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Journal articles on the topic 'Polynomials; Abelian group elements'

Author: Grafiati

Published: 4 June 2021

Last updated: 15 February 2022

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1

Basit, B., and A. J. Pryde. "Polynomials and functions with finite spectra on locally compact Abelian groups." Bulletin of the Australian Mathematical Society 51, no. 1 (February 1995): 33–42. http://dx.doi.org/10.1017/s0004972700013873.

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In this paper we define polynomials on a locally compact Abelian group G and prove the equivalence of our definition with that of Domar. We explore the properties of polynomials and determine their spectra. We also characterise the primary ideals of certain Beurling algebras on the group of integers Z. This allows us to classify those elements of that have finite spectrum. If ϕ is a uniformly continuous function with bounded differences then there is a Beurling algebra naturally associated with ϕ. We give a condition on the spectrum of ϕ relative to this algebra which ensures that ϕ is bounded. Finally we give spectral conditions on a bounded function on ℝ that ensure that its indefinite integral is bounded.

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2

Ramezan-Nassab, M. "Group algebras whose p-elements form a subgroup." Journal of Algebra and Its Applications 16, no. 09 (September 30, 2016): 1750170. http://dx.doi.org/10.1142/s0219498817501705.

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Let [Formula: see text] be a group, [Formula: see text] a field of characteristic [Formula: see text], and [Formula: see text] the unit group of the group algebra [Formula: see text]. In this paper, among other results, we show that if either (1) [Formula: see text] satisfies a non-matrix polynomial identity, or (2) [Formula: see text] is locally finite, [Formula: see text] is infinite and [Formula: see text] is an Engel-by-finite group, then the [Formula: see text]-elements of [Formula: see text] form a (normal) subgroup [Formula: see text] and [Formula: see text] is abelian (here, of course, [Formula: see text] if [Formula: see text]).

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3

Abdollahi, A., A. Azad, A. Mohammadi Hassanabadi, and M. Zarrin. "B.H. Neumann's Question on Ensuring Commutativity of Finite Groups." Bulletin of the Australian Mathematical Society 74, no. 1 (January 2006): 121–32. http://dx.doi.org/10.1017/s000497270004750x.

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This paper is an attempt to provide a partial answer to the following question put forward by Bernhard H. Neumann in 2000: “Let G be a finite group of order g and assume that however a set M of m elements and a set N of n elements of the group is chosen, at least one element of M commutes with at least one element of N. What relations between g, m, n guarantee that G is Abelian?” We find an exponential function f(m,n) such that every such group G is Abelian whenever |G| > f(m,n) and this function can be taken to be polynomial if G is not soluble. We give an upper bound in terms of m and n for the solubility length of G, if G is soluble.

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4

Greenhill, Catherine. "An Algorithm for Recognising the Exterior Square of a Multiset." LMS Journal of Computation and Mathematics 3 (2000): 96–116. http://dx.doi.org/10.1112/s1461157000000231.

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AbstractThe exterior square of a multiset is a natural combinatorial construction which is related to the exterior square of a vector space. We consider multisets of elements of an abelian group. Two properties are defined which a multiset may satisfy: recognisability and involution-recognisability. A polynomial-time algorithm is described which takes an input multiset and returns either (a) a multiset which is either recognisable or involution-recognisable and whose exterior square equals the input multiset, or (b) the message that no such multiset exists. The proportion of multisets which are neither recognisable nor involution-recognisable is shown to be small when the abelian group is finite but large. Some further comments are made about the motivating case of multisets of eigenvalues of matrices.

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5

Hare, K. E., and J. A. Ward. "Finite dimensional H-invariant spaces." Bulletin of the Australian Mathematical Society 56, no. 3 (December 1997): 353–61. http://dx.doi.org/10.1017/s0004972700031142.

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A subset V of M(G) is left H-invariant if it is invariant under left translation by the elements of H, a subset of a locally compact group G. We establish necessary and sufficient conditions on H which ensure that finite dimensional subspaces of M(G) when G is compact, or of L∞(G) when G is locally compact Abelian, which are invariant in this weaker sense, contain only trigonometric polynomials. This generalises known results for finite dimensional G-invariant subspaces. We show that if H is a subgroup of finite index in a compact group G, and the span of the H-translates of μ is a weak*-closed subspace of L∞(G) or M(G) (or is closed in Lp(G)for 1 ≤ p < ∞), then μ is a trigonometric polynomial.We also obtain some results concerning functions that possess the analogous weaker almost periodic condition relative to H.

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6

Neumann, Peter M., and Cheryl E. Praeger. "On Tensor-Factorisation Problems,I: The Combinatorial Problem." LMS Journal of Computation and Mathematics 7 (2004): 73–100. http://dx.doi.org/10.1112/s1461157000001054.

