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Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ 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 www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Outline

1 Introduction

General Aim Reminders

2 Main Results on the group Zn ⋊ Z 3 Main Results on the monoid Z2 ⋊ Z 4 Final Remarks

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks General Aim Reminders

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 www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks General Aim Reminders

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ⑦ ⑥ ✒ ✠ ❄ ✻

GRAPHS

• GEN. FUNCTS.

ALGBRC STRUCT.

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks General Aim Reminders

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 www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks General Aim Reminders

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 www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks General Aim Reminders

Reason of this study

We have not seen any such studies between group (or monoid) presentations and generating functions.

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks General Aim Reminders

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

• btained by considering special generating functions over P.

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks General Aim Reminders

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

• btained 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

• pportunity to make a new classification of infinite groups and

monoids in the meaning of generating functions.

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks General Aim Reminders

Key Point

For group or monoid cases, if we study on an efficient presentation with minimal number of generators,

• r

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 www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks General Aim Reminders

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 − rkZ(H1(G)) + d(H2(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 www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks General Aim Reminders

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 www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks General Aim Reminders

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

• therwise,

Wamsley-1973 , Minimal presentations for finite groups, Bull. London Math. Soc. 5, (1973) 129-144.

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks General Aim Reminders

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

• therwise,

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 www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks General Aim Reminders

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 www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks General Aim Reminders

Minimality for Groups-cont.

Lustig Test ( Lustig-1993 ). Theorem (Minimality Test) For any group G with a presentation P, suppose there is a ring homomorphism ψ from ZG into the matrix ring of all m × m-matrices (m ≥ 1) over some commutative ring R with 1. Suppose also that ψ(1) = Im×m. If ψ maps the second Fox ideal I2(P) to 0 (in other words, if I2(P) is contained in the kernel of ψ), then P is minimal. Lustig-1993 , Fox ideals, N-torsion and applications to groups and 3-monifolds. In Two-dimensional homotopy and combinatorial group theory (C. Hog-Angeloni, W. Metzler and A.J. Sieradski, edts), Cambridge University Press, 219-250 (1993).

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks General Aim Reminders

Minimality for Monoids

(Spherical) monoid pictures. ( Pride-1995 )

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks General Aim Reminders

Minimality for Monoids

(Spherical) monoid pictures. ( Pride-1995 ) Pride Test. (Still unpublished !! since 1995) ( Cevik-2003 - 2007 ) Pride-1995 Low-Dimensional Homotopy Theory for Monoids, Int.

• J. Algebra Comput., (1995).

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Zn ⋊ Z case

Let us consider the split extension G = Zn ⋊θ Z with a presentation PG =

• a, b ; an, aba−kb−1

, (1) where k ∈ Z+, gcd(k, n) = 1 and k < n.

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Zn ⋊ Z case

Let us consider the split extension G = Zn ⋊θ Z with a presentation PG =

• a, b ; an, aba−kb−1

, (1) where k ∈ Z+, gcd(k, n) = 1 and k < n. In Baik-1992 , Y.G. Baik investigated the minimality of PG in terms of pictures. Baik-1992 , Generators of the second homotopy module of group presentations with applications. Ph.D. Thesis. University of

• Glasgow. 1992.

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Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Zn ⋊ Z case

Lemma ( Baik-1992 ) The presentation PG =

• a, b ; an, aba−kb−1

is always minimal but it is efficient if and only if gcd(k − 1, n) = 1.

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Zn ⋊ Z case

Lemma ( Baik-1992 ) The presentation PG =

• a, b ; an, aba−kb−1

is always minimal but it is efficient if and only if gcd(k − 1, n) = 1. Thus PG is minimal while inefficient if k < n, gcd(k, n) = 1 and gcd(k − 1, n) = 1.

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Zn ⋊ Z case

By considering above lemma, the first result is given as in the following. Theorem The presentation PG as in (1), where k < n, gcd(k, n) = 1 and also gcd(k − 1, n) = 1, has a set of generating functions p1(a) = a − 1, p2(b) = kb − 1, p3(a) = φn(a), where φn denotes the n.th cyclotomic polynomial over Q defined by φn(x) = xn − 1 x − 1 having degree n − 1.

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Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Zn ⋊ Z case

Since the sum of the mth powers of the first n positive integers can be expressed as Sm(n) =

n

• k=1

km = 1m + 2m + . . . + nm , the Bernoulli numbers can be written in a formula as Sm(n) = 1 m + 1

m

• k=0

m + 1 k

• Bk nm+1−k ,

where B1 = +1/2.

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Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Zn ⋊ Z case

It is also known that Bernoulli numbers Bn and polynomials Bn(x) are defined by the generating functions as t et − 1 =

∞

• n=0

Bn tn n! and t et − 1 ext =

∞

• n=0

Bn(x)tn n! .

