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APPLIED MATHEMATICS IN OTHER SCIENCES

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TEXT 1


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Applied Mathematics

In modern times mathematics has become an inseparable part of human culture, in which itplays a fundamental role. Throughout the centuries mathematics has been a crucial tool in thehands of mankind. It has allowed us to understand the fundamental principles of the universe, forexample Newton's law of gravity, Einstein's equivalence of mass and energy, Maxwell's equations

of electromagnetism, the laws of quantum mechanics for elementary particles, and even the BigBang theory. The advances in interplanetary exploration and rapid development of computertechnology wouldn't have been possible without mathematics.

Scientists, in their struggle to improve our understanding, have untangled the principalproblems of biology and unveiled the secrets of life. However, the times when it was sufficient fora biologist to know only elementary arithmetic and graphs of functions are long gone. Today, theyneed much more advanced mathematics like linear and multilinear algebras, mathematicalanalysis, the theory of differential and functional equations, statistics and discrete mathematics.Branches of biology like genetics or ecology are considered as parts of mathematics. Mathematicsalso opens new possibilities for medicine. Mathematical models are used to understand our bodiesand to find optimal treatment for diseases. More and more mathematics is used in the social

sciences like economics, psychology, sociology, demography, social epidemiology andcriminology.

Applied mathematics is a collection of theories, techniques, and terminology that havepractical application in various fields of science, including, but not limited to, astronomy,chemistry, dynamics, engineering, physics and even mathematics itself.

A very simple example of the use of applied mathematics in physics is the equation F=kmawhere F is the force to be determined, kis a constant for a particular system of units. kis usuallyset to unity (in other words, k=1) when determining the force exerted by an object withoutconsideration of the pull of the earth's gravity. The variable m is the mass of the object and a is theacceleration the object exhibits at a given moment in time. From this equation, once values for twoof the variables have been determined, and assuming k=1, the remaining value can be determined.This equation, then, provides further desired information for the physical phenomenon beingstudied. The uses of applied mathematics through much more complex equations provide ways toquantify many observed properties of the universe as well as matters of theory that are as yetimpossible to observe directly.

The following is another example of the use of applied mathematics in other fields ofknowledge.

Digital face recognition.

Uses and applications of computer-aided face recognition: FBIs most wanted, criminalapprehension, high profile security.

To begin with, a recognition system has to be unaffected by both external changes, like

environmental light, and the persons position and distance from the camera, and internalvariations, like facial expression, aging, and makeup. Because most commercial applications uselarge databases of faces, recognition systems have to be computationally efficient. This is wheremath comes into play. Most face recognition algorithms fall into one of two main groups: feature-

based and image-based algorithms. Feature-based methods explore a set of geometric features,such as the distance between the eyes or the size of the eyes, and use these measures to representthe given face. These features are computed using simple correlation filters, and are somewhatimmune to changes in light sources, and camera position. However, they are sensitive to aging andfacial expressions.

Image-based systems, the other main approach to face recognition, are based on ideas likeeigenfaces, which are a related set of facial characteristics that a computer uses to recognize a

persons face. Faces actually vary according to a mere 100 factors. The computer must understandwhat these 100 factors are. Each face image is deconstructed into separate set of related facial


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characteristics and an algorithm is created so that the computer can understand the image andanalyze it in comparison to others.

Interesting Fact: during the 2001 Super Bowl in Tampa, Florida the city used facerecognition technology to scan the faces of people in crowds, comparing them with images in adatabase of digital mug shots.

ASSIGNMENTS1. Find in the text the English equivalents of the following Russian words and word-

combinations: , , , , , , , ,, , ,, .

2. Complete each of the following statements with the best ending from the box below:

a) Throughout the centuries mathematics has played a crucial role in human culturebecause

b) The times when it was sufficient for a biologist to know only elementary arithmetic andgraphs of functions are long gone andc) Applied mathematics is a collection of theories, techniques, and terminology that have

practical application in various fields of science, for example

i) astronomy, chemistry, dynamics, engineering, physics and even mathematics itself.ii) now genetics or ecology are considered as parts of mathematics.iii) nowadays, they need much more advanced mathematics like linear and multilinearalgebras, mathematical analysis, the theory of differential and functional equations,statistics and discrete mathematics.

iv) it has untangled the principal problems of biology and unveiled the secrets of life.v) it has helped us to understand the fundamental principles of the universe.vi) the advances in interplanetary exploration and rapid development of computertechnology wouldn't have been possible without mathematics.vii) the social sciences like economics, psychology, sociology, demography, social

epidemiology and criminology.

3. Answer the questions:

a) Is the role of mathematics in the history of mankind really crucial? Why?b) Why is it necessary for every scientist to be an expert in mathematics nowadays?

c) In which fields of knowledge is mathematics used?d) What is applied mathematics? Give the definition.e) Which two groups do most face recognition algorithms fall into?

4. Give a short summary of the text.

TEXT 2

Mathematics in Demography

Leonardo of Pisa (Fibonacci, 1202) proposed one of the earliest mathematical models forpopulation growth. The problem situation stated below is a reworking of Fibonacci's originalproblem which generates an introductory age-specific population model.


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Imagine that we start with one pair of rabbits (one female and one male). After N days, thispair matures to reproductive age and immediately produces a new pair. After N more days, the firstpair again produces offspring. Thus, each pair of rabbits will reproduce two times during theirlifetime (exactly one pair immediately at the start of each new stage, where "pair" always meansone female and one male), at intervals separated by N days, and each new pair of rabbits will go onin a similar fashion.

The problem statement suggests that the rabbit population can be broken down into threegroups: "newly born", "young adults" and "mature adults". Each pair of newly born rabbitssurvives to become young adults and to produce one new pair of offspring at this stage. Each pairof young adults survives to become mature adults and to produce another pair of offspring. Finally,each pair of mature adults moves on to "rabbit heaven"; no survival is allowed after stage 3.

This process of moving through the age-structure and the patterns that emerge can berepresented in two ways:

with diagrams, to break down and understand the dynamics of the problemwith spreadsheets, to capture patterns and create graphs.

Analysis by Diagram

The first step in understanding the model is to find a way to "make sense" of the problemsituation. A chart or diagram like the one shown below is helpful. The columns display the agestructure for each of the first 6 time steps. The rows show the first 6 generations.

Diagram Breakdown of the Rabbit Population

Generation1 NB YA MA2 NB YA MA

NB YA MA

3 NB YA MANB YA MANB YA MA

4 NB YA MANB YANB YANB YANB YA

5 NB YANBNBNBNBNBNBNB

6 NB

Time Step 1 2 3 4 5 6Newly Born 1 1 2 3 5 8Young Adult 0 1 1 2 3 5Mature Adult 0 0 1 1 2 3


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Total 1 2 4 6 10 16

This diagram captures many of the key aspects of the growth process of this rabbitpopulation. Viewing the chart by columns, we can see the age-specific breakdown for each time-step. For example, in the 4th column we see that there are 3 newly born, 2 young adults and 1mature adult. Viewing the chart by rows, we see the progression of the pairs born in a given

generation as they move through the age-specific categories for the rabbit population. For example,the two pairs born in the 3rd generation become young adults in the next column, contributing 2pairs of newly born to the 4th generation below them; they then survive to produce one last time,contributing to the 5th generation.

Analysis by Spreadsheet

The information contained in the diagram can then be summarized in a spreadsheet like theone shown below.

