Topology and geometry of nontrivial rank-one 
convex hulls for two-by-two matrices

Carl-Friedrich Kreiner; Johannes Zimmer

Displaying similar documents to “Topology and geometry of nontrivial rank-one convex hulls for two-by-two matrices”

A complete characterization of invariant jointly rank- convex quadratic forms and applications to composite materials

Vincenzo Nesi, Enrico Rogora (2007)

ESAIM: Control, Optimisation and Calculus of Variations

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The theory of compensated compactness of Murat and Tartar links the algebraic condition of rank- convexity with the analytic condition of weak lower semicontinuity. The former is an algebraic condition and therefore it is, in principle, very easy to use. However, in applications of this theory, the need for an efficient classification of rank- convex forms arises. In the present paper, we define the concept of extremal -forms and characterize them in the rotationally invariant...

Critique of "Two-dimensional examples of rank-one convex functions that are not quasiconvex" by M. K. Benaouda and J. J. Telega

Patrizio Neff (2005)

Annales Polonici Mathematici

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It is noted that the examples provided in the paper "Two-dimensional examples of rank-one convex functions that are not quasiconvex" by M. K. Benaouda and J. J. Telega, Ann. Polon. Math. 73 (2000), 291-295, contain unrecoverable errors.

A note on equality of functional envelopes

Martin Kružík (2003)

Mathematica Bohemica

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We characterize generalized extreme points of compact convex sets. In particular, we show that if the polyconvex hull is convex in $ℝ^{m \times n}$ , $min (m, n) \leq 2$ , then it is constructed from polyconvex extreme points via sequential lamination. Further, we give theorems ensuring equality of the quasiconvex (polyconvex) and the rank-1 convex envelopes of a lower semicontinuous function without explicit convexity assumptions on the quasiconvex (polyconvex) envelope.

Existence of Lipschitz minimizers for the three-well problem in solid-solid phase transitions

Sergio Conti, Georg Dolzmann, Bernd Kirchheim (2007)

Annales de l'I.H.P. Analyse non linéaire

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Linear maps preserving rank 2 on the space of alternate matrices and their applications.

Cao, Chongguang, Tang, Xiaomin (2004)

International Journal of Mathematics and Mathematical Sciences

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Zero-term rank preservers of integer matrices

Seok-Zun Song, Young-Bae Jun (2006)

Discussiones Mathematicae - General Algebra and Applications

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The zero-term rank of a matrix is the minimum number of lines (row or columns) needed to cover all the zero entries of the given matrix. We characterize the linear operators that preserve the zero-term rank of the m × n integer matrices. That is, a linear operator T preserves the zero-term rank if and only if it has the form T(A)=P(A ∘ B)Q, where P, Q are permutation matrices and A ∘ B is the Schur product with B whose entries are all nonzero integers.

Rank three residually connected geometries for $M_{22}$ , revisited.

Leemans, Dimitri, Rowley, Peter (2010)

The Electronic Journal of Combinatorics [electronic only]

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Is it wise to keep laminating ?

Marc Briane, Vincenzo Nesi (2004)

ESAIM: Control, Optimisation and Calculus of Variations

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We study the corrector matrix $P^{ϵ}$ to the conductivity equations. We show that if $P^{ϵ}$ converges weakly to the identity, then for any laminate $det P^{ϵ} \geq 0$ at almost every point. This simple property is shown to be false for generic microgeometries if the dimension is greater than two in the work Briane et al. [Arch. Ration. Mech. Anal., to appear]. In two dimensions it holds true for any microgeometry as a corollary of the work in Alessandrini and Nesi [Arch. Ration. Mech. Anal. 158 (2001) 155-171]....

Trace and determinant in Banach algebras

Bernard Aupetit, H. Mouton (1996)

Studia Mathematica

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We show that the trace and the determinant on a semisimple Banach algebra can be defined in a purely spectral and analytic way and then we obtain many consequences from these new definitions.