Vector Inequalities Research Articles

Inequalities generated by the gamma function

We prove sharp inequalities for arbitrary complex vectors and weights generated by the gamma function. Some limiting cases and applications are discussed. They include general combinatorial, exponential, and integral inequalities.

Stability of solutions to parameterized nonlinear complementarity problems

such that x≥0, F(x,u)-v≥0 , and F(x,u)-v T·x=0 where these are vector inequalities. We characterize the local upper Lipschitz continuity of the (possibly set-valued) solution mapping which assigns solutions x to each parameter pair (v,u). We also characterize when this solution mapping is locally a single-valued Lipschitzian mapping (so solutions exist, are unique, and depend Lipschitz continuously on the parameters). These characterizations are automatically sufficient conditions for the more general (and usual) case where v=0. Finally, we study the differentiability properties of the solution mapping in both the single-valued and set-valued cases, in particular obtaining a new characterization of B-differentiability in the single-valued case, along with a formula for the B-derivative. Though these results cover a broad range of stability properties, they are all derived from similar fundamental principles of variational analysis.

What Is the Economic Meaning of FDH

The central feature of the FDH model is the lack of convexity for its production possibility set, TF. Starting with n observed (distinct) decision making units DMUk , each defined by an input-output vector p k = [y k -x k], domination is defined by ordinary vector inequalities. DMUk is said to dominate DMUj if p k ≥ p j , p k ≠ p j . The FDH production possibility set TF consists of the observed DMUj together with all input-output vectors p=[yk,−xk] with y ≥ 0, x ≥ 0, y ≠ 0, x ≠ 0 which are dominated by at least one of the observed DMUj. DMUk is defined as “FDH efficient” if no DMUj dominates it. In the BCC (or variable return to scale) DEA model the production possibility set TB consists of the observed DMUk together with all input-output vectors dominated by any convex combination of them and DMUk is DEA efficient if it is not dominated by any p in TB. In the DEA model, economic meaning is established by the introduction of (non negative) multiplier (price) vectors w = [u,v]. If DMUk is undominated (in TB) then there exists a positive multiplier vector w for which (a) w T p k = u T y k − v T x k ≥ w T p for every p ∈ TB. In everyday language, the net return (or profit) for DMUk relative to the given multiplier vector w is at least as great as that for any production possibility p. On the other hand, if DMUk is FDH but not DEA efficient then it is proved that there exists no positive multiplier vector >w for which (a) holds, i.e. for any positive w there exists at least one DMUj for which w T p j > w T p k . Since, therefore, FDH efficiency does not guarantee price efficiency what is its economic significance? Without economic significance, how can FDH be considered as being more than a mathematical system however logically soundly it may be conceived?

Iterative algorithms for deblurring and deconvolution with constraints

The deblurring problem is that of recovering the function from (noisy) values . The discrete finite version of the problem is to solve the system of linear equations Hc=d for c, where H is a matrix and d and c are vectors. When the kernel is a function of the difference , the deblurring problem becomes a deconvolution problem. The use of iterative algorithms to effect deblurring subject to non-negativity constraints on c has been presented by Snyder et al for the case of non-negative kernel function h. In this paper we extend these algorithms to include upper and lower bounds on the entries of the desired solution. We show that any linear deblurring problem involving a real kernel h can be transformed into a linear deblurring problem involving a non-negative kernel. Therefore our algorithms apply to general deblurring and deconvolution problems. These algorithms converge to a solution of the system of equations y = Px, with , for , , satisfying the vector inequalities , whenever such a solution exists. When there is no solution satisfying the constraints the simultaneous versions converge to an approximate solution that minimizes a cost function related to the Kullback-Leibler cross-entropy and the Fermi-Dirac generalized entropy.

A Cauchy-Khinchin matrix inequality

We derive a matrix inequality, which generalizes the Cauchy-Schwarz inequality for vectors, and Khinchin's inequality for zero–one matrices. Furthermore, we pose a related problem on the maximum irregularity of a directed graph with prescribed number of vertices and arcs, and make some remarks on this problem.

