Multivariate majorization and rearrangement inequalities with some applications to probability and statistics

Multivariate generalizations of the concept of a Schur convex-function are defined and characterized. These characterizations are shown to be useful in obtaining majorization and rearrangement inequalities. We give simple derivations of known results as well as new ones with applications in probability and statistics.     

Distance majorization and its applications

The problem of minimizing a continuously differentiable convex function over an intersection of closed convex sets is ubiquitous in applied mathematics. It is particularly interesting when it is easy to project onto each separate set, but nontrivial to project onto their intersection. Algorithms based on Newton’s method such as the interior point method are viable for small to medium-scale problems. However, modern applications in statistics, engineering, and machine learning are posing problems with potentially tens of thousands of parameters or more. We revisit this convex programming problem and propose an algorithm that scales well with dimensionality. Our proposal is an instance of a sequential unconstrained minimization technique and revolves around three ideas: the majorization-minimization principle, the classical penalty method for constrained optimization, and quasi-Newton acceleration of fixed-point algorithms. The performance of our distance majorization algorithms is illustrated in several applications.     

Pointwise convergence of some conjugate convolution operators with applications to wavelets

In this paper, we discuss the pointwise convergence of conjugate convolution operators with some applications to wavelets. Some criteria of convergence at (C, 1) continuous points, Lebesgue points and almost everywhere are established.     

Pointwise symmetrization inequalities for Sobolev functions and applications

We develop a technique to obtain new symmetrization inequalities that provide a unified framework to study Sobolev inequalities, concentration inequalities and sharp integrability of solutions of elliptic equations.     

Pointwise products of some Banach function spaces and factorization

The well-known factorization theorem of Lozanovski? may be written in the form L¹≡E⊙E^′$L^1\equiv E\odot E^{\prime }$, where ⊙ means the pointwise product of Banach ideal spaces. A natural generalization of this problem would be the question when one can factorize F through E  , i.e., when F≡E⊙M(E,F)$F\equiv E\odot M(E,F)$, where M(E,F)$M(E,F)$ is the space of pointwise multipliers from E to F  . Properties of M(E,F)$M(E,F)$ were investigated in our earlier paper [41] and here we collect and prove some properties of the construction E⊙F$E\odot F$. The formulas for pointwise product of Calderón–Lozanovski? _Eφ$E_{\phi }$-spaces, Lorentz spaces and Marcinkiewicz spaces are proved. These results are then used to prove factorization theorems for such spaces. Finally, it is proved in Theorem 11 that under some natural assumptions, a rearrangement invariant Banach function space may be factorized through a Marcinkiewicz space.     

Four applications of majorization to convexity in the calculus of variations

The resemblance between the Horn-Thompson theorem and a recent theorem by Dacorogna-Marcellini-Tanteri indicates that Schur-convexity and the majorization relation are relevant for applications in the calculus of variations and its related notions of convexity, such as rank one convexity or quasiconvexity. In Theorem 6.6, we give simple necessary and sufficient conditions for an isotropic objective function to be rank one convex on the set of matrices with positive determinant.Majorization is used in order to give a very short proof of a theorem of Thompson and Freede [R.C. Thompson, L.J. Freede, Eigenvalues of sums of Hermitian matrices III, J. Res. Nat. Bur. Standards B 75B (1971) 115-120], Ball [J.M. Ball, Constitutive inequalities and existence theorems in nonlinear elastostatics, in: R.J. Knops (Ed.), Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, vol. 1, Res. Notes Math., 17, Pitman, 1977, pp. 187-241], or Le Dret [H. Le Dret, Sur les fonctions de matrices convexes et isotropes, CR Acad. Sci. Paris, Série I 310 (1990) 617-620], concerning the convexity of a class of isotropic functions which appear in nonlinear elasticity.Next we prove (Theorem 7.3) a lower semicontinuity result for functionals with the form _∫Ωw(D?(x))dx, with w(F)=h(lnV_F). Here F=R_FU_F=V_FR_F is the usual polar decomposition of F∈gl(n,R), and lnV_F is Hencky’s logarithmic strain.We close this paper with a compact proof of Dacorogna-Marcellini-Tanteri theorem, based only on classical results about majorization. The mentioned resemblance of this theorem with the Horn-Thompson theorem is thus explained.     

