Analytic Coefficients Research Articles

A Classification of Linear Time-Optimal Control Problems in a Neighborhood of the Origin

The main goal of the paper is to classify time-optimal control problems for linear controllable systems with analytic coefficients. The reduction to the Markov moment min-problem is the main tool of this investigation. The constructive solution of the time-optimal control problem using mentioned reduction is obtained under certain conditions.

On the complexity of boundary integral equations with analytic coefficients with logarithmic singularities

We find the exact order of the e-complexity of weakly singular integral equations with periodic and analytic coefficients of logarithmic singularities. This class of equations includes boundary equations for outer boundary-value problems for the two-dimensional Helmholtz equation.

Construction and computation of variable coefficient sylvester differential problems

In this paper, initial value problems for Sylvester differential equations X′( t) = A( t) X( t) + X( t) B( t) + F( t), with analytic matrix coefficients are considered. First, an exact series solution of the problem is obtained. Given a bounded domain Ω and an admissible error ϵ, a finite analytic-numerical series solution is constructed, so that the error with respect to the exact series solution is uniformly upper bounded by ϵ in Ω. An iterative procedure for the construction of the approximate solutions is included.

Classical and generalized solutions of time-dependent linear differential-algebraic equations

A coordinate-free reduction procedure is developed for linear time-dependent differential-algebraic equations that transforms their solutions into solutions of smaller systems of ordinary differential equations. The procedure applies to classical as well as distribution solutions. In the case of analytic coefficients the hypotheses required for the reduction not only are necessary for the validity of the existence and uniqueness results, but even allow for the presence of singularities. Straightforward extensions including undetermined systems and systems with nonanalytic coefficients are also discussed.

On the Holmgren-John uniqueness theorem for the wave equation with piecewise analytic coefficients

In this paper a uniqueness theorem is proved for the wave equation in the domain Q2T=Ω×(0,2T), where Ω is a piecewise analytic Riemannian manifold (Riemannian polyhedron). Initial data are assumed to be given on a part Γ0 × (0, 2T) of the space-time boundary of the cylinder Q2T, Γ0. The uniqueness of a weak solution is proved “in the large,” in a domain formed by the corresponding characteristics of the wave equation. Bibliography:24 titles.

Line profile of H Lyman α from dissociative excitation of H2 with application to Jupiter

Observations of the H Lyman α (Ly α) emission from Jupiter have shown pronounced emissions, exceeding solar fluorescence, in the polar aurora and equatorial “bulge” regions. The H Ly α line profiles from these regions are broader than expected, indicating high‐energy processes producing fast atoms as determined from the observed Doppler broadening. Toward understanding that process a high‐resolution ultraviolet (UV) spectrometer was employed for the first measurement of the H Ly α emission Doppler profile from dissociative excitation of H2 by electron impact. Analysis of the deconvolved line profile reveals the existence of a narrow central peak of 40±4 mÅ full width at half maximum and a broad pedestal base about 240 mÅ wide. Two distinct dissociation mechanisms account for this Doppler structure. Slow H(2p) atoms characterized by a distribution function with peak energy near 80 meV produce the peak profile, which is nearly independent of the electron impact energy. Slow H(2p) atoms arise from direct dissociation and predissociation of singly excited states which have a dissociation limit of 14.68 eV. The wings of H Ly α arise from dissociative excitation of a series of doubly excited states which cross the Franck‐Condon region between 23 and 40 eV. The profile of the wings is dependent on the electron impact energy, and the distribution function of fast H(2p) atoms is therefore dependent on the electron impact energy. The fast atom kinetic energy distribution at 100 eV electron impact energy spans the energy range from 1 to 10 eV with a peak near 4 eV. For impact energies above 23 eV the fast atoms contribute to a slightly asymmetric structure of the line profile. The absolute cross sections of the H Ly α line peak and wings were measured over the range from 0 to 200 eV. Analytic model coefficients are given for the measured cross sections which can be applied to planetary atmosphere auroral and dayglow calculations. The dissociative excitation process, while one contributing process, appears insufficient by itself to explain the line broadening observed at Jupiter.

