Conditions For Positive Definiteness Research Articles

Beta functions of Bruhat-Tits buildings and deformation of l{sup 2} on the set of p-adic lattices

For the space Lat{sub n} of all lattices in an n-dimensional p-adic linear space an analogue of the matrix beta function is constructed; this beta function can degenerate to the Tamagawa zeta function. An analogue of Berezin kernels for Lat{sub n} is proposed. Conditions for the positive-definiteness of these kernels and an explicit Plancherel's formula are obtained.

On non-linear stability in unconstrained non-linear elasticity

A method of obtaining a full three-dimensional non-linear Hadamard stability analysis of inhomogeneous deformations of arbitrary, unconstrained, hyperelastic materials is presented. The analysis is an extension of that given by Chen and Haughton (Proc. Roy. Soc. London A 459 (2003) 137) for two-dimensional incompressible problems. The process that we present replaces the second variation condition expressed as an integral involving a quadratic in three arbitrary perturbations, with an equivalent sixth-order system of ordinary differential equations. The positive definiteness condition is thereby reduced to the simple numerical evaluation of zeros of a well-behaved function. The general theory is illustrated by applying it to the problem of the inflation of a thick-walled spherical shell. The present analysis provides a simpler alternative approach to bifurcation problems approached by using the incremental equations of non-linear elasticity.

Stability and bifurcation of inflation of elastic cylinders

A method of obtaining a full (two-dimensional) nonlinear stability analysis of inhomogeneous deformations of arbitrary incompressible hyperelastic materials is presented. The analysis that we develop replaces the second variation condition expressed as an integral involving two arbitrary perturbations, with an equivalent (third-order) system of ordinary differential equations. The positive-definiteness condition is thereby reduced to the simple numerical evaluation of zeros of a well-behaved function. The general theory is illustrated by applying it to the problem of the inflation of axially stretched thick-walled tubes. The bifurcation theory of such deformations is well known and we compare the bifurcation results with the new stability analysis.

A Multivariate Generalized Autoregressive Conditional Heteroscedasticity Model With Time-Varying Correlations

In this article we propose a new multivariate generalized autoregressive conditional heteroscedasticity (MGARCH) model with time-varying correlations. We adopt the vech representation based on the conditional variances and the conditional correlations. Whereas each conditional-variance term is assumed to follow a univariate GARCH formulation, the conditional-correlation matrix is postulated to follow an autoregressive moving average type of analog. Our new model retains the intuition and interpretation of the univariate GARCH model and yet satisfies the positive-definite condition as found in the constant-correlation and Baba–Engle–Kraft–Kroner models. We report some Monte Carlo results on the finite-sample distributions of the maximum likelihood estimate of the varying-correlation MGARCH model. The new model is applied to some real data sets.

An Interior Point Constrained Trust Region Method for a Special Class of Nonlinear Semidefinite Programming Problems

In this paper, an interior point trust region algorithm for the solution of a class of nonlinear semidefinite programming (SDP) problems is described and analyzed. Such nonlinear and nonconvex programs arise, e.g., in the design of optimal static or reduced order output feedback control laws and have the structure of abstract optimal control problems in a finite dimensional Hilbert space. The algorithm treats the abstract states and controls as independent variables. In particular, an algorithm for minimizing a nonlinear matrix objective functional subject to a nonlinear SDP-condition, a positive definiteness condition, and a nonlinear matrix equation is considered. The algorithm is designed to take advantage of the structure of the problem. It is an extension of an interior point trust region method to nonlinear and nonconvex SDPs, with a special structure which applies sequential quadratic programming techniques to a sequence of barrier problems and uses trust regions to ensure robustness of the iteration. Some convergence results are given, and, finally, several numerical examples demonstrate the applicability of the considered algorithm.

Transformation of quadratic forms to perfect squares for broken extremals

In this paper we study a quadratic form which corresponds to an extremal with piecewise continuous control in variational problems. This form, compared with the classical one, has some new terms connected with the set Θ of all points of discontinuity of the control. Its positive definiteness is a sufficient optimality condition for broken extremals. We show that if there exists a solution to corresponding Riccati equation satisfying some jump condition at each point of the set Θ, then the quadratic form can be transformed to a perfect square, just as in the classical case. As a result we obtain sufficient conditions for positive definiteness of the quadratic form in terms of the Riccati equation and hence, sufficient optimality conditions for broken extremals.