Invariants of equivariant algebraic vector bundles and inequalities for dominant weights

Invariants of equivariant algebraic vector bundles and inequalities for dominant weights

A Linear Programming Approach for Regional Pole Placement Under Pointwise Constraints

This paper considers the problem of control of discrete-time linear systems under several complicating features frequently encountered in practice : uncertain parameters in the model, additive noise, linear symmetrical state and input constraints. The main control design objective, beyond local stabilization, is to locate the closed-loop poles in a ”good” region of the unit disc of the complex plane. The originality of the proposed approach is to use Linear Programming as the basic design tool. It is shown that the conditions derived from positive invariance relations and design constraints can be formulated as linear matrix equalities and vector inequalities. The control design problem can then be solved by Linear Programming, with an objective function describing the search for a trade-off between several design objectives.

Inequalities for stochastic models via supermodular orderings

The aim of this paper is to derive inequalities for random vectors by using the supermodular ordering. The properties of this ordering suggest to use it as a comparison for the “ strength of dependence” in random vectors. In contrast to already established orderings of this type, the supermodular ordering has the advantage that it is not necessary to assume a common marginal distribution for the random vectors under comparison. As a consequence we obtain new inequalities by applying it to multivariate normal distributions, Markov chains and some stochastic models

Vector Analysis of Keyblock Rotations

The mechanical response of rock slopes and excavations depends largely on the geometry of discontinuities and associated rock blocks. Commonly used rigid block models for determining stability, such as the sliding wedge model or the keyblock analysis, assume that initial displacements are pure translations. This paper examines kinematic and kinetic constraints on block rotations. The kinematic constraints include conditions that are independent of the free surface and therefore belong uniquely to the joint pyramid, as well as conditions that depend on the orientation of the free surface. The relevant criteria, expressed in terms of vector inequalities, are represented on the stereographic projection, allowing graphical solution of the kinematical inequalities. Additionally, rotational equilibrium of tetrahedral blocks is considered, the failure modes of block theory are generalized to include rotational modes and procedures for evaluating rotational stability are discussed. The results show that blocks found to be stable with respect to the translational modes of sliding and falling may yet fail by rotation.

Error-energy bounds for adaptive gradient algorithms

The paper establishes robustness, optimality, and convergence properties of the widely used class of instantaneous-gradient adaptive algorithms. The analysis is carried out in a purely deterministic framework and assumes no a priori statistical information. It employs the Cauchy-Schwarz inequality for vectors in an Euclidean space and derives local and global error-energy bounds that are shown to highlight, as well as explain, relevant aspects of the robust performance of adaptive gradient filters (along the lines of H/sup /spl infin// theory).

Cayley's theorem and its application in the theory of vector Padé approximants

Le A be a matrix of even dimension which is anti-symmetric after deletion of its rth row and column and let R, C be the anti-symmetric matrices formed by modifying the rth row and column of A, respectively. In this case, Cayley's (1857) theorem states that det A = Pf R · Pf C, where Pf R denotes the Pfaffian of R. A consequence of this theorem is an explicit factorisation of the standard determinantal representation of the denominator polynomial of a vector Padé approximant. We give a succinct, modern proof of Cayley's theorem. Then we prove a novel vector inequality arising from investigation of one such Pfaffian, and conjecture that all such Pfaffians are nonnegative.

The geometric vector inequality and its properties

The present paper contains properties of the geometric vector inequality, introduced in [4] and [12]. Such properties are of interest not only in itself, but they are also useful for a systematic treatment of special structured vector optimization problems, denoted as geometric vector optimization problems. Especially, a close connection to the Fenchel vector conjugate as well as to the Legendre vector transform will be shown

Vector approximation problems in geometric vector optimization

Using the geometric vector inequality, introduced by Elster/Elster/Go pfert [9], we consider a special case of that inequality which is a basis for treating certain classes of (nonsmooth) vector optimization problems: vector curve fitting problemslp -vector approximation problems and vector regression problems. By the introduced geometric vector inequality we obtain in a natural way dual problems for such special-structured vector optimization problems, which can be solved in an efficient manner, since the dual constraints are linear.

