Progress in Analysis

PREFACE.

CONTENTS.

In the paper some properties of multiparametrical space are investigated. The characterization of behavior of functions from this space near singular point is given via coefficients of generalized polynomial. Nikol'skii-Lizorkin type inequality for functions of this space is obtained, where higher order derivative is estimated via lesser order derivative.

Estimates for Tikhonov regularized solutions of an integral equation of the first kind of convolution type are presented. They are obtained by using anisotropic Nikol'skii-Besov spaces. This article extends the results, obtained by V.I. Burenkov, I.F. Dorofeev, and A.S. Pankratov for isotropic Nikol'skii-Besov spaces, to the anisotropic case.

In this work the Lizorkin-Triebel-Morrey type space is constructed, when the domain G ⊂ Rⁿ satisfies the flexible λ-horn condition and some properties of the functions from the constructed space are studied as with respect to imbedding theory.

Our goal is to obtain weighted imbedding theorems for the Sobolev spaces , where or . With this goal we study the behavior in weighted L_p–spaces of the anisotropic singular integral operators arising when the integral representations of Il'in and Besov are used (see ¹, §7.2).

In this paper the generalized shift operator is considered generated by the Bessel differential operator , by means of which anisotropic Fourier–Bessel singular integral operators (B_n- anisotropic singular integral operators) are investigated. The boundedness of the anisotropic Fourier–Bessel singular integral operators is proved acting boundedly on the space . As well are proved limits embedding theorems on the Sobolev-Bessel space .

Sharp estimates for ‖f‖_Lp(0,h) as h → 0+, where 0 < p < ∞, are given for functions f with prescribed behaviour of ‖f(x + h) – f(x)‖_{Lp(0,a – h)}.

We obtained that on general Carnot group there are John domains which don't satisfy interior homogeneous cone condition (IHCC) and exterior homogeneous cone condition (EHCC) together. On 2-step Carnot group we proved that there are simply connected John domains which don't satisfy (IHCC) and (EHCC) together.

In this short note we shall consider order completion of an ordered linear space at first and secondly its representation as function space. Also, we shall consider relations between above results and generalized supremum.

This paper presents a functional calculus definition of linear fractional (pseudo)-differential operators via generalised Fourier transforms as a natural extension of integer ordered derivatives. First, we describe the extension of our L₂–based functional calculus approach on . Second, we demonstrate that important computational rules as well as properties of integer ordered differential operators are preserved by our approach. This concerns also the –kernel of our operators which belong to the same class as those corresponding to integer derivatives, i.e., they are linear combinations of "polynomials times exponential functions".

By virtue of coefficient inequalities, certain generalized subclasses of normalized analytic functions are introduced. Several integral means inequalities are obtained for higher-order and fractional calculus of functions belonging to the generalized suclasses. In these results, certain hypotheses for the generalized subclasses is weakened than those of the earlier results by T. Sekine, K. Tsurumi and H. M. Srivastava.

The Power Geometry is applied to the boundary problems for the planar and axially-symmetric steady flows of the viscous heat conducting gas. The well-known problem of the boundary layer on the semi-infinite flat plate is considered as example. The asymptotics of the flow at infinity in the conical diffuser is considered here for the first time.

Let be the operator in L²(ℝ^N):

where c_j > -1/2 and j = 1,2,…, N.

We construct a generalized Fourier transform B_c1…cj…cN , which converts the operator into the multiplication operator i y_j, i.e.,

Here is the adjoint operator of B_c1…cj…cN and .

When c₁ = c₂ = ⋯ = c_N = 0, the operator becomes ∂/∂x_j. Hence the transform B_c1…cj…cN coincides with the Fourier transform. We can therefore regard the transform B_c1…cj…cN as a generalized Fourier transform.

On the basis of the transform we explicitly find out the solutions of the Cauchy problems for the heat equation with a strongly singular coefficient and for the Schrödinger equation with a strongly singular potential.

Moreover, we show that there is the Friedrichs extension of -Δ + k/(|x|²), x ∈ ℝ^N as long as k > -N/4.

Using the transform above we define spaces of Sobolev type. Each space is a generalized Sobolev space. We show an embedding theorem for these spaces. We see that the embedding theorem is a generalization of the Sobolev embedding theorem. We finally apply the embedding theorem to the Cauchy problem for the wave equation with a strongly singular coefficient and study some properties of its solution.

Integral transforms involving functions of Bessel type in the kernels are considered. Their representations in the form of H-transforms are established. Results concerning properties of these transforms in weighted spaces of summable functions are characterized. Problems, new trends of research and applications are discussed.

In this paper we consider the L_p estimate of solutions to the generalized Stokes resolvent problem in a bounded and exterior domain of ℝⁿ with Neumann boundary condition.

We expose the results we obtained in the study of regular vectors, as Gevrey vectors and Beurling vectors, of certain systems of partial differential operators.

The sphere is a homogeneous space acted by the rotation group. Using this fact, we can define the convolution for functions on the sphere. By this (non-commutative) convolution, Boehmians can be defined on the sphere.

In this paper we will prove that the Picard group is a two-generator group and a Jørgensen group. Furthermore we will state that the Jørgensen number of the Whitehead link is two.

On construit la solution bornée du second problème de Cousin dans le polydisque-unité D(1) de ℂⁿ, (n ≥ 3), pour la donnée de Cousin Σ telle que .

