We analyze the semiconvex hulls of the subset K in symmetric matrices given by K = fF 2 M 22 : F T = F; jF 11 j = a; jF 12 j = b; jF 22 j = cg that was rst considered by Dacorogna&Tanteri Commun. in PDEs 2001]. We obtain explicit formulae for the polyconvex, the quasiconvex, and the rank-one convex hull for ac ? b 2 0 and show in particular that the quasiconvex and the polyconvex hull are diier...

We analyze the semiconvex hulls of the subset K in symmetric matrices given by K fF M F F jF j a jF j b jF j cg that was rst considered by Dacorogna Tanteri Commun in PDEs We obtain explicit formulae for the polyconvex the quasiconvex and the rank one convex hull for ac b and show in particular that the quasiconvex and the polyconvex hull are di erent if strict inequality holds For ac b we obta...

For a noisy quantum channel, a quantum error correcting code of dimension k exists if and only if the joint rank-k numerical range associated with the error operators of the channel is non-empty. In this paper, geometric properties of the joint rank k-numerical range are obtained and their implications to quantum computing are discussed. It is shown that for a given k if the dimension of the un...

In this paper, the notion of rank−k numerical range of rectangular complex matrix polynomials are introduced. Some algebraic and geometrical properties are investigated. Moreover, for > 0, the notion of Birkhoff-James approximate orthogonality sets for −higher rank numerical ranges of rectangular matrix polynomials is also introduced and studied. The proposed definitions yield a natural general...

In this note, analytic description of V 3 (A) is given for normal matrices of the form A = A 1 ⊕ iA 2 or A = A 1 ⊕ e i 2π 3 A 2 ⊕ e i 4π 3 A 3 , where A 1 , A 2 , A 3 are Hermitian matrices. The new concept " k th roots of a convex set " is used to study the polynomial numerical hulls of order k for normal matrices.

in this note, a generalization of higher rank numerical range isintroduced and some of its properties are investigated

We introduce a new measure of complexity of integer hulls of rational polyhedra called the small Chvátal rank (SCR). The SCR of an integer matrix A is the number of rounds of a Hilbert basis procedure needed to generate all normals of a sufficient set of inequalities to cut out the integer hulls of all polyhedra {x : Ax ≤ b} as b varies. The SCR of A is bounded above by the Chvátal rank of A an...

We study the symmetrised rank-one convex hull of monoclinic-I martensite (a twelve-variant material) in the context of geometrically-linear elasticity. We construct sets of T3s, which are (non-trivial) symmetrised rank-one convex hulls of three-tuples of pairwise incompatible strains. In addition, we construct a fivedimensional continuum of T3s and show that its intersection with the boundary o...

Abstract. It is known that the i-th order laminated microstructures can be resolved by the k-th order rank-one convex envelopes with k ≥ i. So the requirement of establishing an efficient numerical scheme for the computation of the finite order rank-one convex envelopes arises. In this paper, we develop an iterative scheme for such a purpose. The 1-st order rank-one convex envelope R1f is appro...