Representations for the solutions and coefficients of evolution equations

We give new representations for the solutions and coefficients of evolution equations in the linear case. The obtained formulas contain some functional arbitrariness that can be used in identification problems. We also give classes of hyperbolic equations that admit the generalized functionally invariant solutions.     

On some optimal control problems and their finite difference approximations and regularization for quasilinear elliptic equations with controls in the coefficients

Mathematical statements of the optimal control problems for quasilinear elliptic equations with the controls in the variable coefficients of the equation of state are considered. Both local and integral constraints on the controls are considered. The objective functionals correspond to the optimization with respect to a certain number of quality indexes. Finite difference approximations of optimization problems are constructed, and estimates of the approximation error with respect to the state and to the objective functional are established. The weak convergence in control is proved. The approximations are regularized after Tikhonov. Interesting examples of some applied optimization problems that naturally lead to the nonlinear optimal control problems examined in this paper are considered.     

On the Geometry of Point-Plane Constraints on Rigid-Body Displacements

In this paper the rigid-body displacements that transform a point in such a way that it remains on a particular plane are studied. These sets of rigid displacements are referred to as point-plane constraints and are given by the intersection of the Study quadric of all rigid displacements with another quadric in 7-dimensional projective space. The set of all possible point-plane constraints comprise a Segre variety. Two different classes of problems are investigated. First instantaneous kinematics, for a given rigid motion there are points in space which, at some instant, have no torsion or have no curvature to some order. The dimension and degrees of these varieties are found by very simple computations. The corresponding problems for point-sphere constraints are also found. The second class of problems concern the intersections of several given constraints. Again the characteristics of these varieties for different numbers of constraints are found using very simple techniques.     

Computation of Gauss-type quadrature formulas

Gaussian formulas for a linear functional L (such as a weighted integral) are best computed from the recursion coefficients relating the monic polynomials orthogonal with respect to L. In Gauss-type formulas, one or more extraneous conditions (such as pre-assigning certain nodes) replace some of the equations expressing exactness when applied to high-order polynomials. These extraneous conditions may be applied by modifying the same number of recursion coefficients. We survey the methods of computing formulas from recursion coefficients, methods of obtaining recursion coefficients and modifying them for Gauss-type formulas, and questions of existence and numerical accuracy associated with those computations.     

On quadrature formulas with equal coefficients

For the Lipschitz classes, we obtain weight quadrature formulas asymptotically optimal with respect to coefficients.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1217–1223, September, 1995.     

Regularity of solutions of boundary value problems for parabolic equations of arbitrary order in weighted Hölder spaces

We consider initial-boundary value problems for a uniformly parabolic equation of arbitrary order 2m in a noncylindrical domain whose lateral boundary is nonsmooth with respect to t. We assume that the lower-order coefficients and the right-hand side of the equation, generally speaking, grow to infinity no more rapidly than some power function when approaching the parabolic boundary of the domain, all coefficients of the equation are locally Hölder, and their Hölder constants can grow near that boundary. We construct a smoothness scale of solutions of such problems in weighted Hölder classes of functions whose higher derivatives may grow when approaching the parabolic boundary of the domain.     

Intractability Results for Positive Quadrature Formulas and Extremal Problems for Trigonometric Polynomials

Lower bounds for the error of quadrature formulas with positive weights are proved. We get intractability results for quasi-Monte Carlo methods and, more generally, for positive formulas. We consider general classes of functions but concentrate on lower bounds for relatively small classes of trigonometric polynomials. We also conjecture that similar lower bounds hold for arbitrary quadrature formulas and state different equivalent conjectures concerning positive definiteness of certain matrices and certain extremal problems for trigonometric polynomials. We also study classes of functions with weighted norms where some variables are “more important” than others. Positive quadrature formulas are then tractable iff the sum of the weights is bounded.     