Time Newly Young Mature Total ofStep Born Adults Adults Rabbits

1 1 0 0 12 1 1 0 23 2 1 1 44 3 2 1 65 5 3 2 106 8 5 3 167 13 8 5 268 21 13 8 42

9 34 21 13 6810 55 34 21 110

Once the spreadsheet has been created, we can view large amounts of data conveniently,include the data in reports, and easily create graphs. Also, we can vary the assumptions of themodel and explore variations of the problem situation quickly.

According to Eves and Boyer, Fibonacci's original problem was stated as follows: How manypairs of rabbits can be produced from a single pair in a year if every month each pair begins a newpair which from the second month on becomes productive? The sequence of rabbits born eachmonth is the Fibonacci sequence.

ASSIGNMENTS

1. Find in the text the English equivalents of the following Russian words and word-

combinations: , , ,, , .

2. According to the information given in the text, which three of the following are true of

the diagram?

a) The diagram captures many of the key aspects of the growth process of this rabbitpopulation.

b) Viewing the chart by columns, we can see the age-specific breakdown for each time-step.c) In the 5th column we see that there are 3 newly born, 3 young adults and 1 mature adult.d) Viewing the chart by rows, we see the progression of the pairs born in a given generation.


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e) The two pairs born in the 3rd generation become young adults in the next column,contributing 3 pairs of newly born to the 4th generation below them.

3. Answer the questions:a) Who proposed the earliest mathematical models for population growth?

b) How many times does each pair of rabbits reproduce during their lifetime?

c) Into which groups can the rabbit population be broken down?d) In how many ways can the process of moving through the age-structure and the patterns thatemerge be represented? What are they?e) State the original Fibonacci's problem.

4. Speak on the age-specific population model.

TEXT 3

Mathematics in History

I

A. Mathematics is trying to make its contribution in history, where it addresses a very seriousproblem of reliability of the accounts of historical events. How can we be sure that the historicalevents that we learn about in school or from books really took place? Maybe some of them aresimply fairy tales that, because of some mysterious circumstances, are considered now to behistorical facts.

B. The fundamental question that should be asked is what is the origin of our historicalknowledge. We all learned our history at school and generally accepted it as a true description ofthe actual events. However, even in our lifetime some of the recent historical events that wewitnessed are not always described in the way we remember them. How can we be sure that thedescription of the events that took place centuries ago is accurate? Moreover, why should we

believe that these historical events really happened at the time and place that is allocated to them?In order to answer these questions we must look at the history of history.

C. The early historians (for example Thucydides, Herodotus, Ssu-ma Ch'ien and others) weredescribing history of small territories over short periods of time. Ancient and medieval manuscriptsthat are available today usually present accounts of events in separate countries over a time scale ofno more than one or two centuries. The fundamental problem encountered by historians in 16thand 17th centuries working on reconstruction of the global history of mankind was putting togetherin chronological order all of the manuscripts, chronicles and other historical documents to obtain aunified and consistent account of all historical events. This was an extremely difficult problem forthat time. The main obstacle was that most of the manuscripts were not dated, or used an unknownor archaic system of dating, and contained only a description of a sequence of successive events. Itshould be stressed out that the most of historical documents that we have today, related to ancientand medieval times, are not original but only copies made some time ago, often under suspiciouscircumstances.

D. The idea of reconstructing global history emerged during the late Renaissance. Theofficial historical chronology, presently commonly acknowledged, was originated by the Italiantheologian and scientist I. Scaliger (1540-1609). He determined the exact dates of the mostimportant historical events like the Peloponnesian War, Trojan War, founding of Rome, etc., butdid not prove any of his dates. His followers continued this work and it is commonly accepted that

the official chronology was given its final shape by D. Petavius (1583-1652). It is strange thatother historians, in spite of the scientific advantages, very rarely modified the dates of the basichistorical events assigned by Scaliger and Petavius.


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E. According to Scaliger, Petavius and their followers, the events of the ancient world tookplace from about 3,500 years B.C. till the fifth century A.D. As their results were neverindependently confirmed, there is an outstanding question of the credibility of this chronology. Bythe way, not all of the statements made by Scaliger turned out to be true, as for example, hisgeometrical proof of the quadrature of the circle, which he defended ferociously all his life.

F. Even among scholars, not all contemporaries of Scaliger and Petavius, supported their

chronology. For example, in the sixteenth century D. Arcilla, a professor of Salamanca Universityin Spain, claimed that all ancient history was a fabrication made in the middle ages. The director ofthe French Royal Library, Jean Hardouin (1646-1729) declared that practically all the antiquitiesand ancient texts were created (or falsified) after 12th century. The most famous scientist of thatepoch, Sir Isaac Newton (1642-1727), was also against the chronology of Scaliger and Petavius.

Newton published a large monograph entitled "The Chronology of Ancient Kingdoms Amended,"in which he re-dated key ancient events by shifting them several hundreds years forward. Therewere many more scientists, philologists, historians, and jurists who objected to the chronology ofScaliger and Petavius. We should also mention recent and contemporary critics of the conventionalchronology in Germany, including W. Kammeier, H. Illig, U. Topper, H-U. Niemitz and G.Heinsohn.

G. The first scholar who suggested new powerful methods to correct chronological mistakes,was prominent Russian scientist N.A. Morozov (1854-1946). He published a fundamentalmonograph composed of seven large volumes, entitled "Christ. History of Human Culture from theStandpoint of the Natural Sciences". Morozov analyzed in it the conventional chronology using thelatest discoveries in mathematics, astronomy, linguistics, philology and geology. He suggested anew version of the global chronology and a historical reconstruction. According to N.A. Morozovall the ancient events occurred after 3rd century AD.

H. In 1970s at the Moscow State University, a group of young mathematicians undertook thetask of the verification and further development of Morozov's research in global chronology. Oneof them, professor A.T. Fomenko introduced several new methods of independent dating and afterseveral years of investigation he proposed a new version of global chronology, which was evenmore radical that the version of N.A. Morozov. He claimed that the recorded history of mankindstarted not earlier than the year 900 AD, while the majority of historical events, which make

our history, refer to the time after the year 1300 AD.

ASSIGNMENTS


combinations: , , , , .

2. The text has seven paragraphs. Which paragraph mentions the following?i) A.T. Fomenkos introduction of a new version of global chronology.ii) Lots of scientists objected to the chronology of Scaliger and Petavius.iii) It was in the late Renaissance when the idea of reconstructing global history emerged.iv) History should not be accepted as a true description of the actual events without anydoubt.v) New powerful methods to correct chronological mistakes were suggested for the first time

by a prominent Russian scientist.vi) Applied mathematics can be used in history.vii) The most difficult thing to do for historians in 16th and 17th centuries was to puttogether in chronological order all of the manuscripts, chronicles and other historical

documents to obtain a unified and consistent account of all historical events.viii) Scaliger claimed that the events of the ancient world took place from about 3,500 yearsB.C. till the fifth century A.D.


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a) Can we be sure that the historical events that we learn about in school or from books reallytook place?

b) What should we look at the history of history for?c) What was the fundamental problem encountered by historians in 16th and 17th centuries?

d) Who was the first scientist to propose the idea of reconstructing global history?e) Who was the first Russian scientist to suggest new powerful methods to correctchronological mistakes?f) What was professor Fomenkos claim?

4. Retell the text, using the following expressions:

to begin with, next, then, in addition to, as a consequence of this, however, in other words, infact, thus, in summary, therefore, finally.