We give in §3 results concerning the preservation of locally uniformly convergence for sequences and of the normality and compactness for families of 1) continuous mappings and 2) homeomorphisms, by lifting or factorizing in the general case of regular coverings (=coverings, Definition 1). With this aim we present in §2.2. the lifting and factorizing properties by regular coverings pointing out when normal coverings (Definition 2) are needed.

Let be the family of q.c. homeomorphisms between two Riemann surfaces, mapping a given compact subset into (onto) a given compact subset. In the BMO_loc-case, under some restrictions, and are normal, but not necessarily compact. In the BMO-case, and are normal and compact.

Let A be the class of functions f(z) which are analytic in the open unit disk U with f(0) = 0 and f'(0) = 1. The object of the present paper is to consider the subordinations of certain integral operators for functions belonging to the class A.

We construct a bounded holomorphic function in the domain of Cⁿ which has wild boundary behavior at every point of arbitrary discrete subset in the boundary.

In this paper we deduce some properties of Loewner chains and their transition mappings on the unit ball B of ℂⁿ with respect to an arbitrary norm. For this purpose, we deduce that any Loewner chain f(z,t) and its transition mapping v(z,s,t) satisfy some locally Lipschitz conditions in t locally uniformly with respect to z ∈ B. Also we deduce that a mapping f ∈ H(B) has parametric representation if and only if there exists a Loewner chain f(z,t) such that the family {e^-tf(z, t)}_{t ≥ 0} is a normal family on B and f(z) = f(z,0) for z ∈ B. Finally we conclude that the set S⁰(B) of mappings which have parametric representation on B is compact.

We study stability problems of the class ℨ_co(G) of all convex functions g : Δ ⊂ ℝⁿ → ℝ defined on domains Δ ⊂ ℝⁿ and satisfying the differential inclusion

In essence, stability means that local proximity of a mapping f to the mappings of the class ℨ_co(G) implies global proximity of f to them in the C-norm. We prove that the class ℨ_co(G) is stable if the projection of the compact G on each straight line l ⊂ ℝⁿ is a totally disconnected set. The latter condition is also necessary in the case of functions of one variable.

A subspace G of the Hardy space H¹ is said to have the f-property if hI^-1 ∈ G for any h ∈ G and any inner function I which divides h in the sense that hI^-1 ∈ H¹. In this paper we survey some results concerning the mean growth of the derivative of infinite Blaschke products and the f-property and present a new example of a subspace of H¹ which does not have the f-property.

Functions F and G are said to avoid each other if F – G is never zero, and F and G are never simultaneously infinite. Bargman, Bonk, Hinkkanen, and Martin proved that a family of functions meromorphic in the unit disk D is a normal family if there exist three functions G₁, G₂, and G₃, each continuous in D, such that for each the functions f, G₁, G₂, and G₃ all avoid each other. We present some variations and consequences of this result, and give a sufficient condition of the same type for a funcion to be a normal function. We also give some examples of functions that do not avoid normal functions.

Several results on Q_p spaces and their meromorphic counterparts, classes, are mentioned. We study the properties of an important function which show that there are some curious differences between the spaces of analytic functions and their corresponding classes of meromorphic functions in the unit disk.

We discuss unique range sets for polynomials or rational functions. We also discuss related results, including (i) rational functions that share three values, and (ii) sets which are almost (apart from exceptional cases) unique range sets for different classes of meromorphic functions.

We give examples and non-examples of unique range sets for ultrametric entire functions in positive characteristic, in an algebraically closed complete ultrametric field. In particular, if n ≥ 4, we show there exists a unique range set with n elements for the family of non-constant one variable polynomials in any characteristic. As far as a pure existence theorem goes, this is the best one can hope for, as there are no 3 point unique range sets in characteristic three. For all prime powers q = pⁿ ≥ 3, we construct an affinely rigid set of q elements which is not a unique range set for the non-constant one variable polynomials in characteristic p. All our results hold equally well for the family of non-constant one variable non-Archimedean entire functions.

We extend the concept of B^q spaces from the one-dimensional complex function theory to spaces of hyperholomorphic quaternion-valued functions of three real variables. Moreover, we will study some properties of these spaces and prove a characterization of hyperholomorphic Bloch functions in the unit ball of ℝ³ by integral norms.

This is an announcement about our paper: "Bloch-Hardy Pullbacks" which will appear in Acta Sci. Math. (Szeged).

The main purpose of this paper is to define a new region for studying effectiveness of regular functions of several complex variables by basic sets of polynomials of several complex variables in hyperelliptical regions.

In this paper we give some theorems on the propagation of algebraic dependence of meromorphic mappings from an analytic finite covering space X over the complex m-space into complex projective spaces and give their applications. We study this problem under a condition on the existence of meromorphic mappings separating the fibers of X.

Let f be a transcendental and linearly non-degenerate holomorphic curve from C into the n–dimensional complex projective space Pⁿ(C) satisfying

where X is a subset of Cⁿ⁺¹ - {0} in N-subgeneral position for N > n = 2m (m ∈ N). Then, there are at least [(2N - n + 1)/(n + 1)] + 1 vectors a in X such that δ(a, f) = 1.

A description of the possible singularity sets and growth of PF_r generating functions is given.

Let ℘ be the Weierstrass' Pe-function. It is shown that there exists a transcendental meromorphic function g satisfying , where A ≠ 0, B, and are constants.