On the discrepancy of coloring finite sets

Given a subset S of {1, …, n} and a map X : {1, …, n} → {−1, 1}, (i.e. a coloring of {1, …, n} with two colors, say red and blue) define the discrepancy of S with respect to X to be (the difference between the reds and blues on S). Given n subsets of {1, …, n}, a question of Erdos was to find a coloring of {1, …, n} which simultaneously minimized the discrepancy of the n subsets. We give new and simple proofs of some of the results obtained previously on this problem via an inequality for vectors.Corrigendum to “On the discrepancy of coloring finite sets”dx.doi.org/10.1155/S0161171291000534

The CS Language Concept: A New Approach to Robot Motion Design

Current robot control techniques are primarily concerned with position control and, recently, with force control, where a motion planner is used primarily to compute set point changes. The actual control system is fixed and internal and cannot be modified easily through software. Current robot languages are also limited in their ability to specify changes to the control structure. For compliant motion, such as as sembly, more flexibility is needed in the control system, that is, an ability to tailor the controller to the task. This paper develops a high-level control system (CS) language that allows one to specify compliant control tasks as vector equa tions and inequalities that relate sensed and controlled vari ables such as f · v = 0 and f X v = 0 . We show how such requirements can be reformulated as an objective function of an optimization process subject to constraints arising from the dynamic equations involved. Specifically for the two examples above, we show how such requirements can be re formulated as quadratic objective functions and solved using standard linear optimization theory for linear dynamic sys tems. Six examples are presented to illustrate the various ideas.

Multiple Comparisons of Polynomial Distributions

AbstractAsymptotically correct 90 and 95 percentage points are given for multiple comparisons with control and for all pair comparisons of several independent samples of equal size from polynomial distributions. Test statistics are the maxima of the X2‐statistics for single comparisons. For only two categories the asymptotic distributions of these test statistics result from DUNNETT'S many‐one tests and TUKEY'S range test (cf. MILLER, 1981).The percentage points for comparisons with control are computed from the limit distribution of the test statistic under the overall hypothesis H0. To some extent the applicability of these bounds is investigated by simulation. The bounds can also be used to improve Holm's sequentially rejective Bonferroni test procedure (cf. HOLM, 1979).The percentage points for all pair comparisons are obtained by large simulations. Especially for 3×3‐tables the limit distribution of the test statistic under H0 is derived also for samples of unequal size. Also these bounds can improve the corresponding Bonferroni‐Holm procedure.Finally from SKIDÁK's probability inequality for normal random vectors (cf. SKIDÁK, 1967) a similar inequality is derived for dependent X2‐variables applicable to simultaneous X2‐tests.

Elementary Characterization of Classical Minima

many introductory treatments and is consequently ignored). It is therefore surprising that several optima of classical interest usually treated through the calculus of variations can be characterized directly by analysis much more elementary than the Euler-Lagrange theory. We shall substantiate this assertion by using little more than the Cauchy-Schwarz vector inequality to provide direct access to the solutions of several well-known problems, including certain of those associated with the brachistochrone and the minimal surface. We begin with two almost trivial examples from the classical literature [2, pp. 178-180, 201], namely those of determining geodesics in space and on the surface of a sphere. To find the shortest space curve joining fixed points P0(= 0) and PI, we must minimize I jP'(t)ldt

Inequalities for Random Vectors

Inequalities for Random Vectors

Semi-Compactness with Respect to a Euclidean Cone

Our motivation for this note originates with consideration of a subset A of Euclidean w-space, Rn, which contains only part of its boundary. The part contained is t h a t part of the closure of A which cannot be “bettered“ within A with respect to the preference associated with a fixed closed convex cone Γ. Here b is preferred to a if and only if a — b ∊ Γ; if, for instance, Γ is the non-negative orthant of Rn, this preference is ordinary vector inequality.

A Hajek-Renyi Type Inequality for Stochastic Vectors with Applications to Simultaneous Confidence Regions

For a sequence of stochastic vectors forming either a forward or a reverse martingale, a Hajek-Renyi type inequality is derived, and its applications in some problems of simultaneous confidence regions are stressed.