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AbstractA k-multiset is an unordered k-tuple, perhaps with repetitions. If x is an r-multiset {x1, …, xr} and y is an s-multiset {y1, …, ys} with elements from an abelian group A the tensor product x ⊗ y is defined as the rs-multiset {xi yj | 1 ≤ i ≤ r, 1 ≤ j ≤ s}. The main focus of this paper is a polynomial-time algorithm to discover whether a given rs-multiset from A can be factorised. The algorithm is not guaranteed to succeed, but there is an acceptably small upper bound for the probability of failure. The paper also contains a description of the context of this factorisation problem, and the beginnings of an attack on the following division-problem: is a given rs-multiset divisible by a given r-multiset, and if so, how can division be achieved in polynomially bounded time?

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7

Ivanyos, G. "On solving systems of random linear disequations." Quantum Information and Computation 8, no. 6&7 (July 2008): 579–94. http://dx.doi.org/10.26421/qic8.6-7-2.

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An important special case of the hidden subgroup problem is equivalent to the hidden shift problem over abelian groups. An efficient solution to the latter problem could serve as a building block of quantum hidden subgroup algorithms over solvable groups. The main idea of a promising approach to the hidden shift problem is a reduction to solving systems of certain random disequations in finite abelian groups. By a disequation we mean a constraint of the form $f(x)\neq 0$. In our case, the functions on the left hand side are generalizations of linear functions. The input is a random sample of functions according to a distribution which is up to a constant factor uniform over the "linear" functions $f$ such that $f(u)\neq 0$ for a fixed, although unknown element $u\in A$. The goal is to find $u$, or, more precisely, all the elements $u'\in A$ satisfying the same disequations as $u$. In this paper we give a classical probabilistic algorithm which solves the problem in an abelian $p$-group $A$ in time polynomial in the sample size $N$, where $N=(\log\size{A})^{O(q^2)}$, and $q$ is the exponent of $A$.

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8

Matsuda, Ryuki. "Note on integral closures of semigroup rings." Tamkang Journal of Mathematics 31, no. 2 (June 30, 2000): 137–44. http://dx.doi.org/10.5556/j.tkjm.31.2000.405.

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Let $S$ be a subsemigroup which contains 0 of a torsion-free abelian (additive) group. Then $S$ is called a grading monoid (or a $g$-monoid). The group $ \{s-s'|s,s'\in S\}$ is called the quotient group of $S$, and is denored by $q(S)$. Let $R$ be a commutative ring. The total quotient ring of $R$ is denoted by $q(R)$. Throught the paper, we assume that a $g$-monoid properly contains $ \{0\}$. A commutative ring is called a ring, and a non-zero-divisor of a ring is called a regular element of the ring. We consider integral elements over the semigroup ring $ R[X;S]$ of $S$ over $R$. Let $S$ be a $g$-monoid with quotient group $G$. If $ n\alpha\in S$ for an element $ \alpha$ of $G$ and a natural number $n$ implies $ \alpha\in S$, then $S$ is called an integrally closed semigroup. We know the following fact: ${\bf Theorem~1}$ ([G2, Corollary 12.11]). Let $D$ be an integral domain and $S$ a $g$-monoid. Then $D[X;S]$ is integrally closed if and only if $D$ is an integrally closed domain and $S$ is an integrally closed semigroup. Let $R$ be a ring. In this paper, we show that conditions for $R[X;S]$ to be integrally closed reduce to conditions for the polynomial ring of an indeterminate over a reduced total quotient ring to be integrally closed (Theorem 15). Clearly the quotient field of an integral domain is a von Neumann regular ring. Assume that $q(R)$ is a von Neumann regular ring. We show that $R[X;S]$ is integrally closed if and only if $R$ is integrally closed and $S$ is integrally closed (Theorem 20). Let $G$ be a $g$-monoid which is a group. If $R$ is a subring of the ring $T$ which is integrally closed in $T$, we show that $R[X;G]$ is integrally closed in $T[X;S]$ (Theorem 13). Finally, let $S$ be sub-$g$-monoid of a totally ordered abelian group. Let $R$ be a subring of the ring $T$ which is integrally closed in $T$. If $g$ and $h$ are elements of $T[X;S]$ with $h$ monic and $gh\in R[X;S]$, we show that $g\in R[X;S]$ (Theorem 24).

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9

Gutierrez, Jaime, and Carlos Ruiz De Velasco Y Bellas. "Distributive elements in the near-rings of polynomials." Proceedings of the Edinburgh Mathematical Society 32, no. 1 (February 1989): 73–80. http://dx.doi.org/10.1017/s0013091500006921.

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As usual in the theory of polynomial near-rings, we deal with right near-rings. If N = (N, +,·) is a near-ring, the set of distributive elements of N will be denoted by Nd;It is easy to check that, if N is an abelian near-ring (i.e., r + s = s + r, for all r, s∈N), then Nd is a subring of N.

