scholarly journals The small inductive dimension of subsets of Alexandroff spaces

Filomat ◽  
2016 ◽  
Vol 30 (11) ◽  
pp. 3007-3014
Author(s):  
Vitalij Chatyrko ◽  
Sang-Eon Han ◽  
Yasunao Hattori
Keyword(s):  
Simple Description ◽  
Inductive Dimension ◽  
The Family ◽  
Small Inductive Dimension ◽  
Positive Integers

We describe the small inductive dimension ind in the class of Alexandroff spaces by the use of some standard spaces. Then for ind we suggest decomposition, sum and product theorems in the class. The sum and product theorems there we prove even for the small transfinite inductive dimension trind. As an application of that, for each positive integers k,n such that k ? n we get a simple description in terms of even and odd numbers of the family S(k,n) = {S ? Kn : |S|=k+1 and indS=k}, where K is the Khalimsky line.

scholarly journals A Markov model for Selmer ranks in families of twists

2014 ◽  
Vol 150 (7) ◽  
pp. 1077-1106 ◽  
Author(s):  
Zev Klagsbrun ◽  
Barry Mazur ◽  
Karl Rubin
Keyword(s):  
Markov Process ◽  
Markov Model ◽  
Elliptic Curve ◽  
Arbitrary Number ◽  
Equilibrium Distribution ◽  
Two Dimensional ◽  
Absolute Galois Group ◽  
Quadratic Twists ◽  
The Family ◽  
Positive Integers

AbstractWe study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}E$ over an arbitrary number field $K$. Under the assumption that ${\rm Gal}(K(E[2])/K) \ {\cong }\ S_3$, we show that the density (counted in a nonstandard way) of twists with Selmer rank $r$ exists for all positive integers $r$, and is given via an equilibrium distribution, depending only on a single parameter (the ‘disparity’), of a certain Markov process that is itself independent of $E$ and $K$. More generally, our results also apply to $p$-Selmer ranks of twists of two-dimensional self-dual ${\bf F}_p$-representations of the absolute Galois group of $K$ by characters of order $p$.

Intersecting Families are Essentially Contained in Juntas

2009 ◽  
Vol 18 (1-2) ◽  
pp. 107-122 ◽  
Author(s):  
IRIT DINUR ◽  
EHUD FRIEDGUT
Keyword(s):  
Fourier Analysis ◽  
Boolean Functions ◽  
Simple Description ◽  
Discrete Fourier Analysis ◽  
Intersecting Family ◽  
Intersecting Families ◽  
Positive Integers

A family$\J$of subsets of {1, . . .,n} is called aj-junta if there existsJ⊆ {1, . . .,n}, with |J| =j, such that the membership of a setSin$\J$depends only onS∩J.In this paper we provide a simple description of intersecting families of sets. Letnandkbe positive integers withk<n/2, and let$\A$be a family of pairwise intersecting subsets of {1, . . .,n}, all of sizek. We show that such a family is essentially contained in aj-junta$\J$, wherejdoes not depend onnbut only on the ratiok/nand on the interpretation of ‘essentially’.Whenk=o(n) we prove that every intersecting family ofk-sets is almost contained in a dictatorship, a 1-junta (which by the Erdős–Ko–Rado theorem is a maximal intersecting family): for any such intersecting family$\A$there exists an elementi∈ {1, . . .,n} such that the number of sets in$\A$that do not containiis of order$\C {n-2}{k-2}$(which is approximately$\frac {k}{n-k}$times the size of a maximal intersecting family).Our methods combine traditional combinatorics with results stemming from the theory of Boolean functions and discrete Fourier analysis.

scholarly journals Small and large inductive dimension for ideal topological spaces

2021 ◽  
Vol 22 (2) ◽  
pp. 417
Author(s):  
Fotini Sereti
Keyword(s):  
Topological Spaces ◽  
Covering Dimension ◽  
Basic Properties ◽  
Inductive Dimension ◽  
Initial Stage ◽  
Small Inductive Dimension ◽  
Important Field

