Related papers: Extremal sequences for a weighted zero-sum constan…

A sequence $\Big(u_n\Big)_{n=0}^{\infty}$ is said to be convex if it satisfies the following inequality $$ 2u_n\leq u_{n-1}+u_{n+1}\qquad \mbox{for all}\qquad n\in\mathbb{N}. $$ We present several characterizations of convex sequences and…

Let $A$ be a nonempty finite set of $k$ integers. Given a subset $B$ of $A$, the sum of all elements of $B$, denoted by $s(B)$, is called the subset sum of $B$. For a nonnegative integer $\alpha$ ($\leq k$), let \[\Sigma_{\alpha}…

A sequence in an additively written abelian group is called zero-free if each of its nonempty subsequences has sum different from the zero element of the group. The article determines the structure of the zero-free sequences with lengths…

This paper proves the minimum size of a supersequence over a set of eight elements is 52. This disproves a conjecture that the lower bound of the supersequence is the partial sum of the geometric Connell sequence. By studying the internal…

The paper is devoted to characterizing convex trace ranges in finite atomic von Neumann algebras. The main result provides us with the necessary and sufficient condition for the range of a faithful normal trace on a finite atomic von…

Let $G$ be a finite additive abelian group with exponent $n$ and $S=g_{1}\cdots g_{t}$ be a sequence of elements in $G$. For any element $g$ of $G$ and $A\subseteq\{1,2,\ldots,n-1\}$, let $N_{A,g}(S)$ denote the number of subsequences…

We show that a zero-sum-free sequence of length $n$ over an abelian group spans at least $2n$ distinct subsequence sums, unless it possesses a rigid, easily-described structure.

A subset $A$ of an abelian group $G$ is sequenceable if there is an ordering $(a_1, \ldots, a_k)$ of its elements such that the partial sums $(s_0, s_1, \ldots, s_k)$, given by $s_0 = 0$ and $s_i = \sum_{j=1}^i a_i$ for $1 \leq i \leq k$,…

A causal set is a partially ordered set on a countably infinite ground-set such that each element is above finitely many others. A natural extension of a causal set is an enumeration of its elements which respects the order. We bring…

It is well-known that any sequence of at least N integers contains a subsequence whose sum is 0 (mod N). However, there can be very few subsequences with this property (e.g. if the initial sequence is just N 1's, then there is only one…

Let $X,X_1,X_2,\ldots$ be i.i.d. mean zero random vectors with values in a separable Banach space $B$, $S_n=X_1+\cdots+X_n$ for $n\ge1$, and assume $\{c_n:n\ge1\}$ is a suitably regular sequence of constants. Furthermore, let…

In extremal set theory our usual goal is to find the maximal size of a family of subsets of an $n$-element set satisfying a condition. A condition is called chain-dependent, if it is satisfied for a family if and only if it is satisfied for…

The Ulam sequence is defined as $a_1 =1, a_2 = 2$ and $a_n$ being the smallest integer that can be written as the sum of two distinct earlier elements in a unique way. This gives $$1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47,…

A decomposition of a natural number n is a sequence of consecutive natural numbers that sums to n. We construct a one-to-one correspondence between the odd factors of a natural number and its decompositions. We study the decompositions by…

We prove that if $(v_i)$ is a normalized basic sequence and X is a Banach space such that every normalized weakly null sequence in X has a subsequence that is dominated by $(v_i)$, then there exists a uniform constant $C\geq1$ such that…

For a set of positive integers $D$, a $k$-term $D$-diffsequence is a sequence of positive integers $a_1<a_2<\cdots<a_k$ such that $a_i-a_{i-1}\in D$ for $i=2,3,\cdots,k$. For $k\in\mathbb{Z}^+$ and $D\subset \mathbb{Z}^+$, we define…

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\…

The paper is devoted to the study of extremal points of $\mathcal{C}$, the family of all two-variate coherent distributions on $[0,1]^2$. It is well-known that the set $\mathcal{C}$ is convex and weak$^*$ compact, and all extreme points of…

Fix $A$, a family of subsets of natural numbers, and let $G_A(n)$ be the maximum cardinality of a subset of $\{1,2,..., n\}$ that does not have any subset in $A$. We consider the general problem of giving upper bounds on $G_A(n)$ and give…

The subgraph number of a vertex in a graph is defined as the number of connected subgraphs containing that vertex. The graph and its vertex which correspond to the minimum subgraph number among all graphs on $n$ vertices and $k$ cut…