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10

Ashlock, Daniel A. "Permutation polynomials of Abelian group rings over finite fields." Journal of Pure and Applied Algebra 86, no. 1 (April 1993): 1–5. http://dx.doi.org/10.1016/0022-4049(93)90148-m.

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11

Gao, Weidong, Meiling Huang, Wanzhen Hui, Yuanlin Li, Chao Liu, and Jiangtao Peng. "Sums of sets of abelian group elements." Journal of Number Theory 208 (March 2020): 208–29. http://dx.doi.org/10.1016/j.jnt.2019.07.026.

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12

Obad, Alaa Mohammed, Asif Khan, Kottakkaran Sooppy Nisar, and Ahmed Morsy. "q-Binomial Convolution and Transformations of q-Appell Polynomials." Axioms 10, no. 2 (April 19, 2021): 70. http://dx.doi.org/10.3390/axioms10020070.

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In this paper, binomial convolution in the frame of quantum calculus is studied for the set Aq of q-Appell sequences. It has been shown that the set Aq of q-Appell sequences forms an Abelian group under the operation of binomial convolution. Several properties for this Abelian group structure Aq have been studied. A new definition of the q-Appell polynomials associated with a random variable is proposed. Scale transformation as well as transformation based on expectation with respect to a random variable is used to present the determinantal form of q-Appell sequences.

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13

Tran, Tan Nhat, and Masahiko Yoshinaga. "Combinatorics of certain abelian Lie group arrangements and chromatic quasi-polynomials." Journal of Combinatorial Theory, Series A 165 (July 2019): 258–72. http://dx.doi.org/10.1016/j.jcta.2019.02.003.

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14

Westreich, Sara. "Quasitriangular Hopf algebras whose group-like elements form an abelian group." Proceedings of the American Mathematical Society 124, no. 4 (1996): 1023–26. http://dx.doi.org/10.1090/s0002-9939-96-03110-3.

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15

Florentino, Carlos, and Jaime Silva. "Hodge-Deligne polynomials of character varieties of free abelian groups." Open Mathematics 19, no. 1 (January 1, 2021): 338–62. http://dx.doi.org/10.1515/math-2021-0038.

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Abstract Let F F be a finite group and X X be a complex quasi-projective F F -variety. For r ∈ N r\in {\mathbb{N}} , we consider the mixed Hodge-Deligne polynomials of quotients X r / F {X}^{r}\hspace{-0.15em}\text{/}\hspace{-0.08em}F , where F F acts diagonally, and compute them for certain classes of varieties X X with simple mixed Hodge structures (MHSs). A particularly interesting case is when X X is the maximal torus of an affine reductive group G G , and F F is its Weyl group. As an application, we obtain explicit formulas for the Hodge-Deligne and E E -polynomials of (the distinguished component of) G G -character varieties of free abelian groups. In the cases G = G L ( n , C ) G=GL\left(n,{\mathbb{C}}\hspace{-0.1em}) and S L ( n , C ) SL\left(n,{\mathbb{C}}\hspace{-0.1em}) , we get even more concrete expressions for these polynomials, using the combinatorics of partitions.

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16

Campbell, H. E. A., J. C. Harris, and D. L. Wehlau. "On rings of invariants of non-modular Abelian groups." Bulletin of the Australian Mathematical Society 60, no. 3 (December 1999): 509–20. http://dx.doi.org/10.1017/s0004972700036674.

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We study the ring of invariant Laurent polynomials associated to the action of a finite diagonal group G on the symmetric algebra of a vector space over a field F. Here the characteristic p of the field F necessarily does not divide the order q = |G| of the group, so G is said to be non-modular. For certain representations of such groups, we can characterise generators of the ring of invariant polynomials in the original symmetric algebra, extending results of Campbell, Hughes, Pappalardi and Selick. In particular we obtain a recursive formula for the number of minimal generators for these rings of invariants.

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17

Grynkiewicz, David J., Luz E. Marchan, and Oscar Ordaz. "Representation of finite abelian group elements by subsequence sums." Journal de Théorie des Nombres de Bordeaux 21, no. 3 (2009): 559–87. http://dx.doi.org/10.5802/jtnb.689.

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18

Danchev, Peter. "MODULAR ABELIAN GROUP ALGEBRAS." Asian-European Journal of Mathematics 03, no. 02 (June 2010): 275–93. http://dx.doi.org/10.1142/s1793557110000192.

