for many stock.

Parameters component yahoo are on the figure 1.

Figure 1. Window with parameters for changing

After the connection we are able to browse table for the quote that we choose at the component yahoo. These data we use for learning neural networks. The same component we have to use at the end of architecture of neural network as a desired output.



One of the most important parts is to create temporary window as a delayed layer with delayed inputs and we have to set up number of these inputs. Typically we choose from 4 to 15, but it depends on our experience with time series. In the example we used 5 neurons, but the parameter of the number of neuron we can change and try to optimize learning process. All components we ca see on the figure 2.

Figure 2. Main window of GUI editors with components

Before using of time series we have to normalized all data on interval 0 – 1 and for that is prepare component “Normalizer”. The process of normalize data is working automatically when we connect to the server yahoo and get the data. Now we can describe hidden layer of neural network architecture. The number of neurons depends on complexity the all architecture but when you use big number for these (50-100) the number of combination we will very big and computer can optimized process learning.

Figure 3. Chart component with process of learning neural network during 10 000 epochs.

On the figure 3. we can see typical curve of learning process that is use RMSE calculation of error. For controlling of process learning we use control panel. All parameters are shown on the figure 4.



Figure 4. control panel and parameters for learning and testing neural network

During the learning mode we set up number of epochs to achieve minimal error of RMSE. This parameters is important for time of learning neural network. After of learning we can go to testing results of neural network. We have to setup learning mode off in control panels and number of epochs to 1. The results of output layer we have to write to text file or to the excel file and we can analyze this output by graph. The setup of parameters we can see on fig. 5. that we have write results to the file time.xls.

Figure 5. The new output layer Excel Output for testing results of neural network to the excel file

Figure 6. Results of testing neural network of CPST quote on NASDAQ

3. CONCLUSION The using of JOONE systems for analyzing time series is very simple when we use the prepare example and change only some parameters for learning mode and testing mode. Most important is connection by using yahoo financial component to the server finance.yahoo.com that is like data pump to our system. There are historical



prices of quote of NASDAQ and this information we can use for learning mode of neural network. This data are refresh every day and we can use learning mode for new data. This system is free and can be use for explaining principle of neural networks on practical example at financial aerea. LITERATURE: [1] MARRONE, P.; JOONE JavaObject Oriented Neural Engine, The Complete Guide [online]. 2007 [cit.

2009-01-20]. Dostupný z www: <http://heanet.dl.sourceforge.net/sourceforge/joone/JooneCompleteGuide.pdf>.

[2] A GUI Example [online]. 2005 [cit. 2009-01-20]. Dostupný z www: <http://www.jooneworld.com/docs/sampleEditor.html>.

[3] DOSTÁL, P. Pokročilé metody analýz a modelování v podnikatelství a veřejné správě. Akademické nakladatelství CERM: Brno, 2008, ISBN 978-80-7204-605-8.

[4] DOSTÁL, P. Neural Network and Stock Market, In Nostradamus Prediction Conference, UTB Zlín, 1999, s8-13, ISBN 80-214-1424-3.

ADDRESS: Ing. Jindřich Petrucha, Ph.D. Evropský polytechnický institut, s.r.o. Osvobození 699 686 04 Kunovice Tel.: +420572549018 e-mail [email protected]



PARALLEL GRAMMATICAL EVOLUTION FOR OPTIMIZATION OF COMPUTER

NETWORK STRUCTURE

1Pavel Ošmera, 2Imrich Rukovanský

1Brno University of Technology 2European Polytechnic Institute, Ltd. Kunovice

Abstract: In order to find out demonstrably that a computer net is working efficiently, it is essential to monitor its activity via measuring. This process can be carried out by means of monitors (net analyzers). They are able to observe tens of various output parameters (permeability, CPU utilization, utilization of communication lines between nodes/knots, narrow spots) at different net levels. However, the nature of measuring varies from case to case. It depends on what output parameters are being observed, whether we observe only a specific part of the net, or the net as a whole, whether we use hardware or software monitor, or a combination of both. The significance of computer net monitoring as well as the diversity of practical measuring is demonstrated in the following contribution. For optimization of computer net can be used evolutionary algorithms. Parallel grammatical evolution can be use for optimization of networks. This paper describes the possibilities of using genetic algorithms for network design and optimization. The described algorithm – grammatical evolution – has strong generalizing capabilities, which allow it to generate symbolic representation of mathematics models, graphs and other structures. The implementation requirements and the issues, which may arise is discussed together with several methods addressing those issues.

Key words: Grammatical evolution, evolving structures, network optimization, genetic algorithms, grammar, communication lines, nodes, throughput, performance

1. INTRODUCTION Computer network generally consists of a number of interconnected switching nodes. A transmission from one device is routed through these internal nodes to the specified destination device. These nodes (including the boundary nodes) are not concerned with the contents of data; rather their purpose is to provide a switching facility that will move the data from node to node until they reach their destination. Therefore performance improvements may be obtained via node-to-node throughput optimization. The described method – grammatical evolution [1] – has strong generalization capabilities which enable it to evolve and optimize LAN, MAN and other network structures. It is also capable of evolving graphs and other structures which can be used to represent the network structure. To describe the features of grammatical evolution a simple example of generating a mathematical function will be used. 2. GRAMMATICAL EVOLUTION Grammatical evolution GE is based on classic genetic algorithm extended with a context-free grammar. The grammar forms an interface between the data representation of actual solution and the underlying genetic algorithm. Each individual in the population is represented by a sequence of rules of the defined grammar. The particular solution is then generated by translating the chromosome to a sequence of rules, which are then applied in specified order. Therefore the main task of grammatical evolution is the translation of a chromosome to a symbolic representation of a solution, also known as genotype to phenotype translation [2]. After the translation is complete, an individual can be evaluated and its’ fitness can be computed. Then the standard operators (selection, crossover [3] [4], mutation) of genetic algorithms are applied and a new population is generated and a new population cycle is started. Rewriting grammar is defined as a tuple G = (N, T, S, P) where N is a set of non-terminals, T is a set of terminals, S is a starting symbol and P is a set of production rules. The non-terminals are items, which appear in the individuals’ body (the solution) only before or during the translation. After the translation is finished all non-terminals are translated to terminals. Terminals are all symbols which may appear in the generated language, thus they represent the solution. Start symbol is one non-terminal from the non-terminals set, which is used to initialize the translation process. Production rules define the laws under which non-terminals are translated to terminals. Production rules are key part of the grammar definition as they actually define the structure of the



generated solution. The grammar, which was used to generate mathematic functions [5][6] was defined as follows: N = expr, fnc, num, where expr can be expanded to any expression; fnc can be translated to a function terminal; num can be translated into a numeric constant; var can be translated into a variable; T = sin, cos, +, -, /, *, x, 1, 2, 3, 4, 5, 6, 7, 8, 9 and S = <expr>. The start symbol may be interpreted as a general rule of what may appear in the final solution. With the given definition there are no constraints and any expression may appear in the solution. Rules generating mathematical functions are shown in prefix notation, which simplifies the grammar as no parentheses are necessary. 3. PARALLEL GRAMMATICAL EVOLUTION The PGE is based on the grammatical evolution GE [1], where BNF grammars consist of terminals and non-terminals. Terminals are items, which can appear in the language. Non-terminals can be expanded into one or more terminals and non-terminals. Grammar is represented by the tuple N,T,P,S, where N is the set of non-terminals, T the set of terminals, P a set of production rules which map the elements of N to T, and S is a start symbol which is a member of N. For example, below is the BNF used for our problem: N = expr, fnc T = sin, cos, +, -, /, *, X, 1, 2, 3, 4, 5, 6, 7, 8, 9 S = <expr> and P can be represented as 4 production rules: 1. <expr> := <fnc><expr> <fnc><expr><expr> <fnc><num><expr> <var> 2. <fnc> := sin cos + * - U- 3. <var> := X 4. <num> := 0,1,2,3,4,5,6,7,8,9 The production rules and the number of choices associated with each are in Table 1. The symbol U- denotes an unary operation. There are notable differences when compared with [1]. We don’t use two elements <pre_op> and <op>, but only one element <fnc> for all functions with n arguments. There are not rules for parentheses; they are substituted by a tree representation of the function. The element <num> and the rule <fnc><num><expr> were added to cover generating numbers. The rule <fnc><num><expr> is derived from the rule <fnc><expr><expr>. Using this approach we can generate the expressions more easily. For example when one argument is a number, then +(4,x) can be produced, which is equivalent to (4 + x) in an infix notation. The same result can be received if one of <expr> in the rule <fnc><expr><expr> is substituted with <var> and then with a number, but it would need more genes. There are not any rules with parentheses because all information is included in the tree representation of an individual. Parentheses are automatically added during the creation of the text output.If in the GE is not restricted anyhow, the search space can have infinite number of solutions. For example the function cos(2x), can be expressed as cos(x+x); cos(x+x+1-1); cos(x+x+x-x); cos(x+x+0+0+0...) etc. It is desired to limit the number of elements in the expression and the number of repetitions of the same terminals and non-terminals. 4. BACKWARD PROCESSING OF THE PGE The chromosome is represented by a set of integers filled with random values in the initial population. Gene values are used during chromosome translation to decide which terminal or nonterminal to pick from the set. When selecting a production rule there are four possibilities, we use gene value mod 4 to select a rule. However the list of variables has only one member (variable X) and gene value mod 1 always returns 0. A gene is always read; no matter if a decision is to be made, this approach makes some genes in the chromosome somehow redundant. Values of such genes can be randomly created, but genes must be present.Body of the individual is s a linear structure, but in fact it is stored as a one-way tree (child objects have no links to parent objects). In the diagram we use abbreviated notations for nonterminal symbols: f - <fnc>, e - <expr>, n - <num>, v - <var>.



5. GENOTYPE TO PHENOTYPE TRANSLATION At the beginning of the process the body of an individual contains the starting symbol expr, which will be rewritten to an expression. A gene pointer is created and initialized before the first gene. In the first step, the expr symbol is to be translated, thus the according table of productions rules is selected. The selected table has four entries so a decision has to be made on which one to choose. The gene pointer is incremented to the first gene and index of the rule is computed using the modulo operation. The selected rule is then applied to the individual body and the non-terminal is rewritten with the right side of the rule. These steps are repeated until no more non-terminals are present in the individuals’ body. However this basic principle can be further expanded by additional algorithms of translation which improve the overall performance of the process. The backward-processing algorithm translates the non-terminals in the individuals’ body from the end of the body to the beginning. The algorithm produces positive side effects in the operations of crossover and mutation, which can be used to process the population cycles more efficiently. Although this algorithm is very general it requires some modifications of the grammar, which might introduce new issues when a grammar for optimal network generation is introduced. 6. LOGICAL FUNCTION XOR AS A TEST FUNCTION Input values are two integer numbers a and b; a, b 2< 0, 1 >. Output number c is the value of logical function XOR. Training data is a set of triples (a, b, c): P = (0, 0, 0); (0, 1, 1); (1, 0, 1); (1, 1, 0). Thus the training set represents the truth table of the XOR function. The function XOR (+) can be expressed using OR (_), AND (^), NOT (¬) functions: a + b = (a ^ ¬b) _ (¬a ^ b) = (a _ b) ^ (¬a _ ¬b) = (a _ b) ^ ¬(a ^ b) The grammar was simplified so that it does not contain conditional statement and numeric constants, on the other hand three new terminals were added to generate functions _, ^, ¬. Thus the grammar generates representations of the XOR functions using other logical functions with notation ( |, .&, ~ ): 1a) function xxor($a,$b) $result = "no_value"; $result = ($result) | ((((~(~(~(~(~($result)))))) | (($a) & (($a) & (~($b))))) & ($a)) | ((~($a)) & ($b))); return $result; 1b) function xxor($a,$b) $result = "no_value"; $result = ($result) | (((~$result | ($a & ($a & ~$b))) & $a) | (~$a & $b)); return $result; 7. TWO-LEVEL OPTIMISATION Optimization of networks can be solved using a two-level optimization. The first level of the optimization is performed using grammatical evolution. The output can be a function containing several symbolic constants. Such function therefore cannot be evaluated and assigned a fitness value. In order to evaluate the generated function a secondary optimization has to be performed. For secondary optimization differential evolution is used. Input for the second level of optimization is the function with symbolic constants and the output is vector of the values of those constants. Basically it consists of two nested population loops. The inner loop is using standard differential evolution with DE/rand/1 scheme. The outer loop is a single population of parallel grammatical evolution. The resulting Grammatical Differential Evolution (GDE) takes advantage of both the original methods. 8. CONCLUSION We have described the Parallel Grammatical Evolution (PGE) that can map an integer genotype onto a phenotype with the backward processing. PGE has proved successful for creating trigonometric functions. This algorithm can be used for optimisation of computer networks. Our goal was to find very efficient algorithm for it. We first used test task to find suitable structure of PGE. It seems that in future this optimisation tool can be used for such problems as computer networks are.



Parallel GEs with hierarchical structure can increase the efficiency and robustness of systems, and thus they can track better optimal parameters in a changing environment. From the experimental session it can be concluded that modified standard GEs with only two sub-populations can create PGE much better than classical versions of GEs. The parallel grammatical evolution can be used for the automatic generation of programs. We are far from supposing that all difficulties are removed but first results with PGEs are very promising.Both Grammatical evolution [1] and Differential Evolution [8] algorithms are evolutionary computation methods based on genetic algorithms [4], [5]. Using these methods is especially suitable for optimization problems, which are very difficult to solve using classical mathematical methods or where using deterministic computation methods would require unacceptable simplification of the problem. These problems include nonlinear regression and prediction. Most of the methods found in statistical analysis are not eligible since many of the real-world problems have nonlinear character. Linear regression might be used only on short intervals of the measured data or under restricted conditions. Parallel grammatical evolution can be use for optimization of networks. These tasks are therefore suitable for optimisation using genetic algorithms and other evolutions methods [9]. REFERENCES [1] O’NEILL, M.; RYAN, C. Grammatical Evolution: Evolutionary Automatic Programming in an Arbitrary

Language Kluwer Academic Publishers 2003. [2] O’NEILL, M.; BRABAZON, A.; ADLEY, C. The Automatic Generation of Programs for Classification

Problems with Grammatical Swarm, Proceedings of CEC 2004, Portland, Oregon (2004) 104 – 110 [3] PIASECZNY, W.; SUZUKI. H.; SAWAI, H. Chemical Genetic Programming – Evolution of Amino Acid

Rewriting Rules Used for Genotype-Phenotype Translation, Proceedings of CEC 2004, Portland, Oregon (2004) 1639 - 1646.

[4] OŠMERA, P.; ŠIMONÍK, I; ROUPEC, J. Multilevel distributed genetic algorithms. In Proceedings of the International Conference IEE/IEEE on Genetic Algorithms, Sheffield (1995) 505–510.

[5] OŠMERA, P.; ROUPEC, J. Limited Lifetime Genetic Algorithms in Comparison with Sexual Reproduction Based GAs, Proceedings of MENDEL’2000, Brno, Czech Republic (2000) 118 – 126

[6] LI; Z.; HALANG W. A.; CHEN G. Integration of Fuzzy Logic and Chaos Theory; paragraph: Osmera P.: Evolution of Complexity, Springer, 2006 (ISBN: 3-540-26899-5) 527 – 578.

[7] WALDROP, M. M. Complexity – The Emerging Science at Edge of Order and Chaos, Viking 1993 [8] PRICE, K. 1996. Differential evolution: a fast and simple numerical optimizer. 1996 Biennial Conference

of the North American Fuzzy Information Processing Society, NAFIPS, pp. 524-527, IEEE Press, New York, NY, 1996.

[9] POPELKA, O.; RUKOVANSKÝ, I. Optimization of Computer Network with Grammatical Evolution.In ICSC 2007 Kunovice: EPI, s.r.o., 2007, pp. 201- 205.

[10] RUKOVANSKÝ, I. The Significance of Computer Networks Monitoring. In ICSC 2006 Kunovice: EPI, s.r.o , 2006, pp.

[11] RUKOVANSKÝ, I. Evolution of Complex Systems. 8th Joint Conference on Information Sciences. Salt Lake City, Utah, USA. July 21-25, 2005.

ADDRESS: Doc. Ing. Pavel Ošmera, CSc. Institute of Automation and Computer Science Brno University of Technology Technická 2 616 69 Brno Czech Republic E-mail: [email protected] Prof. Ing. Imrich Rukovanský, CSc. European Polytechnic Institute, Ltd. Osvobození 699 686 04 Kunovice Czech Republic E-mail: [email protected]

mailto:[email protected]



SESSION 2



INDUCTION IN MULTI-LABEL TEXT CATEGORIZATION DOMAINS

Sareewan Dendamrongvit, Miroslav Kubat, Zeynel Sendur

University of Miami

Abstract: Modern approaches to automated text categorization often employ machine learning techniques for the induction of the classifiers from preclassified examples. The distinguishing aspect of this application field is that each example-document can fall into two or more classes (some-times many classes) at the same time. This circumstance calls for specific induction algorithms and necessitates the use of less traditional performance criteria in the course of their evaluation. Another idiosyncrasy is that induction of text categorizers tends to be computationally very ex-pensive because text documents are often described by thousands of features. In this paper, we report our recent experience with these issues.

1 INTRODUCTION The goal of the work reported here was to choose a mechanism to support a large multilingual thesaurus that contained thousands of documents from many diverse fields and written in several different languages. To assist the users’ search for relevant documents, an indexing mechanism was needed. However, the sheer size of the database effectively precluded the possibility of creating such indexing scheme manually. By way of an alternative, we decided to look into ways to automate the process by machine learning techniques. The circumstance that the same document may simultaneously belong to two or more categories is in the machine learning not typical, although some initial work has already been done. The simplest solution is to induce a binary classifier separately for each class—mechanisms based on Bayesian classifiers were studied by [1], [2], and [3], the behavior of the instance-based rule was explored by [4], and the currently popular support vector machines were employed by [5] and [6]. The main defect of solutions that induce a separate classifier for each class is that the mutual relations between classes are thus ignored, which can impair classification performance. By way of improvement, [7] modified the methodology of decision trees to make induction from multi-label examples more natural, while [8] and [9] developed several algorithms that handle the multi-label domains in the framework of the “boosting” technique originally developed by [10]. This little survey indicates that the relevant literature has so far approached the problem of induction from multi-label examples from two alternative perspectives: either by a binary classifier for each class or by a general classifier that handles all combinations of classes. Either way, what all these approaches share is the extreme computational intensity, especially in text categorization domains marked by thousands of features needed to describe each document. This makes computational costs a critical performance criterion; before significant reduction of these costs is achieved, full-fledged applications in real-world settings is hard to imagine. The other critical criterion requires that the induced classifiers achieve high accuracy when identifying documents from a given class. In multi-label domains, this calls for specific performance criteria. In the next section, we will formally define the research task and the performance criteria to be used. Then, in Section 3, we report experiments in which we compare two programs that have recently become quite popular. We were interested both in classification performance and in computational costs. Based on the results, the same section offers a brief discussion of which of the two techniques is to be preferred in concrete circumstances. 2 PROBLEM STATEMENT AND PERFORMANCE CRITERIA Let

be an instance space, let

be a finite set of documents, and let be a finite set of classes such that

each belongs to its subset, . The features describing the documents have been obtained from the

relative frequencies of words or terms. Given a set of training examples, , the goal

is to find a classifier to carry out the mapping in a way that optimizes classification performance.

Moreover, the induction of the classifier has to be accomplished in realistic time. Having gained some experience with the development of such classifiers in the work reported in [9], we wanted to take a closer look at the practical behavior of some other approaches known from machine-learning literature.



In particular, we wanted to investigate the behavior—in our specific domain—of two programs: the multi-label C4.5 proposed by [7] and BoosTexter developed by [8]. We found it surprising that no such comparison has so far been made. To establish the criteria to measure classification performance, let us start with those employed by the field of information retrieval for domains where only two class labels are permitted: positive examples and negative examples. Let us denote by TP (true positives) the number of correctly classified positive examples, by FN (false negatives) the number of positive examples misclassified as negative, by FP (false positives) the number of negative examples misclassified as positive ones, and by TN (true negatives) the number of correctly classified negative examples. These four quantities define precision and recall as follows:

(1)

Observing that the user often wants to maximize both of them, while balancing their values, [11] proposed to combine precision and recall in a single formula, , parameterized by the user-specified that

quantifies the relative importance ascribed to either criterion:

(2)

Here, gives more weight to recall and gives more weight to precision; converges to recall if

and to precision if . The situation where precision and recall are deemed equally relevant is

marked by , in which case degenerates to the following:

(3)

Based on these preliminaries, [12] proposed two alternative ways these criteria can be generalized for domains with multi-label examples: (1) macro-averaging, where precision and recall are first computed for each category and then averaged; and (2) micro-averaging, where precision and recall are obtained by summing over all individual decisions. The formulas are summarized in Table 1 where and stand for

the precision, recall and the four basic variables for the i-th class.

Precision Recall

Macro

Micro

Table 1: The macro-averaging and micro-averaging versions of the precision and recall performance criteria for domains with multi-label examples. 3 EXPERIMENTAL RESULTS In the experiments, we wanted to compare BoosTexter with Multi-Label C4.5 along two performance criteria: classification accuracy, and induction time; the former was evaluated in terms of precision- and-recall related criteria, the latter was measured in minutes of CPU induction. Moreover, we wanted to find out how the accuracy and computational costs depend on the number of features used to describe the individual documents. Originally, we intended to apply the programs to the EUROVOC database, a large thesaurus where almost a hundred thousands documents—each described by about 100,000 features—have been preclassified into thousands of hierarchically ordered classes. However, the extreme computational intensity of these programs made it necessary to simplify the data, at least for the experimental purposes reported here. To be more specific, we experimented with a “reduced” database containing 10,000 documents described by 4,000 features, and we used only the 30 top-level class labels. To select the features, we used the document frequency criterion, an



unsupervised feature selection method recommended for the needs of text categorization by [13]: we picked the 4,000 random features from those that appeared in more than 50 documents. To achieve acceptable statistical reliability of the results, we followed the methodology of 5-fold cross validation. This means that in each run a classifier was induced from 8,000 documents and then tested on the remaining 2,000 documents. We repeated this experiment for different numbers of features, running from 500 to all 4,000 features used in the “reduced” database. Figure 1. summarizes the experimental results. The first thing to observe is that BoosTexter systematically outperformed Multi-Label C4.5 in terms of both - and - , especially when a larger number of features was employed. This said, a closer look reveals that each of these methods displayed a somewhat different behavior along the component criteria of : BoosTexter turned out to be better in terms of recall (both micro and macro), whereas Multi-Label C4.5 turned out to be better in terms of precision, especially in situations where only a relatively small subset of features was used. The experiments seem to indicate that the decision-tree based Multi-Label C4.5 is able to get the most from even a very small feature set. This may be due to the fact that BoosTexter considers only isolated features or linear combinations of features, whereas decision trees allow for more flexible representation. The disparate behavior of the two techniques along precision and recall needs to be properly understood before choosing the induction method. For instance, users of automated recommender systems are discouraged when offered a wrong document, even if this happens very rarely. Ability to minimize such cases is measured by precision, and this is why the decision-tree based system will in domains of this kind be preferred. On the other hand, recall is important when we want to make sure that all (or almost all) documents of the requested class have been returned. Then, the recall criterion will be critical, which means that BoosTexter will be given preference. At the same time, we have to be aware of the circumstance that our experiments indicate that a growing number of features seems to mitigate the difference between the two systems' behavior.



Figure 1: Classification performance of the two approaches along different criteria, as measured on independent testing data. Multi-Label C4.5 is better along precision, whereas BoosTexter is better along recall and . BoosTexter seems to gain an edge with the growing number of features employed.

Figure 2: Induction time measured in minutes. The time indicated in the graph is always the sum total of all five

runs of the 5-fold cross validation procedure. Another important aspect to consider, the computational complexity of the algorithms, was the subject of a next round of experiments whose results are summarized by Figure 2 that plots the CPU times consumed by the two programs for growing numbers of features. The reader can see that Multi-Label C4.5 is clearly more expensive than the competing program: note how fast the costs grow with the increasing number of features. We conclude that the practical utility of multi-label decision trees in the complete EUROVOC domain (with hundreds of thousand of documents and tens of thousands of features) is limited. 4 CONCLUSION In this brief communication, we shared with the readers our recent experience with two popular induction algorithms that have been designed for text-categorization applications where each document can belong to two or more classes. We were primarily interested in learning more about their classification performance and computational costs as observed in a concrete real-world application domain. A closer look at the results reveals that BoosTexter is probably the better choice. The exception is the case when the user wants to make sure that the vast majority of the returned documents are relevant to the query, even if many other relevant documents have been overlooked. Then, Multi-Label C4.5 might be preferable. Even so, the high computational costs incurred in similar domains by decision-tree induction are a reason for major concern. More research is needed to speed up the induction process if these techniques are to become practical for the use in the development of large multilabel document thesauri. 5 ACKNOWLEDGMENT The research was partly supported by the NSF grant IIS-0513702. REFERENCES [1] LANGLEY, P. ; IBA, W.; THOMPSON, K. “An analysis of Bayesian classifiers,” in Natl. Conf. on



Artificial Intelligence, 1992, pp. 223-228. [Online]. Available: citeseer.ist.psu.edu/article /langley92analysis.html.

[2] FRIEDMAN, N. ; GEIGER, D.; GOLDSZMIDT, M. “Bayesian network classifiers,” Machine Learning, vol. 29, no. 2-3, pp. 131-163, 1997. [Online]. Available: citeseer.ist.psu.edu/ friedman97bayesian.html.

[3] MCCALLUM A.; NIGAM, K. “A comparison of event models for naive Bayes text classification,” in Proc. Workshop on Learning for Text Categorization (AAAI'98), 1998. [Online]. Available: citeseer.ist.psu.edu/mccallum98comparison.html.

[4] LI, B.; LU, Q.; YU, S. An adaptive k-nearest neighbor text categorization strategy, ACM Trans. on Asian Language Information Processing (TALIP), vol. 3, pp. 215-226, 2004.

[5] JOACHIMS, T. “Text categorization with support vector machines: learning with many relevant features,” in Proc. European Conf. on Machine Learning (ECML'98), C. Nedellec and C. Rouveirol, Eds., no. 1398. Chemnitz, DE: Springer Verlag, Heidelberg, DE, 1998, pp. 137-142. [Online]. Available: citeseer.ist.psu.edu/article/joachims98text.html.

[6] KWOK, J. T. “Automated text categorization using support vector machine,” in Proc. Int'l Conf. on Neural Information Processing (ICONIP'98), Kitakyushu, JP, 1998, pp. 347-351. [Online]. Available: citeseer.ist.psu.edu/kwok98automated.html.

[7] CLARE, A.; KING, R. D. “Knowledge discovery in multi-label phenotype data,” in Proceedings of the 5th European Conference on Principles of Data Mining and Knowledge Discovery (PKDD'01), Freiburg, Germany, 2001.

[8] SCHAPIRE; R. E.; SINGER, Y. “Improved boosting using confidence-rated predictions,” Machine Learning, vol. 37, no. 3, pp. 297-336, 1999. [Online]. Available: citeseer.ist.psu.edu /schapire99improved.html.

[9] SARINNAPAKORN K.; KUBAT, M. “Combining subclassifiers in text categorization: A dst-based solution and a case study,” IEEE Transactions on Knowledge and Data Engineering, vol. 19, no. 12, pp. 1638-1651, 2007.

[10] SCHAPIRE, R. E. “The strength of weak learnability,” Machine Learning, vol. 5, no. 2, pp. 197-227,1990.

[11] VAN RIJSBERGEN, C. J. Information Retrieval, 2nd ed. London: Butterworths, 1979. [Online]. Available: http://www.dcs.gla.ac.uk/iain/keith/index.htm.

[12] YANG, Y. “An evaluation of statistical approaches to text categorization,” Information Retrieval, vol. 1, no. 1/2, pp. 69-90, 1999. [Online]. Available: citeseer.ist.psu.edu/article/ yang98evaluation.html.

[13] YANG, Y. ; PEDERSEN, J. O. “A comparative study on feature selection in text categorization,” in Proceedings of ICML-97, 14th International Conference on Machine Learning, D. H. Fisher, Ed. Nashville, US: Morgan Kaufmann Publishers, San Francisco, US, 1997, pp. 412-420. [Online]. Available: citeseer.ist.psu.edu/yang97comparative.html.

ADDRESS: Sareewan Dendamrongvit Department of Electrical & Computer Engineering University of Miami Coral Gables, FL 33146 U.S.A. [email protected] Asoc. Prof. Dr. Miroslav Kubat, PhD., M.S. Department of Electrical & Computer Engineering University of Miami Coral Gables, FL 33146 U.S.A. [email protected] Zeynel Sendur Department of Electrical & Computer Engineering University of Miami Coral Gables, FL 33146 U.S.A. [email protected]



STATISTICAL AND SOFT COMPUTING METHODS IN CROSS RATES MODELING

Ján Bábel, Zuzana Mečiarová, Lucia Pančíková

University of Žilina

Abstract: This study discusses the analysis of the Currency cross rates using of statistical and soft computing methods. The ARMA and GARCH models represent the standard statistical approach for modeling economic time series and the Neural Network model was estimated for comparing its behavior against these models. The models were applied on cross rate time series provided by the Currency site OANDA. After the estimation, the predictions were finally made and its accuracies were judged by computing the RMSE values.

Keywords: ARMA, GARCH, Neural Network, heteroskedasticity, forecasting, Cross rate, time series analysis

1. INTRODUCTION Over the last decade the Neural Network methods, coming from the brain science, find its place in economic data modeling and forecasting and offer a powerful alternative to linear models. Forecasting simply means understanding, which variables lead to predict other variables and this mean looking at the past to see which variables are significant leading factors of the behavior of the others variables. The premium is on both the precision of the estimates return series or spreads as well as the computational ease and speed with which these estimates can be obtained. In this study we try to answer the question: “Can Neural Network compete against the statistical (econometric) models?” This article is organized into 3 chapters. First chapter briefly discussed the model building process, assuming that reader is familiar with the basic concept of time series modeling. Second chapter contains detailed analysis of models estimation. In next chapter the estimated models were used for forecasting to obtain the prediction for next periods. At the end the conclusion was drawn. 2. MODEL BUILDING 2.1 BUILDING THE ARMA MODEL 1. Compute the ACF and PACF function and choose the correct order of autoregressive part. 2.2 BUILDING THE GARCH MODEL 1. Specify a mean equation by testing for serial dependence in the data and, if necessary, building a statistical

model (e. g., an ARMA model) for the return series. 2. Use the residuals of the mean equation to test for ARCH effects. 3. Specify the volatility model if ARCH effects are statistically significant and perform a joint estimation of the

mean and volatility equations. 4. Check the fitted model carefully and refine it if necessary. 2.3 BUILDING THE NEURAL NETWORK 1. The skip-layer feed-forward neural network have 7 inputs, which represent the AR order in previous models, 1 single-hidden-layer with 2 nodes and 1 output which would be the desired cross rate. 3. MODELS ESTIMATION Time series used in this paper is the Currency cross rate between Czech Koruna (CZK) and Slovak Koruna (SKK)



1.16

1.20

1.24

1.28

1.32

1.36

2007M01 2007M07 2008M01 2008M07

CZK/SKK01/01/2007 --- 31/10/2008

Figure 1 The evolution of cross rate CZK/SKK. The study employs data from 1st of January 2007 to 2nd of

December 2008. Made with quantmod package in R. According to autocorrelation function (ACF) and partial autocorrelation function (PACF), we specified the order of AR model. In this case we got ARMA(7,0) or AR(7) model. The estimation of ARMA(7,0) is

ttt

tttttt

arr

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76

54321

0178.00588.0

0858.01094.0009625.0087.0790.01.246

-.04

-.02

.00

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.04

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1.24

1.28

1.32

1.36

2007M07 2008M01 2008M07

ResidualsActual CZKSKKFitted ARMA

ARMA(7,0)

Figure 2 The estimated ARMA(7,0) model against the original datasets on whole range. (Residuals are at the

bottom). Made in Eviews. Before estimating a full GARCH model, it is usually good practice to test for the presence of ARCH effects in the residuals. The most common test for ARCH effects is ARCH Lagrange Multiplier (LM) test which test the null hypothesis that there are no ARCH effects. If the p-value calculated in R environment is smaller than the conventional 0.05, the null hypothesis is rejected. If there are no ARCH effects in the residuals, then the GARCH model is unnecessary. The ARCH LM test in table 1 shows the presence of ARCH effects or conditional heteroskedasticity so we can go ahead and make a joint estimation with using of volatility GARCH(1,1) model for residuals.

Table 1 ARCH LM test for ARMA(7,0) residuals in R environment Lag 1 7 14

p-value 0.000933 9.122e-05 0.00567 The joint estimation of ARMA(7,0)+ GARCH(1,1) with Gaussian error distribution is



ttt

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arr

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76

54321

0218.005591.0

0651.00821.00843.00929.07269.0 0.0012

21

21

2 3616.03049.00000169.0 ttt a

Next the ARCH LM test was performed for standardized residuals of AR(7)+GARCH(1,1) model to check if the model was successful in removing the ARCH effect. The results of ARCH test in table 2 show no evidence of ARCH effects in standardized residuals so the model was successful in modeling the conditional heteroskedasticity.

Table 2 ARCH test for standardized residuals of the model AR(7) + GARCH(1,1) in R environment Lag 1 7 14

p-value 0.8439 0.5704 0.7441 To make sure if the model was correctly quantified we test the assumption that standardized residuals are IID (independent and identically distributed). For this purpose we use BDS test, which test nonlinear correlation in data and can distinguish between data generated by chaotic systems from data generated by stochastic systems The result of BDS test shows that the standardized residuals are IID. Model is adequate specified and quantified. The fitted model vs. the original dataset is in figure 3

-.04

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.00

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2007M07 2008M01 2008M07

ResidualSActual CZKSKKFitted ARMA+GARCH

ARMA(7,0)+GARCH(1,1)

Figure 3 The estimated ARMA(7,0) + GARCH(1,1) model against the original datasets on whole range.

(Residuals are at the bottom). Made in Eviews.

0 100 200 300 400 500 600

1.2

01.2

51.3

0

Neural Network

t

CZK/S

KK

CZKSKKTrained NN

Figure 4 The estimated 7-2-1 trained network against the original datasets on whole range. Made in R.

Table 3 Coefficient of determination R^2 for different models

models AR(7) AR(7)+GARCH(1,1) Neural Network R^2 0.9694 0.9462 0.9686

(5)



4. PREDICTIONS In previous section the ARMA and GARCH models were estimated. Next, the Neural Network was trained on the same data sample for comparing the ability to fit the given data by computing the coefficient of determination. As we can see in table 3, the best fit was obtained by the ARMA(7,0) closely followed by Neural Network model. The GARCH model ended up last. Next the predictions were made for next 32 trading days and the RMSEs of predictions are in table 4. Again, the ARMA(7,0) produce the best prediction, followed by NN and GARCH. The figure 5 depicts the behavior of these models in forecasting sample.

Table 4 The RMSEs of predictions for different models AR(7) AR(7)+GARCH(1,1) Neural Network

RMSE 0.01067 0.018545 0.01152

1.18

1.20

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PREDICTION-ARMA(7,0)

1.18

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CZKSKKPREDICTED ARMA+GARCH

PREDICTION-ARMA(7,0)+GARCH(1,1,)

Figure 5 Original time series with solid line against the predicted time series with dotted line using the ARMA(7,0) (in the left picture) and ARMA(7,0)+GARCH(1,1) (in the right picture). Made in Eviews

0 5 10 15 20 25 30

1.2

01.2

21.2

41.2

61.2

8

Prediction-Neural Network

t

CZK

/SK

K

CZKSKKPredicted NN

Figure 5 continued: Original time series with solid line against the predicted time series with dotted line using

the 7-2-1 Neural Network. Made in R 5. CONCLUSION In this article we have examined the ARMA and GARCH models for the time series of the Currency cross rate CZK/SKK. Against these statistical models we have formed the Neural Network model. After examining their behavior in estimation as well as in prediction part we came to conclusion that Neural Network provide the efficient alternative against the statistical models. The advantages of NNs include no pre estimation, post estimation model or parameters testing so their application is relative easy even without the solid econometric background. REFERENCES [1] ART, J., ARTLOVÁ, M. Finanční časoví řady, Grada, 2003, ISBN:80-247-0330-0 [2] BOLLERSLEV, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of

Econometrics 31: 307–327.



[3] CARMONA, R. A. Statistical Analysis of Financial Data in S-Plus, Springer, 2004, ISBN: 978-0-387-20286-0

[4] ENGLE, R. GARCH 101: The Use of ARCH/GARCH Models in Applied Econometrics, Journal of Economic Perspectives—Volume 15, Number 4, pages 157–168, 2001

[5] MARČEK, D.; MARČEK, M. Analýza, modelovanie a prognózovanie časových radov s aplikáciami v ekonomike, ES ŽU Žilina, 2001, ISBN:80-7100-870-2

[6] TSAY, RUEY S. Analysis of Financial Time Series, Second Edition, John Wiley & Sons, Inc., 2005, ISBN-13 978-0-471-69074-0

[7] ZIVOT, E.; WANG, J. Modeling Financial Time Series with S-PLUS®, Springer, 2005, ISBN: 978-0-387-27965-7

[8] OANDA, The currency site (2008) [9] http://www.oanda.com/ ACKNOWLEDGEMENTS This work was supported by Slovak grant foundation under the grant VEGA No. 1/0024/08. ADDRESS Ing. Ján Bábel University of Žilina Faculty of Management Science and Informatics Department of Macro and Microeconomics Univerzitná 8215/1 010 26 Žilina E-mail: [email protected] Ing. Zuzana Mečiarová University of Žilina Faculty of Management Science and Informatics Department of Macro and Microeconomics Univerzitná 8215/1 010 26 Žilina E-mail: [email protected] Ing. Lucia Pančíková, PhD. University of Žilina Faculty of Management Science and Informatics Department of Macro and Microeconomics Univerzitná 8215/1 010 26 Žilina E-mail: [email protected]



CELULÁRNÍ PROCESOR LOGICKÝCH FUNKCÍ PRO POUŽITÍ PŘI GENETICKÉM

PROGRAMOVÁNÍ A EVOLUČNÍM VÝVOJI ALGORITMŮ

Petr Skorkovský


Abstract: Some ways are described about preparation of a system based on processes similar to cellular automata, whose behaviors are evolving from a algoritmus described by genetically programming techniques.Usage of logical functions, which may be simple to be coded with a genetic language, is suggested and through several paper parts a simple cellular processor of logical functions, which could be used in the field of evolutionary and genetically programming, is described.