We introduce the space of monogenic functions of bounded mean oscillation (B-MOM) and prove a John-Nirenberg inequality for this space.

We discuss a special system of orthogonal homogeneous monogenic polynomials in the unit ball and on its boundary, respectively. These polynomials and their hypercomplex derivatives are orthogonal in the unit ball and on the boundary. Their norms in L₂ and in the Sobolev space are explicitely known. The completeness of this system is proved in arbitrary Sobolev spaces and the rate of convergence for the best approximation in these spaces can be estimated.

For Ω a sufficently smooth unbounded domain in Rⁿ we develop a decomposition result for the Sobolev space and also, we use modified Cauchy-Green type kernels to construct Clifford analytic-complete function systems in the generalized Bergman space , where D^l is the l-th iterate of the Dirac operator, l is a positive integer less than n and . The modified Cauchy-Green kernels ensure that p lies in this range. Without the modification of the kernels one is restricted to a smaller range. These functions are used to approximate solutions of the equation Δ^ku = 0 with some boundary conditions and with 2k < n. Some similar results are presented for sufficently smooth unbounded domains lying in hyperbolas.

Let Cℓ_n be the (universal) Clifford algebra generated by e₁,…,e_n satisfying e_ie_j + e_je_i = -2δ_ij, i, j = 1,…,n. The Dirac operator is defined by , where e₀ = 1. The modified Dirac operator is introduced by , where ' is the main involution and Qf is given by the decomposition f(x) = Pf (x) + Qf (x) e_n with Pf(x), Qf(x) ∈ Cℓ_n-1. A k + 1-times continuously differentiate function f : Ω → Cℓ_n is called k-hypermonogenic in an open subset Ω of ℝⁿ⁺¹, if M_kf(x) = 0 outside the hyperplane x_n = 0. Note that 0-hypermonogenic are monogenic and n - 1-hypermonogenic functions are hypermonogenic defined by H. Leutwiler and the author. The power function x^m is hypermonogenic. A function f : Ω → Cℓ_n is be called k-hyperbolic harmonic if , where . In the real-valued case, f is k–hyperbolic harmonic, if and only if it satisfies the Laplace-Beltrami equation associated with the hyperbolic metric. We show that f is k-hypermonogenic if and only if both f and xf are k–hyperbolic harmonic. We also prove that a function g is monogenic in the case n odd if and only if locally for a hypermonogenic function f. We prove an integral formula for hypermonogenic functions and also compare hypermonogenic with the holomorphic Cliffordian functions investigated by G. Laville, I. Ramadanoff and L. Pernas.

We prove that using a commutative, associative, unital and finite generated algebra of the symmetry operators one can construct the solution to the Dirac equation in not necessary commutative algebras.

We show that the Maxwell equations for arbitrary inhomogeneous media are equivalent to a single quaternionic equation which can be considered as a generalization of the Vekua equation for generalized analytic functions.

We approximate the Cauchy-Riemann operators in the complex plane by finite differences, such that the factorization of the Laplacian into two adjoint Cauchy-Riemann operators is preserved in the discrete case. Integral representations of the fundamental solutions of these difference operators are given. Furthermore, we define a discrete version of the complex T-operator. Based on this operator a discrete Borel-Pompeiu formula is formulated.

We extend the study of boundary value properties of α-hyperholomorphic functions of two real variables, with α ∈ ℂ, known for domains with the Liapunov boundaries, onto the case of piece-wise Liapunov ones. We present the Sokhotski-Plemelj theorem for the corresponding Cauchy-type integral as well as the necessary and sufficient condition for the possibility to extend a given Hölder function from such a curve up to a null-solution of the α-hyperholomorphic Cauchy-Riemann operator in a domain. The Plemelj-Privalov-type theorem and a formula for the square of the singular Cauchy-type integral are given.

In this paper we study certain properties for nullsolutions of a Dirac-type operator on super-space as introduced in our papers ¹² and ¹³. In particular we solve the inhomogeneous equation. We also discuss analysis on the level of abstract vector variables.

Souček ^1,2 discovered an intriguing connection between the standard twistor correspondence {𝔽_1,2, ℂℙ³, G_2,4, ϕ, ψ}³ and ℂℍℙ¹, the biquaternionic projective line. In particular, the correspondence maps ϕ, ψ are encoded in the non-Hausdorff character of the natural quotient topology on ℂℍℙ¹. Inspired by this result (and algebraic approaches to spacetime physics in general) we describe a program to investigate the connection between issues in relativistic physics and various spacetime-related algebras. As a preliminary result, we show how Souček's result can be written in terms of the Clifford algebra .

The questions defining pure spinors are interpreted as questions of motion of fermion multiplets in momentum spaces which, in a constructive approach based on pure spinors, naturally result lorentzian and ending up at P = R^1,9. The equations found are most of those postulated ad hoc in physics for the interpretation of elementary particle phenomology. It is found in this way that several of the observed elementary particle properties derive from the 3 complex division algebras. Precisely complex numbers generate the U(1) at the origin of charged fermions, steadily appearing in changed-neutral pairs (of fermions or of fermion multiplets); quaternions generate internal symmetry SU(2) and SU(2)_L and are at the origin of the 3 families of leptons; octonions generate SU(3) both flavour and color.

The real division algebra (Majorana spinors) might be correlated with the possible origin of black matter.