TEXT 4


II

In collaboration with G.V. Nosovskij, A.T. Fomenko continued his work on the developmentof new independent scientific methods for dating of ancient events. In 1993-1996, completely newresults were established by them on the chronology of Russia and China. Their work resulted instating the New Chronology, which is a new concept of the global chronology and history. It is

based on the chronological version of A.T. Fomenko, to which new proofs and improvements wereintroduced. It led to the further shifting of the "starting point" of the known history to the 11thcentury AD.

From the point of view of mathematics, the chronology represents an object called afunction. More precisely, we can write it as a function denoted by H(t, x1,x2), which depends onthe three variables: t - the time of a historical event and (x1,x2) - the geographical coordinates(longitude and latitude) of the place where this event occurred, or we can simply say that itsdomain is the Cartesian product of numeric half line and the sphere. The values of the function H(t,x1,x2) represent the fragments of historical recordings describing this particular event.

Figure 1

Figure 1 illustrates the "chronology" function H. On the left hand side of Figure 1 theconcentric spheres represent the domain of H. More precisely, the straight vertical arrow stands for

the time axis where the points correspond to specific dates. For example, the inside colouredsphere illustrates events of the year 1320 at specific locations. The larger spheres on this figurecorrespond to the years 1415 and 1985. In this way, with every date in history we can associate a


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sphere on which the corresponding events are indicated. To every place on the Earth we canassociate a ray originating at its centre to mark the dates of the events that occurred at this place.The books symbolize available descriptions of the historical events. The curved arrows indicate theexact fragments of the available descriptions corresponding to certain concrete events. Briefly, thechronology is a database parameterized by points of the Cartesian product R+ x S2, i.e. the productof the half-axis R+ and the sphere S2. Naturally, this function is not convenient for mathematical

analysis. Clearly the set of values of the function H does not have any natural mathematicalstructure. However, the information contained in the function H allows us, on the one side, toconstruct a variety of scalar (numeric) functions which can be easily analyzed with mathematicalmethods, and on the other side, to provide essential information on the nature of the historicalevents.

An example of a simple scalar function, which can be easily extracted from the historicaldatabase, is the functions of the time-span of the reign of subsequent rulers belonging to a certainspecific dynasty. Such a `dynasty' function can be illustrated by its graph, see Figure 2.

Figure 2

On the horizontal axis the subsequent numbers of the consecutive rulers (or names of kings,emperors, etc.) are placed and on the vertical axis the length of the reign of the corresponding ruleris marked. We will call such a sequence of rulers a numerical dynasty or simply a dynasty. Thedynasty in the above example consists of 12 rulers.

There is another way to analyze chronicles by extracting numerical information from them.For example we can associate with a text X a sequence of integers, which are the numbers ofwords H(X(T)) in the chapter describing the year T (or simply the volume of a year fragment). Wecall H(X(T)) the volume function for X. There are also possibilities for other numerical functionslike the number of references to the year T in subsequent years, the number of all names ofhistorical persons listed in the text, or the frequencies showing how often these names werementioned in the whole text. In his monograph A.T. Fomenko used these functions to analyzesimilarities and differences between documents referring either to the same epoch or two differentepochs. It is clear that for two different documents X and Y the functions H(X(T)) and H(Y(T))can be completely different even if they refer to the same epoch. However, if the functions

H(X(T)) and H(Y(T)) have local maxima practically at the same positions it means that these twochronicles describe the same historical epoch. A.T. Fomenko called it the principle of maximalcorrelation. This principle was empirically checked using the reliable historical data of 16th - 19thcenturies, and its correctness was confirmed. Therefore, the locations of the maxima constitute thenumerical data that can be associated with the text X in order to characterize the epoch it isreferring to.

The methods of Fomenko are based on theoretical and numerical analysis of these and othersimilar functions describing historical data. In particular, he introduces a routine for distinguishingfunctions referring to different dynasties and defines a certain measure of distinctiveness betweenthem (or a probability measure for distinctiveness). In simple words, he found a way to measure a`distance' between the above numerical functions (like for example dynasty functions) in a similar

way to measuring distance between two different locations. Mathematicians say that in such asituation they are dealing with a metric space. The geometry of such metric spaces is definitelydifferent from the geometry we learn in school, but the usual properties related to the measurement


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of distances are still valid in these spaces. If a distance between towns A and B is less than onekilometre we are justified to think that in fact A and B represent the same town. Similarly, if in thespace of functions a distance between two dynasty functions is sufficiently small we may think thatindeed they represent the same dynasty. These methods were extensively tested on the datareferring to well documented. It was proved that if two dynasty functions (for 15 rulers) or volumefunctions were not related, the measure of distinctiveness between numerical functions associated

with these dynasties was between 1 and 10-4

. However, in the case of related events from the sameepoch, the measure of distinctiveness was never higher than 10-8.

ASSIGNMENTS


combinations: , , , , , , , , .

2. Complete the summary below using words from the box:

G.V. Nosovskij and A.T. Fomenko continued their work on the development of new

independent scientific methods for dating of (1)_____ events. In 1993-1996 theyestablished new (2)_____ of Russia and China. On the whole their New Chronologymoved the (3)_____of the known history to the 11th century AD.

Among the methods used by A.T. Fomenko the most famous are: generating of a(4)_____, when time-span of the reign of subsequent rulers belonging to a certain specificdynasty is represented; (5)_____ chronicles by extracting numerical information fromthem; geometry of (6)_____, which differs considerably from the geometry we learn atschool.

starting point, chronology, analyzing, simple scalar function, metric spaces, ancient


a) What is the starting point of the known history according to new chronology of A.T.Fomenko and G.V. Nosovsky?

b) What does the chronology represent from the point of view of mathematics?c) What kind of function can the time-span of the reign of subsequent rulers belonging to a

certain specific dynasty represent?d) What is analyzed with the help of the principle of maximal correlation?e) In which situations do mathematicians deal with a metric space?

4. Name all the methods for dating of ancient events mentioned in the text.

5. Give a short characterization of each method.

TEXT 5


III

It is difficult to imagine that two different dynasties could have identical or almost identical

dynasty functions. The probability of such a coincidence is extremely small already for dynastiescomposed of 10 rulers. Nevertheless, the number of such coincidences, for even longer dynastiesof 15 rulers, turns out to be unexpectedly large. N.A. Morozov, who noticed the coincidence


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between the ancient Rome and the ancient Jewish state, discovered the first examples ofsurprisingly identical pairs of dynasty graphs. A formal method to study such similarities wasintroduced by A.T. Fomenko.

There is another surprise, besides coincidence of the dynasty functions, the other numericalfunctions confirm with very high probability that these dynasties are indeed the same. It brings usto a suspicion that in fact we are dealing with repetitions in the conventional version of the history.

Fomenko discovered dozens of strong coincidences, sometimes between three and more dynasties.But, there are no more such coincidences in the history of the better-documented epochs, forexample starting from the 16th century.

As an example, we would like to discuss two dynasties, one the dynasty of the Holy Roman-German Empire (10th - 13th AD) and another one of the Jewish kings according the Bible (9th -5th BC). On Figure 3, we represent the vertical time line with two graphs of reign durations on itsopposite sides for comparison. On this chart, we start the dates for the dynasty of Jewish kings inthe year zero, which is not a date according to some era but simply indicates the starting "zero"

point for this dynasty. According to the Encyclopedia Britannica, the beginning of this dynasty isaround 922 B.C. Figure 3 was taken from A.T. Fomenko monograph.