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Suppose FG is the F-group algebra of an arbitrary multiplicative abelian group G with p-component of torsion Gp over a field F of char (F) = p ≠ 0. Our theorems state thus: The factor-group S(FG)/Gp of all normed p-units in FG modulo Gp is always totally projective, provided G is a coproduct of groups whose p-components are of countable length and F is perfect. Moreover, if G is a p-mixed coproduct of groups with torsion parts of countable length and FH ≅ FG as F-algebras, then there is a totally projective p-group T of length ≤ Ω such that H × T ≅ G × T. These are generalizations to results by Hill-Ullery (1997). As a consequence, if G is a p-splitting coproduct of groups each of which has p-component with length < Ω and FH ≅ FG are F-isomorphic, then H is p-splitting. This is an extension of a result of May (1989). Our applications are the following: Let G be p-mixed algebraically compact or p-mixed splitting with torsion-complete Gp or p-mixed of torsion-free rank one with torsion-complete Gp. Then the F-isomorphism FH ≅ FG for any group H implies H ≅ G. Moreover, letting G be a coproduct of torsion-complete p-groups or G be a coproduct of p-local algebraically compact groups, then [Formula: see text]-isomorphism [Formula: see text] for an arbitrary group H over the simple field [Formula: see text] of p-elements yields H ≅ G. These completely settle in a more general form a question raised by May (1979) for p-torsion groups and also strengthen results due to Beers-Richman-Walker (1983).

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19

A. Zain, Adnan. "On Group Codes Over Elementary Abelian Groups." Sultan Qaboos University Journal for Science [SQUJS] 8, no. 2 (June 1, 2003): 145. http://dx.doi.org/10.24200/squjs.vol8iss2pp145-151.

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For group codes over elementary Abelian groups we present definitions of the generator and the parity check matrices, which are matrices over the ring of endomorphism of the group. We also lift the theorem that relates the parity check and the generator matrices of linear codes over finite fields to group codes over elementary Abelian groups. Some new codes that are MDS, self-dual, and cyclic over the Abelian group with four elements are given.

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20

Han, Dongchun, Yuan Ren, and Hanbin Zhang. "On ∗-clean group rings over abelian groups." Journal of Algebra and Its Applications 16, no. 08 (August 9, 2016): 1750152. http://dx.doi.org/10.1142/s0219498817501523.

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An associative ring with unity is called clean if each of its elements is the sum of an idempotent and a unit. A clean ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection (∗-invariant idempotent). In a recent paper, Huang, Li and Yuan provided a complete characterization that when a group ring [Formula: see text] is ∗-clean, where [Formula: see text] is a finite field and [Formula: see text] is a cyclic group of an odd prime power order [Formula: see text]. They also provided a necessary condition and a few sufficient conditions for [Formula: see text] to be ∗-clean, where [Formula: see text] is a cyclic group of order [Formula: see text]. In this paper, we extend the above result of Huang, Li and Yuan from [Formula: see text] to [Formula: see text] and provide a characterization of ∗-clean group rings [Formula: see text], where [Formula: see text] is a finite abelian group and [Formula: see text] is a field with characteristic not dividing the exponent of [Formula: see text].

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21

GAO, WEIDONG, DAVID J. GRYNKIEWICZ, and XINGWU XIA. "Onn-Sums in an Abelian Group." Combinatorics, Probability and Computing 25, no. 3 (November 3, 2015): 419–35. http://dx.doi.org/10.1017/s0963548315000255.

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LetGbe an additive abelian group, letn⩾ 1 be an integer, letSbe a sequence overGof length |S| ⩾n+ 1, and let${\mathsf h}$(S) denote the maximum multiplicity of a term inS. Let Σn(S) denote the set consisting of all elements inGwhich can be expressed as the sum of terms from a subsequence ofShaving lengthn. In this paper, we prove that eitherng∈ Σn(S) for every termginSwhose multiplicity is at least${\mathsf h}$(S) − 1 or |Σn(S)| ⩾ min{n+ 1, |S| −n+ | supp (S)| − 1}, where |supp(S)| denotes the number of distinct terms that occur inS. WhenGis finite cyclic andn= |G|, this confirms a conjecture of Y. O. Hamidoune from 2003.

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22

Fisher, Tom. "Visualizing elements of order in the Tate–Shafarevich group of an elliptic curve." LMS Journal of Computation and Mathematics 19, A (2016): 100–114. http://dx.doi.org/10.1112/s1461157016000243.

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We study the elliptic curves in Cremona’s tables that are predicted by the Birch–Swinnerton-Dyer conjecture to have elements of order $7$ in their Tate–Shafarevich group. We show that in many cases these elements are visible in an abelian surface or abelian 3-fold.

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23

Loveys, James. "Abelian groups with modular generic." Journal of Symbolic Logic 56, no. 1 (March 1991): 250–59. http://dx.doi.org/10.2307/2274917.

