On non-negative quasiconvex functions with unbounded zero sets

1997 ◽  
Vol 127 (2) ◽  
pp. 411-422 ◽  
Author(s):  
Kewei Zhang
Keyword(s):  
Dimensional Subspace ◽  
Orthogonal Complement ◽  
Quasiconvex Functions ◽  
Two Dimensional ◽  
Zero Sets ◽  
Rank One ◽  
Lipschitz Constants ◽  
Lipschitz Graphs

We construct nontrivial, non-negative quasiconvex functions denned on M2×2 with p-th order growth such that the zero sets of the functions are Lipschitz graphs of mappings from subsets of a fixed two-dimensional subspace to its orthogonal complement. We assume that the graphs do not have rank-one connections with the Lipschitz constants sufficiently small. In particular, we are able to construct quasiconvex functions which are homogeneous of degree p (p > 1) and ‘conjugating’ invariant.

On connected subsets of M2×2 without rank-one connections

1997 ◽  
Vol 127 (1) ◽  
pp. 207-216 ◽  
Author(s):  
Kewei Zhang
Keyword(s):  
Dimensional Subspace ◽  
Orthogonal Complement ◽  
Two Dimensional ◽  
Lipschitz Extensions ◽  
Rank One ◽  
Lipschitz Graphs

We prove that connected subsets of M2×2 without rank-one connections are Lipschitz graphs of mappings from subsets of a fixed two-dimensional subspace to its orthogonal complement. Under a weaker condition that the set does not have rank-one connections locally, we are able to establish some global results on the set. We also establish some results on Lipschitz extensions of the functions thus obtained.

scholarly journals Sharp Rank-One Convexity Conditions in Planar Isotropic Elasticity for the Additive Volumetric-Isochoric Split

2021 ◽  
Vol 143 (2) ◽  
pp. 301-335
Author(s):  
Jendrik Voss ◽  
Ionel-Dumitrel Ghiba ◽  
Robert J. Martin ◽  
Patrizio Neff
Keyword(s):  
Differential Inequalities ◽  
Two Dimensional ◽  
Ellipticity Condition ◽  
One Dimensional ◽  
Suitable Sense ◽  
Precise Analysis ◽  
Rank One ◽  
Isotropic Elasticity ◽  
Convexity Conditions

AbstractWe consider the volumetric-isochoric split in planar isotropic hyperelasticity and give a precise analysis of rank-one convexity criteria for this case, showing that the Legendre-Hadamard ellipticity condition separates and simplifies in a suitable sense. Starting from the classical two-dimensional criterion by Knowles and Sternberg, we can reduce the conditions for rank-one convexity to a family of one-dimensional coupled differential inequalities. In particular, this allows us to derive a simple rank-one convexity classification for generalized Hadamard energies of the type $W(F)=\frac{\mu }{2} \hspace{0.07em} \frac{\lVert F \rVert ^{2}}{\det F}+f(\det F)$ W ( F ) = μ 2 ∥ F ∥ 2 det F + f ( det F ) ; such an energy is rank-one convex if and only if the function $f$ f is convex.

scholarly journals A characterisation of Hilbert spaces via orthogonality and proximinality

2005 ◽  
Vol 71 (1) ◽  
pp. 107-111
Author(s):  
Fathi B. Saidi
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Real Banach Space ◽  
Nonzero Element ◽  
Dimensional Subspace ◽  
Two Dimensional ◽  
Orthogonal Direction ◽  
Open Problems

In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.

scholarly journals Elliptic Genera of Two-Dimensional $${\mathcal{N} = 2}$$ N = 2 Gauge Theories with Rank-One Gauge Groups

2013 ◽  
Vol 104 (4) ◽  
pp. 465-493 ◽  
Author(s):  
Francesco Benini ◽  
Richard Eager ◽  
Kentaro Hori ◽  
Yuji Tachikawa
Keyword(s):  
Gauge Theories ◽  
Two Dimensional ◽  
Gauge Groups ◽  
Elliptic Genera ◽  
Rank One

scholarly journals AN INTEGRATION APPROACH TO THE TOEPLITZ SQUARE PEG PROBLEM

2017 ◽  
Vol 5 ◽  
Author(s):  
TERENCE TAO
Keyword(s):  
Simple Closed Curve ◽  
Closed Curve ◽  
Finite Sets ◽  
Real Numbers ◽  
Integration Approach ◽  
Sign Patterns ◽  
Lipschitz Constants ◽  
Lipschitz Graphs ◽  
Periodic Version ◽  
Rule Out

The ‘square peg problem’ or ‘inscribed square problem’ of Toeplitz asks if every simple closed curve in the plane inscribes a (nondegenerate) square, in the sense that all four vertices of that square lie on the curve. By a variety of arguments of a ‘homological’ nature, it is known that the answer to this question is positive if the curve is sufficiently regular. The regularity hypotheses are needed to rule out the possibility of arbitrarily small squares that are inscribed or almost inscribed on the curve; because of this, these arguments do not appear to be robust enough to handle arbitrarily rough curves. In this paper, we augment the homological approach by introducing certain integrals associated to the curve. This approach is able to give positive answers to the square peg problem in some new cases, for instance if the curve is the union of two Lipschitz graphs $f$, $g:[t_{0},t_{1}]\rightarrow \mathbb{R}$ that agree at the endpoints, and whose Lipschitz constants are strictly less than one. We also present some simpler variants of the square problem which seem particularly amenable to this integration approach, including a periodic version of the problem that is not subject to the problem of arbitrarily small squares (and remains open even for regular curves), as well as an almost purely combinatorial conjecture regarding the sign patterns of sums $y_{1}+y_{2}+y_{3}$ for $y_{1},y_{2},y_{3}$ ranging in finite sets of real numbers.

Geometry of isometric reflection vectors

2011 ◽  
Vol 61 (5) ◽  
Author(s):  
Francisco García-Pacheco
Keyword(s):  
Dimensional Subspace ◽  
Two Dimensional ◽  
Whole Space ◽  
Open Question

AbstractIn this paper we study the geometry of isometric reflection vectors. In particular, we generalize known results by proving that the minimal face that contains an isometric reflection vector must be an exposed face. We also solve an open question by showing that there are isometric reflection vectors in any two dimensional subspace that are not isometric reflection vectors in the whole space. Finally, we prove that the previous situation does not hold in smooth spaces, and study the orthogonality properties of isometric reflection vectors in those spaces.

Face Recognition using Two-dimensional Subspace Analysis and PNN

2013 ◽  
Vol 72 (6) ◽  
pp. 1-8 ◽  
Author(s):  
Benouis Mohamed ◽  
Tlmesani Redwan ◽  
Senouci Mohamed
Keyword(s):  
Face Recognition ◽  
Dimensional Subspace ◽  
Two Dimensional ◽  
Subspace Analysis

Some remarks on quasiconvexity and rank-one convexity

1996 ◽  
Vol 126 (5) ◽  
pp. 1055-1065 ◽  
Author(s):  
Pablo Pedregal
Keyword(s):  
Necessary Conditions ◽  
Higher Dimensions ◽  
Target Space ◽  
Two Dimensional ◽  
Rank One

We explore some necessary conditions for quasiconvexity in an attempt to show that rank-one convexity does not imply quasiconvexity when the target space for deformations is two- dimensional. An interesting construction is presented, showing how rank-one directions may fit with each other, making the task harder than in higher dimensions.

Sign in / Sign up

Export Citation Format

Share Document