Topological sequence entropy for maps of the interval

A result by Franzová and Smítal shows that a continuous map of the interval into itself is chaotic if and only if its topological sequence entropy relative to a suitable increasing sequence of nonnegative integers is positive. In the present paper we prove that for any increasing sequence of nonnegative integers there exists a chaotic continuous map with zero topological sequence entropy relative to this sequence.

     

Hilbert irreducibility sequences and nonstandard arithmetic

We give sufficient conditions for a sequence of integers to be a Hilbert irreducibility sequence and give such a sequence explicitly.     

A test for conjugacy of partitions of finite sets

In this note we present a simple test, involving a sequence of integers, which assures the conjugacy of a given partition P of a finite set S when our operations lead only to nonnegative integers. If negative integers appear in our operations, our test is inconclusive. The test, when conclusive, and an elementary property of permutations determine a conjugate for P.     

Exhaustive weakly wandering sequences

A sequence of integers {n_i : i = 0, 1…} is an exhaustive weakly wandering sequence for a transformation T if for some measurable set W, X=^∞_i=0TⁿⁱW(disj. We introduce a hereditary Property (H) for a sequence of integers associated with an infinite ergodic transformation T, and show that it is a sufficient condition for the sequence to be an exhaustive weakly wandering sequence for T. We then show that every infinite ergodic transformation admits sequences that possess Property (H), and observe that Property (H) is inherited by all subsequences of a sequence that possess it. As a corollary, we obtain an application to tiling the set of integers with infinite subsets.     

图中的度序列

给出了一个非减的非负整数序列是某个图的度序列的一个新刻划.     

Completeness properties of perturbed sequences

If S is an arbitrary sequence of positive integers, let P(S) be the set of all integers which are representable as a sum of distinct terms of S. Call S complete if P(S) contains all large integers, and subcomplete if P(S) contains an infinite arithmetic progression. It is shown that any sequence can be perturbed in a rather moderate way into a sequence which is not subcomplete. On the other hand, it is shown that if S is any sequence satisfying a mild growth condition, then a surprisingly gentle perturbation suffices to make S complete in a strong sense. Various related questions are also considered.     

Degree sequence of oriented k-hypergraphs

In this paper, we define the degree of a vertex in oriented hypergraphs and give a necessary and sufficient condition for a nondecreasing sequence of integers to be a degree sequence of an oriented 3-hypergraphs.     

Solution to a problem of Nicolas Lichiardopol

We present a solution to a problem posed by Nicolas Lichiardopol, regarding the on-line sorting of a sequence of integers.     

A New Lower Bound on the Potential-Ramsey Number of Two Graphs

A nonincreasing sequence π=(d₁,…,d_n) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices.In this case,G is referred to as a realization of π.Given a graph H,a graphic sequence π is potentially H-graphic if π has a realization containing H as a subgraph.For graphs G₁ and G₂,the potential-Ramsey number r_pot(G₁,G₂) is the smallest integer k such that for every k-term graphic ...     

On the statistical variation of sequences

We introduce a generalization of the notion of sequence of finite variation, using asymptotic density of sets of positive integers. Some approximation results about approaching any sequence by sequences of finite statistical variation are given.     

The potential-Ramsey number of Kn and K t −k

Czechoslovak Mathematical Journal - A nonincreasing sequence π = (d1,…, dn) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. In this...     

The degree sequences of self-complementary graphs

Known necessary conditions for realization of a sequence of integers as the degrees of a self-complementary graph are shown to be sufficient. An algorithm for constructing a realization of such a sequence as degrees of such a graph is illustrated by an example.     

A note on $$AP_3$$ A P 3 -covering sequences

Periodica Mathematica Hungarica - For a given integer $$k\ge 3$$ , a sequence A of nonnegative integers is called an  $$AP_k$$ -covering sequence if there exists an integer $$n_0$$ such...     

Concordant sequences and concordant entire functions

A theorem of Pólya shows that the function 2 ^Z is the ‘smallest’ transcendental entire function that is integer valued on the set ℕ of non-negative integers. Analogous results have been established in which ℕ is replaced by other sets of integers, beginning with the result of Gel’fond for geometric sequences of integers. Other results consider the imposition of additional congruence conditions on the value sequence of the candidate entire function on the subject sequence. The present paper extends the consideration of such congruence conditions from ℕ and geometric sequences to more general sets for which Pólya-type results have been established.     

On the support base of a functional equation arising from multiplication of quantum integers

Given a set of primes P, we determine the necessary and sufficient criterions for the existence of a sequence of polynomials Γ, with support base P, which is a solution of the functional equations arising from multiplication of quantum integers discussed in Melvyn B. Nathanson (2003) [1] and which cannot be generated by quantum integers.     

Topologies on the n-Element Set Consistent with Topologies Close to Discrete on an (n−1)-Element Set

Ukrainian Mathematical Journal - Topologies on a finite set are described by a nondecreasing sequence of nonnegative integers (vector of topologies). We study T0-topologies on the n-element set...     

The C-finite ansatz

A C-finite sequence is a sequence satisfying a linear recurrence equation with constant coefficients. While it is trivial to multiply two C-finite sequences (just like integers), it is not quite so trivial to “factorize” them, or to decide whether they are “prime”. Here we address these problems.     

Fifteen consecutive integers with exactly four prime factors

We describe a successful search for a sequence of fifteen consecutive integers, each the product of exactly four prime factors. Fifteen is best possible.

     

Algebraic Independence of Reciprocal Sums of Binary Recurrences II

 Algebraic independence of the numbers for various d and l, where is a periodic sequence of algebraic numbers and is a sequence of integers satisfying a binary linear recurrence relation, is studied by Mahler’s method.     

On Combinatorial Cubes

A sequence of positive integers with positive lower density contains a Hilbert (or combinatorial) cube size c log log n up to n. We prove this bound is sharp for some thinner sequence.