scholarly journals An update on the sum-product problem

2021 ◽  
pp. 1-20
Author(s):  
MISHA RUDNEV ◽  
SOPHIE STEVENS
Keyword(s):  
Convex Set ◽  
Product Problem

Abstract We improve the best known sum-product estimates over the reals. We prove that \[\max(|A+A|,|A+A|)\geq |A|^{\frac{4}{3} + \frac{2}{1167} - o(1)}\,,\] for a finite $A\subset \mathbb {R}$ , following a streamlining of the arguments of Solymosi, Konyagin and Shkredov. We include several new observations to our techniques. Furthermore, \[|AA+AA|\geq |A|^{\frac{127}{80} - o(1)}\,.\] Besides, for a convex set A we show that \[|A+A|\geq |A|^{\frac{30}{19}-o(1)}\,.\] This paper is largely self-contained.

Enhancement of Video Image Resolution Based on POCS (projection onto convex set)

2012 ◽  
Vol 15 (1) ◽  
pp. 170-174
Author(s):  
Maisaa Husam Al-anizy ◽  
Keyword(s):  
Convex Set ◽  
Image Resolution ◽  
Video Image

Sharpening an Estimate of the Size of the Sumset of a Convex Set

2020 ◽  
Vol 107 (5-6) ◽  
pp. 984-987
Author(s):  
K. I. Ol’mezov
Keyword(s):  
Convex Set

scholarly journals Horosphere slab separation theorems in manifolds without conjugate points

2019 ◽  
Vol 27 (1) ◽  
Author(s):  
Sameh Shenawy
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Existence And Uniqueness ◽  
Convex Set ◽  
Sufficient Conditions ◽  
Conjugate Points ◽  
Separation Theorems ◽  
Farthest Points ◽  
Simply Connected ◽  
Convex Subsets

Abstract Let $\mathcal {W}^{n}$ W n be the set of smooth complete simply connected n-dimensional manifolds without conjugate points. The Euclidean space and the hyperbolic space are examples of these manifolds. Let $W\in \mathcal {W}^{n}$ W ∈ W n and let A and B be two convex subsets of W. This note aims to investigate separation and slab horosphere separation of A and B. For example,sufficient conditions on A and B to be separated by a slab of horospheres are obtained. Existence and uniqueness of foot points and farthest points of a convex set A in $W\in \mathcal {W}$ W ∈ W are considered.

On random divisions of a convex set

1978 ◽  
Vol 15 (3) ◽  
pp. 645-649 ◽  
Author(s):  
Svante Janson
Keyword(s):  
Elementary Proof ◽  
Convex Set ◽  
Limiting Distribution ◽  
Independent Identically Distributed ◽  
Normally Distributed

This paper gives an elementary proof that, under some general assumptions, the number of parts a convex set in Rd is divided into by a set of independent identically distributed hyperplanes is asymptotically normally distributed. An example is given where the distribution of hyperplanes is ‘too singular' to satisfy the assumptions, and where a different limiting distribution appears.

scholarly journals On the existence of a solution for some distributed optimal control hyperbolic system

2000 ◽  
Vol 23 (5) ◽  
pp. 297-311 ◽  
Author(s):  
Dariusz Idczak ◽  
Stanislaw Walczak
Keyword(s):  
Optimal Control ◽  
Hyperbolic System ◽  
Convex Set ◽  
Linear Time ◽  
Time Varying ◽  
Optimal Process ◽  
Existence Of A Solution ◽  
Distributed Optimal Control ◽  
Absolutely Continuous ◽  
State Variable

We consider a Bolza problem governed by a linear time-varying Darboux-Goursat system and a nonlinear cost functional, without the assumption of the convexity of an integrand with respect to the state variable. We prove a theorem on the existence of an optimal process in the classes of absolutely continuous trajectories of two variables and measurable controls with values in a fixed compact and convex set.

scholarly journals Diameter-preserving linear bijections of function spaces

2001 ◽  
Vol 70 (3) ◽  
pp. 323-336 ◽  
Author(s):  
T. S. S. R. K. Rao ◽  
A. K. Roy
Keyword(s):  
Maximal Ideal ◽  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Continuous Functions ◽  
Ideal Space ◽  
Function Algebras ◽  
Compact Convex Set ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractIn this paper we give a complete description of diameter-preserving linear bijections on the space of affine continuous functions on a compact convex set whose extreme points are split faces. We also give a description of such maps on function algebras considered on their maximal ideal space. We formulate and prove similar results for spaces of vector-valued functions.

scholarly journals Towards the fixed point property for superreflexive spaces

2001 ◽  
Vol 64 (3) ◽  
pp. 435-444 ◽  
Author(s):  
Andrzej Wiśnicki
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Unit Sphere ◽  
Convex Set ◽  
Nonexpansive Mappings ◽  
Fixed Point Property ◽  
Point Property ◽  
Geometrical Condition

A Banach space X is said to have property (Sm) if every metrically convex set A ⊂ X which lies on the unit sphere and has diameter not greater than one can be (weakly) separated from zero by a functional. We show that this geometrical condition is closely connected with the fixed point property for nonexpansive mappings in superreflexive spaces.

