Search results for "Combinatorics"
showing 10 items of 1770 documents
Properties of a matrix group associated to a {K,s+1}-potent matrix
2012
In a previous paper, the authors introduced and characterized a new kind of matrices called {K,s+1}-potent. In this paper, an associated group to a {K, s+1}-potent matrix is explicitly constructed and its properties are studied. Moreover, it is shown that the group is a semidirect product of Z_2 acting on Z_{(s+1)^2-1}. For some values of s, more specifications on the group are derived. In addition, some illustrative examples are given.
L∞ estimates in optimal mass transportation
2016
We show that in any complete metric space the probability measures μ with compact and connected support are the ones having the property that the optimal transportation distance to any other probability measure ν living on the support of μ is bounded below by a positive function of the L∞ transportation distance between μ and ν. The function giving the lower bound depends only on the lower bound of the μ-measures of balls centered at the support of μ and on the cost function used in the optimal transport. We obtain an essentially sharp form of this function. In the case of strictly convex cost functions we show that a similar estimate holds on the level of optimal transport plans if and onl…
Banach spaces of general Dirichlet series
2018
Abstract We study when the spaces of general Dirichlet series bounded on a half plane are Banach spaces, and show that some of those classes are isometrically isomorphic between themselves. In a precise way, let { λ n } be a strictly increasing sequence of positive real numbers such that lim n → ∞ λ n = ∞ . We denote by H ∞ ( λ n ) the complex normed space of all Dirichlet series D ( s ) = ∑ n b n λ n − s , which are convergent and bounded on the half plane [ Re s > 0 ] , endowed with the norm ‖ D ‖ ∞ = sup Re s > 0 | D ( s ) | . If (⁎) there exists q > 0 such that inf n ( λ n + 1 q − λ n q ) > 0 , then H ∞ ( λ n ) is a Banach space. Further, if there exists a strictly increasing sequ…
A bilinear version of Orlicz–Pettis theorem
2008
Abstract Given three Banach spaces X, Y and Z and a bounded bilinear map B : X × Y → Z , a sequence x = ( x n ) n ⊆ X is called B -absolutely summable if ∑ n = 1 ∞ ‖ B ( x n , y ) ‖ Z is finite for any y ∈ Y . Connections of this space with l weak 1 ( X ) are presented. A sequence x = ( x n ) n ⊆ X is called B -unconditionally summable if ∑ n = 1 ∞ | 〈 B ( x n , y ) , z ∗ 〉 | is finite for any y ∈ Y and z ∗ ∈ Z ∗ and for any M ⊆ N there exists x M ∈ X for which ∑ n ∈ M 〈 B ( x n , y ) , z ∗ 〉 = 〈 B ( x M , y ) , z ∗ 〉 for all y ∈ Y and z ∗ ∈ Z ∗ . A bilinear version of Orlicz–Pettis theorem is given in this setting and some applications are presented.
On the product of balanced sequences
2011
The product w = u ⊗ v of two sequences u and v is a naturally defined sequence on the alphabet of pairs of symbols. Here, we study when the product w of two balanced sequences u,v is balanced too. In the case u and v are binary sequences, we prove, as a main result, that, if such a product w is balanced and deg ( w ) = 4, then w is an ultimately periodic sequence of a very special form. The case of arbitrary alphabets is approached in the last section. The partial results obtained and the problems proposed show the interest of the notion of product in the study of balanced sequences.
An Extension of Weyl’s Equidistribution Theorem to Generalized Polynomials and Applications
2020
Author's accepted manuscript. This is a pre-copyedited, author-produced version of an article accepted for publication in International Mathematics Research Notices following peer review. The version of record Bergelson, V., Knutson, I. J. H. & Son, Y. (2020). An Extension of Weyl’s Equidistribution Theorem to Generalized Polynomials and Applications. International Mathematics Research Notices, 2021(19), 14965-15018 is available online at: https://academic.oup.com/imrn/article/2021/19/14965/5775499 and https://doi.org/10.1093/imrn/rnaa035. Generalized polynomials are mappings obtained from the conventional polynomials by the use of the operations of addition and multiplication and taking th…
Algebras with intermediate growth of the codimensions
2006
AbstractLet F be a field of characteristic zero and let A be an F-algebra. The polynomial identities satisfied by A can be measured through the asymptotic behavior of the sequence of codimensions and the sequence of colengths of A. For finite dimensional algebras we show that the colength sequence of A is polynomially bounded and the codimension sequence cannot have intermediate growth. We then prove that for general nonassociative algebras intermediate growth of the codimensions is allowed. In fact, for any real number 0<β<1, we construct an algebra A whose sequence of codimensions grows like nnβ.
Varieties of Algebras with Superinvolution of Almost Polynomial Growth
2015
Let A be an associative algebra with superinvolution ∗ over a field of characteristic zero and let $c_{n}^{\ast }(A)$ be its sequence of corresponding ∗-codimensions. In case A is finite dimensional, we prove that such sequence is polynomially bounded if and only if the variety generated by A does not contain three explicitly described algebras with superinvolution. As a consequence we find out that no intermediate growth of the ∗-codimensions between polynomial and exponential is allowed.
A continuing debate in elementary geometry: the Simson–Wallace line and its many generalisations
2016
For some years now the importance has been appraised of demonstrating elementary geometry to pupils and future teachers through interactive geometry software. This fits within a view of the teaching of geometry that stresses a hands-on approach, thanks to which it is possible to teach the subject via historical syllabi, touching on ideas from different origins and of a transversal nature. The debate about the role of elementary geometry in the last 30 years is connected to this, with contributions by scholars such as Yaglom, Scimemi and Betti. In the perspective of following a sequence of elementary geometry constructions historically connected with each other, we suggest a path that analys…
On the order of indeterminate moment problems
2013
For an indeterminate moment problem we denote the orthonormal polynomials by P_n. We study the relation between the growth of the function P(z)=(\sum_{n=0}^\infty|P_n(z)|^2)^{1/2} and summability properties of the sequence (P_n(z)). Under certain assumptions on the recurrence coefficients from the three term recurrence relation zP_n(z)=b_nP_{n+1}(z)+a_nP_n(z)+b_{n-1}P_{n-1}(z), we show that the function P is of order \alpha with 0<\alpha<1, if and only if the sequence (P_n(z)) is absolutely summable to any power greater than 2\alpha. Furthermore, the order \alpha is equal to the exponent of convergence of the sequence (b_n). Similar results are obtained for logarithmic order and for more ge…