The resulting momentum spaces result compact isomorphic to invariant mass spheres imbedded in each other and the mass values are naturally increasing with the dimension of the corresponding fermion multiplets.

Further possible role of pure spinors are presented and discussed.

A real algebra with a generators {e_α}_α=1,…,p-1 satisfying the relations: e_αe_β + e_βe_α = -2δ_αβI(α,β = 1, 2, …, p - 1) is called a pre-Hurwitz algebra, if it has a matrix representation such that ^t(e_α) = - e_α{α = 1,2,…,p - 1). Here we notice that a pre-Hurwitz algebra need not satisfy the irreducibility condition. In this paper, an explicit construction of generators of all pre-Hurwitz algebras is given and their irreducibility is discussed. The main results are given in Theorem I, II and III.

Using the integral formula we give optimal L^p estimates for solutions of the problem on real and complex ellipsoids and the L^p extension of holomorphic functions from submanifolds of bounded strictly pseudoconvex domains in ℂⁿ with non-smooth boundary.

In this note we study removable singularities for analytic functions in Hardy H^p spaces, in BMO and in locally Lipschitz spaces.

In the paper we consider certain classes of matrix quasielliptic operators in the whole space ℝⁿ and special weighted Sobolev spaces . We give theorems on density of finite functions in . We formulate results on isomorphic properties of quasielliptic operators and solvability of quasielliptic systems in , and give applications these results to the theory of Sobolev type equations and systems not Cauchy–Kovalevskaya type.

Klein surfaces, which appear as compactifications of some Riemann surfaces are sought and the relationship between them in terms of their automorphism groups and of the harmonic functions they support is studied.

In this paper, we generalize and strengthen certain contour-solid theorems for analytic functions and normal majorants. The generalization is related to considering meromorphic functions instead of holomorphic, and strengthening is connected with taking into account the multivalence of functions.

We seek to control the solution to a hyperbolic equation in a cylindrical domain by prescribing Dirichlet conditions on some part of the lateral boundary of the domain. We give a new argument, based on geometric optics, for a theorem on control given a convex function, which Lasiecka, Triggiani and Yao have shown via Carleman estimates. We also give an example of a space where control is guaranteed by geometric optics, but no convex function exists.

We prove the following unique continuation property. Let u be a solution of a second order linear parabolic equation and S be a segment parallel to the t-axis. If u has a zero of order faster than any nonconstant polynomial at each point of S then u vanishes in each point (x, t') such that the plane t = t' has nonempty intersection with S.

There has been new interest in the successful application of differential geometric methods in the control of p.d.e.'s. (See for example Contemporary Mathematics #268, AMS 2000, particularly the article by the present authors). Here we describe those results, and some newer results using the methods of curvature flows. We also present an example for which control is possible but cannot be proved by means of any convex function.

We consider the point-invariant classes of the ordinary differential equations. In particular, the point-invariant class of the second order ODE's that contains the Painlevé equations and the point-invariant class of the third order ODE's that contains the Chazy equations are distinguished. Tests for the equivalence of the second order ODE's to the Painlevé I and II equations are indicated. The necessary condition of the equivalence of the second order ODE's to the Painlevé IV and Painlevé III in the case a = 0 are shown.

In a series of papers with the same title we shall give a noncommutative differential geometric method to fractal geometry and show a possiblity of a description of phase transtions in statistical mechanics in terms of fractal geometry. In this paper we consider the representations of Cuntz algebras on proper/improper self similar fractal sets. We can obtain the following results:

1. We give a sufficient condition for the existence of the representations of Cuntz algebras on self similar fractal sets.

2. The Cuntz algebra has representations on proper self similar fractal sets, which are called Hausdorff representations and the criterion whether they are equivalent or not can be given.

3. We can construct representations on improper self similar fractal sets which are called of Hausdorff type and show that they are not equivalent to that on a proper self similar fractal set.

This is an expository note and the full will be given in the forthcoming paper(⁶).

We consider the nonlinear differential equation , where the dissipation and amplitude P are both small. The standard Melnikov analysis allows to detect parameter regions with homoclinic chaos. We show that a resonant parametric perturbation in quadratic and cubic terms may lead to disappearence of transversal homoclinics. This phenomenen can be used to suppress chaos in the equation, which is supported by our numerical experiments.

The following sections are included:

• Introduction

• On polynomial models of dynamical systems

• Classification of separatrix cycles

• Global analysis

• References

Dynamical systems on Riemannian manifolds are examined which are either generated by a semi-flow or by a C¹-smooth map. The exponential growth rate of the k-volumina provides a lower bound of the topological entropy of the system if its phase space is compact and has box-counting dimension k. If the system possesses an equivariant sub-bundle instead of the tangent map its restriction onto the fibers of this sub-bundle can be considered. Inequalities for systems with not necessarily compact phase space are also given. Examples address linear maps in ℝ^m, the time-reversed Lorenz system, and the geodesic flow on a (not necessarily compact) Riemannian manifold without conjugate points.

In this paper we present an abstract approach to inertial manifolds for nonautonomous dynamical systems. Our result on the existence of inertial manifolds requires only two geometrical assumptions, called cone invariance and squeezing property, and two additional technical assumptions, called boundedness and coercivity property.