Local Sensitivity Analysis in Estimation Problems

This article deals with the problem of local sensitivity analysis, that is, how sensitive are the results of a statistical analysis to changes in the data? A general methodology of sensitivity analysis is applied to some statistical problems. The proposed methods are applicable to any statistical problem that can be expressed as an optimization problem or that involves solving a system of equations. As some particular examples, the methodology is applied to the maximum likelihood method, the standard and constrained methods of moments and the standard and constrained probability weighted moments methods. Unlike the standard method of moments, the constrained method of moments ensures that the obtained estimates are always consistent with the data. Closed analytical formulas for the calculation of these local sensitivities are derived. The obtained sensitivities include: (a) the objective function sensitivities to data points and (b) the sensitivities of the estimated parameters to data points. The derived formulas for the sensitivities are based on recent results in the area of mathematical programming. Several examples of parameter estimation problems are used to illustrate the methodology.     

On the asymptotics with respect to a parameter of solutions of differential systems with coefficients in the class <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>

We consider a system of first-order ordinary linear differential equations with coefficients depending on an arbitrary parameter λ. For large λ, if the coefficients are smooth with respect to x, then there are known classical exponentially asymptotic (with respect to λ) formulas for the solution of the system. We generalize such formulas to the case in which the coefficients belong to the class L _q , q > 1. We use a new method for the reduction of problems to an integral system of special form.     

On the error of multidimensional quadrature formulas for certain classes of functions

We obtain regular (with respect to the power scale) estimates of the errors of multidimensional optimal quadrature formulas in spaces of periodic functions with constraints on Fourier coefficients in the ? _p -norm for 1 < p < 2.     

Invariant surfaces of standard two-dimensional systems with conservative first approximation of the third order

We simultaneously study two classes of two-dimensional time-periodic systems of differential equations with a small positive parameter, namely, systems with “slow” or “fast” time whose first-approximation systems are autonomous and conservative and do not contain terms of order higher than three. Thus, the corresponding unperturbed systems have one, two, or three rest points.For the perturbations, we indicate explicit conditions, independent of the small parameter, under which every original system of either class with coefficients three times continuously differentiable with respect to the phase variables and the parameter in a neighborhood of zero has finitely many two-dimensional invariant surfaces homeomorphic to tori for all sufficiently small parameter values. We also give formulas for these surfaces.     

Perturbation Approach to Sensitivity Analysis in Mathematical Programming

This paper presents a perturbation approach for performing sensitivity analysis of mathematical programming problems. Contrary to standard methods, the active constraints are not assumed to remain active if the problem data are perturbed, nor the partial derivatives are assumed to exist. In other words, all the elements, variables, parameters, Karush–Kuhn–Tucker multipliers, and objective function values may vary provided that optimality is maintained and the general structure of a feasible perturbation (which is a polyhedral cone) is obtained. This allows determining: (a) the local sensitivities, (b) whether or not partial derivatives exist, and (c) if the directional derivative for a given direction exists. A method for the simultaneous obtention of the sensitivities of the objective function optimal value and the primal and dual variable values with respect to data is given. Three examples illustrate the concepts presented and the proposed methodology. Finally, some relevant conclusions are drawn. The authors are indebted to the Ministry of Science and Education of Spain, Projects CICYT DPI2002-04172-C04-02 and CICYT DPI2003-01362, and to the Fulbright Commission for partial support. The authors are grateful to the referees for comments improving the quality of the paper.     

On the complexity of parabolic initial-value problems with variable drift

We study the intrinsic difficulty of solving linear parabolic initial-value problems numerically at a single point. We present a worst-case analysis for deterministic as well as for randomized (or Monte Carlo) algorithms, assuming that the drift coefficients and the potential vary in given function spaces. We use fundamental solutions (parametrix method) for equations with unbounded coefficients to relate the initial-value problem to multivariate integration and weighted approximation problems. Hereby we derive lower and upper bounds for the minimal errors. The upper bounds are achieved by algorithms that use Smolyak formulas and, in the randomized case, variance reduction. We apply our general results to equations with coefficients from Hölder classes, and here, in many cases, the upper and lower bounds almost coincide and our algorithms are almost optimal.     