Key words: Cellular automaton, logic functions, evolutionary programming, genetic algorithms, parallel processing.

TEORETICKÝ ÚVOD 1. Pro potřeby automatického vývoje algoritmů s použitím genetického programování je vhodné navrhnout:

Obecný zápis algoritmů genetickým jazykem, softwarové prostředí, které je schopno tímto genetickým jazykem zapsané algoritmy provádět.

2. Cílem návrhu je algoritmus zapsaný genetickým jazykem, který by měl mít tyto vlastnosti:

Jednoduchost zápisu (při nesplnění požadavku se prudce zvyšuje výpočetní zátěž), robustnost (při nesplnění požadavku hrozí zablokování celého algoritmu při jediné chybě v algoritmu), optimální variabilita zápisu (nízká variabilita, nebo příliš vysoká variabilita vede k problémům při

dosažení cíle – nedostatek, nebo velký nárůst kombinací), upřednostňovat paralelní zpracování před sekvenčním nebo procedurálním (Několik paralelních

současně běžících větví algoritmu může dojít k výsledku různými cestami a přitom nehrozí zablokování celého algoritmu chybou jediné větve – viz robustnost.),

realizovaný algoritmus by měl být deterministický, aby bylo možné vždy dojít ke stejnému výsledku při stejných vstupních datech,

zapsaný algoritmus by měl být pokud možno do jisté míry čitelný i pro člověka aby byl schopen zpětně pochopit a analyzovat vnitřní chování algoritmu, nebo jej následně ručně opravovat či dooptimalizovat (nepřehledné a složité zápisy algoritmu neumožňují další ruční zdokonalování, úpravy, nebo jen pouhou analýzu vnitřního chování),

celý systém (zapsaný algoritmus + jeho interpreter) by měl být schopen realizovat libovolný myslitelný (deterministický) algoritmus, včetně virtuální emulace sama sebe, např. měl by být schopen emulovat libovolný známý počítačový mikroprocesor a program na něm běžící v jazyce tohoto mikroprocesoru (v mezích technické realizovatelnosti, dle časové a kapacitní náročnosti – teoretická proveditelnost).

3. Výchozí myšlenkou pro realizaci je princip celulárního automatu, pro který jsou charakteristické tyto

vlastnosti [1]: Paralelismus (výpočet nových hodnot stavů všech prvků probíhá současně, na běžných sériových

počítačích se musí tento postup simulovat), lokalita (nový stav prvku závisí jen na jeho původním stavu a na původních stavech prvků z jeho okolí), homogenita (pro všechny prvky platí stejná lokální přechodová funkce).

VLASTNOSTI A PARAMETRY NAVRŽENÉHO CELULÁRNÍHO AUTOMATU 1. Základní stavební jednotkou celulárního logického procesoru je zvolena jedna buňka Bn,k, která spolu

s ostatními buňkami tvoří jednorozměrný celulární automat: Index „k“ v Bn,k značí číslo kroku při provádění algoritmu, index „n“ v Bn,k značí absolutní adresu

buňky jednorozměrného celulárního automatu v rozsahu < 0, nmax >. Buňka Bn,k je nositelem binární informace:

Obsah buňky Bn,k nabývá okamžité (binární) hodnoty yn,k = (0)2, (1)2,



yn,k je poslední platná hodnota, která je výsledkem výpočtu z předchozího kroku k-1, yn,k-1 je hodnota platná v předchozím kroku k-1 nutná pro výpočet nové hodnoty yn,k v kroku

aktuálním. Buňka Bn,k dále obsahuje informaci o spojení s jinými dvěmi buňkami z blízkého, nebo i vzdáleného

okolí: K tomuto účelu jsou použity dvě relativní adresy an, bn, směřující na jiné dvě (mnohdy sousední)

buňky Bna,k a Bnb,k, které mají absolutní adresy na a nb. Rozsah relativních adres je v ideálním případě an = < -nmax/2 , +nmax/2 >; bn = < -nmax/2 ,

+nmax/2 >, rozsah absolutních adres na = < 0, nmax >; nb = < 0, nmax >.

10

dvou jiných buněk z předchozího kroku použitých jako vstupní

Každá buňka Bn,k může obsahovat jinou přechodovou funkci Fn, než mají ostatní buňky.

cká funkce (popsaná číslem 0 – 255) je realizována stejnou a to současně maximální možnou rychlostí.

2.

rogramem pro vyšší rychlost interpretru, nebo lze užít typ

dálenými adresovými oblastmi pomocí několika na sebe navazujících buněk přes relativní adresování.

ÍHO LOGICKÉHO PROCESORU PRO POUŽITÍ GENETICKÝMI A

ímto způsobem (Binárním kódováním [2] ): = [ [a , b , F ] , [a , b , F ], [a , b , F ], … [a , b , F ] ]

0, a0,1, a0,2, a0,3, a0,4, a , a , a , a0,8, a0,9, a0,10, a0,11], [b0,0, b0,1, b0,2, b0,3, b0,4, b0,5, b0,6, b0,7, b0,8,

, a , a , a , a , a1,5, a1,6, a , a1,8, a1,9, a1,10, a1,11], [b1,0, b1,1, b1,2, b1,3, b1,4, b1,5, b1,6, b1,7,

, a , a , a , a , a2,5, a2,6, a , a2,8, a2,9, a2,10, a2,11], [b2,0, b2,1, b2,2, b2,3, b2,4, b2,5, b2,6, b2,7,

Při realné implementaci musí stačit menší rozsah relativních adres, aby se omezil počet nutných bitů pro uchování informace o adrese v genetickém zápisu,

relativnost adres umožňuje invariantnost vůči posunutí a umístění funkčních celků(blok několika souvisejících funkcí lze umístit kamkoliv bez ztráty funkčnosti, například při křížení, mutacích, posunech v kódu během aplikace genetických a evolučních algoritmů na vývoj hledaného algoritmu).

Výpočet absolutních adres z relativních probíhá tímto způsobem: na = n + an; nb = n + bn. Pro platnost vypočtených adres < 0, nmax >, při přetečení nebo podtečení adresy se uplatňuje nx

mod (nmax + 1) (funkce modulo), z toho vyplývá: o Adresový prostor je uzavřený v obou směrech do sebe, nemá žádný počátek ani konec, o všechny adresy jsou vůči všem ostatním rovnocenné.

Buňka Bn,k dále obsahuje informaci o přechodové funkci Fn kterou buňka provádí v každém kroku „k“

ve formě osmibitové pravdivostní tabulky Fn = [cn0, cn1, cn2, cn3, cn4, cn5, cn6, cn7 ] = < (0)10 , (255)>.

funkce Fn zpracovává tři vstupy: yn,k-1: binární hodnota vlastní buňky Bn,k z předchozího kroku k-1, yna,k-1; ynb,k-1: binární hodnoty

hodnoty v současném kroku k.

yn,k je výstupem z funkce Fn : yn,k = Fn(yn,k-1, yna,k-1, ynb,k-1) = <(0)2, (1)2>. Rozsah 256ti typů logických funkcí bohatě pokrývá možnosti pro sestavování složitějších logických

bloků (a algoritmů), při minimální výpočetní náročnosti. Pouhým kopírováním jednoho bitu z pravdivostní tabulky dle momentální kombinace dosahujeme rychlého a efektivního provádění algoritmů. Nespornou výhodou je i to, že jakkoliv „složitá“ logi

shrnutí: Buňka je tedy zcela popsána následující skupinou informací: Bn,k =[yn,k, yn,k-1, an, bn, Fn]. Pro genetický zápis vlastností jediné buňky Bn,k nám může postačit 32bitů, což je ideální velikost pro zpracování implementací assemblerovým p„integer“ v jazyce C u 32-bitových aplikací: Pro an : 12 bitů = < -2048, + 2047>, bn : 12 bitů = < -2048, + 2047>, Fn : 8 bitů = < (0)10, (255)10 >. Pro realizaci rozsáhlých algoritmů přesahujících možnosti relativního adresování, je nutné zřetězit

přenos informace mezi velmi vz

IMPLEMENTACE CELULÁRNEVOLUČNÍMI ALGORITMY 1. Algoritmus AG je zapsán genetickým kódem například tAG 0 0 0 1 1 1 2 2 2 n n n

Po rozepsání na jednotlivá binární data při použití 32bitů pro jednu buňku Bn,k: AG = [ [ [a0, 0,5 0,6 0,7

b0,9, b0,10, b0,11], [c0,0, c0,1, c0,2, c0,3, c0,4, c0,5, c0,6, c0,7 ] ], [ [a1,0 1,1 1,2 1,3 1,4 1,7

b1,8, b1,9, b1,10, b1,11], [c1,0, c1,1, c1,2, c1,3, c1,4, c1,5, c1,6, c1,7 ] ], [ [a2,0 2,1 2,2 2,3 2,4 2,7

b2,8, b2,9, b2,10, b2,11],



[c2,0, c2,1, c2,2, c2,3, c2,4, c2,5, c2,6, c2,7 ] ], …………………………………………………… [ [an,0 n,1 n,2 n,3 n,4 n,7

b, a , a , a , a , an,5, an,6, a , an,8, an,9, an,10, an,11], [bn,0, bn,1, bn,2, bn,3, bn,4, bn,5, bn,6, bn,7,

[c , c , c , c , c , c , c , c ] ] ]

entován v každém prováděném kroku „k“ p logického procesoru sekvencí bitů:

k = [ B0,k, B1,k, B , … B ]

k = [[y , y , a , b , F0 ], [y , y , a1, b1, F ], [y2,k, y2,k-1, a , b2, F ], … [yn,k, yn,k-1, an, bn, Fn ] ]

, c , c0,3, c0,4, c0,5, c0,6, c ] ],

, c1,2, c1,3, c1,4, c1,5, c1,6, c ] ],

, b ], [c2,0, c2,1, c2,2, c2,3, c2,4, c2,5, c2,6, c2,7 ] ],

[b , b , b , b , b , b , b , b , b , b , b , b ], [c , c , c , c , c , c , c , c ] ] ]

tem k počátečním podmínkám, nebo také ke specifickému zadání úlohy, kter

í úlohy k řešení ku k=0 :

n = <0 , nmax > : S = [s0, s1, s2, … snmax ] :

n,8, bn,9, bn,10, bn,11], n,0 n,1 n,2 n,3 n,4 n,5 n,6 n,7

2. Algoritmus AG je po načtení informace z genetického kódu reprez

im lementujícího celulárního A 2,k n,k

A 0,k 0,k-1 0 0 1,k 1,k-1 1 2 2

Ak = [ [ y0,k, y0,k-1, [a0,0, a0,1, a0,2, a0,3, a0,4, a0,5, a0,6, a0,7, a0,8, a0,9, a0,10, a0,11], [b0,0, b0,1, b0,2, b0,3, b0,4, b0,5, b0,6, b0,7, b0,8, b0,9, b0,10, b0,11], [c0,0, c0,1 0,2 0,7

[ y1,k, y1,k-1, [a1,0, a1,1, a1,2, a1,3, a1,4, a1,5, a1,6, a1,7, a1,8, a1,9, a1,10, a1,11], [b1,0, b1,1, b1,2, b1,3, b1,4, b1,5, b1,6, b1,7, b1,8, b1,9, b1,10, b1,11], [c1,0, c1,1 1,7

[ y2,k, y2,k-1, [a2,0, a2,1, a2,2, a2,3, a2,4, a2,5, a2,6, a2,7, a2,8, a2,9, a2,10, a2,11], [b2,0, b2,1, b2,2, b2,3, b2,4, b2,5, b2,6, b2,7, b2,8, b2,9, b2,10 2,11

…………………………………………………… [ yn,k, yn,k-1, [an,0, an,1, an,2, an,3, an,4, an,5, an,6, an,7, an,8, an,9, an,10, an,11], n,0 n,1 n,2 n,3 n,4 n,5 n,6 n,7 n,8 n,9 n,10 n,11 n,0 n,1 n,2 n,3 n,4 n,5 n,6 n,7

3. Před spuštěním celulárního logického procesoru je nutné načíst do každé buňky Bn,0 na pozici yn,k-1 v A0

výchozí stav algoritmu; je ekvivalenou je potřeba algoritmem řešit: startovní pozice (počáteční podmínka algoritmu, zadán algoritmem) v kro

nkn sy 1, ( = <(0)2, (1)2> )

kroku “k” běžícího celulárního logického procesoru se provede pro každou buňku Bn,k v Ak tato operace:

n = <0 , nmax > : (= <(0)2, (1)2> )

unkce Fn je realizována pravdivostní tabulkou:

p

4. V každém

),,( 1,1,1,, kbnkanknnkn nnyyyFy

F

vstup výstupozice k-1 bn,k-1 yyn, yn+ n+an,k-

1 yn,k

0 0 0 0 cn,0 1 0 0 1 cn,1 2 0 1 0 cn,2 3 0 1 1 cn,3 4 1 0 0 cn,4 5 1 0 1 cn,5 6 1 1 0 c n,6

7 1 1 1 cn,7 5. Po uplynutí m kroků celulárního logického procesoru, kdy k = m, každá buňka B na pozici yn,m v Am

obsahuje cílo

řešení algoritmu (konečný stav rního kého procesoru) v kroku k = m:

n,m

vý stav zn, který je ekvivalentem hledaného řešení algoritmu:

celulá logic

n = <0 , nmax > : Z = [z0, z1, z2, … znmax ] : mnn yz , (= <(0)2, (1)2> )

6. Celý průběh algoritmu implementovaného celulárním logickým procesorem od počátku až do konce jeho

běhu lze zapsat ve tvaru:

. Při hledání očekávaného cílového algoritmu AG genetickým algoritmem se hledá takový AG, který po

k = <0 , m > : PCLP = [A0, A1, A2, … Am ]

7



implementaci celulárním logickým procesorem PCLP

kódovaného v A0,

u

ospěje k očekávanému cíli Z = [z , z , z , … z ] kódovaného v A .

POPIS CELULÁRNÍHO LOGICKÉHO PROCESORU S POUŽITÍM

Stavová matice Yk celulárního logického procesoru obsahující aktuální binární hodnoty všech buněk v kroku „k“:

od počátečního zadání S = [s0, s1, s2, … snmax ]

po provedení m kroků algoritm PCLP = [A0, A1, A2, … Am ], d 0 1 2 nmax m

MATEMATICKÝ MATICOVÉHO POČTU 1. Přehled definic:

knkkkk yyyyY max,,2,1,0 , 22, 1,0kny

Matice F obsahující logické funkce všech buněk vyjádřené pravdivostními tabulkami v matici F:

,

6max,

5max,

4max,

3max,

2max,

1max,

0max,

6,2

5,2

4,2

3,2

2,2

1,2

0,2

6,1

5,1

4,1

3,1

2,1

1,1

0,1

6,0

5,0

4,0

3,0

2,0

1,0

0,0

max210

n

n

n

n

n

n

n

n

c

c

c

c

c

c

c

c

c

c

c

c

c

c

c

c

c

c

c

c

c

c

c

c

c

c

c

c

CCCCF

7max,7,27,17,0 ncccc

22, 1,0inc

Čtvercová matice V obsahující vazby na okolní buňky zakódované v koeficientech uvnitř matice, sloužící pro sběr výstupů z okolních buněk:

, pro vn,j platí:

ň á další vazbu (maximum jsou 2 vazby, minimum je 0 vazeb) =>

Matice Pk obsahující vypočtené indexy pro výběr nové aktuální binární hodnoty všech buněk z matice logických funkcí F pravdivostních t k:

2

2max,2

2,12,0

1max,1,22

1,0

0max,0,20,12

2

2

2

2

n

n

n

vvv

vvv

vvv

vvv

V

max,2max,1max,0 nnn

buňka na pozici n přijímá výstup z jiné buňky na pozici na => 0, 2nnv ,

a

buňka na pozici n přijímá výstup z jiné buňky na pozici nb => 12v ,

bu ka nem žádnou

, bnn

0, jnv .

abule

knkkkk iiiiP max,,2,1,0 , 1010101010101010, 7,6,5,4,3,2,1,0kni

Transformační matice Tn,k(i) obsahující výpočet koeficientů pro převod indexů in,k z desítkové soustavy (0,1,2, … ,7) na konkrétní vybranou hodnotu z pravdivostní tabulky:



1

1

1

1

1

1

1

1

,

1!7!

654321

1!7!

754321

1!7!

764321

1!7!

765321

1!7!

765421

1!7!

765431

1!7!

765432

1!7!

7654321

)(

i

i

i

i

i

i

i

i

kn

ii

iiiiiiiii

iiiiiiiii

iiiiiiiii

iiiiiiiii

iiiiiiiii

iiiiiiiii

iiiiiiiii

iiiiiii

iT , nebo

7

6

5

4

3

,,

,

,

,

,

,

0

0

0

0

0

0

)(

kn

kn

kn

kn

kn

i

i

i

i

i

knkn iT

2

1

,

,

,

0

0

kn

kn

kn

i

i

i

Všechny výsledné matice Tn,k lze sdružit transponovaně do celkové matice

Tkn

Tk

Tk

Tkk TTTTT max,,2,1,0

k pozdějšímu použití pro skalární součin matic. 2. Průběh výpočtu přechodu z předchozího stavu celulárního logického procesoru Yk do nového stavu Yk+1:

Pro výpočet všech indexů (které obsahuje matice Pk) výběru nového výstupu Yk+1 z pravdivostních

tabulek Pk v kroku „k“ se provede maticová operace násobení dvou matic: VYP kk ,

pro každý ze vzniklých indexů in,k matice Pk se vypočítá transformační matice Tn,k(in,k) a sdruží se jako transponované prvky Tn,k

T v matici Tk, Dle vypočítaných indexů in,k matice převedených na transformační matice v matici Tk s pomocí

skalárního součinu matic Tn,k s pravdivostními tabulkami logických funkcí Cn vyjádřených v matici F se provede výběr nových hodnot yn,k+1

nT

knknk CTyY ,1,1 ::

ZÁVĚR Uvedené poznatky jsem se pokusil ověřit vytvořením jednoduchého programu, který implementuje výše popsaný celulární procesor logických funkcí. S jeho pomocí jsem ručně (bez genetických a evolučních technik) vytvořil několik jednoduchých algoritmů. Jako příklad mohu uvést fungující tříbitový čítač. Dále jsem se pokusil zmapovat různé typy logických funkcí, které lze realizovat osmibitovou pravdivostní tabulkou. Je zde široká škála možností při současně snadné realizaci. V mé další práci bych se rád věnoval případnému využití uvedeného celulárního logického procesoru pro evoluční a genetické programování, což je velmi obsáhlá oblast s velkým tvůrčím potenciálem. LITERATURA: [1] MAŘÍK, V.; ŠTĚPÁNKOVÁ, O. ; LAŽANSKÝ, J. a kolektiv, Umělá inteligence, Academia : 2001. [2] HYNEK, J. Genetické algoritmy a genetické programování, Grada Publishing, a.s. 2008.



ADDRESS Ing. Petr Skorkovský Faculty of Electrical Engineering and Communication Brno University of Technology Údolní 244/53 602 00 Brno 2 GSM: +420 777 084 778 e-mail: [email protected]



INSPECTION OF STEEL DEGRADATION BY MAGNETIC ADAPTIVE TESTING

1Gábor Vértesy, 2Ivan Tomáš

1Research Institute for Technical Physics and Materials Science 2Academy of Sciences of the Czech Republic

Abstract Three series of low carbon steel samples, plastically deformed by cold rolling to different stages of deformation, were investigated by the method of Magnetic Adaptive Testing. Magnetically closed and magnetically open samples were used. Results of the non-destructive magnetic tests were compared with the destructive mechanical measurements of Vickers hardness and ductile-brittle transition temperature. Linear, sensitive correlation with low scatter of values was found between the magnetic parameters and the two mechanical ones for any shape of the investigated specimens. Based on these results, Magnetic Adaptive Testing is suggested as a highly promising non-destructive alternative of destructive Charpy impact tests for monitoring irradiation embrittlement of surveillance samples in pressure vessels of nuclear reactors.

INTRODUCTION Magnetic approach is an obvious candidate for non-destructive testing, for detection and characterization of any defects in materials and products made of such materials [1, 2]. A novel, sensitive and experimentally friendly method, Magnetic Adaptive Testing (MAT) has been considered recently, based on systematic measurement of magnetic minor hysteresis loops [3]. MAT introduces general magnetic descriptors to diverse variations in non-magnetic properties of ferromagnetic materials, optimally adapted to the just investigated property and the material. Nowadays, steel degradation of reactor pressure vessels presents a very important and urgent problem of nuclear industry. No effective nondestructive method for reliable detection of the steel ductile-brittle transition was commonly established and accepted yet. Degradation of the pressure vessel steel is caused by joint influence of the long time exposure of the material to elevated temperature, to neutron irradiation and to mechanical pressure. Action of each of those factors contributes to modification of microstructure of the steel, and magnetic methods, like e. g. MAT, are expected to bring an acceptable solution of the problem. In the present work we applied MAT for a series of degraded steel, and the measurements were performed both on magnetically closed and open samples. We examined structural degradation of the compressed (cold rolled) steel samples and studied correlation of the nondestructively measured MAT results, with the Vickers hardness (HV) and the ductile-brittle transition temperature (DBTT) determined destructively, in the traditional way. The series of low carbon steel samples is investigated in the frame of a chain of magnetic non-destructive measurements on round robin samples, organized by the Universal Network for Magnetic Non-Destructive Evaluation (UNMNDE) [4]. EXPERIMENT Three series of low carbon steel samples, each containing five samples, were prepared for the magnetic measurement. The material of them is low-carbon steel with 0.16 wt. % of C, 0.20 wt. % of Si, 0.44 wt. % of Mn, with the rest of Fe. Convenient pieces of the material were cold-rolled down to plastic deformation, , of 0, 5, 10, 20 and 40 %, and then the samples of desired dimensions and shapes were carefully machined from the deformed steel. Three types of shapes were prepared: picture-frames, rectangular plates and rectangular bars. Vickers hardness, HV, of the deformed material was measured with the standard Vickers indentation technique. Charpy impact test was performed on bar samples in the 201363 K temperature range. The ductile-brittle transition temperature, DBTT, was defined as the midpoint between the low toughness brittle- and the high toughness ductile-fracture regimes. A specially designed Permeameter was applied for measurement of families of minor loops of the magnetic circuit differential permeability. The picture-frame samples were magnetized and the induced signal from the



time-change of their magnetization was recorded with the aid of magnetizing and pick-up coils, respectively, wound directly on each of the samples. The magnetically open plate- and bar-samples were measured by attached magnetizing/measuring heads, namely by U-shaped yokes equipped with magnetizing and pick-up coils each. The samples are magnetized with step-wise increasing magnetic field and the signal of pick-up coil is used for characterization of the sample. The Permeameter works under full control of a PC computer. The computer registers data-files for each measured family of the minor “permeability” loops. The recorded signal data are processed by a data-evaluation program, one family of minor loops for each measured sample. The data-pools of the minor loops determined for every sample made possible to calculate the optimal MAT descriptors. RESULTS AND DISCUSSIONS The most sensitive and reliable MAT descriptors were taken for each series of samples. A representative set of such optimal MAT descriptors is plotted in Fig. 1 for the frame-, for the plate- and for the bar-shaped samples. In each measurement series, all the descriptors were normalized by data of the virgin (not rolled) sample, which made it possible to compare numbers measured on the different series As expected, the highest sensitivity was obtained in the case of magnetically closed samples, which follows from the most perfect magnetizing and sensing conditions in this measurement. But it is also evident that qualitatively the same relationships are obtained for all the three sample series.

0 10 20 30 40 50 600

2

4

6

8

10

12

14

Hc of major loop

Bar

Plate

Frame

Opt

imal

MA

T d

escr

ipto

rs

Rolling reduction [%]

Fig. 1. Optimal MAT descriptors for the three series of differently shaped samples

In the case of frame samples it is possible to measure the traditional characteristics of major loop, too. For comparison, the coercivity, Hc, of major loop is also shown in Fig. 1. It is seen very well, that MAT descriptors, determined from the family of minor hysteresis loops, are much more sensitive to the mechanical degradation even on magnetically open samples that those of magnetically closed samples determined from the major loop. By direct comparison of the traditionally, destructively measured parameters with the magnetic quantities, correlation can be found between the optimally chosen MAT descriptors and the ductile-brittle transition temperature, and/or the Vickers hardness. This can be seen in Figs. 2 and 3, where the MAT descriptors are shown as functions of the HV and the DBTT, respectively, for the different shape samples.

140 160 180 2000

2

4

6

8

10

12

14

Bar

Plate

Frame

Op

tima

l MA

T d

escr

ipto

rs

Vickers hardness

Fig. 2. Optimal MAT descriptors versus Vickers hardness for different shape samples



As it is seen from results of the measurements, the consecutive series of the nondestructively determined MAT descriptors well describe magnetic reflection of the investigated material modifications. The presented figures demonstrate, that in the case of the low carbon steel samples, plastically deformed by cold rolling, MAT feels the difference between the most compressed and the not compressed samples approximately as about 12:1 in the case of frames and about 6:1, in the case of bar samples, in contrast to about 1:1.5 and about 1:1.8 sensitivity as it is felt by the Vickers hardness and ductile-brittle transition temperature measurement, respectively. The conventional magnetic description of the compression by the coercive field values, HC, ended up also with the top sensitivity of about 1:2 only.

240 250 260 270 2800

2

4

6

8

10

12

14

Bar

Plate

FrameO

ptim

al M

AT

de

scri

pto

rs

Ductile-brittle transition temperature [K]

Fig. 3. Optimal MAT descriptors versus DBTT for different shape samples

Another advantage (compared to traditional magnetic methods) of the MAT method is the not required magnetic saturation of the samples, and possibility of application of a small simple yoke for magnetization and measurement of magnetically open samples. The method does not give absolute values of the traditional magnetic quantities, but evidently it is able to serve as a powerful tool for comparative measurements, and for detection of changes, which occur in structure of the inspected samples during their lifetime or during a period of their heavy-duty service. SUMMARY Magnetic Adaptive Testing yields highly sensitive and reliable correlation with plastic deformation and brittleness of the investigated material, which can be obtained from measurements typical by its low required magnetization of the samples. The presented results give a good chance to determine the level of embrittlement of ferromagnetic steel objects (e.g. of the nuclear pressure vessel surveillance specimens) due to their heavy-duty industrial service period, with the aid of the non-destructive method of Magnetic Adaptive Testing. ACKNOWLEDGEMENTS The work was supported by projects No.1QS100100508 and AVOZ 10100520 of the Academy of Sciences of the Czech Republic, by Hungarian Scientific Research Fund (project K-62466) and by the Czech-Hungarian and Japanese-Hungarian Bilateral Intergovernmental S&T Cooperation. REFERENCES [1] JOHNSON, M. J.; LO, C. C. H.; ZHU, B.; CAO, H.; JILES, D.C. J. Nondestruct.Eval., 20 (2000), 11. [2] JILES, D. C. Encyclopedia of Materials Science and Technology, Oxford, Elsevier Press, p.6021., 2001. [3] TOMÁŠ, I. J.Magn.Magn.Mat. 268 (2004), 178. [4] http://www.ndesrc.eng.iwate-u.ac.jp/UniversalNetwork/html/main.html. ADDRESS: Dr. Gábor Vértesy, Dr.Sc. Research Institute for Technical Physics and Materials Science Hungarian Academy of Sciences H-1525 Budapest P.O.B. 49 Hungary



RNDr. Ivan Tomáš, CSc. Institute of Physics Academy of Sciences of the Czech Republic Na Slovance 2 18221 Praha Czech Republic



ON AN EXAMPLE OF NON-LINEAR PROGRAMMING

Marie Tomšová

Vysoké učení technické v Brně

Abstract: Master study programmes at Faculty of Electrical Engineering and Communication, Brno University of Technology include a subject called MPSO (abbr. of Master, Probability, Statistics and Operational research). Elements of operational analysis are included in this subject. This contribution deals with one of its topics – non-linear programming.

1. INTRODUCTION Non-linear programming – a generalization of linear programming (for details cf. [2]) – can be used to solve a greater number of problems than linear programming. One of its variants is called quadratic programming. In this case the utility function )(xf is a quadratic function and the border function )(xgi of restricting conditions

is linear. Linear functions are concave up. If also the utility function is concave up, Kuhn-Tucker conditions hold (for

details cf. [2]). Lagrange function is quadratic in x j and linear in . Therefore, derivatives used in Kuhn-Tucker

conditions are linear. This means that the problem has been thus turned into a linear problem, i.e. it can be solved by a simplex method. (When looking for a maximum of a concave down utility function, we minimize the problem by negating the sign of the given utility function thus turning the function into a concave up one.) Problems which are more complicated than a concave up quadratic problem cannot be solved in a general way.

il

Two questions are therefore important: is the class of concave up quadratic problems of such an importance as to justify its study? how can one easily recognize that a quadratic function is concave up? The answer to the first question is positive – there are a great many practical problems which may be reformulated in the language of quadratic programming. The second question may be answered using the theory of quadratic forms, which has been introduced and developed for the study of conic sections and quadrics. 2. QUADRATIC FORMS Definition 1: Quadratic form f( x ) is an expression

ji

n

i

n

jij xxcxf

1 1

)( , where

nx

x

x 1

Example 1: 2221

2121 62),( xxxxxxf

Every quadratic form may be written in the following matrical form: xCxxf T **)( , where C is a

suitable square matrix. In our example the matrix C will be a 2x2 matrix:

).1(*6*2*)(

) ,(),(

2221

2122

2221122111

21

222221122121

2111

2

1222112221111

2

1

2221

121121

xxxxcxccxxcx

xcxxcxxcxcx

xxcxcxcxc

x

x

cc

ccxx

The condition is valid for infinitely many matrices, e.g. for . 62112 cc

13

32,

133

392,

16

02



However, symmetric matrices (the last one) are important in this respect. In symmetric matrices . jiij cc Definition 2: Quadratic form is called positive definite if its value is 0 only if all x i equal 0 while in all other

cases 0x )(xf 0. Quadratic form is called positive semidefinite if

0x )(xf 0, i.e. if it is always non-negative.

Theorem 1. Positive semidefinite quadratic forms are concave up.

Proof: Suppose xCxf T is positive semidefinite. We have to prove that

0)()1()(])1([ 2121 xfxfxxf . In the matrical form:

)]())[(1(

)1(2])1)[(1()(

)1()1(2)1(

)1())1(())1((

2121

212222112

221121222

112

22112121

xxCxx

xCxxCxxCxxCx

xCxxCxxCxxCxxCx

xCxxCxxxCxx

T

TTTT

TTTTT

TTT

The product in front of the brackets is negative, form in the brackets is non-negative, therefore the total product is always negative or equal to zero. The test of positive definiteness of a quadratic form has the form of the following theorem:

Theorem 2: (Sylvestr criterion): Let C be a symmetric matrix of a quadratic form f, i.e. xCxf T . If all main

minors of C are positive, the form is a positive definite one. Example 2: Use the above theorem to test the positive definiteness of the form

322123

22

21 4272 xxxxxxx .

The matrix C is ; for the first minor we have that |1|>0, for the second one we have that

720

221

011

C

012

1

1

1

and for the third one 03

720

221

011

. All minors are positive, the form is therefore a

positive definite one. Example 3: The form x2 is a positive definite quadratic form. An arbitrary function x2 + bx + C=0, e.g x2–2x–1=0 is concave up (only shifted). Concavity of the quadratic function is determined by the quadratic term. The other terms influence the position of the function in the plane. Example 4: Let us show this using a function

yxyxyxyxf 4243),( 22 Let us try to find a linear transformation, which transforms the function (with the exception of a possible constant) into a quadratic form with the same matrix. Functions x2 and x2-2x-1, byvaxu ,

)

seem to

be suitable. Let us look for the quadratic form in the form

. We get (**4 22 dvucvu

dcabcaycbxcxybbyyaaxx 2222 4842

When comparing coefficients of the respective terms we get (for coefficients at xy) , (for coefficients at 3c



x) , (for coefficients at y) 22 bca 48 bac8,3

and (for the absolute term) .

In total we get that

04 22 dcabba49/,7/2,7/4 dc

vu 34 22

ba

u **

0x

. The original function has a minimum in [a,b]=[4/7,2/7]. When shifting the origin [0,0] into this point

[a,b]=[4/7,2/7] we get a quadratic form , which has its minimum in the new origin. The linear part has been cancelled; the new quadratic form has the same matrix as the original quadratic part. The absolute term does not influence concavity, therefore it is unimportant in this respect.

v

Corollary: The quadratic function f which satisfies Theorem 1 satisfies also the following Kuhn-Tucker theorem.

Theorem 3: Vector

is a solution of convex (concave up) programming if and only if there exists a vector 0l

such that the following six (Kuhn-Tucker) conditions are valid:

(KT1) 0), 0

l(

x

xF 0

j

for all j

(KT2) 0))(

(),( 0

10

00

m

i j

ii

n

j jj x

xgl

x

xfx

x

lx )(

1

00

T Fx

00

x

(KT3) (usual condition of non-negativity of the solution)

(KT4) 0)(),( 0

l

xF

i0

0 xgl

i

(restriction)

(KT5) 0)(),(

01

000

0

Fl T

xgll

lxi

m

ii

i

00

l

(KT6)

3. FORMULATION OF THE QUADRATIC PROGRAMMING PROBLEM

n

jk

n

jjjkkj xpxxcxf1, 1

min)(

i = 1,2,,m ijij bxa

j = 1,2,,n 0jx

Matrical form: min xpxCx T

bxA

nx

x

x

x2

1

mb

b

b 1

0x Suppose that C is symmetric and positive semidefinite, i.e. the problem is a concave up one. Using Kuhn-Tucker conditions we reformulate the problem into a form suitable for simplex method.

Lagrange function: F x l x Cx px l Ax bT( , ) ( ) l =( l1,,ln )

02 AlpCxx

F T

(KT1)

xF

xx x C p lAT

( )2 0 (KT2) [0 is a scalar zero]



Ax b 0 (restrictio )

n

x 0 , l 0

Denote vx

F F

x xn

T

1

,K and substitute into KT1: F

2x C lA v pT

bdxA (complement of variable d)

x vT 0

x 0 , l 0 , v 0 And the problem can be solved by a modified simplex method.

ROBLEM he complete form of a quadratic programming problem is as follows:

4. SOLUTION OF THE QUADRATIC PROGRAMMING PT

1) When maximizing, we change the problem max)(ˆ xf into its opposite, i.e. min)(ˆ:)( xfxf .

2) Using the Sylvestr criterion we find out whether the quadratic part f is positive definite (or at least semidefinite). If so, 3) we can find out the absolute minimum f x( he domain of admissibl h) . If it is in t e solutions, the problem as

been solved successfully. If not (or if we have not looked for the absolute minimum), we use the Wolfe or the one-phase method, which find out minimum inside a domain.

Example 3: For the problem max422),( 2

even

0, , 5 2121

21

xxxx2 find out (if necessary)

hn-Tucker theorem.

First we find out if the extreme lies in the domain of admissible solutions:

-2x -2x +4 = 0 (partial derivatives with respect to x )

ain of admissible solutions. We turn the problem into a lem: f= f(x1,x2)= x1

2+2x x +2x 2+4x x2 =min. Using the Sylvestr criterion we easily verify

nvex) problem

x12+2x1x2+2x2

2+4x1x2+( x1+x2 5)

212121 xxxxxxxxf

the default model based on the Ku

1 2 1

-2x1-4x2+1= 0 The solution x = 3.5, x1 2= -1.5 does not lie in the domminimizing prob 1 2 2 1

that the quadratic form with the matrix

21

11C is positive definite (the first and second minor is 1).

Therefore, we get a concave up (co . We assemble the Lagrange function. Since we have one non-trivial restricting condition g(x1,x2)= x1+x2 5 0 , Lagrange function has one multiplier only: F(x ,x ,1 2 )= f(x1,x2)+* g(x1,x2) =

Denote v the vector of partial derivatives of the Lagrange function with respect to : 21, xx

),(),( vvFFF

v

. In our case 2121 xx x

142 1 xx , 422 vxxv 22211

We sim e absolute termof

plify the two equalities so that th is at the right-hand side. We get the first two equations the model:

(r1) 422 121 vxx

(r2) 2 1 vxx 14 22



The restriction – in our case the only non trivial one – will be the third equation. If we introduce the new variable d, we get

5(r3) 21 dxx

The second Kuhn-Tucker condition will be the fourth equation

(r4) 0 * vx T The fifth equation is the fourth Kuhn-Tucker condition. We include it in the form

(r5) 0 * d (gT eneral case), i.e. in our case 0* d Finally we include the trivial non-negativity conditions

0

x 0 0v 0d

bled the model. The m ltiplicator has Thus we have assem u

been denoted in order not to mistake it for 1.

] NOVÁK, M. Probability theory in combined form of study at FEEC BUT, Mezinárodní konference EPI Kunovice : 2006.