We develop some fixed point theory of maps and iterated maps from the point of view of real algebraic geometry in order to provide sufficent conditions for the existence of chaotic sets (in the sense of Takens, ⁵ for discrete dynamical systems induced by continuous maps f : Rⁿ → Rⁿ. Furthermore, the well known Conley-Moser conditions for proving topological conjugacy to the shift map can be weakened if a coarser topology is chosen.

In the local dynamics of Newton's method, a generic double root of a holomorphic function of two variables has a Cantor family of holomorphic superstable manifolds.

A major challenge in applying the Conley index theory to a given dynamical system is to find isolating neighborhoods of the invariant sets of interest. In this paper we describe a computational method which allows for the efficient construction of isolating neighborhoods. As an example we compute an isolating neighborhood of a heteroclinic orbit connecting a fixed point to a period two point in the Hénon map.

We discuss the band-gap structure and the integrated density of states for periodic elliptic operators in the Hilbert space L₂(ℝ^m), for m ≥ 2. We specifically consider situations where high contrast in the coefficients leads to weak coupling between the period cells. Weak coupling of periodic systems frequently produces spectral gaps or spectral concentration. Our examples include Schrödinger operators, elliptic operators in divergence form, Laplace-Beltrami-operators, Schrödinger and Pauli operators with periodic magnetic fields. There are corresponding applications in heat and wave propagation, quantum mechanics, and photonic crystals.

We propose a simple method for constructing asymptotics of eigenfunctions for the Schrödinger equation with a shallow potential well and its generalization to the problem of water waves trapped by an underwater ridge.

The long-time asymptotics is analyzed for all finite energy solutions to the 1D Klein-Gordon equation coupled to a nonlinear oscillator. The coupled system is invariant with respect to the rotation group U(1). Each finite energy solution converges to the "eigenfunctions" ψ_±(x)e^iω±t as t → ±∞. The problem is inspired by Schrödinger's identification of the quantum stationary states to the eigenfunctions in the quantum electrodynamics which is invariant with respect to the (global) gauge group U(1).

This paper deals with global behaviour for k–outgoing solutions of -Δu - k²u = f with Dirichlet boundary condition. Especially, we aim at getting two things. On the one hand, we consider scattering by one or many obstacles of which global capacity converges to zero. On the other hand, we expand an approach for which knowledge of Green kernel is not required. Indeed, that method could be developed for more complicated operators.

For homothetical obstacles, more precise consequences are obtained (connection between scattering amplitude and capacity). We also consider the case of many tiny spheres: roughly speaking, we prove that the scattering amplitude is approximately the sum of scattering amplitudes related to each isolated sphere (we recognize the first Born approximation).

This is a survey of our recent work on the degenerate mixed boundary value problem

where Ω is a domain in ℝⁿ, P is a second order linear elliptic operator with real coefficients, Γ ⊆ ∂Ω is a closed set, and B is an oblique boundary operator defined only on ∂Ω\Γ which is assumed to be a smooth part of the boundary. We discuss the solvability of the above boundary value problem in the class of nonnegative functions, and properties of the generalized principal eigenvalue, the ground state, and the Green function associated with this problem. The notion of criticality and subcriticality for this problem is introduced and a criticality theory for this problem is established. The analogs for the generalized Dirichlet boundary value problem, where Γ = ∂Ω, were examined intensively by many authors.

Consider the oscillations of the harmonic crystal in ℝⁿ with an arbitrary n ≥ 1. We study the distribution μ_t of the random solution at time t ∈ ℝ. The initial measure μ₀ has translation-invariant correlation matrices, zero mean and a finite mean energy density. It also satisfies Rosenblatt- or Ibragimov-Linnik type mixing condition. The main result is the convergence of μ_t to a Gaussian measure as t → ∞. The proof is based on the long time asymptotics of the Green function and on the S.N.Bernstein's "room-corridors" method.

We are interested in the asymptotic behavior of waves propagating in periodic thin graph-like high contrast 2D acoustic media. A differential problem is derived, whose spectrum provides the asymptotics of the frequency spectrum of acoustic waves in such a material in the limit of thin structure of high contrast.

In a domain Ω = ℝ² × (0,1) the equations of linear elasticity are considered. The spatial operators self adjoint extension with respect to Dirichlets boundary condition and its spectral family {P_λ} are studied. The derivative exists almost everywhere and has singularities at discrete lying points λ₁,λ₂,… with λ_j → ∞ as j → ∞. These singularities are connected to resonance effects of the solution to the wave equation of linear elasticity.

We describe continuity and regularity properties of pseudo-differential operators with anti-Wick symbols in the space of tempered distributions on R²ⁿ. We consider in particular symbols in the Sobolev Spaces and show that, independently of s ∈ R, the corresponding operators belong to the Schatten-Von Neumann Classes S_p.

We show that on the affine group U = {(b, a) : b ∈ ℝ, a > 0}, localization operators associated to admissible wavelets and separable symbols are paracommutators. We show that if the symbol is a function of b only, then the localization operator can be expressed in terms of a paraproduct; and if the symbol is a function of a only, then the localization operator is a Fourier multiplier.

Some new approaches to wavelet-based image denoising will be described. These new approaches improve upon the classic wavelet shrinkage denoising methods. They exploit tree-based correlations between parent coefficients and their descendants, which are present in images of natural scenes. A method of the author, tree-adapted wavelet shrinkage, is well-suited for use in combination with image compression procedures. It performs particularly well in preserving edge details in images.

