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TRANSCRIPT
XII. Antalya Cebir Günleri
Antalya, Türkiye
– Mayıs
Welcome to Antalya Algebra Days XII,
meeting coordinator
AAD XII Contents
Contents
Invited talks
. Wilfrid Hodges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mahmut Kuzucuoğlu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ömer Küçüksakallı . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . James D. Lewis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christian Lomp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Daniel Panario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexander Pott . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peter Roquette. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Hans-Georg Rück . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Amin Shokrollahi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Patrick F. Smith . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Arne Winterhof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contributed Talks
. Ahmet Arıkan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. A. O. Asar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inês Margarida Rodrigues Pais da Silva Borges . . . . . . . . . . . . . . . . . . . . . . Engin Büyükaşık . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ahmet Sinan Çevik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yılmaz Durgun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Betül Gezer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pedro A. Guil Asensio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Ayşe Dilek Güngör . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Serpil Güngör . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Erhan Gürel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sevgi Harman. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Eylem Güzel Karpuz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fatih Koyuncu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Engin Mermut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figen Öke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hakan Özadam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ferruh Özbudak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dilek Pusat-Yılmaz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contents AAD XII
. Aslıhan Sezgin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ebru Solak. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Funda Taşdemir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Eylem Toksoy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Posters
. Akleylek, Cenk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Nurdagül Anbar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Cam Vural . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Münevver Çelik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ayça Çeşmelioğlu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deveci, Duman. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Şükrü Uğur Efem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mehmet İnan Karakuş . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dilber Koçak. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Azadeh Neman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ata Firat Pir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elif Saygı . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zülfükar Saygı . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Seher Tutdere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Personnel
Acknowledgements
Participants
Notes on Turkish
Session chairs
Timetable
AAD XII INVITED TALKS
Invited talks
Definability versusdefinability-up-to-isomorphism, in groups and
fields
Wilfrid Hodges
Many algebraic constructions F , where F (A) is a structure built on thestructure A, are defined up to isomorphism over A. (A typical example isalgebraic closure, where F (A) is the algebraic closure of a field A.) Suppose weask whether such a construction F can be defined outright and not just up toisomorphism. In general the answer calls on some interactions of cohomologyand set theory. The talk will survey examples from groups and fields, andwill mention recent joint work with Saharon Shelah.
Herons Brook, Sticklepath, Okehampton EX PY, England
http://wilfridhodges.co.uk
References AAD XII
Universal Groups
Mahmut Kuzucuoğlu
A group is called a locally finite group if every finitely generated sub-group is a finite group. A locally finite group U is called universal if
(a) every finite group can be embedded into U ,
(b) any two isomorphic finite subgroups of U are conjugate in U .
Existence and basic properties of countable universal locally finite groups aregiven by P. Hall in [] see also in []. For any given uncountable cardinalityκ, existence of 2κ non-isomorphic universal locally finite groups of cardinalityκ is given by S. Shelah and A. J. Macintyre in [].
We are interested in centralizers of finite subgroups in simple non-linearlocally finite groups. In particular the following question. Is the centralizer ofevery finite subgroup in a non-linear locally finite simple group infinite ? Weanswer this question for direct limit of finite alternating groups. Particularcase gives an answer to the centralizers of finite subgroups in universal groups.
Theorem . (Ersoy-Kuzucuoğlu) Let G be a simple locally finite group whichis a direct limit of finite alternating groups, and F be a finite subgroup of G.Then CG(F ) contains an abelian subgroup A which is isomorphic to Dr
p primeZp.
We also mention universal groups that are not necessarily locally finitegroups constructed by O. H. Kegel in []. We will discuss basic properties ofthis regular limit group Sλ of symmetric groups. We also discuss the followingresult.
Lemma . (O. H. Kegel, M. Kuzucuoğlu) Let B be a bounded subgroup of aregular limit group Sλ with trivial center. Then C
Sλ(B) is isomorphic to Sλ.
References
[] Ersoy, K; Kuzucuoğlu M. Centralizers of subgroups in simple locallyfinite groups, Submitted.
[] Hall, P. Some constructions for locally finite groups, J. London Math.Soc. () –.
[] Kegel, O. H., Wehrfritz, B. () Locally Finite Groups, North-HollandPublishing Company.
AAD XII References
[] Kegel, O. H. Regular limits of infinite symmetric groups, Ischia GroupTheory , World Scientific
[] Macintyre, A.J. and Shelah, S. Uncountable Universal Locally FiniteGroups, J. Algebra () -.
Middle East Technical UniversityDepartment of Mathematics Ankara, TURKEY
metu.edu.tr/~matmah
References AAD XII
Computing class numbers via elliptic units
Ömer Küçüksakallı
The class number is a powerful invariant in algebraic number theory whichcan be used to investigate the integer solutions of polynomials, such as Fer-mat’s Equation. It can be computed for extensions with small degree anddiscriminant, however computations take a very long time for higher exten-sions. In this talk, we will describe a heuristic method to compute the classnumbers of some abelian extensions of imaginary quadratic fields. This isthe elliptic analogue of an algorithm of Schoof used for cyclotomic fields.We will use elliptic units analytically constructed by Stark and the Galoisaction on them given by Shimura’s reciprocity. In the end we will give acounter-example to Vandiver’s conjecture in the elliptic curve case.
References
[] Class numbers of ray class fields of imaginary quadratic fields. To appearin Mathematics of Computation.
Middle East Technical University, Department of Mathematics, AnkaraTurkey
www.metu.edu.tr/~komer/
AAD XII References
The Bloch–Kato theorem and Hodge typeconjectures
James D. Lewis
The Bloch–Kato conjecture was recently proven by V. Voevodsky and hiscollaborators. It is a generalization of the Merkurjev–Suslin theorem andthe Milnor conjecture [theorem]. This conjecture [theorem] turns out to beunder the same general umbrella as the Hodge conjecture and its generaliza-tions (due to Beilinson). We will explain the Bloch–Kato theorem and itsconnection to the Hodge type conjectures.
University of Alberta
http://www.math.ualberta.ca/Lewis_JD.html
References AAD XII
Injective hulls of simple modules over Down-UpAlgebras
Christian Lomp
The module theoretical properties of indecomposable injective modules overa Noetherian ring R are important for the structure theory of R. For a com-mutative Noetherian ring R, Eben Matlis showed in [] that any injectivehull of a simple R-module is Artinian, a property that, in general does nothold for non-commutative rings. However Randall Dahlberg showed in []that injective hulls of simple modules over U(sl2) are locally Artinian. Theenveloping algebra U(sl2) is an instance of a larger class of Noetherian do-mains, the Down-Up algebras, introduced by Georgia Benkart and Tom Robyin []. The Down-Up algebras A(α, β, γ) form a three parameter family ofassociative algebras. For any parameter set (α, β, γ) ∈ C
3 one defines a C-algebra, denoted by A(α, β, γ), generated by two elements u and d subject tothe relations
d2u = αdud+ βud2 + γd
du2 = αudu+ βu2d+ γu
which is a Noetherian domain if and only if β 6= 0. In particular A(2, 1, 1) =U(sl2) holds.
During the X. Antalya Algebra Days, Patrick Smith asked in a private con-versation with Paula Carvalho, which Noetherian Down-Up algebras satisfythe condition that their injective hulls of simple modules are locally Artinian.
I will present the findings on Patrick’s question from our joint work [] withPaula Carvalho and Dilek Pusat-Yilmaz, namely that a Noetherian Down-Up algebra A(α, β, γ) has the desired property if the roots of the polynomialX2 −αX − β are distinct roots of unity or both equal to 1. If time permits Iwill also report on the progress made by Paula Carvalho and Ian Musson in[].
References
[] Matlis, E., Modules With Descending Chain Condition, Trans. Amer.Math. Soc. (), - ()
[] Dahlberg, R.L., Injective Hulls of Simple sl(, C) Modules are LocallyArtinian, Proc. Amer. Math. Soc. (), - ()
AAD XII References
[] Benkart, G. and Roby, T., Down-up algebras., J. Algebra (), no., -.
[] Carvalho, P.A.A.B, Lomp C. and Pusat-Yilmaz, D., Injective Modulesover Down-Up Algebras, to appear in Glasgow Math. J.
[] Carvalho, P.A.A.B and Musson, I., Monolithic modules over NoetherianRings, arXiv:.
University of Porto
www.fc.up.pt/mp/clomp
References AAD XII
Normal bases in finite fields
Daniel Panario
This talk surveys normal bases and normal elements in finite fields. Theseconcepts were defined, and their existence proved, years ago. However,due to their many recent applications, they have been vastly studied in thelast years.
Let q be a prime power. An element α in a finite field Fqn is called normalif N = α, αq, . . . , αqn−1
is a basis of Fqn over Fq. In this case, the basis Nis called a normal basis of Fqn over Fq.