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AbstractLet G be a stable abelian group with regular modular generic. We show that either1. there is a definable nongeneric K ≤ G such that G/K has definable connected component and so strongly regular generics, or2. distinct elements of the division ring yielding the dependence relation are represented by subgroups of G × G realizing distinct strong types (when regarded as elements of Geq).In the latter case one can choose almost 0-definable subgroups representing the elements of the division ring. We find a bound ((G: G0)) for the size of the division ring in case G has no definable subgroup K so that G/K is infinite with definable connected component. We show in case (2) that the group G/H, where H consists of all nongeneric points of G, inherits a weakly minimal group structure from G naturally, and Th(G/H) is independent of the particular model G as long as G/H is infinite.

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24

Angiono, Iván, and César Galindo. "Pointed finite tensor categories over abelian groups." International Journal of Mathematics 28, no. 11 (October 2017): 1750087. http://dx.doi.org/10.1142/s0129167x17500872.

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We give a characterization of finite pointed tensor categories obtained as de-equivariantizations of the category of corepresentations of finite-dimensional pointed Hopf algebras with abelian group of group-like elements only in terms of the (cohomology class of the) associator of the pointed part. As an application we prove that every coradically graded pointed finite braided tensor category is a de-equivariantization of the category of corepresentations of a finite-dimensional pointed Hopf algebras with abelian group of group-like elements.

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25

LEE, GREGORY T., and ERNESTO SPINELLI. "LIE METABELIAN SKEW ELEMENTS IN GROUP RINGS." Glasgow Mathematical Journal 56, no. 1 (August 13, 2013): 187–95. http://dx.doi.org/10.1017/s0017089513000165.

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AbstractLet F be a field of characteristic p ≠ 2 and G a group without 2-elements having an involution ∗. Extend the involution linearly to the group ring FG, and let (FG)− denote the set of skew elements with respect to ∗. In this paper, we show that if G is finite and (FG)− is Lie metabelian, then G is nilpotent. Based on this result, we deduce that if G is torsion, p > 7 and (FG)− is Lie metabelian, then G must be abelian. Exceptions are constructed for smaller values of p.

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26

Espuelas, Alberto. "On fixed-point-free elements." Mathematical Proceedings of the Cambridge Philosophical Society 103, no. 2 (March 1988): 207–11. http://dx.doi.org/10.1017/s0305004100064781.

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In Theorem 1·2 of [3] the following extension of the classical theorem of Shult is proved.Theorem. Let G be a p-solvable group and let V be a faithful KG-module, where char. Assume that G contains an element x of order pn acting fixed-pointfreely on V. If p is a Fermat prime suppose further that the Sylow 2-subgroups of G are abelian. If p = 2 assume that the Sylow q-subgroups of G for each Mersenne prime q less than 2n are abelian. Then

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27

Danchev, Peter. "The number of idempotents in abelian group rings." Filomat 26, no. 4 (2012): 719–23. http://dx.doi.org/10.2298/fil1204719d.

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Suppose that R is a commutative unitary ring of arbitrary characteristic and G is a multiplicative abelian group. Our main theorem completely determines the cardinality of the set id(RG), consisting of all idempotent elements in the group ring RG. It is explicitly calculated only in terms associated with R, G and their divisions. This result strengthens previous estimates obtained in the literature recently.

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28

Diniz, Diogo, Claudemir Fidelis, and Sérgio Mota. "Identities and central polynomials for real graded division algebras." International Journal of Algebra and Computation 27, no. 07 (November 2017): 935–52. http://dx.doi.org/10.1142/s0218196717500436.

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Let [Formula: see text] be a finite dimensional simple real algebra with a division grading by a finite abelian group [Formula: see text]. In this paper, we provide a finite basis for the [Formula: see text]-ideal of graded polynomial identities for [Formula: see text] and a finite basis for the [Formula: see text]-space of graded central polynomials for [Formula: see text].

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29

Decker, T., J. Draisma, and P. Wocjan. "Efficient quantum algorithm for identifying hidden polynomials." Quantum Information and Computation 9, no. 3&4 (March 2009): 215–30. http://dx.doi.org/10.26421/qic9.3-4-3.

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We consider a natural generalization of an abelian Hidden Subgroup Problem where the subgroups and their cosets correspond to graphs of linear functions over a finite field $\F$ with $d$ elements. The hidden functions of the generalized problem are not restricted to be linear but can also be $m$-variate polynomial functions of total degree $n\geq 2$. The problem of identifying hidden $m$-variate polynomials of degree less or equal to $n$ for fixed $n$ and $m$ is hard on a classical computer since $\Omega(\sqrt{d})$ black-box queries are required to guarantee a constant success probability. In contrast, we present a quantum algorithm that correctly identifies such hidden polynomials for all but a finite number of values of $d$ with constant probability and that has a running time that is only polylogarithmic in $d$.

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30

Hirasawa, Mikami, and Kunio Murasugi. "Twisted Alexander polynomials of 2-bridge knots associated to dihedral representations." Journal of Knot Theory and Its Ramifications 27, no. 02 (February 2018): 1850015. http://dx.doi.org/10.1142/s0218216518500153.