Figure 3

There are many more examples of similar dynasty pairs in the conventional chronology. Forinstance, the parallel between the first period of the Roman episcopate in 141-314 A.D. and thesecond period of the Roman episcopate in 314-532 A.D. is shown in Figure 4.

Figure 4


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On Figure 5, we present another pair of graphs, this time without annotations. All thesegraphs were also taken from the monograph.

Figure 5

These parallels suggest that the traditional history of ancient times consist of multiplerecounts of the same events scattered in many locations at various times. The first scientist whorealized it was N.A. Morozov. Further progress was made by A.T. Fomenko who succeeded todecipher the principle structure of these duplicates in Roman and Biblical history. On Figure 6, weshow a graphical representation of his result related to the Roman and European history. Thechronological blocks annotated by the same letters represent duplicates in the conventionalchronology.

Figure 6


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We will discuss some of typical arguments against the New Chronology. One of the mostpopular arguments in support of the conventional chronology is that the carbon-14 dating methodsupports it. But in fact it is not true. The carbon-14 method, which was discovered by WillardLibby, is based on the measurement of the radiocarbon level in organic samples. It assumesessentially uniform level of the isotope carbon-14 in every living material, but it is now clear that

carbon-14 was never homogeneously distributed. In fact, in order to improve its "accuracy," thecarbon-14 method was calibrated using samples of "known" age. It was done by constructing theso-called calibration curves, which are dependent on the conventional chronology. That means thecarbon-14 dating method is secondary and is not able either confirm or discard any chronologicaltheory. In addition, the errors induced by this method exceed all reasonable time intervals. Wewould like to point out that if the global chronology was changed, the carbon-14 dating methodwould also work nicely with the new dating system. It is not possible to present here a completediscussion of this complicated problem.

There are other arguments, of different type, claiming that there is nothing abnormal incoincidence of dynasty functions for different dynasties. For instance, we know that the probabilityof having winning lottery is very small but still there are communities that have one or more

lottery winners. So, even very unlike events could happen. Critics of the New Chronology oftenmention that biographies of certain rulers, like Napoleon and Hitler (both dictators) are quitesimilar, so by applying the method of Morozov and Fomenko we should consider them to be thesame person and ultimately make a senseless statement that the first 20 years of the 19th centuryare simply the years thirties and forties of the 20th century. There are many more similararguments, but all of them miss the point that extremely rare events only happen in large samples.For example, although the chances of having a winning lottery ticket are extremely small,nevertheless the probability that somebody wins is one. But, this is not the case with the unrelateddynasty functions, for which the coincidence in the whole sample is even less probable than thecoincidence of two random fingerprints.

There is also a claim that the "strange" coincidences between dynasty functions could beremoved by making appropriate corrections of the historical data. However, even with modifieddates the probability arguments still hold.

Regarding the archaeological dating, we should point out that it is closely dependent on theconventional chronology. The usual dating procedure in archaeology is based on the comparison ofthe excavated objects with objects already dated. In this procedure, finding some objects ofidentifiable style or origin can lead to a conclusion of the age of the whole site. The whole processis highly subjective and cannot be considered as a proof of the conventional chronology.

ASSIGNMENTS


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combinations: , , , , , , , , , .

2. The list below gives some of the typical arguments against the New Chronology.

Which three of the arguments are mentioned in the text?a) The first 20 years of the 19th century are simply the years thirties and forties of the 20thcentury.

b) The "strange" coincidences between dynasty functions could be removed by makingappropriate corrections of the historical data.c) The carbon-14 dating method supports conventional chronology.d) The probability of winning a lottery is quite small.e) Even very unlike events could happen.g) Identical pairs of dynasty graphs mean that these dynasties are indeed the same.h) There is nothing abnormal in coincidence of dynasty functions for different dynasties.i) All the methods proposed by A.T. Fomenko are highly subjective and devoid of

scientific approach.


a) Who introduced a formal method to study similarities and coincidences in history?b) Are there any coincidences in the history of the better-documented epochs, starting fromthe 16th century?c) What do parallels representing reign durations for comparison suggest?d) What are the typical arguments against the New Chronology?e) What chronology do you personally believe: the conventional or the new one?

4. Give a short summary of the text.

TEXT 6

Mathematics in Lexicography

I

Ambiguity is ubiquitous in natural language. The most common form of ambiguity concernsthe meanings of individual words, as in the following examples:

The minister decided to leave the party. (church minister/government minister, drinks

party/political party)Hes a curious individual. (odd/nosey)The last example involves homographs (different words which happen to be spelled the

same). However, it should be noted that only a small percentage of word sense ambiguity is due tohomography. Many words have gained multiple senses by metonymy or by figurative ormetaphorical uses. The resulting senses are sufficiently different to be considered bylexicographers as distinct concepts (e.g., political party/drinks party).

This text describes a stochastic model of the creation of word senses. This model not onlyexplains the near-exponential rule (the number of senses per word in a monolingual dictionary hasan approximately exponential distribution), but also provides a deeper insight into the process ofnaming.

Let LDbe a language as defined by the set of (word, sense)pairs in a dictionary D. Weconsider the evolution of the language LDover time. We must always bear in mind thatLDis, ofcourse, only an approximate representation of the semantics of the corresponding natural language.


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For example, the compiler of a dictionary may choose to include archaic words as a historicalrecord or to exclude whole categories of words such as slang or technical terms.

Consider the evolution ofLD as a stochastic process in which each step is either (a) theelimination of a word sense (by obsolescence), (b) the introduction of a new word (by creation,

borrowing, word-formation, or any other mechanism), or (c) the addition of a new sense for anexisting word (by association with an existing sense).

Let tbe the probability of a step of type (a), u the probability of a step of type (b), and v theprobability of a step of type (c). Note that t+ u + v = 1. The parameters of our model t, u, v areunknowns which will be estimated from the observed values ofNs (the number of words with ssenses).

We make the following simplifying assumptions:1. New-word single-sense assumption: When a neologism enters the language LD, it has a

single sense.2. Independence of obsolescence and number of senses: The probability that a (word,

sense) pair leaves the language LDby obsolescence is independent of the number of senses thisword has inLD.

The new-word single-sense assumption is an essential part of our model. To test it we require

two editions of the same dictionary. The 1994 edition of the Dictionnairede lacadmie franaiseindicates which words are new compared to the 1935 edition.

Less than 17% of these words are polysemic. Furthermore, this corresponds, according to ourmodel and to within-sample error, to the proportion of originally monosemic words entering thelanguage that can be expected to acquire new senses during the period between the publication ofthe two editions. Assumption 2 above is not as important as assumption 1, since later we restrictourselves to a no-obsolescence model.

The set ofs senses of an ambiguous word may correspond to a number c of essentiallydistinct concepts, where c is some number between one and s. For example, the plumbing andanatomy senses ofjointcorrespond to the same concept, since they could inspire the same newsenses by association. The group of criminals sense ofjointclearly corresponds to a differentconcept, since it could inspire a very different set of new senses by association. Associationsinspired by distinct concepts are assumed to occur independently. We assume that a word with ssenses inLDrepresents on average 1 + (s 1) concepts. We call the concept creation factor(since, in a no-obsolescence model, is simply the probability that a new sense for a word w can

be considered a new concept compared to the existing senses for w).We can now state a thirdassumption:

3. Associations are with concepts: The probability that a concept gives rise to a new sensefor a word w by association is proportional to the number of concepts represented by w in LD,which is assumed to be on average 1 + (s 1), wheres is the number of senses ofw and is aconstant.

