A new approach to connect Algebra with Analysis ... · Introduction Main Results on the group Zn o...
Transcript of 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
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal 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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
General AimReminders
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Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
General AimReminders
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
General AimReminders
Reason of this study
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).
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
General AimReminders
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 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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
General AimReminders
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 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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
General AimReminders
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 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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
General AimReminders
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
General AimReminders
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
General AimReminders
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)
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
General AimReminders
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 !!
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
General AimReminders
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 !!
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
General AimReminders
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
General AimReminders
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).
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
General AimReminders
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 Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
General AimReminders
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 Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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!.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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)
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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).
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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,
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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).
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
Z2 o Z case
The above corollary can also stated for the presentation
PM =⟨
b1, b2, a ; b1b2 = b2b1, b1a = abdet M1 , b2a = ab1b2
⟩.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
Remarks
'
&
$
%(Simple) GRAPHS
Chemical
E N E R G Y
-
The chemical energy is one of the most important applicationareas of graph theory (cf. Gutman-2001 ,
Gungor et al.-2010 , Bozkurt et al.-2010 ).
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
Remarks
'
&
$
%(Simple) GRAPHS
Chemical
E N E R G Y
-
The chemical energy is one of the most important applicationareas of graph theory (cf. Gutman-2001 ,
Gungor et al.-2010 , Bozkurt et al.-2010 ).
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
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
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal 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) =[
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 |.
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
Remarks
It is worth to study whether this chemical energy can also beobtained from group or monoid pictures.
'
&
$
%E N E R G Y
-P I C T U R E S
Chemical
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
Remarks
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Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions
IntroductionMain Results on the group Zn o Z
Main Results on the monoid Z2 o ZFinal Remarks
THANK YOU!
Ahmet Sinan Cevik www.ahmetsinancevik.com A new approach to connect Algebra with Analysis: Relationships and Applications between Presentations and Generating Functions