A new approach to connect Algebra with Analysis ... · Introduction Main Results on the group Zn o...

IntroductionMain Results on the group Zn o Z

Main Results on the monoid Z2 o ZFinal Remarks

A new approach to connect Algebra with Analysis:Relationships and Applications betweenPresentations and Generating Functions

Ahmet Sinan Cevik www.ahmetsinancevik.com

Selcuk University, Konya/[email protected]

Questions, Algorithms, and Computations in Abstract GroupTheory

May 21-24, 2013

Braunschweig

Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions



Outline

1 IntroductionGeneral AimReminders

2 Main Results on the group Zn o Z

3 Main Results on the monoid Z2 o Z

4 Final Remarks




General AimReminders

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 newapproach to connect Algebra with Analysis: Relationships andApplications between Presentations and Generating Functions,Boundary Value Problems, 2013, 2013:51doi:10.1186/1687-2770-2013-51.





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GRAPHS

GEN. FUNCTS.

ALGBRC

STRUCT.





Reason of this study

In the literature, there are so many studies about figuring out therelationship between algebraic structures and special generatingfunctions (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-Eulernumbers 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.






There exists a connection between graphs and generatingfunctions since the number of vertex-colorings of a graph is givenby a polynomial on the number of used colors (see Birkhoff-1946 ,

Cardoso-2012 ). Based on this polynomial, one can define thechromatic number as the minumum number of colors such thatthe chromatic polynomial is positive.

Birkhoff-1946 , Chromatic polynomials, Trans. Am. Math. Soc.60, (1946) 355-451.

Cardoso-2012 , A generalization of chromatic polynomial of agraph subdivision, J. Math. Scien. 183(2), (2012).






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 worthto study similar connections. In here, we actually assume Psatisfies either efficiency or inefficiency while it is minimal. Then itwill be investigated whether some generating functions can beapplied, and then studied what kind of new properties can beobtained by considering special generating functions over P.Since the results in Cardoso-2012 imply a new studying area forgraphs in the meaning of representation of parameters bygenerating functions, we hope that this study will give anopportunity to make a new classification of infinite groups andmonoids in the meaning of generating functions.






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 worthto study similar connections. In here, we actually assume Psatisfies either efficiency or inefficiency while it is minimal. Then itwill be investigated whether some generating functions can beapplied, and then studied what kind of new properties can beobtained by considering special generating functions over P.

Since the results in Cardoso-2012 imply a new studying area forgraphs in the meaning of representation of parameters bygenerating functions, we hope that this study will give anopportunity to make a new classification of infinite groups andmonoids in the meaning of generating functions.






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 worthto study similar connections. In here, we actually assume Psatisfies either efficiency or inefficiency while it is minimal. Then itwill be investigated whether some generating functions can beapplied, and then studied what kind of new properties can beobtained by considering special generating functions over P.Since the results in Cardoso-2012 imply a new studying area forgraphs in the meaning of representation of parameters bygenerating functions, we hope that this study will give anopportunity to make a new classification of infinite groups andmonoids in the meaning of generating functions.





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. Thissituation effects very positively using the generating functions forthis type of presentations since we have a great advantage to studywith quite limited number of variables in such a generatingfunction.





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 andd(.) 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.





Efficiency (Deficiency)-cont.

P is called minimal if χ(P) 6 χ(P ′) for all presentations P ′.P is called efficient if χ(P) = δ(G ).

G is called efficient if χ(G ) = δ(G ), whereχ(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 since1994)





According to the Key Point, if P is

efficient, then we need to assure that the minimal number ofgenerators !! Wamsley-1973 Not be considered unless statedotherwise,

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 showthat it is

MINIMAL !!





Minimality for Groups-cont.

(Spherical) pictures ( Rourke-1979 , J.Howie-1989 ,

Pride-1991 )

Rourke-1979 , Presentations and the trivial group, Topology oflow dimensional manifolds (ed. R. Fenn), Lecture Notes inMathematics 722 (Springer, Berlin, 1979), 134-143.

J.Howie-1989 , The Quotient of a Free Product of Groups by aSingle 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.





Minimality for Groups-cont.

Lustig Test ( Lustig-1993 ).

Theorem (Minimality Test)

For any group G with a presentation P, suppose there is a ringhomomorphism ψ from ZG into the matrix ring of allm ×m-matrices (m ≥ 1) over some commutative ring R with 1.Suppose also that ψ(1) = Im×m. If ψ maps the second Fox idealI2(P) to 0 (in other words, if I2(P) is contained in the kernel ofψ), then P is minimal.

Lustig-1993 , Fox ideals, N -torsion and applications to groups and3-monifolds. In Two-dimensional homotopy and combinatorialgroup theory (C. Hog-Angeloni, W. Metzler and A.J. Sieradski,edts), Cambridge University Press, 219-250 (1993).





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).




Zn o Z case

Let us consider the split extension G = Znoθ 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. Baikinvestigated 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 ofGlasgow. 1992.




Zn o Z case

Let us consider the split extension G = Znoθ 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. Baikinvestigated 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 ofGlasgow. 1992.




Zn o 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) 6= 1.

Thus PG is minimal while inefficient if k < n, gcd(k , n) = 1 andgcd(k − 1, n) = 1.




Zn o Z case

By considering above lemma, the first result is given as in thefollowing.