The concept creation factor is another unknown which will be estimated from the values ofNs.

Table 1

NumberNs of words with s senses in samples from the 1933 and the 1993 edition of theShorter Oxford English Dictionary.

N1 N2 N3 N4 N5 N6 N7 N8 N91933 427 186 104 49 24 15 22 6 81993 403 176 86 44 32 16 14 7 1

We make a fourth hypothesis in order to render the problem mathematically tractable:


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4. Stationary-state hypothesis:LDconsidered as a stochastic process is in a stationary state,in the sense that the probability P(s) that an arbitrary word ofLD has exactly s senses does notchange asLDevolves.

To test the validity of the stationary-state hypothesis, we compared the 1933 and 1993editions of the Shorter Oxford English Dictionary (SOED). In the space of 60 years, the number ofwords in the SOED increased by 24%. Nevertheless the values ofP(s) (s = 1, 2, . . . , 9) remained

almost constant. A chi-square test revealed that the differences in the values ofP(s) (s = 1, 2, . . .)could be accounted for by sampling error.The corresponding values ofNs are given in Table 1.

ASSIGNMENTS


combinations: , , , , , , , , , , , , , -, .

2. Match:

Stationary-state hypothesis The probability that a concept gives rise to anew sense for a word w by association is

proportional to the number of conceptsrepresented by w inLD, which is assumed to beon average 1 + (s 1), wheres is the numberof senses ofw and is a constant.

New-word single-sense assumption LD considered as a stochastic process is in astationary state, in the sense that the probability

P(s) that an arbitrary word ofLD has exactly ssenses does not change asLDevolves.

Associations are with concepts The probability that a (word, sense) pair leavesthe languageLDby obsolescence is independentof the number of senses this word has inLD.

Independence of obsolescence and

number of senses

When a neologism enters the language it has asingle sense.


a) What is the most common form of ambiguity?b) What steps of language evolution are mentioned in the text?c) How many senses does a neologism have when it enters the language, according to thefirst simplifying assumption?d) How can the new-word single-sense assumption be tested?e) How do we test the validity of the stationary-state hypothesis?

4. Name all the simplifying assumptions mentioned in the text.

5. Speak on the way mathematics is used in lexicography.

TEXT 7


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Mathematics in Lexicography

II

Let mbe the expected number of senses per word inLD. Since

m = sP(s) (1)s=1

and the values ofP(s) are constant by the stationary-state hypothesis, m is also a constant.The expected net increase in the number of word senses in LDduring one step of the process

is t+ (1 t) = 1 2t, since the probability that a word sense is lost by obsolescence is tand theprobability that a word sense is gained is 1 t. Ifrdenotes the expected net increase in the numberof words in LD during one step of the process, then we must have (12t)/r = m, since m is aconstant.

Thus r= (1 2t)/m (2)

Note that the number of words inLDwould be constant if and only ift= 0.5.Letpout(s) represent the probability that the next change in the languageLDis that a word with

s senses loses one of its senses by obsolescence. Let pin(s) represent the probability that the nextchange inLDis that a word withs senses gains a new sense.

Note that pout(s) = t and pin(s) = v, by the definitions oftand v.

s=1 s=1By the stationary-state hypothesis, the expected net increase in Ns (the number of words in

LDwith exactly s senses) during one step must be proportional to P(s). Denote the expected netincrease inNsby s= dP(s), for some constant d.

We then have s= d, since p(s) = 1. But s= rsince the totals=1 s=1 s=1

expected increase in the number of words is r. Thus s = rP(s) = (1 2t)P(s)/m (by equation (2)).We can also express s, the expected net increase in Ns, in terms of the probabilities pin(s)

andpout(s), which gives the following equation:

(1 2t)P(s)/m = pin(s) pout(s) +pin(s 1) +pout(s + 1) (3)

since Ns is decremented when a word with s senses gains or loses a sense and Ns is

incremented when a word withs 1 senses gains a sense or a word withs + 1 senses loses a sense.From the assumption of the independence of obsolescence and number of senses, it followsdirectly thatpout(s) is proportional tosP(s). Letpout(s) =KsP(s), for some constantK.

Then, since pout(s) = t, we have t= KsP(s) =Kmby equation (1). s=1 s=1

ThusK= t/m andpout(s) = tsP(s)m

Under the assumption that associations are with concepts, pin(s) is proportional to both 1 +(s 1) andP(s).Suppose thatpin(s) =KP(s)(1 + (s 1)).


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Since pin(s) =v, P(s) = 1, and sP(s) = m,

s=1 s=1 s=1 we have v = KP(s)(1 + (s 1)) =K(1 ) +Km.

s=1

ThusK= v/(1 + m), and hence, fors = 1, 2, . . .

pin(s) = v(1 + (s 1))P(s)1 + m

Note that the creation of a new word with a single sense is a special case. By definition ofuas the probability that the next step of the process is the creation of a new word, pin(0) = u.Summing equation (3), fors = 1, 2, . . ., gives

1 2t = -Pout(1) +Pin(0) = -tP(1) + um m

Thus u = 1 2t+ tP(1) (4)m

and, since by definition v = 1 t u, v = 1 t 1 2t+ tP(1)m

Plugging in the formulas forpin(s), pout(s), and v, our basic equation (3) becomes, aftersimplification, fors > 1:

t(s + 1)(1 + m)P(s + 1) {(m mt 1 + 2t tP(1))(1 + s)+(1 2t+ ts)(1 + m)}P(s) + (m mt 1 + 2t tP(1))(1 2 + s)P(s 1) =0 (5)

Empirical evidence indicates thatP(s) is a near-exponential function. In fact, ifP(s) were anexponential function,

then since P(s) = 1 and sP(s) = m, we can easily deduce that s=1 s=1

P(s) would be equal to m1(1 m1)s1. The proof of the following result is simple but rathertedious and hence is omitted.

ASSIGNMENTS


combinations: , , , , , , ().

7. Get ready to read aloud all the formulae given in the text.



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a. What is since the probability that a word sense gained if the probability that aword sense is lost by obsolescence is t?

b. Would the number of words inLDbe constant ift= 0.5?c. What must the expected net increase in Ns (the number of words in LD with

exactly s senses) during one step be proportional to according to the stationary-state hypothesis?

d. IsP(s) a near-exponential function?e. Can you deduce the second given in the text formula without peeping at the text?

TEXT 8

Mathematics in Psychodiagnostics

I

This articlepresents a formalization for employee competencies which is based on a

psychological framework separating the overt behavioural level from the underlying competencelevel. On the competence level, employees draw on action potentials (knowledge, skills andabilities) which in a given situation produce performance outcomes on the behavioural level. Thisconception is based on the competence performance approach by Korossy which usesmathematical structures to establish prerequisite relations on the competence and the performancelevel. From this framework, a methodology for assessing competencies in dynamic work domainsis developed which utilizes documents employees have created to assess the competencies theyhave been acquiring. From the resulting structures, employee competency profiles can be derivedand development planning can be supported. The structures also provide the means for makinginferences within the competency assessment process which in turn facilitates continuous updatingof competency profiles and maintenance of the structures.

A theoretical framework is the Competence Performance Theory. It belongs to a family oftheories that originated from research into knowledge spaces.