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Let [Formula: see text] be a non-abelian semi-direct product of a cyclic group [Formula: see text] and an elementary abelian [Formula: see text]-group [Formula: see text] of order [Formula: see text], [Formula: see text] being a prime and [Formula: see text]. Suppose that the knot group [Formula: see text] of a knot [Formula: see text] in the [Formula: see text]-sphere is represented on [Formula: see text]. Then we conjectured (and later proved) that the twisted Alexander polynomial [Formula: see text] associated to [Formula: see text] is of the form: [Formula: see text], where [Formula: see text] is the Alexander polynomial of [Formula: see text] and [Formula: see text] is an integer polynomial in [Formula: see text]. In this paper, we present a proof of the following. For a [Formula: see text]-bridge knot [Formula: see text] in [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text] is written as [Formula: see text], where [Formula: see text] is the set of [Formula: see text]-bridge knots whose knot groups map on that of [Formula: see text] with [Formula: see text] odd.

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31

Deaconescu, Marian, and Desmond MacHale. "Odd order groups with an automorphism cubing many elements." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 46, no. 2 (April 1989): 281–88. http://dx.doi.org/10.1017/s1446788700030743.

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AbstractWe determine the structure of a nonabelian group G of odd order such that some automorphism of G sends exactly (1/p)|G| elements to their cubes, where p is the smallest prime dividing |G|. These groups are close to being abelian in the sense that they either have nilpotency class 2 or have an abelian subgroup of index p.

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32

Velni, Komeil Babaei, and Ali Jalali. "Abelian S-matrix elements of D-brane and T-duality." International Journal of Modern Physics A 33, no. 17 (June 19, 2018): 1850106. http://dx.doi.org/10.1142/s0217751x18501063.

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Recently, it has been suggested that the elements of the S-matrix on the worldvolume of an Abelian D-brane might be in accordance with the Ward identity associated with the T-duality. This shows that by applying linear T-duality, a group of S-matrix elements could be found invariant under such transformations. In this work, we apply the T-duality transformations on the S-matrix elements of one B-field and three NS Abelian gauge fields and find some other Abelian S-matrix elements of one closed and three open string states. Also, we will show that the predicted S-matrix elements are reproduced exactly by explicit calculations.

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33

Kahn, Bruno. "Relatively unramified elements in cycle modules." Journal of K-theory 7, no. 3 (April 11, 2011): 409–27. http://dx.doi.org/10.1017/is011003002jkt147.

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AbstractIn a recent paper, Merkurjev showed that for a smooth proper variety X over a field k, the functor M* ↦ A0(X, M0) from cycle modules to abelian groups is corepresented by a cycle module constructed on the Chow group of 0-cycles of X. We show that if “proper” is relaxed, the result still holds by replacing the Chow group of 0-cycles by the 0-th Suslin homology group of X.

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34

Koelink, H. T. "Addition Formula For Big q-Legendre Polynomials From The Quantum Su(2) Group." Canadian Journal of Mathematics 47, no. 2 (April 1, 1995): 436–48. http://dx.doi.org/10.4153/cjm-1995-024-8.

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AbstractFrom Koornwinder's interpretation of big q-Legendre polynomials as spherical elements on the quantum SU(2) group an addition formula is derived for the big g-Legendre polynomial. The formula involves Al-Salam-Carlitz polynomials, little q-Jacobi polynomials and dual q-Krawtchouk polynomials. For the little q-ultraspherical polynomials a product formula in terms of a big q-Legendre polynomial follows by q-integration. The addition and product formula for the Legendre polynomials are obtained when q tends to 1.

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35

SAMOTIJ, WOJCIECH, and BENNY SUDAKOV. "The number of additive triples in subsets of abelian groups." Mathematical Proceedings of the Cambridge Philosophical Society 160, no. 3 (January 26, 2016): 495–512. http://dx.doi.org/10.1017/s0305004115000821.

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AbstractA set of elements of a finite abelian group is called sum-free if it contains no Schur triple, i.e., no triple of elementsx,y,zwithx+y=z. The study of how large the largest sum-free subset of a given abelian group is had started more than thirty years before it was finally resolved by Green and Ruzsa a decade ago. We address the following more general question. Suppose that a setAof elements of an abelian groupGhas cardinalitya. How many Schur triples mustAcontain? Moreover, which sets ofaelements ofGhave the smallest number of Schur triples? In this paper, we answer these questions for various groupsGand ranges ofa.

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36

Thom, Andreas. "Convergent Sequences in Discrete Groups." Canadian Mathematical Bulletin 56, no. 2 (June 1, 2013): 424–33. http://dx.doi.org/10.4153/cmb-2011-155-3.

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AbstractWe prove that a finitely generated group contains a sequence of non-trivial elements that converge to the identity in every compact homomorphic image if and only if the group is not virtually abelian. As a consequence of the methods used, we show that a finitely generated group satisfies Chu duality if and only if it is virtually abelian.