First we briefly give an account of basic properties and results of normalelements including existence and number of normal elements.
Then we focus on how to operate with normal basis. As Hensel noted, ina normal basis qth powers are for free. This can be exploited to have fastexponentiation algorithms. As a consequence, normal elements are importantin cryptographic applications where exponentiation and discrete logarithmcomputations are employed.
Next we discuss how to find normal elements. It turns out that not all nor-mal elements behave in the same way, the so called optimal normal elementsbeing preferable for most computations with normal elements. These specialelements are directly related to Gauss periods in finite fields and have beencharacterized by Gao and Lenstra. Unfortunately, optimal normal elementsonly exists for some extension fields. This makes the study of low complex-ity normal elements relevant. We comment on several old and new resultsto produce low complexity normal elements. We conclude giving some openproblems.
Carleton University, Canada
AAD XII References
APN and PN functions: Differences andSimilarities
Alexander Pott
Motivated by cryptography, one is interested in functions
f : F np → F
np
such that the equations (given a 6= 0 and b)
f(x+ a)− f(x) = b
have only a few solutions δ(a, b). More precisely, the maximum value of allthe numbers |δ(a, b)| should be small. It is easy to see that the maximum is2 if p = 2, and it is 1 if p is odd. In the case p = 2, functions which achievethis minimum are called almost perfect nonlinear (APN), in the case podd perfect nonlinear (PN).
Both for the PN and the APN case, some (but not too many) examples areknown: infinite families as well as sporadic examples. It seems that there aremore APN functions known since the defining property for APN functionsis less restrictive: Some of the δ(a, b) are 0, some are 2. In the PN case,all these numbers must be 1. Similarly, the absolute values of the Walshcoefficients in the APN case are not determined by the APN property, butthey are determined in the PN case.
Another important difference seems to be the underlying algebraic struc-ture: Quadratic PN functions (and all PN functions except those constructedby Coulter and Matthews are quadratic) give rise to a strong algebraic struc-ture (semifields). Nothing comparable seems to be true for quadratic APNfunctions.
PN functions can be used to construct finite projective planes. APN func-tions also describe certain incidence structures, but these have, in general,less structure than projective planes.
However, there are also similarities between APN and PN functions: Someconstructions, described in terms of polynomials in Fpn , work both for thePN and the APN case. Moreover, a switching construction which has beenshown to be quite powerful in the APN case has the potential to be usefulalso in the PN case.
In my talk, I will discuss these similarities and differences between APNand PN functions. In particular, I will cover the following topics:
• Incidence structures defined by PN and APN functions.
References AAD XII
• Automorphism groups of these incidence structures.
• Semifields and the equivalence of functions.
• The switching construction of PN and APN functions.
Otto-von-Guericke-University Magdeburg
AAD XII References
News on the Arf invariant
Peter Roquette
The -Lira note of Turkish currency carries the portrait of the mathe-matician Cahit Arf, accompanied with a formula for the Arf invariant of aquadratic form. I shall explain the notion of Arf invariant and its place withinthe general theory of quadratic forms. The talk is relying not only on pub-lished papers but also on letters and other documents of Arf’s time. Recentlyit has turned out that some of Arf’s results have to be modified.
References
[] Roquette, Peter and Lorenz, Falko: On the Arf invariant in historicalperspective. Math. Semesterberichte () -.
University of Heidelberg, Germany
roquette.uni-hd.de
References AAD XII
Elliptic curves and Drinfeld modules
Hans-Georg Rück
This is an introductory talk to the theory of Drinfeld modules. We want toexplain how Drinfeld modules can be defined analogously to elliptic curves,following the path from an analytic torus to an algebraic structure.
References
[] David Goss: Basic Structures of Function Field Arithmetic, Springer,.
[] Hans-Georg Rück, Ulrich Tipp: Heegner points and L-series of auto-morphic cusp forms of Drinfeld type, Documenta Mathematica (),-.
Universität Kassel, Germany
http://www.mathematik.uni-kassel.de/~rueck
AAD XII References
Efficiency of Random Matrices over FiniteFields
Amin Shokrollahi
A “random” m × n matrix over a field K is a matrix sampled from someprobability distribution over the space of such matrices. In this talk, wewill investigate properties of such matrices over finite fields using severalinteresting probability distributions. The final goal is to construct matricesthat behave like uniform random matrices (where the probability distributionis uniform) as far as their rank properties are concerned, and at the same timeallow for very fast algorithms for solving systems of linear equations. Suchmatrices are used in the design of state of the art codes which allow forrecovery of data in the face of data erasures.
École Polytechnique Fédérale de Lausanne (EPFL) - ALGO
http://algo.epfl.ch/
References AAD XII
Homological properties and chain conditions
Patrick F. Smith
There are many results linking homological properties of rings and moduleswith other properties, in particular chain conditions. The famous Auslander-Buchsbaum-Serre Theorem is one such. We shall investigate some of theseresults starting with the Auslander-Buchsbaum-Serre Theorem and includingthe work of Cohn on free ideal rings and more generally hereditary rings.Cohn’s work is related to a theorem of Schreier on free groups.
University of Glasgow
AAD XII References
Exponential sums and linear complexity ofnonlinear pseudorandom number generators
Arne Winterhof
Let p be a prime, r a positive integer, q = pr and denote by Fq the finitefield of q elements. Given a polynomial f(X) ∈ Fq[X] of degree d > 2, wedefine the nonlinear pseudorandom number generator (µn) by the recurrencerelation
µn+1 = f(µn), n = 0, 1, . . . , (∗)
with µ0 ∈ Fq such that (µn) is purely periodic with period T 6 q.Niederreiter and Shparlinski developed a method to study the exponential
sums
Sa,N (f) =
N−1∑
n=0
χ
s−1∑
j=0
αjµn+j
, 1 6 N 6 T,
where χ is a nontrivial additive character of Fq and a = (α0, . . . , αs−1) ∈Fsqr0, see also the survey []. In general this method leads only to a nontrivial
bound if d = qo(1).For a nonnegative integer we define its p-weight as
σ
(
l∑
i=0
nipi
)
=
l∑
i=0
ni if 0 6 ni < p.
For 0 6= f(X) =∑d
i=0 γiXi ∈ Fq[X] we define its p-weight degree as
w(f) = maxσ (i) |γi 6= 0, 0 6 i 6 d.
Therefore, w(f) 6 deg(f). Under certain restrictions on f(X) we proved in[] a bound on Sa,N (f) which is nontrivial if w(f) is small enough but thedegree can be large.
We also use the p-weight to bound the N th linear complexity of the sequencedefined in (∗). The linear complexity is a measure for the unpredictabilityand thus suitability in cryptography.
References
[] Alvar Ibeas and Arne Winterhof. Exponential sums and linear complex-ity of nonlinear pseudorandom number generators with polynomials ofsmall p-weight degree. Unif. Distrib. Theory, to appear.
References AAD XII
[] Alev Topuzoğlu and Arne Winterhof. Pseudorandom sequences. In Top-ics in geometry, coding theory and cryptography, volume of Algebr.Appl., pages –. Springer, Dordrecht, .
Austrian Academy of Sciences (Linz)
http://www.ricam.oeaw.ac.at/people/page.cgi?firstn=Arne;lastn=Winterhof
AAD XII References
Contributed Talks
Some progress on minimal non X-groups
Ahmet Arıkan
Let X be a class of groups. A group G is called a minimal non X-group ifevery proper subgroup of G is an X-group but G itself is not.
We consider certain classes like “minimal non-solvable groups”, “minimalnon-Baer groups” and some others here, and give some recent results relevantto them.
Infinite perfect groups will be under consideration and the results will bedisplayed mostly in Fitting p-group case.
References
[] Arıkan, A., Characterizations of minimal non-solvable Fitting p-groups.J. Group Theory , no. , - ().
[] Arıkan, A., Sezer, S. and Smith, H., Locally finite minimal non-solvablegroups, to appear in Central European Journal of Mathematics.
[] Arıkan, A., Trabelsi N., Perfect minimal non-Baer groups, submitted.
[] Asar, A. O., Locally nilpotent p-groups whose proper subgroups are hyper-central or nilpotent-by-Chernikov. J. London Math. Soc. () (),no. , -.
