scholarly journals Anisotropic tubular neighborhoods of sets

2021 ◽  
Author(s):  
Antonin Chambolle ◽  
Luca Lussardi ◽  
Elena Villa
Keyword(s):  
Convex Body ◽  
Finite Union ◽  
Compact Set ◽  
Second Derivative ◽  
Volume Function ◽  
Tubular Neighborhoods ◽  
The Right ◽  
Topological Boundary

AbstractLet $$E \subset {{\mathbb {R}}}^N$$ E ⊂ R N be a compact set and $$C\subset {{\mathbb {R}}}^N$$ C ⊂ R N be a convex body with $$0\in \mathrm{int}\,C$$ 0 ∈ int C . We prove that the topological boundary of the anisotropic enlargement $$E+rC$$ E + r C is contained in a finite union of Lipschitz surfaces. We also investigate the regularity of the volume function $$V_E(r):=|E+rC|$$ V E ( r ) : = | E + r C | proving a formula for the right and the left derivatives at any $$r>0$$ r > 0 which implies that $$V_E$$ V E is of class $$C^1$$ C 1 up to a countable set completely characterized. Moreover, some properties on the second derivative of $$V_E$$ V E are proved.

RESTRICTED UNIVERSALITY OF POWER SERIES

2001 ◽  
Vol 33 (5) ◽  
pp. 543-552 ◽  
Author(s):  
JEAN-PIERRE KAHANE ◽  
ANTONIOS D. MELAS
Keyword(s):  
Power Series ◽  
Finite Union ◽  
Radius Of Convergence ◽  
Nonempty Interior ◽  
Partial Sums ◽  
Universal Approximation ◽  
Compact Set ◽  
Approximation Properties

We prove the existence of a power series having radius of convergence 0, whose partial sums have universal approximation properties on any compact set with connected complement that is contained in a finite union of circles centred at 0 and having rational radii, but do not have such properties on any compact set with nonempty interior. This relates to a theorem of A. I. Seleznev.

scholarly journals Uniform bounds on sup-norms of holomorphic forms of real weight

2016 ◽  
Vol 12 (05) ◽  
pp. 1163-1185 ◽  
Author(s):  
Raphael S. Steiner
Keyword(s):  
Modular Forms ◽  
Orthonormal Basis ◽  
Fourier Coefficients ◽  
Compact Set ◽  
Index Subgroup ◽  
Uniform Bounds ◽  
Order Of Magnitude ◽  
The Right ◽  
Finite Index Subgroup ◽  
Right Order

We establish uniform bounds for the sup-norms of modular forms of arbitrary real weight [Formula: see text] with respect to a finite index subgroup [Formula: see text] of SL2(ℤ). We also prove corresponding bounds for the supremum over a compact set. We achieve this by extending to a sum over an orthonormal basis [Formula: see text] and analyzing this sum by means of a Bergman kernel and the Fourier coefficients of Poincaré series. Under some weak assumptions, we further prove the right order of magnitude of [Formula: see text]. Our results are valid without any assumption that the forms are Hecke eigenfunctions.

scholarly journals THE THEOREM OF AVERAGING FOR THE ALMOST-PERIODIC FUNCTIONS

2017 ◽  
Vol 18 (6) ◽  
pp. 100-112
Author(s):  
O.P. Filatov
Keyword(s):  
Differential Inclusion ◽  
Initial Conditions ◽  
Dimensional Space ◽  
Compact Set ◽  
Finite Dimensional Space ◽  
Almost Periodic ◽  
Finite Dimensional ◽  
All Solutions ◽  
Independent Variable ◽  
The Right

It is proved that the limit of maximal mean is an independent variable of initial conditions if a vector exists from the convex hull of a compact set out of a finite-dimensional space and the components of vector are independent variables with respect to the spectrum of almost-periodic function. The compact set is the right hand of differential inclusion. The limit of maximal mean is taken over all solutions of the Couchy problem for the differential inclusion.

On transfinite diameters in $\mathbb{C}^{d}$ for generalized notions of degree

2021 ◽  
Vol 127 (2) ◽  
pp. 337-360
Author(s):  
Norman Levenberg ◽  
Franck Wielonsky
Keyword(s):  
Convex Body ◽  
Unit Ball ◽  
General Formula ◽  
Plurisubharmonic Function ◽  
Integral Formula ◽  
Compact Set ◽  
Transfinite Diameter ◽  
Robin Function ◽  
Euclidean Unit Ball ◽  
Definition Of

We give a general formula for the $C$-transfinite diameter $\delta_C(K)$ of a compact set $K\subset \mathbb{C}^2$ which is a product of univariate compacta where $C\subset (\mathbb{R}^+)^2$ is a convex body. Along the way we prove a Rumely type formula relating $\delta_C(K)$ and the $C$-Robin function $\rho_{V_{C,K}}$ of the $C$-extremal plurisubharmonic function $V_{C,K}$ for $C \subset (\mathbb{R}^+)^2$ a triangle $T_{a,b}$ with vertices $(0,0)$, $(b,0)$, $(0,a)$. Finally, we show how the definition of $\delta_C(K)$ can be extended to include many nonconvex bodies $C\subset \mathbb{R}^d$ for $d$-circled sets $K\subset \mathbb{C}^d$, and we prove an integral formula for $\delta_C(K)$ which we use to compute a formula for $\delta_C(\mathbb{B})$ where $\mathbb{B}$ is the Euclidean unit ball in $\mathbb{C}^2$.

