Riemannian convexity of functionals

This paper extends the Riemannian convexity concept to action functionals defined by multiple integrals associated to Lagrangian differential forms on first order jet bundles. The main results of this paper are based on the geodesic deformations theory and their impact on functionals in Riemannian setting. They include the basic properties of Riemannian convex functionals, the Riemannian convexity of functionals associated to differential m-forms or to Lagrangians of class C ¹ respectively C ², the generalization to invexity and geometric meaningful convex functionals. Riemannian convexity of functionals is the central ingredient for global optimization. We illustrate the novel features of this theory, as well as its versatility, by introducing new definitions, theorems and algorithms that bear upon the currently active subject of functionals in variational calculus and optimal control. In fact so deep rooted is the convexity notion that nonconvex problems are tackled by devising appropriate convex approximations.     

Informativeness of Linear Functionals

In this paper, we study the informativeness of linear functionals in reconstruction problems and obtain exact orders of the informativeness of linear functionals in the Besov and Sobolev classes W and SW.     

An overdetermined problem in nonlinear elliptic equations of second order

We define some functionals involvingu(x) andx'u _i , whereu(x) is a classical solution of the equation (q ^p-2 u _i ) _i +k(u)q ^p =0,p > 1, and prove that such functionals satisfy a second order elliptic differential equation. By a suitable choice of such functionals we investigate an overdetermined problem.Partially supported by Regione Autonoma della Sardegna.     

The Computational Power of ℳ︁ω

We prove that the Kleene schemes for primitive recursion relative to the μ‐operator, relativized to some nondeterministic objects, have the same power to express total functionals when interpreted over the partial continuous functionals and over the Kleene‐Kreisel continuous functionals. Relating the former interpretation to Niggl's ℳ︁^ω we prove Nigg's conjecture that ℳ︁^ω is strictly weaker than Plotkin's PCF + PA.     

The gauge theorem for a class of additive functionals of zero energy

Summary In earlier works, the gauge theorem was proved for additive functionals of Brownian motion of the form ₀ ^t q(B _s )ds, whereq is a function in the Kato class. Subsequently, the theorem was extended to additive functionals with Revuz measures in the Kato class. We prove that the gauge theorem holds for a large class of additive functionals of zero energy which are, in general, of unbounded variation. These additive functionals may not be semi-martingales, but correspond to a collection of distributions that belong to the Kato class in a suitable sense. Our gauge theorem generalizes the earlier versions of the gauge theorem.Research supported in part by NSA grant MDA-92-H-30324     

Fully symmetric functionals on a Marcinkiewicz space are Dixmier traces

As a consequence of the exposition of Dixmier type traces in the book of A. Connes (1994) [2], we were led to ask how general is this class of functionals within the space of all unitarily invariant functionals on the corresponding Marcinkiewicz ideal Mψ. In this paper we prove the surprising result that the set of all Dixmier traces on Mψ coincides with the set of all fully symmetric functionals on this space.     

Basic properties of scalarizmg functionals for multiobjective optimization

If a partial ordering or preordering induced by a cone D defines a multi objective optimization problem, then scalarizing functionals for this problem shall posses two basic properties. D – monotonicity and D _e approximation. Several ways of constructing functionals with these properties are discussed in the paper.     

Distribution of functionals of bridges for diffusions with jumps

The paper deals with a method of calculation of distributions for functionals of bridges of a process which is a generalization of a diffusion with jumps. The approach to calculation of distributions for integral functionals of bridges is the same as for the diffusion itself. This approach is based on calculation of the Laplace transform of distributions of the integral functionals. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 34–47.     

Implementation of the recurrence relations of biorthogonality

The aim of this work is the implementation of general algorithms for computing some generalized Padé-type approximants [1]. The quality of these approximants depends on the choice of some functionals. In the classical example of the seriesf(t)=log(1+t)/t we found functionals that give better results than the usual ones.     

Groups of Isometries of Hardy-Type Spaces of Analytic Functionals in the Unit Ball of the Space~L p

The tensor structure of spaces L _p (R ⁿ ) of summable functions of several variables is described. A scale of Hardy-type spaces of analytic functionals defined in the unit ball of the space L _p (R ¹) of summable functions of one variable is introduced. One-parameter groups of isometries of such spaces of analytic functionals are investigated.     

The representations of two types of functionals on L∞(Ω, ?) and L∞(Ω, ?, ?)