The Skolemization of prenex formulas in intermediate logics

The skolem class of a logic consists of the formulas for which the derivability of the formula is equivalent to the derivability of its Skolemization. In contrast to classical logic, the skolem classes of many intermediate logics do not contain all formulas. In this paper it is proven for certain classes of propositional formulas that any instance of them by (independent) predicate sentences in prenex normal form belongs to the skolem class of any intermediate logic complete with respect to a class of well-founded trees. In particular, all prenex sentences belong to the skolem class of these logics, and this result extends to the constant domain versions of these logics.     

Eigenvalue perturbation theory of structured real matrices and their sign characteristics under generic structured rank-one perturbations

An eigenvalue perturbation theory under rank-one perturbations is developed for classes of real matrices that are symmetric with respect to a non-degenerate bilinear form, or Hamiltonian with respect to a non-degenerate skew-symmetric form. In contrast to the case of complex matrices, the sign characteristic is a crucial feature of matrices in these classes. The behaviour of the sign characteristic under generic rank-one perturbations is analyzed in each of these two classes of matrices. Partial results are presented, but some questions remain open. Applications include boundedness and robust boundedness for solutions of structured systems of linear differential equations with respect to general perturbations as well as with respect to structured rank perturbations of the coefficients.     

Sufficient optimality conditions for extremal controls based on functional increment formulas

We consider the optimal control problem without terminal constraints. With the help of nonstandard functional increment formulas we introduce definitions of strongly extremal controls. Such controls are optimal in linear and quadratic problems. In the general case, the optimality property is provided with an additional concavity condition of Pontryagin’s function with respect to phase variables.     

SOME COMBINATORIAL PROBLEMS ON ORDERED TREES

Some formulas are established to calculate the number of leaves on all structurallydifferent ordered trees with n nodes,and the total leaf path length and the total node pathlength for those trees.Certainly,respective average numbers are obtained(the average pathlength may be considered as the average height of random ordered trees with n nodes),Actually,this paper presents a general method dealing with some classes of combinaorialproblems on ordered trees.     

Some Properties of Functional Integrals with Respect to the Bogoliubov Measure

We consider problems related to integration with respect to the Bogoliubov measure in the space of continuous functions and calculate some functional integrals with respect to this measure. Approximate formulas that are exact for functional polynomials of a given degree and also some formulas that are exact for integrable functionals belonging to a broader class are constructed. An inequality for traces is proved, and an upper estimate is derived for the Gibbs equilibrium mean square of the coordinate operator in the case of a one-dimensional nonlinear oscillator with a positive symmetric interaction.     

Mayer and optimal stopping stochastic control problems with discontinuous cost

We study two classes of stochastic control problems with semicontinuous cost: the Mayer problem and optimal stopping for controlled diffusions. The value functions are introduced via linear optimization problems on appropriate sets of probability measures. These sets of constraints are described deterministically with respect to the coefficient functions. Both the lower and upper semicontinuous cases are considered. The value function is shown to be a generalized viscosity solution of the associated HJB system, respectively, of some variational inequality. Dual formulations are given, as well as the relations between the primal and dual value functions. Under classical convexity assumptions, we prove the equivalence between the linearized Mayer problem and the standard weak control formulation. Counter-examples are given for the general framework.     

Diffraction in periodic structures and optimal design of binary gratings. Part I: direct problems and gradient formulas

The aim of the paper is to provide the mathematical foundation of effective numerical algorithms for the optimal design of periodic binary gratings. Special attention is paid to reliable methods for the computation of diffraction efficiencies and of the gradients of certain functionals with respect to the parameters of the non-smooth grating profile. The methods are based on a generalized finite element discretization of strongly elliptic variational formulations of quasi-periodic transmission problems for the Helmholtz equation in a bounded domain coupled with boundary integral representations in the exterior. We prove uniqueness and existence results for quite general situations and analyse the convergence of the numerical solutions. Furthermore, explicit formulas for the partial derivatives of the reflection and transmission coefficients with respect to the parameters of a binary grating profile are derived. Finally, we briefly discuss the implementation of the generalized finite element method for solving direct and adjoint diffraction problems and present some numerical results. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.