The parallel X-ray of a convex set K ⊂ R in a direction u is the function which associates to each line l parallel to u the length of K ∩ l. The problem of nding a set of directions such that the corresponding X-rays distinguish any two convex bodies has been widely studied in geometric tomography. In this paper we are interested in the restriction of this problem to convex cones, and we are mo...

A language L over an alphabet Σ is prefix-convex if, for any words x, y, z ∈ Σ, whenever x and xyz are in L, then so is xy. Prefix-convex languages include right-ideal, prefixclosed, and prefix-free languages. We study complexity properties of prefix-convex regular languages. In particular, we find the quotient/state complexity of boolean operations, product (concatenation), star, and reversal,...

We study succinct representations of a convex univariate function φ over a finite domain. We show how to construct a succinct representation, namely a piecewise-linear function φ̄ approximating φ when given a black box access to an L-approximation oracle φ̃ of φ (the oracle value is always within a multiplicative factor L from the true value). The piecewise linear function φ̄ has few breakpoints (...

Let C (fl), fl > 0, denote the family of no rmal i zed c loseto-convex funct ions of o rder ft. F o r fl = 1 this is the usual set of c loseto-convex functions, which had been defined by Kap lan . In a prev ious pape r [3] we solved the Fekete-Szeg6 p r o b l e m of maximiz ing l a 3 2 a2l, 2 ~ [0, 1], for c loseto-convex functions. The largest n u m b e r 20 for which [a a 20 a2[ is max imized...

Let K,L ⊂ IRn be two convex bodies with non-empty interiors and with boundaries ∂K, ∂L, and let χ denote the Euler characteristic as defined in singular homology theory. We prove two translative integral formulas involving boundaries of convex bodies. It is shown that the integrals of the functions t 7→ χ(∂K ∩ (∂L + t)) and t 7→ χ(∂K ∩ (L + t)), t ∈ IRn, with respect to an ndimensional Haar mea...

Abstract Let $$\mathcal {B}$$ B be a homogeneous differential operator of order $$l=1$$ l = 1 or $$l=2$$ 2 . We show that sequence functions the form $$(\mathcal {B}u_j)_j$$ ( u</mml:...

in this paper, downlink resource allocation for a sparse code multiple access (scma) based heterogeneous  cellular networks (hcn) is investigated. the main aim of this paper is to obtain joint power allocation and codebook assignment to achieve the least total system transmit power. in this regard, the proposed objective function is formulated as total transmit power subject to users minimum ra...

in this paper, an optimization problem with a linear objective function subject to a consistent finite system of fuzzy relation inequalities using the max-product composition is studied. since its feasible domain is non-convex, traditional linear programming methods cannot be applied to solve it. we study this problem and capture some special characteristics of its feasible domain and optimal s...

We present in detail Thomas Royen’s proof of the Gaussian correlation inequality which states that μ(K ∩ L) ≥ μ(K)μ(L) for any centered Gaussian measure μ on R and symmetric convex sets K,L in R.

The goal of this paper is to study the effect of inexact first-order information on the first-order methods designed for smooth strongly convex optimization problems. It can be seen as a generalization to the strongly convex case of our previous paper [1]. We introduce the notion of (!,L,μ)-oracle, that can be seen as an extension of the (!,L)-oracle (previously introduced in [1]), taking into ...