On Associated and Co-Associated Complex Differential Operators

The paper deals with initial value problems of the form \frac{\partial u}{\partial t} = \mathcal L u, \ \ \ u = u_0 \ \mathrm {for} \ t = 0 in [0,T] \times G \subset \mathbb R^+_0 \times \mathbb R^n where \mathcal L is a linear first order differential operator. The desired solutions will be sought in function spaces defined as kernel of a linear differential operator l being associated to \mathcal L . Mainly two assumptions are required for such initial value problems to be solvable: Firstly, the operators have to be associated, i.e. lu = 0 implies l(\mathcal L u) = 0 . Secondly, an interior estimate \| \mathcal L u \|_{G'} ≤ c(G,G’) \| u \|_G (with G' \subset G ) must be true. Moreover, operators \mathcal L are investigated possessing a family of associated operators l_k (which then are said to be co-associated). The present paper surveys the use of associated and co-associated differential operators for solving initial value problems of the above (Cauchy–Kovalevskaya) type. Discussing interior estimates as starting point for the construction of related scales of Banach spaces, the paper sets up a possible framework for further generalizations. E.g., that way a theorem of Cauchy–Kovalevskaya type with initial functions satisfying a differential equation of an arbitrary order k (with not necessarily analytic coefficients) is obtained.

Unique continuation for solutions to pde's; between hörmander's theorem and holmgren' theorem

We prove a new unique continuation result for solutions to partial differential equations, “interpolating” between Holmgren's Theorem and Hormander's Theorem. More precisely, under some partial analyticity assumptions on the coefficients we obtain an intermediate unique continuation result which is in general weaker than Holmgren's Theorem (which applies to problems with analytic coefficients) but stronger than Hormander's Theorem (which applies to problems with C1 coefficients). Some applications to the wave and the Schroedinger equation are considered next. In particular we obtain a result conjectured by Hormander, namely that for the wave equation with C1 but time independent coefficients one has unique continutaion across any noncharacteristic surface.

Global Analytic Hypoellipticity of a Class of Degenerate Elliptic Operators on the Torus

In this paper we prove global analytic hypoellipticity on the torus for certain classes of operators in the form of a sum of squares of vector fields with real-valued and real analytic coefficients. The main conditions are that every point of the torus is of finite type for the vector fields and that the coefficients are independent of some of the variables. We show, for instance, that the well known not locally analytic hypoelliptic operator of Baouendi and Goulaouic [1] is globally analytic hypoelliptic on the torus. Also, we show that the operators that were proved in Hanges-Himonas [11] and in Christ [6], [7] to be not locally analytic hypoelliptic, are globally analytic hypoelliptic. Global analytic hypoellipticity of sum of squares operators on a compact analytic manifold M is an open problem except for certain cases. From the complex analysis point of view, the most important situation is when M is the boundary of a domain in C n and the operator is Kohn’s Laplacian, b, for M. In the symplectic case the global analytic hypoellipticity of b follows from the fundamental result of Tartakoff [24], and Treves [26]. In

Analytic variations on quadtrees

Quadtrees constitute a hierarchical data structure which permits fast access to multidimensional data. This paper presents the analysis of the expected cost of various types of searches in quadtrees -- fully specified and partial-match queries. The data model assumes random points with independently drawn coordinate values. The analysis leads to a class of "full-history" divide-and-conquer recurrences. These recurrences are solved using generating functions, either exactly for dimensiond=2, or asymptotically for higher dimensions. The exact solutions involve hypergeometric functions. The general asymptotic solutions rely on the classification of singularities of linear differential equations with analytic coefficients, and on singularity analysis techniques. These methods are applicable to the asymptotic solution of a wide range of linear recurrences, as may occur in particular in the analysis of multidimensional searching problems.

The differential field associated to a general analytic nonlinear dynamical system

A rigorous and detailed construction of the differential field K/sub Sigma /(u,y) associated, in a differential-algebraic context, to a nonlinear system Sigma described in state-space form with analytic coefficients is presented. The field K/sub Sigma /(u,y) represents an input-output description of Sigma , and its study is of key importance in problems concerning the input-output behavior of Sigma . Technically, the basic step of the construction of K/sub Sigma /(u,y) consists in proving that the differential ideal associated to the state-evolution equation of Sigma is prime. >

Perturbation theory by the moment method applied to coupled anharmonic oscillators

Perturbation theory according to the moment method is a powerful tool to treat coupled anharmonic oscillators because it leads to simple recurrence relations for the perturbation corrections. We calculate analytic energy coefficients for the lowest states of an exemplary model and illustrate how to overcome the problem posed by degenerate states that are connected through the perturbation.

A generalized hard-sphere model for Monte Carlo simulation

A new molecular model, called the generalized hard-sphere, or GHS model, is introduced. This model contains, as a special case, the variable hard-sphere (VHS) model of Bird [Rarefied Gas Dynamics, edited by S. S. Fisher (AIAA, New York, 1981), Part 1, p. 239] and is capable of reproducing all of the analytic viscosity coefficients available in the literature that are derived for a variety of interaction potentials incorporating attraction and repulsion. In addition, a new procedure for determining interaction potentials in a gas mixture is outlined. Expressions needed for implementing the new model in the direct simulation Monte Carlo (DSMC) methods are derived. This development makes it possible to employ interaction models that have the same level of complexity as used in Navier–Stokes calculations.

