scholarly journals On the tensor spectral p-norm and its dual norm via partitions

2020 ◽  
Vol 75 (3) ◽  
pp. 609-628 ◽  
Author(s):  
Bilian Chen ◽  
Zhening Li
Keyword(s):  
Dual Norm

scholarly journals The Cauchy problem for the Finsler heat equation

2020 ◽  
Vol 13 (3) ◽  
pp. 257-278 ◽  
Author(s):  
Goro Akagi ◽  
Kazuhiro Ishige ◽  
Ryuichi Sato
Keyword(s):  
Cauchy Problem ◽  
Linear Operator ◽  
Heat Equation ◽  
Laplace Operator ◽  
Sufficient Condition ◽  
Smooth Functions ◽  
Dual Norm ◽  
The Cauchy Problem

AbstractLet H be a norm of {\mathbb{R}^{N}} and {H_{0}} the dual norm of H. Denote by {\Delta_{H}} the Finsler–Laplace operator defined by {\Delta_{H}u:=\operatorname{div}(H(\nabla u)\nabla_{\xi}H(\nabla u))}. In this paper we prove that the Finsler–Laplace operator {\Delta_{H}} acts as a linear operator to {H_{0}}-radially symmetric smooth functions. Furthermore, we obtain an optimal sufficient condition for the existence of the solution to the Cauchy problem for the Finsler heat equation\partial_{t}u=\Delta_{H}u,\quad x\in\mathbb{R}^{N},\,t>0,where {N\geq 1} and {\partial_{t}:=\frac{\partial}{\partial t}}.

scholarly journals On a Characterisation of Differentiability of the Norm of a Normed Linear Space

1971 ◽  
Vol 12 (1) ◽  
pp. 106-114 ◽  
Author(s):  
J. R. Giles
Keyword(s):  
Banach Space ◽  
Linear Space ◽  
Normed Linear Space ◽  
Dual Norm ◽  
Uniform Rotundity ◽  
Strong Differentiability ◽  
Differentiability Properties

The purpose of this paper is to show that the various differentiability conditions for the norm of a normed linear space can be characterised by continuity conditions for a certain mapping from the space into its dual. Differentiability properties of the norm are often more easily handled using this characterisation and to demonstrate this we give somewhat more direct proofs of the reflexivity of a Banach space whose dual norm is strongly differentiable, and the duality of uniform rotundity and uniform strong differentiability of the norm for a Banach space.

scholarly journals Dual Norm Based Iterative Methods for Image Restoration

2011 ◽  
Vol 44 (2) ◽  
pp. 128-149 ◽  
Author(s):  
Miyoun Jung ◽  
Elena Resmerita ◽  
Luminita A. Vese
Keyword(s):  
Image Restoration ◽  
Iterative Methods ◽  
Dual Norm

scholarly journals Uniformly weak differentiability of the norm and a condition of Vlasov

1976 ◽  
Vol 21 (4) ◽  
pp. 393-409 ◽  
Author(s):  
J. R. Giles
Keyword(s):  
Banach Space ◽  
Smoothness Condition ◽  
Dual Norm ◽  
Weak Differentiability

AbstractIn determining geometrical conditions on a Banach space under which a Chebychev set is convex, Vlasov (1967) introduced a smoothness condition of some interest in itself. Equivalent forms of this condition are derived and it is related to uniformly weak differentiability of the norm and rotundity of the dual norm.

SHARP CONSTANTS BETWEEN EQUIVALENT NORMS IN WEIGHTED LORENTZ SPACES

2010 ◽  
Vol 88 (1) ◽  
pp. 19-27 ◽  
Author(s):  
SORINA BARZA ◽  
JAVIER SORIA
Keyword(s):  
Lorentz Space ◽  
Lorentz Spaces ◽  
Best Constants ◽  
Sharp Constants ◽  
Dual Norm ◽  
Equivalent Norms ◽  
Weighted Lorentz Spaces ◽  
Weighted Lorentz Space

AbstractFor an increasing weight w in Bp (or equivalently in Ap), we find the best constants for the inequalities relating the standard norm in the weighted Lorentz space Λp(w) and the dual norm.

scholarly journals BIDUAL OCTAHEDRAL RENORMINGS AND STRONG REGULARITY IN BANACH SPACES

2019 ◽  
pp. 1-17 ◽  
Author(s):  
Johann Langemets ◽  
Ginés López-Pérez
Keyword(s):  
Banach Space ◽  
Studia Math ◽  
Direct Consequence ◽  
Baire Class ◽  
Dual Norm ◽  
Separable Predual ◽  
Separable Case ◽  
Strongly Regular ◽  
First Baire Class

We prove that every separable Banach space containing an isomorphic copy of $\ell _{1}$ can be equivalently renormed so that the new bidual norm is octahedral. This answers, in the separable case, a question in Godefroy [Metric characterization of first Baire class linear forms and octahedral norms, Studia Math. 95 (1989), 1–15]. As a direct consequence, we obtain that every dual Banach space, with a separable predual and failing to be strongly regular, can be equivalently renormed with a dual norm to satisfy the strong diameter two property.

On the sensitivity analysis of eigenvalues

2015 ◽  
Vol 29 ◽  
pp. 223-235 ◽  
Author(s):  
Rafikul Alam
Keyword(s):  
Condition Number ◽  
Accuracy Assessment ◽  
Simple Eigenvalue ◽  
Condition Numbers ◽  
Companion Matrix ◽  
Dual Norm ◽  
Small Perturbations ◽  
Left And Right ◽  
Numer Math ◽  
Roots Of A Polynomial

Let $ \lam$ be a simple eigenvalue of an $n$-by-$n$ matrix $A.$ Let $y$ and $ x$ be left and right eigenvectors of $A$ corresponding to $\lam,$ respectively. Then, for the spectral norm, the condition number $\cond(\lam, A) := \|x\|_2\, \|y\|_2 /{|y^*x|}$ measures the sensitivity of $\lam$ to small perturbations in $A$ and plays an important role in the accuracy assessment of computed eigenvalues. R. A. Smith [Numer. Math., 10(1967), pp.232-240] proved that $ \cond(\lam, A) = \|x\|_2\|y\|_2/{|y^*x|} = \|\adj(\lam I -A)\|_2/{|p'(\lam)|}$, where $ \adj(A)$ is the ``adjugate" of $A$ and $p'(\lam)$ is the derivative of $ p(z) :=\det(z I- A)$ at $\lam.$ We extend Smith's condition number to any matrix norm $\|\cdot\|$ and show that $$\cond(\lam, A) = \frac{\|yx^*\|_*}{|y^*x|} = \frac{\|\adj(\lam I - A)^*\|_*}{|p'(\lam)|}$$ measures the sensitivity of $\lam$ to small perturbations in $A,$ where $\norm_*$ is the dual norm of $\|\cdot\|.$ The {\sc matlab} command {\tt roots} computes roots of a polynomial $p(x)$ by computing the eigenvalues of a companion matrix $C_p$ associated with $p.$ We analyze the sensitivity of $\lam$ as a root of $p(x)$ as well as the sensitivity of $\lam$ as an eigenvalue of $C_p$ and compare their condition numbers.

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