We analyze the relation between the concept of auxiliary variables and the Inverse problem of the calculus of variations to construct a Lagrangian from a given set of equations of motion. The problem of the construction of a consistent second order dynamics from a given first order dynamics is investigated. At the level of equations of motion we find that this reduction process is consistent pr...

It is shown that there exist a continuous function f and a regulated function g defined on the interval [0, 1] such that g vanishes everywhere except for a countable set, and the K-integral of f with respect to g does not exist. The problem was motivated by extensions of evolution variational inequalities to the space of regulated functions.

This paper sheds new light on several interrelated topics of second-order variational analysis, both in finite and infinite-dimensional settings. We establish new relationships between second-order growth conditions on functions, the basic properties of metric regularity and subregularity of the limiting subdifferential, tilt-stability of local minimizers, and positive-definiteness/semidefinite...

This paper stresses the role of co-coercivity and related notions in the convergence of iterative schemes for solving monotone, but not necessarily strongly or even strictly monotone variational inequalities. The analysis will be conducted within the conceptual framework of the \auxiliary problem principle".

In this paper the author considers a continuum model of a transportation network in the presence of capacity constraints on the flow. The equilibrium conditions are expressed in terms of a Variational Inequality for which an existence theorem is enunciated.

It is now well known that curvature conditions à la Bakry-Émery are equivalent to contraction properties of the heat semigroup with respect to the classical quadratic Wasserstein distance. However, this curvature condition may include a dimensional correction which up to now had not induced any strenghtening of this contraction. We first consider the simplest example of the Euclidean heat semig...

A novel treatment of non-adiabatic couplings is proposed. The derivation starts from the long-known, but not well-known, fact that the wave function of the complete system of elctrons and nuclei can be written, without approximation, as a Born-Oppenheimer-type product of a nuclear wavefunction, X(R), and an electronic one, ΦR(r), which depends parametrically on the nuclear configuration R. From...

In this paper we deal with the construction of gap functions for variational inequalities by using an approach which bases on the conjugate duality. Under certain assumptions we also investigate a further class of gap functions for the variational inequality problem, the so-called dual gap functions.

The so-called Ritz–Galerkin method is one of the most fundamental tools of modern computing. Its origins lie in Hilbert’s “direct” approach to the variational calculus of Euler– Lagrange and in the thesis of Walther Ritz, who died 100 years ago at the age of 31 after a long battle with tuberculosis. The thesis was submitted in 1902 in Göttingen, during a period of dramatic developments in physi...

Abstract. The discrete, the quantum, and the continuous calculus of variations, have been recently unified and extended by using the theory of time scales. Such unification and extension is, however, not unique, and two approaches are followed in the literature: one dealing with minimization of delta integrals; the other dealing with minimization of nabla integrals. Here we review a more genera...