Convex set theoretic adaptor control systems

2002 ◽  
Author(s):  
Tung-Ching Tsao ◽  
M.G. Safonov
Keyword(s):  
Control Systems ◽  
Convex Set

On topological properties of weakly m-convex sets

2021 ◽  
Vol 34 ◽  
pp. 75-84
Author(s):  
Tetiana Osipchuk
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Topological Classification ◽  
Connected Components ◽  
Topological Properties ◽  
Convex Domains ◽  
Closed Set ◽  
Dimensional Plane ◽  
Open Set ◽  
Plane Passing

The topological properties of classes of generally convex sets in multidimensional real Euclidean space $\mathbb{R}^n$, $n\ge 2$, known as $m$-convex and weakly $m$-convex, $1\le m<n$, are studied in the present work. A set of the space $\mathbb{R}^n$ is called \textbf{\emph{$m$-convex}} if for any point of the complement of the set to the whole space there is an $m$-dimensional plane passing through this point and not intersecting the set. An open set of the space is called \textbf{\emph{weakly $m$-convex}}, if for any point of the boundary of the set there exists an $m$-dimensional plane passing through this point and not intersecting the given set. A closed set of the space is called \textbf{\emph{weakly $m$-convex}} if it is approximated from the outside by a family of open weakly $m$-convex sets. These notions were proposed by Professor Yuri Zelinskii. It is known the topological classification of (weakly) $(n-1)$-convex sets in the space $\mathbb{R}^n$ with smooth boundary. Each such a set is convex, or consists of no more than two unbounded connected components, or is given by the Cartesian product $E^1\times \mathbb{R}^{n-1}$, where $E^1$ is a subset of $\mathbb{R}$. Any open $m$-convex set is obviously weakly $m$-convex. The opposite statement is wrong in general. It is established that there exist open sets in $\mathbb{R}^n$ that are weakly $(n-1)$-convex but not $(n-1)$-convex, and that such sets consist of not less than three connected components. The main results of the work are two theorems. The first of them establishes the fact that for compact weakly $(n-1)$-convex and not $(n-1)$-convex sets in the space $\mathbb{R}^n$, the same lower bound for the number of their connected components is true as in the case of open sets. In particular, the examples of open and closed weakly $(n-1)$-convex and not $(n-1)$-convex sets with three and more connected components are constructed for this purpose. And it is also proved that any compact weakly $m$-convex and not $m$-convex set of the space $\mathbb{R}^n$, $n\ge 2$, $1\le m<n$, can be approximated from the outside by a family of open weakly $m$-convex and not $m$-convex sets with the same number of connected components as the closed set has. The second theorem establishes the existence of weakly $m$-convex and not $m$-convex domains, $1\le m<n-1$, $n\ge 3$, in the spaces $\mathbb{R}^n$. First, examples of weakly $1$-convex and not $1$-convex domains $E^p\subset\mathbb{R}^p$ for any $p\ge3$, are constructed. Then, it is proved that the domain $E^p\times\mathbb{R}^{m-1}\subset\mathbb{R}^n$, $n\ge 3$, $1\le m<n-1$, is weakly $m$-convex and not $m$-convex.

scholarly journals Characterizations and properties of graphs of Baire functions

2018 ◽  
Vol 68 (4) ◽  
pp. 789-802
Author(s):  
Balázs Maga
Keyword(s):  
Banach Space ◽  
Topological Space ◽  
Convex Set ◽  
Vertical Section ◽  
Fréchet Space ◽  
Frechet Space ◽  
Similar Question ◽  
Baire Functions

Abstract Let X be a paracompact topological space and Y be a Banach space. In this paper, we will characterize the Baire-1 functions f : X → Y by their graph: namely, we will show that f is a Baire-1 function if and only if its graph gr(f) is the intersection of a sequence $\begin{array}{} \displaystyle (G_n)_{n=1}^{\infty} \end{array}$ of open sets in X × Y such that for all x ∈ X and n ∈ ℕ the vertical section of Gn is a convex set, whose diameter tends to 0 as n → ∞. Afterwards, we will discuss a similar question concerning functions of higher Baire classes and formulate some generalized results in slightly different settings: for example we require the domain to be a metrized Suslin space, while the codomain is a separable Fréchet space. Finally, we will characterize the accumulation set of graphs of Baire-2 functions between certain spaces.

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