Vector Inequalities Research Articles

A Cauchy--Khinchin--van Dam matrix inequality

We derive a matrix inequality, which generalizes the Cauchy inequality for vectors, Khinchin's inequality for zero-one matrices and van Dam's inequality for matrices.

Some inequalities in inner product spaces related to the generalized triangle inequality

In this paper we obtain some inequalities related to the generalized triangle and quadratic triangle inequalities for vectors in inner product spaces. Some results that employ the Ostrowski discrete inequality for vectors in normed linear spaces are also obtained.

Weak and strong moments of random vectors

We discuss a conjecture about comparability of weak and strong moments of log-concave random vectors and show the conjectured inequality for unconditional vectors in normed spaces with a bounded cotype constant.

Chebyshev type inequalities for synchronous vectors in Banach spaces

Chebyshev type inequalities for synchronous vectors in Banach spaces

Some Trapezoidal Vector Inequalities for Continuous Functions of Selfadjoint Operators in Hilbert Spaces

On utilising the spectral representation of selfadjoint operators in Hilbert spaces, some trapezoidal inequalities for various classes of continuous functions of such operators are given.

Some Vector Inequalities for Continuous Functions of Self-Adjoint Operators in Hilbert Spaces

On utilizing the spectral representation of self-adjoint operators in Hilbert spaces, some inequalities for the composite operator , where and for various classes of continuous functions are given. Applications for the power function and the logarithmic function are also provided.

Dynamic system methods for solving mixed linear matrix inequalities and linear vector inequalities and equalities

A novel idea is proposed for solving a system of mixed linear matrix inequalities and linear vector inequalities and equalities. First, the problem is converted into an unconstrained minimization problem with a continuously differentiable convex objective function. Then, a continuous-time dynamic system and a discrete-time dynamic system are proposed for solving it. Under some mild conditions, the proposed dynamic systems are shown to be globally convergent to a solution of the problem. The merits of the methods refer to their simple numerical implementations and capability for handling nonstrict LMIs easily. In addition, the methods are promising in neural circuits realization, and therefore have potential applications in many online control problems. Several numerical examples are presented to illustrate the performance of the methods and substantiate the theoretical results.

Some inequalities for power series of selfadjoint operators in Hilbert spaces via reverses of the Schwarz inequality

In this paper, we obtain some operator inequalities for functions defined by power series with real coefficients and, more specifically, with nonnegative coefficients. In order to obtain these inequalities, some recent reverses of the Schwarz inequality for vectors in inner product spaces are utilized. Natural applications for some elementary functions of interest are also provided.

Relations on generalized degree sequences

We study degree sequences for simplicial posets and polyhedral complexes, generalizing the well-studied graphical degree sequences. Here we extend the more common generalization of vertex-to-facet degree sequences by considering arbitrary face-to-flag degree sequences. In particular, these may be viewed as natural refinements of the flag f -vector of the poset. We investigate properties and relations of these generalized degree sequences, proving linear relations between flag degree sequences in terms of the composition of rank jumps of the flag. As a corollary, we recover an f -vector inequality on simplicial posets first shown by Stanley.

Vector inequalities for powers of some operators in Hilbert spaces

Vector inequalities for powers of some operators in Hilbert spaces with applications for operator norm, numerical radius, commutators and self-commutators are given. .

Q-norm inequalities for sequences of Hilbert space operators

We give some inequalities related to a large class of operator norms, the so-called Qnorms, for a (not necessary commutative) family of bounded linear operators acting on a Hilbert space that are related to the classical Schwarz inequality. Applications for vector inequalities are also provided.

Some inequalities for the norm and the numerical radius of linear operators in Hilbert spaces

In this paper various inequalities between the operator norm and its numerical radius are provided. For this purpose, we employ some classical inequalities for vectors in inner product spaces due to Buzano, Goldstein-Ryff-Clarke, Dragomir-S ´andor and the author.