. Nelineární programování, 6. mezinárodní matematický workshop, Brno 2007, 117 -118.

erační analýza. SKRIPTORIUM VOŠ : Kunovice 1995.

gr. Marie Tomšová stav matematiky, Fakulta elektrotechniky a komunikačních technologií

í technické v Brně vutbr.cz

BIBLIOGRAPHY: [1

[2] TOMŠOVÁ, M[3] TYC, O. Operační analýza. MZLU Brno : 2002. [4] ZAPLETAL, J. Op[5] DUDORKIN, N. Operační analýza, FEL ČVUT, Praha : 1997. ADDRESS MÚVysoké učene-mail: tomsova@feec.



MODELING OF HIGH ORDER TRANSFER FUNCTION IN CURRENT MODE

Tomáš Dostál

European Polytechnic Institute, Ltd. Kunovice

Abstract: Active circuits in current mode of several orders and types (low-pass, high-pass, band-pass, band-reject and all-pass) are presented in this paper. These circuits are based on state-variable multiple-loop feedback models in the form of signal flow graphs. The graphs are modified and transformed in current mode, to be easily implemented using modern multi-output transconductors.

Keywords: Active circuits, current mode, multi-loop feedback structures, signal flow graphs, multi-output transconductor.

INTRODUCTION Active circuits operating in current mode (CM) have received considerable attention due to the advantages over the classical voltage mode (VM). The CM circuits offer higher frequency performances, can operate at lower DC supply with higher dynamic range. The CM is known as a very attractive approach [1], which is inherent in simplicity implementing such mathematical operations as current summation, subtraction, integration and multiplication by constant. There can be also simple realized current replicas and multi outputs with independent loading. It is well known that in the CM is better to employ new modern functional blocks in place of the standard operational amplifier. From them, the transconductors (OTA) have received particular interest, e.g. [2] - [5]. In this paper the multi-loop feedback state variable structure of the universal nth-order circuit in standard voltage mode is modified, transformed to the CM and non-conventionally realized by modern building blocks - OTA’s with more outputs. These filters have advantageous over the cascade and ladder structures, namely in universality of function, simplicity in direct design and independently adjustable coefficients. HIGH ORDER FILTERS The current transfer function of any nth-order circuit can be generally expressed as

011

1

011

1

...

...)(

bsbsbsb

asasasa

I

IsK

nn

nn

nn

nn

inp

out

. (1)

The formula (1) is usually normalized and has the form with bn = 1. The other coefficients (a, b) are real numbers or zeros. The denominator keeps the full shape of the polynomial in (1), but the numerator has simpler and concrete form, what depends on the desired type of the filter as follows. For the low-pass (LP) all-pole filters (Butterworth, Bessel, Chebyshev)

.,...3,2,1for,0,00 niaAa i (2)

For the high-pass (HP) all-pole filter ).1,...(3,2,1for,0, niaAa inn (3)

For the band-pass (BP) all-pole filter if the order n is even

.,...12

,12

...,1,0for ,0,2

for, nnn

jan

iAa jii

(4) For the notch LP (LPN) and HPN filters (Cauer, inverse Chebyshev) and also for the band-reject (BR) filters the numerator has

.,...5,3,1for ,0,...,4,2,0for, jaiAa jii (5) Note that the Ai is concrete real number, other then for the coefficient b i. The all-pass filter (APF) has similar numerator as denominator, except in signs, thus

odd.,...5,3,1for ,,even,...4,2,0for, njbaniba jjii (6)



-b1

-b0

-bn-1

a1

1/s

a2

1/s

1IinpIouta01/s 1/s

-bn-2

an

an-1

...

...

-b1

-b0

-bn-1

a11/s

a2

1/s

1IinpIout

a0

1/s 1/s

-bn-2

an

an-1

...

...

an-2

a) b)

Fig. 1. Signal flow graph of the follow the leader feedback structure in current mode. a) with the output summation, b) with the input distribution.

CANONICAL STATE-VARIABLE MULTI-LOOP STRUCTURES IN CURRENT MODE The transfer function (1) can be directly implemented by one from the state-variable multiple-loop feedback structures (MLFS), well known in the classical voltage mode (VM) [2]. As an acceptable model of the MLFS the signal flow graph (SFG) will be used. We can recall the canonical analogue structure, usually titled follow-the-leader feedback (FLF) and the inverse one IFLF. The basic SFG’s in VM has been retransformed to the CM using the adjoint VM CM transformation known from [1] to obtain the desired current CM-SFG’s, namely of the structure FLF in Fig. 1. The both structures FLF and IFLF can be modified with summation of the output signal (OS) (Fig. 1a), or with the input signal distribution (ID) (Fig. 1b). All these MLFS’s (FLF-OS, FLF-ID, IFLF-OS and IFLF-ID) can be directly implemented using current integrators, current amplifiers (multipliers by constant), current summers and current distributors, what is a little complicated and therefore the MLFS’s will be firstly ingeniously modified.

Fig. 2. Modified signal flow graph of the structure FLF-OS-CM.



MODIFICATION OF THE MULTI-LOOP STRUCTURES For the CM most suitable is the FLF-OS-CM (Fig. 1a), where is not any multi-current distributor and summation of the currents is given virtually by the node only (applying CKL). Nevertheless the circuit realization of this SFG (Fig. 1a) needs a lot of multipliers by coefficients (a i, b i), what is reason of the following modification. The transfer function (1) is rearranged in this form

nn

nn

n

n

n

n

nn

nn

n

n

n

n

n

n

bs

b

bs

b

bs

b

sb

bbs

a

bs

a

bs

a

sb

a

b

a

sK0

11

221

011

221

...1

...

)(

. (7)

The SFG corresponding with the formula (7) is shown in Fig. 2. There some of branches have the unity transfers and half multipliers can be omitted and replaced by direct connections. Note that this general SFG (Fig. 2) will be any more simplified in concrete filter realization, what depends on the given type, as is shown bellow. ILLUSTRATIVE EXAMPLE To illustrate the given structure FLF-OS-CM the 4th-order all-pass filter operating in the CM was taken, with following specification: the constant group delay g = 600 ns, in the pass-band with cut-off frequency fc = 1 MHz and Bessel approximation. The desired transfer function (1) has now the concrete form, where the numerator and denominator are the same except the signs, namely

44

33

2210

44

33

2210)(

sbsbsbsbb

sbsbsbsbbsK

. (8)

The values of the coefficients (b i) have been given by program NAFID [6] for the filter design as follows: b0 = 1.29715 1028, b1 = 3.89065 1021, b2 = 5.00157 1014, b3 = 3.33398 107, b4 = 1., (9)

Fig. 3. Signal flow graph of the all-pass filter with the structure FLF-OS-CM.

The SFG of the all-pass filter with the structure FLF-OS-CM and corresponding with the transfer function (8), is shown in Fig. 3. There a lot of branches have the unity transfers 1, what can be simply realized by circuit diagram given in Fig. 4, using transconductors (OTA) with three current outputs. There is not any multiplier, because the multiplying by the fraction of the coefficients b i is along the integration. There are four current integrators (subcircuits OTA-1, 2, 3, 4, with the capacitors Cn) and one current distributor (subcircuit CDTA-5). This circuit (Fig. 4) has been symbolically analyzed by computer tool SNAP [7], to obtain the current transfer function with the form of the formula (8) and with the following expressions for particular coefficients

.1,,,, 41

13

21

212

321

3211

1321

43210 a

C

ga

CC

gga

CCC

ggga

CCCC

gggga (10)

Substituting (9) to (10), the design equations are obtained. Then choosing the transconductances

g

C

1 = g2 = g3 = g4 = g5 = g = 1 mS, (11) the resulting values of the capacitances are

1 =30 pF, C2 = 67 pF, C3 = 129 pF, C4 = 300 pF. (9)



To verify the functionality of the proposed circuit (Fig. 4) , the PSpice [8] simulation has been carried out, using an adequate ABM model of the ideal OTA’s. The resulting frequency characteristic with constant group delay has confirmed the symbolical analysis and theoretical assumptions. Additional studying of the parasitic influences and modelling of the real components will be done. CONCLUSIONS The paper introduced the higher order circuits in the current mode, based on the canonical structure follow-the-leader feedback with output summation using multiple output transconductors. To illustrate and confirm the given structures the all-pass filter was designed and simulated by PSpice. The results have confirmed the design assumptions. Furthermore these structures enable an easy and direct implementation of the other types (LP, BP, HP) of the nth-order filters operating in current mode and the universal multifunctional filter can be easy implemented too. All mentioned circuits are fittingly electronically controllable by the transconductances (g) and auxiliary DC currents. The given model in the form of signal flow graph, its modification and transformation into the current mode provide a really good way how to obtain the nth-order circuit for high-frequency application in the current mode. An advantage of this circuit is its function universality, independently adjustable coefficients, and simplicity in direct uniform design.

Fig. 4. Circuit diagram of the current mode 4th-order all-pass filter with the OTA’s, based on the SFG given in

Fig. 3. REFERENCES [1] TOUMAZOU, C.; LIDGEY, F. J.; HAIGH, D. G. Analogue IC design: The current-mode approach.

Peter Peregrinus Ltd., London, 1990. [2] CHEN, W. K. The circuits and filters handbook. CRC Press, Florida, 1995. [3] SUN, Y.; FIDLER, J. K. Current-mode OTA-C realization of arbitrary filter characteristics. Electronics

Letters, 1996, vol. 32, no. 13, p. 1181 -1182. [4] SUN, Y.; FIDLER, J. K. Current-mode multiple-loop filters using dual-output OTA’s and grounded

capacitors. International Journal of circuit theory and application, 1997, vol. 25, no. 1, p. 69 - 80. [5] ACAR, C.; ANDAY, F.; KUNTMAN, H. On the realization of OTA-C filters. International Journal of

circuit theory and application, 1993, vol. 21, no. 3, p. 331 - 341. [6] HÁJEK, K.; SEDLAČEK, J. NAFID - program as powerful tool in filter education area. In: Proceedings

of the conference CIBLIS’97, Leicester (UK), 1997, p. PK-4 1-10. [7] BIOLEK, D. Program SNAP for symbolical analysis of the circuits. On line http://snap.webpark.cz. [8] Network simulator PSpice A/D. On line http://orcad.com.

http://snap.webpark.cz/

http://orcad.com/



ADDRESS: Prof. Ing. Tomáš DOSTÁL, DrSc. European Polytechnic Institute, Ltd., Osvobození 699, 686 04 Kunovice, Czech Republic, Tel: +420 777194307 Email: dostalt@edukomplex. cz



ANALOG BEHAVIORAL MODELING OF THE MULTIFUNCTIONAL FILTER WITH

CURRENT CONVEYORS

Josef Slezák, Roman Šotner

The Faculty of Electrical Engineering and Communication, Brno University of Technology

Abstract: Computer modeling of frequency response of active blocks used in current mode multifunctional frequency filter is described in this article. Described method is used for modeling of active blocks with unavailable OrCAD simulation macromodels, where active blocks are modeled using only ideal parts. This paper is focused on design and simulation of circuit using CCII+ and CCII-.

Keywords: current conveyor, CCII, PSpice, analog filter

1 INTRODUCTION Current conveyors (CC) are perspective active blocks of present [1], [2], [13]. In academic area this parts are well-known in contrast to practical field where widespread is very small. Basic CC includes three gates. Also CCs with more than three gates exist but are not manufactured. Occasionally multi gate CCs are manufactured in small series for science purpose. The main advantage of CC is higher working frequency. Three gates CCs are included in some modern operational amplifiers like AD 844 [5] and OPA 860 [6]. Some of CCs are manufactured as individual parts like current multipliers EL 2082 [7]. Realization of frequency filter using current mode is suitable solution for higher working frequency but also have many disadvantages. One of these is necessity of transfer from voltage to current on the input of the filter and transfer from current to voltage on the output of filter. The reason of this is that most of electronic circuits are working using current mode. Also current multi-outputs of active parts are required. Multi-output current part can be used as a current distributor which distributes the same current to couple of paths. One of advantages of the current mode [4] is easy realization of summation which is realized by simple connecting concrete paths to a junction. The principle of positive and negative CC of second generation (CCII) and adjusting of current gain B is shown in Fig. 1.

Fig. 1. Principle of second generation current conveyor (CCII)

2 MODELING Behavioral model describes outer behavior of part. Types of possible analysis of a part are determined by level of modeling of the part [8]. Basic building blocks for behavior modeling are controlled sources [9] (in this case current controlled current sources, voltage controlled voltage sources and voltage controlled current sources), which can be used for modeling of all active blocks. Model of level 1 represents only basic feature of active block. In the case of CC its current gain B = 1 (Fig. 2a). Model of level 2 (Fig. 2b) includes input and output resistance of simulated part. Using frequency depend parts, frequency depend model of level 3 is obtained (Fig. 2c). In this case capacitors representing input and output capacitance and circuit ensuring falling of gain are added. This model is useable for simulation of frequency characteristics. For simulation in time domain the model is not suitable because dynamics of input signal is not taken into account. For input voltage of any value the model has linear response. Level 4 allows describing un-linearity of circuit so features like limited dynamic range can be modeled. Un-linearity is modeled using diodes and voltage and current sources. Level 5 and higher levels models are combination of behavioral modeling and detailed modeling at transistor level. This level is used by manufacturer of integrated circuits for modeling of their parts. This type of model is the most accurate way of description of behavior of a part and it is suitable for more than one analysis. Disadvantages of this model are high complexity of the model and long simulation time. For simulation used in this paper, the modeling level



3 is sufficient. Modified level 3 model is present in Fig. 3. This model is suitable for evaluation of sensitivity of some CCII parameter or for tolerance analysis.

Z Z

-+

+-

E1

VCVS

GAIN = 1

F1

CCCS

GAIN = 1

X

Y

0

RoutRinpY

RinpX

Z

-+

+-

E1

VCVS

GAIN = 1

XF1

CCCS

GAIN = 1Y

0

0

RoutRinpY

RinpXCinpX

CinpY

Cout

(b) level 2(a) level 1 (c) level 3

-+

+-

E1

VCVS

GAIN = 1

F1

CCCS

GAIN = 1

X

Y

Fig. 2. Levels of modeling of CCII

U1

CCII3lev

CINP = 2pROUTZ = 0.5megRINPX = 95RINPY = 1megCOUT = 5pSIGN = 1B = 1

GN

D

X

Y

Z Z

Y

X

0

-+

+-

E1

E

F1

F

+-

G1

G

GAIN = @signRinpX

@RinpX

RinpY@RinpY

B@B

RoutZ@RoutZ

Cinp1@Cinp

Cout@Cout

Z

Y

X

GND

Cinp2@Cinp

Fig. 3. Level 3 model of CC used for simulation of changing of current gain

3 DESIGN AND ANALYSIS OF FREQUENCY FILTER USING CCII MODELS Multifunctional second order Kervin-Hueslman-Newcomb filter [10] using current mode [4] is assembled using input current amplifier with CC1 (voltage to current converter) and two current integrators (CC4 and CC7). Similar circuits with different active blocks are presented in [11], [12], [13]. Block CC2 and CC5 ensure current outputs of the filter and their response is corresponding to responses of block CC1 and CC4. Output of low pass is ensured using CC7. Summation is realized at input junction where summation of copies of BP and PL currents are connected. For this feedback blocks CC3 and CC6 are used. Current gain of all CCII+ is B = 1. Model CCII+ is based on parameters of real AD 844 where gain B = 1 is fixed. Models of CCII- (CC3, CC6) are based on real EL 2082 where current gain is adjustable in the range from 0 to 2. Adjusting current gain of CC3, quality factor of filter is varying. B1 and B2 used in formulas (1) to (3) are current transfer of integrators and B3 is current transfer of CC3 in feedback path. Filter realizes high pass filter

212111

32

2

2121

21

11

32

2

22 1

)()(

CCRRs

CR

Bs

s

CCRR

BBs

CR

Bs

s

sQ

s

sNsK

CC

HP

, (1)

band pass filter

212111

32

11

2121

21

11

32

11

1

1

1

)(

CCRRs

CR

Bs

sCR

CCRR

BBs

CR

Bs

sCR

B

sK BP

, (2)



Fig. 4. Multifunctional 2nd-order filter using current mode with CCII. and low pass filter

212111

32

2121

2121

21

11

32

2121

21

1

1

)(

CCRRs

CR

Bs

CCRR

CCRR

BBs

CR

Bs

CCRR

BB

sK LP

. (3)

Requirements for design: cut off frequency fC = 1 MHz, quality factor Q = 5, Butterwoth approximation. Simplification: R1 = R2 = R, C1 = C2 = C a B1 = B2 = B = 1 values of resistors and capacitors can be easily calculated. Resistors R0 are set to 330 and capacitors C = 470 pF, then

33910.470.10.1..2

1

...2 12621 Cf

BRRR

C

, (4)

2.05

113

QQ

BB . (5)

Note that values of all resistors connected to invert current input of AD 844 have to be re-counted with regard to input resistance of current input of AD 844. Fig. 5. present simulations of frequency responses of filter using models of level 3 and original macromodels of AD844 and EL 2082.

-60

-55

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

15

1,0E+04 1,0E+05 1,0E+06 1,0E+07 1,0E+08

f [Hz]

K [dB]

filter with ABM models of 3. level

filter with macromodels from PSpice library

HP

BP

LP

Fig. 5. Magnitude responses of simulated filter with ABM models and macromodels of CCII-s.

3 CONCLUSION In frequency domain accuracy of presented model of level 3 of conveyors AD 844 and EL 2082 compared to original OrCAD macromodels is very good. Models are used for simulation of multifunctional 2nd-order filter. Simulations of comparison of presented models and original OrCAD macromodels show that deviation of both



characteristic is more significant from 30 MHz. The reason of this is difference between both simulation models. The level 3 model is relatively simple compared to original OrCAD macromodel which models more than just one pole. Due to linear behavior of level 3 model the response of simulation in time domain may product bad results. Presented method offers easy way for modeling of active blocks. Attention must be paid to choose the simulation model of correct level for desired analysis. 4 ACKNOWLEDGMENTS Research described in the paper was supported by the Grant Agency of the Czech Republic under grant No. 102/08/H027 and by the Czech Ministry of Education under the research program MSM 0021630513. REFERENCES [1] BIOLEK, D.; VRBA, K.; CAJKA, J.; DOSTAL, T. General three-port current conveyor: a useful tool for

network design. Journal of Electrical Engeneering, 2000, Vol. 51, No. 1- 2, pp. 36-39. ISSN 1335-3632. [2] CHEN, W. K. The circuits and filters handbook. CRC Press, Florida, 1995. [3] JERABEK, J.; VRBA, K. Vybrané vlastnosti univerzálního proudového konvejoru, ukázka návrhu

aplikace. Elektrorevue. 2006, č. 2006/41, s. 1 – 9, ISSN 1213 – 1539 on www: http://www.elektrorevue.cz.

[4] TOUMAZOU, C.; LIDGEY, E. J.; HAIGH, D. G. Analogue IC design: The current mode approach, Peter Peregrinus Ltd., London, 1990.

[5] Analog Device. Monolithic Op Amp AD 844 Data Sheets. 2003, 16 s., Dostupné z WWW: http://www.analog.com/.

[6] Texas Instruments Inc. OPA 860 Wide Bandwidth Operational Transconductance Amplifier and Buffer. 2006, 32 s. dostupné z WWW: http://www.ti.com.

[7] Intersil (Elantec). EL 2082CN Current-Mode Multiplier, 1996, 16 s., dostupné z WWW: http://www.intersil.com

[8] DOSTAL, T. Modelling of Modern Active Devices for simulation of Analog Circuits in PSpice. Radioengineering. 2000, Vol. 9, No. 3, ISSN 1210 – 2512.

[9] PUNCOCHAR, J. Operační zesilovače-historie a současnost. BEN Praha 2002, ISBN 80-7300-047-4 [10] KERVIN, W. J.; HUELSMAN, L. P.; NEWCOMB, R. W. State variable synthesis for insensitive

integrated circuit transfer functions. IEEE-SC, 1967, vol. 2, no. 2, pp. 87-92. [11] BIOLEK, D.; BIOLKOVA, V.; KOLKA, Z. Universal Current-Mode OTA-C KHN Biquad, International

Journal of Electronics, Circuits and Systems, vol. 1, no. 4, p. 214-217, 2007, ISSN 1307-4156. [12] DOSTAL, T. Filters with Multi-Loop Feedback Structure in Current Mode. Radioengineering. 2003, vol.

12, no. 3, p. 6 - 11, ISSN 1210 – 2512. [13] BIOLEK, D.; SENANI, R.; BIOLKOVA, V.; KOLKA, Z. Active elements for analog signal processing:

Classification, Review, and New Proposal. Radioengineering. 2008, vol. 17, no. 4, p. 15 – 32, ISSN 1210 – 2512.

ADDRESS: Ing. Josef Slezák The Faculty of Electrical Engineering and Communication Brno University of Technology Purkyňova 118 61200 Brno Czech Republic E-mail: [email protected], Ing. Roman Šotner The Faculty of Electrical Engineering and Communication Brno University of Technology Purkyňova 118 61200 Brno Czech Republic E-mail: [email protected]



NONLINEAR CIRCUIT SYNTHESIS USING MATHCAD

Jiří Petržela


Abstract: This article shows the possible utilization of Mathcad for the synthesis of the nonlinear circuits. Assuming that the mathematical model is given the entire process of developing the associated electronic realization can be done by using Mathcad build-in procedures and functions. This program is especially useful from the viewpoint of studying eigenvalue migration with respect to the parameters of the dynamical system, if making a tolerance analysis and circuit synthesis, calculating Ljapunov exponents, performing one or two-dimensional FFT, viewing Poincaré sections and one-dimensional bifurcation diagrams, etc. These routines will be described by means of the practical example, namely for the design of third-order chaotic oscillator.

Keywords: Bifurcation diagram, circuit synthesis, nonlinear oscillator, Mathcad.

1 INTRODUCTION Suppose that the designer´s aim is to construct an oscillator based on the following third-order differential equation

xqpxqpxqphxqxqxqx 332211321 , (1)

where dots denote the time derivatives. It is well known that such system can produce a very complicated behavior including chaos [1]. It is evident that the most straightforward design procedure is based on the integrator block schematic. Only three basic building blocks are needed, namely inverting integrator, summing amplifier and two-port with nonlinear transfer characteristic. Unfortunately this approach leads to the large amount of the active circuit elements. From the viewpoint of final simplicity assume the parallel connection of linear third-order admittance function and passive resistor with piecewise linear (PWL) ampér-voltage (AV) curve, see Fig. 1. There are two basic PWL characteristics suitable for the synthesis of the chaotic oscillators given in Fig. 2. Note that in each segment a nonlinear resistor consumes energy from the rest of the circuit such that it can be realized using resistors and two ideal diodes. Moreover, the chaotic behavior is preserved also in the case of non-ideal diodes. It is not hard to derive the following conditions for the individual segments of the vector field

sDGsNGsYsD

sN

bsbsb

asasassY baba ,,

012

2

012

23

, (2)

where s is a complex frequency. Solving this equation yields

00322

13

00112

223 bbbb GpspspsGbasGbasGbas , (3)

for the inner segment of the vector field and similarly

00322

13

00112

223 aaaa GqsqsqsGbasGbasGbas , (4)

for both outer segments of the vector field. Although it is possible to obtain symbolical decomposition [2] it is much better to leave the symbols and use the numbers. Having the concrete admittance network the task is to find the normalized values of the circuit elements. On the contrary a designer can look for the admittance network using standard methods such as decomposition into partial fractions or use the technique known as a continued division (from highest or lowest power). The numerical values of the equivalent eigenvalues p i and q i (coefficients of the characteristic polynomials) can be taken from the publication [3]. The Mathcad symbolic toolbox can be used for the purpose of decomposition. It allows an user to quickly solve for the variable in complex nonlinear equation, intuitively operate with the forward or backward Laplace transforms or make a symbolical operations with the square matrices.



In our case the decompositions are shown in Fig. 3.

Fig. 1: Classical circuit synthesis of the nonlinear oscillator, network based on the continued fraction.

Fig. 2: Two possible shapes of PWL AV characteristics.

Fig. 3: Using Mathcad for the admittance network decomposition.



Fig. 4: Using Mathcad for the searching of network component numerical values.

Fig. 5: Using Mathcad for the numerical integration.



Fig. 6: Utilizing Mathcad for bifurcation diagram visualization.

Sometimes, especially if dealing with a large network, it is impossible to find a symbolical expression for the norma-lized circuit components such in the closed form, for example Rb=(p1, p2, p3). The searching leads through the solving of the system of nonlinear algebraic equations. This obstacle can be removed using the GivenFind routine as it is demonstrated in Fig. 4. If the process fails to converge the calculation precision should be lowered or initial guess values have to be changed. The proper function of a final oscillator can be proved by means of the numerical integration. The fourth-order Runge-Kutta method is available as a build-in procedure but others can be implement-ted using fundamental definition formulas. This is shown in Fig. 5 where the so-called double-scroll chaotic attractor is presented. It is worth nothing that also parasitic properties of the used active building blocks can be considered. This is essential if a high-frequency chaotic oscillator is designed and thus low values of capacitors are used. To trace different routes to multiple-periodic motion or chaos in the given general dynamical system the one-dimensional bifurcation diagram is an useful tool. The most straightforward method to implement associated procedure is given in Fig. 6 where current gain of CCII+ is varied from =0.9 to =1. Mentioned problem is solved by recording the trajectory intersections with the carefully chosen Poincaré plane. 2 ADVANCED FUNCTIONS Another very important procedure suitable for the quantitative analysis of the nonlinear dynamical systems is the spectrum of the Ljapunov exponents (LE). These real numbers measure the average ration of the exponential divergency of the two neighbourhood trajectories in the state space. To perform this process twelve differential



equations instead of three are integrated simultaneously. The equations in variation are added to the original dynamical system since it defines the volume element in the state space. After a couple of integration steps the Gram-Smith orthogonalization is evaluated. It is also important to omit the transition event and calculate after a fiducidal point is on the attractor. The Kaplan-Yorke metric dimension can be established directly from the spectrum of LE. Mathcad is capable to compute capacity dimension of the attractor, but this gives even more demands on the performance of the personal computer. The whole procedure implemented in Matlab can be found in the book [5]. For the nonlinear oscillator studied above the natural bifurcation parameters could be the slopes of the PWL resistor. Possible solutions are graphically illustrated in Fig. 7.

Fig. 7: Contour plot of the largest Ljapunov exponent calculated by means of Mathcad.

For chaos it is necessary to have just one positive LE. If each LE is negative then the associated solution is a fixed point represented by a black color. For one LE equal zero there is a limit cycle marked by a dark grey color. Finally, chaotic motion is light grey color and white denotes the unbounded solution. The significance of LE is demonstrated in the publication [6] where is this chaos quantifier utilized for the searching for irregular behavior using stochastic optimization. 3 CONCLUSSION Some basic nonlinear circuit design problems have been addressed in this contribution. From the viewpoint of advanced circuit synthesis the most important is the possibility to optimize the final structure of the admittance network. The mystery behind this optimization is still an unanswered question and it is a topic of further study. ACKNOWLEDGEMENT The research presented in this brief article has been supported by a project of Ministry of Education, Czech Republic, number MSM 0021630513. REFERENCES: [1] POSPÍŠIL, J.; KOLKA, Z.; HORSKÁ, J.; BRZOBOHATÝ, J. Simplest ODE equivalents of Chua´s



circuit family. International Journal of Bifurcation and Chaos, 2000, vol. 10, no. 1, pp. 1–23. ISSN 0218-1274.

[2] GOTZ, M.,;FELDMAN, U.; SCHWARZ, W. Synthesis of higher dimensional Chua circuits. IEEE Trans. on CAS I, 1993, vol. 40, no. 11, pp. 854–860. ISSN 1057-7122.

[3] CHUA, L.,;LIN, G. N. Canonical realization of the Chua´s circuit family. IEEE Trans. on CAS I, 1990, vol. 37, no. 7, pp. 885–902. ISSN 1057-7122.

[4] ITOH, M. Synthesis of electronic circuits for simulating nonlinear dynamics. International Journal of Bifurcation and Chaos, 2001, vol. 11, no. 3, pp. 605–653. ISSN 0218-1274.

[5] WYK, M. A.; STEEB, W. H. Chaos in Electronics. Kluwer Academic Publishers, 1997. 481 pages. ISBN 0-7923-4576-2.

[6] PETRŽELA, J.; HANUS, S.; LANGER, T. Searching for chaos in the dynamical systems using genetic algorithm. In Proceedings of the 14th Electrotechnical and Computer Science Conference ERK 2006. Portorož (Slovinsko): IEEE Section, 2006, pp. 246–249. ISSN 1581-4572.

ADRESS: Ing. Jiří Petržela, PhD. Faculty of Electrical Engineering and Communication Brno University of Technology 612 00 Brno Czech Republic E-mail: [email protected]



STRUCTURE OF GRAVITATION

Pavel Ošmera


Abstract: We would like to find some acceptable structure of gravitation as vortex-fractal-coil structure. It is known that planetary model of hydrogen is not right and the quantum model is too abstract. Our imagination is that the hydrogen is a levitation system of the proton and the electron. In the axes of the levitation could be gravitation lines.

Keywords: structure of graviation, structure of the electron, the proton and neutron.

1. INTRODUCTION Gravitation is a universal force of attraction acting between all matter. It is by far the weakest known force in nature and thus plays no role in determining the internal properties of everyday matter. Due to its long reach and universality, however, gravity shapes the structure and evolution of stars, galaxies, and the entire universe. The trajectories of bodies in the solar system are determined by the laws of gravity, while on Earth all bodies have a weight, or downward force of gravity, proportional to their mass, which the Earth's mass exerts on them. Gravity is measured by the acceleration that it gives to freely falling objects. At the Earth's surface, the acceleration of gravity is about 9.8 metres (32 feet) per second per second. Thus, for every second an object is in free fall, its speed increases by about 9.8 metres per second. Newton's laws of motion: 1st law: a body remains at rest or moves in a straight line of constant velocity as long as no external forces

acts on it 2nd law: a body acted on by a force will accelerate such that force equals mass times acceleration (F=ma) 3rd law: for every action there is an equal and opposite reaction Although Newtonian mechanics was the grand achievement of the 1700's, it was by no means the final answer. For example, the equations of orbits could be solved for two bodies, but could not be solved for three or more bodies. The three body problem puzzled astronomers for years until it was learned that some mathematical problems suffer from deterministic chaos, where dynamical systems have apparently random or unpredictable behavior (see below). There is three body problem and complexity: Deterministic laws, such as Newton's laws of motion, imply predictability only in the idealized limit of infinite precision. The Universe itself cannot know its own workings with absolute precision, and there cannot predict what will happen next in every detail. Deterministic chaos seems random because we are necessarily ignorant of the ultrafine details and so is the Universe itself. The behavior of complex systems is not truly random but it is just the final state that is so sensitive to the initial conditions. It is impossible to predict the future behavior without infinite knowledge of all the motions and energy (i.e. a butterfly in South America influences storms in the North Atlantic). Even games with simple rules can produce complex behavior. Although this is `just' a mathematical game, there are many examples of the complex behavior occurring in Nature. 2. HISTORY OF GRAVITY a) Early ideas The very earliest ideas regarding gravity must have been based on every day experience. For example: Objects fall unless they are supported. "Down" is different from "across". Climbing a hill is harder work than walking on a level. We still use these very basic ideas and properties of objects every day without further thought. Because we live on a large Earth and are not familiar with gravitation from any other object, we tend to equate gravity with "down".



b) Ancient Greeks: Aristotle It was with the Ancient Greeks, and in particular Aristotle, that these disparate observations began to be unified into one idea. For Aristotle, physics was the investigation of "causes" in the widest possible sense. While "Gravity" was not yet a concept in itself, Aristotle realized that these various properties of objects must be related. To Aristotle, the cause of falling was heaviness. The heavier the object, the more it falls - a large rock plummets to the earth, a leaf ambles along downward slowly and a dandelion fluff barely falls at all and frequently rises higher into the sky instead. But what was the connection between heavy objects and falling? To understand this, one must have an idea of the Aristotelian worldview. To Aristotle, all Matter was made of four elements, Earth, Water, Air and Fire. Earth, the basest and least noble, was in the center. This was the ground we walk on. Next was the sphere of Water, followed by the sphere of Air and then of Fire. The eternal and perfect Celestial spheres of the stars and planets surrounded and limited the Universe. Everything under the moon was composed of some mixture of the four elements. Clouds for example were considered to be mostly air with a bit of water and fire. The natural place for heavy objects (made principally of Earth) was back in the center. If one removed a heavy object, say a stone, from its natural place (i.e. by lifting it) it would tend to return to its "proper" place. In a similar way, fire tended to rise, because it was trying to return to its natural place above the sphere of air. Intermediate objects, such as the leaf or dandelion fluff, were made of less Earth and more Air (or Fire) and hence fell more slowly, or perhaps not at all. c) Middle Ages Medieval physics (and astronomy) was largely based on the ideas of Aristotle. Since the teachings of Aristotle had been given the seal of approval by the Church, they were taken to be the revealed truth and essentially unquestioned in the Universities. In general, this didn't work too badly: the most objects work more or less in the manner described by Aristotle. This is not surprising since his work was essentially the codification of day to day experience and is very commonsensical. In certain cases, however, serious discrepancies might have been noted. One example of this was the medieval view of cannonball trajectories. In the Aristotelian view, when a cannonball's initial upward and forward impetus was exhausted it would fall vertically to the Earth, its natural place. This indeed should be true of any object shot or thrown into the air. Anyone who had watched a rock thrown into the air could tell this was not true. The trouble was that the people teaching the theories and the people dealing with the real world objects were different. Furthermore, until the beginning of the Renaissance, experimentation was discouraged and considered beneath the dignity of philosophers. The way to truth was considered to be pure thought and the scriptures. d) Renaissance: Galileo Galileo's work represents the beginnings of a modern understanding of Gravity. Ironically, to achieve this, Galileo began by disavowing any interest in "causes". Instead of trying to answer the question "why do objects fall?" he explored "how do objects fall?" This is an extremely important step. Even today we do not fully understand the "why" of gravity although we understand the "how" very well indeed. Galileo began his exploration of how objects fall by comparing the rates at which objects fall. He also tried to figure out how fast they fall. His basic conclusions were the following:

objects of different weight fall at the same speed, falling starting from a complete stop, objects move more and more quickly the longer they have been

falling. the distance an object falls is proportional to the square of the elapsed time.

He arrived at these conclusions through a beautiful series of experiments. The first thing he realized was that he would need to slow down the motion of objects to be able to measure their fall. He did this by allowing the objects he studied to roll down a tilted board instead of falling straight down. He had to assume that this procedure was valid... fortunately it was. Second, he knew that he had no accurate clocks for measuring times. Instead he used his natural sense of rhythm (he was a very musical man). In the path of the objects he rolled down the plane he placed little bumps. Every time the object went over a bump it made a click. By arranging the bumps so that the clicks came in a regular series he knew the time between the bumps was the same. The story of Galileo dropping cannonballs of different weights off the Leaning Tower of Pisa is probably apocryphal. If he did this, it was certainly less important to him than his controlled experiments. Galileo's contribution to the understanding of gravity was threefold. First, he subtly changed the question being asked. Second he based his answers on careful experimentation and measurement. Third, he gave a mathematical



quantitative description of his results and gave the limits within which he had verified this description. e) Enlightenment: Newton Newton, born in the year Galileo died, developed the modern concept of gravity. Instead of simply exploring how objects fall, he posited a force of gravity that was responsible for a variety of effects. Newton started from Galileo's law of falling objects and applied it to an unlikely object: the Moon. Why, he asked, did the moon not fall to the earth? Other unsupported objects (like rocks, sticks etc.) fall immediately to the ground. The Moon seems to flout the law of gravity. That's the trick, however. The moon only seems to be immune to gravity. Newton realized that the Moon is not immune to gravity. It is continuously falling towards the Earth, but it keeps missing it! A little explanation of this somewhat outrageous sounding statement is in order. Imagine standing on a tower on a flat earth. Throw a rock sideways out the window. Eventually, the rock will fall to the ground. Now throw the rock harder. It will hit the ground farther from the tower. On a flat earth this can be continued, throwing the rock harder and harder with the impact farther and farther away. But not so on a spherical earth. On a spherical Earth, the earth curves away under the falling rock. When the rock is thrown with only a little speed, the distance is small and the surface of the Earth the distance spans is almost flat. But if the rock is thrown hard enough, the ground will drop a great deal. In fact, if the rock is thrown very, very hard indeed it will never hit the ground because the earth keeps receding beneath it! This is what is called being in orbit. The secret to flying is falling but missing the ground! Newton thus realized that gravity was not something special to the Earth. Gravity also acts in space. This was a profound, even revolutionary idea. According to Aristotle, the laws governing the heavens were considered to be completely different from the laws of physics here on Earth. Now, however, if the moon was affected by gravity, then it made sense that the rest of the Solar System should also be subject to gravity. Newton found that he could explain the entire motion of the Solar System from the planets to the moons to the comets with a single law of Gravity: All bodies attract all other bodies, and the strength of the attraction is proportional to the masses of the two bodies and inversely proportional to the square of the distance between the bodies. A modern mathematical way of saying this is:

where G (Newton's Constant) is a constant value equal to 6.67x10-11 m3/s2/kg, M is the mass of one object, m is the mass of the other object, R (radius) is the distance between the objects and F is the resulting gravitational force pulling the objects together. This is called the Universal Law of Gravity. Universal because it applies to all bodies in the Universe regardless of their nature. Gravity is not just about falling, it is about attraction. As I write this, the

keyboard in front of me pulls ever so slightly on the phone to my right, the penny I left at home gently tugs at the umbrella I lost in San Diego and the flight of a bird above the Adler makes me fractionally lighter! All objects pull on all other objects! What a fantastic statement! Of course, for most objects, the force of attraction is incredibly tiny and not noticeable, but it is always there. Despite its power in explaining the orbits of the Solar System, Newton (and his critics) were unhappy with the lack of a mechanism by which gravity worked. Until then, all forces were believed to be "contact" forces. That is to say, to push an object one had to be touching it. I push a pen across the table using my hand directly. Even if I blow a piece of paper, I am really moving the air with my lungs which then moves across to the paper and pushes it along. Almost everything in our experience works this way - except for gravity. The Newtonian concept of "action-at-a-distance" was profoundly disturbing to his opponents, who attacked his theory as "occult" and explaining nothing. f) Post Enlightenment - 1700s, 1800s From the period immediately following Newton's discovery of his Universal Law of Gravitation, to about the turn of the last century (1900), the theory of gravitation stayed essentially unchanged. More sophisticated mathematical tools for understanding the interplay of the planets were developed, but the underlying theory remained stable. As in the earlier Aristotelian world-view, gravitation was intricately connected with the structure of the Universe. The moons revolved around the planets, the planets revolved around the Sun, the Sun



floated through space passing other stars, all with clockwork precision. The Universe was orderly and controlled by gravity and the laws of motion. The excitement during this period mainly came from the systematic application of the theory of gravity to the heavens. For example: 1) Comets: an understanding of how objects orbited the Sun allowed predictions of the path of comets. The

best known case of this was Halley's prediction of the return of the comet that now bears his name. 2) Discovery of Neptune: In the time of Newton, only six of the nine planets had been discovered. While the

discovery of the planet Uranus was by and large accidental, the discovery of Neptune was a triumph of the Newtonian theory of gravity. After the discovery of Uranus, great attention was paid to this newest of planets. Its orbit was carefully mapped out in great detail. And something strange was found... The orbit of Uranus did not seem to follow Newton's laws precisely! The motion across the skies was just slightly different from the motion predicted on the basis of the theory of gravity. Astronomers were presented with a choice. Either Newton was wrong, or their calculations were somehow incomplete. Many influential scientists thought that perhaps the law of gravitation did not apply so far from the Sun.