General edge asymptotics of solutions of second-order elliptic boundary value problems I

SynopsisThis is the first of two papers in which we study the singularities of solutions of second-order linear elliptic boundary value problems at the edges of piecewise analytic domains in ℝ3. When the opening angle at the edge is variable, there appears the phenomenon of “crossing” of the exponents of singularities. For this case, we introduce the appropriate combinations of the simple tensor product singularities that allow us to give estimates in ordinary and weighted Sobolev spaces for the regular part of the solution and for the coefficients of the singularities. These combinations appear in a natural way as sections of an analytic bundle above the edge. Their behaviour is described with the help of divided differences of powers of the distance to the edge. The class of operators considered includes second-order elliptic operators with analytic complex-valued coefficients with mixed Dirichlet, Neumann or oblique derivative conditions. With our description of the singularities we are able to remove some restrictive hypotheses that were previously made in other works. In this first part, we prove the basic facts in a simplified framework. Nevertheless the tools we use are essentially the same in the general situation.

The piecewise parabolic finite analytic method. Part I: Theory

To meet both the qualitative and quantitative requirements of the convective diffusion process, a numerical scheme, called the piecewise parabolic finite analytic method, is established by using a piecewise parabolic function as a boundary condition. This scheme incorporates the local analytic solution of the convective diffusion equation into the numerical scheme. It is free from numerical diffusion and automatically incorporates the convective diffusion effects. In this first part, a detailed study of the piecewise parabolic finite analytic scheme is presented. Initially, the maximum and minimum principle of the convective diffusion equation is used to determine the properties of the piecewise parabolic finite analytic coefficients. Then, based upon the truncation error and properties of the coefficients, the existence, uniqueness, stability, convergence, and error estimates of the numerical solution are obtained. For nonlinear problems the existence and error estimates are also obtained.

Accurate computation and interpolation technique of finite analytic coefficients

The finite analytic method (FA) developed in the last decade is an effective numerical method for solving fluid flow problems. However, because of the limitation in the present computer, large round-off errors are found in calculating FA coefficients when Reynolds number is large. This paper investigates the cause of this difficulty and presents a special programming technique in making an accurate computation of FA coefficients. Then a fundamental function known as “Pe” is tabulated by the accurate computation. In practical application the interpolation technique is employed so that the FA coefficients can be obtained reliably and quickly.

On integral operators of bergman type for systems of linear partial differential equations in two complex variables

We consider a system of formally hyperbolic linear partial differential equations in two complex variables with analytic coefficients. For the representation of solutions of this system we use integral operators of Bergman type that transform analytic functions of a complex variable into solutions. The integral operators are defied and conditions are investigated under which such operators exist. In particular for some special systems integral operators are given explicitly as examples.

ON THE SMOOTHNESS OF SOLUTIONS OF DIFFERENTIAL EQUATIONS AT SINGULAR POINTS OF THE BOUNDARY OF THE DOMAIN

Second-order elliptic equations with analytic coefficients and right sides in a domain with piecewise smooth boundary are studied. It is assumed that the boundary is characteristic at all points. Both Lipschitz and non-Lipschitz singularities of the boundary are admitted. It is proved that for large values of the spectral parameter, solutions possess high smoothness even at those points where the boundary has singularities. The results are based on the study of a constructive representation of solutions of the equations in the form of series of analytic functions.

On the connection between the Malliavin covariance matrix and Hörmander's condition

A celebrated theorem of Hörmander gives a sufficient condition for a second order differential operator to be hypoelliptic. For operators with analytic coefficients this condition turns out to be also necessary but this is not true for general smooth coefficients. On the other hand Malliavin conceived a probabilistic approach to the same problem, known as “Malliavin calculus,” in which a key role is played by the “Malliavin covariance matrix.” The aim of our paper is to give several characterizations of the Malliavin covariance matrix which are equivalent to Hörmander's condition (and consequently imply the hypoellipticity). In this way the distance between Hörmander's condition and the hypoellipticity property is clearly pointed out in probabilistic terms.

Vecteurs analytiques réduits et analyticité

Let P be a differential operator with analytic coefficients and elliptic, or more generally of principal type and hypoelliptic, in an open Ω of R n and let ( k j ) be an increasing sequence of positive integers. We give a necessary and sufficient condition on the sequence ( k j ) in order that we can characterize the analyticity of functions u in Ω thanks to estimates on the sequence ( P k j u). A same result is given for the system D 1, …, D n of first derivatives.