The Amitsur–Levitski Theorem tells us that the standard polynomial in 2n non-commuting indeterminates vanishes identically over the matrix algebra Mn(K). For K = R or C and 2≤ r ≤ 2n−1, we investigate how big Sr(A1, . . . ,Ar) can be when A1, . . . ,Ar belong to the unit ball. We privilegiate the Frobenius norm, for which the case r = 2 was solved recently by several authors. Our main result is...

T he properties of large conductance C a2+-dependent K + channels in smooth muscle cells (smc) isolated from norm al and atherosclerotic hum an aorta were studied using the patch-clam p technique. It was shown tha t smc from norm al hum an aorta possess a homogeneous population of norm al C a2+-dependent K + channels. In atherosclerotic aorta two kinetically different types of these channels co...

K is a centrally symmetric convex body with nonempty interior and fK(·) is also called the distance function of K because fK(x) = min{ρ ∈ R≥0 : x ∈ ρK}. The Euclidean norm is denoted by fB(·), where B is the n-dimensional unit ball, and the associated inner product is denoted by 〈·, ·〉. Finally, we denote by C the cube with edge length 2 and center 0, and thus fC(·) denotes the maximum norm. As...

In this paper we study the structure of set ${\\rm Hom}(X,\\mathbb{R})$ all lattice homomorphisms from a Banach $X$ into $\\mathbb{R}$. Using relation among and disjoint families, prove that topological dual free FBL}(A)$ generated by $A$ contains family cardinality $2^{|A|}$, answering question B. de Pagter A. W. Wickstead. We also deal with norm-attaining homomorphisms. For classical lattices...

Rank K Binary Matrix Factorization (BMF) approximates a binary matrix by the product of two binary matrices of lower rank, K, using either L1 or L2 norm. In this paper, we first show that the BMFwithL2 norm can be reformulated as an Unconstrained Binary Quadratic Programming (UBQP) problem. We then review several local search strategies that can be used to improve the BMF solutions obtained by ...

12 because the conditions formulated in Corollary 1 are satissed for the problem (??). Therefore we have for every z 2 C jjjI + zBjjj k jjjIjjj k : Hence for every unitarily invariant norm we have by the properties of the unitarily invariant norms jjI + zBjj jjIjj: This completes the proof. 2 The above considerations imply that the characterization of a zero-trace matrix by means of the problem...

denote its Euclidean operator norm (often called the 2-norm). If  is nonsingular, then its condition number () is defined by () = kk°°−1°° = 1() () where 1 ≥ 1 ≥    ≥  ≥ 0 are the singular values of . The s constitute lengths of the semi-axes of the hyperellipsoid  = { : kk = 1} in -dimensional space; thus  measures elongation of  at its extreme [1]. The role that  ...

Scaled iterates associated with the serial Kogbetliantz method for computing the singular value decomposition (SVD) of complex triangular matrices are considered. They are defined by B (k) S = |diag(B )| B |diag(B)|, where B are matrices generated by the method. Sharp estimates are derived for the Frobenius norm of the off-diagonal part of B (k) S , in the case of simple singular values. This n...

The paper considers parametric uncertain systems of the form ẋ t Mx t ,M ∈ M,M ⊂ Rn×n, where M is either a convex hull, or a positive cone of matrices, generated by the set of vertices V {M1,M2, . . . ,MK} ⊂ Rn×n. Denote by μ‖ ‖ the matrix measure corresponding to a vector norm ‖ ‖. When M is a convex hull, the condition μ‖ ‖ Mk ≤ r < 0, 1 ≤ k ≤ K, is necessary and sufficient for the existence ...

Observe that H 0 (M) = L p(M). Also, Hk := H2 k is a Hilbert space under the L2-inner product. F k contains only smooth functions. In general, a sequence in F k will not converge in the H k norm to a function in F k , so we need to complete the space to have anything useful. An alternate approach would have been to start with functions in Lp rather than completing the space of smooth functions ...