A relation of weak majorization and its applications to certain inequalities for means

A relation of weak majorization for n-dimensional real vectors is established, the result is then used to derive some inequalities involving the power mean, the arithmetic mean and the geometric mean in n variables.     

Sums of powers and majorization

We study certain sequences involving sums of powers of positive integers and in connection with this, we give examples to show that power majorization does not imply majorization.     

Pointwise estimates for some kernels on compact Lie groups

Si ottengono delle stime per il nucleo di Gauss-Weierstrass su gruppi di Lie compatti e se ne indicano alcune applicazioni.     

Pointwise direct estimate for combinations of some exponential-type operators

The purpose of this paper is to give the pointwise direct estimate for the combinations of some exponential-type operators.     

Pointwise direct estimate for combinations of some exponential-type operators

The purpose of this paper is to give the pointwise direct estimate for the combinations of some exponential-type operators. Supported by the Zheijiang Provincial Natural Science Foundation of China.     

The extreme points of some convex sets in the theory of majorization

Let (A, %plane1D;49C;, μ) be a finite measure space, and let Ω_{µ, w}⁺f denote the set of all nonnegative real-valued %plane1D;49C;-measurable functions on A weaklymajorized by a nonnegative function f, in the sense of Hardly, Littlewood and Pólya. For a nonatomic µ, the extreme points ofΩ_{µ, w} ⁺f are shown to be the nonnegativefunctions obtained by taking a fraction (1−θ) of the largest values of and arranging them in any way on any subset of A of measure(1−θ), with values elsewhere set equal to zero. Topological properties of these extreme points are given.     

Inverse Problems with Pointwise Overdetermination for some Quasilinear Parabolic Systems

In the article, we examine well-posedness questions in the Sobolev spaces of the inverse source problem in the case of a quasilinear parabolic system of the second order. The main part of the operator is linear. The overdetermination conditions are values of a solution at some collection of interior points. It is demonstrated that, in the case of at most linear growth of the nonlinearity, there exists a unique global (in time) solution and the problem is well-posed in the Sobolev classes. The conditions on the data are minimal and the results are sharp.     

On majorization and Schur products

Suppose A, D₁,…,D_m are n × n matrices where A is self-adjoint, and let $\text{X = Σ}{}^{\mathrm{m}}{}_{\mathrm{k\; =\; 1}}\text{D}{}_{\mathrm{k}}\text{AD}{}^1{}_{\mathrm{k}}$. It is shown that if $\text{ΣD}{}_{\mathrm{k}}\text{D}{}^1{}_{\mathrm{k}}\text{= ΣD}{}^1{}_{\mathrm{k}}\text{D}{}_{\mathrm{k}}\text{= I}$, then the spectrum of X is majorized by the spectrum of A. In general, without assuming any condition on D₁,…,D_m, a result is obtained in terms of weak majorization. If each D_k is a diagonal matrix, then X is equal to the Schur (entrywise) product of A with a positive semidefinite matrix. Thus the results are applicable to spectra of Schur products of positive semidefinite matrices. If A, B are self-adjoint with B positive semidefinite and if b_ii = 1 for each i, it follows that the spectrum of the Schur product of A and B is majorized by that of A. A stronger version of a conjecture due to Marshall and Olkin is also proved.     

Idempotent Boolean matrices and majorization

We obtain a new structural characterization of idempotent Boolean matrices. This characterization allows us to describe all Boolean matrices that are majorized by a given idempotent. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 1, pp. 11–29, 2007.     

Substochastic matrices and von Neumann majorization

Several classes of substochastic matrices are introduced in higher dimensions, and some theorems about their extreme points are presented. An extension of a theorem of von Neumann concerning doubly substochastic matrices is discussed, and the classes for which this extension remains valid are determined.     

Majorization versus power majorization

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