Related papers: On approximately Convex and Affine Sequences

We consider mesh functions which are discrete convex in the sense that their central second order directional derivatives are positive. Analogous to the case of a uniformly bounded sequence of convex functions, we prove that the uniform…

For a sequence $S$ of terms from an abelian group $G$ of length $|S|$, let $\Sigma_n(S)$ denote the set of all elements that can be represented as the sum of terms in some $n$-term subsequence of $S$. When the subsum set is very small,…

We develop a theory of limits for sequences of dense abstract simplicial complexes, where a sequence is considered convergent if its homomorphism densities converge. The limiting objects are represented by stacks of measurable [0,1]-valued…

We introduce the concept of strong high-order approximate minimizers for nonconvex optimization problems. These apply in both standard smooth and composite non-smooth settings, and additionally allow convex or inexpensive constraints. An…

We show weak lower semi-continuity of functionals assuming the new notion of a "convexly constrained" $\mathcal A$-quasiconvex integrand. We assume $\mathcal A$-quasiconvexity only for functions defined on a set $K$ which is convex.…

Given a function $f$ defined on a nonempty and convex subset of the $d$-dimensional Euclidean space, we prove that if $f$ is bounded from below and it satisfies a convexity-type functional inequality with infinite convex combinations, then…

For a sequence $\{a_n\}_{n\geq 0}$ of real numbers and for a parameter $0<p<1$, we define the sequence of its arithmetic means $\{a^*_n\}_{n\geq 0}$ and the sequence of its $p$-binomial means $\{a^p_n\}_{n\geq 0}$ as \begin{align*}…

Let $\mathcal{I} \subset \mathbb{N}$ be an infinite subset, and let $\{a_i\}_{i \in \mathcal{I}}$ be a sequence of nonzero real numbers indexed by $\mathcal{I}$ such that there exist positive constants $m, C_1$ for which $|a_i| \leq C_1…

Similarly to $\beta$-adic van der Corput sequences, abstract van der Corput sequences can be defined for abstract numeration systems. Under some assumptions, these sequences are low discrepancy sequences. The discrepancy function is…

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…

Let $\beta\in(1,2)$ and $x\in [0,\frac{1}{\beta-1}]$. We call a sequence $(\epsilon_{i})_{i=1}^\infty\in\{0,1\}^{\mathbb{N}}$ a $\beta$-expansion for $x$ if $x=\sum_{i=1}^{\infty}\epsilon_{i}\beta^{-i}$. We call a finite sequence…

For a fixed $c > 0$ we construct an arbitrarily large set $B$ of size $n$ such that its sum set $B+B$ contains a convex sequence of size $cn^2$, answering a question of Hegarty.

Slowly convergent series and sequences as well as divergent series occur quite frequently in the mathematical treatment of scientific problems. In this report, a large number of mainly nonlinear sequence transformations for the acceleration…

This article is concerned with the approximation of unbounded convex sets by polyhedra. While there is an abundance of literature investigating this task for compact sets, results on the unbounded case are scarce. We first point out the…

Given $\beta\in(1,2)$ and $x\in[0,\frac{1}{\beta-1}]$, a sequence $(\epsilon_{i})_{i=1}^{\infty}\in\{0,1\}^{\mathbb{N}}$ is called a $\beta$-expansion for $x$ if $$x=\sum_{i=1}^{\infty}\frac{\epsilon_{i}}{\beta^{i}}.$$ In a recent article…

In this work we introduce the concept of an Underestimate Sequence (UES), which is motivated by Nesterov's estimate sequence. Our definition of a UES utilizes three sequences, one of which is a lower bound (or under-estimator) of the…

Sequence transformations accomplish an acceleration of convergence or a summation in the case of divergence by detecting and utilizing regularities of the elements of the sequence to be transformed. For sufficiently large indices, certain…

In this paper we build provably near-optimal, in the minimax sense, estimates of linear forms and, more generally, "$N$-convex functionals" (the simplest example being the maximum of several fractional-linear functions) of unknown "signal"…

Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and optimizing functionals arising from some…

We consider minimization of functions that are compositions of convex or prox-regular functions (possibly extended-valued) with smooth vector functions. A wide variety of important optimization problems fall into this framework. We describe…