Introduction Main Results on the group Z n ⋊ Z Main Results on the monoid Z 2 ⋊ Z Final Remarks A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions Ahmet Sinan C ¸evik www.ahmetsinancevik.com Sel¸ cuk University, Konya/Turkey sinan.cevik@selcuk.edu.tr Questions, Algorithms, and Computations in Abstract Group Theory May 21-24, 2013 Braunschweig Ahmet Sinan C ¸evik A new approach to connect Algebra with Analysis: Relationships www.ahmetsinancevik.com
Introduction Main Results on the group Z n ⋊ Z Main Results on the monoid Z 2 ⋊ Z Final Remarks Outline 1 Introduction General Aim Reminders 2 Main Results on the group Z n ⋊ Z 3 Main Results on the monoid Z 2 ⋊ Z 4 Final Remarks Ahmet Sinan C ¸evik A new approach to connect Algebra with Analysis: Relationships www.ahmetsinancevik.com
Introduction Main Results on the group Z n ⋊ Z General Aim Main Results on the monoid Z 2 ⋊ Z Reminders Final Remarks This talk is based on the joint work Cevik et al.- 2013 . Cevik et al.-2013 A.S. Cevik, I.N. Cangul, Y. Simsek, A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions, Boundary Value Problems , 2013, 2013 :51 doi:10.1186/1687-2770-2013-51. Ahmet Sinan C ¸evik A new approach to connect Algebra with Analysis: Relationships www.ahmetsinancevik.com
Introduction Main Results on the group Z n ⋊ Z General Aim Main Results on the monoid Z 2 ⋊ Z Reminders Final Remarks ✬ ✩ ⑥ GRAPHS ✬ ✩ ✫ ✪ ⑦ ✻ GEN. FUNCTS. ✫ ✪ ✬ ✩ ❄ ✒ ALGBRC ✠ ✫ STRUCT. ✪ Ahmet Sinan C ¸evik A new approach to connect Algebra with Analysis: Relationships www.ahmetsinancevik.com
Introduction Main Results on the group Z n ⋊ Z General Aim Main Results on the monoid Z 2 ⋊ Z Reminders Final Remarks Reason of this study In the literature, there are so many studies about figuring out the relationship between algebraic structures and special generating functions (cf., for instance, Woodcock-1979 , Simsek-2004 , Srivastava-2011 ). Woodcock-1979 , Convolutions on the ring of p -adic integers, J. Lond. Math. Soc. 20(2), (1979) 101-108. Simsek-2004 , An explicit formula for the multiple Frobenius-Euler numbers and polynomials, JP J. Algebra Number Theory Appl. 4, (2004) 519-529. Srivastava-2011 , Some generalizations and basic (or q -) extensions of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. Inform. Sci. 5, (2011) 390-444. Ahmet Sinan C ¸evik A new approach to connect Algebra with Analysis: Relationships www.ahmetsinancevik.com
Introduction Main Results on the group Z n ⋊ Z General Aim Main Results on the monoid Z 2 ⋊ Z Reminders Final Remarks Reason of this study There exists a connection between graphs and generating functions since the number of vertex-colorings of a graph is given by a polynomial on the number of used colors (see Birkhoff-1946 , Cardoso-2012 ). Based on this polynomial, one can define the chromatic number as the minumum number of colors such that the chromatic polynomial is positive. Birkhoff-1946 , Chromatic polynomials, Trans. Am. Math. Soc. 60, (1946) 355-451. Cardoso-2012 , A generalization of chromatic polynomial of a graph subdivision, J. Math. Scien. 183(2), (2012). Ahmet Sinan C ¸evik A new approach to connect Algebra with Analysis: Relationships www.ahmetsinancevik.com
Introduction Main Results on the group Z n ⋊ Z General Aim Main Results on the monoid Z 2 ⋊ Z Reminders Final Remarks Reason of this study We have not seen any such studies between group (or monoid) presentations and generating functions . Ahmet Sinan C ¸evik A new approach to connect Algebra with Analysis: Relationships www.ahmetsinancevik.com
Introduction Main Results on the group Z n ⋊ Z General Aim Main Results on the monoid Z 2 ⋊ Z Reminders Final Remarks Reason of this study We have not seen any such studies between group (or monoid) presentations and generating functions . So, by considering a group or a monoid presentation P , it is worth to study similar connections. In here, we actually assume P satisfies either efficiency or inefficiency while it is minimal. Then it will be investigated whether some generating functions can be applied, and then studied what kind of new properties can be obtained by considering special generating functions over P . Ahmet Sinan C ¸evik A new approach to connect Algebra with Analysis: Relationships www.ahmetsinancevik.com
