Analysis of Nonlinear Quadratic Control in Infinite Time

This study refers to minimization of quadratic functionals in infinite time. The coefficients of the quadratic form are quadratic matrices, function of the state variable. Dynamic constraints are represented by a bilinear differential systems of the form. The necessary extremum conditions determine the adjoint variables λ and the control variables u as functions of state variable, respectively the adjoint system corresponding to those functions. Thus it will be obtained a matrix differential equation where the solution representing the positive defined symmetric matrix P ( x ), verifies the Riccati algebraic equation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quadratic forms of a matric-t variate

Summary Crowther [2] studied the distribution of a quadratic form in a matrix normal variate. This, in some sense, is extended by De Waal [4]. They represented the density function of this quadratic form in terms of generalized Hayakawa polynomials. Application of some specific results of these authors facilitates the derivation of distributions of quadratic forms of the matric-t variate. Attention is also given to the distributions of the characteristic roots and the trace of this quadratic matrix. Special cases are considered and some useful integrals are formulated. Financially supported by the CSIR and the University of the Orange Free State     

Remarks on a generalized beta function

We show that a certain generalized beta function B(x,y;b) which reduces to Euler's beta functions B(x,y) when its variable b vanishes and preserves symmetry in its parameters may be represented in terms of a finite number of well known higher transcendental functions except (possibly) in the case when one of its parameters is an integer and the other is not. In the latter case B(x,y;b) may be represented as an infinite series of either Wittaker functions or Laguerre polynomials. As a byproduct of this investigation we deduce representations for several infinite series containing Wittaker functions, Laguerre polynomials, and products of both.     

Analytic,Computational, and Approximate Forms for Ratios of Noncentral and Central Gaussian Quadratic Forms

Many useful statistics equal the ratio of a possibly noncentral chi-square to a quadratic form in Gaussian variables with all positive weights. Expressing the density and distribution function as positively weighted sums of corresponding F functions has many advantages. The mixture forms have analytic value when embedded within a more complex problem. The mixture forms also have computational value. The expansions work well with quadratic forms having few components and small degrees of freedom. A more general algorithm from earlier literature can take longer or fail to converge in the same setting. Many approximations have been suggested for the problem. A positively weighted noncentral quadratic form can always have two moments matched to a noncentral chi-square. For a single quadratic form, the noncentral form performs neither uniformly more or less accurately than older approximations. The approach also gives a noncentral F approximation for any ratio of a positively weighted noncentral form to a positively weighted central quadratic form. The method provides better accuracy for noncentral ratios than approximations based on a single chi-square. The accuracy suffices for many practical applications, such as power analysis, even with few degrees of freedom. Naturally the approximation proves much faster and simpler to compute than any exact method. Embedding the approximation in analytic expressions provides simple forms which correctly guarantee only positive values have nonzero probabilities, and also automatically reduce to partially or fully exact results when either quadratic form has only one term.     

Formula for the Laplace transform of the projection of a distribution on the positive semiaxis and some of its applications

We obtain a formula for the Laplace transform of the restriction of an arbitrary probability distribution on the positive semiaxis in the form of a Cauchy-type integral in infinite limits of the characteristic function of this distribution. Using this result and the estimates of the concentration function of the sum of independent random variables, we derive a representation for the Laplace transform of the restriction of the harmonic measure on the positive semiaxis. In conclusion, we present an estimate of the lower ladder height distribution for the case in which the distribution of the value of the jump in a random walk is normal.     

Integral quadratic forms and Dirichlet series

A Dirichlet series with multiplicative coefficients has an Euler product representation. In this paper we consider the special case where these coefficients are derived from the numbers of representations of an integer by an integral quadratic form. At first we suppose this quadratic form to be positive definite. In general the representation numbers are not multiplicative. Instead we consider the average number of representations over all classes in the genus of the quadratic form. And we consider only representations of integers of the form tk ² with t square-free. If we divide the average representation number for these integers by a suitable factor, we do get a multiplicative function. Using results from Siegel (Ann. Math. 36:527–606, 1935), we derive a uniform expression for the Euler product expansion of the corresponding Dirichlet series. As a special case, we consider the standard quadratic form in n variables corresponding to the identity matrix. Here we use results from Shimura (Am. J. Math. 124:1059–1081, 2002). For 2≤n≤8, the genus of this particular quadratic form contains only one class, and this leads to a rather simple expression for the Dirichlet series, where the coefficients are just the number of representations of a square as the sum of n squares. Finally we consider the indefinite case, where we can get results similar to the definite case.     