Gazi University
http://websitem.gazi.edu.tr/~arikan
References AAD XII
Barely Transitivity and hypercentrality inLocally Finite p-Groups
A. O. Asar
In this work it is shown that if there exists a perfect locally finite barelytransitive p-group, then it has a finite subgroup whose centralizer has fi-nite exponent. As an application of this result it follows that there doesnot exist a totally imprimitive p-subgroup of FSym(Ω) which is a minimalnon-FC-group , where Ω is infinite. This result together with the earlierresults answers the following question in the negative: Does there exist aperfect locally finite minimal non-FC-group? Of course an imperfect locallyfinite minimal non-FC-group exist. It is an extension of its commutator sub-group which is a divisible abelian q-group of finite rank by a cyclic p-group.Furthermore it is shown that the existence of a perfect locally finite minimalnon-hypercentral p-group satisfying certain properties implies the existence ofa perfect locally finite barely transitive p-group. Finally a sufficient conditionis given for a perfect locally finite countable minimal non-hypercentral andnon-(residually finite) group to contain a finite subgroup whose centralizerhas finite exponent.
AAD XII References
Irreducible actions of Hopf algebras
Inês Margarida Rodrigues Pais da Silva Borges
A theorem by Bergen, Cohen and Fishman states that if a Hopf algebra Hacts finitely on a module algebra A with finite Goldie dimension, such that Ais a simple A#H-module, then A has finite vector space dimension over AH .At the heart of its proof is the Jacobson’s Density Theorem. In this talk weextend this theorem to certain operator algebras using Julius Zelmanowitz’density theorems.
ISCAC, Coimbra
References AAD XII
Rings over which flat covers of simple modulesare projective
Engin Büyükaşık
Throughout, R is a ring with a unit element and all modules are unitalright R-modules. In [] L. Bican et al. proved that all modules have flatcovers over arbitrary rings. It is known that, a ring R is right perfect if andonly if flat cover of any right R-module is projective. The rings over whichflat covers of finitely generated modules are projective are characterized in[] and [].
The aim of this talk is to introduce and give several characterization ofthe rings R over which flat covers of simple right R-modules are projective.A ring R is said to be right B-perfect if Hom(F, R) → Hom(F, R/I) is anepimorphism for every flat right module F and maximal right ideal I of R.
Theorem . For a ring R the following are equivalent.. R is right B-perfect.. Flat covers of simple modules are projective.. R is semiperfect and flat covers of simple modules are local.
Theorem . For a ring R the following are equivalent.. R is right B-perfect.. Every right ideal of R containing J(R) is cotorsion.. R is semilocal and J(R) is right cotorsion.
References
[] B. Amini, A. Amini and M. Ershad, Almost-perfect rings, Comm. Alg.,,, -.
[] A. Amini, M. Ershad and H. Sharif, Rings over which flat covers offinitely generated modules are projective, Comm. Alg., , , -.
[] L. Bican, R. El Bashir and E. Enochs, All modules have flat covers, Bull.London. Math. Soc., , , -.
[] P. A. Guil Asensio, I. Herzog, Left cotorsion rings, Bull. London Math.Soc., , , -.
[] J. Xu, Flat covers of modules, Lecture notes in mathematics, vol.,Springer, .
AAD XII References
Izmir Institute of Technology
http://web.iyte.edu.tr/~enginbuyukasik/
References AAD XII
Subgroup Separability and Efficiency
Ahmet Sinan Çevik
Let G be a group and let H be a subgroup of G. Then G is said to beH-separable if, for each x ∈ G−H, there exists N ⊳ G with finite index suchthat x /∈ NH. Moreover G is called subgroup separable if G is H-separablefor all finitely generated subgroups H of G. The newest known results aboutsubgroup separability can be found, for instance, in a joint paper “(Cyclic)Subgroup Separability of HNN and Split Extensions” written by Ateş andÇevik (published in Math. Slovaca, Vol () (), -). Furthermorelet S be a generating set for G. We also recall that the Cayley graph ofG, denoted by ΓG, with respect to S has a vertex for every element of G,with an edge g to gs for all elements g ∈ G and s ∈ S. Thus the initialvertex of the edge is g and the terminal is gs. Finally we let remind thedefinition of “efficiency” on finitely presented groups. So let us suppose thatG is such a group with a finite presentation PG = 〈X;R〉. Then the Eulercharacteristic of PG is defined by χ(P) = 1 − |X| + |R|, where |.| denotesthe number of elements in the related set. Also there exists an upper boundδ(G) = 1− rkZ(H1(G)) + d(H2(G)), where rkZ(.) denotes the Z-rank of thetorsion-free part and d(.) denotes the minimal number of generators. In fact,by a paper written by Epstein in , it always true that χ(PG) > δ(G). Wethen define χ(G) = minχ(P) : P is a finite presentation for G. Hence thepresentation PG is called efficient if χ(PG) = δ(G). In addition, G is calledefficient if χ(G) = δ(G).
In this talk we are mainly interested in separability and efficiency on groupsunder standard wreath products. To do that we will first give a new geometricway to get a presentation for the standard wreath product in terms of Cayleygraphs. Then we will express the first result of the talk about efficiency.Moreover, by considering the standard wreath product G of any finite groupsB by A, we will give the relationship between B-separability and efficiencyon G as another result of the talk. We note that these two results have beenobtained by Çevik and Ateş in a joint work which was published in the RockyMountain J. Math. () (), -.
References
[] F. Ateş and A.S. Çevik, Separability and Efficiency under StandardWreath Product in terms of Cayley Graphs, Rocky Mountain J. Math.() (), -.
AAD XII References
[] D.B.A. Epstein, Finite Presentations of Groups and -manifolds, Quart.J. Math. Oxford Ser(), (), -.
Selcuk University, Konya-Turkey
http://asp.selcuk.edu.tr/asp/personel/web/goster.asp?sicil=8857
References AAD XII
The least proper class containing weaksupplements
Yılmaz Durgun
This study deals with the classes Small, S and WS of short exact sequenceof R-modules determined by small, supplement and weak supplement sub-modules respectively, and the class WS which is the least proper class con-tain all of them over a hereditary ring R. Small is the class of all shortexact sequences 0 −→ A −→α B −→ C −→ 0 where Im(α) ≪ B, WS isthe class of all short exact sequences 0 −→ A −→α B −→ C −→ 0 whereIm(α) has(is) a weak supplement in B. S is the class of all short exact se-quence 0 −→ A −→α B −→ C −→ 0 where Im(α) has a supplement in Bdefined by Zöschinger in [] may not form proper classes. The classes aredifferent from each other, in general. On the other hand the proper classesgenerated by these classes, that is the least proper classes containing theseclasses are equivalent: 〈Small〉 = 〈S〉 = 〈WS〉 (The least proper class con-taining a class A is denoted by 〈A 〉 see []). WS-elements are preservedunder Ext(g, f) : Ext(C,A) −→ Ext(C ′, A′) with respect to the second vari-able, they are not preserved with respect to the first variable. We extendthe class WS to the class WS, which consists of all images of WS-elementsof Ext(C,A′) under Ext(f, 1A) : Ext(C ′, A) −→ Ext(C,A) for all homomor-phism f : C −→ C ′.
To prove that WS is a proper class we will use the result of [] that statesthat a class P of short exact sequences is proper if ExtP(C,A) is a subfunctorof ExtR(C,A), then ExtP(C,A) is a subgroup of ExtR(C,A) for every R-modules A,C and the composition of two P-monomorphism(epimorphism) isa P-monomorphism(epimorphism).We obtain the following results:
Lemma . If f : A −→ A′, then f∗ : Ext(C,A) −→ Ext(C,A′) preservesWS-elements.
Lemma . If g : C ′ −→ C, then g∗ : Ext(C,A) −→ Ext(C ′, A) preservesWS-elements.
Corollary . The WS-elements of Ext(C,A) form a subgroup.
Lemma . Let R be hereditary ring. For a WS class of short exact se-quences of R modules, the composition of an Small-epimorphism and a WS-epimorphism is a WS-epimorphism.
Lemma . Let R be hereditary ring. For a WS class of short exact se-quences of R modules, the composition of two WS monomorphism is a WSmonomorphism.
AAD XII References
Theorem . If R is a hereditary ring, WS is a proper class.
Corollary . If R is hereditary ring, then 〈Small〉 = 〈S〉 = 〈WS〉 = WS.
Joint work with: Prof. Rafail Alizade.
References
[] Butler, M. C. R. and G. Horrocks. . Classes of Extensions and Res-olutions. Philos. Trans. R. Soc. London (A):-.
[] Nunke, R. J. . Purity and Subfunctor of the Identity. Topics inAbelian groups (Proc. Sympos., New Mexico State Univ., ) -.
[] Pancar, A. . Generation of Proper Classes of Short Exact Sequences.Internat. J. Math. and Math. Sci. ():-.
[] Zöschinger, H. . Über Torsions- und κ-Elemente von Ext(C,A). Jour-nal of Algebra :-.
IZMIR INSTITUTE OF TECHNOLOGY
References AAD XII
Squares and Cubes in Elliptic DivisibilitySequences
Betül Gezer
Elliptic divisibility sequences (EDSs) are generalizations of a class of integerdivisibility sequences called Lucas sequences. There has been much interestin cases where the terms of Lucas sequences are squares and cubes. But thequestion of when a term of an EDS can be a square has not been answeredyet. We answer this question by using the general terms of these sequences.In this work, we give the general terms of the elliptic divisibility sequenceswith zero terms and then we determine which terms of these are squares andcubes.