Introduction Main Results on the group Z n ⋊ Z General Aim Main Results on the monoid Z 2 ⋊ Z Reminders Final Remarks Reason of this study We have not seen any such studies between group (or monoid) presentations and generating functions . So, by considering a group or a monoid presentation P , it is worth to study similar connections. In here, we actually assume P satisfies either efficiency or inefficiency while it is minimal. Then it will be investigated whether some generating functions can be applied, and then studied what kind of new properties can be obtained by considering special generating functions over P . Since the results in Cardoso-2012 imply a new studying area for graphs in the meaning of representation of parameters by generating functions, we hope that this study will give an opportunity to make a new classification of infinite groups and monoids in the meaning of generating functions. Ahmet Sinan C ¸evik A new approach to connect Algebra with Analysis: Relationships www.ahmetsinancevik.com
Introduction Main Results on the group Z n ⋊ Z General Aim Main Results on the monoid Z 2 ⋊ Z Reminders Final Remarks Key Point For group or monoid cases, if we study on an efficient presentation with minimal number of generators, or an inefficient but minimal presentation then we clearly have a minimal number of generators . This situation effects very positively using the generating functions for this type of presentations since we have a great advantage to study with quite limited number of variables in such a generating function. Ahmet Sinan C ¸evik A new approach to connect Algebra with Analysis: Relationships www.ahmetsinancevik.com
Introduction Main Results on the group Z n ⋊ Z General Aim Main Results on the monoid Z 2 ⋊ Z Reminders Final Remarks Efficiency For a group (or a monoid) presentation P = � x ; r � , the Euler characteristic is defined by χ ( P ) = 1 − | x | + | r | . By Epstein-1961 , there exists a lower bound δ ( G ) = 1 − rk Z ( H 1 ( G )) + d ( H 2 ( G )) ≤ χ ( P ) , where rk ( . ) denotes the Z -rank of the torsion-free part and d ( . ) denotes the minimal number of generators. Epstein-1961 , Finite presentations of groups and 3-manifolds, Quart. J. Math. Oxford Ser. 12(2), 1961 205-212. Ahmet Sinan C ¸evik A new approach to connect Algebra with Analysis: Relationships www.ahmetsinancevik.com
Introduction Main Results on the group Z n ⋊ Z General Aim Main Results on the monoid Z 2 ⋊ Z Reminders Final Remarks Efficiency (Deficiency)-cont. P is called minimal if χ ( P ) � χ ( P ′ ) for all presentations P ′ . P is called efficient if χ ( P ) = δ ( G ). G is called efficient if χ ( G ) = δ ( G ), where χ ( G ) = min { χ ( P ) : P is a finite presentation for G } . Some authors just consider | r | − | x | and call it deficiency of P . δ ( G ) ≤ χ ( P ) for monoids (S.J. Pride - unpublished since 1994) Ahmet Sinan C ¸evik A new approach to connect Algebra with Analysis: Relationships www.ahmetsinancevik.com
Introduction Main Results on the group Z n ⋊ Z General Aim Main Results on the monoid Z 2 ⋊ Z Reminders Final Remarks According to the Key Point , if P is efficient, then we need to assure that the minimal number of generators !! Wamsley-1973 Not be considered unless stated otherwise, Wamsley-1973 , Minimal presentations for finite groups, Bull. London Math. Soc. 5, (1973) 129-144. Ahmet Sinan C ¸evik A new approach to connect Algebra with Analysis: Relationships www.ahmetsinancevik.com
Introduction Main Results on the group Z n ⋊ Z General Aim Main Results on the monoid Z 2 ⋊ Z Reminders Final Remarks According to the Key Point , if P is efficient, then we need to assure that the minimal number of generators !! Wamsley-1973 Not be considered unless stated otherwise, Wamsley-1973 , Minimal presentations for finite groups, Bull. London Math. Soc. 5, (1973) 129-144. inefficient, then to catch the aim in here, we need to show that it is MINIMAL !! Ahmet Sinan C ¸evik A new approach to connect Algebra with Analysis: Relationships www.ahmetsinancevik.com
Introduction Main Results on the group Z n ⋊ Z General Aim Main Results on the monoid Z 2 ⋊ Z Reminders Final Remarks Minimality for Groups-cont. (Spherical) pictures ( Rourke-1979 , J.Howie-1989 , Pride-1991 ) Rourke-1979 , Presentations and the trivial group, Topology of low dimensional manifolds (ed. R. Fenn), Lecture Notes in Mathematics 722 (Springer, Berlin, 1979), 134-143. J.Howie-1989 , The Quotient of a Free Product of Groups by a Single High-Powered Relator. I. Pictures. Fifth and Higher Powers. Proc. London Math. Soc. 59(3) (1989), 507-540. Pride-1991 , Identities among relations of group presentations. Group theory from a geometrical viewpoint ( Trieste , 1990), 687-717, World Sci. Publ., River Edge , NJ, 1991. Ahmet Sinan C ¸evik A new approach to connect Algebra with Analysis: Relationships www.ahmetsinancevik.com
Recommend
More recommend
