Abstract:
Being Omega an open and bounded Lipschitz domain of R^n, we consider the generalized Willmore functional on Omega defined, for smooth functions, as the p-Willmore energy of each isolevel set integrated over all levels. We propose a new framework, that combines varifolds and Young measures, to study the relaxation of this functional in BV(Omega) with respect to the strong topology of L^1.

Abstract:
This article explores the generalized analysis-of-variance or ANOVA dimensional decomposition (ADD) for multivariate functions of dependent random variables. Two notable properties, stemming from weakened annihilating conditions, reveal that the component functions of the generalized ADD have \emph{zero} means and are hierarchically orthogonal. By exploiting these properties, a simple, alternative approach is presented to derive a coupled system of equations that the generalized ADD component functions satisfy. The coupled equations, which subsume as a special case the classical ADD, reproduce the component functions for independent probability measures. To determine the component functions of the generalized ADD, a new constructive method is proposed by employing measure-consistent, multivariate orthogonal polynomials as bases and calculating the expansion coefficients involved from the solution of linear algebraic equations. New generalized formulae are presented for the second-moment characteristics, including triplets of global sensitivity indices, for dependent probability distributions. Furthermore, the generalized ADD leads to extended definitions of effective dimensions, reported in the current literature for the classical ADD. Numerical results demonstrate that the correlation structure of random variables can significantly alter the composition of component functions, producing widely varying global sensitivity indices and, therefore, distinct rankings of random variables. An application to random eigenvalue analysis demonstrates the usefulness of the proposed approximation.

Abstract:
It is proved that generalized excursion measures can be constructed via time change of Ito's Brownian excursion measure. A tightness-like condition on strings is introduced to prove a convergence theorem of generalized excursion measures. The convergence theorem is applied to obtain a conditional limit theorem, a kind of invariance principle where the limit is the Bessel meander.

Abstract:
We consider quantum dynamics for which the strict adiabatic approximation fails but which do not escape too far from the adiabatic limit. To treat these systems we introduce a generalisation of the time dependent wave operator theory which is usually used to treat dynamics which do not escape too far from an initial subspace called the active space. Our generalisation is based on a time dependent adiabatic deformation of the active space. The geometric phases associated with the almost adiabatic representation are also derived. We use this formalism to study the adiabaticity of a dynamics surrounding an exceptional point of a non-hermitian hamiltonian. We show that the generalized time dependent wave operator can be used to correct easily the adiabatic approximation which is very unperfect in this situation.

Abstract:
An information theory description of finite systems explicitly evolving in time is presented for classical as well as quantum mechanics. We impose a variational principle on the Shannon entropy at a given time while the constraints are set at a former time. The resulting density matrix deviates from the Boltzmann kernel and contains explicit time odd components which can be interpreted as collective flows. Applications include quantum brownian motion, linear response theory, out of equilibrium situations for which the relevant information is collected within different time scales before entropy saturation, and the dynamics of the expansion.

Abstract:
In the present paper we define a generalized measure of discrimination between two past lifetime distributions of a system. We also charecterize a propotional reversed hazard model and study its important properties.

Abstract:
The paper introduces a novel Ito's formula for time dependent tempered generalized functions. As an application, we study the heat equation when initial conditions are allowed to be a generalized tempered function. A new proof of the Ustunel- Ito's formula for tempered distributions is also provided.

Abstract:
We discuss the extension of the Lewis and Riesenfeld method of solving the time-dependent Schr\"odinger equation to cases where the invariant has continuous eigenvalues and apply it to the case of a generalized time-dependent inverted harmonic oscillator. As a special case, we consider a generalized inverted oscillator with constant frequency and exponentially increasing mass.

Abstract:
The present letter finds the complete set of exact solutions of the time-dependent generalized Cini model by making use of the Lewis-Riesenfeld invariant theory and the invariant-related unitary transformation formulation and, based on this, the general explicit expression for the decoherence factor is therefore obtained. This study provides us with a useful method to consider the geometric phase and topological properties in the time-dependent quantum decoherence process.

Abstract:
We investigate the relation between the invariant operators satisfying the quantum Liouville-von Neumann and the Heisenberg operators satisfying the Heisenberg equation. For time-dependent generalized oscillators we find the invariant operators, known as the Ermakov-Lewis invariants, in terms of a complex classical solution, from which the evolution operator is derived, and obtain the Heisenberg position and momentum operators. Physical quantities such as correlation functions are calculated using both the invariant operators and Heisenberg operators.