A short proof of the separable reduction theorem

AbstractWe present a simple proof of the separable reduction theorem, a crucial result of nonsmooth analysis which allows to extend to Asplund spaces the results known for separable spaces dealing with Fréchet subdifferentials. It relies on elementary results in convex analysis and avoids certain technicalities.

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A Short Proof of an Interpolation Theorem

Canadian Mathematical Bulletin ◽

10.4153/cmb-1974-025-2 ◽

1974 ◽

Vol 17 (1) ◽

pp. 127-128 ◽

Cited By ~ 1

Author(s):

Edward Hughes

Keyword(s):

Half Plane ◽

Complex Plane ◽

Simple Proof ◽

Real Line ◽

Short Proof ◽

Interpolation Theorem ◽

Operator Interpolation ◽

Upper Half Plane ◽

Positive Functions ◽

Class Of Functions

In this note we give a simple proof of an operator-interpolation theorem (Theorem 2) due originally to Donoghue [6], and Lions-Foias [7].Let be the complex plane, the open upper half-plane, the real line, ℛ+ and ℛ- the non-negative and non-positive axes. Denote by the class of positive functions on which extend analytically to —ℛ-, and map into itself. Denote by ’ the class of functions φ such that φ(x1/2)2 is in .

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The Engulfing Property from a Convex Analysis Viewpoint

Journal of Optimization Theory and Applications ◽

10.1007/s10957-020-01755-1 ◽

2020 ◽

Vol 187 (2) ◽

pp. 408-420 ◽

Cited By ~ 1

Author(s):

Andrea Calogero ◽

Rita Pini

Keyword(s):

Simple Proof ◽

Convex Functions ◽

Convex Analysis ◽

Strict Convexity

Abstract In this note, we provide a simple proof of some properties enjoyed by convex functions having the engulfing property. In particular, making use only of results peculiar to convex analysis, we prove that differentiability and strict convexity are conditions intrinsic to the engulfing property.

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A simple proof of the sum formula

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700019389 ◽

2001 ◽

Vol 63 (2) ◽

pp. 337-339 ◽

Cited By ~ 1

Author(s):

A. Verona ◽

M. E. Verona

Keyword(s):

Simple Proof ◽

Convex Functions ◽

Short Proof ◽

Sum Formula

In this note we present a simple, short proof of the sum formula for subdifferentials of convex functions.

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A Simple Proof of the Multinomial Theorem using Multivariate Calculus

10.31219/osf.io/2t7wn ◽

2019 ◽

Author(s):

Sumit Kumar Jha

Keyword(s):

Simple Proof ◽

Short Proof ◽

Multinomial Theorem ◽

Multivariate Calculus

We give a simple and short proof of the multinomial theorem using multivariate calculus.

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An Upper Bound for the Size of $s$-Distance Sets in Real Algebraic Sets

The Electronic Journal of Combinatorics ◽

10.37236/9712 ◽

2021 ◽

Vol 28 (3) ◽

Author(s):

Gábor Hegedüs ◽

Lajos Rónyai

Keyword(s):

Simple Proof ◽

Upper Bound ◽

Quadratic Forms ◽

Short Proof ◽

Algebraic Method ◽

Upper Bounds ◽

Additive Combinatorics ◽

Algebraic Sets ◽

Distance Sets ◽

Real Quadratic

In a recent paper, Petrov and Pohoata developed a new algebraic method which combines the Croot-Lev-Pach Lemma from additive combinatorics and Sylvester’s Law of Inertia for real quadratic forms. As an application, they gave a simple proof of the Bannai-Bannai-Stanton bound on the size of $s$-distance sets (subsets $\mathcal{A}\subseteq \mathbb{R}^n$ which determine at most $s$ different distances). In this paper we extend their work and prove upper bounds for the size of $s$-distance sets in various real algebraic sets. This way we obtain a novel and short proof for the bound of Delsarte-Goethals-Seidel on spherical s-distance sets and a generalization of a bound by Bannai-Kawasaki-Nitamizu-Sato on $s$-distance sets on unions of spheres. In our arguments we use the method of Petrov and Pohoata together with some Gröbner basis techniques.

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A Simple Proof of the Chuang’s Inequality

Annals of West University of Timisoara - Mathematics and Computer Science ◽

10.1515/awutm-2017-0016 ◽

2017 ◽

Vol 55 (2) ◽

pp. 85-89

Author(s):

Bikash Chakraborty

Keyword(s):

Simple Proof ◽

Short Proof

AbstractThe purpose of this paper is to present a short proof of the Chuang’s inequality.

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A simple proof of Whittle's bridging condition in dynamic programming

Journal of Applied Probability ◽

10.1017/s0021900200097424 ◽

1980 ◽

Vol 17 (04) ◽

pp. 1114-1116

Author(s):

Roger Hartley

Keyword(s):

Dynamic Programming ◽

Simple Proof ◽

Short Proof ◽

Special Treatment ◽

Optimal Value ◽

Optimal Policies

We offer a short proof that the bridging condition introduced by Whittle is sufficient for regularity in negative dynamic programming. We exploit concavity of the optimal value operator and do not need a special treatment of the case when optimal policies do not exist.

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