A CHEBYSHEV INEQUALITY FOR MULTIVARIATE NORMAL DISTRIBUTION

Chebyshev inequality estimates the probability for exceeding the deviation of a random variable from its mathematical expectation in terms of the variance of the random variable. In modern probability theory, the Chebyshev inequality is the most frequently used tool for proving different convergence processes; for example, it plays a fundamental role in proofs of various forms of laws of large numbers. The mathematical expression of the bound on the probability in the Chebyshev inequality is very simple and can be modified easily for different kinds of sequence of random variables (e.g., for the case of sums of independent random variables). This fact lies behind these frequent applications. In this setting, the Chebyshev inequality has pure theoretical “applications” in probability theory and its role is to provide “a guarantee” of convergence but not to give a bound on concrete probability content.In the present article we consider the Chebyshev inequality as a probability bound that is essential for the translation from its conventional theoretical applications to the practical setting if easy-to-compute multivariate generalizations are derived.Such an inequality for the random vectors having multivariate Normal distribution is proved. The new inequality gives a lower bound in terms of variances on the probability that the random vector in question falls into an Euclidean ball with center at mean vector. The need and importance of consideration of this kind of multivariate Chebyshev inequality stemmed from several problems in engineering and informational sciences (Hassibi and Boyd [9], Jeng [10], Jeng and Woods [11], Molina, Katseggelos, Mateos, Hermoso, and Segall [13]). Jeng [10] derived an inequality that gives an upper bound for the probability in question. The simultaneous application of the established multivariate Chebyshev inequality and Jeng's inequality is useful in practical problems by providing lower and upper bounds on the probability content.The inequality is attractive by its being easy to compute and its similarity to the original Chebyshev inequality, in contrast to well-known complicated multivariate Chebyshev inequalities. The present article also gives some insights into the very origin of the Chebyshev inequality, which makes the article self-contained.

G -majorization inequalities for linear maps, II

In this article, we give a unified approach to vector inequalities of the form , where L and K are linear maps on a linear space equipped with a G-majorization preordering ⪯ G , and (·)↓ is a special map corresponding to the preordering. The results obtained are interpreted for various classes of matrices. Some inequalities for singular values of matrices are discussed.

Mean-risk optimization for index tracking

SUMMARY This paper presents an analysis of the tracking problems of multiple indices with multidimensional performance criterion consisting of mean wealth and the tracking errors. We evaluate the performance of portfolios via the vector inequalities defined by convex cones, which enable us to describe various preference relations for investors. In Brownian market models with deterministic coefficients, we completely determine the set of efficient portfolios as well as the efficient frontier in our context. As a product of our analysis, we exhibit a version of Tobin's mutual fund theorem.

Coexistence, permanence and resilience in biological reaction-diffusion systems

Reaction-diffusion systems are widely used to describe spatio-temporal phenomena in a variety of scientific fields, including population ecology. In this paper, I demonstrate that existing results for coexistence and permanence of general Lotka-Volterra systems with absorbing boundaries can be applied in a complementary manner to address a variety of boundary conditions, including the insulating problem. Furthermore, the condition is applicable even to systems containing positive feedback mechanisms in the dynamics. A single (vector) inequality, the first iterate condition, is derived which serves as a sufficient condition for coexistence, permanence and resilience. Additionally, I demonstrate that this inequality condition is but the first in a series of conditions that can be used to describe the behaviour of such systems. Finally, I provide a comparison between the iterate conditions and an alternative test for solution resiliency.

Moment inequalities for B-valued random vectors with applications to the strong limit theorems

Analogues of the Marcinkiewicz–Zygmund, Rosenthal and Acosta inequalities for Banach space valued random vectors are proved. As an application some results on the complete convergence and the strong law of large numbers are obtained. Some well-known results are improved and extended.

Relationships Among Flag f -Vector Inequalities for Polytopes

We examine linear inequalities satisfied by the flag $f$-vectors of polytopes. One source of these inequalities is the toric $g$-vector; convolutions of its entries are non-negative for rational polytopes. We prove a conjecture of Meisinger about a redundancy in these inequalities. Another source of inequalities is the {\bf cd}-index; among all $d$-polytopes, each {\bf cd}-index coefficient is minimized on the $d$-simplex. We show that not all of the {\bf cd}-index inequalities are implied by the toric $g$-vector inequalities, and that not all of the toric $g$-vector inequalities are implied by the {\bf cd}-index inequalities. Finally, we show that some inequalities from convolutions of {\bf cd}-index coefficients are implied by other {\bf cd}-index inequalities.

On the tightness of the Zhang-Yeung inequality for Gaussian vectors

On the tightness of the Zhang-Yeung inequality for Gaussian vectors

Schur majorization inequalities for symmetrized sums with applications to tensor products

We show that if w≺y and x≺z are four vectors in Rn, then a number of Schur majorizations hold between “symmetrized” vector functions of w,x,y and z, e.g., (wi+xj)i,j≺(yi+zj)i,j where the left-hand expression means the vector of dimension n2 consisting of all sums wi+xj of the co-ordinates of w and x, arranged in lexicographic order. Among other things, we get vector and matrix versions of Muirhead’s theorem for scalar inequalities. From the vector inequalities follow many scalar inequalities for “symmetrized” sums, some of which are scattered through the inequality literature.In Section 2, applications are given to matrix inequalities for tensor products, e.g., if A,B and C are Hermitian and λ(A)≺λ(B), then λ(A⊗C)≺λ(B⊗C).