On the converse Jensen inequality

We give a survey on the converse Jensen inequality and we show that several recently published inequalities are simple consequences of certain long time known results. We also give a new refinement of the converse Jensen inequality as well as improvements of some related results.     

Jensen matrix inequalities and direct sums

Let A, X and Y be n-by-n complex matrices such that A is positive semi-definite and X, Y are contractions. We prove that if f is an increasing convex function on [0, ∞) such that f(0) ≤ 0, then the eigenvalues of f(|X*AY|) are dominated by those of X*f(A)X⊕Y* f(A)Y. Several related results are considered.     

Singular value inequalities for matrix sums and minors

This paper gives new proofs for certain inequalities previously established by the author involving sums of singular values of matrices A, B, C = A + B, and also sums of singular values of A, B, and C when A, B are complementary submatrices of C. Some new facts concerning these inequalities are also included.     

A Harnack inequality for Dirichlet eigenvalues

We prove a Harnack inequality for Dirichlet eigenfunctions of abelian homogeneous graphs and their convex subgraphs. We derive lower bounds for Dirichlet eigenvalues using the Harnack inequality. We also consider a randomization problem in connection with combinatorial games using Dirichlet eigenvalues. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 247–257, 2000     

A matrix trace inequality and its application

In this short paper, we give a complete and affirmative answer to a conjecture on matrix trace inequalities for the sum of positive semidefinite matrices. We also apply the obtained inequality to derive a kind of generalized Golden-Thompson inequality for positive semidefinite matrices.     

Optimal young's inequality and its converse: a simple proof

We give a new proof of the sharp form of Young's inequality for convolutions, first proved by Beckner [Be] and Brascamp-Lieb [BrLi]. The latter also proved a sharp reverse inequality in the case of exponents less than 1. Our proof is simpler and gives Young's inequality and its converse altogether. Submitted: March 1997, Final version: April 1997     

Principal minors, Part I: A method for computing all the principal minors of a matrix

An order O(2ⁿ) algorithm for computing all the principal minors of an arbitrary n × n complex matrix is motivated and presented, offering an improvement by a factor of n³ over direct computation. The algorithm uses recursive Schur complementation and submatrix extraction, storing the answer in a binary order. An implementation of the algorithm in MATLAB® is also given and practical considerations are discussed and treated accordingly.     

A cone programming approach to the bilinear matrix inequality problem and its geometry

We discuss an approach for solving the Bilinear Matrix Inequality (BMI) based on its connections with certain problems defined over matrix cones. These problems are, among others, the cone generalization of the linear programming (LP) and the linear complementarity problem (LCP) (referred to as the Cone-LP and the Cone-LCP, respectively). Specifically, we show that solving a given BMI is equivalent to examining the solution set of a suitably constructed Cone-LP or Cone-LCP. This approach facilitates our understanding of the geometry of the BMI and opens up new avenues for the development of the computational procedures for its solution. Research supported in part by the National Science Foundation under Grant CCR-9222734.     

On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues

We prove that the Hersch-Payne-Schiffer isoperimetric inequality for the nth nonzero Steklov eigenvalue of a bounded simply connected planar domain is sharp for all n ⩾ 1. The equality is attained in the limit by a sequence of simply connected domains degenerating into a disjoint union of n identical disks. Similar results are obtained for the product of two consecutive Steklov eigenvalues. We also give a new proof of the Hersch-Payne-Schiffer inequality for n = 2 and show that it is strict in this case.     

Ona-Wright convexity and the converse of Minkowski's inequality

Summary Leta (0, 1/2] be fixed. A functionf satisfying the inequalityf(ax + (1 – a)y) + f((1 – a)x + ay) f(x) + f(y), called herea-Wright convexity, appears in connection with the converse of Minkowski's inequality. We prove that every lower semicontinuousa-Wright convex function is Jensen convex and we pose an open problem. Moreover, using the fact that 1/2-Wright convexity coincides with Jensen convexity, we prove a converse of Minkowski's inequality without any regularity conditions.     