A knowledge space is a mathematical structure consisting of all the knowledge states withina certain domain that a person may be in. A knowledge state is formalized as the subset of tasks ofthe domain that a person is capable of accomplishing. Of great importance are the dependencieswithin the set of tasks which can be interpreted as meaning if a person is capable ofaccomplishing task a, then he or she will also be able to accomplish task b. These dependenciesrestrict the number knowledge states than can be expected to appear within a certain population oflearners. These (and other) characteristics of the structures have been useful for creating adaptivetests. Adaptivity then means that an individual will be presented with those tasks that are

maximally suited to his or her current state of knowledge and therefore are neither too demanding,nor too easy.A possible limitation of knowledge space theory which has been put forward by Korossy is

its sole focus on the behavioural (or performance) level. The theory thereby neglects progress incognitive psychology or educational sciences that have advanced theoretical understanding of thereasons for different levels of performance. Such cognitive theories of the underlying skills thatshape performance offer better ways to suggest training and development measures. Therefore,Korossy has suggested an extension of the theory of knowledge spaces, which takes into accounttwo sets of concepts, namely the set of tasks (or more generally, the set ofperformance outcomes)and the set ofcompetencies (knowledge, skills and abilities) that are necessary to accomplish thetasks. If the elements of the two sets are related to one another, structures can be derived which can

be interpreted in terms of prerequisite relations or learning paths on the performance and on thecompetence level. Accordingly, the advantage of the competence-performance approach is that the


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competencies help to predict performance outcomes and provide an explanation for discrepanciesin performance.

For example, missing competencies can help to explain why an employee was not able toaccomplish a certain task. Hence, development programs can be created that focus on thesesunderlying competencies. Since its inception, knowledge space theory has been applied in manydifferent contexts and in different kinds of knowledge assessment procedures.

Figure 7 shows part of this structure with the tasks for the position (1.1, 2.1, 4.2 and so forth)and the competencies required to accomplish these tasks (A, B, I, J, and so forth). The resultingcompetence structure (on the right) shows learning paths for the employees that proceed from the

bottom to the top of the structure. Each step involves learning additional competencies so that newtasks can be mastered. For illustration purposes, it only shows part of the overall structure and

provides a simplified view of Korossys approach.There is a set Pof tasks, in the example those are the tasks that have to be carried out in a

certain position. Subsets of P are called Performance States, if they contain the tasks(performances) a person is able to accomplish. A collection of Performance States closed underunion is called aPerformance Space P. Of course the tasks are not independent of one another: If a

person is able to accomplish a certain task, one can surmise that he or she will be able to

accomplish a certain other tasks as well. This relationship is formalized by the surmise functions:P ( (P)) which assigns to each taskp Pa set of Performance Statess(p) in each of whichpis included, and which are minimal with respect to , meaning that if a person is able toaccomplish taskp, that person is at least in one of the Performance States ins(p).

Tasks

1.1 2.1 4.2 5.4

Competencies

A X

B X X

I X X X

J X X XL X

M X

O X X X

P X X X

Figure 7: Part of a Competence-Performance Structure for a Human Resource

A B I J L M O P

2.1 4.2 1.1

5.4

B I J L O P

2.1 1.1 5.4

A B I J M O P

4.2 1.1 5.4I J O

P

1.1

5.4

I J

5.4

O

P

1.1

I J O

P

1.1

5.4


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Manager

There is a one-to-one correspondence between the surmise function s and the PerformanceSpace P, meaning that Ponly consists of those states which can be expected on the basis of thesurmise function and vice versa. In the above structure for instance, a Performance State thatcontains task 2.1, but doesnt contain task 1.1, cannot be expected, as 1.1 is a prerequisite for 2.1.

The second set is the set of competencies which should be of smaller magnitude than theset of tasks, in order to provide some additional explanatory power:Performance in many different tasks should be determined by the combination of a few

underlying competencies. Eis structured in a similar way as P: A surmise function assigns toeach competency e Ea set of Competence States (e) in each of which e is included, and whichare minimal with respect to , meaning that if a person is possesses competency e, that person is atleast in one of the Competence States in (e). The surmise function establishes a CompetenceSpaceK. The two spacesPandKare connected by an interpretation function k: P (K), whichassigns to each taskp P a set of Competence States kx K in which the tasks can beaccomplished.

When the two sets Eand Pare related to one another in a matrix such as in Figure 7, acompetence-performance structure can be derived. First of all, Competence states can be derivedas subsets of competencies that are necessary for accomplishing the tasks (the columns of thematrix, e.g. {I;J}, {B;I;J;L;O;P} and so forth). By closing the collection of competence statesunder union, a Competence Space is derived. After that, all tasks that can be expected in a certaincompetence state are assigned to this state (see Figure 7).

ASSIGNMENTS


combinations: , , , , , ,

(), , ( ).2. Give definitions of the following notions: a knowledge space, a knowledge state,competence performance approach, adaptivity.

3. Name a few: action potentials,competencies,knowledge states.


a) What does the competence performance approach by Korossy use?b) What is a knowledge space?c) How can the dependencies within the set of tasks be interpreted?

d) What does adaptivity mean?e) In what way has Korossy extended the theory of knowledge spaces?f) How can Competence states be derived?

5. Complete the summary below using words from the box.

The Competence Performance Theory belongs to a family of theories that originatedfrom research into (1) _____.

A knowledge space is a mathematical structure consisting of all the (2) _____within acertain domain that a person may be in. A knowledge state is formalized as the subset oftasks of the domain that a person is capable of(3) _____ . Of great importance are the (4)

_____within the set of tasks which can be interpreted as meaning if a person is capable of

accomplishing task a, then he or she will also be able to accomplish task b. Thesedependencies restrict the number knowledge states than can be expected to appear within acertain population of learners. These (and other) characteristics of the structures have been


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useful for creating (5) _____ . Adaptivity then means that an individual will be presentedwith those tasks that are maximally suited to his or her current state of knowledge andtherefore are neither too demanding, nor too easy.

A possible limitation of knowledge space theory which has been put forward byKorossy is its sole focus on the (6) _____level. The theory thereby neglects progress in (7)

_____psychology or educational sciences that have advanced theoretical understanding of

the reasons for different levels of performance.

accomplishing, adaptive tests, behavioural, knowledge spaces, cognitive, dependencies,

knowledge states

6. Retell the summary of the text.

APPENDIX

Reading of Mathematical Expressions

1) a + ba plus b2) a ba minus b3) a ba multiplied by b4) a ba divided by b5) a a overb, ora divided by b b6) a = ba equals b, ora is equal to b7) m ab m divided by a multiplied by b8) ax The square root ofax9) a4 a fourth, a fourth power ora exponent 4

10) an

a nth, a nth power, ora exponent n11) nb The nth root ofb12) (a + b)2 = a2 + 2ab + b2 The square of the sum of two numbers is equal to the square of thefirst number, plus twice the product of the first and second, plus the square of the second13) (a b)2 = a2 2ab + b2 The square of the difference of two numbers is equal to the square of thefirst number minus twice the product of the first and second, plus the square of the second14) x Increment ofx15) Summation of 16) dx Differential ofx17) dy/dx Derivative ofy with respect tox18) d2y/dx2 Second derivative ofy with respect tox

19) dn

y/dxn

nth derivative ofy with respect tox20) dy/dx Partial derivative ofy with respect tox21) dny/dxn nth partial derivative ofy with respect tox22) Integral of 23)a

Integral between the limits a and b b

24) 5dnThe fifth root ofdto the nth power25) a + b

a bThe square root ofa plus b overa minus b26) a3 = logcda cubed is equal to the logarithm ofdto the base c

27)tf[S, (S)] ds The integral offofSand ofS, with respect to Sfrom to t


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28) d2y + (l+ b (S))y = 0 The second derivative ofy with respect tos, plusy times the quantity lds2 plus b ofs, is equal to zero

29)Xa b= etlXsub a minus b is equal to e to the powerttimes l30)f(z) =Kabfofzis equal toKsub ab31) 2u = 0 The second partial (derivative) ofu with respect to tis equal to zero t2

Glossary

Ambiguity is the property ofwords, terms, notations and concepts (within a particularcontext) as being undefined, indefinable, or without an obvious definition and thus having anunclearmeaning.

