Related papers: A Note on Minimal zero-sum sequences over ${\mathb…

Using Schmidt's Subspace Theorem, this paper improves and extends an existing transcendence result for sequences of algebraic numbers. The theorems thus produced correspond to a central theorem on the irrationality of sequences due to…

We show that if $\lambda_1,\ldots,\lambda_k$ are algebraic numbers, then $$|A+\lambda_1\cdot A+\dots+\lambda_k\cdot A|\geq H(\lambda_1,\ldots,\lambda_k)|A|-o(|A|)$$ for all finite subsets $A$ of $\mathbb{C}$, where…

Let $D$ be a dictionary in a Hilbert space $H$, that is, a set of unit elements whose linear combinations are dense in $H$. We consider the least $m$-term deviation $\sigma_m(x)$ of an element $x\in H$: this is the distance of $x$ from the…

It is proven that, contrarily to the common belief, the notion of zero is not necessary for having positional representations of numbers. Namely, for any positive integer $k$, a positional representation with the symbols for $1, 2, \ldots,…

A sequence of nonnegative integers \pi =(d_1,d_2,...,d_n) is graphic if there is a (simple) graph G with degree sequence \pi. In this case, G is said to realize or be a realization of \pi. Degree sequence results in the literature generally…

We present a locale that abstracts over the necessary ingredients for constructing a minimal bad sequence, as required in classical proofs of Higman's lemma and Kruskal's tree theorem.

Consider the space of sequences of k letters ordered lexicographically. We study the set M({\alpha}) of all maximal sequences for which the asymptotic proportions {\alpha} of the letters are prescribed, where a sequence is said to be…

The document tries to put focus on sequences with certain properties and periods leading to the first value smaller than the starting value in the Collatz problem. With the idea that, if all starting numbers lead ultimately to a smaller…

In the field of pattern mining, a negative sequential pattern is specified by means of a sequence consisting of events to occur and of other events, called negative events, to be absent. For instance, containment of the pattern $\langle a\…

Let $G$ be a finite abelian group with exponent $n$. Let $\eta(G)$ denote the smallest integer $\ell$ such that every sequence over $G$ of length at least $\ell$ has a zero-sum subsequence of length at most $n$. We determine the precise…

A numeral system is defined by three closed $\lambda$-terms : a normal $\lambda$-term $d_0$ for Zero, a $\lambda$-term $S_d$ for Successor, and a $\lambda$-term for Zero Test, such that the $\lambda$-terms $({S_d}^{i} ~ d_0)$ are…

We prove that, if $G$ is a finite almost simple group and $H$ is a maximal subgroup of $G$, then the $10$th term of the derived series of $H$ is perfect. The same is true if $G$ is perfect and $H$ is core-free. The constant $10$ is best…

Let $\chi$ be a real and non-principal Dirichlet character, $L(s,\chi)$ its Dirichlet $L$-function and let $p$ be a generic prime number. We prove the following result: If for some $0\leq \sigma<1$ the partial sums $\sum_{p\leq…

A sequence of positive integers is complete if every positive integer is a sum of distinct terms. A positive linear recurrence sequence (PLRS) is a sequence defined by a homogeneous linear recurrence relation with nonnegative coefficients…

In this paper, we study minimal (with respect to inclusion) zero forcing sets. We first investigate when a graph can have polynomially or exponentially many distinct minimal zero forcing sets. We also study the maximum size of a minimal…

A permutation $\sigma\in S_n$ is said to be $k$-universal or a $k$-superpattern if for every $\pi\in S_k$, there is a subsequence of $\sigma$ that is order-isomorphic to $\pi$. A simple counting argument shows that $\sigma$ can be a…

We prove that the sequence $(N_k)_k$, where each $N_k$ is defined as the smallest positive integer $n$ for which the $n$th term $g_{k,n}$ of the $k$-G\"obel sequence is not an integer, is unbounded.

A numerical semigroup is an additive subsemigroup of the natural numbers that contains zero and has finite complement. A numerical semigroup is irreducible if it cannot be written as an intersection of numerical semigroups properly…

Let $h\geq 2$ be a positive integer. For any subset $\mathcal{A}\subset \mathbb{Z}_n$, let $h^{\wedge}\mathcal{A}$ be the set of the elements of $\mathbb{Z}_n$ which are sums of $h$ distinct elements of $\mathcal{A}$. In this paper, we…

Let $A, B\subseteq \mathbb{Z}$ be finite, nonempty subsets with $\min A=\min B=0$, and let $$\delta(A,B)={\begin{array}{ll} 1 & \hbox{if} A\subseteq B, 0 & \hbox{otherwise.} If $\max B\leq \max A\leq |A|+|B|-3$ and \label{one}|A+B|\leq…