This is joint work with Osman Bizim, [email protected].
References
[] A. Bremner, N. Tzanakis, Lucas sequences whose th or th term is asquare, J. Number Theory (), -.
[] A. Bremner, N. Tzanakis, On squares in Lucas sequences, J. NumberTheory (), -.
[] M. Einsiedler, G. Everest, T. Ward, Primes in elliptic divisibility se-quences, LMS J. Comput. Math. (), -, (electronic).
[] G. Everest, A. van der Poorten, I. Shparlinski, T. Ward, Recurrence Se-quences, Mathematical Surveys and Monographs, , AMS, Providence,RI, .
[] G. Everest, T. Ward, Primes in divisibility sequences, Cubo Mat. Educ. (), -.
[] B.Gezer, O. Bizim, Squares in Elliptic divisibility sequences, Acta Arith-metica, to appear.
[] A.Pethö, Full cubes in the Fibonacci sequence, Publ. Math. Debrecen, (),-.
[] P. Ribenboim, Pell numbers, squares and cubes, Publ. Math. Debrecen, (), -.
[] P. Ribenboim, W. McDaniel, The square terms in Lucas sequences, J.Number Theory (), -.
AAD XII References
[] P. Ribenboim, W. McDaniel, Squares in Lucas sequences having an evenfirst parameter, Colloquium Mathematicum ,(), -.
[] R. Shipsey, Elliptic divisibility sequences, PhD thesis, Goldsmith’s (Uni-versity of London), .
[] C. S. Swart, Elliptic curves and related sequences, PhD thesis, RoyalHolloway (University of London), .
[] M. Ward, The law of repetition of primes in an elliptic divisibility se-quences, Duke Math. J. (), -.
[] M. Ward, Memoir on elliptic divisibility sequences, Amer. J. Math. (), -.
Uludag University, Faculty of Science, Department of Mathematics, Görükle. Bursa-TÜRKIYE
References AAD XII
Model Category Structures Arising fromDrinfeld Vector Bundles
Pedro A. Guil Asensio
We present a general construction of model category structures on the cat-egory C(Qco(X)) of unbounded chain complexes of quasi-coherent sheaveson a semi-separated scheme X. This construction is based on making com-patible the filtrations of individual modules of sections at open affine subsetsof X. We apply this to describe the homotopy category K(C(Qco(X))) viavarious model structures on C(Qco(X)). As particular instances, we recoverrecent results on the flat model structure for quasi-coherent sheaves. Our ap-proach also includes the case of (infinite-dimensional) vector bundles, and ofrestricted flat Mittag-Leffler quasi-coherent sheaves, as introduced by Drin-feld. However, we show that the unrestricted case does not induce a modelcategory structure as above.
University of Murcia, Spain
AAD XII References
The Bounds for Distance Estrada Index
Ayşe Dilek Güngör
Let G be a connected graph on n vertices, and let µ1, µ2, · · · , µn be theD-eigenvalues of its distance matrix D. In this talk, we will present thedefinition and some properties of the distance Estrada index
DEE = DEE(G) =
n∑
i=1
eµi
of the graph G (see []). We further present lower and upper bounds forDEE(G) and relations between DEE(G) and the distance energy.
References
[] De la Peña, J.A., Gutman, I. and Rada, J. Estimating the Estrada Index,Linear Algebra Appl. (), -.
[] Deng H., Radenković S. and Gutman I. The Estrada index, in:Cvetković, D., Gutman I. (Eds.), Applications of Graph Spectra, Math.Inst., Belgrade, pp. -.
[] A. D. Güngör, Ş. B. Bozkurt, On the Distance Estrada Index of Graphs,Hacettepe J. Math. Stats. () (), -.
[] Indulal, G., Gutman, I. and Vijaykumar, A. On the Distance Energy ofa Graph, MATCH Commun. Math. Comput. Chem. (), -.
Selcuk University, Konya-Turkey
http://asp.selcuk.edu.tr/asp/personel/web/goster.asp?sicil=4568
References AAD XII
Co-coatomically Supplemented Modules
Serpil Güngör
M will mean an R-module where R is an arbitrary ring with identity. Amodule M is called coatomic if every submodule is contained in a maximalsubmodule of M . A proper submodule N of M is called co-coatomic if M/N iscoatomic. A module M is co-coatomically supplemented if every co-coatomicsubmodule U of M has a supplement V , i.e. V is minimal in the collectionof submodules L of M such that M = N + L.We have the following results.
Proposition . Let M be a co-coatomically supplemented module. ThenM/N is co-coatomically supplemented.
Proposition . Let M be a co-coatomically supplemented R-module. Thenevery co-coatomic submodule of the module M/Rad(M) is a direct summand.
Theorem . Let R be any ring. The following are equivalent for an R-module M .. Every co-coatomic submodule of M is a direct summand of M .. Every maximal submodule of M is a direct summand of M .. M/Soc(M) does not contain a maximal submodule.
Joint work with Prof. Dr. Rafail Alizade
References
[] Robert Wisbauer, Foundations of Module and Ring Theory, Gordon andBreach Science Publishers, ().
[] R. Alizade, G. Bilhan, P.F. Smith, Modules Whose Maximal SubmodulesHave Supplements,Communications in Algebra, :, -, ().
İzmir Institute of Technology
AAD XII References
A Note on the Products (1µ+1)(2µ+1) . . . (nµ+1)Erhan Gürel
Let Ωµ(n) = (1µ + 1)(2µ + 1) . . . (nµ + 1) where µ > 2 is an integer. Weprove that Ω3(n) is never squarefull, and in particular never a square, usingarguments similar to those in [], where Cilleruelo proves that Ω2(n) is nota square for n 6= 3. In [], among many other results, Amdeberhan, Medinaand Moll claim that Ωµ(n) is not a square if µ is an odd prime and n > 12.However, we have found a gap in the proof of this statement in [], which weillustrate by giving counterexamples.
References
[] Amdeberhan, T. , Medina, L. A. , Moll, V. H. Arithmetical propertiesof a sequence arising from an arctangent sum, J. Number Theory (), –.
[] Cilleruelo, J. Squares in (12 + 1) . . . (n2 + 1), J. Number Theory (), –.
Middle East Technical University, N.C.C., SZ-, Güzelyurt, KKTC, Mersin, Turkey
www.ncc.metu.edu.tr/cv.php?id=85
References AAD XII
Finite generation of ideals in rings of finitecharacter
Sevgi Harman
A ring R is said to be of finite character if each nonzero element of it iscontained in only finitely many maximal ideals. Let R be a ring and I anideal of R. Then by the J-radical of the ideal I we mean the intersection of allmaximal ideals of R containing I, and the Jmax-radical of I the intersectionof all maximal ideals of R of maximal height that contains I. It is shown thatover a finite dimensional integral domain R of finite character, each maximalideal of R[X] of maximal height is the J-radical of an ideal generated by threeelements and Jmax-radical of an ideal generated by two elements. We alsoshow that over a one dimensional S-domain R of finite character each primeideal of R[X] that does not contract to the zero ideal of R is the radical ofan ideal generated by at most two elements.
References
[] V. Erdoğdu, Efficient Generation of prime ideals in polynomial rings upto radical, Communications in Algebra (to appear).
[] I. Kaplansky, Commutative Rings, University of Chicago Press, Boston,.
Istanbul Technical University
AAD XII References
Generalized Bruck-Reilly *-extension ofmonoids
Eylem Güzel Karpuz
This is a joint work with Firat Ateş and Ahmet Sinan Çevik. Let M bea monoid and θ : M → M be an endomorphism. Then the Bruck-Reillyextension BR(M, θ) is the set
N0 ×M × N
0 = (p,m, q) : p, q > 0,m ∈ M
with multiplication
(p1,m1, q1)(p2,m2, q2) = (p1 − q1 + t, (m1θt−q1)(m2θ
t−p2), q2 − p2 + t),
where t = max(q1, p2). BR(M, θ) is a monoid with identity (0, 1M , 0). If Mis defined by the presentation < A;R >, then BR(M, θ) is defined by
< A, b, c ; R, bc = 1, ba = (aθ)b, ac = c(aθ) (a ∈ A) >,
in terms of generators (0, a, 0) (a ∈ A), (0, 1M , 1) and (1, 1M , 0) []. Thisextension is considered a fundamental construction in the theory of semi-groups. In [], the author defined a monoid, namely generalized Bruck-Reilly*-extension and studied some Green’s relations on it.