A strong converse inequality for left gamma quasi-interpolants in L p-spaces

Recently Müller [6] introduced the left Gamma quasi-interpolants and obtained an approximation equivalence theorem for them. We show the strong converse inequality of type B for the left Gamma quasi-interpolants.     

A correlational inequality for linear extensions of a poset

Suppose 1, 2, and 3 are pairwise incomparable points in a poset onn≥3 points. LetN (ijk) be the number of linear extensions of the poset in whichi precedesj andj precedesk. Define λ by $$\lambda = \frac{{N(213)N(312)}}{{\left[ {N(123) + N(132)} \right]\left[ {N(231) + N(321)} \right]}}$$ Two applications of the Ahlswede-Daykin evaluation theorem for distributive lattices are used to prove that λ?(n?1)²/(n+1)² for oddn, and λ?(n?2)/(n+2) for evenn. Simple examples show that these bounds are best-possible. Shepp (Annals of Probability, 1982) proved thatP(12)?P(12/13), the so-calledxyz inequality, whereP(ij) is the probability thati precedesj in a randomly chosen linear extension of the poset, thus settling a conjecture of I. Rival and B. Sands. The preceding bounds on λ yield a simple proof ofP(12)<P(12/13), which had also been conjectured by Rival and Sands.     

Some inequalities related to the Stam inequality

Zamir showed in 1998 that the Stam classical inequality for the Fisher information (about a location parameter)

Strong converse inequalities for Meyer-König and Zeller operators

In this paper we give the strong converse inequality of type B for Meyer-König and Zeller operators.     

A computational method for eigenvalues and eigenvectors of a matrix with real eigenvalues

This paper describes a new computational procedure for calculating eigenvalues and eigenvectors of a square matrix. The method is based on a matrix function, the sign of a matrix. Eigenvalues and eigenvectors of matrices with distinct eigenvalues and nondefective matrices with repeated roots can be determined in a straightforward manner. Defective matrices require additional calculations.     

Contact equations, Lipschitz extensions and isoperimetric inequalities

We characterize locally Lipschitz mappings and existence of Lipschitz extensions through a first order nonlinear system of PDEs. We extend this study to graded group-valued Lipschitz mappings defined on compact Riemannian manifolds. Through a simple application, we emphasize the connection between these PDEs and the Rumin complex. We introduce a class of 2-step groups, satisfying some abstract geometric conditions and we show that Lipschitz mappings taking values in these groups and defined on subsets of the plane admit Lipschitz extensions. We present several examples of these groups, called Allcock groups, observing that their horizontal distribution may have any codimesion. Finally, we show how these Lipschitz extensions theorems lead us to quadratic isoperimetric inequalities in all Allcock groups.     

Distribution of eigenvalues and eigenvectors of Wishart matrix when the population eigenvalues are infinitely dispersed and its application to minimax estimation of covariance matrix

We consider the asymptotic joint distribution of the eigenvalues and eigenvectors of Wishart matrix when the population eigenvalues become infinitely dispersed. We show that the normalized sample eigenvalues and the relevant elements of the sample eigenvectors are asymptotically all mutually independently distributed. The limiting distributions of the normalized sample eigenvalues are chi-squared distributions with varying degrees of freedom and the distribution of the relevant elements of the eigenvectors is the standard normal distribution. As an application of this result, we investigate tail minimaxity in the estimation of the population covariance matrix of Wishart distribution with respect to Stein's loss function and the quadratic loss function. Under mild regularity conditions, we show that the behavior of a broad class of tail minimax estimators is identical when the sample eigenvalues become infinitely dispersed.     

The determinant of the scattering matrix and its relation to the number of eigenvalues

For a large class of scattering systems we study the behavior of the determinant of the scattering matrix as a function on the spectrum of the unperturbed operator. The variation of this determinant is related to the number of eigenvalues due to the perturbation. This relation generalizes results of Levinson and others. The range of physical systems to which these results apply is thus considerably extended.