This article gives the representations of two types of real functionals on L ^∞(Ω, Ƒ) or L ^∞(Ω, Ƒ, ℙ) in terms of Choquet integrals. These functionals are comonotonically subadditive and comonotonically convex, respectively.     

On geometric properties of functors of positive-homogenous and semiadditive functionals

We investigate the functor OH of positive-homogenous functionals and the functor OS of semiadditive functionals. We prove that OH(X) is an absolute retract if and only if X is an open-generated compactum, and OS(X) is an absolute retract if and only if X is an opengenerated compactum of weight ≤ ω₁. We investigate the softness of mappings of multiplication of monads generated by these functors.     

Necessity of a Wiener type condition for boundary regularity of quasiminimizers and nonlinear elliptic equations

We use variational methods to obtain a pointwise estimate near a boundary point for quasisubminimizers of the p-energy integral and other integral functionals in doubling metric measure spaces admitting a p-Poincaré inequality. It implies a Wiener type condition necessary for boundary regularity for p-harmonic functions on metric spaces, as well as for (quasi)minimizers of various integral functionals and solutions of nonlinear elliptic equations on R ⁿ .     

Co-generators for algebras over fields and commutative applications

Let R be an algebra over a field k. We say that an ideal I of R is co-generated by a set F of k-linear functionals R → k if I is the largest ideal of R that is contained in the kernels of all the functionals in F;. We will state sufficient conditions for an ideal of a free k-algebra to be co-generated by finitely many functionals. We then get as a corollary that every ideal of the polynomial algebra over a field in finitely many variables is finitely co-generated. This is a known result, but the way we construct the co-generators in the general case, leads to that we get a tight bound on how many co-generators are needed for such ideals.     

A characterization of reflexive spaces

We prove that a Banach space is reflexive if for every equivalent norm, the set of norm attaining functionals has non-empty norm-interior in the dual space. It is also proved that the set of norm attaining functionals on a Banach space that is not a Grothendieck space is not a w*-G _δ subset of the dual space.     

Completely almost periodic functionals

Using the notion of complete compactness introduced by H.  Saar, we define completely almost periodic functionals on completely contractive Banach algebras. We show that, if (M, Γ) is a Hopf–von Neumann algebra with M injective, then the space of completely almost periodic functionals on M _* is a C*-subalgebra of M.     

Curves of minimal action over metric spaces

Given a metric space X, we consider a class of action functionals, generalizing those considered in Brancolini et al. (J Eur Math Soc 8:415–434, 2006) and Ambrosio and Santambrogio (Atti Accad Naz Lincei Cl Sci Fis Mat Natur Rend Lincei Mat Appl 18: 23–37, 2007), which measure the cost of joining two given points x ₀ and x ₁, by means of an absolutely continuous curve. In the case X is given by a space of probability measures, we can think of these action functionals as giving the cost of some congested/concentrated mass transfer problem. We focus on the possibility to split the mass in its moving part and its part that (in some sense) has already reached its final destination: we consider new action functionals, taking into account only the contribution of the moving part.     

Distributions of Additive Functionals of Brownian Motion Stopped at Moments of Maxima and Minima of Random Times

We study distributions of additive functionals of a Brownian motion stopped at moments constructed by maxima and minima operations from the inverse range time, moments inverse to some additive functionals, and the first exit time. Some interesting examples of application are considered. Bibliography: 6 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 298, 2003, pp. 36–53.     

A non-local approximation of free discontinuity problems in SBV and SBD

We consider a class of functionals which are defined in the spaces SBV and SBD and which do not depend on the traces u^± on the set of discontinuity points. In this work we prove that it is possible to approximate these energies, in the sense of Γ-convergence, by means of a family of non-local functionals defined in Sobolev spaces. Moreover we illustrate some applications for image processing and mechanics.     

Lipschitzian Properties of Integral Functionals on Lebesgue Spaces Lp, 1 \leqslant p < ￥\bf{L_p, 1 \leqslant {p} < \infty}

We give complete characterizations of integral functionals which are Lipschitzian on a Lebesgue space L _p with p ≠ ∞. When the measure is atomless, we characterize the integral functionals which are locally Lipschitzian on such Lebesgue spaces. In every cases, the Lipchitzian properties of the integral functional can be described by growth conditions on the subdifferentials of the integrand which are equivalent to Lipschitzian properties of the integrand.