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Relevant bibliographies by topics / Polynomials; Abelian group elements

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Academic literature on the topic 'Polynomials; Abelian group elements'

Author: Grafiati

Published: 4 June 2021

Last updated: 15 February 2022

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

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 (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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Ramezan-Nassab, M. "Group algebras whose p-elements form a subgroup." Journal of Algebra and Its Applications 16, no. 09 (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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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 (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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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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Hare, K. E., and J. A. Ward. "Finite dimensional H-invariant spaces." Bulletin of the Australian Mathematical Society 56, no. 3 (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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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 (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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Matsuda, Ryuki. "Note on integral closures of semigroup rings." Tamkang Journal of Mathematics 31, no. 2 (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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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 (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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Ashlock, Daniel A. "Permutation polynomials of Abelian group rings over finite fields." Journal of Pure and Applied Algebra 86, no. 1 (1993): 1–5. http://dx.doi.org/10.1016/0022-4049(93)90148-m.

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Dissertations / Theses on the topic "Polynomials; Abelian group elements"

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Greenhill, Catherine. "From multisets to matrix groups : some algorithms related to the exterior square." Thesis, University of Oxford, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.360301.

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Ngcibi, Sakhile Leonard. "Studies of equivalent fuzzy subgroups of finite abelian p-Groups of rank two and their subgroup lattices." Thesis, Rhodes University, 2006. http://hdl.handle.net/10962/d1005230.

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We determine the number and nature of distinct equivalence classes of fuzzy subgroups of finite Abelian p-group G of rank two under a natural equivalence relation on fuzzy subgroups. Our discussions embrace the necessary theory from groups with special emphasis on finite p-groups as a step towards the classification of crisp subgroups as well as maximal chains of subgroups. Unique naming of subgroup generators as discussed in this work facilitates counting of subgroups and chains of subgroups from subgroup lattices of the groups. We cover aspects of fuzzy theory including fuzzy (homo-) isomorphism together with operations on fuzzy subgroups. The equivalence characterization as discussed here is finer than isomorphism. We introduce the theory of keychains with a view towards the enumeration of maximal chains as well as fuzzy subgroups under the equivalence relation mentioned above. We discuss a strategy to develop subgroup lattices of the groups used in the discussion, and give examples for specific cases of prime p and positive integers n,m. We derive formulas for both the number of maximal chains as well as the number of distinct equivalence classes of fuzzy subgroups. The results are in the form of polynomials in p (known in the literature as Hall polynomials) with combinatorial coefficients. Finally we give a brief investigation of the results from a graph-theoretic point of view. We view the subgroup lattices of these groups as simple, connected, symmetric graphs.

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Books on the topic "Polynomials; Abelian group elements"

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Norman, Christopher. Finitely Generated Abelian Groups and Similarity of Matrices over a Field. Springer London, 2012.

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Norman, Christopher. Finitely Generated Abelian Groups and Similarity of Matrices over a Field. Springer, 2012.

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Book chapters on the topic "Polynomials; Abelian group elements"

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Chabert, Jean-Luc. "The Picard Group of the Ring of Integer-Valued Polynomials on a Valuation Domain." In Rings, Modules, Algebras, and Abelian Groups. CRC Press, 2004. http://dx.doi.org/10.1201/9780824750817.ch6.

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Higgins, Peter M. "10. Vector spaces." In Algebra: A Very Short Introduction. Oxford University Press, 2015. http://dx.doi.org/10.1093/actrade/9780198732822.003.0010.

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‘Vector spaces’ discusses the algebra of vector spaces, which are abelian groups with an additional scalar multiplication by a field. Every finite abelian group is the direct product of cyclic groups. Any finite abelian group can be represented in one of two special ways based on numerical relationships between the subscripts of the cyclic groups involved. In one representation, all the subscripts are powers of primes; in the alternative, each subscript is a divisor of its successor. It concludes by bringing together the ideas of modular arithmetic, the construction of the complex numbers, factorization of polynomials, and vector spaces to explain the existence of finite fields.

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3

Brubaker, Ben, Daniel Bump, and Solomon Friedberg. "Knowability." In Weyl Group Multiple Dirichlet Series. Princeton University Press, 2011. http://dx.doi.org/10.23943/princeton/9780691150659.003.0012.

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This chapter introduces the Knowability Lemma, which explains when products of Gauss sums associated to elements of a preaccordion are explicitly evaluable as polynomials in q, the order of the residue class field. It considers an episode in the cartoon associated to the short Gelfand-Tsetlin pattern and the three cases that apply according to the Knowability Lemma, two of which are maximality and knowability. Knowability is not important for the proof that Statement C implies Statement B. The chapter discusses the cases where ε‎ is Class II or Class I, leaving the remaining two cases to the reader. It also describes the variant of the argument for the case that ε‎ is of Class I, again leaving the two other cases to the reader.

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4

Clay, Matt. "Automorphisms of Free Groups." In Office Hours with a Geometric Group Theorist. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691158662.003.0006.

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This chapter discusses the automorphisms of free groups. Every group is the collection of symmetries of some object, namely, its Cayley graph. A symmetry of a group is called an automorphism; it is merely an isomorphism of the group to itself. The collection of all of the automorphisms is also a group too, known as the automorphism group and denoted by Aut (G). The chapter considers basic examples of groups to illustrate what an automorphism is, with a focus on the automorphisms of the symmetric group on three elements and of the free abelian group. It also examines the dynamics of an automorphism of a free group and concludes with a description of train tracks, a topological model for the free group, and the Perron–Frobenius theorem. Exercises and research projects are included.

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