We study a class of pseudodifferential operators in ℝⁿ in the frame of the spaces of type S of Gelfand and Shilov.

By means of a Lizorkin - Marcinkiewicz theorem about the L^p-continuity of Fourier multipliers, we introduce a family of L^p-bounded pseudodifferential operators with symbol in the Hörmander classes.

In this frame some L^p regularity results for hypoelliptic operators are stated.

This paper deals with the newly created singularities of the solutions of several specific examples of weakly hyperbolic semilinear systems in the plane. Two or three characteristics are supposed to be mutually tangential at the origin only and the initial data are either continuous or non-smooth. The exact strength of the emergent singularities is also investigated.

In this paper we develop an approach to the study of meromorphic connections with logarithmic poles along a Saito free divisor. In particular some properties of Christoffel symbols of such a connection are established. We also compute the sets of all integrable homogeneous meromorphic connections with logarithmic poles along the discriminant of the minimal versal deformation of an A₃-singularity and along Sato's hypersurface occurred in the context of the theory of prehomogeneous spaces.

In this article we first consider the composition operator T on certain Banach spaces of holomorphic functions on a domain V ⊂ Cⁿ, which gives us the "Schröder" equation. Then we will consider a general operator, namely, a weighted composition operator which induces a generalization of the "Schröder" equation. We will study the problem of eigennumbers (or eigenfunctions) of the operator T.

The Riemann-Hilbert-Poincare problem for holomorphic functions in polydiscs is considered. The solvability conditions and the explicit solution are given.

This paper is devoted_to the study of a system of differential equations:

The existence of local solutions is proved. To the solutions, the representation theorem, the identity theorem, and the maximum modulus principle are formulated without proof.

In this paper, we generalize the results of¹, on theory of hyperholomorphic analysis to the case for several variables, that is, Clifford analysis. These results are not only of theoretical importance but also parctical interest. In fact, we also generalize³, the Dirac operator to . When the base i ={i₁, …,i_n} is given, the Dirac operator D is determined, while ^ψD may consider as the rotation transform on the vector ψ = {ψ¹, … , ψⁿ}. Such a generalization is similar to Descartes coordinate system to the moving coordinate system, therefore, it is very convenient to build some basic formulas on Clifford analysis and to find some analytic representations of differential equations. From this idea, we obtain a very important representation of Bergman kernel function:

this is a key formula for proving Bergman kernel function ^ψB to be a projector of L_p(Ω,C(V_n,0)) onto _ψA^p(Ω,C(V_n,0)), where ^φ,ψC is a compact operator. Moreover, we build up some results on Clifford analysis, such as Stokes formula, Borel-Pompeiu formula and Cauchy formula, propreties of Bergman kernel function. Moreover as their application, we discuss the Dirichlet problem for related bi-analytic functions of several variables on the bounded simply-connected domain with smooth boundary, and then representations of solution are obtained.

In this paper, the Haseman boundary value problems for bianalytic functions are discussed. Using the Cauchy type integral of bianalytic functions and the singular integral equation method, we have not only established an explicit form for general solutions of the Haseman problems, but also found the conditions for the solvability of the above problems. And the relation between the index and the number of the linear independent solutions have been obtained.

The boundary value problem for harmonic functions outside cuts lying on the arcs of a circumference is considered. The Dirichlet condition is given on one side of each cut and Neumann condition is specified on the other side. The problem is reduced to the Riemann-Hilbert problem for complex analytic function, which is solved in a closed form. An explicit solution of the original problem is obtained.

A Haseman boundary value problem is studied in a cut plane for analytic vectors.

A survey on modern constructive results developed in the framework of boundary value problems for analytic functions is presented. Applications of these results in some branches of mathematics and mechanics are also on the discussion.

A general linear elliptic second order system in a simply connected plane domain with continuous coefficients is investigated in connection with the linear oblique derivative boundary value problem using some generalized analytic function theory.

Some problems for complex differential equations of Fuchsian type are studied, which include the Vekua equation with irregular coefficients and the spectrum problem for Riemann-Hilbert-Poincaré problem of analytic functions.

We introduce a concept of lines of catastrophe which determine those sets where a given vector field or complex function, particularly a solution of a system of equations, has a "bad" behavior. These lines are studied for new wide classes of generalized analytic functions. The obtained results are in some extend similar to the deficiency relation in Nevanlinna theory and in the theory of Γ-lines.

In this paper, the convergence of the trigonometric interpolation for 2π-periodic analytic functions on [0,2π] with a sequence of preassigned nodal sets are discussed.

As a generalization of the decomposition of the Hilbert space L₂ for the unit disk an explicit decomposition for domains with Green functions is given. For this the kernels occuring in the well–known higher–order Cauchy–Pompeiu formula are expressed through suitable derivatives of a respective Green function.

The operator curl (or rot) is a very important first-order differential operator, which appears in various physical models, for example in Maxwell equations of electricity and magnetism. In fluid and plasma physics rot appears to measure the velocity of various flows. In recent paper we present some investigations dedicated to teh basis of the mathematical theory of the boundary value problems for rot with lower order terms and, in particular, to the study of eigenvalue problem (1), (2) from a rigorous mathematical point of view.

A certain second order equation not of parabolic type together with proper initial and boundary values is shown to be uncorditionally uniquely solvable. Problems of this kind are treated in ¹.