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37

GONÇALVES, JAIRO Z., and ÁNGEL DEL RÍO. "BASS CYCLIC UNITS AS FACTORS IN A FREE GROUP IN INTEGRAL GROUP RING UNITS." International Journal of Algebra and Computation 21, no. 04 (June 2011): 531–45. http://dx.doi.org/10.1142/s0218196711006327.

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Marciniak and Sehgal showed that if u is a non-trivial bicyclic unit of an integral group ring then there is a bicyclic unit v such that u and v generate a non-abelian free group. A similar result does not hold for Bass cyclic units of infinite order based on non-central elements as some of them have finite order modulo the center. We prove a theorem that suggests that this is the only limitation to obtain a non-abelian free group from a given Bass cyclic unit. More precisely, we prove that if u is a Bass cyclic unit of an integral group ring ℤG of a solvable and finite group G, such that u has infinite order modulo the center of U(ℤG) and it is based on an element of prime order, then there is a non-abelian free group generated by a power of u and a power of a unit in ℤG which is either a Bass cyclic unit or a bicyclic unit.

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38

FRANKS, JOHN, MICHAEL HANDEL, and KAMLESH PARWANI. "Fixed points of abelian actions on S2." Ergodic Theory and Dynamical Systems 27, no. 5 (October 2007): 1557–81. http://dx.doi.org/10.1017/s0143385706001088.

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AbstractWe prove that if ${\mathcal F}$ is a finitely generated abelian group of orientation preserving C1 diffeomorphisms of $\mathbb {R}^2$ which leaves invariant a compact set then there is a common fixed point for all elements of ${\mathcal F}$. We also show that if ${\mathcal F}$ is any abelian subgroup of orientation preserving C1 diffeomorphisms of S2 then there is a common fixed point for all elements of a subgroup of ${\mathcal F}$ with index at most two.

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39

ABDELGHANI, LEILA BEN, and DANIEL LINES. "INVOLUTIONS ON KNOT GROUPS AND VARIETIES OF REPRESENTATIONS IN A LIE GROUP." Journal of Knot Theory and Its Ramifications 11, no. 01 (February 2002): 81–104. http://dx.doi.org/10.1142/s0218216502001482.

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We prove the existence of a rationalisation [Formula: see text] of a classical or high-dimensional knot group Π which admits an involution if the Alexander polynomials of the knot are reciprocal. Using the group [Formula: see text] and its involution, we study the local structure, in the neighbourhood of an abelian representation, of the space of representation of the knot group Π in a a Lie group. We apply these results to the groups of classical prime knots up to 10 crossings.

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40

Fulman, Jason. "Hall polynomials and the fixed space of an automorphism of a finite abelian p-group." Archiv der Mathematik 73, no. 1 (July 1999): 1–10. http://dx.doi.org/10.1007/s000130050012.

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41

Chandler, K. A. "Groups Formed by Redefining Multiplication." Canadian Mathematical Bulletin 31, no. 4 (December 1, 1988): 419–23. http://dx.doi.org/10.4153/cmb-1988-061-5.

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AbstractLet G be a group with elements 1,…, n such that the group operation agrees with ordinary multiplication whenever the ordinary product of two elements lies in G. We show that if n is odd, then G is abelian.

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42

Bryce, R. A., and L. J. Rylands. "A note on groups with non-central norm." Glasgow Mathematical Journal 36, no. 1 (January 1994): 37–43. http://dx.doi.org/10.1017/s0017089500030524.

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The norm K(G) of a group G is the subgroup of elements of G which normalize every subgroup of G. Under the name kern this subgroup was introduced by Baer [1]. The norm is Dedekindian in the sense that all its subgroups are normal. A theorem of Dedekind [5] describes the structure of such groups completely: if not abelian they are the direct product of a quaternion group of order eight and an abelian group with no element of order four. Baer [2] proves that a 2-group with non-abelian norm is equal to its norm.

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43

Bryce, R. A. "Subgroups like Wielandt's in finite soluble groups." Mathematical Proceedings of the Cambridge Philosophical Society 107, no. 2 (March 1990): 239–59. http://dx.doi.org/10.1017/s0305004100068511.

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In 1935 Baer[1] introduced the concept of kern of a group as the subgroup of elements normalizing every subgroup of the group. It is of interest from three points of view: that of its structure, the nature of its embedding in the group, and the influence of its internal structure on that of the whole group. The kern is a Dedekind group because all its subgroups are normal. Its structure is therefore known exactly (Dedekind [7]): if not abelian it is a direct product of a copy of the quaternion group of order 8 and an abelian periodic group with no elements of order 4. As for the embedding of the kern, Schenkman[13] shows that it is always in the second centre of the group: see also Cooper [5], theorem 6·5·1. As an example of the influence of the structure of the kern on its parent group we cite Baer's result from [2], p. 246: among 2-groups, only Hamiltonian groups (i.e. non-abelian Dedekind groups) have nonabelian kern.