3) In the period 1843-1846 John Adams and Urbain Leverrier independently came to the conclusion that the perturbations of Uranus's orbit were due to an eighth planet. Shortly thereafter, the new planet was discovered precisely where Adams and Leverrier had predicted. Newton was spectacularly vindicated!

4) Binary Stars: William Hershel's observations of binary stars during the early 1800s showed that the Newtonian laws of gravity also applied to the stars. They also allowed, for the first time, the calculation of the mass of stars other than our Sun.

5) Rings of Saturn: The law of gravitation also illuminated the origin and nature of the rings of Saturn. The rings could not be thin solid sheets as previously thought. James Maxwell showed that such rings would break apart under the combined actions of their own motion and the gravity of Saturn. He suggested instead that the rings were made up of many individual particles.

Of course, other advances were made. Among the most important were the experiments of Cavendish. Cavendish directly demonstrated the gravitational force between two objects in the laboratory. Indirectly, this was equivalent to the first measurement of the mass of the Earth. g) Twentieth Century: Einstein The twentieth century was a time of tremendous progress in physical science. For the understanding of gravity, the century began with two puzzles. The first of these puzzles concerned the orbit of the planet Mercury. In the Newtonian theory of gravity, the orbit of a single planet around the Sun should be a perfect ellipse. In the real world however, the planet is subject to the gravitational forces from the other planets in the Solar System, and hence, does not move in a perfect ellipse (This is how Neptune was discovered). In the case of Mercury, the motion was expected to look almost like an ellipse, but the point of closest approach to the Sun (perihelion) was expected to slowly revolve around the Sun. This is called the "perihelion advance of Mercury". Astronomers carefully measuring the position of Mercury over a period of time came to a startling conclusion: The perihelion advance was there, but it was occurring too quickly. At first, astronomers assumed this was due to the influence of another undiscovered planet. But after extensive searches, no new planet was found. What was going on? Nobody knew. The second puzzle was related to a series of experiments performed by the Hungarian physicist Roland Eotvos at the end of the 19th century. Eotvos was intrigued by a curious fact about Newton's laws of gravity and motion. Newton's laws can be written: Newton's law of gravity says that the gravitational force felt by an object is proportional to its "mass". Newton's law of motion also involves a "mass". But why should both laws involve the same quantity? After all, motion and gravity seem to be two very different things. Why should they both depend on the same property of an object? One might imagine a world where the force of gravity depended on how green an object was, or perhaps some other property. Scientists call the mass in the law of gravity "gravitational" mass and the mass involved in motion "inertial" mass. Amazingly, Eotvos' experiments showed that the gravitational mass was the same as the inertial mass to at least a few parts in a hundred million. One consequence of this is that all objects fall towards the Earth at the same rate. A larger mass is pulled with a larger force, but a larger mass also needs a larger force to get it moving. If one calculates the acceleration of an object, the mass cancels out entirely. Nobody had any idea why this should be the case. While black holes are very weird objects, they are perhaps not the strangest consequence of General Relativity. Stranger still are the singularities hidden deep in the heart of every black hole. Singularities are regions of space



where the density of matter becomes infinite, and the very concepts of matter, space and time lose their meaning. In their vicinity, time travel becomes possible, and the laws of physics break down completely. Luckily for us, black holes' event horizons shield us from the hidden singularities. They clothe the singularities in a one-way surface that would allow hapless astronauts in, but not out. As long as an object remains outside the event horizon, it would be possible to get pulled back out of the vicinity of the black hole. But the event horizon marks the point of no return. Thus, there is no way for the madness of the singularity to get out to infect the rest of the Universe. But what if a singularity could be formed without a surrounding black hole? This would be a Bad Thing! It is an open question whether such "naked" singularities can be formed. Relativists conjecture that the formation of a naked singularity is forbidden by "Cosmic Censorship", but no one has proved this to be the case. h) Future Directions General Relativity is perhaps the most beautiful physical theory yet created. It is powerful, pleasing to the aesthetic sense and well-tested. It is one of the crowning glories of modern physics. At about the same time General Relativity was born, another theory was being created. This was Quantum Mechanics. If General Relativity deals with very massive objects, then Quantum Mechanics deals with the interactions of very small objects, such as electrons and protons. Quantum Mechanics has been verified to a stunning degree of accuracy. It is perhaps the most successful theory in all of physics. So what would happen if one had a very massive, but small, object? Both GR and QM would apply. This seems reasonable... until one tries to do the math! It turns out that the two theories are incompatible. I don't mean that they predict different results (that would be straightforward to test), but rather that we don't even know how to express a theory that combines both GR and QM! The usual method for obtaining a quantum theory of a physical process is to take the classical theory and to "quantize" it. But if one does this to General Relativity, the answers to all calculations become infinite! Nothing makes sense anymore. Most physicists believe that a true combination of GR and QM is possible, but it won't be found as merely an extension of GR. The search for a theory that combines GR and QM is called the search for the Theory of Everything (TOE). Recently a theory known as "string theory" has gained a lot of support as a candidate TOE. What is different about string theory? Normal Quantum Mechanics treats all particles as points of zero size. This leads to a lot of problems when distances get small or energies get large. String theory says that particles are not points after all, but instead small little loops. The sizes of these loops are about 10-34 cm-- so very, very small indeed. But not zero! Most of the problems reconciling GR and QM go away when one uses this theory. The full consequences of string theory have not been worked out yet (the mathematics is incredibly complex) but so far it seems very promising. But we don't know yet, and the final theory of gravity may be something else entirely. Whatever it is, however, we can be certain that the attempts to understand it will have profound consequences for our understanding of the Universe. Mainstream quantum gravity work is called string theory and assumes that the particles which masses exchange to produce gravity (gravitons) have spin-2 which is a complex spin assumed to be needed so that two masses will attract when exchanging them. This spin-2 assumption requires 10 dimensions in string theory, and because 6 dimensions are too small to be seen (yet crucially affect the predictions of the theory), string theory can’t be checked. It has maybe a hundred unknown parameters concerning 6 invisible compactified dimensions, which leads to 10500 different possibilities which can never be investigated even by a fast computer running for the age of the universe. The spin-2 graviton argument on which string theory is built simply ignores almost all of the mass involved, which is in the immense masses of galaxies in the surrounding universe!



Fig.1 Vortex-fractal structure of the neutron and the proton [15]

Fig. 2 Vortex-fractal ring structure of the electron [15]

3. CONCLUSIONS The exact analysis of real physical problems is usually quite complicated, and any particular physical situation may be too complicated to analyze directly by solving the differential equations. Ideas as the field lines (magnetic and electric lines) are for such purposes very useful. We think they are created from organized subparts of vacuum [6], [9], [10]. A physical understanding is a completely unmathematical, imprecise, and inexact, but it is absolutely necessary for a physicist [1]. It is necessary combine an imagination with a calculation. Our approach is given by developing gradually the physical ideas – by starting with simple situations and going on more and more complicated situations. But the subject of physics has been developed over the past 200 years by some very ingenious people, and it is not easy to add something new that is not in discrepancy with them. We used the knowledge from experiments. Now we realize that the phenomena of chemical interaction and, ultimately, of life itself are to be understood in terms of electromagnetism. Maxwell’s discovery of the laws of electrodynamics will be judged as very significant event of the 19th century. The electron structure is a pure fractal-ring structure with a vortex bond between rings. This vortex bond is electromagnetic field with two complementary vortex structures; electric and magnetic vortex structures. The proton structure is pure fractal-coil structure. Both are created from the same subsubrings [15]. It seems that gravitation lines are in the same axes as levitatiing electrons. Gravitation lines repel each other and due to the two bodies are attracted. The gravitation lines are created from gravitons. Space that is full of



gravitons we can call gravitonum (or shortly gravum) to distinquish from the terms: vaccum, ether or aether etc. ACKNOWLEDGMENT: This work has been supported by the Czech Grant Agency; Grant No: MSM21630529. REFERENCES [1] FEYNMAN, R. P.; LEIGHTON, R. B.; SANDS, M. The Feynman Lectures on Physics, volume I, II, III

Addison-Wesley publishing company, 1977. [2] DUNCAN, T. Physics for today and tomorrow, Butler & Tanner Ltd., London, 1978. [3] HUGGETT, S. A.; JORDAN, D. A Topological Aperitif, Springer-Verlag, 2001. [4] OŠMERA, P. Evolution of universe structures, Proceedings of MENDEL 2005, Brno, Czech Republic

(2005) 1-6. [5] OŠMERA, P. The Vortex-fractal Theory of the Gravitation, Proceedings of MENDEL’2005, Brno, Czech

Republic (2005) 7-14. [6] OŠMERA, P. The Vortex-fractal Theory of Universe Structures, Proceedings of the 4th International

Conference on Soft Computing ICSC2006, January 27, 2006, Kunovice, Czech Republic, 111-122. [7] OŠMERA ,P. Vortex-fractal Physics, Proceedings of the 4th International Conference on Soft Computing

ICSC2006, January 27, 2006, Kunovice, Czech Republic, 123-129 . [8] OŠMERA, P. Evolution of Complexity in Li Z., Halang W. A., Chen G.: Integration of Fuzzy Logic and

Chaos Theory; Springer, 2006 (ISBN: 3-540-26899-5). [9] OŠMERA, P. The Vortex-fractal Theory of Universe Structures, CD Proceedings of MENDEL 2006,

Brno, Czech Republic (2006) 12 pages. [10] OŠMERA, P. Vortex-fractal Physics, CD Proceedings of MENDEL 2006, Brno, Czech Republic (2006)

14 pages. [11] OŠMERA, P. Electromagnetic field of Electron in Vortex-fractal Structures, CD Proceedings of

MENDEL 2006, Brno, Czech Republic (2006) 10 pages. [12] OŠMERA, P. Vortex-ring Modelling of Complex Systems and Mendeleev’s Table, WCECS2007,

proceedings of World Congress on Engineering and Computer Science, San Francisco, 2007, 152-157. [13] OŠMERA, P. From Quantum Foam to Vortex-ring Fractal Structures and Mendeleev’s Table, New

Trends in Physics, NTF 2007, Brno Czech Republic, 2007, 179-182. [14] OŠMERA, P. Vortex-fractal-ring Structure of Electron, Proceedings of the 6th International Conference

on Soft Computing ICSC2008, January 25, 2008, Kunovice, Czech Republic. [15] OŠMERA, P. Vortex-fractal-ring Structure of Electron, Proceedings of the International Conference on

Soft Computing MENDEL2008, June 18-20, 2008, Brno, Czech Republic, 78-85. [16] PAULING, L. General Chemistry, Dover publication, Inc, New York, 1988. [17] OŠMERA, P. Evolution of nonliving Nature, Kognice a umělý život VIII, Prague, Czech Republic,

(2008), 231-244. ADDRESS: Doc. Ing. Pavel Ošmera, CSc. Brno University of Technology Technická 2 616 69 Brno Czech Republic E-mail: osmera @fme.vutbr.cz



IMAGE PROCESSING BY MEANS OF LABVIEW ENVIRONMENT

Jiří Liška, Jiří Šťastný, Miroslav Cepl


Abstract: This contribution describes the possibility of using of National Instruments tool LabVIEW in version 8.5 for image processing applications. As the example there are some basic algorithms for image processing chosen. For scanning images in a real time ASUS webcam is used. The used ASUS webcam is a standard webcam connected into PC via USB interface.

Key-Words: Image processing, threshold, linear filter, correlation.

INTRODUCTION Laboratory Virtual Instrumentation Engineering Workbench (LabVIEW) is a platform and development environment for a visual programming language from National Instruments. The graphical language is named G. Originally released for the Apple Macintosh in 1986, LabVIEW is commonly used for data acquisition, instrument control and industrial automation on a variety of platforms including Microsoft Windows, various flavors of UNIX, Linux and Mac OS. The latest version of LabVIEW is version 8.6, released in August of 2008. The programming language used in LabVIEW, also referred to as G, is a dataflow programming language. Execution is determined by the structure of a graphical block diagram (the LV-source code) on which the programmer connects different function-nodes by drawing wires. These wires propagate variables and any node can execute as soon as all its input data become available. Since this might be the case for multiple nodes simultaneously, G is inherently capable of parallel execution. Multi-processing and multi-threading hardware is automatically exploited by the built-in scheduler, which multiplexes multiple OS threads over the nodes ready for execution. LabVIEW ties the creation of user interfaces (called front panels) into the development cycle. LabVIEW programs/subroutines are called virtual instruments (VIs). Each VI has three components: a block diagram, a front panel, and a connector panel. The last is used to represent the VI in the block diagrams of other, calling VIs. Controls and indicators on the front panel allow an operator to input data into or extract data from a running virtual instrument. However, the front panel can also serve as a programmatic interface. Thus a virtual instrument can either be run as a program with the front panel serving as a user interface, or, when dropped as a node onto the block diagram, the front panel defines the inputs and outputs for the given node through the connector pane. This implies each VI can be easily tested before being embedded as a subroutine into a larger program. The graphical approach also allows non-programmers to build programs simply by dragging and dropping virtual representations of lab equipment with which they are already familiar. The LabVIEW programming environment, with the included examples and the documentation, makes it simple to create small applications. This is a benefit on one side, but there is also a certain danger of underestimating the expertise needed for good quality "G" programming. For complex algorithms or large-scale code, it is important that the programmer possesses an extensive knowledge of the special LabVIEW syntax and the topology of its memory management. The most advanced LabVIEW development systems offer the possibility of building stand-alone applications. Furthermore, it is possible to create distributed applications, which communicate by a client/server scheme, and are therefore easier to implement due to the inherently parallel nature of G-code PREPARE LABVIEW FOR IMAGE PROCESSING There are some necessary steps to capable image processing in LabVIEW: Standard installation of LabVIEW 8.5 During standard installation must be included module Vision and Motion In case that we want to use cameras directly from National Instruments companies for grabs images the

above mentioned two steps are sufficient. In other case when we want to use some other cameras with different drivers it is necessary to make the following two steps.



National Instruments Vision Assistant 8.5 [3] - Vision Assistant is a tool for prototyping and testing image processing applications. To prototype an image processing application, build cystom algorithms with the Vision Assistant scripting feature. The scripting feature records every step of the processing algorithm. After completing the algorithm, you can test it on other images to make sure it works.

Fig. 1 Screen of Vision Assistant

NI-IMAQ for USB Cameras driver [4] - software allows users to configure any DirectShow imaging

device and acquire images into LabVIEW. Devices that support this feature include USB cameras, webcams, microscopes, scanners, and many consumer-grade imaging products.

This product is freeware and we can dowload it from website of National Instruments company. After installing this package we are able to use new functions from Vision and Motion category. These new functions are called IMAQ USB that serve for communication between LabVIEW and standard periphery via USB interface. TOOLS FOR IMAGE PROCESSING IN LABVIEW This is a list of some chosen algorithms for image processing included into LabVIEW. Vision and Motion->Image Processing->Processing-> IMAQ Equalize produces a histogram equalization of an image. This VI redistributes the pixel values of an image to linearize the accumulated histogram. The precision of the VI is dependent on the histogram precision, which in turn is dependent on the number of classes used in the histogram. IMAQ Theshold applies a threshold to an image. IMAQ MultiThreshold performs thresholds of multiple intensity ranges to an image IMAQ Label labels the particles in a binary image. IMAQ Watershed Transform computes the watershed transform on an image IMAQ Inverse inverts the pixel intensities of an image to compute the negative of an image IMAQ AutoMThreshold cmputes the optimal values to threshold an image into a given number of classes. This algorithm uses a variant of the classification by clustering method. Starting from a random sort, a clustering algorithm is iterated until a stable and reliable result is found IMAQ Local Threshold thresholds an image into a binary image based on the specified local adaptive thresholding Metod Vision and Motion->Image Processing->Filters-> IMAQ Convolute filters an image using a linear filter. The calculations are performed with either integers or floating points, depending on the image type and the contents of the kernel.



IMAQ Correlate computes the normalized cross correlation between the source image and the template image

AQ LowPass calculates the inter-pixel variation between the pixel being processed and those pixels

XAMPLE le there are some basic algorithms for image processing chosen. For scanning images in a real time

IMsurrounding it. If the pixel being processed has a variation greater than a specified percentage, it is set to the average pixel value as calculated from the neighboring pixels EAs the exampASUS webcam is used. In the window in fig. 2 there are six images of control units which serve as the example of possibilities of the given application for image processing. Further there is also a ListBox which contains the list of all available (connected) cameras in the system. As well as there is a control unit for setting parameters of thresholding.

Fig. 2 Exemplary window of application



Fig. 3 Block diagram of the example

CONCLUSION LabVIEW is a very user friendly tool. By means of intuitive control it is possible to make functional examples. The programming is realized via graphic interface what is acceptable for less advanced users too. The efficiency of LabVIEW environment is very good for image processing applications. In the example presented in this article there are five different algorithms applied to one input image with resolution 320 x 240. This example was tested on notebook Dell Latitude D505 with processor Intel Pentium 1.5 GHz and its computing speed was three shots per second. ACKNOWLEDGEMENT This research was supported by the grants: MSM 0021630529 Intelligent Systems in Automation (Research design of Brno University of Technology) No 102/07/1503 Advanced Optimisation of Communications Systems Design by Means of Neural Networks. The Grant Agency of the Czech Republic (GACR) MSM 6215648904/03 Development of relationships in the business sphere as connected with changes in the life style of purchasing behaviour of the Czech population and in the business environment in the course of processes of integration and globalization (Research design of Mendel University in Brno). REFERECES [1] LabVIEW Help, http://www.ni.com/pdf/manuals/372228k.pdf [2] LabVIEW Wikipedia, http://en.wikipedia.org/wiki/LabVIEW [3] NI Vision Assistant tutorial, http://www.ni.com/pdf/manuals/372228k.pdf [4] NI IMAQ ror USB cameras overview, http://zone.ni.com/devzone/cda/epd/p/id/5030 ADDRESS: Ing. Jiří Liška Department of Automation and Computer Science Brno University of Technology Technická 2 616 69 Brno, Czech Republic E-mail: [email protected],

http://www.ni.com/pdf/manuals/372228k.pdf

http://en.wikipedia.org/wiki/LabVIEW

http://www.ni.com/pdf/manuals/372228k.pdf

http://zone.ni.com/devzone/cda/epd/p/id/5030



Doc. RNDr. Ing. Jiří Šťastný, CSc. Department of Automation and Computer Science Brno University of Technology Technická 2 616 69 Brno Czech Republic E-mail: [email protected] Ing. Miroslav Cepl Ústav informatiky PEF Mendlova zemědělská a lesnická univerzita v Brně Zemědělská 1 61300 Brno E-mail: [email protected]



PRIORITY SWITCHING BY MEANS OF NEURAL NETWORK

Jiří Stastny, P. Pokorny, O. Popelka


Abstract: The paper compares two neural networks used for optimization of priority packet switching. Packet has to be delivered from a sender to a receiver through packet switch with minimum time delay. Each packet can have a self priority which says how important speed of transfer is. For a fast choice of preferential packets in the packet switch and their sending neural network can be used. Two neural networks are tested here. One of them is the Hopfield neural network (NN) and the second one is the Self-organizing network (SON) with Competitive learning (Competitive NN).

Key-Words: Neural network, priority, optimal selection, Hopfield neural network, Self-organizing network.

1 INTRODUCTION Network elements like a network switches are most important elements for IT communication. An IP6 Ethernet (TCP/IP) protocol comes with packet priority. Each packet has a priority and can be sent to the packet switch. Packet switch as a network connecting point solves problem with priority switching. There should be fast way to decide for optimal packet selection. For finding out an optimal resolution the neural networks are used. The first step for using NN is its learning on a set of examples. It consists of setting-up weights and biases of NN for each neuron. Every time there is a new set of examples so that new learning is necessary. In the case of packet switching there is one invariable array of examples so just one learning process is needed. Learning is made just once and weights are set to the packet switch manually. 2 HOPFIELD NN Hopfield NN is classified as a historical NN. It means, it is one of the oldest NN (1982) but in some case it is powerful tool (as we can see later). Learning of Hopfield NN is based on Hebb weight learning function [6]. Network floats from initial conditions to equilibrium points in a few steps. Each step creates result in output which goes recurrently to network’s input. It’s recurrent NN with one layer.

Fig. 1 The architecture for a Hopfield NN and shape of a Satlin function.

a(0) = p and then for k= 1,2,… a(k) = satlin(LW * a(k-1) + b) As note, the input p is an initial condition. In the first step is p used, next steps are using recurrent values. The Satlin is a function displayed on the right corner of picture Fig. 1. LW box (in the picture) includes arrays of weights, each is multiple with input a(k). For m packet switch inputs (then m outputs) are there m^2 mentioned weight arrays (m neurons). So, of this finding we can work out the arithmetical difficulty, it is in the table Tab. 1. Hopfield NN is deciding between example states. Initial conditions gives initial state to NN and NN progresses in agreement with minimal Energetic function to the equilibrium state.



Arithmetical difficulty

m Dimension States one step steps in all

4 16 24 256 19 4 864

5 25 120 625 25 15 625

6 36 720 1 296 31 40 176

7 49 5 040 2 401 32 76 832

8 64 40 320 4 096 33 135 168 Tab. 1 Table of the arithmetical difficulty for Hopfield’s NN.

The m symbol in the table says number of packet switch’s inputs (outputs). Dimension is number of rows which is NN computing with. States is number of example states. As was mentioned, Hopfield NN is recurrent network and final result is ready after a few steps. Number of steps which needs NN for final result is in the table column named steps and time for one step is in column named one step. Last column says total time needed for final result solving. 3 SELF-ORGANIZING NET WITH COMPETITIVE LEARNING (SON) SON (Competitive NN) is network which divide input elements to the clusters in agreement with the example elements. Competitive learning is used for set-up weights and biases (originating Competitive layer) [6]. In this problem are packet switch’s inputs (outputs) like an input elements and examples are states of problem. The architecture for a competitive network is shown in Fig. 2.

Fig. 2 The architecture for a Competitive NN.

As we can see, there is no recurrent linkage so final product is done in one step. The LI box (in Fig. 1) includes arrays of weights. Each weight’s vector income to the box named ndist. This makes negative of the Euclidean distance. It means:

ndist .

The C box is a Competitive transfer function which selects maximum of array argument and on this position puts number 1, other array members are 0. There are values of Competitive NN arithmetical difficulty in the table Tab. 2.

Arithmetical difficulty

m Dimension States one step steps in all

4 16 24 384 1 384

5 25 120 3 000 1 3 000

6 36 720 25 920 1 25 920

7 49 5 040 246 960 1 246 960

8 64 40 320 2 580 480 1 2 580 480 Tab. 2 Table of the arithmetical difficulty for Competitive NN.

Sense of the columns is explains below table Tab. 1. 4 COMPARE HOPFIELD NN WITH COMPETITIVE NN In the priority packet switching transfer speed is the most important. In the table Tab. 3 there are numbers that



say how difficult calculations are for each NN. For packet switch with 4 inputs (outputs) Competitive NN is 13 times faster than Hopfield NN. For 5 and 6 inputs Competitive NN is still faster but with higher dimension (number of inputs) arithmetical difficulty increases manifold faster than with Hopfield NN. For 7 and higher number of inputs Hopfield NN is faster than Competitive NN.

A. d. for Hopfield NN A. d. for Competitive NN

m Dimension States one step steps in all one step steps in all

4 16 24 256 19 4 864 384 1 384

5 25 120 625 25 15 625 3 000 1 3 000

6 36 720 1 296 31 40 176 25 920 1 25 920

7 49 5 040 2 401 40 96 040 246 960 1 246 960

8 64 40 320 4 096 48 196 608 2 580 480 1 2 580 480 Tab. 3 Table of Hopfield NN and Competitive NN arithmetical difficulties.

For information: There was one more method tested for pocket switch with 4 inputs. It was method of a serial computing with no NN used. This method is based on gradual findings and matching with each example. This method is 20 times slower than Hopfield NN method. In higher dimensions (more inputs) serial computing was not tested because the arithmetical difficulty would be huge. 5 CONCLUSION Neural networks are modern tools for fast choose of example states. Every problem has to be carefully explored and the neural network that perfectly fits to the problem has to be chosen. In the above mentioned example SON with Competitive learning will be used for packet switches with 4 to 6 inputs and Hopfield NN will be better for 7 and more inputs. Neural networks use parallel computing. In serial computing arithmetical difficulty is much higher. The use of NNs for packet switching is a good way how to speed up network connection. But there is one problem with NNs - for high dimensions (especially for SON) it is necessary to use a higher memory. In order to eliminate this problem other methods can be used for example evolutional methods based on genetic algorithm. ACKNOWLEDGEMENT This research was supported by the grants: MSM 0021630529 Intelligent Systems in Automation (Research design of Brno University of Technology) No 102/07/1503 Advanced Optimisation of Communications Systems Design by Means of Neural Networks. The Grant Agency of the Czech Republic (GACR) MSM 6215648904/03 Development of relationships in the business sphere as connected with changes in the life style of purchasing behaviour of the Czech population and in the business environment in the course of processes of integration and globalization (Research design of Mendel University in Brno). 6 REFERENCES [1] The MathWorks, Inc. Neural Network Toolbox™ 6 , User's Guide. [Document] 2008. [2] MEDSKER, L. R. A.; JAIN, L. C. Recurrent Neural Network , Design and Applications. London CRC

Press, 2001. [3] KASABOV, N. K. Foundations of Neural Networks, Fuzzy Systems and Knowledge Engineering.

London, The MIT Press, 1995. [4] VOLNA, E. Neuronové sítě 1. [Document] Ostrava, 2002. [5] DENNIS, S.; MCAULEY, D. Connectionist Models of Cognition,

http://www.itee.uq.edu.au/~cogs2010/cmc/index.html . [6] KOHONEN, T. Self-Organized Formation of Topologically Correct Feature Maps. In: Biological

Cybernetics 43, pp. 59-69, 1982.



ADDRESS: Doc. RNDr. Ing. Jiří Šťastný, CSc. Department of Automation and Computer Science Brno University of Technology Technická 2 616 69 Brno Czech Republic E-mail: [email protected] Ing. Pavel Pokorný Department of Automation and Computer Science Brno University of Technology Technicka 2 616 69 Brno Czech Republic http://www.vutbr.cz/ E-mail: [email protected] Ing. Ondřej Popelka Department of Automation and Computer Science Brno University of Technology Technicka 2 616 69 Brno Czech Republic http://www.vutbr.cz/

EXISTENCE OF POSITIVE SOILUTIONS OF DISCRETELINEAR DELAYED EQUATIONS

Jaromır Bastinec, Josef Diblık, Zdenek Smarda


Abstract: In the paper we develop a method of inequalities to prove the existence of positivesolutions to linear difference equations with negative coefficients and with delays. A series ofcomparison results for solutions of this class of equations is derived. These are used to provethe existence of positive solutions of a particular and very frequently investigated class of linearequation with only one delay. The relevant result is given in the form of an inequality (witha suitable auxiliary function) for the equation coefficient. Comparisons to known results areincluded as well.

Key words: Retraction method, discrete delayed equation, oscillating solution, positive solu-tion.

1. INTRODUCTION AND THE PROBLEM CONSIDERED

We use the following notation: for integers s, q, s ≤ q, we define

Zqs := s, s + 1, . . . , q

where s = −∞ and q = ∞ are admitted, too.The topic of our study is a linear scalar discrete equation of k-th order

∆x(n) = −k∑

i=0

pi(n)x(n− i), (1)

where p0 : Z∞a → R, pi : Z∞a → R+ := [0,∞), i = 1, . . . , k, k ≥ 1, a is an integer and n ∈ Z∞a . Letϕ : Za

a−k → R. Together with discrete equation (1), we consider an initial problem: determine a solutionx = x(n) of equation (1) satisfying the initial conditions

x(n) = ϕ(n), n ∈ Zaa−k (2)

with prescribed constants ϕ(n) ∈ R. The solution of initial problem (1), (2) is defined as an infinitesequence of numbers xn∞n=−k with xn = x(a + n), i.e.,

x−k = ϕ(a− k), . . . , x0 = ϕ(a), x1 = x(a + 1), . . . , xn = x(a + n), . . .

such that, for any n ∈ Z∞a , equality (1) holds. If convenient, we denote the solution x = x(n) of theinitial problem (1), (2) by x(n) = x(n; a, ϕ). We recall an obvious fact that a solution x = x(n) of initialproblem (1), (2) depends continuously on the initial data. The solution x = x(n) of initial problem (1), (2)is called positive (negative) if x(n) > 0 (x(n) < 0) on Z∞a−k. The solution x = x(n) of initial problem (1),(2) being neither positive nor negative on Z∞a−k is called oscillatory.

Our aim is to find sufficient conditions with respect to the right-hand side of equation (1) to guaranteethe existence of at least one initial function

x(n) = ϕ∗(n), n ∈ Zaa−k

with ϕ∗ : Zaa−k → (0,∞) such that the solution x∗ = x∗(n; a, ϕ∗) of the initial problem (1), (2) with

ϕ ≡ ϕ∗ remains positive on Z∞a−k.The paper is organized as follows. In section 2 an auxiliary nonlinear result is formulated. Then it is

applied to (1) in section 3 where the main results on positive solutions are proved. Some comparisons toknown results and remarks finalize the paper in section indicating future directions that may be pursuedin the context of this research as well.

2. NONLINEAR PRELIMINARIES

Let us consider the scalar discrete equation

∆u(n) = f(n, u(n), u(n− 1), . . . , u(n− k)), (3)

where f : Z∞a × Rk+1 → R and k ≥ 1 is an integer. Let ϕ : Zaa−k → R be a given function. Together with

the discrete equation (3) we consider an initial problem: find the solution u = u(n), n ∈ Z∞a−k of (3)satisfying initial conditions

u(a−m) = ϕ(a−m), m = 0, 1, . . . , k. (4)

The notion of a solution of the initial problem (3), (4) can be adapted easily from section 1 and thereforewe do not recall it. The existence and uniqueness of the solution of the initial problem (3), (4) is obviousas well.

Let two functions b, c : Z∞a−k → R be given such that b(n) < c(n) for each n ∈ Z∞a−k. For any n ∈ Z∞a−k

we define a setω(n) := t ∈ R, b(n) < t < c(n). (5)

In addition to this, we define a set

Ω := (n, t) : n ∈ Z∞a−k, t ∈ ω(n).

Obviously∂ω(n) = t ∈ R, (b(n)− t)(t− c(n)) = 0 = b(n), c(n)

and∂Ω = (n, t) : n ∈ Z∞a−k, t ∈ ∂ω(n).

We will formulate an auxiliary nonlinear result on the existence of a solution u = u(n), n ∈ Z∞a−k

of (3) with its graph (n, u(n)), n ∈ Z∞a−k remaining in Ω. In other words, this means that, under certainassumptions, there exists at least one initial function ϕ such that

b(n) < ϕ(n) < c(n)

for n ∈ Zaa−k and

b(n) < u(n; a, ϕ) < c(n) (6)

for every n ∈ Z∞a−k. It is easy to see that, from inequalities (6), we can deduce the existence of a positivesolution of equation (3) if our sufficient conditions are be valid for the choice: b(n) ≡ 0 and c(n) > 0,n ∈ Z∞a−k. This idea will be applied to equation (1).

We divide ∂Ω into two nonempty disjoint subsets B1 and B2 where

B1 := (n, t) ∈ ∂Ω, t = b(n),B2 := (n, t) ∈ ∂Ω, t = c(n).

Now we are ready to formulate a nonlinear result, necessary for our investigation, concerning the ex-istence of a solution of (3) with its graph lying in the set Ω (see [4, 8], for related results we refer to [7, 10]).

Theorem 1Let the function f : Z∞a × Rn+1 → R be continuous. If, moreover, inequalities

f(n, b(n), u1, . . . , uk)− b(n + 1) + b(n) < 0, (7)f(n, c(n), u1, . . . , uk)− c(n + 1) + c(n) > 0 (8)

hold for every n ∈ Z∞a , u1 ∈ ω(n− 1), . . . , uk−1 ∈ ω(n− k − 1) and uk ∈ ω(n− k), then there exists aninitial problem

u(a−m) = ϕ(a−m), m = 0, 1, . . . , k

with ϕ : Zaa−k → R, ϕ(n) ∈ ω(n), n ∈ Za

a−k such that the corresponding solution u = u(n, a, ϕ) ofequation (3) satisfies the inequalities

b(n) < u(n; a, ϕ) < c(n)

for every n ∈ Z∞a−k.

3. RESULTS

In this part we investigate equation (1). The main result is given by Theorem 4.

Theorem 2Let

k∑i=1

pi(n) > 0 (9)

for any n ∈ Z∞a−k.Then, for the existence of a positive solution x = x(n) of (1), the existence of a function ν : Z∞a−k →R+ := (0,∞) such that

∆ν(n) ≤ −k∑

i=0

pi(n)ν(n− i), (10)

for n ∈ Z+∞a is sufficient and necessary.

Moreover, x(n) < ν(n) holds on Z∞a−k.

Proof. Necessity. This is obvious since it is possible to put, e.g., ν := 2x, where x is a positive solutionof (1).Sufficiency. We will use Theorem 1 with

f(n, u(n), u(n− 1), . . . , u(n− k)) := −k∑

i=0

pi(n)u(n− i),

b(n) := 0, c(n) := ν(n).

In such a case, the set ω defined by (5) turns into ω(n) ≡ t ∈ R, 0 < t < ν(n). We verify inequalities (7),(8). Regarding to (7) we have

f(n, b(n), u1, . . . , uk)− b(n + 1) + b(n) = f(n, 0, u1, . . . , uk) = −k∑

i=1

pi(n)u(n− i).

It is easy to see that ui > 0 if ui ∈ ω(n− i), i = 1, . . . , k. Then (we use (9) as well)

f(n, b(n), u1, . . . , uk)− b(n + 1) + b(n) ≤ −k∑

i=1

pi(n) < 0

and (7) holds. With respect to (8) we have

f(n, c(n), u1, . . . , uk)− c(n + 1) + c(n)=f(n, ν(n), u1, . . . , uk)− ν(n + 1) + ν(n)

=− p0(n)ν(n)−k∑

i=1

pi(n)ui − ν(n + 1) + ν(n).

Since ui ∈ ω(n− i), then ui < ν(n− i), i = 1, . . . , k, and, due to (9), (10), we get

f(n, c(n), u1, . . . , uk)− c(n + 1) + c(n)

>− p0(n)ν(n)−k∑

i=1

pi(n)ν(n− i)− ν(n + 1) + ν(n)

=−k∑

i=0

pi(n)ν(n− i)−∆ν(n) ≥ 0.

Inequality (8) is valid. We conclude that all the assumptions of Theorem 1 are valid as well. With regardto equation (1), this means that there exists an initial function ϕ : Za

a−k → R, ϕ(n) ∈ ω(n), n ∈ Zaa−k

such that x = x(n, a, ϕ) satisfies the inequalities

0 ≡ b(n) < u(n; a, ϕ) < c(n) ≡ ν(n) (11)

for every n ∈ Z∞a . Inequality (11) coincides with the conclusion of Theorem 2.

For the proof of the main result, we need a comparison result for the equation (1) and an equation

∆w(n) = −k∑

i=0

Pi(n)w(n− i) (12)

where P0 : Z∞a → R, Pi : Z∞a → R+, i = 1, . . . , k, k ≥ 1 under the assumption Pi(n) ≤ pi(n), i = 1, . . . , k,n ∈ Z∞a .

Theorem 3Let

k∑i=1

Pi(n) > 0 (13)

for any n ∈ Z∞a−k. Assume that equation (1) admits a positive solution x = µ(n) on Z∞a−k and

Pi(n) ≤ pi(n), (14)

i = 1, . . . , k, n ∈ Z∞a .Then the equation (12) has a positive solution w = w(n) on Z∞a−k and, moreover, w(n) < µ(n).