In this talk, we give a presentation for generalized Bruck-Reilly *-extensionof monoids and then by using Bruck-Reilly and and this generalized Bruck-Reilly *-extensions we answer the following question negatively.
Question. Does the group of units of a finitely presented monoid have tobe finitely generated?
References
[] U. Asibong-Ibe, ∗-Bisimple type A w-semigroups-I, Semigroup Forum, (), -.
[] J. M. Howie, N. Ruskuc, Constructions and presentations for monoids,Communications in Algebra, () (), -.
[] Y. Shung, L. M. Wang, ∗-Bisimple type A w2-semigroups as generalizedBruck-Reilly ∗-extensions, Southeast Asian Bulletin of Math., (),-.
AAD XII References
Polytope method over rings containing nonzero-divisors
Fatih Koyuncu
For any field F, there is a relation between the factorization of a polynomialf ∈ F [x1, ..., xn] and the integral decomposition, with respect to Minkowskisum, of the Newton polytope of f . We extended this result to polynomialrings R[x1, ..., xn] for an arbitrary ring R containing non zero-divisors.
References
[] S. Gao, Absolute irreducibility of polynomials via Newton polytopes,Journal of Algebra (), No. -.
Muğla University, Department of Mathematics
References AAD XII
P-pure submodules and its relation with neatand coneat submodules
Engin Mermut
Let R be an arbitrary ring with unity. Take all modules to be left R-modules.
A subgroup A of an abelian group B is said to be a neat subgroup if A∩pB =pA for all prime numbers p ([], [, p. ]). This is a weakening of thecondition for being a pure subgroup. There are several reasonable ways togeneralize this concept to modules.
Following Stenström ([, .] and [, §]), we say that a submodule Aof an R-module B is neat in B if for every simple module S, the sequenceHom(S,B) −→ Hom(S,B/A) −→ 0 obtained by applying the functor Hom(S,−)to the canonical epimorhism B −→ B/A is exact.
Another natural generalization of neat subgroups is what is called P-purity.Denote by P the collection of all left primitive ideals of the ring R. We saythat a submodule A of an R-module B is P-pure in B if A∩PB = PA for allP ∈ P. In [], the relation of P-purity with complements and supplementshave been used to describe the structure of c-injective modules over Dedekinddomains.
A natural question to ask is when neatness and P-purity coincide. Supposethat the ring R is commutative. Then P is the collection of all maximal idealsof R. Recently Fuchs ([]) has characterized the commutative domains forwhich these two notions coincide. Fuchs calls a ring R to be an N-domainif R is a commutative domain such that neatness and P-purity coincide.Unlike expected, Fuchs shows that N-domains are not just Dedekind domains.For a commutative domain R, Fuchs proves that R is an N-domain if andonly if all maximal ideals of R are projective (and so all maximal ideals areinvertible ideals and finitely generated). We slightly generalize this resultby taking instead of domains commutative rings R such that every maximalideal contains a regular element so that the ideals of R that are invertiblein the total quotient ring of R will be just projective ones as in the case ofcommutative domains. On the way, we also obtain some properties for adual concept to neat: coneat submodules. A monomorphism f : K → L iscalled coneat if each module M with RadM = 0 is injective with respectto it, that is, the Hom sequence Hom(L, M) → Hom(K, M) → 0 obtainedby applying the functor Hom(−,M) to the monomorphism f : K → L isexact. We use a description of coneat short exact sequences to show thatover commutative small rings (these are the rings such that the radical of
AAD XII References
every injective module is itself), the splitting of every coneat short exactsequence ending with a simple module implies that all simple modules haveprojective dimension 6 1 (that is every maximal ideal is projective).
References
[] L. Fuchs. Infinite Abelian Groups, volume . Academic Press, New York,.
[] L. Fuchs. Neat submodules over integral domains. . Preliminarynotes, in preparation.
[] K. Honda. Realism in the theory of abelian groups I. Comment. Math.Univ. St. Paul, :–, .
[] Engin Mermut, Catarina Santa-Clara, and Patrick F. Smith. Injectivityrelative to closed submodules. J. Algebra, ():–, .
[] Bo T. Stenström. High submodules and purity. Arkiv för Matematik,():–, .
[] Bo T. Stenström. Pure submodules. Arkiv för Matematik, ():–,.
Dokuz Eylül University, İzmir/TURKEY
http://kisi.deu.edu.tr/engin.mermut/
References AAD XII
On Extensions of a valuation on K to K(x)
Figen Öke
Let v be a valuation of a field K, Gv its value group and kv its residuefield and w be an extension of v to K(x). w is called residual transcendentalextension of v if kw/kv is a transcendental extension and w is called residualalgebraic extension of v if kw/kv is an algebraic extension. In this studyresidual transcendental and residual algebraic extensions of v to K(x) arerepresented.
References
[] V. Alexandru, N.Popescu, A.Zaharescu, A theorem of characterizationof residual trancendental extensions of a valuation, J. Math. Kyoto Univ.(JMKYAZ) () –.
[] V.Alexandru, N.Popescu, A.Zaharescu, Minimal pair of definition of aresidual transcendental extension of a valuation, J. Math. Kyoto Univ(JMKYAZ) () –.
[] V.Alexandru, N.Popescu, A.Zaharescu, All valuations on K(x) J. Math.Kyoto Univ. - () –.
Trakya University, Department of Mathematics, Edirne, TURKEY
AAD XII References
An application of strong Groebner basis tocoding theory
Hakan Özadam
Let GR(pa,m) be the Galois ring with characteristic pa and cardinalitypam. Since GR(pa,m)[x] is not a principal ideal ring, the study of the idealstructure of GR(pa,m)[x]
〈xN−1〉, and therefore the study of cyclic codes of length N
over GR(pa,m), is much more complicated compared to the case of cycliccodes over finite fields. This motivates applying the theory of Groebner basisto cyclic codes over Galois rings. It is well-known that the classical theory ofGroebner basis can be extended to the theory of Groebner basis over rings. In[], the authors introduce a special type of Groebner basis over principal idealrings which they call strong Groebner basis. Given a cyclic code of length N
over GR(pa,m), which is an ideal of GR(pa,m)[x]〈xN−1〉
, it has been explained in []and [] how to determine the minimum Hamming distance of C, using strongGroebner basis. Recently, Lopez-Permouth, Özadam, Özbudak and Szabodetermined the minimum Hamming distance of certain constacyclic codes oflength nps over GR(pa,m) via strong Groebner basis in []. In this talk, Iwill give an overview of this result.
References
[] G. Norton and A. Sălăgean, “Strong Gröbner bases for polynomials overa principal ideal ring”, Bull. Austral. Math. Soc. , no. , pp. -,.
[] G. H. Norton and A. Sălăgean, “Cyclic codes and minimal strong Gröbnerbases over a principal ideal ring”, Finite fields and their app. , pp. -, .
[] A. Sălăgean, “Repeated-root cyclic and negacyclic codes over a finitechain ring", Discrete App. Math, vol , no. , pp. -, .
[] S. R. López-Permouth, H. Özadam, F. Özbudak and S. Szabo, “Poly-cyclic codes and repeated-root constacyclic codes”, preprint, .
Middle East Technical University
References AAD XII
Quadratic forms of codimension over finitefields containing F4 and Artin-Schreier type
curves
Ferruh Özbudak
Let Fq be a finite field containing F4. Let k > 2 be an integer. We give afull classification of quadratic forms over Fqk of codimension provided thatcertain three coefficients are from F4. As an application of this we obtain newresults on the classification of maximal and minimal curves over Fqk . We alsogive some nonexistence results on certain systems of equations over Fqk .
*This is a joint work with Elif Saygı and Zülfükar Saygı.
Middle East Technical University
AAD XII References
Cotorsion Modules and Pure-Injectivity
Dilek Pusat-Yılmaz
A module C is called cotorsion if Ext1(F,C) = 0 for any flat module F .We show that any cotorsion module satisfies a compactness condition oncertain finite definition subgroups. Namely, those associated to divisibilityconditions on pp-formulae in the First Order Logic of Modules. Using thischaracterization, we obtain a new proof of the fact that the endomorphismring of any flat cotorsion module is f-semiperfect and idempotents lift modulothe Jacobson radical, thus completing some characterizations obtained in [].We apply these results to the particular case of hereditary rings and obtainconditions that force a hereditary ring to be semiperfect in terms of thepresentation of its cotorsion envelope.
Joint work with Deniz Erdemirci and Pedro Guil Asensio.
References
[] P.A. Guil Asensio and I. Herzog, Pure-injectivity in the category of flatmodules, Contemporary Mathematics American Mathematical Society, ().
[] B.L. Osofsky, A generalization of quasi-Frobenius rings, J. Algebra () -.
[] J. Xu, Flat Covers of Modules, Lecture Notes in Mathematics; ,Springer .
