On the Measurability of the Conjugate and the Subdiierential of a Normal Integrand

Interchange of minimization and integration with measurability constraints

In [9], Theorem 14.60 shows that without measurability constraints, to minimize an integral functional with respect to all measurable mappings is equivalent to integrate the minimum over the constant mappings of the underlying cost function. We prove here the analogous result under measurability and almost sure constraints. In our case, the interchange introduces a conditional expectation, and ...

Existence and Measurability of the Solution of the Stochastic Differential Equations Driven by Fractional Brownian Motion

Additive maps on C$^*$-algebras commuting with $|.|^k$ on normal elements

Let $mathcal {A} $ and $mathcal {B} $ be C$^*$-algebras. Assume that $mathcal {A}$ is of real rank zero and unital with unit $I$ and $k>0$ is a real number. It is shown that if $Phi:mathcal{A} tomathcal{B}$ is an additive map preserving $|cdot|^k$ for all normal elements; that is, $Phi(|A|^k)=|Phi(A)|^k $ for all normal elements $Ainmathcal A$, $Phi(I)$ is a projection, and there exists a posit...

On X ̃-frames and conjugate systems in Banach spaces

The generalization of p-frame in Banach spaces is considered in this paper. The concepts of an $tilde{X}$-frame and a system conjugate to $tilde{X}$-frame were introduced. Analogues of the results on the existence of conjugate system were obtained. The stability of $tilde{X}$-frame having a conjugate system is studied.

Accountability of Modern Adiabatic Connection-Based Double-Hybrids Constructed from Cubic and Quadratic Integrand Functions for Dipole Polarizabilities of Water Nanoclusters

In this work, we dissect the performance of  two modern Perdew-Burke-Ernzerhof (PBE)-based double-hybrid (DH) density functionals to predict the isotropic and anisotropic polarizabilities of water nanoclusters (H2O)n [n = 2-6]. The considered models include the cubic integrand (CI) and quadratic integrand (QI) functions as well as the spin-opposite-scaled...