The Convergence of a Levenberg–Marquardt Method for Nonlinear Inequalities

In this paper, we consider the least l ₂-norm solution for a possibly inconsistent system of nonlinear inequalities. The objective function of the problem is only first-order continuously differentiable. By introducing a new smoothing function, the problem is approximated by a family of parameterized optimization problems with twice continuously differentiable objective functions. Then a Levenberg–Marquardt algorithm is proposed to solve the parameterized smooth optimization problems. It is proved that the algorithm either terminates finitely at a solution of the original inequality problem or generates an infinite sequence. In the latter case, the infinite sequence converges to a least l ₂-norm solution of the inequality problem. The local quadratic convergence of the algorithm was produced under some conditions.     

On the equivalence of quadratic APN functions

Establishing the CCZ-equivalence of a pair of APN functions is generally quite difficult. In some cases, when seeking to show that a putative new infinite family of APN functions is CCZ inequivalent to an already known family, we rely on computer calculation for small values of n. In this paper we present a method to prove the inequivalence of quadratic APN functions with the Gold functions. Our main result is that a quadratic function is CCZ-equivalent to the APN Gold function x^2r+1{x^{2^r+1}} if and only if it is EA-equivalent to that Gold function. As an application of this result, we prove that a trinomial family of APN functions that exist on finite fields of order 2 ⁿ where n ≡ 2 mod 4 are CCZ inequivalent to the Gold functions. The proof relies on some knowledge of the automorphism group of a code associated with such a function.     

The high-temperature relaxation function of a spin system with a quadratic contribution of fluctuations to the resonance frequency

We investigate the high-temperature relaxation function of a spin system with quadratic coupling of the resonance frequency to the Gaussian random process. In the general case, this function is expressed as an integral of an infinite auxiliary series. For theN-exponential Gauss Markov process, the problem is reduced to solving a system of 2N linear equations. For brevity, we analyze the effect of fluctuations on the form of the magnetic resonance line (the Fourier image of the relaxation function). For both the one- and multiexponential processes in a crystal with dynamics of a relaxation type in the continuous phase transition domain, we find a nonmonotonic dependence of the asymmetrical homogeneously widened resonance line on the rate of fluctuations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 2, pp. 316–328, November, 1999.     

Multivariate quadratic forms of random vectors

We obtain the distribution of the sum of n random vectors and the distribution of their quadratic forms: their densities are expanded in series of Hermite and Laguerre polynomials. We do not suppose that these vectors are independent. In particular, we apply these results to multivariate quadratic forms of Gaussian vectors. We obtain also their densities expanded in Mac Laurin series or in the form of an integral. By this last result, we introduce a new method of computation which can be much simpler than the previously known techniques. In particular, we introduce a new method in the very classical univariate case. We remark that we do not assume the independence of normal variables.     

Dispersion relations for normal waves in a cylinder made of orthorhombic monocrystal with pyro- and piezoelectric properties

By integrating the system of differential equations of coupled thermoelectroelastic vibrations of a pyroactive crystal medium of the orthorbombic system in special nonclassical vector-valued functions of ordinary and generalized complex variables we construct the dispersion equation for normal waves in a cylindrical waveguide of circular cross section. The dispersion function is obtained in the form of a determinant of infinite order subject to reduction in numerical investigations. Bibliography: 3 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 118–123.     

Optimizing an objective function under a bivariate probability model

The motivation of this paper is to obtain an analytical closed form of a quadratic objective function arising from a stochastic decision process with bivariate exponential probability distribution functions that may be dependent. This method is applicable when results need to be offered in an analytical closed form without double integrals. However, the study only applies to cases where the correlation coefficient between the two variables is positive or null. A stochastic, stationary objective function, involving a single decision variable in a quadratic form is studied. We use a primitive of a bivariate exponential distribution as first expressed by Downton [Downton, F., 1970. Bivariate exponential distributions in reliability theory. Journal of Royal Statistical Society B 32, 408–417] and revisited in Iliopoulos [Iliopoulos, George., 2003. Estimation of parametric functions in Downton’s bivariate exponential distribution. Journal of statistical planning and inference 117, 169–184]. With this primitive, optimization of objective functions in Operations Research, supply chain management or any other setting involving two random variables, or calculations which involve evaluating conditional expectations of two joint random variables are direct. We believe the results can be extended to other cases where exponential bivariates are encountered in economic objective function evaluations. Computation algorithms are offered which substantially reduce computation time when solving numerical examples.     

k-正态分布及其应用

近本文研究了截断随机变量和k-正态分布.利用对数凹函数理论,获得了涉及截断随机变量和截断随机变量的函数的方差的不等式链,推广了涉及正态分布和分层教学模型的一些经典结论.同时在附录部分给出了仿真结果.     