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Zn ⋊ Z case

Corollary ( Cevik et al.-2013 ) The generating function p3(a) = φn(a) = φn(a) = an − 1 a − 1 , where a is the generator of Zn, is actually expressed in the meaning of (twisted) Bernoulli numbers and polynomials. We may refer Srivastava et al.-2005 , Jang et al.-2010 for (twisted) Bernoulli numbers and polynomials.

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Zn ⋊ Z case

Srivastava et al.-2005 , q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series, Russ.

• J. Math. Phys.

12(2), (2005) 241-268. Jang et al.-2010 , A note on symmetric properties of twisted q-Bernoulli polynomials and the twisted generalized q-Bernoulli polynomials, Adv. Diff. Equa. ID. 801580, (2010) 13 pages.

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Z2 ⋊ Z case

Let K be a free abelian monoid of rank 2 (i.e. K = Z2) presented by PK = y1, y2 ; y1y2 = y2y1, and let ψ be the endomorphism ψM, where M is the matrix    α α′ β β′    (α, α′, β, β′ ∈ Z+) given by [y1] − → [yα

1 yα′ 2 ] and [y2] −

→ [yβ

1 yβ′ 2 ]. Further, let A be the infinite

cyclic monoid Z with a presentation PA = x ; . Then the semidirect product M = K ⋊θ A has a presentation PM =

• y1, y2, x ; y1y2 = y2y1, y1x = xyα

1 yα′ 2 , y2x = xyβ 1 yβ′ 2

• .

(2)

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Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Z2 ⋊ Z case

Lemma ( Cevik-2003 ) The presentation PM in (2) is efficient if and only if det M ≡ 1 mod p. On the other hand it is minimal but inefficient if det M = 2. Cevik-2003 , Minimal but inefficient presentations of the semidirect products of some monoids, Semigroup Forum 66, 1-17 (2003).

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Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Z2 ⋊ Z case

The array polynomials Sn

k (x) are defined by means of the

generating function (et − 1)ketx x! =

∞

• n=0

Sn

k (x)tn

n!. Array polynomials can also be defined in the form Sn

k (x) = 1

k!

k

• j=0

(−1)k−j k j

• (x + j)n.

(3) Since the coefficients of array polynomials are integers, they have very huge applications, specially in the system control of engineering.

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

In fact these integer coefficients give us an opportunity to use these polynomials in our case since we are working on presentations.

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

In fact these integer coefficients give us an opportunity to use these polynomials in our case since we are working on presentations. There also exist some other polynomials, namely Dickson, Bell, Abel, Mittag-Leffler etc., which have integer coefficients.

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Z2 ⋊ Z case

Theorem Let us consider the monoid M = Z2 ⋊θ Z with a presentation PM =

• b1, b2, a ; b1b2 = b2b1, b1a = ab2

1, b2a = ab1b2

• .

Then PM has a set of generating functions p1(a) = Sn

n (a) − 2S1 0(a),

p2(b1) = Sn

n (b1) − S1 0(b1),

p3(b2) = S1

0(b2) − Sn n (b2),

where Sn

k (x) is defined as in (3).

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Z2 ⋊ Z case

Theorem Let us consider the monoid M = Z2 ⋊θ Z with a presentation PM =

• b1, b2, a ; b1b2 = b2b1, b1a = ab2

1, b2a = ab1b2

• .

Then PM has a set of generating functions p1(a) = Sn

n (a) − 2S1 0(a),

p2(b1) = Sn

n (b1) − S1 0(b1),

p3(b2) = S1

0(b2) − Sn n (b2),

where Sn

k (x) is defined as in (3).

The above PM is inefficient but minimal.

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Z2 ⋊ Z case

Theorem Let us consider the presentation PM =

• b1, b2, a ; b1b2 = b2b1, b1a = abdet M

1

, b2a = ab1b2

• for the monoid M = Z2 ⋊θ Z. Then PM has a set of generating

functions p1(a) = Sn

n (a) − det M S1 0(a),

p2(b1) = Sn

n (b1) − S1 0(b1),

p3(b2) = S1

0(b2) − Sn n (b2),

where det M = 2 and Sn

k (x) is defined as in (3).

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Z2 ⋊ Z case

Theorem Let us consider the presentation PM =

• b1, b2, a ; b1b2 = b2b1, b1a = abdet M

1

, b2a = ab1b2

• for the monoid M = Z2 ⋊θ Z. Then PM has a set of generating

functions p1(a) = Sn

n (a) − det M S1 0(a),

p2(b1) = Sn

n (b1) − S1 0(b1),

p3(b2) = S1

0(b2) − Sn n (b2),

where det M = 2 and Sn

k (x) is defined as in (3).

The above PM is efficient on minimal number of generators.

