Characterizations are given for the positive and completely positive maps on n n complex matrices that leave invariant the diagonal entries or the kth elementary symmetric function of the diagonal entries, 1 < k n. In addition, it was shown that such a positive map is always decomposable if n 3, and this fails to hold if n > 3. The real case is also considered.

We determine the asymptotic behaviour (as k →∞, up to multiplicative constants not depending on k) of the entropy numbers ek (Dσ : lp → lq), 1 ≤ p ≤ q ≤ ∞, of diagonal operators generated by logarithmically decreasing sequences σ = (σn). This complements earlier results by Carl [2] who investigated the case of power-like decay of the diagonal. 2000 Mathematics Subject Classification: 47B06, 46B...

A diagonal Latin square is a Latin square whose main diagonal and back diagonal are both transversals. In this paper we give some constructions of pairwise orthogonal diagonal Latin squares. As an application of such constructions we obtain some new infinite classes of pairwise orthogonal diagonal Latin squares which are useful in the study of pairwise orthogonal diagonal Latin squares.

The Lifting Problem for Commuting Subnormals (LPCS) asks for necessary and sufficient conditions for a pair of subnormal operators on Hilbert space to admit commuting normal extensions. Given a 2-variable weighted shift T with diagonal core, we prove that LPCS is soluble for T if and only if LPCS is soluble for some power Tm (m ∈ Z+,m ≡ (m1,m2),m1,m2 ≥ 1). We do this by first developing the bas...

We propose a fast structured selected inversion method for extracting the diagonal blocks of the inverse of a sparse symmetric matrix A, using the multifrontal method and rank structures. When A arises from the discretization of some PDEs and has a low-rank property (the intermediate dense matrices in the factorization have small off-diagonal numerical ranks), structured approximations of the d...

We consider distributed optimization problems where networked nodes cooperatively minimize the sum of their locally known convex costs. A popular class of methods to solve these problems are the distributed gradient methods, which are attractive due to their inexpensive iterations, but have a drawback of slow convergence rates. This motivates the incorporation of second-order information in the...

(1) For every set X such that ∇X ⊆ idX holds X is trivial. Let A be a relational structure. We say that A is diagonal if and only if: (Def. 1) The internal relation of A ⊆ idthe carrier of A. Let A be a non trivial set. Observe that 〈A,∇A〉 is non diagonal. Next we state the proposition (2) For every reflexive relational structure L holds idthe carrier of L ⊆ the internal relation of L. One can ...

Let H(t) = (1−t/T )H0+(t/T )H1, t ∈ [0, T ], be the Hamiltonian governing an adiabatic quantum algorithm, where H0 is diagonal in the Hadamard basis and H1 is diagonal in the computational basis. We prove that H0 and H1 must each have at least two large mutually-orthogonal eigenspaces if the algorithm’s running time is to be subexponential in the number of qubits. We also reproduce the optimali...

is a Hessenberg matrix and its determinant is F2n+2. Furthermore, a Hessenberg matrix is said to be a Fibonacci-Hessenberg matrix [2] if its determinant is in the form tFn−1 + Fn−2 or Fn−1 + tFn−2 for some real or complex number t. In [1] several types of Hessenberg matrices whose determinants are Fibonacci numbers were calculated by using the basic definition of the determinant as a signed sum...