Simultaneously stabilizing controller design for a class of MIMO systems

It is shown that a class of linear, time-invariant, multi-input multi-output plants can be simultaneously stabilized. This class of plants all have the same number of zeros at infinity, at zero, or both, but no other zeros in the unstable region. If they have zeros at zero or infinity, then their gain matrices at zero and infinity also satisfy a positive-definiteness condition. There is no restriction on the poles of the plants considered in this class. An explicit design procedure is proposed to achieve simultaneously stabilizing controllers. All simultaneously stabilizing controllers for this class of plants are also characterized in terms of a parameter matrix that satisfies a unimodularity condition.

Solving the General Truncated Moment Problem by the r-Generalized Fibonacci Sequences Method

We give in this paper a new method for solving the generalized truncated power moment problem. To this aim we use r-generalized Fibonacci sequences and their associated minimal polynomials. We provide an algorithm of construction of solutions in a short method. This method allows us to avoid any appeal to Hankel matrices or any positive definiteness conditions as in the Flessas–Burton–Whitehead (FBW) approach. Examples and general cases are discussed.

A Multivariate GARCH Model with Time-Varying Correlations

In this paper we propose a new multivariate GARCH model with time-varying correlations. We adopt the vech representation based on the conditional variances and the conditional correlations. While each conditional-variance term is assumed to follow a univariate GARCH formulation, the conditional-correlation matrix is postulated to follow an autoregressive moving average type of analogue. By imposing some suitable restrictions on the conditional-correlation-matrix equation, we construct a MGARCH model in which the conditional-correlation matrix is guaranteed to be positive definite during the optimisation. Thus, our new model retains the intuition and interpretation of the univariate GARCH model and yet satisfies the positive-definite condition as found in the constant-correlation and BEKK models. We report some Monte Carlo results on the finite-sample distributions of the MLE of the varying-correlation MGARCH model. The new model is applied to some real data sets. It is found that extending the constant-correlation model to allow for time-varying correlations provides some interesting time histories that are not available in a constant-correlation model.

Conditions for positive definiteness of M-matrices

The present paper concentrates on conditions that are necessary and sufficient for M-matrices to be positive definite. The obtained results can be used in the analysis of productivity of the Leontief input-output model.

Uniqueness and continuous data dependence in the elastic cylinder

Selected continuous data dependence is established for the linearised elastic cylinder under relaxed positive-definite and homogeneity conditions on elasticities in the superposed linear deformation and for constrained sets of the solution. The proofs are based on those in [Math. Proc. Cambridge Philos. Soc. 103 (1988) 535] and employ a Lagrange identity obtained from Betti's reciprocal theorem. The treatment is intended only to be indicative of the method and is restricted to dependence on body force, certain base data, and the superposed elasticities in the half-length of the finite cylinder. Estimates are also presented for the semi-infinite cylinder under alternative boundary conditions. Uniqueness of the solution is also established.

Linear Hamiltonian systems on time scales: Positivity of quadratic functionals

In this work, we consider a linear Hamiltonian system x Δ = A t x σ + B t u, u Δ =− C t x σ − A t T u (H) on an arbitrary time scale ℸ, which allows one (among others) • to treat both continuous and discrete linear Hamiltonian systems (as the special cases for T = R and T = Z ) within one theory; • to explain the discrepancies between these two theories while studying systems of form (H). As a main result, we prove that disconjugacy of system (H) is a sufficient condition for positive definiteness of the quadratic functional associated with (H), The principal tool is the Picone identity on ℸ. We derive also the corresponding Wronskian identity, Riccati equation in this general setting on time scales.

Impact of correlation and induced correlation on the estimation of project cost of buildings

Treatment of correlation between variables is necessary for deriving any theoretical distribution of the project cost of buildings. This paper highlights some often ignored theoretical requirements necessary for a rigorous treatment of correlations. The condition for a positive definite correlation matrix is described, along with an analytical procedure and a computer program developed to verify the positive definite condition when correlation coefficients between input variables are estimated using historical data. The analytical procedure and the developed computer program can be used in any application that obtains correlation information from historical data or as subjective judgements to be used in a functional relationship. A new concept called induced correlation is suggested to define and to treat correlation between derived variables that arise from the common (shared) primary variables in their functional forms. A published numerical example is used to highlight the stages where correlation between variables can have an impact on the estimation of moments (estimated expected value and estimated standard deviation) for the project cost of buildings, and to demonstrate the improvement in the estimation of the standard deviation of project cost as a result of treating correlations in risk analysis.