Proof. We will use Theorem 1 with

f(n, u(n), u(n− 1), . . . , u(n− k)) := −k∑

i=0

Pi(n)u(n− i),

b(n) := 0, c(n) := µ(n).

Then the set ω defined by (5) turns into ω(n) ≡ t ∈ R, 0 < t < µ(n). Now we begin to verifyinequalities (7), (8). In the case of (7), we proceed in much the same way as in the proof of Theorem 2.We get

f(n, b(n), u1, . . . , uk)− b(n + 1) + b(n) = f(n, 0, u1, . . . , uk) = −k∑

i=1

Pi(n)u(n− i).

Since ui ∈ ω(n− i) we have ui > 0, i = 1, . . . , k and (we use (13))

f(n, b(n), u1, . . . , uk)− b(n + 1) + b(n) ≤ −k∑

i=1

Pi(n) < 0.

Inequality (7) holds. With regard to (8), we have

f(n, c(n), u1, . . . , uk)− c(n + 1) + c(n)=f(n, µ(n), u1, . . . , uk)− µ(n + 1) + µ(n)

=− P0(n)µ(n)−k∑

i=1

Pi(n)ui − µ(n + 1) + µ(n).

Since ui ∈ ω(n− i), then ui < µ(n− i), i = 1, . . . , k and, due to (13), (14),

f(n, c(n), u1, . . . , uk)− c(n + 1) + c(n)

>− P0(n)µ(n)−k∑

i=1

Pi(n)µ(n− i)− µ(n + 1) + µ(n)

≥−k∑

i=0

pi(n)µ(n− i)−∆µ(n) = ∆µ(n)−∆µ(n) = 0.

Inequality (8) is valid and all the assumptions of Theorem 1 are valid as well. Regarding the equation (12),this means that there exists an initial function ϕ : Za

a−k → R, ϕ(n) ∈ ω(n), n ∈ Zaa−k such that w =

w(n, a, ϕ) satisfies the inequalities

0 ≡ b(n) < w(n; a, ϕ) < c(n) ≡ µ(n)

for every n ∈ Z∞a . Consequently, the conclusion of Theorem 3 holds.

Before formulating of the main result, we need auxiliary results on asymptotic decompositions of someauxiliary expressions.

Definition 1 Let us define the expression lnq t, q ≥ 1, by lnq t = ln(lnq−1 t), ln0 t ≡ t where t > expq−2 1and exps t = exp

(exps−1 t

), s ≥ 1, exp0 t ≡ t and exp−1 t ≡ 0 (instead of ln0 t, ln1 t, we will only write t

and ln t).

The following lemmas (necessary for the proof of the main result) can be proved in an elementary wayby the method of induction. Therefore, their proofs are omitted. The symbol “ o ” used in formulas (15)and (16) stands for the Landau order symbol.

Lemma 1For fixed r, σ ∈ R \ 0, the asymptotic representation

(n− r)σ = nσ

[1− σr

n+

σ(σ − 1)r2

2n2− σ(σ − 1)(σ − 2)r3

6n3+ o

(1n3

)](15)

holds for n →∞.

Lemma 2For fixed r, σ ∈ R \ 0 and fixed integer q ≥ 1, the asymptotic representation

lnσq (n− r)lnσ

q n=1− rσ

n lnn . . . lnq n− r2σ

2n2 lnn . . . lnq n

− r2σ

2(n lnn)2 ln2 n . . . lnq n− · · · − r2σ

2(n lnn . . . lnq−1 n)2 lnq n

+r2σ(σ − 1)

2(n lnn . . . lnq n)2− r3σ(1 + o(1))

3n3 lnn . . . lnq n(16)

holds for n →∞.

Let ` ≥ 0 be a fixed integer. We define auxiliary functions p`, ν` as

p`(n) :=(

k

k + 1

)k

×[

1k + 1

+k

8n2+

k

8(n lnn)2+ · · ·+ k

8(n lnn . . . ln` n)2

](17)

and

ν`(n) :=(

k

k + 1

)n

·√

n lnn ln2 n . . . ln` n (18)

which play an important role in the investigation of positive solutions of an equation

∆x(n) = −p(n)x(n− k), (19)

being a particular case of equation (1) (with p0 ≡ p1 ≡ · · · ≡ pk−1 ≡ 0 and pk ≡ p). We assume that nin (17) and (18) is sufficiently large for p` and ν` to be well defined.

Lemma 3Let ` ≥ 0 be a fixed integer. Then the inequality

∆ν(n) ≤ −p`(n)ν(n− k) (20)

has a solution ν ≡ ν` provided that n is sufficiently large.

Proof. We consider the left hand side of (20) and asymptotically decompose ∆ν` for n →∞.

∆ν`(n) =ν`(n + 1)− ν`(n)

=(

k

k + 1

)n+1

·√

(n + 1) ln(n + 1) ln2(n + 1) . . . ln`(n + 1)

−(

k

k + 1

)n

·√

n lnn ln2 n . . . ln` n

=(

k

k + 1

)n

·√

n lnn ln2 n . . . ln` n×[(

k

k + 1

)· V1 − 1

]where

V1 =

√(n + 1) ln(n + 1) ln2(n + 1) . . . ln`(n + 1)√


=√

n + 1√n

·√

ln(n + 1)√lnn

·√

ln2(n + 1)√ln2 n

· · · · ·√

ln`(n + 1)√ln` n

The decomposition of ∆ν` will be finished if we give an asymptotic decomposition of V1. Since, byformula (15) with σ = 1/2 and r = −1,

√n + 1√

n= 1 +

12n

− 18n2

+1

16n3+ o

(1n3

),

by formula (16) with σ = 1/2, q = 1 and r = −1,√ln(n + 1)√

lnn= 1 +

12n lnn

− 14n2 lnn

− 18(n lnn)2

+1 + o(1)6n3 lnn

,

by formula (16) with σ = 1/2, q = 2 and r = −1,√ln2(n + 1)√

ln2 n= 1 +

12n lnn ln2 n

− 14n2 lnn ln2 n

− 14(n lnn)2 ln2 n

− 18(n lnn ln2 n)2

+1 + o(1)

6n3 lnn ln2 n,

etc., and, by formula (16) with σ = 1/2, q = ` and r = −1,√ln`(n + 1)√

ln` n= 1 +

12n lnn . . . ln` n

− 14n2 lnn . . . ln` n

− . . .

− 14(n lnn . . . ln`−1 n)2 ln` n

− 18(n lnn . . . ln` n)2

+1 + o(1)

6n3 lnn . . . ln` n,

we have

V1 = 1 +12n

+1

2n lnn+

12n lnn ln2 n

+ · · ·+ 12n lnn . . . ln` n

− 18n2

− 18(n lnn)2

− 18(n lnn ln2 n)2

− · · · − 18(n lnn . . . ln` n)2

+1

16n3+ o

(1n3

). (21)

Now we asymptotically decompose for n →∞ the right-hand side of (20) with ν := ν`. We get

−p`(n)ν`(n− k) =

−(

k

k + 1

)k [1

k + 1+

k

8n2+

k

8(n lnn)2+ · · ·+ k

8(n lnn . . . ln` n)2

]

×(

k

k + 1

)n−k

·√

(n− k) ln(n− k) ln2(n− k) . . . ln`(n− k)

=−(

k

k + 1

)n

·√


×[

1k + 1

+k

8n2+

k

8(n lnn)2+ · · ·+ k

8(n lnn . . . ln` n)2

]× V2

where

V2 =

√(n− k) ln(n− k) ln2(n− k) . . . ln`(n− k)√


=√

n− k√n

·√

ln(n− k)√lnn

·√

ln2(n− k)√ln2 n

· · · · ·√

ln`(n− k)√ln` n

.

Since, by formula (15) with σ = 1/2 and r = k,√

n− k√n

= 1− k

2n− k2

8n2− k3

16n3+ o

(1n3

),

by formula (16) with σ = 1/2, q = 1 and r = k,√ln(n− k)√

lnn= 1− k

2n lnn− k2

4n2 lnn− k2

8(n lnn)2− k3 + o(1)

6n3 lnn,

by formula (16) with σ = 1/2, q = 2 and r = k,√ln2(n− k)√

ln2 n= 1− k

2n lnn ln2 n− k2

4n2 lnn ln2 n

− k2

4(n lnn)2 ln2 n− k2

8(n lnn ln2 n)2− k3 + o(1)

6n3 lnn ln2 n,

etc., and by formula (16) with σ = 1/2, q = ` and r = k,√ln`(n− k)√

ln` n= 1− k

2n lnn . . . ln` n− k2

4n2 lnn . . . ln` n− . . .

− k2

4(n lnn . . . ln`−1 n)2 ln` n− k2

8(n lnn . . . ln` n)2

− k3 + o(1)6n3 lnn . . . ln` n

,

we have

V2 = 1− k

2n− k

2n lnn− k

2n lnn ln2 n− · · · − k

2n lnn . . . ln` n

− k2

8n2− k2

8(n lnn)2− k2

8(n lnn ln2 n)2− · · · − k2

8(n lnn . . . ln` n)2

− k3

16n3+ o

(1n3

).

Then [1

k + 1+

k

8n2+

k

8(n lnn)2+ · · ·+ k

8(n lnn . . . ln` n)2

]× V2

=[

1k + 1

+k

8n2+

k

8(n lnn)2+ · · ·+ k

8(n lnn . . . ln` n)2

]×

[1− k

2n− k

2n lnn− k

2n lnn ln2 n− · · · − k

2n lnn . . . ln` n

− k2

8n2− k2

8(n lnn)2− k2

8(n lnn ln2 n)2− · · · − k2

8(n lnn . . . ln` n)2

− k3

16n3+ o

(1n3

)]=

1k + 1

[1− k

2n− k

2n lnn− k

2n lnn ln2 n− · · · − k

2n lnn . . . ln` n

]+

k

k + 1

[1

8n2+

18(n lnn)2

+1

8(n lnn ln2 n)2+ · · ·+ 1

8(n lnn . . . ln` n)2

]−

(1 +

k

k + 1

)k2

16n3+ o

(1n3

). (22)

Now we see that for the inequality

∆ν`(n) ≤ −p`(n)ν`(n− k)

to be valid if n →∞, it is sufficient(k

k + 1

)· V1 − 1

≤ −[

1k + 1

+k

8n2+

k

8(n lnn)2+ · · ·+ k

8(n lnn . . . ln` n)2

]× V2

or (using (21) and (22))(k

k + 1

) [1 +

12n

+1

2n lnn+

12n lnn ln2 n

+ · · ·+ 12n lnn . . . ln` n

− 18n2

− 18(n lnn)2

− 18(n lnn ln2 n)2

− · · · − 18(n lnn . . . ln` n)2

+1

16n3

]− 1 + o

(1n3

)≤− 1

k + 1

[1− k

2n− k

2n lnn− k

2n lnn ln2 n− · · · − k

2n lnn . . . ln` n

]− k

k + 1

[1

8n2+

18(n lnn)2

+1

8(n lnn ln2 n)2+ · · ·+ 1

8(n lnn . . . ln` n)2

]+

(1 +

k

k + 1

)k2

16n3+ o

(1n3

).

We see that the last inequality turns into

0 ≤ k(2k2 + k − 1)16n3(k + 1)

+ o

(1n3

)and holds if n →∞.

The following result is an interesting consequence of Lemma 3.

Lemma 4Let ` ≥ 0 be a fixed integer. Then the equation

∆x(n) = −p`(n)x(n− k)

has a positive solution x = x(n) < ν`(n) provided that n is sufficiently large.

Proof. Since the inequality (20) has a (positive) solution ν ≡ ν` provided that n is sufficiently large,the proof is a straightforward consequence of Theorem 2 (we assume a to be sufficiently large) withp0 ≡ p1 ≡ · · · ≡ pk−1 ≡ 0 and pk ≡ ν`.

Theorem 4 (Main result)Let ` ≥ 0 be a fixed integer and 0 < p(n) ≤ p`(n) for n → +∞.Then the equation (19) has a positive solution x = x(n) < ν`(n) provided that n is sufficiently large.

Proof. This is a direct consequence of Theorem 3 (we assume a to be sufficiently large) with P0 ≡ P1 ≡· · · ≡ Pk−1 ≡ 0 and Pk ≡ p(n) and Lemma 4 if we put p0 ≡ p1 ≡ · · · ≡ pk−1 ≡ 0 and pk ≡ p`(n) in (1).

4. CONCLUSION AND FUTURE DIRECTIONS

The following result is well-known (see [11, p. 192]).

Theorem 5Let k ∈ Z∞1 be fixed, p(n) > 0 for n ≥ 0, and

p(n) ≤ kk

(k + 1)k+1. (23)

Then the difference equation (19) where n = 0, 1, 2, . . . has a positive solution.

Comparing Theorem 5 with Theorem 4, we see that the latter is a substantial improvement of the aboveresult because positive solutions exist even if inequality (23) is changed to a general one: p(n) ≤ p`(n),and obviously

p`(n) >kk

(k + 1)k+1

for every fixed ` and every n belonging to the domain of ln` n. Theorem 4 substantially improves Theorem 2in [2] and Theorem 3.2 in [3] as well. Theorem extends to time varying coefficients the statement a) ofCorollary 7.4.1. in [11, p. 176].

As noted in [11, p. 179], for p(n) ≡ p = const, the inequality (23) is sharp since, in this case, thenecessary and sufficient condition for the oscillation of all the solutions of (19) is given by the inequality

p >kk

(k + 1)k+1.

Therefore, the value

pc =kk

(k + 1)k+1

is called a critical value since it separates the case of all the solutions being oscillating from the one ofthere existing a positive solution. The function p(n) we defined by formula (17) can be called a criticalfunction since

limn→∞

p`(n)

= limn→∞

(k

k + 1

)k [1

k + 1+

k

8n2+

k

8(n lnn)2+ · · ·+ k

8(n lnn . . . ln` n)2

]=

kk

(k + 1)k+1= pc.

We consider a similar situation in the the continuous case. The forthcoming theorem for the continuouscase states that, if an opposite inequality with respect to the inequality guaranteeing the existence of apositive solution holds, then all solutions oscillate. We consider a scalar linear differential equation withconstant delay

x(t) = −a(t)x(t− τ), (24)

where t ∈ I := [t0,∞), t0 ∈ R, a : I → R+ := (0,∞) is a continuous function and τ > 0. A final result(in a certain sense) concerning the case of there existing a positive solution is given in [9]. This can beformulated as

Theorem 6I ) Let ` ∈ Z∞0 be fixed and a(t) ≤ a`(t) with

a`(t) :=1eτ

+τ

8et2+

τ

8e(t ln t)2

+τ

8e(t ln t ln2 t)2+ · · ·+ τ

8e(t ln t ln2 t . . . ln` t)2

if t →∞. Then (24) has an eventually positive solution x = x(t).II ) Let ` ∈ Z∞2 be fixed Let us assume that

a(t) > ak−2(t) +θτ

8e(t ln t ln2 t . . . lnk−1 t)2

if t →∞, k ≥ 2 is an integer and θ > 1 is a constant.Then every solution of (24) oscillates for t →∞.

Our main result unfortunately gives no answer to the question whether all solutions of (19) are oscil-lating in the case of an opposite inequality with respect to the inequality indicated in Theorem 4 beingtrue. Let us formulate (as an open question) the following natural conjecture which is an addendum toTheorem 4.

Conjecture 1Let ` ≥ 2 be a fixed integer, θ > 1 be a fixed constant and

p(n) > p`−2(n) +kθ

8(n lnn . . . ln`−1 n)2

for n →∞.Then every solution of (19) oscillates for t →∞.

For related results, the reader is referred also to [1, 5, 6, 12]–[17].

Acknowledgement. This work was supported by the Grant 201/07/0145 of Czech Grant Agency(Prague), by the Grant 201/08/0469 of Czech Grant Agency (Prague) by the Council of Czech Gov-ernment MSM 00216 30503, by the Council of Czech Government MSM 00216 30519 and by the Councilof Czech Government MSM 00216 30529.

Reference

[1] AGARWAL, R.P., BOHNER, M., WAN-TONG LI: Nonoscillation and Oscillation: Theory forFunctional Differential Equations, Marcel Dekker, Inc., 2004.

[2] BASTINEC, J., DIBLIK, J.: Remark on positive solutions of discrete equation ∆u(k + n) =−p(k)u(k), Nonlinear Anal., 63 (2005), e2145–e2151.

[3] BASTINEC, J., DIBLIK, J.: Subdominant positive solutions of the discrete equation ∆u(k + n) =−p(k)u(k), Abstr. Appl. Anal., 2004:6 (2004), 461–470.

[4] BASTINEC, J., DIBLIK, J., ZHANG, B.: Existence of bounded solutions of discrete delayed equa-tions, Proceedings of the Sixth International Conference on Difference Equations, CRC, Boca Raton,FL, 359–366, 2004.

[5] BEREZANSKY, L., BRAVERMAN, E. : On existence of positive solutions for linear differenceequations with several delays, Adv. Dyn. Syst. Appl., 1 (2006), 29–47.

[6] CHATYARAKIS, G.E., STAVROULAKIS, I.P.: Oscillations of first order linear delay differenceequations, Aust. J. Math. Anal. Appl., 3, No 1, Art. 14 (2006), 11 pp. (electronic).

[7] DIBLIK, J.: Anti-Lyapunov method for systems of discrete equations, Nonlinear Anal., 57 (2004),1043–1057.

[8] DIBLIK, J.: Asymptotic behavior of solutions of discrete equations, Funct. Differ. Equ., 11 (2004),37–48.

[9] DIBLIK, J.: Positive and oscillating solutions of differential equations with delay in critical case, J.Comput. Appl. Math. 88 (1998), 185–202.

[10] DIBLIK, J., RUZICKOVA, I., RUZICKOVA, M.: A general version of the retract method for discreteequations, Acta Math. Sin., 23 (2007), No 2, 341–348.

[11] GYORI, I., LADAS, G.: Oscillation Theory of Delay Differential Equations, Clarendon Press (1991).

[12] GYORI, I., PITUK, M.: Asympotic formulae for the solutions of a linear delay difference equation,J. Math. Anal. Appl. 195 (1995), 376–392.

[13] KARAJANI, P., STAVROULAKIS, I.P.: Oscillation criteria for second-order delay and functionalequations, Stud. Univ. Zilina, Math. Ser., 18, No 1, (2004), 17–26.

[14] KIKINA, L.K., STAVROULAKIS, I.P.: A survey on the oscillation of solutions of first order delaydifference equations, Cubo, 7, No 2, (2005), 223–236.

[15] MEDINA, R., PITUK, M.: Nonoscillatory solutions of a second order difference equation of Poincaretype, Appl. Math. Lett. (in the print).

[16] STAVROULAKIS, I.P.: Oscillation criteria for first order delay difference equations, Mediterr. J.Math., 1, No 2, (2004), 231–240.

[17] STAVROULAKIS, I.P.: Oscillation criteria for delay and difference equations, Stud. Univ. Zilina,Math. Ser., 17, No 1, (2003), 161–167.

ADDRESS:

doc. RNDr. Jaromır Bastinec, CSc.Department of MathematicsFaculty of Electrical Engineering and Communication,Brno University of TechnologyTechnicka 8 , 61600 BRNO , Czech [email protected]

prof. RNDr. Josef Diblık, DrSc.Brno University of TechnologyBRNO , Czech [email protected], [email protected]

doc. RNDr. Zdenek Smarda, CSc.Department of MathematicsFaculty of Electrical Engineering and Communication,Brno University of TechnologyTechnicka 8 , 61600 BRNO , Czech [email protected]

GENERATED FUZZY IMPLICATORS

BIBA, Vladislav

Faculty of Electrical Engineering and Communication, Brno University of Technology,[email protected]

Abstract.Conjunctors in MV-logic with truth values range [0, 1] are monotone extensions of the classical con-

junction. Let f : [0, 1] → [0,∞] be a strictly decreasing function, such that f(1) = 0, then we can defineconjunctor C : [0, 1]2 → [0, 1] by

C(x, y) = f (−1)(f(x) + f(y)),

where the pseudo-inverse f (−1) is given by f (−1)(x) = supt ∈ [0, 1]; f(t) > x, f is called an additivegenerator of C. Dual operator to the conjunctor C called the disjunctor. Commonly used disjunctors inMV-logic are the triangular conorms S. A function I : [0, 1]2 → [0, 1] is said to an implicator if and onlyif I(1, 0) = 0 and I(0, 0) = I(0, 1) = I(1, 1) = 1 and I is non-increasing in its first component and non-decreasing in its second component. The unary operator n : [0, 1] → [0, 1] is called negator if for anya, b ∈ [0, 1] holds

a ≤ b ⇒ n(b) ≤ n(a),

n(0) = 1, n(1) = 0.

Starting with the triangular conorm S and standard negation Ns(x) = 1−x, we can introduce the implicationoperator in [0, 1]− valued logic as follows: I(x, y) = S(N(x), y). Another way of extending the classical binaryimplication operator to the unit interval [0, 1] uses the residuation I with respect to the left-continuousconjunctor C

I(x, y) = supz ∈ [0, 1];C(x, z) ≤ y.

There exists several constructions of implicators. We will investigate some genereted implicators and theirproperties.

Key words and phrases. fuzzy implicators, pseudoinverse, generators.

Mathematics Subject Classification. Primary 03B52, 03E72; Secondary 39B99.

1 Preliminaries

We briefly recall definitions of the most important connectives in MV-logic.

Definition 1.1 An unary operator n : [0, 1] → [0, 1] is called a negator if, for any a, b ∈ [0, 1],

• a < b ⇒ n(b) ≤ n(a), and

• n(0) = 1, n(1) = 0.

The negator n is called a strong negator if and only if the mapping n is one to one. Evidently, a strongnegator is continuous and its inverse n−1 is a strong negator too. The negator n is called involutive negatorif and only if n(n(a)) = a for all a ∈ [0, 1]. It can be easily proved that an involutive negator is strong andn−1(a) = n(a).

Definition 1.2 A non-decreasing mapping C : [0, 1]2 → [0, 1] is called a conjunctor if, for any a, b ∈ [0, 1],it holds

• C(a, b) = 0 whenever a = 0 or b = 0, and

• C(1, 1) = 1.

Commonly used conjunctors in MV-logic are the triangular norms.

__________________________________________________________________________________________________________________ ,,ICSC - SEVENTH INTERNATIONAL CONFERENCE ON SOFT COMPUTING APPLIED IN COMPUTER AND ECONOMIC ENVIRONMENTS" EPI Kunovice, Czech Republic. January 23, 2009 175

Definition 1.3 A triangular norm (t-norm for short) is a binary operation on the unit interval [0, 1], i.e.,a function T : [0, 1]2 → [0, 1] such that for all x, y, z ∈ [0, 1], the following four axioms are satisfied:(T1) Commutativity T (x, y) = T (y, x),(T2) Associativity T (x, T (y, z)) = T (T (x, y), z),(T3) Monotonicity T (x, y) ≤ T (x, z) whenever y ≤ z,(T4) Boundary Condition T (x, 1) = x.

Remark 1.4 Note that the dual operator to the conjunctor C, defined by a non-decreasing mapping D :[0, 1]2 → [0, 1] such that D(a, b) = 1 whenever a = 1 or b = 1 and D(0, 0) = 0, is called the disjunctorD. Commonly used disjunctors in MV-logic are the triangular conorms. Triangular conorms (also calledS−norms) are dual to t−norms under the order reversing operation which assigns 1− x to x on [0, 1].

In the literature, especially at the beginnings, we can find several different definitions of fuzzy implications.In this article we will use the following one, which is equivalent to the definition introduced by Fodor andRoubens in [5].

Definition 1.5 A function I : [0, 1]2 → [0, 1] is called a fuzzy implicator if it satisfies the following conditons:

(I1) I is decreasing in its first variable,

(I2) I is increasing in its second variable, and

(I3) I(1, 0) = 0, I(0, 0) = I(1, 1) = 1.

Now, we recall definitions of some important properties of implicators which we will investigate.

Definition 1.6 A fuzzy implicator I : [0, 1]2 → [0, 1] is said to satisfy:

(NP) the left neutrality property, or is said to be left neutral, if

I(1, y) = y; y ∈ [0, 1],

(EP) the exchange principle if

I(x, I(y, z)) = I(y, I(x, z)) for all x, y, z ∈ [0, 1],

(IP) the identity principle ifI(x, x) = 1; x ∈ [0, 1],

(OP) the ordering property ifx ≤ y ⇐⇒ I(x, y) = 1; x, y ∈ [0, 1],

(CP) the contrapositive symmetry with respect to a given negator n if

I(x, y) = I(n(y), n(x)); x, y ∈ [0, 1].

Definition 1.7 Let I : [0, 1]2 → [0, 1] be a fuzzy implication. The function NI defined by NI(x) = I(x, 0)for all x ∈ [0, 1], is called the natural negation of I.

Definition 1.8 A function I : [0, 1]2 → [0, 1] is called an (S, N)−implication if there exist a t-conorm Sand fuzzy negation N such that

I(x, y) = S(N(x), y), x, y ∈ [0, 1].

If N is a strong negation, then I is called a strong implication.

The following characterization of (S, N)−implications is from [1].

Theorem 1.9 For a function I : [0, 1]2 → [0, 1], the following statements are equivalent:

• I is an (S, N)-implication generated from some t-conorm and some continuous (strict, strong) fuzzynegation N.

• I satisfies (I2), (EP) and NI is a continuous (strict, strong, respectively) fuzzy negation.


Another way of extending the classical binary implication operator to the unit interval [0, 1] uses the resid-uation I with respect to a left-continuous triangular norm T

I(x, y) = maxz ∈ [0, 1];T (x, z) ≤ y.

Theorem 1.10 For a function I : [0, 1]2 → [0, 1], the following statements are equivalent:

• I is an R-implication based on some left-continuous t-norm T.

• I satisfies (I2), (OP), (EP), and I(x, .) is a right-continuous for any x ∈ [0, 1].

Our constructions of implicators will make use of extending the classical inverse of function. One way ofextending is described in next definitions.

Definition 1.11 Let ϕ : [0, 1] → [0, 1] be a non-decreasing function. The function ϕ(−1) which is defined by

ϕ(−1)(x) = supz ∈ [0, 1];ϕ(z) < x,

is called the pseudo-inverse of the function ϕ, with the convention sup ∅ = 0.

Definition 1.12 Let f : [0, 1] → [0, 1] be a non-increasing function. The function f (−1) which is defined by

f (−1)(x) = supz ∈ [0, 1]; f(z) > x,

is called the pseudo-inverse of the function f, with the convention sup ∅ = 0.

One of the main contributions of our paper are, in fact, corrollaries of the following technical result.

Proposition 1.13 Let c be a positive real number, then for pseudo-inverse of positive multiple of any left-continuous function f we get

(c · f(x))(−1) = f (−1)(x

c

).

2 New generated implicators

It is possible to generate implicators in a similar way as t-norms. Two of theese posibilities are in the nextexamples.

Example 2.1 Let f1, f2, f3 : [0, 1] → [0, 1] be functions defined as follows:

f1(x) =

1− x if x ≤ 0.5,

0.5− 0.5x otherwise,

f2(x) =1x− 1,

f3(x) = − ln(x).

Now, we will investigate operator If (x, y) : [0, 1]2 → [0, 1] which is given for decreasing function f(x) by

If (x, y) = f (−1)(f(y+)− f(x)).

For our functions f1(x), f2(x), f3(x) we get

If1(x, y) =

1 if x ≤ y,

1− 2x + 2y if x ≤ 0.5, y < 0.5, x− y ≤ 0.25,

0.5 if x ≤ 0.5, y < 0.5, x− y > 0.25,

0.5 if x > 0.5, y < 0.5, x ≤ 2y,

0.5 + y − 0.5x if x > 0.5, y < 0.5, x > 2y,

1− x + y if x > 0.5, y ≥ 0.5,

If2(x, y) =

1 if x ≤ y,

11y−

1x +1

otherwise,

If3(x, y) =

1 if x ≤ y,yx otherwise.

Note that If3 is well-known Godel implicator. We can see that previous mappings are implicators:


Let’s turn our attention to the next example:

Example 2.2 Let g1, g2 : [0, 1] → [0, 1] be given by

g1(x) =

x if x ≤ 0.5,

0.5 + 0.5x otherwise,

g2(x) = − ln(1− x).

Now we will investigate operator Ig : [0, 1]2 → [0, 1], which is given for increasing function g(x) by

Ig(x, y) = g−1(g(1− x) + g(y)).

For functions g1 and g2 we get

Ig1(x, y) =

1− x + y if x ≥ 0.5, y ≤ 0.5, x− y ≥ 0.5,

0.5 if x ≥ 0.5, y ≤ 0.5, 0.25 ≤ x− y < 0.5,

1− 2x + 2y if x ≥ 0.5, y ≤ 0.5, x− y < 0.25,

min(1− x + 2y, 1) if x < 0.5, y ≤ 0.5,

min(2− 2x + y, 1) if x ≥ 0.5, y > 0.5,

1 if x < 0.5, y > 0.5,

Ig2(x, y) = 1− eln(x(1−y)) = 1− x + xy.

Note that Ig2 is Kleene-Dienes implicator. As in previous example, also Ig1 and Ig2 are implicators.

Our investigations we can use and form a theorem generalizing previous examples.

Theorem 2.3 Let f : [0, 1] → [0,∞] be a strictly decreasing function such that f(1) = 0 and g : [0, 1] →[0,∞] be a strictly increasing function such that g(0) = 0. Then functions If and Ig defined as follows

If (x, y) = f (−1)(f(y+)− f(x)),

Ig(x, y) = g(−1)(g(1− x) + g(y)),

are implicators.

We are able to generalize the previous theorem:

Theorem 2.4 Let g : [0, 1] → [0,∞] be a strictly increasing function such that g(0) = 0 and n be a fuzzynegator. Then the function Ig

n:Ign(x, y) = g(−1)(g(n(x)) + g(y)),

is an implicator.

3 Basic properties of generated implicators

First, we will investigate the properties of If implicators:

Proposition 3.1 Let f : [0, 1] → [0,∞] be a left-continuous, strictly decreasing function such that f(1) = 0.Then If satisfies ordering and neutrality properties. Moreover, if f is continuous strictly decreasing functionsuch that f(1) = 0, then implicator If satisfies exchange principle.

It is well known that generators of continuous Archimedean t-norms are unique up to a positive multiplicativeconstant, and this is also true for the f generators of If implicators. The next theorem is a corrollary ofProposition 1.13.

Theorem 3.2 The f generator of an If implicator is uniquely determined up to a positive multiplicativeconstant.

Second, we turn our attention to the implicators Ig and their properties.


Proposition 3.3 Let g : [0, 1] → [0,∞] be a left-continuous, strictly increasing function such that g(0) = 0.Then Ig satisfies neutrality property and it is a contrapositive implicator. Moreover, if g is a continuous,strictly increasing function such that g(0) = 0, then the implicator Ig satisfies the exchange principle.

Theorem 3.4 The generator g of an Ig implicator is uniquely determined up to a positive multiplicativeconstant.

Remark 3.5 Note, that if f(x) = g(1− x), then the implicators If and Ig are identical.

For implicators Ign and their properties we get next propositions.

Proposition 3.6 Let n be a negator, and g be a left-continuous, strictly increasing function such thatg(0) = 0. Then the function Ig

n : [0, 1]2 → [0, 1] which is defined by

Ign(x, y) = g(−1)(g(n(x)) + g(y)),

satisfies the neutrality property.

Proposition 3.7 Let n be an involutive negator, g be a left-continuous, strictly increasing function suchthat g(0) = 0. Then implicator Ig

n is a contrapositive implicator with respect to the negator n.

Proposition 3.8 Let g be a continuous, strictly increasing function such that g(0) = 0. Then implicator Ign

satisfies the exchange principle.

Theorem 3.9 The g generator of an Ign implicator is uniquely determined up to a positive multiplicative

constant.

4 Other properties of our generated implicators

In this section we will investigate properties of natural negations, which are based on our generated impli-cators.

Example 4.1 Let f1(x), f2(x), f3(x) be functions which were mentioned in Example 2.1. Then the naturalnegators NI are given by:

NIf1(x) =

1− 2x if x ≤ 0.25,

0.5 if 0.25 < x ≤ 0.5,

0.5− 0.5x otherwise,

NIf2(x) = NIf3

(x) =

1 if x = 0,

0 otherwise.

Remark 4.2 Note, that Nf2 and Nf3 coincide with Godel negator. More, it is easy to proof, that if f :[0, 1] → [0,∞] is a strictly decreasing function such that f(0) = ∞ and f is right-continuous in x = 0, thenNI(x) given by If (x, y) is Godel negation NG1(x).

Proposition 4.3 Let f : [0, 1] → [0,∞] be a left-continuous, strictly decreasing function, such that f(1) = 0.Then If is a R-implicator based on a conjunctor C, which is given by the additive generator f .

Remark 4.4 Note that Ig = RC∗ , where C∗ is the conjunctor generated by the additive generator f∗ andf∗(x) = g(1− x).

Example 4.5 Let f1(x) = 1 − x, f2(x) = 1 − x2. Pseudoinverse of these functions are: f(−1)1 (x) = 1 − x,

f(−1)2 (x) =

√1− x. Then

NIf1(x) = f

(−1)1 (1− (1− x)) = 1− x,

NIf2(x) = f

(−1)2 (1− (1− x2)) =

√1− x2,

These two negators are involutive and are generated by continuous functions f1, f2. This we can generalizein next theorem.


Theorem 4.6 Let f : [0, 1] → [0, c] (where c is possitive real number) be continuous decreasing functionsuch that f(1) = 0 and f(0) = c. Then negator NIf

is involutive.

Proposition 4.7 Let N(x) be continuous negator, then function N (−1)(1−N(x)) is involutive negator.

This proposition is a corollary of previuos theorem.

Theorem 4.8 The generator f(x) is determined up to a possitive multiplicative constant.

Because of this fact we can divide negators generated by function f(x) into two classes:

1 Negators where f(0) = 1

2 Godel negator (only negator where f(0) = ∞).

Proposition 4.9 Let n be a negator, and g be a left-continuous, strictly increasing function such thatg(0) = 0. Then for Ig

n holds:∀x ∈ [0, 1]; NIg

n(x) = n(x).

Theorem 4.10 Let g : [0, 1] → [0,∞] be a strictly increasing, continuous function, g(0) = 0, and n be acontinuous (strict, strong) negation. Then Ig

n(x, y) = g(−1)(g(n(x)) + g(y)) is (S, N)-implication.

Example 4.11 Let g1(x) = x, g2(x) = − ln(1 − x), then g(−1)1 (x) = min(x, 1), g

(−1)2 (x) = 1 − e−xand Ig

will be:Ig1(x, y) = g

(−1)1 (1− x + y) = min(1− x + y, 1) = ISL

Ig2(x, y) = g(−1)2 (− ln(x)− ln(1− y)) = 1− x + xy = ISP

Let n(x) =√

1− x2, we will get following implicators:

Ig1n (x, y) = g

(−1)1 (

√1− x2 + y) = min(

√1− x2 + y, 1) = SL(n(x), y)

Ig2n (x, y) = g

(−1)2 (− ln(1−

√1− x2)− ln(1− y)) =

√1− x2 + y − y

√1− x2 = SP (n(x), y)

Theorem 4.12 Let g : [0, 1] → [0,∞] be continuous increasing function g(0) = 0 and n(x) be the negator.Then Ig

n(x, y) = S(n(x), y), where S-norm S is generated by g(x).

References

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EXISTENCE AND UNIQUENESS OF SOLUTIONS OFFRACTIONAL DELAY DIFFERENTIAL EQUATIONS

Bretislav Fajmon

Faculty of Electrical Engineering and Communication, Brno University of Technology

Zdenek Smarda


Abstract: The paper deals with the existence and uniqueness of solutions of fractional delaydifferential equations. Moreover, using the Banach fixed point theorem and the Caputo’s frac-tional derivative there are determined conditions of existence of unique monotonic increasingsolution of an investigated equation .

1 Introduction

The definition of the fractional differentiation of the Riemann-Liouville or Grunwald-Letnikov type playedan important role in the development of the theory of fractional derivatives and integrals and for itsapplications in pure mathematics (see [1,4-7]). However, the demands of modern technology require acertain revision of the well-established pure mathematical approach. There have appeared a number ofworks , especially in the theory of viscoelasticity and in hereditary solid mechanics, where fractionalderivatives are used for a better description of material properties.

Applied problems require definitions of fractional derivatives allowing the utilization of physicallyinterpretable initial conditions which contain f(a), f ′(a), etc. Unfortunately, the Riemann-Liouville ap-proach leads to initial conditions containing the limit values of the Riemann-Liouville fractional derivativesat the lower terminal t = a. Here we observe a conflict between the well-established and polished math-ematical theory and practical needs. A certain solution to this conflict was proposed by M.Caputo firstin his paper [2]and two years later in his book [3]. The Caputo’s integral and derivative we will apply toinvestigation of the following initial value problem

d

dtx(t)− f(t, x(t), Dα1

r1x(t− r1), . . . , Dαn

rnx(t− rn))

= g(t, x(t), Dβ1σ1

x(t− σ1), . . . , Dβmσm

x(t− σm)), t > 0, (1)

x(t) = x0, t ≤ 0, (2)

where αi ∈ (0, 1], βj ∈ (0, 1], i = 1, . . . , n, j = 1, . . . ,m and ri, σj are positive constants.

2 Preliminaries

Let L1[a, b] denote the space of all Lebesque integrable functions on the interval [a, b], 0 ≤ a < b < ∞.Let C[0, T ] denote the space of continuous functions on the interval [0, T ]. For x ∈ C[0, T ] we will use thenorm

||x||1 = supt

e−Nt|x(t)|, N > 0.

Definition 2.1. The fractional order integral of the function f ∈ L1[a, b] of order β ∈ R+ is defined by

Iβa f(t) =

∫ t

a

(t− s)β−1

Γ(β)f(s)ds.