On closed sets of relational constraints and classes of functions closed under variable substitutions

Pippenger’s Galois theory of finite functions and relational constraints is extended to the infinite case. The functions involved are functions of several variables on a set A and taking values in a possibly different set B, where any or both of A and B may be finite or infinite. Received April 30, 2004; accepted in final form February 8, 2005.     

Subdifferentials of value functions and optimality conditions for DC and bilevel infinite and semi-infinite programs

The paper concerns the study of new classes of parametric optimization problems of the so-called infinite programming that are generally defined on infinite-dimensional spaces of decision variables and contain, among other constraints, infinitely many inequality constraints. These problems reduce to semi-infinite programs in the case of finite-dimensional spaces of decision variables. We focus on DC infinite programs with objectives given as the difference of convex functions subject to convex inequality constraints. The main results establish efficient upper estimates of certain subdifferentials of (intrinsically nonsmooth) value functions in DC infinite programs based on advanced tools of variational analysis and generalized differentiation. The value/marginal functions and their subdifferential estimates play a crucial role in many aspects of parametric optimization including well-posedness and sensitivity. In this paper we apply the obtained subdifferential estimates to establishing verifiable conditions for the local Lipschitz continuity of the value functions and deriving necessary optimality conditions in parametric DC infinite programs and their remarkable specifications. Finally, we employ the value function approach and the established subdifferential estimates to the study of bilevel finite and infinite programs with convex data on both lower and upper level of hierarchical optimization. The results obtained in the paper are new not only for the classes of infinite programs under consideration but also for their semi-infinite counterparts.     

Noncentral quadratic forms of the skew elliptical variables

In this paper the quadratic forms in the skew elliptical variables are studied. A family of the noncentral generalized Dirichlet distributions is introduced and their distribution functions and probability density functions are obtained. The moment generating functions of the quadratic forms in the skew normal variables are obtained. Sufficient and necessary conditions for the quadratic forms in the skew normal variables to have the noncentral generalized Dirichlet distributions are obtained. This leads to the noncentral Cochran's Theorem for the skew normal distribution.     

Some new examples in the theory of quadratic forms

We construct a 6-dimensional anisotropic quadratic form and a 4-dimensional quadratic form over some fieldF such that becomes isotropic over the function field but every proper subform of is still anisotropic over . It is an example of non-standard isotropy with respect to some standard conditions of isotropy for 6-dimensional forms over function fields of quadrics, known previously. Besides of that, we produce an 8-dimensional quadratic form with trivial determinant such that the index of the Clifford invariant of is 4 but can not be represented as a sum of two 4-dimensional forms with trivial determinants. Using this, we find a 14-dimensional quadratic form with trivial discriminant and Clifford invariant, which is not similar to a difference of two 3-fold Pfister forms. The proofs are based on computations of the topological filtration on the Grothendieck group of certain projective homogeneous varieties. To do these computations, we develop several methods, covering a wide class of varieties and being, to our mind, of independent interest. Received November 11, 1997; in final form June 24, 1999 / Published online May 8, 2000     

Family of multivariate generalized t distributions

In this paper, we introduce a new family of multivariate distributions as the scale mixture of the multivariate power exponential distribution introduced by Gómez et al. (Comm. Statist. Theory Methods 27(3) (1998) 589) and the inverse generalized gamma distribution. Since the resulting family includes the multivariate t distribution and the multivariate generalization of the univariate GT distribution introduced by McDonald and Newey (Econometric Theory 18 (11) (1988) 4039) we call this family as the “multivariate generalized t-distributions family”, or MGT for short. We show that this family of distributions belongs to the elliptically contoured distributions family, and investigate the properties. We give the stochastic representation of a random variable distributed as a multivariate generalized t distribution. We give the marginal distribution, the conditional distribution and the distribution of the quadratic forms. We also investigate the other properties, such as, asymmetry, kurtosis and the characteristic function.