Theorem

The presentation PG as in (1), where k < n, gcd(k , n) = 1 andalso 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.




Zn o Z case

Since the sum of the mth powers of the first n positive integerscan 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.




Zn o 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

Bntn

n!

andt

et − 1ext =

∞∑n=0

Bn(x)tn

n!.




Zn o 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 themeaning of (twisted) Bernoulli numbers and polynomials.

We may refer Srivastava et al.-2005 , Jang et al.-2010 for

(twisted) Bernoulli numbers and polynomials.




Zn o Z case

Srivastava et al.-2005 , q-Bernoulli numbers and polynomialsassociated 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 twistedq-Bernoulli polynomials and the twisted generalized q-Bernoullipolynomials, Adv. Diff. Equa. ID. 801580, (2010) 13 pages.




Z2 o Z case

Let K be a free abelian monoid of rank 2 (i.e. K = Z2) presentedby 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] 7−→ [yα1 yα′

2 ] and [y2] 7−→ [yβ1 yβ′

2 ]. Further, let A be the infinitecyclic monoid Z with a presentation PA = 〈x ; 〉. Then thesemidirect product M = K oθ A has a presentation

PM =⟨

y1, y2, x ; y1y2 = y2y1, y1x = xyα1 yα′

2 , y2x = xyβ1 yβ′

2

⟩.

(2)




Z2 o 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 thesemidirect products of some monoids, Semigroup Forum 66, 1-17(2003).




Z2 o Z case

The array polynomials Snk (x) are defined by means of the

generating function

(et − 1)ketx

x!=∞∑n=0

Snk (x)

tn

n!.

Array polynomials can also be defined in the form

Snk (x) =

1

k!

k∑j=0

(−1)k−j(

kj

)(x + j)n. (3)

Since the coefficients of array polynomials are integers, they havevery huge applications, specially in the system control ofengineering.




In fact these integer coefficients give us an opportunity to usethese polynomials in our case since we are working onpresentations.

There also exist some other polynomials, namely Dickson,Bell, Abel, Mittag-Leffler etc., which have integer coefficients.




Z2 o Z case

Theorem

Let us consider the monoid M = Z2 oθ 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) = Snn (a)− 2S1

0 (a),p2(b1) = Sn

n (b1)− S10 (b1),

p3(b2) = S10 (b2)− Sn

n (b2),

where Snk (x) is defined as in (3).

The above PM is inefficient but minimal.




Z2 o Z case

Theorem

Let us consider the presentation

PM =⟨

b1, b2, a ; b1b2 = b2b1, b1a = abdet M1 , b2a = ab1b2

⟩for the monoid M = Z2 oθ Z. Then PM has a set of generatingfunctions

p1(a) = Snn (a)− det M S1

0 (a),p2(b1) = Sn

n (b1)− S10 (b1),

p3(b2) = S10 (b2)− Sn

n (b2),

where det M 6= 2 and Snk (x) is defined as in (3).

The above PM is efficient on minimal number of generators.




Z2 o Z case

There also exist Stirling numbers of the second kind which aredefined 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(

kj

)(k − j)n,




Z2 o Z case

and they satisfy the well known properties

S(n, k) =

1 ; k = 1 or k = n(n2

); k = n − 1,

δn,0 ; k = 0,

where δn,0 denotes the Kronecker symbol.




Z2 o Z case

It is known that Stirling numbers are used in combinatorics, innumber theory, in discrete probability distributions for findinghigher order moments, etc. We finally note that since S(n, k) isthe 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

(xk

)k!S(n, k)

as a formula for Stirling numbers. We then have the followingresult.




Z2 o 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 =0∑

k=0

(ak

)k!S(0, k)− 2

1∑k=0

(ak

)k!S(1, k),

b01 − b1

1 =0∑

k=0

(b1

k

)k!S(0, k)−

1∑k=0

(b1

k

)k!S(1, k),

b12 − b0

2 =1∑

k=0

(b2

k

)k!S(1, k)−

0∑k=0

(b2

k

)k!S(0, k).




Z2 o Z case

The above corollary can also stated for the presentation

PM =⟨

b1, b2, a ; b1b2 = b2b1, b1a = abdet M1 , b2a = ab1b2

⟩.




Remarks

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

More specify, by using generating functions (used in here orsome others) whether it is possible to obtain a new minimalitytest for groups and monoids.

The material in this talk and the above notes can also beinvestigated for semigroups.




Remarks

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Chemical

E N E R G Y

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The chemical energy is one of the most important applicationareas of graph theory (cf. Gutman-2001 ,

Gungor et al.-2010 , Bozkurt et al.-2010 ).




Remarks

Gutman-2001 The energy of a graph: Old and new results. In: A.Betten, A. Kohnert, R. Laue, A. Wassermann (Eds.), AlgebraicCombinatorics and Applications, Springer-Verlag, Berlin, (2001).

Gungor et al.-2010 On the Harary Energy and Harary EstradaIndex 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




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) =[

1dij

]Harary (square) matrix, where dij is the lenght

of the shortest path between vertices vi and vj .ρ1, ρ2, . . ., ρn eigenvalues of H(Γ)

Harary Energy HE (Γ) =n∑

i=1

|ρi |.




Remarks

It is worth to study whether this chemical energy can also beobtained from group or monoid pictures.

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Remarks

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