Definition 2.2. The Caputo’s definition fractional-order derivative Dα of order α ∈ (0, 1] of the function

g(t) is defined as (see[6])

Dαa g(t) = I1−α

a

d

dtg(t), t ∈ [a, b].

The following properties are some of the main ones of the fractional derivatives and integrals. Letβ, γ ∈ R+ and α ∈ (0, 1). Then

(i) Iβa : L1 → L1, and if f(x) ∈ L1 then Iγ

a Iβa f(x) = Iγ+β

a f(x).

(ii) limβ→n

Iβa f(x) = In

a f(x) uniformly on [a, b], n = 1, 2, . . . , where I1af(x) =

∫ x

af(s)ds.

(iii) If f(x) is absolutely continuous on [a, b] then limα→1

Dαa f(x) = df(x)

dx .

(iv) If f(x) = k 6= 0, then Dαa k = 0.

3 Existence and uniqueness of solution.

Theorem 3.1. Assume that

f : [0, T ]×Rn+1 → R+, g : [0, T ]×Rm+1 → R+

and

|f(t, x0, .x1, . . . , xn)− f(t, y0, y1, . . . , yn ≤ K

n∑i=0

|xi − yi|, K > 0,

(3)

|g(t, x0, .x1, . . . , xm)− g(t, y0, y1, . . . , yn ≤ Lm∑

i=0

|xi − yi|, L > 0.

Then the initial value problem (1),(2) has a unique monotonic increasing solution x ∈ C[0, T ] and dxdt ∈

C[0, T ].

Proof. Put y(t) = ddtx(t) then x(t) = x0 + Iy(t), which implies

x(t− ri) = x0 +∫ t

ri

y(θ − ri)dθ, i = 0, 1, . . . , n.

Thusx(t− ri) = x0 + Iri

y(t− ri),

from which we obtainI1−αiri

d

dtx(t− ri) = I1−αi

riy(t− ri), i = 0, 1, . . . , n.

Similarly we get

I1−βjσj

d

dtx(t− σj) = I1−βj

σjy(t− σj), j = 0, 1, . . . ,m.

Now equation (1) can be written as

y(t) = f(t, x0 + Iy(t), I1−α1r1

y(t− r1), . . . , I1−αnrn

y(t− rn)

+ g(t, x0 + Iy(t), I1−β1σ1

y(t− σ1), . . . , I1−βmσm

y(t− σm). (4)

Define the operator F : C[0, T ] → C[0, T ] by

Fy(t) = f(t, x0 + Iy(t), I1−α1r1

y(t− r1), . . . , I1−αnrn

y(t− rn)

+ g(t, x0 + Iy(t), I1−β1σ1

y(t− σ1), . . . , I1−βmσm

y(t− σm),


then

Fy(t)− Fz(t) = f(t, x0 + Iy(t), I1−α1r1

y(t− r1), . . . , I1−αnrn

y(t− rn)

− f(t, x0 + Iz(t), I1−α1r1

z(t− r1), . . . , I1−αnrn

z(t− rn)

+ g(t, x0 + Iy(t), I1−β1σ1

y(t− σ1), . . . , I1−βmσm

y(t− σm),

− g(t, x0 + Iz(t), I1−β1σ1

z(t− σ1), . . . , I1−βmσm

z(t− σm)

and

e−Nt|Fy(t)− Fz(t)| = e−Nt|f(t, x0 + Iy(t), I1−α1r1

y(t− r1), . . . , I1−αnrn

y(t− rn)

− f(t, x0 + Iz(t), I1−α1r1

z(t− r1), . . . , I1−αnrn

z(t− rn)|+ e−Nt|g(t, x0 + Iy(t), I1−β1

σ1y(t− σ1), . . . , I1−βm

σmy(t− σm),

− g(t, x0 + Iz(t), I1−β1σ1

z(t− σ1), . . . , I1−βmσm

z(t− σm)|

≤ (K + L)∫ t

0

e−Nt|y(s)− z(s)|ds

+ Kn∑

i=1

∫ t

ri

e−Nt (t− s)−αi

Γ(1− αi|y(s− ri)− z(s− ri)|ds

+ L

m∑j=1

∫ t

σj

e−Nt (t− s)−βj

Γ(1− βj)|y(s− σj)− z(s− σj)|ds

which implies that

≤ (K + L)||y − z||1∫ t

0

e−N(t−θ)dθ

+ K||y − z||1n∑

i=1

e−Nri

∫ t−ri

0

e−N(t−ri−θ) (t− ri − θ)−αi

Γ(1− αi)dθ

+ L||y − z||1m∑

j=1

e−Nσj

∫ t−σj

0

e−N(t−σj−θ) (t− σj − θ)−βj

Γ(1− βj)dθ

≤ K + L

N||y − z||1

+ K||y − z||1n∑

i=1

∫ N(t−ri)

0

e−uu−αi

N1−αiΓ(1− αi)du

+ L||y − z||1m∑

j=1

∫ N(t−σj)

0

e−uu−βj

N1−βj Γ(1− βj)du

≤

K + L

N+ K

n∑i=1

1N1−αi

+ Lm∑

j=1

1N1−βj

||y − z||1 < q||y − z||1,

where

q =

K + L

N+ K

n∑i=1

1N1−αi

+ Lm∑

j=1

1N1−βj

.

Now we choose N sufficiently large such that q < 1 then

||fy − Fz||1 < ||y − z||1.

Hence the map f is contraction and has a unique positive fixed point y ∈ C[0, T ] which proves the exis-tence of a unique solution x ∈ C[0, T ] of the intial problem (1),(2). Now the solution of (1) can be writtenas x(t) = x0 + Iy(t) , this implies that dx

dt = y > 0 which proves that dxdt ∈ C[0, T ] and x is monotonic

increasing.


Acknowledgement

This research has been supported by the Czech Ministry of Education in the frames of MSM002160503Research Intention MIKROSYN New Trends in Microelectronic Systems and Nanotechnologies andMSM0021630529 Research Intention Inteligent Systems in Automatization.

Reference

[1] EL-SAYED, A.M.A., ABD EL-SALAM, S.A.: On the stability of some fractional-order non-autonomous systems, EJQTDE, No. 6, (2007), 1-14.

[2] CAPUTO, M.: Linear model of dissipation whose Q is almost frequency independent -II, Geophys.J.R. Astr. Soc., vol. 13, 1967, 529-539.

[3] CAPUTO, M.: Elasticita e Dissipazione, Zanichelli, Bologna, 1969.

[4] KRUPKOVA, V., SMARDA, Z.: On Some Properties of Fractional Calculus. In Proceedings ofIV. International Conference on Soft Computing Applied in Computer and Economic Enviroment,European Polytechnical Institute Kunovice, (2006), 157-162.

[5] KRUPKOVA, V., SMARDA, Z.: Discretization of the Gruunwald-Letnikov fractional derivative andintegral, In Proceedings of Abstracts and reviewed contributions on CDROM , XXIV. InternationalConference on Education Process, UNOB Brno (2006).

[6] PODLUBNY, I.: Fractional differential Equations, Mathematics in Science and Engineering , Aca-demic Press, vol.198, 1999.

[7] YU,G., GAO, G.: Existence of fractional differential equations, J.Math. Anal. Appl.310, (2005),26-29.


ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF CERTAINCLASSES OF NONLINEAR SINGULARINTEGRODIFFERENTIAL EQUATIONS

Olga Filippova


Zdenek Smarda


Abstract: The paper deals with the asymptotic behaviour of solutions of nonlinear integrod-ifferential equations with a separable kernel in a neighbourhood of the singular point [0+, 0].Solutions of above mentioned equations are constructed in the form of asymptotic expansionsfor x→ 0+ with respect to a general solution of a certain ordinary differential equation.

1 Introduction

The existence and asymptotic behaviour of solutions of ordinary differential equations and integrodif-ferential equations in a neighbourhood of a singular point have been devoted many papers (see [1-10]).Diblık[2] investigated a singular initial problem for implicit ordinary differential equations. The aim ofthis paper is to show that results of paper [2] for ordinary differential equations is possible to extend onintegrodifferential equations with a separable kernel in the form

g(x)y′ = y +∫ x

0+

N∑i+j=1

uij(x)vij(s)yi(x)yj(s)

ds, (1)

where uij(x), vij(x) ∈ C(J), J = (0, x0], x0 > 0 is a sufficiently small. First we recall some importantnotions which we will use in the following considerations:

(i) f(x) = O(g(x)) as x → x+0 denotes that there exists a constant K > 0 such that

∣∣∣ f(x)g(x)

∣∣∣ < K onsome right-hand neighbourhood of the point x0.

(ii) f(x) = o(g(x)) as x→ x+0 denotes that lim

x→x+0

f(x)g(x) = 0.

(iii) f(x) ∼ g(x) denotes that limx→x+

0

f(x)g(x) = 1.

(iv) A sequence of functions (φn(x))∞1 will be called an asymptotic sequence if for all n there is validφn+1 = o(φn(x)) as x→ x+

0 .

(v) A formal series∑anφn(x), n ∈ N, an ∈ R will be called an asymptotic expansion of a function

h(x) as x→ x+0 to the M -th member if the following conditions hold:

a) (φn(x))∞1 is the asymptotic sequence as x→ x+0 .

b)[h(x)−

∑Mn=1 anφn(x)

]= O(φn(x)) as x→ x+

0 .

The asymptotic expansion of the function h(x) will be indicated h(x) ≈∑M

n=1 anφn(x).

(vi) A point [x0, y0] in which conditions of existence and uniqueness of a solution of equation (1) arenot valid will be called a singular point with respect to equation (1).


2 Asymptotic expansion of a solution of equation (1).

We will construct the solution of (1) in the form of one parametric asymptotic expansion

y(x,C) =∞∑

h=1

fh(x)ϕh(x,C), (2)

where ϕ(x,C) is the general solution of the differential equation

g(x)y′ = y

so that

ϕ(x,C) = C exp[∫ x

x0

dt

g(t)

]and f1(x) ≡ 1, fh(x), h ≥ 2 are unknown functions, C 6= 0 is a constant.Consider the following differential equation

g(x)y′ = qy + p(x). (3)

Diblık[2] proved asymptotic estimates of the solution of (3) which we can formulate as follows:

Theorem 2.1. Assume that

I) Let q be a constant, q < 0, g(x) ∈ C1(J), g(x) > 0, limx→x+

0

g(x) = 0, g′(x) ∼ ψ1(x)gλ1(x) as

x→ x+0 ,

λ1 > 0, limx→x+

0

ψ1(x)gτ (x) = 0, τ is any positive number.

II) p(x) ∈ C(J), p(x) = b0(x)gλ(x) + O(b1(x)gλ+ε(x)), ε > 0, limx→x+

0

bi(x)g′(x) = 0, i = 0, 1, b0(x) ∈

C(J),b0(x) 6= 0, b′0(x) ∼ ψ2(x)gλ2(x) as x→ x+

0 , λ2+1 > 0, limx→x+

0

ψ2(x)gτ (x) = 0, limx→x+

0

gτ (x)(b0(x))−1 =

0.

Then equation (3) has a unique solution on J satisfying asymptotic estimates

y(x) =−1qb0(x)gλ(x) +O(gν(x)), y′(x) = O(gν−1(x)), (4)

where ν ∈ (λ, λ+ minλ1, λ2 + 1, ε).

Now we will show that results in Theorem 2.1. concerning only differential equation (3) we can apply tointegrodifferential equation (1).Consider integrodifferential equation (1) in the form

x3y′ = y +∫ x

0+xy2(x)y(s)ds. (5)

In this case i = 2, j = 1, u21(x) = x, v21(s) = 1, ϕ(x,C) = C exp(

12x2

0− 1

2x2

).

Let us demonstrate a construction of an asymptotic expansion of a solution of equation (5) only to thethird order. Then

y = ϕ(x,C) + f2(x)ϕ2(x,C) + f3(x)ϕ3(x,C). (6)

,

y′ =ϕ(x,C)g(x)

+ ϕ2(x,C)(f ′2(x) + 2

f2(x)g(x)

)+ ϕ3(x,C)

(f ′3(x) + 3

f3(x)g(x)

). (7)

y2(x)y(s) = ϕ2(x,C)ϕ(s, C). (8)


Substituting (6-8) into (5) we get

ϕ+ (x3f ′2 + 2f2)ϕ2 + (x3f ′3 + 3f3)ϕ3 = ϕ+ f2ϕ2 + f3ϕ

3 +∫ x

0+xϕ2(x,C)ϕ(s, C)ds. (9)

Comparing of coefficients at the same powers of ϕ we obtain:

ϕ1 : 1 = 1 (10)ϕ2 : x3f ′2 + 2f2 = f2 (11)

ϕ3 : x3f ′2 + 3f3 = f3 + ϕ−3(x,C)∫ x

0+xϕ2(x,C)ϕ(s, C)ds (12)

For uknown coefficients f2(x), f3(x) we get the following equation

x3f ′2 = −f2. (13)

x3f ′3 = −2f3 + xϕ−1(x,C)∫ x

0+ϕ(s, C)ds. (14)

From (13) we get f2(x) = [ϕ(x,C)]−1. Put

u = ϕ−1(x,C)∫ x

0+ϕ(s, C)ds. (15)

Differentiating (15) we obtain the differential equation

x3u′ = −u+ x3

which has the form of equation (3). Now we verify assumptions of Theorem 2.1. It is obvious that

g(x) = x3, q = −1, p(x) = x3

so thatg′(x) = 3x2 ∼ ψ1(x)gλ1(x) = 3(x3)

23 ⇒ λ1 =

23, ψ1(x) = 3

limx→0+

ψ1(x)gτ (x) = limx→0+

3(x3)τ = 0

where τ is any positive constant. Now we verify assumption II):

p(x) = b0(x)gλ(x) + 0(b1(x)gλ+ε).

Hence x3 = 1(x3)1 ⇒ b0(x) = 1, λ = 1, b1(x) = 0, so we can choose ε > 23 and as b′0(x) = 0 then ψ2(x) = 0

so that we can choose a constant λ2 + 1 > 23 .

Thence it is clear that limx→x+

0

ψ2(x)gτ (x) = 0, limx→x+

0

gτ (x)(b0(x))−1 = 0. Hence ν ∈ (1, 1 + min2/3, λ2 +

1, ε.From here we get ν ∈ (1, 5

3 ) and

u = x3 + 0(x3ν), ν ∈ (1,53)

Now equation (14) has the form

x3f ′3 = −2f3 + x4 +O(x3ν+1), ν ∈ (1,53) (16)

Applying Theorem 2.1. to equation (16) we obtain λ1 = 2/3, furthermore p(x) = x4 +O(x3ν+1) so thatb0(x) = 1, λ = 4

3 , b1(x) = 1. Put ε = ν − 1, ψ2(x) = 0. We can choose λ2 + 1 > 23 . Hence ν1 ∈ ( 4

3 , 2) and

f3(x) =12x4 +O(x3ν1),


where ν1 is a constant with respect to equation (16). Then the asymptotic expansion of the third orderwith respect to equation (5) has the form

y(x,C) ≈ 2ϕ(x,C) +(

12x4 +O(x3ν1)

)ϕ3(x,C).

Acknowledgement


Reference

[1] DIBLIK, J., The singular Cauchy-Nicoletti problem for system of two ordinary differential equations,Math. Bohem. 117 (1), (1992), 55-67.

[2] DIBLIK, J., Asymptotic behaviour of solutions of certain differential equations partially solvable withrespect to derivative. Sib. Mat. Journal, No.5, 1982, 80-91. (in Russian)

[3] DIBLIK, J., RUZICKOVA, M., Existence of positive solutions of a singular initial problem for anonlinear system of differential equations, Rocky Mountain Journal of Mathematics 34, (2004),923-944.

[4] DIBLIK, J., NOWAK, C., A nonuniqueness criterion for a singular system of two ordinary differ-ential equations, Nonlinear Analysis, 64 (2006), 637-656.

[5] DIBLIK, J., On existence of solutions of singular Cauchy problem for system of differential equationsnonsolvable to a derivation, Publ. Inst. Math. 37(51), (1985), 73-80.

[6] KIGURADZE, I.T., Some singular boundary value problems for ordinary differential equations, Izd.Tbilisi Univ., 1975 (in Russian).

[7] KONJUCHOVA, N.B., Singular Cauchy problems for systems of ordinary differential equations,Zurnal Vycisl. Matematiki i Fiziki, No 5, (1981), 629-645 (in Russian).

[8] SMARDA Z., On an initial value problem for singular integrodifferential equations, DEMONSTRA-TIO MATHEMATICA, Vol. XXXV, No 4, (2002), 803-811.

[9] SMARDA, Z., Existence and uniqueness of solutions of nonlinear integrodifferential equations, Jour-nal of Applied Mathematics, Statistics and Informatics, 2, (2005), 73-79.

[10] SMARDA Z., On solutions of an implicit singular system of integro-differential equations dependingon a parameter, DEMONSTRATIO MATHEMATICA, Vol.XXXI, No 1, (1998), 125-130.


P-SEMIHYPERGROUPS OF PREFERENCE RELATIONSAND ASSOCIATED BITOPOLOGIES

Jan Chvalina, Michal NovakFaculty of Electrical Engineering and Communication, Brno University of Technology

Abstract: Preference relations on sets of alternatives of a very general nature have beenstudied in a number of papers. This contribution is devoted to constructions of semihyper-groups and hypergroups of preference relations, in particular tolerances, based on certain P–hyperoperations which generalize one valued sandwich operations on semigroups. Basic separa-tion axioms in bitopological spaces formed by left and right topologies associated to preferencesare also studied.

Key words: bitopological space, hypergroup, preference relation, semihypergroup, tolerance.

The concept of preference is the foundation of all choice theory in economics. In particular in microe-conomics, preferences of consumers and other entities are modelled with preference relations. Much ofmodern microeconomic theory arises from thinking about preferences over things like political parties,environmental policies, business strategies, local decisions etc., where the underlying set M of possiblealternatives could be very general (cf e.g [24]). Preference relations are sometimes considered in a rathergeneral background as arbitrary binary relations on sets of alternatives. A list of interesting papers onthis topic includes [1], [3], [8], [9], [13], [20], [21].When forming the concept of preference relation a number of classical constructions – those used in algebraof relations and their modifications – turn to be useful. Multistructures formed by preferences have alsobeen studied. As has been mentioned in paper [23], a (weak) preference relation on a nonempty choice setX is traditionally defined as a complete, reflexive and transitive binary relation, i.e. a complete preorder onX. All of these assumptions are commonly considered rationality postulates, the later being consistencyrequirements and the former a decisiveness prerequisite. As it is of interest to model the behaviourof boundedly rational individuals in economics, the implications of relaxing the axioms are naturallymotivated. Moreover, the paper [23] reads: ”there is another reason why incomplete preferences can beviewed as interesting. Such preferences provide a conservative modelling technique in those situations inwhich the modeller has only partial information about the preferences of an agent.” It is also to be notedthat there are many economic instances in which a decision maker is in fact a committee, i.e. individualchoices are based on social preferences which are naturally modelled as incomplete.In the contribution we are going to use concepts from the hyperstructure theory. Therefore, recall nowsome of its basic definitions and concepts. A hypergrupoid is a pair (H, •), where H 6= 0 and • : H×H →P∗(H) is a binary hyperoperation on H. Symbol P∗(H) denotes the system of all nonempty subsets ofH. If the associativity axiom a • (b • c) = (a • b) • c holds for all a, b, c ∈ H, then the pair (H, •) is called asemihypergroup. If moreover the reproduction axiom: for any element a ∈ H equalities a•H = H = H •ahold, is satisfied, then the pair (H, •) is called a hypergroup.In the algebraic theory of hyperstructures there were introduces and studied so called P–semihypergroupsand P–hypergroups (cf. [29] and related papers). The concept is a generalization of the notion of a vari-ant of a semigroup or a sandwich semigroup. In the case of sandwich semigroups of binary relations ona set there exists a close connection to the concept of a relator (cf. [27]). Relators are simply nonvoidcollections of reflexive relations on sets. Theory of relators (essentially identical to the generalized uni-formities of I. Konishi – 1952 and V. S. Krishnan – 1955) generalizing various uniformities (A. Csaszarand R. Z. Domiaty – 1979/80, P. Fletcher and W. F. Lindgren – 1982, W. F. Lindgren and P. Fletcher –1978, N. Levine – 1970, 1973, M. G. Murdeshwar and S. A. Naimpally – 1966, W. Page – 1978) has beenintensively studied by Arpad Szas since the end of 1980s in a series of papers and in his monographyRelators, nets and integrals. We explain the above mentioned concepts on the example of a semigroup ofpreference relations on a set of some alternatives.


Let R be a fixed binary relation on a set X. If for any two relations A and B in B(X), which is the setof all binary relations on X, we define A ∗ B as ARB, where juxtaposition is the usual composition ofrelations, we obtain the sandwich semigroup (BR(X), ∗) of binary relations on the set X with sandwichrelation R (cf. [28]). These semigroups were studied by K. Chase in 1978/9 so that he could apply themin the automata theory. Now, considering a nonempty subset P ⊆ B(X) – in the case that all relationsA ∈ P are reflexive, the set P is called relator (cf. [10], [27]) – and defining a binary hyperoperation onB(X) by A P B = APB = ARB;R ∈ P we obtain a P–hypergroup (cf. [29]). Indeed, it is easy to seethat for any triad of relations A,B, C ∈ B(X) there holds

A P (B P C) = APBPC = ASBTC;S, T ∈ P = (A P B) P C,

i.e. that the so called central P–hyperoperation is associative.The contribution [16] ends with the following remark which is relevant in our further considerations:

Remark 1. Let us conclude with a remark that the arbitrary set R(M) ⊂ P(M × M) of preferencerelations on a commodity set M can be endowed with a commutative hypergroup. Being a subset of apower set P(M ×M) of all binary relations on M , the system R(M) of preference relations is naturallypartially ordered by set inclusion ”⊆”. Then defining

R ∗ S = T ;T ∈ R(M), R ⊆ T or S ⊆ T

we get easily that the pair (R(M), ∗) is a commutative hypergroup (cf also [12]).

Let us further denote by Toc(X) a system of commuting tolerances on a set X. Notice that commutingrelations have been studied in some papers by Tamas Glavosits and Arpad Szasz in e.g. [10], [27] andin particular in the paper Characterizations of commuting relations (Institute of Mathematics, DebrecenUniversity – preprint, 9pp) by Szasz, where Theorem 2 states that if R,S are tolerances, i.e. reflexive andsymmetric relations, on X, then the following assertions are equivalent:

(1) RS = SR,

(2) RS is a tolerance,

(3) R(x) ∩ S(y) 6= ∅ implies S(x) ∩R(y) 6= ∅ for all x, y ∈ X.

In fact the author of the first result concerning commutativity of equivalences is Frantisek Sik, Spisyvyd. Prır. fak. Masarykovy Univ. 3 (1954), 97–102. Some generalizations have been obtained by LadislavKosmak in Acta Math. Univ. Comenianae 1980. In what follows we suppose that Toc(X) is a commutativesemigroup of tolerances and P is a subsemigroup of Toc(X). Denote by S the semiring of all evenpositive integers. Define on Toc(X) a binary relation ρP in the following way: For R,S ∈ Toc(X) we put[R,S] ∈ ρP or simply R ρP S whenever there exists an even integer n ∈ S such that

S ∈ Rn2 P R

n2 = Rn P = Rn T ;T ∈ P = P Rn.

Proposition 1. Let Toc(X) be an abelian semigroup of tolerances on X and P its subsemigroup. Thenthe binary relation ρP ⊂ Toc(X)× Toc(X) is transitive.

Proof. Suppose R,S, T ∈ Toc(X) are tolerances such that RρPS, SρPT , i.e. there exist positive evenintegers m,n ∈ S such that

S ∈ RmP, T ∈ SnP.

This means that for a suitable pair of tolerance relations U, V ∈ P we have

S ∈ RmU, T ∈ SnU,

which implies

T = (RmU)nV = RmURmU . . . RmUV = (Rm)nUnV = RmnUnV ∈ RmnP.

Since m,n ∈ S, we have that RρPT , hence the relation ρP is transitive.


Further, denoting 4T = [R,R];R ∈ Toc(X) – the identity (or diagonal) relation on the semigroupToc(X) – we obtained (using Theorem 2.1 from [12]) the following result:

Theorem 1. Let Toc(X) be an abelian semigroup of tolerance relations on X, P be its subsemigroupand ρP be the above defined relation on Toc(X). Denote σP = ρP ∪ 4T and for any pair of tolerancesR,S ∈ Toc(X) define

R ∗ S = σP(R) ∪ σP(S) = T ∈ Toc(X);RρPT or SρPT ∪ R,S.

Then (Toc(X), ∗) is a commutative, i.e. abelian, hypergroup of tolerances.

In a certain more general approach, let us consider a commutative semigroup S and ∅ 6= P ⊂ S.

Theorem 2. Let (S, ·) be a commutative monoid (with the unit e) of idempotent elements, i.e. (S, ·) is acommutative band. Suppose P ⊂ S is a submonoid of S (i.e. its carrier set) and RP ⊂ S × S is a binaryrelation defined by [x, y] ∈ RP iff y ∈ x · P · x = x · P. Then the relation RP is reflexive and transitive,i.e. it is a quasi-order on the set X.

Proof. Suppose x, y, z ∈ S are arbitrary elements. Since x = x·e ∈ x·P = RP(x), we have that [x, y] ∈ RP ,which means that the relation RP is reflexive. Now suppose [x, y] ∈ RP , [y, z] ∈ RP , i.e. y ∈ x·P, z ∈ y ·P.Then there exist elements u, v ∈ P such that y = x · u, z = y · v = x · u · v ∈ x · P, thus [x, z] ∈ RP .Consequently the relation RP is a quasi-order on S.

As far as the theory of representation of preferences is concerned, continuous representations are also im-portant. Therefore, in a substantial part of the theory of preferences, metric spaces or generally topologicalspaces serve as structural background.In what follows now we present some results concerning separation properties of topologies determined byreflexive preferences on sets of alternatives (we suppose no restricting conditions concerning cardinalitiesof the underlying sets). If X is such a set, R ⊂ X×X, then the pair (X, R) called a monorelation systemcan be also regarded as a directed graph, or a digraph. For an element x ∈ X we put (as usually)

R(x) = y ∈ X; [x, y] ∈ RR−(x) = y ∈ X; [y, x] ∈ R.

In the terminology of the graph theory in any ordered pair [x, y] x is called initial vertex and y is calledterminal vertex. An R–path (of the length n) is a finite chain γ = (u1, u2, . . . , un) of arcs in which theterminal vertex of arc ui is the initial vertex of arc ui+1 for all i < n. In our case of the monorelationalstructure (X, R), which is a 1–graph any path is completely determined by the sequence of verticesx1, x2, . . . xn+1. A path with initial vertex x and terminal vertex y will be denoted by γ[x, y]. An R–circuit is a path γ = u1, . . . , un such that no arc appears more than once in the sequence, u1 = un,and for all i < n the terminal vertex of ui is the initial vertex of ui+1.Now we are going to define two topologies on X determined by the transitive closure R of the preferenceR. A subset A ⊆ X is said to be R−–saturated if for every pair a, b ∈ X we have that b ∈ A, a 6∈ Aimplies a 6∈ R−(b). Put now

T −R = A ⊆ X;A is R−–saturated ∪ ∅.

It is easy to verify that T −R is a topology on the set X with a totally additive closure operation (i.e., theintersection of any system of open subsets is an open set). The topological space (X, T −R ) is called leftassociate of the preference R, where for A ⊆ X

d−(A) = b ∈ X;∃ a path γ[a, b] for some a ∈ A,

i.e. the closure of the set A in the space (X, T −R ). Denote by T +R the so called dual topology to T −R . Then

closures d+(A) are the least T −R –open sets containing the set A, i.e.

d+(A) = b ∈ X;∃ a path γ[b, c] for some c ∈ A.

The topology T+R formed by R–saturated sets is said to be a right associated topology of the preference

relation R. In paper [15] some separation properties of spaces (X, T −R ) and (X, T +R ) are characterized. Now


we are going to consider an important interesting topological structure, which has been intensively studiedsince 1960 (cf. [7]) – bitopological spaces created by the above topologies T −R , T +

R . A bitopological spaceis a set X endowed with two topologies T1, T2, i.e. a triad (X, T1, T2). A nice survey of separation axiomsfor bitopological spaces is given in [25] and [26]. Since there exist various modifications of generalizationsof separation axioms for bitopological spaces, we now recall necessary concepts.A bitopological space (X, T1, T2) is said to be pairwise T0 if for each pair of distinct points there existseither a T1– or a T2– neighboroughood of one point not containing the other point. A space (X, T1, T2) issaid to be pairwise T1 if for every pair of distinct points x, y there exists a T1– or a T2– neighboroughhoodof x not containing y; it is called pairwise TD, if the intersection of the T1–derivative and the T2–derivativeof a singleton x is T1–closed as well as T2–closed for each x ∈ X. Obviously being pairwise T1 impliesbeing pairwise TD, which in turn implies being pairwise T0. A bitopological space (X, T1, T2) is said tobe pairwise T2 (or pairwise Hausdorff ) if for each pair of distinct points x, y ∈ X there exists a T1–neighboroughood U and a T2–neighboroughood V such that x ∈ U, y ∈ V and U ∩ V = ∅. The followinggeneralizations of this separation axiom can be found in related works:A bitopological space (X, T1, T2) is called pairwise semi T2 if for every pair of distinct points x, y ∈ Xthere exists a T1–semi open set U and a T2–semi open set V such that x ∈ U, y ∈ V and U ∩ V = ∅. Thebitopological space is called weakly pairwise T2 if for every pair of distinct points x, y ∈ X there existsa T1–open set U and a T2–open set V such that U ∩ V = ∅ and either x ∈ U, y ∈ V or x ∈ V, y ∈ U .Furthermore, the bitopological space is called quasi-Hausdorff if for every pair of distinct points x, y ∈ Xthere exists a pair of disjoint quasi-open sets U, V such that x ∈ U, y ∈ V . Notice that a subset Aof (X, T1, T2) is said to be quasi-open if for every x ∈ A there exists a T1–open neighboroughood Ux

contained in A or a T2–open neighboroughood Vx contained in A. For details on the bitopological spacestheory cf. [7], [25], [26].

Proposition 2. Every bitopological space which is weakly pairwise T2 is quasi-Hausdorff and every quasi-Hausdorff bitopological space is pairwise T1.

For proof cf. [15]. It is to be noted that in general the converse implications are not valid. This followsfrom two examples included in [15].

Theorem 3. Let (X, R) be a digraph, T +R , T −R associated topologies. The following statements are equiv-

alent:

1. (X, R) does not contain any circuit.

2. (X, T +R , T −R ) is pairwise T0.

3. (X, T +R , T −R ) is pairwise TD.

4. (X, T +R , T −R ) is pairwise T1.

5. (X, T +R , T −R ) is quasi-Hausdorff.

6. (X, T +R , T −R ) is weakly pairwise T2.

Proof. When applying Proposition 2 we get that statement 6 implies 5 and 5 implies 4. From the definitionof corresponding separation axioms we also get that 4 implies 3 and 3 implies 2. Therefore it is sufficientto prove that 1 implies 6 and 2 implies 1.Let (X, R) be a digraph without circuits, and x, y ∈ X distinct vertices. If x, y belong to differentcomponents of (X, R), then evidently these components form a pair of disjoint quasi-open subsets of thespace (X, T +

R , T −R ). Suppose both vertices x, y belong to the same component. The following three casesare possible: (i) y ∈ R

+(x), (ii) y ∈ R

−(y), (iii) y 6∈ R(x).

Suppose (i) holds. Since (X, R) does not contain any circuits, we have that y 6∈ R−

(x), x 6∈ R+(y) and

R−

(x) ∩ R+(y) = ∅. Put A = R

−(x) ∪ x, B = R

+(y) ∪ y. Then A ∩ B = ∅ and by the definition

of associated topologies we have that A is T −R –open and B is T +R –open. Suppose now that either (ii) or

(iii) holds. Putting A = R+(x) ∪ x, B = R

−(y) ∪ y we have that A ∩ B = ∅, A is T +

R –open and Bis T −R –open. Consequently (X, T +

R , T −R ) is a weakly pairwise T2 space, hence the implication 1 =⇒ 6 isvalid.Assume that (X, T +

R , T −R ) is a pairwise T0 space and admit that the digraph (X, R) contains a circuit,which is not a loop. Denote the circuit C. Consider vertices x, y ∈ C such that y ∈ R+(x). Then x ∈ R

+(y)


and the smallest T −R –neighboroughood of x contains y. The same holds for the other point y. This is acontradiction, though. Therefore the implication 2 =⇒ 1 holds too.

Remark 2. If a bitopological space (X, T +R , T −R ) is pairwise T2, then T1 and T2 are T1–topologies

(cf. [26], p. 280). Thus (X, T +R , T −R ) is pairwise T2 if and only if it is discrete. Since for a quasi-discrete

space the concepts of a semi-open set and an open set coincide, we may conclude that the bitopologicalspace (X, T +

R , T −R ) is pairwise semi T2 if and only if it is discrete.

Remark 3. The issue of some special properties of the bitopological space with the carrier set Toc(X)associated with the quasi-order σP depending on properties of P ⊂ Toc(X) seems to be an open question.

ACKNOWLEDGEMENT

The authors were supported by the Council of Czech Government grant MSM 0021630529.

Reference

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[3] ARROW, K. J. Rational choice functions and orderings. Econometrica 26 (1959), 21–127.

[4] CORSINI, P. Prolegomena of Hypergroup Theory. Tricesimo, Aviani Editore, 1993.

[5] CORSINI, P.; LEOREANU, V. Applications of Hyperstructure Theory. Dordrecht, Kluwer AcademicPublishers, 2003.

[6] DRESHER, M.; ORE, O. Theory of multigroups. Amer. J. Math. 60 (1938), 705–733.

[7] DVALISHVILLI, B. L. Bitopological Spaces Theory. Relations with Generalized Algebraic Structuresand Applications. North Holland Mathematical Studies 199, Amsterdam, Boston; Elsevier 2005.

[8] FIALA, P., JABLONSKY, J.; MANAS, M. Multikriterialnı rozhodovanı. VSE v Praze, Fakultainformatiky a statistiky, Praha 1994.

[9] FODOR, J. Preference relations in decision models. OTKA TO 46762 and BSTC Flanderes – Hun-gary BIL 00/51. 2000, 7pp. Preprint.

[10] GLAVOSITS, T.; SZASZ, A. Decomposition of commuting relations. Acta Math Inform. Univ. Os-trava 11 (2003), 25–28.

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ADDRESS:

Prof. RNDr. Jan Chvalina, DrSc.Department of MathematicsFaculty of Electrical Engineering and Communication,Brno University of TechnologyTechnicka 8, 61600 BRNO, Czech [email protected]

RNDr. Michal Novak, Ph.D.Department of MathematicsFaculty of Electrical Engineering and Communication,Brno University of TechnologyTechnicka 8, 61600 BRNO, Czech [email protected]


ASYMPTOTIC BEHAVIOR OF ONE AUXILIARY SYSTEM

Zdenek Svoboda


Abstract: The first Lyapunof method is very useful tool of investigation of the differentialequation with righthand side in the polynomial form. The solution of the differential equationis expressed as power series with coefficients which are solutions of suitable auxiliary system ofdifferential equations. This contribution deals with the asymptotic properties of this auxiliarysystem which is obtained for one type functional differential equation with delay.

Key words: Delayed differential equation, characteristic equation

1 MOTIVATION

In this contribution some properties of one auxiliary system of functional differential equation with delayare studied. This system we obtain at the application of the first Lyapunov method to one differentialequation with delay which has polynomial righthand side. This method is a well known technique for studyof asymptotic behavior of ordinary differential equations in the form of a linear system with perturbation.In this method we suppose the solution in the form of convergent power series and the coefficients atthe power of solution of the linear equation without perturbation are given as solutions of the auxiliarysystem of linear differential equations, for details see [1]. Modifying this method such that we supposethe solution in the form a formal power series is possible obtain the results for equations in the implicitform [2] or for integro-differential equations [10]. The existence of solutions with concrete asymptoticform were proved in cited results using the Wazewski’s topological method. Analogous representationsof solutions for retarded differential equation were derived in [7], [9]. In these papers were obtain resultsfor equations which contain no delay linear part. The aim of this contribution is describe the solutions ofthe auxiliary system of equation which contain the delayed linear part i.e. equation

y′(t) = −a(t)y(t) + b(t)y(t− r) +N∑

i+j=2

cij(t)yi(t)yj(t− r), (1)

where N ≥ 2 is an integer, r is a positive constant and functions a, b and cij are continuous on theinterval [0,∞). Moreover, a and b are positive and there exists a constant ε > 0 such that a(t) ≥ ε+λ(r)where

λ(r) = lim supn→∞

n√

An(r), An(r) =n∑

k=0

(n− k)krk

k!,

a(t) = o(∫

a(t)dt)

, |b(t)| ≤ exp

− t∫t−r

a(s)ds

.

We note that this assumption assure that the solution of the linear equation without the perturbation

y′(t) = −a(t)y(t) + b(t)y(t− r)

is possible estimate by solution of equation

y′(t) = −a(t)y(t) + exp

− t∫t−r

a(s)ds

y(t− r). (2)


This solution has form y(t) = exp(−∫ t

0a(s)ds

)z(t) where z(t) is the solution of equation z′(t) = z(t−r).

For this type of equation is known, that for all unbounded solution z1(t), z2(t) on [0,∞) holds relation

z2(t) = Kz1(t) + o(z1(t))

for t → ∞, where K is constant. The function z1(t) = exp(λt) is monotone solution, where λ is realpositive solution of semicharacteristic equation:

λ = exp(−λr). (3)

This solution is single and has form:

λr = lim supn→∞

n√

An(r), An(r) =n∑

k=0

(n− k)krk

k!.