Convergence analysis of the multiple-channel filtered-U recursive LMS algorithm for active noise control

The active noise control in large-dimension ducts usually requires the use of a multiple-channel adaptive system with several error sensors, secondary sources and reference sensors. Here we present the convergence analysis of the multiple-channel filtered-U recursive LMS algorithm, which employs adaptive IIR filters to improve the cancellation capabilities with respect to those of the corresponding FIR case, namely, the multiple-channel filtered-X LMS. A positive-definite condition is found to be sufficient to ensure convergence. Such a condition includes also the effects of the error path modeling mismatch, extending and generalizing previous results existing in the literature.

An area modulus of elasticity: Definition and properties

A new modulus of elasticity is defined to be the ratio of an equibiaxial stress to the relative area change in the planes in which the stress acts. This area modulus of elasticity is intermediate in properties between Young's modulus and the bulk modulus. Expressions for the area modulus are computed in isotropic elasticity. A simple, convenient expression for the compliance tensor of transverse isotropy is found in terms of, amongst others, the longitudinal (axial) area modulus and this leads to a new, concise condition for positive definiteness of the compliance tensor. The limits of incompressibility, inextensibility and constant area are briefly considered.

Portfolio Variance and Correlation Matrices

Executive Summary. The fact that portfolio variances must be positive implies that the correlation matrix for asset returns should be positive definite. As such, a test for positive definiteness in the correlation matrix should be routinely implemented prior to any portfolio optimization exercise. A program is developed to test for positive definiteness in correlation matrices and to detect correlation coefficients that violate the positive definiteness condition. The correlation matrices from five randomly selected papers are tested and violations revealed in larger correlation matrices. Where violations are detected, the portfolio manager or analyst should proceed with the optimization exercise only if the alternative bounds on the correlation coefficients computed by our program are acceptable.

State-space solution to the periodic multirate H/sub ∞/ control problem: a lifting approach

A two-Riccati equation solution to the H/sub /spl infin// control problem for periodic multirate systems is derived via the lifting method. The solution is expressed in terms of two algebraic Riccati equations. The causality constraints are represented by a set of positive definiteness conditions and coupling criteria. As a by-product, the study shows that there is a close connection between the solution of the causally constrained lifted problem and the solution obtained by solving the problem directly as a periodic H/sub /spl infin// problem defined for the multirate system.

Positive-definite conditions of elastic constants of two-dimensional quasicrystals with noncrystallographic symmetries.

In the present work, the positive-definite conditions of the elastic constants of two-dimensional (2D) quasicrystals with non-crystallographic symmetries have been derived. These results are necessary for studying elastic and mechanical properties of two-dimensional quasicrystals both in theory and in experiment. Project supported by the National Natural Science Foundation of China.

Nonparametric cross-covariance modeling as exemplified by soil heavy metal concentrations from the Swiss Jura

Geostatistics is used in soil science to map spatial distributions of soil properties from limited samples. Covariance models provide the basic measure of spatial continuity which is used to weight the information available at different sample locations, as in kriging. Traditionally, a closed-form analytical model is fitted to allow for interpolation of sample covariance values while ensuring the positive definiteness condition. For cokriging where several different properties are cross correlated with each other, the simultaneous modeling of several (cross) covariances is made more difficult because of the restrictions imposed by the linear coregionalization model. An algorithm for automatic joint modeling of multiple (cross) covariance tables is proposed, building on an extension of Bochner's positive definiteness theorem and eigenvalue correction. The objective of this paper is to present the new methodology and demonstrate its application to modeling large cross-covariance matrix. Such task is not easily done or, more bluntly, seldom done in the conventional way when the number of coregionalized variables becomes large. The data set used for the case study relates to heavy metal pollution in soil.

STRICTLY POSITIVE DEFINITE FUNCTIONS ON THE COMPLEX HILBERT SPHERE

We study strictly positive definite functions on the complex Hilbert sphere. A link between strict positive definiteness and (harmonic) polynomial interpolation on finite‐dimensional spheres is investigated. Sufficient conditions for strict positive definiteness are presented.