Behavioural based on the proposition that all things which organisms do, including acting,thinking and feeling, can and should be regarded asbehaviors.

Carbon-14 dating method dating method based on the measurement of the radiocarbonlevel in organic samples.

Cartesian product is a direct product of sets.Chi-square test is any statisticalhypothesis test in which the test statistic has a chi-square

distribution when the null hypothesis is true, or any in which theprobability distribution of the teststatistic (assuming the null hypothesis is true) can be made to approximate a chi-square distributionas closely as desired by making the sample size large enough.

Cognitive psychology is the school of psychology that examines internal mental processessuch as problem solving, memory, and language.

Monosemic word is the word that has only one meaning.Near-exponential rule the number of senses per word in a monolingual dictionary has an

approximately exponential distribution.Neologism is a word, term, orphrase which has been recently created ("coined") often

to apply to new concepts, to synthesize pre-existing concepts, or to make older terminology soundmore contemporary.

Polysemy is the capacity for a sign (e.g. a word, phrase, etc.) or signs to have multiplemeanings.

Quadrature of the circle is a problem proposed by ancientgeometers. It is the challengeto construct a square with the same area as a given circle by using only a finite number of stepswith compass and straightedge.

Semantics refers to the aspects ofmeaning that are expressed in a language, code, or otherform of representation ofinformation.

Stochastic model is a tool for estimating probability distributions of potential outcomes byallowing for random variation in one or more inputs over time.

Scientists Reference

BoyerC.B. (1906 1976) was a historian ofmathematics. He wrote the booksHistory ofAnalytic Geometry,History of the Calculus,A History of Mathematics, and The Rainbow: FromMyth to Mathematics.

Einstein A. (1879 1955) - contributed more than any other scientist to the modern vision ofphysical reality. His special and general theories of relativity are still regarded as the mostsatisfactory model of the large-scale universe that we have.

Fomenko A.T. (born 13 March 1945) is a Russian mathematician, professor of Moscow

State University, well-known as a topologist, and a full member of the Russian Academy ofSciences. He was born in Donetsk, Ukraine.Hardouin J. (1646 - 1729), French classical scholar, was born at Quimperin Brittany.
http://en.wikipedia.org/wiki/Wordshttp://en.wikipedia.org/wiki/Meaning_(linguistic)http://en.wikipedia.org/wiki/Behaviorhttp://en.wikipedia.org/wiki/Direct_producthttp://en.wikipedia.org/wiki/Statisticalhttp://en.wikipedia.org/wiki/Hypothesis_testhttp://en.wikipedia.org/wiki/Chi-square_distributionhttp://en.wikipedia.org/wiki/Chi-square_distributionhttp://en.wikipedia.org/wiki/Null_hypothesishttp://en.wikipedia.org/wiki/Probability_distributionhttp://en.wikipedia.org/wiki/Wordhttp://en.wikipedia.org/wiki/Terminologyhttp://en.wikipedia.org/wiki/Phrasehttp://en.wikipedia.org/wiki/Sign_(semiotics)http://en.wikipedia.org/wiki/Classical_antiquityhttp://en.wikipedia.org/wiki/Geometershttp://en.wikipedia.org/wiki/Square_(geometry)http://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Compass_and_straightedgehttp://en.wikipedia.org/wiki/Meaning_(linguistic)http://en.wikipedia.org/wiki/Languagehttp://en.wikipedia.org/wiki/Codehttp://en.wikipedia.org/wiki/Informationhttp://en.wikipedia.org/wiki/1906http://en.wikipedia.org/wiki/1976http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/March_13http://en.wikipedia.org/wiki/1945http://en.wikipedia.org/wiki/Russiahttp://en.wikipedia.org/wiki/Mathematicianhttp://en.wikipedia.org/wiki/Moscow_State_Universityhttp://en.wikipedia.org/wiki/Moscow_State_Universityhttp://en.wikipedia.org/wiki/Topologyhttp://en.wikipedia.org/wiki/Russian_Academy_of_Scienceshttp://en.wikipedia.org/wiki/Russian_Academy_of_Scienceshttp://en.wikipedia.org/wiki/Donetskhttp://en.wikipedia.org/wiki/Ukrainehttp://en.wikipedia.org/wiki/1646http://en.wikipedia.org/wiki/1729http://en.wikipedia.org/wiki/Francehttp://en.wikipedia.org/wiki/Quimperhttp://en.wikipedia.org/wiki/Brittanyhttp://en.wikipedia.org/wiki/Wordshttp://en.wikipedia.org/wiki/Meaning_(linguistic)http://en.wikipedia.org/wiki/Behaviorhttp://en.wikipedia.org/wiki/Direct_producthttp://en.wikipedia.org/wiki/Statisticalhttp://en.wikipedia.org/wiki/Hypothesis_testhttp://en.wikipedia.org/wiki/Chi-square_distributionhttp://en.wikipedia.org/wiki/Chi-square_distributionhttp://en.wikipedia.org/wiki/Null_hypothesishttp://en.wikipedia.org/wiki/Probability_distributionhttp://en.wikipedia.org/wiki/Wordhttp://en.wikipedia.org/wiki/Terminologyhttp://en.wikipedia.org/wiki/Phrasehttp://en.wikipedia.org/wiki/Sign_(semiotics)http://en.wikipedia.org/wiki/Classical_antiquityhttp://en.wikipedia.org/wiki/Geometershttp://en.wikipedia.org/wiki/Square_(geometry)http://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Compass_and_straightedgehttp://en.wikipedia.org/wiki/Meaning_(linguistic)http://en.wikipedia.org/wiki/Languagehttp://en.wikipedia.org/wiki/Codehttp://en.wikipedia.org/wiki/Informationhttp://en.wikipedia.org/wiki/1906http://en.wikipedia.org/wiki/1976http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/March_13http://en.wikipedia.org/wiki/1945http://en.wikipedia.org/wiki/Russiahttp://en.wikipedia.org/wiki/Mathematicianhttp://en.wikipedia.org/wiki/Moscow_State_Universityhttp://en.wikipedia.org/wiki/Moscow_State_Universityhttp://en.wikipedia.org/wiki/Topologyhttp://en.wikipedia.org/wiki/Russian_Academy_of_Scienceshttp://en.wikipedia.org/wiki/Russian_Academy_of_Scienceshttp://en.wikipedia.org/wiki/Donetskhttp://en.wikipedia.org/wiki/Ukrainehttp://en.wikipedia.org/wiki/1646http://en.wikipedia.org/wiki/1729http://en.wikipedia.org/wiki/Francehttp://en.wikipedia.org/wiki/Quimperhttp://en.wikipedia.org/wiki/Brittany


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Herodotus (484 BCca. 425 BC) is regarded as the "FatherofHistory". He is almostexclusively known for writing The Histories, a record of his 'inquiries' into the origins of theGreco-Persian Wars which occurred in 490 and 480-479 BCE especially since he includes anarrative account of that period, which would otherwise be poorly documented, and many longdigressions concerning the various places and peoples he encountered during wide-ranging travelsaround the lands of the Mediterranean and Black Sea.