This facts are arguments for selection of the function

ϕ(t, C) = C exp(−∫ t

0(a(s)− λr)ds

). (4)

Due to a(t) ≥ ελr for positive ε and a(t) = o(∫

a(t)dt) holds a(t)− λr = o(∫

a(t)− λrdt) and

ϕ(t− r, C) ∼ ϕ(t, C).

2 Auxiliary system

Demand of the exactly description of functions in the auxiliary system is reason for imposition thenotation below.

2.1 Basic notation

For specification of the coefficients of power series raised to the power are suitable sequences of nonnegativeintegers with finite sum. This idea is generalization of well-known concept of multiindexes and for theproduct of two power series raised to the power are suitable an ordered couple of these sequences.

We denote : s = (s1, s2) is an ordered couple of sequences sj =skj

∞k=1

of nonnegative integers with

finite sum |sj | =∞∑

k=1

skj and moreover |s| denotes the couple of integers |s| = (|s1|, |s2|). For any couple

of sequences (of numbers or functions) C = (c1, c2), where cj = ckj ∞k=1 and s is defined

Cs =2∏

j=1

∞∏k=1

(ckj

)skj , where

(ckj

)0 = 1 for every ckj .

Now we denote two numbers depending on the couple of sequences s = (s1, s2) :

V (s) =2∑

j=1

∞∑k=1

kskj , s! =

2∏j=1

∞∏k=1

skj !.

Moreover the symbol∑

|s|=(i,j)V (s)=k

denotes the sum over all couples of sequences s satisfying :

V (s) = k, |s| = (i, j)

. Acceptance this notation enables to relative simply expressed the coefficients of product of two powerseries raised to a power. Then it is possible to write:( ∞∑

k=1

ck1xk

)i( ∞∑k=1

ck2xk

)j

=∞∑

k=i+j

xk∑

|s|=(i,j)V (s)=k

i!j!s!Cs,


2.2 Application of the first Lyapunov method

Bellow be studied the equation which has the linear part of righthand side in the form of the equation (2)i.e.

y′(t) = −a(t)y(t) + exp

(t∫

t−r

− a(s)ds

)y(t− r) +

N∑i+j=2

cij(t)yi(t)yj(t− r). (5)

Now we assume that the solution of this equation has the form of formal series

y(t) =∞∑

n=1

fn(t)ϕn(t, C). (6)

The function ϕ(t, C) = C exp(−∫ t

0(a(s) + λr)ds

)is solution of the equation (2) without perturbation

and f1(t) ≡ 1. The other functions fn(t) are unknown functions. For these we obtain the auxiliary systemof functional differential equations with delay such that we substitute the series (6) in the equation (2)andafter rearrangement we compare coefficients at the same powers of ϕk(t, C).

f ′n(t) = ((n− 1)a(t)− nλr)fn(t) + exp

(t∫

t−r

((n− 1)a(t)− nλr)ds

)fn(t− r) +

∑n(t), (7n)

where∑

n(t) =N∑

i+j=2

ci,j(t)∑

|s|=(i,j)V (s)=n

i!j!s!Fs, and F(t) =

(fn(t)∞n=1,

fn(t− r)

ϕn(t− r, C)ϕn(t, C)

∞n=1

).

At the first we note that this auxiliary system is recurrent with respect the system of solutions fn(t)due to fact that the conditions s1 = s2 ≥ 2 where s = (s1, s2) and V (s) = n imply si

j = 0 for i = 1, 2 andj ≥ n.

At the second is possible reduce these equations after substitution

fn(t) = exp

t∫0

((n− 1)a(s)− nλr)ds

gn(t)

to recently studied type of functional differential equation. For the function g(t) we obtain the equation

gn(t) = gn(t− r) + exp

− t∫0

((n− 1)a(s)− nλr)ds

∑n(t). (8)

3 Solutions of the auxiliary system

In [4] are studied systems of delayed differential equation with constant coefficients. In In this paper isdefined delayed exponential matrix as the generalization of exponential matrix. Recall this notion. Let Bis constant square matrix and r is a positive constant we define defined delayed exponential matrix as

eBtr =

0, for t < −r;I, for 0− r ≤ t < 0;I + B t

1! + B2 (t−r)2

2! + · · ·+ Bk (t−(k−1)r)k

k! , for (k − 1)r ≤ t < kr,

where 0 and I denote the zero matrix and identity. In this paper is next assertion.Let the square matrix A, B are permutable i.e. holds AB = BA. Then the solution of the time delay

systemx(t) = Ax(t) + Bx(t− r) + f(t),

satisfying initial condition x(t) = ϕ(t) , −r ≤ t ≤ 0 has form

x(t) = eA(t+r)eB1tr ϕ(−r) +

0∫−r

eA(t−s)eB1(t−r−s)r (ϕ′(s)−Aϕ(s)) ds +

0∫eA(t−r−s)eB1(t−r−s)f(s)ds,


where the matrix B1 is defined as B1 = e−ArB.In our case we have scalar time delay equation and in this formula the matrices A, B are real numbers.

We put A = 0 and B = 1 and B1 = B = 1 and then the symbol etr denotes the function

etr =

0, for t < −r;1, for 0− r ≤ t < 0;1 + t

1! + (t−r)2

2! + · · ·+ (t−(k−1)r)k

k! , for (k − 1)r ≤ t < kr,

Applying this fact we obtain the next theorem.

Theorem 1. For any function ϕ ∈ C1[−r, 0] and the equation (7n) the solution of this Cauchy problemhas form:

fn(t) = eR t0 ((n−1)a(s)−nλr)dset

rϕ(−r) +

0∫−r

et−r−sr ϕ′(s)ds +

t∫0

et−r−sr e

R s0 −((n−1)a(u)−nλr)du∑

n(s)ds

, (9n)

For applying the first Lyapunov method we put the initial function ϕ ≡ 0 and the solution of thesystem (7n) have form

fn(t) = eR t0 ((n−1)a(s)−nλr)ds

t∫0

et−r−sr e

R s0 −((n−1)a(u)−nλr)du∑

n(s)ds. (10)

The asymptotic properties of integral (10) follows from the fact that erkr =

k∑i=0

(k − i)i

i!ri and using

the estimation in [8] limk→∞

er(k+1)r

erkr

= erλr . we may state the asymptotic properties of solutions of the

system (7n).

ACKNOWLEDGMENT

This paper was supported by the Grant 201/08/0469 of the Czech Grant Agency (Prague) and by theCzech Ministry of Education in the frame of project MSM002160503 Research Intention MIKROSYNNew Trends in Microelectronic Systems and Nanotechnologies.

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[7] Z. Svoboda, Asymptotic behaviour of solutions of one differential equation with delay. DEMONSTRATIOMATHEMATICA, 1995, XXVIII,n. 1, 9-18


[8] Z. Svoboda, Asymptotic estimate of unbounded solutions of one retarded functional differential equation.In Sbornk pspvk 5. konference o matematice a fyzice ma vysokch kolch technickch s mezinrodn ast. Brno,UNOB Brno 2007. p. 351 - 356. ISBN 978-80-7231-274-0.

[9] Z. Svoboda, Remark on asymptotic properties of one differentila equuation with unbounded delay. In Pro-ceedings of 7th interantional coference Aplimat. 7th International Conference APLIMAT. Bratislava: Facultyof Mechanical Engineering Slovak University of Technology in Bratislava, 2008, s. 303-307, ISBN 978-80-89313-03-7, 303-307

[10] Z. Smarda, The existence and asymptotic behaviour of solutions of the singular integor-differential equation.Archivum Mathematicum, 26(1) 1990, 7-18

ADDRESS:

RNDr. Zdenek Svoboda, CSc.Department of MathematicsFaculty of Electrical Engineering and Communication,Brno University of TechnologyTechnicka 8 , 61600 BRNO , Czech [email protected]


DECOMPOSITION METHOD OF SOLVING OFINTEGRODIFFERENTIAL EQUATIONS

Zdenek Smarda


Abstract: The paper deals with the decomposition method of solving of certain classes of theVolterra and Fredholm integrodifferential equations either with a separable kernel or a differ-ence kernel. There are specified initial conditions to determine uknown constants as well.

1 Introduction

In the past few years many papers are devoted to methods of solving of integral and integrodifferentialequations (see[1-12]). Scientests and engineers come across the integral and integro-differential equationsthrough their research work in heat and mass diffusion processes, electric circuit problems, neutrondiffusion (see[5,6,11]). In the electrical RLC circuit, one encounters an integrodifferential equation todetermine the instantaneous current flowing through the circuit with a given voltage E(t) , and it iswritten as

LdI

dt+ RI +

1C

∫ t

0

I(τ)dτ = E(t)

with the initial condition I(0) = I0. To solve this problem we need to know the appropriate technique(see [6,11]). One quick source of integrodifferential equations can be clearly seen when we convert thedifferential equation to an integral equation by using Leibnitz rule. The integrodifferential equation canbe viewed in this case as an intermediate stage when finding an equivalent Volterra integral equation tothe given differential equation.

In this paper we will derive the Adomian decomposition method for certain classes of the Volterraand Fredholm integrodifferential equations and we will illustrate one on concrete examples.

2 Decomposition method for Volterra integrodifferential equa-tions

In this section we will introduce the decomposition method to solve the intial problem

un(x) = f(x) +∫ x

0

K(x, t)u(t)dt, uk(0) = bk, k = 0, 1, . . . , n− 1, (1)

where bk are constants . It is natural to seek an expression for u(x) that will be derived from (1). Thiscan be done by integrating both sides of equation (1) from 0 to x as many times as the order of derivativeinvolved.Consequently we obtain

u(x) =n−1∑k=0

1k!

bkxk + L−1(f(x)) + L−1

(∫ x

0

K(x, t)u(t)dt

), (2)

wheren−1∑k=0

1k!

bkxk


is obtained by using the initial conditions and L−1 is an n−fold integration operator. Now we apply thedecomposition nethod by defining the solution u(x) of equation (2) as a decomposed series

u(x) =∞∑

n=0

un(x). (3)

Substituting (3) into both sides of equation (2) we get

∞∑n=0

un(x) =n−1∑k=0

1k!

bkxk + L−1(f(x)) + L−1

(∫ x

0

K(x, t)

( ∞∑n=0

un(t)

)dt

)(4)

This equation can be explicitly written as

u0(x) + u1(x) + u2(x) . . . =n−1∑k=0

1k!

bkxk + L−1(f(x)) (5)

+L−1

(∫ x

0

K(x, t)u0(t)dt

)+L−1

(∫ x

0

K(x, t)u1(t)dt

)+L−1

(∫ x

0

K(x, t)u2(t)dt

)+ . . . (6)

The components u0(x), u1(x), u2(x), . . . of the uknown function u(x) are determined in a recursive mannerif we set

u0(x) =n−1∑k=0

1k!

bkxk + L−1(f(x)),

u1(x) = L−1

(∫ x

0

K(x, t)u0(t)dt

),

u2(x) = L−1

(∫ x

0

K(x, t)u1(t)dt

),

and so on. The above equations can be written in a recursive form as

u0(x) =n−1∑k=0

1k!

bkxk + L−1(f(x)), (7)

un(x) = L−1

(∫ x

0

K(x, t)un−1(t)dt

), n ≥ 1. (8)

In view of equations (7),(8), the components u0(x), u1(x), u2(x) . . . are immediately determined. Thenthe solution u(x) of equation (1) is obtained as a series form using (3). The series solution may be putinto an exact closed-form solution which can be clarified by some illustration as follows. It is to be notedhere that the phenomena of self-cancelling noise terms (see[1,2,4]) may be applied here if the noise termsappear in u0(x) and u1(x) . The following example will explain how we can use decomposition method.

Example 1.

Consider the following initial problem

u′′(x) = x +∫ x

0

(x− t)u(t)dt, u(0) = 0, u′(0) = 1. (9)

Applying the two-fold integration operator

L−1(.) =∫ x

0

∫ x

0

(.)dxdx (10)


to both sides of equation (9) we get

u(x) = x +x3

3!+ L−1

(∫ x

0

(x− t)u(t)dt

). (11)

Following the decomposition scheme , i.e. equations (7) and (8) we have

u0(x) = x +x3

3!,

u1(x) = L−1

(∫ x

0

(x− t)u0(t)dt

)=

x5

5!+

x7

7!,

u2(x) = L−1

(∫ x

0

(x− t)u1(t)dt

)=

x9

9!+

x11

11!.

Now the final solution can be written as

u(x) = x +x3

3!+

x5

5!+

x7

7!+

x9

9!+ . . . (12)

and this leads to u(x) = sinhx the exact solution in closed form.

3 Decomposition method for the Fredholm integrodifferentialequations

Consider the Fredholm integrodifferential equation with a separable kernel

u(n)(x) = f(x) + g(x)∫ 1

0

h(t)u(t)dt, u(k)(0) = bk, k = 0, 1, . . . , n− 1. (13)

Equation (13) can be written in the operator form as

Lu(x) = f(x) + g(x)∫ 1

0

h(t)u(t)dt, (14)

where the differential operator is given by L = dn

dxn . Then the integral operator L−1 is an n-fold integrationoperator. Applying L−1 to both sides of equation (13) yields

u(x) = b0 + b1x +12!

b2x2 + + · · ·+ 1

(n− 1)!bn−1x

n−1 + L−1(f(x)) +(∫ 1

0

h(t)u(t)dt

)L−1(g(x)). (15)

In other words, we integrate equation (13) n times from 0 to x and we use the initial conditions at everystep of integration. It is important to note that equation (15) is a standart Fredholm integral equation.Now we define the solution of (13) by series (3). Substituting (3) into both sides of equation (15) we get

∞∑n=0

un(x) =n−1∑k=0

1k!

bkxk + L−1(f(x)) +(∫ 1

0

h(t)u(t)dt

)L−1(g(x)). (16)

This can be written explicitly as follows:

u0(x) + u1(x) + u2(x) . . . =n−1∑k=0

1k!

bkxk + L−1(f(x))

+(∫ 1

0

K(x, t)u0(t)dt

)L−1(g(x))

+(∫ 1

0

K(x, t)u1(t)dt

)L−1(g(x))

+(∫ 1

0

K(x, t)u2(t)dt

)L−1(g(x))

+ . . . (17)


The components u0(x), u1(x), u2(x), . . . of uknown function u(x) are determined ina recurrent manner,in a similar fashion as discussed before, if we set

u0(x) =n−1∑k=0

1k!

bkxk + L−1(f(x))

u1(x) =(∫ 1

0

K(x, t)u0(t)dt

)L−1(g(x))

u2(x) =(∫ 1

0

K(x, t)u1(t)dt

)L−1(g(x))

and so on . The above scheme can be written in compact form

u0(x) =n−1∑k=0

1k!

bkxk + L−1(f(x))

un(x) =(∫ 1

0

K(x, t)un−1(t)dt

)L−1(g(x)), n ≥ 0.

The series solution is proven to be convergent. Sometimes the series gives an exact expression for u(x). The decomposition method avoids massive computational work and difficulties that arise from othermethods. The computational work can be minimized, sometimes, by observing the so-called self-cancellingnoise terms phenomena which was introduced by Adomian and Rach [2] and it was proved that the exactsolution of any integral or integrodifferential equation, for some cases, may be obtained by consideringthe first two components u0, u1 only. Instead of evaluating several components, it is useful to examine thefirst two components. If we observe the appearance of like terms in both the components with oppositesigns, then by cancelling these terms, the remaining noncancelled terms of u0 may in some cases providethe exact solution. This can be justified through substitution. The self-cancelling terms between thecomponents u0, u1 are called the noise terms. However, if the exact solution is not attainable by usingthis phenomena, then we should continue determining other components of u(x) to get a closed formsolution or an approximate solution.Now we will demonstrate this method by an example.

Example 2.1.


u′′′(x) = sinx− x−∫ π/2

0

xtu′(t)dt, u(0) = 1, u′(0) = 0, u′′(0) = −1. (18)

Integrating both sides of equation (18) from 0 to x three times and using the initial conditions we obtain

u(x) = cosx− x4

4!− x4

4!

∫ π/2

0

tu′(t)dt. (19)

We use the series solution given by

u(x) =∞∑

n=0

un(x). (20)

Substituting (20) into both sides of equation (19) we get

∞∑n=0

un(x) = cosx− x4

4!− x4

4!

∫ π/2

0

t

( ∞∑n=0

u′n(x)

)dt. (21)

This can be explicitly written as

u0(x) + u1(x) + u2(x) . . . = cos x− x4

4!− x4

4!

∫ π/2

0

tu′0(t)dt

−x4

4!

∫ π/2

0

tu′1(t)dt− x4

4!

∫ π/2

0

tu′2(t)dt

− . . . (22)


Put

u0(x) = cosx− x4

4!, (23)

u1(x) = −x4

4!

∫ π/2

0

t

(− sin t− t3

3!

)dt =

x4

4!+

π5

5!3!32. (24)

Considering the first two components u0(x) and u1(x) in equations(23) and (24), we observe that theterm x4

4! appears in both components with opposite sign. Thus according to the noise phenomena theexact solution is u(x) = cosx. And this can be easily verified to be true.

4 Converting to Fredholm integral equations

This section deals with a technique that reduces Fredholm integrodifferential equation to an equivalentFredholm integral equation. This can be done by integrating both sides of the integro differential equationsas many times as the order of the derivative involved in the equation from 0 to x and using the giveninitial cinditions. Having established the transformation to a standart Fredholm integral equation, wemay proceed using any of the alternative method, namely the decomposition method, direct compositionmethod, the successive approximation method or the method of successive substitutions.

Example 3.1


u′′(x) = ex − x + x

∫ 1

0

tu(t)dt, u(0) = 1, u′(0) = 1. (25)

Integrating both sides of (25) twice from 0 to x and using the initial conditions we obtain

u(x) = ex − x3

3!+

x3

3!

∫ 1

0

tu(t)dt, (26)

which is a typical Fredholm integral equation. By the direct computational method this eaquation canbe written as

u(x) = ex − x3

3!+ α

x3

3!, (27)

where a constant α is determined by

α =∫ 1

0

tu(t)dt. (28)

Substituting (27) into (28) we get

α =∫ 1

0

t

(et − x3

3!+ α

x3

3!

)dt,

which reduces to yield α = 1 . Thus, the solution can be written as u(x) = ex.

Acknowledgement


Reference

[1] ADOMIAN, G.: A review of the decomposition method and some recent results for nonlinear equation,Math. Comput. Modelling, 13, 1992, 17-43.


[2] ADOMIAN, G., RACH, R.: Noise terms in decomposition series solution, Computers Math. Appl.,24, 1992, 61-64.

[3] FILIPPOVA, O., SMARDA, Z.: Singular Initial Value Problem for Volterra Integrodifferential Equa-tions, In Proceedings of the First International Forum of Young Researchers , Izhevsk, PublishingHouse of ISTU, 2008, p.331-336.

[4] CHERRUAULT, Y., SACCOMANDI, G.: New results for convergence of Adomian’s method appliedto integral equations, Math. Comput. Modelling, 16, 1993, 85-93.

[5] RAHMAN, M.: Integral equations and their applications, WIT Press, Southampton, UK, 2007.

[6] RAHMAN, M.: Mathematical methods with applications, WIT Press, Southampton, UK, 2000.

[7] SMARDA, Z., FAJMON, B.: Estimates of solutions of Volterra Integral equations with singularkernels, In Proceedings of electronic version of reviewed contribution. Brno, UNOB Brno, 2007,10-15.

[8] SMARDA, Z.: Integral equations with difference kernels on finite intervals, In Proceedings of elec-tronic version of reviewed contribution, Brno, University of Defence, 2007, 25-30.

[9] SMARDA, Z.: Existence principles for integral equations on compact intervals, In Thesis of Confer-ence reports, Kyiv , Ukrajina, Kyivskij nacionalnyj universitet, 2007, 158-159.

[10] SMARDA, Z.: Existence and uniqueness of solutions of Volterra integrodifferential equations withunbounded delay II, Journal of Applied Mathematics, 2008(1), 235-240.

[11] SMARDA, Z., FILIPPOVA, O.: Current characteristic of a simple electric circuit described by differ-ential and integrodifferential equation, In Proceedings of electronic version of reviewed contribution.Brno, UO Brno, 2008 1-5.

[12] WAZWAZ, A.M.: A first course in integral equations, World Scientific, Singapore, 1997.


ON THE TOLERANCE RELATION T ON (β) LATTICES

Josef Zapletal

European Polytechnical Institut, LLC.686 04 Kunovice, Czech Republic,

e-mail: [email protected]

Abstract: This paper is dedicated to the studies of te relation of tolerance. Chajda and Zelinkaproved that the relation of tolerance on groups and Boolean algebras which is compatible withthe incident structure merges in the relation of congruence in [2] and [21]. Both this struc-tures have further property which studies ensue from the enquiry of languages especially fromthat, that both this structures contain one-point disjoint subset. But this second property havealso (β) semilattices. Hence it is quite natural to investigate the relation of tolerance on (β)semilattices and (β) lattices. I did not prove the affirmation of Chajda and Zelinka genarallyfor all (β) semilattices and lattices but I prove this affirmation for one particular subset of(β) lattices it is for strong (β) lattices wehich will be studiet in the second part of this paper.

Key words. (β) lattice, strong (β) lattice, disjunctive subset, reducible elelent, irredudibleelement,

1 Basic properties of (β) lattices

1.1 Definition Let S be a lattice with the greatest element 1 [the least element 0 ].We say that the lattice S has the property (β)∨ [(β)∧] if for every two different elementsx, y ∈ S, x ∨ y 6= 1 there exists an element z ∈ S such that either x < z, y ‖ z ory < z, x ‖ z [ if for everz two different elemnts x, y ∈ S, x ∧ y 6= 0 there exists an elementz ∈ S such that either z < x, y ‖ z or z < y, x ‖ z ].If S with the greatest and least elements has both properies (β)∨, (β)∧ we denominate it(β) lattice or we say that the lattice has the property (β).

1.2 Lemma Let S be a (β)∧lattice with the least element 0 satisfying the minimumcondition. Let a, x ∈ S, 0 < a < x. Then there exists an element z ∈ S such that z < aand a < y ∨ a ≤ x.

Proof. There is a ∧ x = a > 0. Therefore there exists z such that z < x, z ‖ a.

1.3 Theorem Let S be a (β)∧lattice with the least element 0 satisfying the mini-mum condition. Then for every element X ∈ S which is not atomic element there exista < x, z < x such that a ∨ z = x, it is every element in S other from the least elementand atoms is reducible with respect to the operation ∨.

Proof. According to the assumption the element x is not atom, therefore there existsan element a0 < x. According to lemma 1.2 there exists z0 < x such that z0 ∨ a0 ≤ x. Ifz0 ∨ a0 = x then we put z0 = z and a0 = a hence the proof is trough. Let z0 ∨ a0 < xWe put z0 ∨ a0 = a1 For the elements x and a1 there exists again an element z1 such thatz1 < x and a1 < z1 ∨ a1 ≤ x. The lattice S satisfies the minimum condition thereforethere exists a natural number n such that z0 ∨ a0 < z1 ∨ a1 < . . . < zn ∨ an = z ∨ a = xand the theorem holds true.


1.4 Corollary Let S be a (β)∧lattice with the least element 0 satisfying the minimumcondition. Then every irreducible element with respect to the operation ∨ asside from theleast element is an atom.

1.5 Remark Every atom in arbitrary lattice is irreducible according to the join oper-ation ∨.

1.6 Theorem Let S be a (β)∧lattice with the least element 0 satisfying the minimumcondition. Then atoms and join-irreducible elements of S blend.

Proof. The affirmation of the theorem follovs from corollary 1.4 and from the remark1.5.

1.7 Theorem In a lattice satisfying the minimum condition every one of its elementscan be represented as the union of a finite number of join-irreducible elements.

Proof. The theorem is dual affirmation of the theorem 14. of the bibliography [14].

1.8 Corollary Let S be a (β)∧lattice with the least element 0 satisfying the minimumcondition. Then every element x ∈ S, x 6= 0 is supremum of the set of atoms.

Proof. Every element x ∈ S is supremum of the set of the join-irreducible elementsaccording to the theorem 1.7. Further with respect to the theorem 1.6 the sets of atomsand join-irreducible elements blend.

1.9 Theorem Let S be a (β)∧lattice with the least element 0 satisfying the minimumcondition in which every element excepting the least element is the supremum of a set ofatoms. Them S is (β)∧ lattice.

Proof. Let x, y ∈ S, x 6= y and x ∧ y > 0. Then thete exist an atom a which is underone of the elements x, y and does not lye under the second one. In the opposite case theywould be under both elementshe same atoms and it would be x = y. Let a ≤ x and a 6≤ y.If a = x it would be 0 < x ∧ y ≤ x = a and hence x ∧ y = a for the element a is atomand hence y ≥ a which is a contradiction.Thus a < x. It remains to prove that a ‖ y. It issufficiente to prove that a > y does not hold. Let us suppose contrary that a > y. But theelement a istom and then y = 0 and hence x ∧ y = 0 which is a contradiction. Thereforea 6> y which implies a ‖ y.

1.10 LemmaLet S be a lattice with the least element 0 satisfying the minimum con-dition. Then the following assertions are equivalent:(A) Every element x ∈ S with the exception of the least element 0 is the supremum ofthe set

of atoms.(B) The lattice S has the property (β)∧.

Proof. (A) ⇒ (B) holds from the theorem 1.9(B) ⇒ (A) holds from he corollary 1.8.

1.11 Lemma Let S be a lattice with the greatest element 1 satisfying the maximumcondition. Then the following assertions are equivalent:(A) Every element x ∈ S with the exception of the greatest element 1 is the infimum

of the set of dual atoms.


(B) The lattice S has the property (β)∨.

Proof. Lemma 11 is dual affirmation of the lemma 10.

1.12 Theorem Let S be a lattice with the least element 0 and the greatest element1 satisfying the minmum and maximum conditions, it is the lattice of finite length. Thenthe following assertions are equivalent:(A) Every element x ∈ S, x 6= 0 is the supremun of the set of atoms and every ele-

ment y ∈ S, y 6= 1 is the infimum of the set of dual atoms.Mak(B) The lattice S has the property (β).(C) Every element x ∈ S, x 6= 0 with the exception of atoms has at least two predecesors

and every element y ∈ S, y 6= 1 with the exception of dual atoms has at least twosuccessors.

Proof. (C) is only the other formulation of (A). The whole theorem follows from lem-mas 10 and 11.

1.13 Lemma Let S be (β) lattice of finite length, x, y ∈ S, x 6= y. Then there existsatom a [dual atom d] which is less or equal to one of the elements x, y [greater or equalto one of elements x, y] and is not under the second-one [and is not over the second-one].

Proof. If one of the elements x, y is equal to the least 0 the the affirmation is ob-vious. Let x 6= 0 6= y. According to lemma 1.10 the elemnts x, y are supremas of thesets of atoms. It is x 6= y which implies the existence of an atom a which is less or equalto one of them and does not lie under the second. The second affirmation of lemma is dual.

1.14 Theorem Let S be a β lattice of finite length with the least and greatest el-ements. Then for every two different elements x, y ∈ S there exist element u ∈ S suchthat:

either y ≤ u, u ∨ x = 1 and u ∨ y 6= 1 or x ≤ u, u ∨ x 6= 1 and u ∨ y = 1and element v ∈ S such that

either v ≤ y, v ∧ x = 0 and v ∧ y 6= 0 or v ≤ x, v ∧ x 6= 0 and v ∧ y = 0

Proof. i) Let x ∨ y = 1. For x 6= y then either x 6= 1 or y 6= 1. Let us suppose that thefirst case occures,the second is analogous. Then it is sufficiente to put u = x.

ii) Let x ∨ y < 1 With respect to the previous lemma

1.15 Definition Let S be a latticeof finite length with the least and greatest elements0 and 1. By complement of x is meant any element u of S satisfying the equations

x ∧ u = 0 (1) x ∨ u = 1 (2)If element x of S has exactly one complement it is called a uniquely complemented ele-ment in S. If all elements of S are complemented (uniquely complemented) S is said tobe a complemented (uniquely complemented) lattice.

1.16 Definition For lattices boundedbellow the complement concept maz also be gen-eralized as follows¿ By the semicomplement of an element x of a lattice S bounded bellowwe mean every element u ∈ S satisfying (1). Evidently at the same time x is also thesemicomplement of u. If we want to emphasize the symmetricity of this connection, wesay that x and u are disjoint elements.

1.17 Definition A lattice S bounded bellow with the least element 0 is called weakly


complemented if for any pair a, b(abof elements of S the element a has a semicomplementu, which is not semicomplement of the second element b it is

a ∧ u = 0, b ∧ u 6= 0.

1.18 Corollary Let S be β lattice of finite length. Then S is weakly complemented.

Proof. The affirmation folows from the Theorem 1.14 and the Definition 1.17 and itsduality.

1.19 Corollary Let S be β lattice of finite length. Then S is semicomplemented lattice.

1.20 Theorem Let S be β lattice of finite length. Then for every double of elementsx, y ∈ S, x ‖ y for which x ∨ y 6= 0 [x ∧ y 6= 0] there exist elents u, v, [u′, v′] in S suchthat x < u, u ‖ y and y < v, v ‖ x [x > u′, u′ ‖ y and y > v′, v′ ‖ x].

Proof. We will suppose thw existence of two elemnts x, y ∈ S, x ∨ y 6= 1 for which donot exist both elemnts u, v ∈ S with the property given in the Theorem. We will studyelements x.x ∨ y now. From Lemma 1.13 follows the existence of a dual atom d1 whichis ovwer one of tham and with the second incomparable. It is obvious that d1 > x andd1 ‖ x ∨ y and hence d1 ‖ y. Similarly for the double y, x ∨ y follows the existence of adual atom d2 for which d2 > y, d2 ‖ x ∨ y and thus d2 ‖ x. It is sufficiente to put u = d1

and v = d2.The second affirmation of the theorem is dual to the first-one.

1.21 Corollary Let S be β lattice of finite length. Then for every double x, y ∈ S ofincomparabled elements for which x ∨ y 6= 1 [x ∧ y 6= 0] there exist u, v ∈ S [u′, v′ ∈ S]such that u ∨ xx = 1, u ∨ y 6= 1 and v ∨ x 6= 1, v ∨ y = 1 [u′ ∧ x = 0, u′ ∧ y 6= 0 andv′ ∧ x 6= 0, v′ ∧ y = 0].

Proof. The affirmation folows the previous Theorem and from the Theorem 1.14.

1.22 Theorem Let S be a finite (β) lattice with the greatest and least elements.Then for every atom [dual atom] at least one complement in S exists.

Proof. Let a ∈ S be an atom. Let us suppose the elements the elements a and theleast-one 0. From Lemma 1.13 follows the existence od dual atom d which is over one ofthem and incomparable with other. We have for all x, x > a the relation x > 0.Therefored ‖ a, d > 0. Hence d ∨ a = 1 and simultaneously d ∧ a = 0. Hence follows the assertionof the theorem, that the elements a and d are mutually complementary.

2 Strong (β) lattices

2.1 Definition Let S be a lattice with the greatest element 1 [ least element 0]. Wesay that the lattice S has the strong property (β)∨ [(β)∧] when for all different elementsx, y ∈ S, x ∨ y 6= 1 there exists an element z ∈ S such that either x−≺ z, y ‖ z ory−≺ z, x ‖ z.[ When for all different elements x, y ∈ S, x ∨ y 6= 1 there exists an elementz ∈ S such that either z−≺ x, y ‖ z or z−≺ y, x ‖ z ].The semilattice S which has both properties (β)∨, (β)∧ is called the strong (β) lattice orthe lattice with strong property β.


2.2 Remark The following symbol between twoo elemnts x−≺ z means that the ele-ment x is the predecessor of the element y, it is the opened interval (x.y) is a void set.

2.3 Lemma Let A be full atomic Boolean algebra, x ∈ A non least, nonatomic andnondualatomic element. Then the element x has at least two predecessors.

Proof. For element x there exists just one complement x′.The element x is not atom and not dual atom, this impliesat least two dual atoms d1, d2 for which x′ < d1−≺ 1and x′ < d2−≺ 1. We suppse d1. Then d1 ∨ x′ = d1,simultaneously d1 ‖ x and d1 ∧ x 6= 0. If d1 ∧ x = 0 thentwo complements for the element x would be existing. Wewill suppose the existence of x0 for which x∧d1 < x0 < x.Hence x′ ∨ (x∧ d1) = 1∧ (x′ ∨ d1) = 1∧ d1 = d1. We haved1 ≤ x′ ∨ x0. We suppose now that x′ ∨ x0 = d1 then

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x0 = x ∧ d1. In the contrary occurence A will include a sublattice as the sublattice atthe Figure 1. and this is a contradiction with distributivity of A. Let us suppose nowthat x = ∨x0 > d1 then x′ ∨ x = 1 and hence x0 = x in the other way the element x′

would have two complements x and x0. Hence the element x∧d1 is the predecessor of theelement x. The second predecessor x ∧ d2 will be determined analogously.

2.4 Lemma Let S be a lattice with the greatest element 1 and the least element 0.Let for every two elements x, y ∈ S, x < y 6= 1 exists u ∈ S such that x−≺ u, y ‖ u andfor every two elements s, t ∈ S, 0 6= s < t exists an element v such that v−≺ t, v ‖ s.Then S is strong (β) lattice.

Proof. For doubles of comparable elements are the conditions of the Definition 2.1 sat-isfied.Let w, z ∈ S, w ‖ z and w ∨ z 6= 1. Evidently w < w ∨ z. With respect to presumptionsof the Lemma there exists successor u of the element w such that u ‖ w ∨ z ¿From therelation w ‖ z follows u 6≤ z. Let us permit that u > z. Then for elements w, z exist twosuprems u and w ∨ z which are incomparable. This givs a contradiction for S is a lattice.The proof for incomparable elements for which w ∧ z 6= 0 is analogous.

2.5 Theorem The full atomic Boolean algebra A is a strong (β) lattice.

Proof. We prove that A has the strong property (β)∧. The strong property (β)∨ canbe proved analogously.

Let x, y ∈ A, x ∧ y 6= 0, x ‖ y. There exists a predecessor u of the element x. If u ‖ ythis part of the proof is finished. Therefore let us contrary suppose that the elements u, yare comparable. The relation y ≤ u does not hold. We will suppose that u < y Accordingto previous Lemma there exists an element v−≺ x and v ‖ u.We prove that v ‖ y. In thecontrary case then it would be u < y, v < y. Simultaneously u and also v are predecessorsof the element x which implies x < y which is the contradiction with the assumption ofthis part of the proof.

Let x < y, x 6= 0. Let us entertain all predecessors of the element y. Simultaneouslywe will suppose that the element x lies under every of them. The interval [0, y] is wholeatomic Boolrean algebra. All elements covered by element y are dual atoms of this algebra


and their intersection is equal to the least elemnt 0. But this is a contradiction with theassumption that x > 0. Hence there exists the predecessor of the element y, we denoteit by v such that v ‖ x. If the element x is a predecessor of y then the existence of thepredecessor v of the element y for which v ‖ x is obvious.

We proved that the whole atomic Boolean algebra A has the property (β)∧.

2.6 Remark There exist strong (β) lattices which are not Boolean algebras.

Proof. Let S be a lattice described by the diagram of Haase as the Figure 2. We provethat this lattice is strong (β) lattice. We create the table of successors for elements forwhich x < y 6= 1 and of predecessors for elements for which 0 6= s < t. In the table isrelated always only one of possible elements.

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x < y 6= 1 x−≺ u, u ‖ y. 0 6= s < t v−≺ t, v ‖ s

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2.7 Theorem There exist strong (β)lattices which are not Birkhoff lattices.

Proof. From the Corollary of the Theorem 65 [14] the lattice of finite length is Birkhofflattice if and only if it is semimodular lattice. Wewill investigate the lattice S inscribedwith the Haase diagram on the Figure 3. We earmark the triplet of elements a, b, x for


which a ‖ b, a∧b < x < a. For every element t > a∧b for which t ≤ b holds t∨x = 1. Hence(t ∨ x) ∧ a 6= x. Ther edoes not exist an aelement t : a∧, t ≤ b such that (t ∨ x) ∧ a = x.Analogously there does not existt in the lattice S an element t′ < a′∨ b′, t′ ≥ b′ such that(t′ ∧ x′) ∨ a′ = x′. Hence the given lattice is not semimodular and hence according to theCollorary of the theorem 65 is not Birkhoff lattice.

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2.8 Remark Every strong (β) lattice is (β) lattice.

2.9 Remark Tere exist (β) lattices which are not strong (β) lattices.

Proof. It is sufficiente tu suppose the lattice the lattice perscribed by Haase diagramat the Figur 4. We werify readily that the lattice S is the (β) lattice. We see that for theelements a, b ∈ S the relation a < b holds. Simultaneously does not exist a successor ofthe element a which will be incomparable with the element b.

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2.10 Remark Let S be the strong (β) lattice. Every dual ideal D[x], x ∈ S is not


(β) lattice and thus is also not strong (β) lattice.

Proof. The lattice perscribed by Haase diagram at the Figure 5 is strong (β) lattice.We suppose principal end P defined by the element x. The relation a ∧ b 6= x holds true.For elements a, b does not exist an element u for which u < b and u ‖ a in P .

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2.11 Theorem Let S be a finite strong (β) lattice. Then every element x ∈ S has atleast one complement x′ ∈ S.

Proof. The elements 0, 1 are mutually complementary. Let x ∈ S, 0 6= x 6= 1. Thelattice S is atomic. We denote by A the set of all dual atoms in S and as I[x] theprincipal ideal of the element x. For the element x 6= 0 with respect to Corollary 1.8exists an dual atom a0 ∈ A such that a0 6∈ I[x]. Let us suppose the element x ∨ a0 = b0.If b0 = 1then the element a0 is a complement of the elemnt x and the proof is finished.

