We study the operator-valued free Fisher information of random matrices in an operator-valued noncommutative probability space. We obtain a formula for Φ M2(B) (A,A,M2(B), η), where A ∈ M2(B) is a 2 × 2 operator matrix on B, and η is linear operators on M2(B). Then we consider a special setting: A is an operator-valued semicircular matrix with conditional expectation covariance, and find that Φ...

Theory and implementation for the global optimization of a wide class of algorithms is presented via convex/affine relaxations. The basis for the proposed relaxations is the systematic construction of subgradients for the convex relaxations of factorable functions by McCormick [Math. Prog., 10 (1976), pp. 147–175]. Similar to the convex relaxation, the subgradient propagation relies on the recu...

In the setting of a real Hilbert space H, we investigate the asymptotic behavior, as time t goes to infinity, of trajectories of second-order evolution equations ü(t) + γu̇(t) +∇φ(u(t)) +A(u(t)) = 0, where ∇φ is the gradient operator of a convex differentiable potential function φ : H → IR, A : H → H is a maximal monotone operator which is assumed to be λ-cocoercive, and γ > 0 is a damping param...

We consider the nonselfadjoint Sturm-Liouville operator with regular but not strongly regular boundary conditions. We examine the basis property of the root function system of the mentioned operator. In the present paper we study eigenvalue problems for the nonselfadjoint Sturm-Liouville operator Lu = u′′ − q(x)u (1) defined on the interval (0, 1), where q(x) is an arbitrary complex-valued func...

We prove that for the relative entropy of faithful normal states φ and ω on the von Neumann algebra M the formula S(φ,ω) = sup{ω(Λ)-logφ(/): h = h*eM] holds. In general von Neumann algebras the relative entropy was defined and investigated by Araki [1, 3]. After Lieb had proved the joint convexity of the relative entropy in the type / case [10] several proofs appeared in the literature and they...

Let H(d) be the space of complex hermitian matrices of size d×d and let H+(d) ⊂ H(d) be the cone of positive semidefinite matrices. A linear operator Φ : H(d1) → H(d2) is said to be positive if Φ[H+(d1)] ⊂ H+(d2). The concurrence C(Φ; ·) of a positive operator Φ : H(d1) → H(d2) is a real-valued function on the cone H+(d1), defined as the largest convex function which coincides with 2 q σ d2 2 (...

Belief Calculus, ABC, see Darwiche, Ginsberg (1992), and Stern (2003). 〈Φ,⊕, 〉 , Support Structure, Φ , Support Function, for statements on U. Null and full support values are 0 and 1. ⊕ , Support Summation operator, , Support Scaling or Conditionalization, 〈Φ,⊕〉 , Partial Support Structure. ⊕, gives the support value of the disjunction of any two logically disjoint statements from their indivi...

Let H be a complex Hilbert space. Denote by B(H)+ the set of all positive bounded linear operators on H. A bijective map φ : B(H)+ → B(H)+ is said to preserve Lebesgue decompositions in both directions if for any quadruple A,B,C,D of positive operators, B = C +D is an A-Lebesgue decomposition of B if and only if φ(B) = φ(C)+φ(D) is a φ(A)-Lebesgue decomposition of φ(B). It is proved that every ...

Let φ(z) = (φ 1 (z),. .. , φ n (z)) be a holomorphic self-map of U n and ψ(z) a holomorphic function on U n , where U n is the unit polydisk of C n. Let p ≥ 0, q ≥ 0, this paper gives some necessary and sufficient conditions for the weighted composition operator W ψ,φ induced by ψ and φ to be bounded and compact between p-Bloch space B p (U n) and q-Bloch space B q (U n).