Leonardo of Pisa (Fibonacci) (1170 1250) - played an important role in reviving ancientmathematics and made significant contributions of his own.Liber abaci introduced the Hindu-Arabic place-valued decimal system and the use of Arabic numerals into Europe.

Maxwell J.C. (1831 1879) did revolutionary work on electricity and magnetism and onthe kinetic theory of gases.Morozov N.A.

Newton I. (1643 1727) - the greatest English mathematician of his generation. He laid thefoundation for differential and integral calculus. His work on optics and gravitation make him oneof the greatest scientists the world has known.

Petavius D. (1583 1652) one of the most brilliant scholars in a learned age. Carrying on

and improving the chronological labours of Joseph Justus Scaliger, he published in 1627 an Opusde doctrina temporum, which has been often reprinted. An abridgment of this work,Rationariumtemporum, was translated into French and English, and has been brought down to the year 1849.

Scaliger I. (1540 1609) a French religious leader and scholar, known for expanding thenotion of classical history from Greek and Ancient Roman history to include Persian, Babylonian,Jewish and Ancient Egyptian history.

Ssu-ma Ch'ien (Sima Qian) (ca. 14590 BC) was a Prefect of the Grand Scribes of the HanDynasty. He is regarded as the father of Chinese historiography because of his highly praisedwork,Records of the Grand Historian an overview of the history of China covering more than twothousand years from the Yellow Emperorto Emperor Han Wudi. His work laid the foundation forlaterChinese historiography.

Thucydides (ca. 460 BC ca. 395 BC) was an ancient Greekhistorian, and the author of theHistory of the Peloponnesian War, which recounts the 5th century BC war between Sparta andAthens to the year411 BC. Thucydides is considered by many to be a scientific historian becauseof his efforts in his History to describe the human world in terms of cause and effect, his strictstandards of gathering evidence, and his neglect of the gods in explaining the events of the past.Other scholars lay greater emphasis on the Historys elaborate literary artistry and the powerfulrhetoric of its speeches and insist that its author exploited non-"scientific" literary genres no lessthan newer, rationalistic modes of explanation.

:

1. Robert A. Herrmann Ph. D. Mathematics. Mathematics department, U.S. NavalAcademy, 527 Holloway Road, Annapolis, MD 1402-5002.

2. A Dictionary of the English Language, Samuel Johnson, facsimile edition, Times Books,London, 1979 (original edition published 1755) (c. 40,000 words).

3. Newman, Mark E. J. The structure and function of complex networks. SIAM Review 45,2003

4. Pagel Mark, Onions, C. T., editor The Oxford Dictionary of English Etymology. OxfordUniversity Press, Oxford, 2000.

5. www.siam.org

6. The history, rate and pattern of world linguistic evolution. In Chris Knight, MichaelStuddert-Kennedy, and James R. Hurford editors, The Evolutionary Emergence of Language.Cambridge University Press, Cambridge, pages 391416, 2001
http://en.wikipedia.org/wiki/484_BChttp://en.wikipedia.org/wiki/425_BChttp://en.wikipedia.org/wiki/List_of_people_known_as_father_or_mother_of_somethinghttp://en.wikipedia.org/wiki/Historyhttp://en.wikipedia.org/wiki/Histories_(Herodotus)http://en.wikipedia.org/wiki/Greco-Persian_Warshttp://en.wikipedia.org/wiki/Mediterraneanhttp://en.wikipedia.org/wiki/Black_Seahttp://en.wikipedia.org/wiki/Joseph_Justus_Scaligerhttp://en.wikipedia.org/wiki/145_BChttp://en.wikipedia.org/wiki/90_BChttp://en.wikipedia.org/wiki/Prefecthttp://en.wikipedia.org/w/index.php?title=Grand_Scribes&action=edithttp://en.wikipedia.org/wiki/Han_Dynastyhttp://en.wikipedia.org/wiki/Han_Dynastyhttp://en.wikipedia.org/wiki/Chinese_historiographyhttp://en.wikipedia.org/wiki/Records_of_the_Grand_Historianhttp://en.wikipedia.org/wiki/History_of_Chinahttp://en.wikipedia.org/wiki/Yellow_Emperorhttp://en.wikipedia.org/wiki/Emperor_Wu_of_Han_Chinahttp://en.wikipedia.org/wiki/Chinese_historiographyhttp://en.wikipedia.org/wiki/460_BChttp://en.wikipedia.org/wiki/395_BChttp://en.wikipedia.org/wiki/Greekshttp://en.wikipedia.org/wiki/Historyhttp://en.wikipedia.org/wiki/History_of_the_Peloponnesian_Warhttp://en.wikipedia.org/wiki/5th_century_BChttp://en.wikipedia.org/wiki/Spartahttp://en.wikipedia.org/wiki/Athenshttp://en.wikipedia.org/wiki/411_BChttp://en.wikipedia.org/wiki/484_BChttp://en.wikipedia.org/wiki/425_BChttp://en.wikipedia.org/wiki/List_of_people_known_as_father_or_mother_of_somethinghttp://en.wikipedia.org/wiki/Historyhttp://en.wikipedia.org/wiki/Histories_(Herodotus)http://en.wikipedia.org/wiki/Greco-Persian_Warshttp://en.wikipedia.org/wiki/Mediterraneanhttp://en.wikipedia.org/wiki/Black_Seahttp://en.wikipedia.org/wiki/Joseph_Justus_Scaligerhttp://en.wikipedia.org/wiki/145_BChttp://en.wikipedia.org/wiki/90_BChttp://en.wikipedia.org/wiki/Prefecthttp://en.wikipedia.org/w/index.php?title=Grand_Scribes&action=edithttp://en.wikipedia.org/wiki/Han_Dynastyhttp://en.wikipedia.org/wiki/Han_Dynastyhttp://en.wikipedia.org/wiki/Chinese_historiographyhttp://en.wikipedia.org/wiki/Records_of_the_Grand_Historianhttp://en.wikipedia.org/wiki/History_of_Chinahttp://en.wikipedia.org/wiki/Yellow_Emperorhttp://en.wikipedia.org/wiki/Emperor_Wu_of_Han_Chinahttp://en.wikipedia.org/wiki/Chinese_historiographyhttp://en.wikipedia.org/wiki/460_BChttp://en.wikipedia.org/wiki/395_BChttp://en.wikipedia.org/wiki/Greekshttp://en.wikipedia.org/wiki/Historyhttp://en.wikipedia.org/wiki/History_of_the_Peloponnesian_Warhttp://en.wikipedia.org/wiki/5th_century_BChttp://en.wikipedia.org/wiki/Spartahttp://en.wikipedia.org/wiki/Athenshttp://en.wikipedia.org/wiki/411_BC


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7. Tobias Ley Math matters. Apply it. Journal of Universal Computer Science, vol. 9, no.12 (2003), 1500-1518

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