The existence of support sets in a real topological vector spaces depends upon the existance of certain types of linear continuous functionals, which are called support functionals. In this paper we will show some applications of the support functionals in connection with approximation on C*–algebra.

A new approach to the quantization of the Lie bialgebras based on the Chrednik r-matrices associated to the root systems is presented. It is shown that the number of parameters of quantization is defined by the type of the underlying Lie algebra. Two-parametric quantization of the Lie bialgebra of the B_n type is considered in details and the statement of the theorem concerning the existence of quantization is given.

Some discretization techniques of Kemmoku and Saito are explored further leading to a new star product and some clarification of the role of spectral variables as a phase space. Connections of KP, qKP, and Moyal type dKP are developed.

A pair of abstract equations of a general form with unknowns from a ring, having a factorization pair of subrings are considered. Assuming the correctness of factorization of some elements, built by factors, a theorem of existence and uniqueness is established.

A matrix equation with projectors and an unknown triangular matrix is specified arising, in particular, in mechanics. In case of correct factorization of the matrix coefficients a formula for matrix solutions is established. An illustrative problem from mechanics simulating one case of the considered matrix equation is presented.

We discuss analytic continuation for a certain fourth order elliptic problem. Hörmander proved analytic continuation for solutions to certain higher order elliptic problems ⁶. His more general theorem ⁷ does not apply to this case. Both proofs are based on Carleman estimates. We show that a recently developed Carleman estimate with two large parameters can be used to prove both results.

The problem of the condenser in an annular domain is considered, and the shape of the boundary is investigated under certain overdetermined conditions relating the gradient of the solution to the distance from the origin, as well as to the boundary curvatures. Some model cases are shown, where the domain of the problem is proved to be radially symmetric.

In this article, we announce and summarize two results we obtained on the symmetry of certain heat conductors having a stationary isothermic surface or a stationary hot spot.^5,6 Both results are based on a balance law for solutions of the heat equation and on an asymptotic formula for certain integral means.

Many results have appeared in the literature for equations of the form Δ^pu = f(u), p ≥ 1, in particular for f(u) = e^u. We obtain some explicit radial solutions which are entire functions as well as some which are non-entire functions. The solutions are obtained as a consequence of a differential (in)equality or by a judicious choice of form for the solution.

In this note we discuss a generalized mean value property

for given functions f and μ, and all integrable caloric functions h in Ω.

The particular case of f ≡ 1 and μ being the Dirac delta function supported in would give rise to the corresponding mean value property for harmonic functions, provided the above integral identity holds for some Ω ⊂ ℝⁿ⁺¹ and caloric functions h in Ω.

A phenomenon of propagation of electromagnetic waves through layered medium is considered. The asymptotic formula for the impedance in a complex frequency domain is derived and is applied to the inverse problem of the theory of ground penetrating radars. A new method for processing the data of ground penetrating radars is proposed. The method is based on an analytical continuation of the given data to a domain of complex frequencies. To establish the limits of the applicability of the method an error estimate is derived which shows that the method is numerically efficient if thickness of the layers is not less than some resolution threshold (which is of an order of wave length). Numerical efficiency of the method is illustrated by successful numerical testing.

The well posedness of the Cauchy problems for the 2+1 dimensional wave equation with the two dimensional sphere as target is investigated. We prove that the solutions are not unique in Sobolev spaces below the critical exponent and also in suitable Besov spaces of critical regularity. We also investigate ill-posedness in the critical Sobolev space of order 1, showing that the problem is not well-posed in presence of a forcing term.

We investigate the asymptotic behaviour of solutions to the initial-boundary value problem for the nonlinear dissipative wave equation in the whole space or the exterior domain outside a star-shaped obstacle. We shall treat the nonlinear dissipative term like a₁(1 + |x|)^-δ|u_t|^βu_t (a₁, β, δ > 0) and prove that the energy does not in general decay. Further, we can deduce that the classical solution is asymptotically free and the local energy decays at a certain rate as t goes to infinity.

We shall consider a 1-dimensional semilinear wave equation in a finite interval (0, l). The semilinear term f(t, ·) is odd and increasing in a neighborhood of 0. We shall show that the solution u(x, t) oscillates on the time t, that is, let x be any fixed in (0,l), then u(x, t) changes its sign infinitely many times as t evolves.

We shall deal with nonlinear 3-dimensional wave equations defined in sphere-symmetric domain with time-periodically oscillating boundaries. We shall show the existence of sphere-symmetric time-periodic solutions of BVP for the wave equations with small nonlinear time-periodic term. Quite recently, it is shown in ² that for a nonlinear one-dimensional wave equation, there exists a time-periodic classical solution with small amplitude. However, as far as we know, there are no works on the existence of periodic solutions of BVP for nonlinear multi-dimensional wave equations in domain with the time-periodically oscillating boundaries. We shall show our results, combining the methods developed in ² with the methods in ³. Our methods strongly depend on the character of 3-space dimension.

For solutions to one-dimensional Schrödinger equation estimates are obtained provided the real part of the potential is greater than some positive constant.

This lecture is divided in two parts.

In the first part we consider weak equations and systems (linear), that is to say, 2-evolution linear operators with multiple real characteristic roots in the sense of Petrowsky and scalar conditions involving principal part and also involving part with lower order of the operator so as H^+∞ Cauchy problems to be well posed, in analogy of Levi conditions for weakly hyperbolic operators (M. Ghedamsi, D. Gourdin, J. Takeuchi, M. Mechab ⁴).

