Rank 1 convex hulls of isotropic functions in dimension 2 by 2

Miroslav Šilhavý

Displaying similar documents to “Rank 1 convex hulls of isotropic functions in dimension 2 by 2”

On semiconvexity properties of rotationally invariant functions in two dimensions

Miroslav Šilhavý (2004)

Czechoslovak Mathematical Journal

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Let $f$ be a function defined on the set $𝐌^{2 \times 2}$ of all $2$ by $2$ matrices that is invariant with respect to left and right multiplications of its argument by proper orthogonal matrices. The function $f$ can be represented as a function $\tilde{f}$ of the signed singular values of its matrix argument. The paper expresses the ordinary convexity, polyconvexity, and rank 1 convexity of $f$ in terms of its representation $\tilde{f} .$

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.

Rank and perimeter preserver of rank-1 matrices over max algebra

Seok-Zun Song, Kyung-Tae Kang (2003)

Discussiones Mathematicae - General Algebra and Applications

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For a rank-1 matrix $A = a \otimes b^{t}$ over max algebra, we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the linear operators which preserve the rank and perimeter of rank-1 matrices over max algebra. That is, a linear operator T preserves the rank and perimeter of rank-1 matrices if and only if it has the form T(A) = U ⊗ A ⊗ V, or $T (A) = U \otimes A^{t} \otimes V$ with some monomial matrices U and V.

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.

Finite-rank perturbations of positive operators and isometries

Man-Duen Choi, Pei Yuan Wu (2006)

Studia Mathematica

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We completely characterize the ranks of A - B and $A^{1 / 2} - B^{1 / 2}$ for operators A and B on a Hilbert space satisfying A ≥ B ≥ 0. Namely, let l and m be nonnegative integers or infinity. Then l = rank(A - B) and $m = r a n k (A^{1 / 2} - B^{1 / 2})$ for some operators A and B with A ≥ B ≥ 0 on a Hilbert space of dimension n (1 ≤ n ≤ ∞) if and only if l = m = 0 or 0 < l ≤ m ≤ n. In particular, this answers in the negative the question posed by C. Benhida whether for positive operators A and B the finiteness of rank(A - B) implies that...

Chern rank of complex bundle

Bikram Banerjee (2019)

Commentationes Mathematicae Universitatis Carolinae

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Motivated by the work of A. C. Naolekar and A. S. Thakur (2014) we introduce notions of upper chern rank and even cup length of a finite connected CW-complex and prove that upper chern rank is a homotopy invariant. It turns out that determination of upper chern rank of a space $X$ sometimes helps to detect whether a generator of the top cohomology group can be realized as Euler class for some real (orientable) vector bundle over $X$ or not. For a closed connected $d$ -dimensional complex manifold...

Rank α operators on the space C(T,X)

Dumitru Popa (2002)

Colloquium Mathematicae

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For 0 ≤ α < 1, an operator U ∈ L(X,Y) is called a rank α operator if $x ₙ \to^{τ_{α}} x$ implies Uxₙ → Ux in norm. We give some results on rank α operators, including an interpolation result and a characterization of rank α operators U: C(T,X) → Y in terms of their representing measures.