Solutions of Homogeneous Elliptic Equations

1970 ◽  
Vol 13 (1) ◽  
pp. 99-104 ◽  
Author(s):  
E. Dubinsky ◽  
T. Husain
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Elliptic Equations ◽  
Elliptic Partial Differential Equation ◽  
Compact Set ◽  
Polynomial Operator ◽  
Exponential Functions ◽  
Sum Of Products ◽  
Constant Coefficients ◽  
The Right

We consider an elliptic partial differential equation with constant coefficients and zero on the right hand side. It is well known [1] that every solution of such an equation can be approximated uniformly on each compact set by a sum of products of polynomials and exponential functions which satisfy the equation. Furthermore, if one assumes that the polynomial operator is homogeneous, then the approximation can be made with polynomials alone. It is our purpose to show, in the latter case, when the number of variables is two, that each solution can be written as an infinite series in certain specific polynomials. Our method is to factor the polynomial and build up the solution in terms of solutions of first degree equations.

On the Inequality for Volume and Minkowskian Thickness

2006 ◽  
Vol 49 (2) ◽  
pp. 185-195 ◽  
Author(s):  
Gennadiy Averkov
Keyword(s):  
Convex Body ◽  
Geometric Inequality ◽  
Symmetric Convex Body ◽  
Symmetric Convex ◽  
Centrally Symmetric ◽  
Arbitrary Convex ◽  
Simple Corollary ◽  
The Difference ◽  
The Right ◽  
Finite Dimensional Banach Space

AbstractGiven a centrally symmetric convex body B in , we denote by ℳd(B) the Minkowski space (i.e., finite dimensional Banach space) with unit ball B. Let K be an arbitrary convex body in ℳd(B). The relationship between volume V(K) and the Minkowskian thickness (= minimal width) ΔB(K) of K can naturally be given by the sharp geometric inequality V(K) ≥ α(B) · ΔB(K)d, where α(B) > 0. As a simple corollary of the Rogers-Shephard inequality we obtain that with equality on the left attained if and only if B is the difference body of a simplex and on the right if B is a cross-polytope. The main result of this paper is that for d = 2 the equality on the right implies that B is a parallelogram. The obtained results yield the sharp upper bound for the modified Banach–Mazur distance to the regular hexagon.

scholarly journals Refinements of Hermite-Hadamard Inequalities for Functions When a Power of the Absolute Value of the Second Derivative IsP-Convex

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
A. Barani ◽  
S. Barani ◽  
S. S. Dragomir
Keyword(s):  
Second Derivative ◽  
Absolute Value ◽  
Right Hand ◽  
Special Means ◽  
Second Derivatives ◽  
The Absolute ◽  
The Right

We extend some estimates of the right-hand side of Hermite-Hadamard-type inequalities for functions whose second derivatives absolute values areP-convex. Applications to some special means are considered.

Investor-State Dispute Settlement under NAFTA Chapter 11: The Shape of Things to Come?

1998 ◽  
Vol 35 ◽  
pp. 263-290
Author(s):  
J. Anthony VanDuzer
Keyword(s):  
Free Trade Agreement ◽  
Dispute Settlement ◽  
Trade Agreement ◽  
Minimum Standards ◽  
The North ◽  
State Behaviour ◽  
Nafta Chapter 11 ◽  
To Come ◽  
The Right

SummaryRecently, there has been a proliferation of international agreements imposing minimum standards on states in respect of their treatment of foreign investors and allowing investors to initiate dispute settlement proceedings where a state violates these standards. Of greatest significance to Canada is Chapter 11 of the North American Free Trade Agreement, which provides both standards for state behaviour and the right to initiate binding arbitration. Since 1996, four cases have been brought under Chapter 11. This note describes the Chapter 11 process and suggests some of the issues that may arise as it is increasingly resorted to by investors.

What face familiarity feelings say about the lateralization of specific entities within the core system

2019 ◽  
Vol 42 ◽  
Author(s):  
Guido Gainotti
Keyword(s):  
Temporal Lobe ◽  
Memory System ◽  
Core System ◽  
The Core ◽  
Memory Traces ◽  
Specific Memory ◽  
Target Article ◽  
Temporal Structures ◽  
The Right

Abstract The target article carefully describes the memory system, centered on the temporal lobe that builds specific memory traces. It does not, however, mention the laterality effects that exist within this system. This commentary briefly surveys evidence showing that clear asymmetries exist within the temporal lobe structures subserving the core system and that the right temporal structures mainly underpin face familiarity feelings.

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