<p>Undoubtedly, the small inductive dimension, ind, and the large inductive dimension, Ind, for topological spaces have been studied extensively, developing an important field in Topology. Many of their properties have been studied in details (see for example [1,4,5,9,10,18]). However, researches for dimensions in the field of ideal topological spaces are in an initial stage. The covering dimension, dim, is an exception of this fact, since it is a meaning of dimension, which has been studied for such spaces in [17]. In this paper, based on the notions of the small and large inductive dimension, new types of dimensions for ideal topological spaces are studied. They are called *-small and *-large inductive dimension, ideal small and ideal large inductive dimension. Basic properties of these dimensions are studied and relations between these dimensions are investigated.</p>

Mixed covering arrays on graphs of small treewidth

2021 ◽  
pp. 2150085
Author(s):  
Soumen Maity ◽  
Charles J. Colbourn
Keyword(s):  
Weighted Graph ◽  
Minimum Size ◽  
Index Formula ◽  
Covering Array ◽  
Covering Arrays ◽  
Combinatorial Objects ◽  
The Family ◽  
Positive Integers ◽  
Basic Graph ◽  
Test Suites

Covering arrays are combinatorial objects that have been successfully applied in design of test suites for testing systems such as software, hardware, and networks where failures can be caused by the interaction between their parameters. Let [Formula: see text] and [Formula: see text] be positive integers with [Formula: see text]. Two vectors [Formula: see text] and [Formula: see text] are qualitatively independent if for any ordered pair [Formula: see text], there exists an index [Formula: see text] such that [Formula: see text]. Let [Formula: see text] be a graph with [Formula: see text] vertices [Formula: see text] with respective vertex weights [Formula: see text]. A mixed covering array on[Formula: see text] , denoted by [Formula: see text], is a [Formula: see text] array such that row [Formula: see text] corresponds to vertex [Formula: see text], entries in row [Formula: see text] are from [Formula: see text]; and if [Formula: see text] is an edge in [Formula: see text], then the rows [Formula: see text] are qualitatively independent. The parameter [Formula: see text] is the size of the array. Given a weighted graph [Formula: see text], a mixed covering array on [Formula: see text] with minimum size is optimal. In this paper, we introduce some basic graph operations to provide constructions for optimal mixed covering arrays on the family of graphs with treewidth at most three.

scholarly journals CYCLIC SYSTEMS OF SIMULTANEOUS CONGRUENCES

2010 ◽  
Vol 06 (02) ◽  
pp. 219-245 ◽  
Author(s):  
JEFFREY C. LAGARIAS
Keyword(s):  
Diophantine Equations ◽  
Extremal Solutions ◽  
Infinitely Many Solutions ◽  
Cyclic System ◽  
Positivity Condition ◽  
The Family ◽  
Integer Solutions ◽  
Almost All ◽  
Positive Integers ◽  
Simultaneous Congruences

This paper considers the cyclic system of n ≥ 2 simultaneous congruences [Formula: see text] for fixed nonzero integers (r, s) with r > 0 and (r, s) = 1. It shows there are only finitely many solutions in positive integers qi ≥ 2, with gcd (q1q2 ⋯ qn, s) = 1 and obtains sharp bounds on the maximal size of solutions for almost all (r, s). The extremal solutions for r = s = 1 are related to Sylvester's sequence 2, 3, 7, 43, 1807,…. If the positivity condition on the integers qi is dropped, then for r = 1 these systems of congruences, taken ( mod |qi|), have infinitely many solutions, while for r ≥ 2 they have finitely many solutions. The problem is reduced to studying integer solutions of the family of Diophantine equations [Formula: see text] depending on three parameters (r, s, m).

scholarly journals Juggler's Exclusion Process

2012 ◽  
Vol 49 (01) ◽  
pp. 266-279
Author(s):  
Lasse Leskelä ◽  
Harri Varpanen
Keyword(s):  
Markov Process ◽  
Gibbs Measure ◽  
Equilibrium Distribution ◽  
Exclusion Process ◽  
Jump Height ◽  
The Family ◽  
System Of Particles ◽  
Unit Speed ◽  
Positive Integers ◽  
Special Case

Juggler's exclusion process describes a system of particles on the positive integers where particles drift down to zero at unit speed. After a particle hits zero, it jumps into a randomly chosen unoccupied site. We model the system as a set-valued Markov process and show that the process is ergodic if the family of jump height distributions is uniformly integrable. In a special case where the particles jump according to a set-avoiding memoryless distribution, the process reaches its equilibrium in finite nonrandom time, and the equilibrium distribution can be represented as a Gibbs measure conforming to a linear gravitational potential.

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