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Z2 ⋊ Z case

There also exist Stirling numbers of the second kind which are defined as the generating function (et − 1)k k! =

∞

• n=0

S(n, k)tn n! . These Stirling numbers can also be defined by S(n, k) = 1 k!

k

• j=0

(−1)j k j

• (k − j)n,

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Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Z2 ⋊ Z case

and they satisfy the well known properties S(n, k) =          1 ; k = 1 or k = n n 2

• ;

k = n − 1, δn,0 ; k = 0, where δn,0 denotes the Kronecker symbol.

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Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Z2 ⋊ Z case

It is known that Stirling numbers are used in combinatorics, in number theory, in discrete probability distributions for finding higher order moments, etc. We finally note that since S(n, k) is the number of ways to partition a set of n objects into k groups, these numbers find an application area in the theory of partitions.

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Z2 ⋊ Z case

It is known that Stirling numbers are used in combinatorics, in number theory, in discrete probability distributions for finding higher order moments, etc. We finally note that since S(n, k) is the number of ways to partition a set of n objects into k groups, these numbers find an application area in the theory of partitions. In addition to the above formulas for S(n, k), we have xn =

n

• k=0

x k

• k!S(n, k)

as a formula for Stirling numbers. We then have the following result.

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Z2 ⋊ Z case

Corollary PM =

• b1, b2, a ; b1b2 = b2b1, b1a = ab2

1, b2a = ab1b2

• has a set of generating functions in terms of Stirling numbers as

a0 − 2a1 =

• k=0

a k

• k!S(0, k) − 2

1

• k=0

a k

• k!S(1, k),

b0

1 − b1 1 =

• k=0

b1 k

• k!S(0, k) −

1

• k=0

b1 k

• k!S(1, k),

b1

2 − b0 2 = 1

• k=0

b2 k

• k!S(1, k) −
• k=0

b2 k

• k!S(0, k).

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Z2 ⋊ Z case

The above corollary can also stated for the presentation PM =

• b1, b2, a ; b1b2 = b2b1, b1a = abdet M

1

, b2a = ab1b2

• .

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Remarks

As we noted in Key Point, to study with the minimal presentations has an advantage for our aim. Conversely, useage of generating functions whether imply a presentation having minimal number of generators.

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Remarks

As we noted in Key Point, to study with the minimal presentations has an advantage for our aim. Conversely, useage of generating functions whether imply a presentation having minimal number of generators. More specify, by using generating functions (used in here or some others) whether it is possible to obtain a new minimality test for groups and monoids.

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Remarks

As we noted in Key Point, to study with the minimal presentations has an advantage for our aim. Conversely, useage of generating functions whether imply a presentation having minimal number of generators. More specify, by using generating functions (used in here or some others) whether it is possible to obtain a new minimality test for groups and monoids. The material in this talk and the above notes can also be investigated for semigroups.

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Remarks

✬ ✫ ✩ ✪

(Simple) GRAPHS Chemical E N E R G Y

✲

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Remarks

✬ ✫ ✩ ✪

(Simple) GRAPHS Chemical E N E R G Y

✲

The chemical energy is one of the most important application areas of graph theory (cf. Gutman-2001 , Gungor et al.-2010 , Bozkurt et al.-2010 ).

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Remarks

Gutman-2001 The energy of a graph: Old and new results. In: A. Betten, A. Kohnert, R. Laue, A. Wassermann (Eds.), Algebraic Combinatorics and Applications, Springer-Verlag, Berlin, (2001). Gungor et al.-2010 On the Harary Energy and Harary Estrada Index of a Graph, MATCH-Commun. Math. Comput. Chemist. 64(1), (2010) 281-296 Bozkurt et al.-2010 Randic Matrix and Randic Energy, MATCH-Commun. Math. Comput. Chemist. 64(1), (2010) 239-250

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Γ simple graph. A(Γ) adjacency (square) matrix. λ1, λ2, . . ., λn eigenvalues of A(Γ) (Distance) Energy E(Γ) =

n

• i=1

|λi|. H(Γ) =

• 1

dij

• Harary (square) matrix, where dij is the lenght
• f the shortest path between vertices vi and vj.

ρ1, ρ2, . . ., ρn eigenvalues of H(Γ) Harary Energy HE(Γ) =

n

• i=1

|ρi|.

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Remarks

It is worth to study whether this chemical energy can also be

• btained from group or monoid pictures.

✬ ✫ ✩ ✪

E N E R G Y

✲

P I C T U R E S Chemical

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

Remarks

✚✙ ✛✘ ✚✙ ✛✘ ✒✑ ✓✏ ✒✑ ✓✏ ✒✑ ✓✏

• ⑦

✲ ✸ ✟ ✟ ✯ ✲ ◗ ◗ s ❄ ❄ ✒ ■

• +

+ + + a a a a a a b b b b

✲ ✤ ✣ ✜ ✢

ENERGY

???

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships

Introduction Main Results on the group Zn ⋊ Z Main Results on the monoid Z2 ⋊ Z Final Remarks

THANK YOU!

Ahmet Sinan C ¸evik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships