On Γ-convergence of pairs of dual functionals

The paper considers a slightly modified notion of the Γ-convergence of convex functionals in uniformly convex Banach spaces and establishes that under standard coercitivity and growth conditions the Γ-convergence of a sequence of functionals {Fj} to implies that the corresponding sequence of dual functionals converges in an analogous sense to the dual to functional .     

A class of Adams–Fontana type inequalities and related functionals on manifolds

A class of Adams–Fontana type inequalities are established on compact Riemannian manifolds without boundary via the Young inequality together with the usual Adams–Fontana inequality (Comment Math Helv 68:415–454, 1993). As an application, a sequence of functionals are defined on manifolds, a sufficient condition on which the Palais–Smale condition holds is given and the existence of critical points of the functionals is also considered in the spirit of Adimurthi (Ann Scuola Norm Sup Pisa Cl Sci 17:393–413, 1990) and Adimurthi and Sandeep (Nonlinear Differ Equ Appl 13:585–603, 2007).     

Bessel pairs and optimal Hardy and Hardy–Rellich inequalities

We give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in R ⁿ , n ≥ 1, so that the following inequalities hold for all \({u \in C_{0}^{\infty}(B)}\) :

$\label{one} \int\limits_{B}V(x)|\nabla u |^{2}dx \geq \int\limits_{B} W(x)u^2dx,$

$\label{two} \int\limits_{B}V(x)|\Delta u |^{2}dx \geq\int\limits_{B} W(x)|\nabla u|^{2}dx+(n-1)\int\limits_{B}\left(\frac{V(x)}{|x|^2}-\frac{V_r(|x|)}{|x|}\right)|\nabla u|^2dx.$

This characterization makes a very useful connection between Hardy-type inequalities and the oscillatory behaviour of certain ordinary differential equations, and helps in the identification of a large number of such couples (V, W)—that we call Bessel pairs—as well as the best constants in the corresponding inequalities. This allows us to improve, extend, and unify many results—old and new—about Hardy and Hardy–Rellich type inequalities, such as those obtained by Caffarelli et al. (Compos Math 53:259–275, 1984), Brezis and Vázquez (Revista Mat. Univ. Complutense Madrid 10:443–469, 1997), Wang and Willem (J Funct Anal 203:550–568, 2003), Adimurthi et al. (Proc Am Math Soc 130:489–505, 2002), and many others.

     

? 1-regularized linear regression: persistence and oracle inequalities

We study the predictive performance of ? ₁-regularized linear regression in a model-free setting, including the case where the number of covariates is substantially larger than the sample size. We introduce a new analysis method that avoids the boundedness problems that typically arise in model-free empirical minimization. Our technique provides an answer to a conjecture of Greenshtein and Ritov (Bernoulli 10(6):971–988, 2004) regarding the “persistence” rate for linear regression and allows us to prove an oracle inequality for the error of the regularized minimizer. It also demonstrates that empirical risk minimization gives optimal rates (up to log factors) of convex aggregation of a set of estimators of a regression function.     

Δ20-categoricity in Boolean algebras and linear orderings

We characterize Δ₂⁰-categoricity in Boolean algebras and linear orderings under some extra effectiveness conditions. We begin with a study of the relativized notion in these structures.     

Stability of isoperimetric type inequalities for some Monge–Ampère functionals

We investigate the stability of some inequalities of isoperimetric type related to Monge–Ampère functionals. In particular, firstly we prove the stability of a reverse Faber–Krahn inequality for the Monge–Ampère eigenvalue and its generalization. Then we give a stability result for the Brunn–Minkowski inequality and for a consequent Urysohn’s type inequality for the so-called \(n\) -torsional rigidity, a natural extension of the usual torsional rigidity.     

Functional inequalities and subordination: stability of Nash and Poincaré inequalities

We show that certain functional inequalities, e.g. Nash-type and Poincaré-type inequalities, for infinitesimal generators of C ₀ semigroups are preserved under subordination in the sense of Bochner. Our result improves earlier results by Bendikov and Maheux (Trans Am Math Soc 359:3085–3097, 2007, Theorem 1.3) for fractional powers, and it also holds for non-symmetric settings. As an application, we will derive hypercontractivity, supercontractivity and ultracontractivity of subordinate semigroups.     

Evolutionary derivation of Runge–Kutta pairs for addressing inhomogeneous linear problems

Numerical Algorithms - Two new Runge–Kutta (RK) pairs of orders 6(4) and 7(5) are presented for solving numerically the inhomogeneous linear initial value problems with constant coefficients....     

Methods for solving systems of linear and convex inequalities based on the Fejér principle

The paper contains a brief review of the author’s results concerning the technique of constructing Fejér contraction mappings, which are used in iterative processes of solving linear and convex systems of inequalities as well as accompanying optimization problems. The general approach is based on the notion of M-Fejér step “p → q” defined by the property

$$\left| {q - y} \right| < \left| {p - y} \right|,\forall y \in M$$

. This property (postulate) assumes the existence of a point p ∈ \(\overline {convM} \) and of a proper sufficiently arbitrary point q. Some of the problems considered in the paper are illustrated by schemes reflecting the analytics of these problems.

     

Invariance of functionals and related Euler–Lagrange equations

We establish a connection between symmetries of functionals and symmetries of the corresponding Euler–Lagrange equations. A similar problem is investigated for equations with quasi-B _u -potential operators.     

Fatou’s lemma and lower epi-limits of integral functionals

In this work, with the introduction in the $\sigma$-finite case of a modulus of equi-integrability, we prove some new extensions of Fatou’s lemma and some of its consequences in the convergence theory of integral functionals. We present the case of a sequence of integral functionals using an analog of Ioffe’s criterion for a sequence of integrands.     

Interpolation and Φ-moment inequalities of noncommutative martingales

This paper is devoted to the study of Φ-moment inequalities for noncommutative martingales. In particular, we prove the noncommutative Φ-moment analogues of martingale transformations, Stein’s inequalities, Khintchine’s inequalities for Rademacher’s random variables, and Burkholder–Gundy’s inequalities. The key ingredient is a noncommutative version of Marcinkiewicz type interpolation theorem for Orlicz spaces which we establish in this paper.     

Limit distribution “in the large” of linear functionals over a random process of the second order

An infinite-dimensional analogue of the theorem on the existence of an -typical distribution of linear functionals in a vector space of high dimension does not exist. There are given some additional conditions ensuring, nevertheless, the existence of a typical distribution of linear functionals in the infinite-dimensional case.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 97, pp. 199–202, 1980.The author is grateful to S. B. Makarov for pointing out and corresponding these errors.     

Asymptotic equivalence of linear stochastic Itô systems and oscillation of solutions of linear second-order equations

We obtain conditions for the asymptotic equivalence of linear stochastic and deterministic systems and analyze the oscillation of solutions of the Itô stochastic equation of the second order of the form \(\ddot x + (p(t) + q(t)\dot W(t))x = 0\) on the half-line.     

Nonlinear functionals of wavelet expansions – adaptive reconstruction and fast evaluation

Summary. This paper is concerned with the efficient evaluation of nonlinear expressions of wavelet expansions obtained through an adaptive process. In particular, evaluation covers here the computation of inner products of such expressions with wavelets which arise, for instance, in the context of Galerkin or Petrov Galerkin schemes for the solution of differential equations. The central objective is to develop schemes that facilitate such evaluations at a computational expense exceeding the complexity of the given expansion, i.e., the number of nonzero wavelet coefficients, as little as possible. The following issues are addressed. First, motivated by previous treatments of the subject, we discuss the type of regularity assumptions that are appropriate in this context and explain the relevance of Besov norms. The principal strategy is to relate the computation of inner products of wavelets with compositions to approximations of compositions in terms of possibly few dual wavelets. The analysis of these approximations finally leads to a concrete evaluation scheme which is shown to be in a certain sense asymptotically optimal. We conclude with a simple numerical example. Received June 25, 1998 / Revised version received June 5, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000     

Operator inequalities and reverse inequalities related to the Kittaneh–Manasrah inequalities

This work is concerned with exploring more refinement forms of the Young inequalities and the Kittaneh–Manasrah inequalities. We deduce the Operator version inequalities and reverse version inequalities related to the Kittaneh–Manasrah inequalities.