Izmir Institute of Technology
References AAD XII
Soft Near-rings
Aslıhan Sezgin
This is a joint work with Akın Osman Atagün. Soft set theory, which can beused as a new mathematical tool for dealing with uncertainty was introducedby Molodtsov. In this paper, we indicate the study of soft near-rings by usingMolodtsov’s definition of the soft sets. The notions of soft near-rings, softsubnear-rings, soft (left, right) ideals, (left, right) idealistic soft near-ringsand soft near-ring homomorphisms are introduced. Moreover, several relatedproperties are investigated and illustrated by a great deal of examples.
References
[] L.A. Zadeh, Fuzzy sets, Inform. Control. () -.
[] L.A. Zadeh, Toward a generalized theory of uncertainty (GTU)-an out-line, Inform. Sci. () -.
[] Z. Pawlok, Rough sets, Int. J. Inform. Comput. Sci. () -.
[] Z. Pawlok, A. Skowron, Rudiments of soft sets, Inform. Sci. ()-.
[] W.L. Gau, D.J. Buehrer, Vague sets, IEEE Tran. Syst. Man. Cybern. () -.
[] M.B. Gorzalzany, A method of inference in approximate reasoning basedon interval-valued fuzzy sets, Fuzzy Sets and Systems () -.
[] K. Atanassov, Operators over interval valued intuitionistic fuzzy sets,Fuzzy Sets and Systems () -.
[] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems ()-.
[] L. Zhou, W.Z. Wu, On generalized intuitionistic fuzzy rough approxima-tion opeartors, Inform. Sci. () () -.
[] D. Molodtsov, Soft set theory-first results, Comput. Math. Appl. () -.
[] P.K. Maji, A.R. Roy, R. Biswas, An application of soft sets in a decisionmaking problem, Comput. Math. Appl. () -.
AAD XII References
[] A.R. Roy, P.K. Maji, A fuzzy soft set theocratic approach to decisionmaking problem, J. Comp. Appl. Math. () -
[] D. Chen, E.C.C. Tsang, D.S. Yeung, X. Wang, The parametrization re-duction of soft sets and its applications, Comput. Math. Appl. ()-.
[] Z. Kong, L. Gao, L. Wang, S. Li, The parameter reduction of soft setsand algorithm, Comput. Math. Appl. () () -.
[] P.K. Maji, R. Biswas, A.R. Roy, Soft set theory, Comput. Math. Appl. () -.
[] C.F. Yang, A note on soft set theory, Comput. Math. Appl. ()-. [Comput. Math. Appl. (-) () -].
[] M.I. Ali, F. Feng, X. Liu, W.K. Min, M. Shabir, On some new operationsin soft set theory, Comput. Math. Appl. () () -.
[] W. Xu, J. Ma, S. Wang, G. Hao, Vague soft sets and their properties,Comput. Math. Appl., () () -.
[] H. Aktaş, N. Çagman, Soft sets and soft groups, Inform. Sci. ()-.
[] H. Aktaş, N. Çagman, Erratum to "Soft sets and soft groups", Inform.Sci. () () . [Inform. Sci. () -].
[] F. Feng, Y.B. Jun, X. Zhao, Soft semirings, Comput. Math. Appl. () -.
[] G. Pilz, Near-rings, North Holland Publishing Company, Amsterdam-New York-Oxford, .
Department of Mathematics, Bozok University, , Yozgat, Turkey
References AAD XII
Decomposability of (1, 2)-Groups
Ebru Solak
A torsion free abelian group of finite rank is called almost completely de-composable if it has a completely decomposable subgroup of finite index. Ap-local, p-reduced almost completely decomposable group of type (1, 2) isbriefly called a (1, 2)-group. Almost completely decomposable groups can berepresented by matrices over the ring Zh = Z/hZ, where h is the exponent ofthe regulator quotient. This particular choice of representation allows for abetter investigation of the decomposability of the group. Arnold and Dugasshowed in several of their works that (1, 2)- groups with regulator quotientof exponent at least p7 allow infinitely many isomorphism types of indecom-posable groups. It is not known if the exponent 7 is minimal.
References
[] D.Arnold, Pure subgroups of finite rank completely decomposable groups,In Abelian Group Theory, Lecture Notes in Mathematics, volume ,pages -. Springer Verlag, New York, ().
[] D. Arnold, M. Dugas, Representation type of posets and finite rank Butlergroups. In Coll. Math. , (), -.
Middle East Technical University, Department of Mathematics, AnkaraTurkey
www.metu.edu.tr/~esolak/
AAD XII References
Equiprime Ideals of Near-ring Modules
Funda Taşdemir
This is a joint work with Akın Osman Atagün and Hüseyin Altındiş. Inthis paper we introduce the notion of equiprime N -ideals (ideals of near-ring modules) where N is a near-ring. We consider the interconnections ofequiprime, -prime and completely prime N -ideals. The relationship betweenan equiprime N -ideal P of an N -group Γ and the ideal (P : Γ) of the near-ringN is also investigated.
References
[] Atagün, A. O. IFP Ideals in Near-rings, Hacet. J. of Math. and Stat. (), -, .
[] Atagün, A. O. and Groenewald, N. J. Primeness in Near-rings withMultiplicative Semi-Group Satisfying ’The Three Identities’, J. Math.Sci. Adv. Appl. (), -, .
[] Birkenmeier, G. and Heatherly, H. Medial Near-rings, Monatsh. Math., -, .
[] Birkenmeier, G. and Heatherly, H. Left Self Distributive Near-rings, J.Austral. Math. Soc.(Series A) , -, .
[] Booth, G. L., Groenewald, N. J. and Veldsman, S. A Kurosh-AmitsurPrime Radical for Near-rings, Comm. Algebra (), -, .
[] Booth, G. L. and Groenewald, N. J. Equiprime Left Ideals and EquiprimeN -groups of a Near-ring, Contributions to General Algebra , -,.
[] Çallıalp, F. and Tekir, Ü. On the Prime Radical of a Module over aNoncommutative ring, Taiwanese Journal of Mathematics (), -, .
[] Groenewald, N. J. Different Prime Ideals in Near-rings, Comm. Algebra (), -, .
[] Holcombe, W. L. M. Primitive near-rings, Doctoral Dissertation, Uni-versity of Leeds, .
References AAD XII
[] Juglal, S., Groenewald, N. J. and Lee, E. K. S. Different prime R-ideals,To appear Algebra Colloquium.
[] Lomp, C. and Peña, A. J. A Note on Prime Modules, DivulgacionesMatemáticas (), -, .
[] Pilz, G. Near-rings (North-Holland, ).
[] Ramakotaiah, D. and Rao, G. K. IFP Near-rings, J. Austral. Math.Soc.(Series A) , -, .
[] Veldsman, S. On Equiprime Near-rings, Comm. Algebra (), -, .
[] Wisbauer, R. On prime modules and rings, Comm. Algebra (),-, .
Department of Mathematics, Bozok University, , Yozgat, Turkey
AAD XII References
On Cofinitely Supplemented Lattices
S. Eylem Toksoy
L will mean a complete modular lattice with smallest element 0 and greatestelement 1. A lattice L is said to be supplemented if every element a of L hasa supplement in L, i.e. an element b such that a∨ b = 1 and a∧ b ≪ b/0. Anelement c of a complete lattice L is said to be compact if for every subset Xof L with c 6
∨
X there exists a finite subset F of X such that c 6∨
F . Alattice L is called a compact lattice if is compact and a compactly generatedlattice if each of its elements is a join of compact elements. An element a of alattice L is said to be cofinite in L if the quotient sublattice 1/a is compact.A lattice L is called a cofinitely supplemented lattice if every cofinite elementof L has a supplement in L.
For compactly generated compact lattices a supplement of an element iscompact (see [, Proposition . ()]). In the following proposition we showthat for an arbitrary lattice L a supplement of a cofinite element is compact.
Proposition . Let a be a cofinite element of a lattice L and b be a supple-ment of a. Then b/0 is compact.
Theorem . (cf. [, Theorem ..]) A lattice L is a cofinitely supplementedlattice if and only if every maximal element of L has a supplement in L.
Theorem is used in the proof of the following theorem which gives a newresult for modules.
Theorem . If a/0 is a cofinitely supplemented sublattice of L and 1/a hasno maximal element, then L is also a cofinitely supplemented lattice.
Corollary . Let M be a module, N be a cofinitely supplemented submoduleof M . If Rad(M/N) = M/N , then M is cofinitely supplemented.
Joint work with: Refail Alizade, İzmir Institute of Technologye-mail: [email protected]
References
[] Alizade R., Bilhan G., Smith P.F., "Modules whose Maximal Sub-modules have Supplements", Communications in Algebra. Vol., No:,pp.- ().
[] Calugareanu G., Lattice Concepts of Module Theory, Kluwer AcademicPublishers, Dordrecht, Boston, London ().