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44

Svaba, Pavol, Tran van Trung, and Paul Wolf. "LOGARITHMIC SIGNATURES FOR ABELIAN GROUPS AND THEIR FACTORIZATION." Tatra Mountains Mathematical Publications 57, no. 1 (November 1, 2013): 21–33. http://dx.doi.org/10.2478/tmmp-2013-0033.

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ABSTRACT Factorizable logarithmic signatures for finite groups are the essential component of the cryptosystems MST1 and MST3. The problem of finding efficient algorithms for factoring group elements with respect to a given class of logarithmic signatures is therefore of vital importance in the investigation of these cryptosystems. In this paper we are concerned about the factorization algorithms with respect to transversal and fused transversal logarithmic signatures for finite abelian groups. More precisely we present algorithms and their complexity for factoring group elements with respect to these classes of logarithmic signatures. In particular, we show a factoring algorithm with respect to the class of fused transversal logarithmic signatures and also its complexity based on an idea of Blackburn, Cid and Mullan for finite abelian groups.

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45

Okay, Cihan. "Spherical posets from commuting elements." Journal of Group Theory 21, no. 4 (July 1, 2018): 593–628. http://dx.doi.org/10.1515/jgth-2018-0008.

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AbstractIn this paper, we study the homotopy type of the partially ordered set of left cosets of abelian subgroups in an extraspecial p-group. We prove that the universal cover of its nerve is homotopy equivalent to a wedge of r-spheres where {2r\geq 4} is the rank of its Frattini quotient. This determines the homotopy type of the universal cover of the classifying space of transitionally commutative bundles as introduced in [2].

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46

Jedlička, Přemysl, Agata Pilitowska, and Anna Zamojska-Dzienio. "Indecomposable involutive solutions of the Yang–Baxter equation of multipermutational level 2 with abelian permutation group." Forum Mathematicum 33, no. 5 (August 26, 2021): 1083–96. http://dx.doi.org/10.1515/forum-2021-0130.

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Abstract We present a construction of all finite indecomposable involutive solutions of the Yang–Baxter equation of multipermutational level at most 2 with abelian permutation group. As a consequence, we obtain a formula for the number of such solutions with a fixed number of elements. We also describe some properties of the automorphism groups in this case; in particular, we show they are regular abelian groups.

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47

Saeed-ul-Islam, M. "Representations of finite abelian groups Cnm,p." Glasgow Mathematical Journal 26, no. 2 (July 1985): 133–40. http://dx.doi.org/10.1017/s0017089500005899.

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Let C= CmXCmX…XCm be the finite abelian group of order mn generated by n elements w1…,wn of order m. Let C be the field of complex numbers and P a projective representation of G with factor set α over C (see Morris [2]). Further letand

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48

ALEMANY, ELENA, ANTONIO BELTRÁN, and MARÍA JOSÉ FELIPE. "FINITE GROUPS WITH TWO p-REGULAR CONJUGACY CLASS LENGTHS II." Bulletin of the Australian Mathematical Society 79, no. 3 (April 17, 2009): 419–25. http://dx.doi.org/10.1017/s0004972708001287.

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AbstractLet G be a finite group. We prove that if the set of p-regular conjugacy class sizes of G has exactly two elements, then G has Abelian p-complement or G=PQ×A, with P∈Sylp(G), Q∈Sylq(G) and A Abelian.

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49

Wang, Zhihua. "The corepresentation ring of a pointed Hopf algebra of rank one." Journal of Algebra and Its Applications 17, no. 12 (December 2018): 1850236. http://dx.doi.org/10.1142/s0219498818502365.

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Let [Formula: see text] be an arbitrary pointed Hopf algebra of rank one and [Formula: see text] the group of group-like elements of [Formula: see text]. In this paper, we give the decomposition of a tensor product of finite dimensional indecomposable right [Formula: see text]-comodules into a direct sum of indecomposables. This enables us to describe the corepresentation ring of [Formula: see text] in terms of generators and relations. Such a ring is not commutative if [Formula: see text] is not abelian. We describe all nilpotent elements of the corepresentation ring of [Formula: see text] if [Formula: see text] is a finite abelian group or a particular Hamiltonian group. In this case, all nilpotent elements of the corepresentation ring form a principal ideal which is either zero or generated by a nilpotent element of degree 2.

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50

An, Lijian, and Juan Peng. "Finite p-Groups in Which Any Two Noncommutative Elements Generate an Inner Abelian Group of Order p4." Algebra Colloquium 20, no. 02 (April 3, 2013): 215–26. http://dx.doi.org/10.1142/s1005386713000199.

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