Let b0 6= 1. The lattice S is strong (β) lattice. Hence follows the existence of suchelement z1 which satisfies the twoo following conditions¿ b0 ‖ z1 and a0−≺ z1, We provethat x ∧ z1 = 0. Let us contrary suppose that x ∧ z1 = c1 6= 0. Then c1 ∨ a0 ∈ I[b0] andc1 ∨ a0 6= z1 for z1 6∈ I[b0]. Such we have a0 < c1 ∨ a0 < z1 for c1 ‖ a0 else it wouldbe c1 = 0. But this is a contradiction according to the assumption that the element z1

covers the element a0. Hence x∧ z1 = 0 holds true. Therefore a0 < z1 and simultaneouslyz1 ‖ x ∨ a0 = b0 and hence b0 < x ∨ z1 = b1

If b1 = 1 the element z1 complement of the element xand the proof is trough. Letb1 6= 1. For the elements b1, z1 exists an element z2 ∈ S such that b1 ‖ z2 and z1−≺ z2,.Again x∧z2 = 0. The opposite acount x∧z2 = c2 6= 0 implies c2∨z1 ∈ I[b1] and c2∨z1 6= z2

and hence follows z1 < c2 ∨ z1 < z2 which is a contradiction with the assumption that z2

covers z1. We have again z1 < z2 and z2 ‖ b1. Therefore b1 < x ∨ z2 = b2.The lattice S is bounded. Continuing in the previous construction we can finde for every

x ∈ S the sequence a0−≺ z1−≺ z2 . . . −≺ zk with the property x ∨ z = 0 for i = 1, 2, ..., k.Further there exists a sequence bi = x ∨ zi for which b1 < b2 < ... < bk−1 < bk = 1 Suchwe found the complement zk of the element x in S.


3 The relation of tolerace on strong (β) lattices

3.1 Lemma Let S be a strong β lattice of the finite length with the elements a0, b0,for which a0 > b0. Then there exists a natural number n and a sequence ain

i=0 in S suchthat ai −ai+1, i = 0, 1, . . ., n− 1 and at it the sequence bin

i=0

b0 for i = 0bi =

bi−1 ∧ ai for i = 1, 2, . . . , n

with the properties ai+1 ‖ bi for i = 0, 1, . . . , n− 1, bn = 0 and an ∨ b0 = a0.Proof. 1) S is strong β latticeThere exists for the element a0 > b0 the predecessor a1

such that a1 ‖ b0. Hence b1 = a1 ∧ b0 < b0. It is a1 > b1 and there exists the predecessora2 of the element a1 for which a2 ‖ b1 and b2 = a2 ∧ b1 < b1. The lattice s is of the finitelength therefore finally exists such index n that bn = 0.

2) Let 0 < i ≤ n−1. Let us further suppose the triplet ai+1, ai, bi. It holds goodai > bi, ai > ai+1 Hence ai ≥ bi∨ ai+1. We will suppose that ai > bi∨ ai+1. Both elementsai+1, bi lie under the element ai. Hence also their supremum is under the element ai whichimplies the existence of the element bi ∨ ai between the elements ai and ai+1 But this isa contradiction with the assumption that ai −ai+1.

3) Now we take the elements an, bn−1. From the second part of this proof fol-lows an ∨ bn−1 = an−1. Quite analogosly for an−1 and bn−2 holds the equalityrelationan−1∨ bn−2 = an−2 and hence an∨ bn−1∨ bn−2 = an−2. Using the associativity of the oper-ation ∨ we obtain after another n steps the equality relation an∨bn−1∨bn−2∨ . . . ∨b1∨b0 =an ∨ b0 = a0.

3.2 Definition Let M be arbitrary set. The reflexive and symmetric binary relationT on M will be called the tolerance relation videlicet

1) a T a2) If a T b then b T a

for arbitrary a, b ∈ M .

3.3 Definition Let S be a lattice, ρ binary relation onto S. We say that the relation ρis compatible with the operations of S if for every quaternion of elements a1, a2, b1, b2 ∈ Sfor which (a1, b1) ∈ ρ and (a2, b2) ∈ ρ the following relations hold :

(a1 ∨ a2, b1 ∨ b2) ∈ ρ(a1 ∧ a2, b1 ∧ b2) ∈ ρ.

3.4 Lemma Let S be a lattice, a, b ∈ S and T a tolerance relation compatible withlattice operations onto S. Let a T b. Then for every double from the closed interval[a ∧ b, a ∨ b] ⊂ S the relation x T y holds true.

Proof. The proof is given in [1].

3.5 Lemma Let S be a strong β lattice of the finite length, c < b < a elements in S.Let T is a tolerance relation compatible with the lattice operations of S and c T b, b T a.Then c T a.

Proof. S is strong β latticewhich imlies the existence of the element a1 for whichai −a, a1 ‖ c and a1 ∧ a > a1 ∧ b ≥ a1 ∧ c.


1) If a1 ∧ b = a1 ∧ c then from the relations (a1 ∧ a) T (a1 ∧ b) = (a1 ∧ c) andc T c and from the compatibility of the relation T with the lattice operations of S follows((a1 ∧ a) ∨ c) T ((a1 ∧ b) ∨ c) and hence a T c.

2) Let a1∧b > a1∧c. We put a1∧b = b1 and a1∧c = c1. There exists again the el-ement a2 −a1, a2 ‖ c1 for which a2∧a1 > a2∧b1 ≥ a2∧c1. If a2∧b1 = a2∧c1 then from therelations (a2∧a1)T (a2∧b1) = (a2∧c1) and cT c follows ((a2∧a1)∨c1)T ((a2∧b1)∨c1) whichgives a1 T c1. Now using cT c and the compatibility of T in S we receive (a1∨ c)T (c1∨ c)and hence a T c.

3) Let a2 ∧ b1 > a2 ∧ c. We put a2 ∧ b1 = b2, a2 ∧ c1 = c2. There exists again theelement a3 −a2, a3 ‖ c2 and simultaneously a3∧a2 > a3∧ b2 ≥ a3∧ c2. If a3∧ b2 = a3∧ c2

then we prove as in the part 2) the relation a T c. In the opposite case there we constructsequences a − a1 − a2 . . . − ak, c − c1 − c2 . . . − ck−1 such that eitherak ∧ bk−1 = ak ∧ ck−1 and similarly as in the part 2) we prove aT c or ak ∧ bk−1 > ak ∧ ck−1

and for the elements of sequences hold a1 ‖ c, a2 ‖ c1, . . . ak−1 ‖ ck−2, ck−1 = 0. Simulta-neously a1 > b1 > c1, a2 > b2 > c2, . . . , ak−1 > bk−1 > ck−1. That holds for S is of finatelength. From the compatibility of the relation T with the operations of S and from thedefinition of the relation of tolerance T we have (ak−1 ∨ b) T (ak−1 ∨ ck−1) and hencebk−1 T 0.

4) From Lemma 3.1 follows for the elements ak−1, bk−1 the existence of naturalnumber s and as such that ak−1 > as for which ak−1∧as = a and bk−1∧as = 0, bk−1∨as =ak−1. Similarly as in 3) as T 0.

5) Thus it is bk−1T 0 and asT 0. From the comfactificity of the relation T with theoperations of S and using 4) we have (bk−1∨as)T 0. From this imply (ak−1∨ck−2)T (0∨ck−2)which is from Lemma 3.1 equal to ak−2 T ck−2 and hence from the same Lemma the rela-tion (ak−2 ∨ c) T (ck−2 ∨ c) implies a T c and from the symmetry also c T a.

3.6 Theorem Let S be a strong β lattice of the finite length. Then the relation oftolerance T compatible with the operations of S is the congruence.

Proof. We will suppose the existence of tolerance relation T which is compatible withthe operations of S and is not a congruence relation. Then there exist tree different ele-ments a, b, c for which a T b, b T c hold and a T c does not hold.

From the properties of the tolerance relation follows aT b, cT b and bT b. Thisimplies (a ∨ c) T b and (a ∧ c) T b and hence (a ∨ c ∨ b) T b and (a ∧ c ∧ b) T b.

One of the following cases turns up :1) b = a ∧ b ∧ c, b < a ∨ b ∨ c2) b > a ∧ b ∧ c, b = a ∨ b ∨ c3) b > a ∧ b ∧ c, b < a ∨ b ∨ c

In the first case the elements a, c lye in the interval [b, a ∨ b ∨ c], in the sec-ond one in the interval [a ∧ b ∧ c, ] and simultaneously (a ∧ c) T b and (a ∧ c ∧ b) T b.Hence using Lemma 3.4 we have a T c. Let the third case turns up. Thena∧b∧c < b < a∨b∨c and togrether with (a∨c∨b)T b and (a∧c∧b)T b the assumptionsof Lemma 3.5 are satisfied. Therefore it holds (a ∧ c ∧ b) T (a ∨ b ∨ c). The elements a, clye in the interval [a ∧ b ∨ c, a ∨ b ∨ c] and a T c again.

In all three situations we get a contradiction with the assumption that a T cdoes not hold. Such is the Theorem proved.

3.7 Remark Contrary to the conception of the trivial congruence relation of an algre-bra we will distinguish the concepts of trivial and universal congruences.


3.8 Lemma Let S be a strong β lattice of the finite length. Let θ be a netrivial con-gruence on S. Then exists an atom a ∈ S such that a θ 0.

Proof. θ is not trivial. This implies the existence of two different elements r, s ∈ S forwhich r θ s. There existsn element u ∈ S such that either u ∧ r = v 6= 0 and u ∧ s = 0 oru ∧ r = 0 and u ∧ sv′ 6= 0 We suppose the first case. For every atom lying in prime idealof the element v holds a ∧ r = a, a ∧ s = 0. Finally a θ 0 which follows from stability ofthe congruence θ.

3.9 Theorem Let S be a strong β lattice of the finite length. Let θ be a nontrivialcongruence on S. Let a ∈ S be atomic element, a θ 0. Let there exists an element u ∈ Ssuch that u ‖ a and u ‖ a−1. Then the congruence θ is universal.

Proof. From the assumptions u ‖ a, u ‖ a1 and according to 1.13 follows the existenceof dual-atom v, v ‖ a, v ‖ a−1, v > u and of dual atom w, w ‖ a−1, w ‖ a, u > w.From the relations a θ 0, a−1 θ a−1 and from the stability of the congruence θ there follows(a ∨ a−1) θ a−1 hence 1 θ a−1 and also a−1 θ 1. Simultaneously the element v is dual atomnoncomparable with a, therefore a ∨ v = 1 and hence 1 θ v and similarly for the atom wholds a−1 ∧ w = 0, 1 ∧ w = w and hence 0 θ w. We inscribe with g1 the element v ∨ a−1

and with h1 the element w∧ a. We obtain from the relations v θ 1, a−1 θ 1 and w θ 0, a θ 0the following relations: g1 θ 1 and h1 θ 0. If either g1 = 0 or h1 = 1 Theorem is proved. Letg1 6= 0 and h1 6= 1 then for elements a−1, g1 follows the existence of an atom z1 for whichz1 < a−1, y1 ‖ g1 simultaneously z1 ‖ w and z1 θ 0. Hence for h2 = h1 ∨ z1 the trelationh2 θ 0 holds. For h2 = 1 Theorem is proved. We suppose contrary that h2 6= 1. For theelements a, h2 exists a dual atom t1 such that t1 > a, t1 ‖ h2 The element t1 lies in dualideal of the element a, what implies t1 ‖ v and t1 ‖ a−1 and simultaneously t1 θ 1 Hencefor g2 = t1 ∧ g1 follows g2 θ 1 and g2 < g1. If h2 = 0 Theorem is proved again. Let g2 6= 0.Then we have for elements g1, g2 an atom z2 for which z2 < g1 and z2 ‖ g2. This impliesz2 θ 0 and simultaneously z2 ‖ z1, z2 ‖ w. Hence h3 = z2 ∨ h2 > h2 and h3 θ 0. If h3 = 1the proof is finished. In the opposite case ther exists for h2 and h3 an element t2 for whichT2 > h2, t2 ‖ h3 and t2 ‖ t1 and simultaneosly t2 lies in dual ideal of the element a andhence t2 ‖ v. Again t2 θ 1. Hence g3 = t2 ∧ g2 < g2 and g3 θ 1. If g3 = 0 then the Theoremis proved.In the opposite case we find after the finite number of steps either k1 such thathk1 = 1 or k2 such that gk2 = 0 or finally such k that zk is atom and tk dual atom for whozk θ 1and tkθ0. Simultaneously yk ‖ u and tk ‖ u otherwise it would be zk < g1 and tk > h1 whichis a contradiction with assumptions. Therefore we have (u∧zk)θ (u∧1 which implies 0θ uand similarly (u∨tk)θ(u∨0) which implies 1θv and hence 0θ1 and the Theorem holds true.

3.10 Corollary Let S be a strong β lattice of the finite length and θ a nontrivialcongruence on S. Let a ∈ S be atomic element such that aθ0 for which exist two differentcomplements a−1

1 ., a−12 . in S. Then θ is universal congruence on S.

Proof. If a−11 ‖ a−1

2 . then the proof follows directly from the previous Theorem fora−1

1 ‖ a and a−12 ‖ a too.

Let a−11 > a−1

2 . then a−12 is not dual-atomic element and there exists over it an

element a3 which is simultaneosly complement of a. In the opposite case we have a < a3

and supremum of the elements a−12 and a would be smaller than the greatest element 1

which is a contradiction with the asumption that a−12 is complement of a. Hence there

exist to the element a two incomparable complements and the assumptions of the previous


Theorem are stisfied.

3.11 Remark There exist strong β lattices wich szstem of congruences does notcontent only two elements, it is only trivial and universal congruence.

Proof. Let us suppose the lattice S given by Haase diagram at Figure 2. Then therelation ρ defined as folows :

ρ : (0, a); (b, g); (c, m); (d, f); (h, 1); (f, d); (1, h); (m, c); (g, b); (a, 0)is a congruence.

Reference

[1] BIRKHOFF, G. Lattice Theory. Amer. Math. Soc. Colloquium Publ. 25 revised edi-tion, New York 1948.

[2] BORUVKA, O. Grundlagen der Grupoid und Gruppentheorie. Berlin, (1960).

[3] CROISOT, R. Equivalences principales bilateres de finies dans un demi-groupe. J.Mat. pures et app. (N.S) 36, 1957, 373-417.

[4] CHAJDA,I., ZELINKA B. Tolerance relations on Lattices, Casopis pest. matem., 99,1974, 394-399.

[5] JURGENSEN, H. Inf-halbverbande als syntaktische Halbgruppen. Acta Math.Academie Sci. Hungaricae T 31 (1-2), 37-41,

[6] KARASEK, J., NOVAK, V., SLAPAL, J., ZAPLETAL, J. Cardinal Characteristicsof Ordered Sets, Distinguishing Subsets on Lattices and n-ary Ordered Sets.kshop97 Technical University in Brno and Czech Technical University in Prague, January1997. Part Mathematical 41-42.

[7] NIEMINEN, J. Remarks on Distinguishing Subsets of Join-semilattices. Demonstra-tio Math. Vol. IX. N o2,1976.

[8] VOVAK, V. On the well Dimension of ordered Sets. Cz. Math. J. V. 19(94),N o1,1969, 1-16.

[9] NOVOTNY, M. On some Relations defined by Languages. Prague Studies in Math-ematical Linguistics, 4, Prague 1972, APH of Cz. Academi of Sci. 157-170.

[10] NOVOTNY, M. Uber endlich charakterisierbare Sprachen.,Publ. Fac.Sci. Univ. J.EPurkyne, Brno, N o468, 1965 495-502.

[11] PERROT, J.F. Contribution a 1’etude des monoıdes syntactiques et de certainsgroupes associes aux automates finis. THESE, UNIVERSITE PARIS VI, 1972.

[12] PIERCE, R. Homomorphism of Semigroups. Am. of Math. 59 919540, 287-291,

[13] SAKAROVITCH, I. Monoıdes syntactiques et langages algebriques., THESE 3‘eme

CYCLE, UNIVERSITE PARIS VII, 1976.

[14] SCHEIN, B.M. Homomorphisms and subdirect decompositions of semigroups., PAcificJ. of Math, 17, 1966,529-547.


[15] SCHEIN, B.M. On some Problems in the Theory of PArtial Automata. Kybernetika,1, (5), 1969, 44-49.

[16] SZASZ, G. Introduction to Lattice Theory. Akademiai Kiado, Budapest 1963.

[17] TEISSIER, M. Sur les equivalences regulieres dans demigroupes. C. R. Acad. Sci.Paris 232 (1951) 1987-1989.

[18] ZAPLETAL, J. Distinguishing subsets of Semigroups and Groups. Arch. Mat. Brno(1968), T4, Fasc. 4, 241-250.

[19] ZAPLETAL, J. On some Properties of Congruences on Semigroups. Sbornık VUT vBrne 1970, c 1-4 213-216.

[20] ZAPLETAL, J. Distinguishing Subsets in Lattices. Arch. Math. 2; IX, Brno (1973),73-82.


Investigation on Optimization of Chemical Process

Tran Trong Dao Ivan Zelinka

Department of Applied Informatics, Tomas Bata University Nad Stranemi 4511, Zlin 760 05, Czech Republic

E-mail: [email protected], [email protected]

ABSTRACT

In this paper, the modeling of a dynamic chemical engineering process is presented in a highly understandable way using a unique combination of the simplified fundamental theory and direct hands-on computer simulation. The main aim is to use them for analysis of behavior of dynamical system, especially of a given chemical reactor. A non-linear mathematical model is required to describe the dynamic behaviour of batch process; this justifies the use of evolutionary method of the EAs to deal with this process. The new algorithm - differential evolution (DE) from the artificial intelligence field is used to find the optimal parameters. Differential Evolution is an evolutionary optimization technique which is exceptionally simple, significantly faster & robust at numerical optimization and is more likely to find a function’s true global optimum. For all algorithms, each simulation was repeated 100 times to show and check robustness of used methods. All data were processed and used in order to get summarizing results and graphs.

KEYWORDS: Simulation; Optimization; Evolutionary algorithms; Differential evolution.

1. INTRODUCTION

Designing optimal reactor parameters including control constitutes is one of the most complex tasks in process engineering. The situation is particularly complicated by the fact that the precise mechanism of chemical reaction kinetics is very often unknown. For this reason it is necessary to carry out extensive measurements of input and output concentration dependencies of components on time, temperature, etc. The optimization of dynamic process has received growing attention in recent years because it is essential for the process industry to strive for more efficient and agile manufacturing in face of saturated market and global competition (T. Backx, O. Bosgra 2000). Evolutionary algorithms derived by observing the process of biological evolution in nature, have proven to be a powerful and robust optimizing technique in many cases (Gross B.; Roosen P. 1998). In computer science evolutionary computation is a subfield of artificial intelligence (more particularly computational

intelligence) that involves combinatorial optimization problems (Wikipedia encyclopedia). Since the 60s, several approaches (genetic algorithms, evolution strategies etc.) have been developed which apply evolutionary concepts for simulation and optimization purposes. Also in the area of multiobjective programming, such approaches (mainly genetic algorithms) have already been used (Evolutionary Computation 3(1), 1–16)(Thomas Hanne 2000). Evolutionary algorithms such as evolution strategies and genetic algorithms have become the method of choice for optimization problems that are too complex to be solved using deterministic techniques such as linear programming or gradient (Jacobian) methods. The large number of applications (Beasley (1997)) and the continuously growing interest in this field are due to several advantages of EAs compared to gradient based methods for complex problems ( Ivo F. Sbalzarini, Sibylle Muller and Petros Koumoutsakos 2000).


In this paper, the methods of artificial intelligence by evolutionary algorithms - DE is presented for optimizing chemical engineering processes, particularly those in which the evolutionary algorithm is used for static optimization of a chemical batch reactor to improve its parameter. Consequently, it is used to design geometry technique equipments for chemical reaction. The method was used to optimize the design of the growth chamber, and was found to be in good agreement with the observed growth rate results.

2. DESCRIPTION OF A REACTOR

Reactor disposes by two physical inputs. First input denoted ”Input Chemical FK” is chemical dosing into reaction about mass flow rate FKm& , temperature FKT

and specific heat FKc . Second input denoted “Input cooling medium” is water drain into the reactor double side with mass flow rate Vm& , temperature VPT and

specific heat Vc . This coolant further traverses among - jacketed through space of reaction and his total weight in this space is VRm . Coolant after it gets off the exit reaction denoted “output cooling medium” about mass flow rate Vm& , temperature VT and specific heat Vc . At the beginning of the process there is an initial batch inside the reactor with parameter mass Pm . Reactionary mixture then has total mass m , temperature T , specific heat Rc and stirs till the time chemicals FK described by parameter concentration

FKa . Non-linear model of reactor

Description of the reactor applies a system of four balance equations (1). The first one expresses a mass balance of reaction mixture inside the reactor, the second a mass balance of the chemical FK, and the last two formulate entalpic balances, namely balances of reaction mixture and cooling medium. Equation (1), where for simplified notation of basic equations (2) is represented by term “k”.

][tmmFK ′=& (1)

][][][][ tatmktatmm FKFKFK +′=&

][][])[][(][][

tTctmtTtTSKtatmkHTcm

RV

FKrFKFKFK

′+−==Δ+&

][][])[][(

tTcmtTcmtTtTSKTcm

VVVRVVV

VVPVV

′+=−+

&

&

k = Ae− E

R T [t ] (2)

After modification into the standard form, the balance equations are obtained in form (3)

FKmtm &=′ ][ (3)

][][

][ ][ taeAtm

mta FKtTR

EFK

FK

−

−=′ &

R

V

RR

ArtTR

E

R

FKFKFK

ctmtTSK

ctmtTSK

ctaHeA

ctmTcm

tT

][][

][][][

][

][

][

+−Δ

+

=′

−&

VR

VV

VVR

V

VVRVR

VPVV m

tTmcm

tTSKcm

tTSKm

TmtT ][][][][&&

−−+=′

The design of the reactor was based on standard chemical-technological methods and gives a proposal of reactor physical dimensions and parameters of chemical substances. These values are called in this participation expert parameters. The objective of this part of the work is to perform a simulation and optimization of the given reactor. Therefore into system equations (3) were instated constants:

A = 219,588 s-1, E = 29967,5087 J.mol-1, R = 8,314

J.mol-1.K-1, cFK = 4400 J.kg.K-1, cV = 4118 J.kg.K-1,

cR = 4500 J.kg.K-1, ΔHr = 1392350 J.kg-1,

K = 200 kg.s-3.K-1,

Next parameters, that are important for calculations are: • Geometric dimension of the reaction: r[m] , h[m] • Density of chemicals: Pρ = 1203 kg.m-3 ,

FKρ = 1050 kg.m-3 • Stoicheiometric rate chemical: Pm = 2,82236.mFK

3. OPTIMIZATION OF PROCESS PARAMETERS AND THE REACTOR GEOMETRY

We illustrate the design approach using the batch reaction system shown in Fig. 1. The main aim in this example is finding the optimization of process parameters and the reactor geometry. Here, it is a optimization of batching value FKm& together with process parameters of the cooling medium and including also reactor geometry and cooling area. Mathematical problems


In this optimization was founded optimized parameters with one another linked ,so that heat transfer surface, volume, and hence also mass mixtures of reaction was mutually in relation. Heat transfer surface S have relation:

22 rrhS ππ += (4)

Where r is radius and h is high of the space reactor

Volume of vessel of rector applies to relation: hrV 2π= (5)

Total mass of mixtures in the reaction is initial batch inside the reactor with parameter mass Pm a mass

input chemical FK FKm , that:

FKp mmm += (6) The stechiometric ratio is given by (7).

FKP mm 82236,2= (7)

Total volume of mixtures in the reaction equal sum of volume initial mixtures in the reaction and volume of FK:

FK

FK

p

pFKp

mmVVV

ρρ+=+=

(8)

The relationship between the optimized volume of reactor and the mass of added chemical FK is given by (8). Then substituting to (7) gives the mass of the initial batch in the reactor.

pFK

FKpFK

Vm

ρρρρ

+=

82236,2 (9)

In this example, the optimization was then added parameter thickness d of vessel, which have relation that:

Sdm VVR ρ= (10)

In this optimization the point was to minimize the area arising as a difference between the required and real temperature profile of the reaction mixture in a selected time interval, which was the duration of a batch cycle. The required temperature was 97°C (370.15 K). The cost function that was minimized is given in (11):

[ ]∑=

−=t

tt tTwf

0cos (11)

Static optimization of reactor

The above described reactor, in the original set-up, gives unsatisfactory results. To improve reactor behavior, static optimization was performed using the algorithm DE. In this work the optimization was performed by the following optimization of batching value reactor’s parameters geometry.

4 .USED ALGORITHM AND PARAMETER SETTING

For the experiments described here, stochastic optimisation algorithms, such as Differential Evolution (DE) (Price, 1999) had been used. Alternative algorithms, like Genetic Algorithms (GA) and Simulated Annealing (SA), are now in process, and results are hoped to be presented soon. Main reason why DE has been sed comes from contemporary state in chemical engineering and EAs use. Since now has been done some research with attention on use of EAs in chemical engineering optimization, including DE. This participation has to show that applicability of relatively new algorithms is also positive and can lead to applicable results, as was shown for example in Zelinka (2001), which has been done under 5th EU project RESTORM (acronym of Radically Environmentally Sustainable Tannery Operation by Resource Management) and main aim was to use EAs in chemical engineering processes. Differential Evolution (Price, 1999) is a population-based optimization method that works on real-number coded individuals. For each individual xi,G in the current generation G, DE generates a new trial individual x’i,G by adding the weighted difference between two randomly selected individuals xr1,G and xr2,G to a third randomly selected individual xr3,G. The resulting individual x’i,G is crossed-over with the original individual xi,G. The fitness of the resulting individual, referred to as perturbated vector ui,G+1, is then compared with the fitness of xi,G. If the fitness of ui,G+1 is greater than the fitness of xi,G, xi,G is replaced with ui,G+1, otherwise xi,G remains in the population as xi,G+1. Deferential Evolution is robust, fast, and effective with global optimization ability. It does not require that the objective function is differentiable, and it works with noisy, epistatic and time-dependent objective functions. Pseudocode of DE is:

( ) ( ) ( )

[ ] ( )

⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

+=

⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

⎩⎨⎧ ≤

=

⎪⎩

⎪⎨

⎧

=∨<

−⋅+

=≤∀

∈≠≠≠∈

≤∀

<

⎪⎩

⎪⎨⎧

∈===

−•+=≤∀∧≤∀

∈+∈≥

+++

+

=

1

otherwise )()( if

Select 5.

otherwise )]1,0[( if

)(

, 3.4

each once selectedrandomly ,,...,2,1 4.2 :except selected,randomly ,,....,2,1,, 4.1

:recombine and Mutate .4 While.3

]1,0[]1,0[ ,0 ,,...,2,1 ,,...,2,11,0:

:Initialize .2

.,:bounds initial and ],1,0[,1,0,4,, :Input .1

,

,1,1,1,

,,

,,,,,,

1,,

321321

max

)()()(0,,

max

213

GG

xxfufu

x

xjjCRrand

xxFx

uDj

iDjirrrNPrrr

NPi

GG

randGDjNPixxrandxxDjNPi

xxCRFNPGD

Gi

GiGiGiGi

Gij

randj

GrjGrjGrj

Gij

rand

j

loj

hijj

lojGji

hilo

r

rrrr

rr

The randomly generated binary perturbation vector controls the allowed dimensions for an individual. If an element of the perturbation vector is set to zero, then the individual is not allowed to change its position in the corresponding dimension. An individual will travel a certain distance (called the path length) towards the


leader in n steps of defined length. If the path length is chosen to be greater than one, then the individual will overshoot the leader. This path is perturbed randomly. For an exact description of use of the algorithms see (Price, 1999).

The control parameter settings have been found empirically and are given in Tab. 1. The main criterion for this setting was to keep the same setting of parameters as much as possible and of course the same number of cost function evaluations as well as population size (parameter NP). Individual length represents number of optimized parameters, see Tab. 2.

Tab.1 DE parameter setting

A

NP 20

F 0.9

CR 0.2 Generations 200

Individual Length 6

CF Evaluations 4000

5. EXPERIMENTAL RESULTS

Due to the fact that EAs are partly of stochastic nature, a large set of simulations has to be done in order to get data for statistical data processing. Algorithms DE have been applied 100 times in order to find the optimum of process parameters and the reactor geometry. All important data has been visualized directly or/and processed for graphs demonstrating performance DE algorithms. Estimated parameters and their diversity (minimum, maximum and average) are depicted in Fig. 1 & Fig. 2 . For the demonstration are graphically the best solutions shown in Fig. 4, 6, 8 and Fig. 10. The best values of parameter setting are recorded in Tab. 3. On Fig. 3, 5, 7 and Fig. 9 are for example shown records of all 100 simulations and the best solutions of all 100 simulations.

Tab. 2 Optimized reactor parameters and their range inside which has been optimization done

Parameter Range

FKm& [kg.s-1] 0 – 500

r [m] 0.3 – 3.0

h [m] 0.5 – 3.5

VPT [K] 273.15 – 323.15

Vm& [kg.s-1] 0-10

d [m] 0.03 – 0.1

Tab. 3 The best values of optimized parameters

Parameter DE

FKm& [kg.s-1] 0.042049

r [m] 0.543594

h [m] 1.86628

VPT [K] 294.718

Vm& [kg.s-1] 9.94168

d [m] 0.0563635

Parameter diversity for repeated simulations

Results visualized on Fig. 3 – 10 are also numerically recorded in Tab.4.

Tab. 4 Estimated parameters

Parameter Min Avg Max

FKm& [kg.s-1] 0.020 0.154 0.460

r [m] 0.318 1.754 2.996

h [m] 0.512 1.740 3.492

VPT [K] 293.219 304.565 322.652

Vm& [kg.s-1] 2.349 9.227 9.997

d [m] 0.0300 0.0410 0.0834

Fig.1 Parameter variation

Fig.2 Parameter variation – detail


Fig. 3 100 simulations for m

Fig. 4 The best solution for m

Fig. 5 100 simulations for aFK

Fig. 6 The best solution for aFK

Fig. 7 100 simulations for T

Fig. 8 The best solution for T

Fig. 9 100 simulations for TV

Fig. 10 The best solution for TV


6. CONCLUSION

This paper has presented a systematic procedure to derive a solution model for operation of a dynamic chemical reactor process. The results produced by the optimizations depend not only on the problem being solved but also on the way how to define a given function.

Calculation was 100 times repeated and the best, worst and average result (individual) was recorded from the last population in each simulation. All one hundred triplets (best, worst, average) were used to create Fig. 1 & Fig. 2.

From the graphs, it is evident that the courses of DE algorithm are densities in a thin specter. Alongside it, sometime few values drift out of the actual solution (see Fig.3, 5, 7, 9). But Parameter diversity for repeated simulations we have obtained best value for optimization (see Fig. 4, 6, 8, 10). From these results we may conclude, that this model is not always stable, but the optimization by the cost function show that the EAs are used successfully in the process optimization.

Basic optimizations presented here were based on a relatively simple functional. Unless the experimenter is limited by technical issues when searching for optimal geometry parameters, there is no problem in defining more complex functional including as subcriteria e.g., stability, costs, time-optimal criteria, controllability, etc. or their arbitrary combinations.

Finally, on the basis of presented results, it may be stated that the reactor parameters have been found by optimization which demonstrates performance superior to that of reactor set up by an expert. Future work objective is to formalize this process in order support the control system design to the best extent possible. According to all results obtained during time it is planned, the main activities would be focused expanding of this comparative study for genetic algorithms and simulated annealing.

7. ACKNOWLEDGEMENT

This work was supported by grant No. MSM 7088352101 of the Ministry of Education of the Czech Republic and by grants of the Grant Agency of the Czech Republic GACR 102/06/1132.

8. REFERENCES

A.K. Qin, P.N. Suganthan. Self-adaptive differential evolution algorithm for numerical optimization, in: Proceedings of the 2005 IEEE Congress on Evolutionary Computation, vol. 2, 2005, pp. 1785-1791.

Aziz N., Hussain M.A., Mujtaba I.M. (2000). Performance of different types of controllers in tracking optimal temperature in batch reactors. Computers and Chemical.

Beasley, D. 1997 Possible applications of evolutionary computation. In B¨ack, T., Fogel, D. B. &Michalewicz, Z., editors, Handbook of Evolutionary Computation. 97/1, A1.2, IOP Publishing Ltd. and Oxford University Press

Back, T., Fogel, D.B., Michalewicz, Z., 1997. Handbook of Evolutionary Computation. Institute of Physics, London.

Gamperle, R., S.D. Muller, Koumoutsakos, P. A parameter study for differential evolution, in: Proceedings WSEAS International Conference on Advances in Intelligent Systems, Fuzzy Systems, Evolutionary Computation, 2002, pp. 293-298.

Gross B.; Roosen P. Total process optimization in chemical engineering with evolutionary algorithms. Computers & Chemical Engineering. Volume 22, Supplement 1, 15 March 1998, Pages S229-S236. European Symposium on Computer Aided Process Engineering-8.

Hanne, T. Global multiobjective optimization using evolutionary algorithms. Journal of Heuristics, 2000.

Ivo F. Sbalzarini, Sibylle Muller and Petros Koumoutsakos 2000. Multiobjective optimization using evolutionary algorithms. Center for Turbulence Research, Proceedings of the Summer Program 2000.

Kirkpatrick S., Gelatt Jr., C.D., Vecchi, M.P., Optimization by Simulated Annealing, Science, vol. 220, no. 4598, 671-680, (1983).

Mansour, M. J.E. Ellis, Methodology of on-line optimisation applied to a chemical reactor Applied Mathematical Modelling, Volume 32, Issue 2, February 2008, Pages 170-184.

T. Backx, O. Bosgra, W. Marquardt, Integration of model predictive control and optimization of processes, in: Proceedings ADCHEM 2000, vol. 1, 2000, pp. 249–260.

Thomas Hanne, Global Multiobjective Optimization Using Evolutionary Algorithms , Journal of Heuristics, 6: 347–360 (2000).

Zelinka, I., 2004. SOMA - self organizing migrating algorithm. In: Babu, B.V., Onwubolu, G. (Eds.), New Optimization Techniques in Engineering. Springer, New York, pp. 167–218.

Zelinka I. (2004) SOMA. In: B.V. Babu, G. Onwubolu Eds., New optimization techniques in engineering, Springer- Verlag, chapter 7, ISBN 3-540-20167-X

Zelinka, I., Nolle, L., 2006. Plasma reactor optimizing using differential evolution. In: Price, K.V., Lampinen, J., Storn, R. (Eds.), Differential Evolution: A Practical Approach to Global Optimization. Springer, New York, pp. 499–512.

Thomas Bäck, Evolutionary algorithms in theory and practice: evolution strategies, evolutionary programming, genetic algorithms, Oxford University Press, Oxford, 1996.



ENVIRONMENTS” EPI Kunovice, Czech Republic, January 23, 2009 227

A ARTEMENKO, O................................................45 B BÁBEL, J. ....................................................29, 107 BAŠTINEC, J. ...................................................163 BĚŤÁK, P. ...........................................................75 BIBA, V. ............................................................175 BRZOBOHATÝ, J...............................................75 C CEPL, M. ...........................................................153 D DAO, T. T. .........................................................221 DENDAMRONGVIT, S. ...................................101 DIBLÍK, J. .........................................................163 DOSKOČIL, R.....................................................83 DOSTÁL, P....................................................11, 19 DOSTÁL, T. ......................................................129 F FAJMON, B.......................................................181 FILIPOVÁ, O. ...................................................185 CH CHVALINA, J. ..................................................189 K KRATOCHVÍL, O...............................................19 KUBAT, M. .......................................................101 L LIŠKA, J. ...........................................................153 LUDÍK, P.............................................................69 LYAKHOV, S......................................................25 M MARČEK, D........................................................51 MARČEK, M.......................................................57 MEČIAROVÁ, Z. ........................................29, 107 MUSIL, V. ...........................................................75 N NOVÁK, M........................................................189 O OŠMERA, P.................................................95, 145 P PANČÍKOVÁ, L..........................................29, 107 PETRUCHA, J. ..............................................63, 91 PETRŽELA, J. ...................................................139 POKORNY, P. ...................................................159 POPELKA, O.....................................................159 R RUKOVANSKÝ, I. .............................................95

S SENDUR, Z. ......................................................101 SKORKOVSKÝ, P. ...........................................113 SLEZÁK, J.........................................................135 SLOVÁČEK, D. ..................................................63 SVOBODA, Z....................................................195 Š ŠMARDA, Z. .....................................163, 185, 201 ŠOTNER, R. ......................................................135 ŠŤASTNÝ, J ........................................69, 153, 159 ŠTENCL, M.........................................................69 T TOMÁŠ, I. .........................................................119 TOMŠOVÁ, M. .................................................123 V VERBITSKAYA, J..............................................35 VÉRTESY, G.....................................................119 VYKLYUK, Y. ....................................................39 Z ZAPLETAL, J....................................................207 ZELINKA, I. ......................................................221

AUTOR INDEX


ENVIRONMENTS” EPI Kunovice, Czech Republic, January 23, 2009 228

Název:

ICSC 2009 – Seventh International Conference on Soft Computing Applied in Computer and Economic Environments

Autor:

Kolektiv autorů

Vydavatel, nositel autorských práv, vyrobil:

Evropský polytechnický institut, s.r.o.

Osvobození 699, 686 04 Kunovice

Náklad: 100 ks

Počet stran: 228

Vydání: první

Rok vydání: 2009

ISBN 978-80-7314-163-9

seventh international conference on...

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