References AAD XII
[] Çetindil Y., Generalizations of Cofinitely Supplemented Modules to Lat-tices, M.Sc. Thesis, İzmir Institute of Technology, İzmir ().
İzmir Institute of Technology
www.iyte.edu.tr/$\sim$eylemtoksoy
AAD XII References
Posters
Speeding Up Montgomery ModularMultiplication For Prime Fields
Sedat Akleylek, Murat Cenk
We give faster versions of Montgomery modular multiplication algorithmwithout pre-computational phase for GF (pm), where p is prime and m > 1,which can be considered as a generalization of [], [] and []. We propose setsof moduli which can be used in public key cryptographic applications. Weeliminate pre-computational phase with proposed sets of moduli. We showthat these methods are easy to implement for hardware.
References
[] M. Knezevic, L. Batina and I. Verbauwhede, Modular Reduction with-
out Precomputational Phase, IEEE International Symposium on Circuitsand Systems (ISCAS ), IEEE, pages, .
[] M. Knezevic, J. Fan, K. Sakiyama, and I. Verbauwhede, Modular Reduc-
tion in GF (2n) without Pre-Computational Phase, International Work-shop on the Arithmetic of Finite Fields (WAIFI ), LNCS , Ç.K. Koç, J. Luis Imana, and J. Von zur Gathen (eds.), Springer-Verlag,pp. -, .
[] M. Knezevic, F. Vercauteren, and I. Verbauwhede, Faster Interleaved
Modular Multiplication Based on Barrett and Montgomery Reduction
Methods, COSIC internal report, pages, .
Middle East Technical University and Ondokuz Mayıs University
Middle East Techical University
References AAD XII
On ramification in extensions of rationalfunction fields
Nurdagül Anbar
Let K (x) be a rational function field, which is a finite separable extensionof the rational function field K (z). In the first part, we have studied thenumber of ramified places of K (x) in K (x) /K (z). Then we have given aformula for the ramification index and the different exponent in the extensionF (x) over a function field F , where x satisfies an equation f (x) = z for somez ∈ F and separable polynomial f (x) ∈ K [x]. In fact, this generalizes thewell-known formulas for Kummer and Artin-Schreier extensions.
References
[] H. Stichtenoth, Algebraic Function Fields and Codes, Springer, Berlin,.
[] R. Lidl, H. Niederreiter, Finite Fields, . edition, Cambrigge UniversityPress, Cambridge, .
[] A. Garcia, Lectures notes on Algebraic Curves, Sabancı University, .
[] H. Hasse, Theorie der relativ-zyklischen algebraischen Funktionnkörper,insbesondere bei endlichem Konstantenkörper, J. Reine Angew. Math., , pp. -.
Sabancı University
AAD XII References
Drinfeld modular curves with many rationalpoints over finite fields
Cam Vural
For some kinds of reasons one is interested to construct curves which havemany ratioal points over a finite field. Drinfeld modular curves can be usedto construct that kinds of curves over a finite field. In my work I am usingreductions of the Drinfeld modular curves X0(n) with suitable primes to getsuch nice curves. The main idea is to divide the Drinfeld modular curves byan Atkin-Lehner involution which has many fixed points to obtain a quotientwith a better ratio number of rational points/genus.
Metu
References AAD XII
Quantum Groups
Münevver Çelik
R-matrices are solutions of the Yang-Baxter equation. They give rise tolink invariants. R-matrices are derived from a special kind of Hopf algebra,namely quantum group. In this work, I will define quantum groups andpresent the way to derive link invariants from R-matrices.
METU
AAD XII References
A Remark on Permutations with Full Cycle
Ayça Çeşmelioğlu
For q > 2, Carlitz proved in [] that the group of permutation polynomials(PPs) over Fq is generated by the linear polynomials and xq−2. Based on thisresult, we point out a simple method for representing all PPs with full cycleover the prime field Fp, where p is an odd prime. We use the isomorphismbetween the symmetric group Sp of p elements and the group of PPs overFp, and the well-known fact that permutations in Sp have the same cyclestructure if and only if are conjugate.
References
[] L. Carlitz, “Permutations in a finite field ”, Proc. Amer. Math. Soc. ,, .
Sabancı Üniversitesi
http://myweb.sabanciuniv.edu/acesmelioglu
References AAD XII
On The Basic k-nacci Sequences in The DirectProduct Dn × Z2i
Ömür Deveci & Erdal Karaduman
AbstractIn this work, defining basic k-nacci sequences and the basic periods of thesesequences in finite groups then we obtain the basic periods of basic k -naccisequences and the periods of k -nacci sequences in the direct product Dn×Z2i .
Department of Mathematics, Faculty of Science and Letters, Kafkas Univer-sity, Kars, TURKEY
Department of Mathematics, Faculty of Science, Atatürk University, Erzurum, TURKEY
AAD XII References
On o-minimal structures
Şükrü Uğur Efem
A linearly ordered structure is said to be o-minimal if every definable subsetof it is a finite union of intervals and points. The motivating example is theordered field of reals. The notion of o-minimality was implicitly introducedin the eighties by Lou van den Dries who observed that many non-trivialproperties of semi-algebraic sets follow from those simple axioms, and thendeveloped further by Pillay and Steinhorn. In this poster we survey someknown results and applications of o-minimality.
References
[] L. van den Dries, o-Minimal Structures, in: Logic: from Foundations toApplications, Clarendon Press, Oxford (), pp -.
[] A. Pillay and C. Steinhorn. Definable sets in ordered structures. I. Trans-actions of American Math. Society, :-, .
[] L. Lipshitz and Z. Robinson. Overconvergent real closed quantifier elim-ination. Bull. London Math. Soc. (), no. , –
Sabancı Üniversitesi
References AAD XII
Maximal Subgroups in Hall Universal Groups
Mehmet İnan Karakuş
In an infinite group, it is a difficult task to decide whether a maximalsubgroup exist or not.It is a well known trivial example that the p-quasicyclicgroup (or Prüfer p-group, or C∞
p ) has no maximal subgroup. A locally finitegroup G is called a universal group(see []) if
. Every finite group can be embedded in G
. Any two isomorphic finite subgroups of G are conjugate in G
We discuss the existence of maximal subgroups in locally finite universalgroups. In particular there is a construction of a maximal p-subgroup whichis also maximal subgroup of the group.The construction is due to M. DalleMolle (see [])
References
[] M. Dalle Molle, Sylow subgroups which are maximal in the universallocally finite group of Philip Hall, J. Algebra (), no. , –.
[] P. Hall, Some constructions for locally finite groups, J. London Math.Soc. (), -.
Middle East Technical University
AAD XII References
Classification of Finitary Linear Simple LocallyFinite Groups
Dilber Koçak
A group is called locally finite if every finitely generated subgroup is a finitegroup. G is called a linear group if it is a subgroup of GL(n, F ) for some fieldF.
The classification of simple locally finite linear groups is completed in-dependently by Belyaev, Borovik, Hartley-Shute and Thomas. They haveproved:
Theorem . (BBHST: Belyaev, Borovik, Hartley, Shute, and Thomas [],[], [] and []). Each locally finite simple group that is not finite but hasa faithful representation as a linear group in finite dimension over a field isisomorphic to a Lie type group Φ(K), where K is an infinite, locally finitefield, that is, an infinite subfield of Fp, for some prime p.
A group of linear transformations is called finitary if each element minusthe identity is an endomorphim of finite rank. Then observe that every linearlocally finite simple group is a finitary linear simple locally finite group. Re-cently the classification of finitary simple locally finite groups are completedby J. I. Hall in [].
Theorem . (Hall). A locally finite simple group that has a faithful repre-sentation as a finitary linear group is isomorphic to one of:
() a linear group in finite dimension;() an alternating group Alt(Ω) with Ω infinite;() a finitary symplectic group FSpK(V, s);() a finitary special unitary group FSUK(V, u);() a finitary orthogonal group FΩK(V, q);() a finitary special linear group FSLK(V,W,m).
Here K is a (possibly finite) subfield of Fp, the algebraic closure of the primesubfield Fp. The forms s,u, and q are nondegenerate on the infinite dimen-sional K-space V; and m is a nondegenerate pairing of the infinite dimensionalK-spaces V and W. Conversely, each group in ()-() is locally finite, simple,and finitary but not linear in finite dimension.
References AAD XII
References
[] V. V. BELYAEV, Locally finite Chevalley groups, in Studies in GroupTheory, Urals Scientific Centre of the Academy of Sciences of USSR,Sverdlovsk (), - (in Russian).
[] A. V. BOROVIK, Periodic linear groups of odd characteristic, Dokl.Akad. Nauk. SSSR (), -.
