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We study representations of integers as sums of the form $\pm a_1\pm a_2\pm \dotsb \pm a_n$, where $a_1,a_2,\ldots$ is a prescribed sequence of integers. Such a sequence is called an Erd\H{o}s-Sur\'anyi sequence if every integer can be…

We show that almost split sequences in the category of comodules over a coalgebra with finite-dimensional right-hand term are direct limits of almost split sequences over finite dimensional subcoalgebras. In previous work we showed that…

This paper is a continuation of the paper "Numerical Semigroups: Ap\'ery Sets and Hilbert Series". We consider the general numerical AA-semigroup, i.e., semigroups consisting of all non-negative integer linear combinations of relatively…

We study the approximability of general convex sets in $\mathbb{R}^n$ by intersections of halfspaces, where the approximation quality is measured with respect to the standard Gaussian distribution $N(0,I_n)$ and the complexity of an…

We introduce statistically $p$-upward quasi-Cauchy sequences, defined by the condition $\lim_{n\to\infty}\frac{1}{n}|\{k\leq n: x_k - x_{k+p}\geq\varepsilon\}|=0$ for every $\varepsilon>0$, and develop the corresponding notions of…

We consider the higher order Tur\'an inequality and higher order log-concavity for sequences $\{a_n\}_{n \ge 0}$ such that \[ \frac{a_{n-1}a_{n+1}}{a_n^2} = 1 + \sum_{i=1}^m \frac{r_i(\log n)}{n^{\alpha_i}} + o\left( \frac{1}{n^{\beta}}…

The paper deals with a problem of Additive Combinatorics. Let ${\mathbf G}$ be a finite abelian group of order $N$. We prove that the number of subset triples $A,B,C\subset {\mathbf G}$ such that for any $x\in A$, $y\in B$ and $z\in C$ one…

Nuclear $C^*$-algebras having a system of completely positive approximations formed with convex combinations of a uniformly bounded number of order zero summands are shown to be approximately finite dimensional.

In this article we prove that an "isometric multiple HNN-extension" of a group satisfying the falsification by fellow traveler property is almost convex. As a corollary, Wise's example of a CAT(0) non-Hopfian group is Almost convex.

We show that every planar triangulation on $n>10$ vertices has a dominating set of size $n/7=n/3.5$. This approaches the $n/4$ bound conjectured by Matheson and Tarjan [MT'96], and improves significantly on the previous best bound of…

Let $(r_{n})_{n \in \mathbb{N}}$ be the sequence of subdiagonal Pad\'{e} approximations of the exponential function. We prove that for $-A$ the generator of a uniformly bounded $C_{0}$-semigroup $T$ on a Banach space $X$, the sequence…

We describe an action of the concordance group of knots in the three-sphere on concordances of knots in arbitrary 3-manifolds. As an application we define the notion of almost-concordance between knots. After some basic results, we prove…

There is a hierarchy of structure conditions for convex sets. In this paper we study a recently defined [3, 8, 9] condition called locally nonconical convexity (abbreviated LNC). Is is easy to show that every strictly convex set is LNC, as…

A quasi-kernel of a digraph $D$ is an independent set $Q$ such that every vertex can reach $Q$ in at most two steps. A 48-year conjecture made by P.L. Erd\H{o}s and Sz\'ekely, denoted the small QK conjecture, says that every sink-free…

We study the complexity of S-adic sequences corresponding to a family of 216 multi-dimensional continued fractions maps, called Triangle Partition maps (TRIP maps), with an emphasis on those with low upper bounds on complexity. Our main…

The $3x+k$ function $T_{k}(n)$ sends $n$ to $(3n+k)/2$ resp. $n/2,$ according as $n$ is odd, resp. even, where $k \equiv \pm 1~(\bmod \, 6)$. The map $T_k(\cdot)$ sends integers to integers, and for $m \ge 1$ let $n \rightarrow m$ mean that…

A weakly distance-regular digraph is quasi-thin if the maximum value of its intersection numbers is 2. In this paper, we focus on commutative quasi-thin weakly distance-regular digraphs, and classify such digraphs with valency more than 3.…

Let G be a Lie group, $g = Lie(G)$ - its Lie algebra, $g*$ - the dual vector space and $\widehat G$ - the set of equivalence classes of unitary irreducible representations of $G$. The orbit method [1] establishes a correspondence between…

For a finite set $A\subset \mathbb{R}^d$, let $\Delta(A)$ denote the spread of $A$, which is the ratio of the maximum pairwise distance to the minimum pairwise distance. For a positive integer $n$, let $\gamma_d(n)$ denote the largest…

F. Wehrung has asked: Given a family $\mathcal{C}$ of subsets of a set $\Omega$, under what conditions will there exist a total ordering on $\Omega$ under which every member of $\mathcal{C}$ is convex? <p> Note that if $A$ and $B$ are…