[] J. I. HALL, Periodic simple groups of finitary linear transformations,Annals of Mathematics (),-
[] B. HARTLEY and G. SHUTE, Monomorphisms and direct limits offinite groups of Lie type, Quart. J. Math. (), -.
[] S. THOMAS, The classification of the simple periodic linear groups,Arch. Math. (), -.
METU
AAD XII References
On Stability of Free Products in Bounded Balls
Azadeh Neman
In a series of papers, Z. Sela proved that free groups, and more generallytorsion-free hyperbolic groups, have a stable first-order theory. It has beenconjectured by E. Jaligot [] that the free product of two arbitrary stablegroups is stable. However, a full answer seems to become a very large projectof generalization, from free groups to free products, of the famous articles ofSela. Until this monumental task is done, we provide a very preliminary resultin the direction of the stability of free products of stable groups, restrictingourselves to quantifer-free definable sets and to bounded balls of free productsand including finite amalgamation.
References
[] A. Neman. Stability and bounded balls of free products, Accepted
[] E. Jaligot. Groups of finite dimension in model theory. In C. Glymour,W. Wang, and D. Westerstahl, editors, Proceedings from the th In-ternational Congress of Logic, Methodology, and Philosophy of Sciences,Beijing, august . Studies in Logic and the Foundations of Mathe-matics, King’s College Publications, London, .
[] W. Magnus, A. Karrass and D. Solitar Combinatorial group theory.Presentations of groups in terms of generators and relations
Istanbul Bilgi University
References AAD XII
Monomial Gotzmann sets in a quotient by apure power
Ata Firat Pir
We study Gotzmann sets in a quotient R = F [x1, . . . , xn]/(xa1) of a polyno-
mial ring over a field F . These are monomial sets whose sizes grow minimallywhen multiplied with the variables. We partition the set of monomials ina Gotzmann set with respect to the multiplicity of x1 and show that if thesize of a component in a partition is sufficiently large, then this componentis a multiple of a Gotzmann set in F [x2, . . . , xn]. Otherwise we derive lowerbounds on the size of a component depending on neighboring components.For n = 3, we classify all Gotzmann sets in R and for a given degree, wecompute all integers j such that the only Gotzmann set in that degree is thelexsegment set of size j. We also we note down adoptions of some propertiesconcerning the minimal growth of the Hilbert function in F [x1, . . . , xn] to R.
Bilkent University
AAD XII References
On the number of Boolean functions satisfyingstrict avalanche criteria
Elif Saygı
Boolean functions play an important role in the design of both block andstream ciphers. In this work, the number of Boolean functions satisfyingstrict avalanche criteria are considered. Also, the number of functions withparticular difference distribution vectors is studied. The exact formula for aspecial case is given. Results of some statistical observations are comparedto the exact values.
This is a joint work with Ali Doğanaksoy and Zülfükar Saygı.
Hacettepe University
References AAD XII
Quadratic Feedback Shift Registers andMaximum Length Sequences
Zülfükar Saygı
In this work, the properties of the quadratic feedback shift registers generatingmaximum length sequences are considered. Some necessary conditions fora quadratic feedback function f of a feedback shift register to generate amaximum length sequence is given. Also a method generalizing this conditionis presented. Instead of searching all the sequence, looking at the algebraicnormal form of the function f one can understand if the corresponding shiftregister generates a sequence having short period.
This is a joint work with Elif Saygı and Ali Doğanaksoy.
TOBB University of Economics and Technology
http://zsaygi.etu.edu.tr/
AAD XII References
Recursive Towers of Function Fields over FiniteFields
Seher Tutdere
In Garcia and Stichtenoth gave explicit constructions of towers offunction fields over the finite field Fq. Moreover, in the case that q = pk (fork > 2 and p is a prime) they have given some examples of towers havingpositive limit which are called asymptotically good and optimal towers (see[, ]). Now we deal with the following problem: Are there any such towersof function fields over the prime fields Fp for any prime p? If so, then how todefine polynomials which give such nice towers?
References
[] A. Garcia, H. Stichtenoth, On the asymptotic behaviour of some towersof function fields over finite fields, J. Number Theory () -.
[] H. Stichtenoth, Algebraic Function Fields and Codes, Springer, Berlin,().
[] S. Tutdere, A Recursive Tower of Function Fields over F2, Ms.c. Thesis,Sabancı University, .
Sabancı University
ACKNOWLEDGEMENTS AAD XII
Personnel
Acknowledgements
AAD XII PARTICIPANTS
Participants
NOTES ON TURKISH AAD XII
Notes on Turkish
In alphabetical order, the letters of the Turkish alphabet are:
A B C Ç D E F G Ğ H I İ J K L M N O Ö P R S Ş T U Ü V Y Za b c ç d e f g ğ h ı i j k l m n o ö p r s ş t u ü v y z
Words are spelled as they are spoken. Except in loanwords and as noted justbelow, there is no variation between long and short vowels. There is hardlyany variation between stressed and unstressed syllables.
The consonants that need mention are: c, pronounced like English j; ç,like English ch; ğ, which lengthens the vowel that precedes it (and neverbegins a word); j, as in French; and ş, like English sh. Doubled consonantsare held longer.
ı backi front
unround#
u backclose
ü frontround
a back@e front
unround
o backopen
ö frontround
As for the vowels, the a is like uh inEnglish; ö and ü are as in German, orare like the French eu and u; and u islike the short English oo. Diphthongsare obtained by addition of y: so, ay isEnglish long ı, and ey is English long a.
The eight vowels exhibit three bi-nary distinctions: back/front, un-round/round, and close/open. Inuse, many words feature a stem followedby one or more suffixes. Often the vowelin a suffix harmonizes with preceding vowel: then it resembles the preced-ing vowel as far as possible while remaining hard, or while remaining unroundopen. So one might introduce special symbols for the variable vowels in suf-fixes, as suggested in the table.
The question Avrupalılaştıramadıklarımızdan mısınız? can be analyzed as astem with suffixes:
Avrupa0lı1la2ş3tır4ama5dık6lar7ımız8dan9 mı10sınız11?
The suffixes translate mostly as separate words in English, in almost thereverse order: Are you11 one-of 9 those7 whom6 we8 could-not 5 Europeanize(make4 be2come3 Europe0an1)?10 If we change Europeanize to Turkify, we getTürkleştiremediklerimizden misiniz? Detached, the suffixes might be writtenl@ş, t#r, @m@, d#k, l@r, #m#z, d@n, m#, s#n#z.
AAD XII SESSION CHAIRS
Useful expressions
Lütfen / Teşekkürler / Bir şey değil Please / Thanks / It’s nothing.Evet/hayır Yes/no. Var/yok There is / there isn’t. Affedersiniz Excuse me.Efendim Madam or Sir.∗ Merhaba Hello. Günaydın Good morning.Hoş geldiniz / Hoş bulduk Welcome / (its response).İyi günler/akşamlar/geceler Good day/evening/night.Güle güle Fare well (said to the person leaving);Allaha ısmarladık or Hoşça kalın Good bye (said to the person staying behind).Bay/Bayan Mr/Ms, or gentlemen’s/ladies’. Beyefendi/Hanımefendi Sir/Madam.
İtiniz/çekiniz Push/pull; giriş/çıkış entrance/exit;sol/sağ left/right; soğuk/sıcak cold/hot.
Nasılsınız? / İyiyim; siz? / Ben de iyiyim. How are you? / I’m fine; you? /I’m also fine.Elinize sağlık Health to your hand (the chef’s, who replies: Afiyet olsun Mayit be healthy).Kolay gelsin May [your work] come easy.Geçmiş olsun May [your sickness, difficulty, &c.] be over.İnşallah If God wills; Maşallah or Allah korusun May God protect.Rica ederim I request, or Estağfurullah, can be used with the sense of I don’tdeserve such praise! or Don’t say such [bad] things about yourself !
Sıfır, bir, iki, üç, dört, beş, altı, yedi, sekiz, dokuz , , , , , , , , , ;on, yirmi, otuz, kırk, elli, altmış, yetmiş, seksen, doksan , , , . . . , ;yüz, bin, milyon, milyar 102, 103, (103)2, (103)3;yüz kırk dokuz milyon beş yüz doksan yedi bin sekiz yüz yetmiş ,,.
Daha/en more/most; az less, en az least.
Al-/sat-/ver- take, buy / sell / give;alış/satış/alışveriş buying (rate)/selling (rate)/shopping.
İn-/bin-/gir-/çık go: down, off / onto / into / out, up;aşağı/yukarı lower/upper; alt/üst bottom/top.
Kim, ne, ne zaman, nerede, nereye, nereden, niçin, nasıl, kaç, ne kadar?who, what, when, where, whither, whence, why, how, how many, how much?
Session chairs
∗Efendi is from the Greek αὐθέντης, whence also English authentic.
TIMETABLE AAD XII
Timetable