In the second part we give smooth solutions of global Cauchy problem for nonlinear perturbation of iterated linear Schrödinger type equations which have linearized associated Cauchy problem satisfying scalar condition of the first part (M. Ghedamsi, D. Gourdin ³).

In this note, we shall discuss the smoothing effect of dispersive equations with constant coefficients. The main result is the high frequency estimate for the inhomogeneous initial value problem. The detail of the proof and the further results will be given in the subsequent paper ⁴.

We consider the Gevrey smoothing effects of the solutions to the Cauchy problem for Schrödinger-type equations.

We prove that if the initial data decay like e^-c〈x〉κ, where c > 0 and 0 < κ < 1, in a neighborhood of the x-projection of the backward bicharacteristic issuing from a point (y₀,η₀), then (y₀, η₀) does not belong to the Gevrey wave front set of order 1/κ of the solution.

In this note we discuss the existence of global in time solutions to the mixed problem for quasi-linear characteristic symmetric hyperbolic systems with strong dissipation assuming that the data are small enough.

We consider the existence and some decay estimates of solutions of the following initial boundary value problem for some nonlinear degenerate parabolic equations:

Here u(x,t) is a scalar function of the spatial variable x(∈ Ω) and time t(> 0), and Ω is a regular bounded domain in R^N (N > 2) with the smooth boundary ∂Ω, R⁺ = (0,+∞), Δ denotes the N-dimensional Laplace operator, m > 1, p > 0, and f(x,t) and u₀(x) satisfy some hypotheses which will be given later.

We study the initial value problem for a general nonlinear hyperbolic equation of first order . Suppose that Hamiltonian H(t, x, u,p) belongs to W^{2, ∞}((0,N) × Rⁿ × (-N,N) × (-N,N)ⁿ) for any natural number N. First we establish a lifting principle for first order hyperbolic equations, which decomposes the nonlinear hyperbolic partial differential equation into the corresponding Hamilton system of ordinary differential equations. From this fact the uniqueness theorem of the solution in W^2,∞((0,T) × Rⁿ) directly follows. Denote the set W^k,∞(Rⁿ) also by for k ∈ N. However we introduce a different topology into , which also makes this space complete. We can construct the solution of evolutional type in the space , i.e. , if the initial data u₀ = u(0, ·) is supposed to belong to W^2,∞(Rⁿ). At the same time we obtain an estimate T of the lifespan of the solution, to show that is positive for above H and any initial data u₀ ∈ W^2,∞(Rⁿ). Moreover is verified to be equal to the lifespan of the corresponding Hamilton flow X(t,y).

In this work we study the well-posedness of the Cauchy Problem for weakly hyperbolic systems in inhomogeneous Gevrey classes, that extend the standard Gevrey functions. Particular cases are represented by anysotropic Gevrey classes and generalized Gevrey classes defined in terms of a complete polyhedron. Well-posedness is obtained by imposing some restrictions on the lower order terms, depending on the Gevrey class and on the order of the operator. The main tools of the proof are the technique of quasi-symmetrization and approximated energy estimates.

We describe a way to solve locally the Cauchy problem for nonlinear hyperbolic equations with characteristic roots of constant multiplicity. A universal reduction to a first order system gives the optimal space of well posedness under different assumptions on the lower order terms or on the regularity of the equation. The method gives also the propagation of the analytic regularity of the solution in domains of influence.

We prove C^∞ and Gevrey well posedness of the Cauchy problem for a strictly hyperbolic operator with non-absolutely continuous coefficients in the time variable.

We study the Cauchy problem for hyperbolic systems with multiple characteristics. Our purpose is to obtain necessary conditions on both the principal symbol and the lower order terms in order that the Cauchy problem has a unique distribution solution.

In this paper we prove the non-solvability, in a Gevrey frame, of a class of two variables partial differential operators.

The motion of viscous incompressible fluids is described by the Navier-Stokes equations in a bounded domain in R³. A modification of the Navier-Stokes equations is investigated. In the physical fluid dynamics, the Navier-Stokes equations have been formulated under the assumption that the rate of deformation of fluids is sufficiently small and therefore the relation between the viscous stress and the rate of deformation is linear. The rate of deformation depends on the velocity gradient in the fluids. As a result the Navier-Stokes equations are found to be well representing the motion of fluids for small velocity gradients. From such a consideration as this, we have found out a modified equation by taking the motion of fluids for large velocity gradients also into account. The flow based on the modified Navier-Stokes equations varies from the Navier-Stokes flow into a non-Newtonian flow, namely the solution of the modified Navier-Stokes equations is switched so as to satisfy a non-Newtonian equation at all the times when the velocity gradient gets larger.

a(ξ) is a strongly hyperbolic matrix m × m of reduced dimension greater or equal to ; we state that a(ξ) is presymmetric. The schedule of the proof is given when m = 5.

There are many ways to give a rigorous meaning to the Feynman path integral. In my talk especially the method of the time-slicing approximation determined through broken line paths is studied. It has been proved that these time-slicing approximate integrals of the Feynman path integral in configuration space and also in phase space converge in L² space as the discretization parameter tends to zero. In my talk it is shown that these approximate integrals of the Feynman path integral and more general form of the Feynman path integral converge in some weighted Sobolev spaces as well. In addition, as an application of this convergence result in the weighted Sobolev spaces, a rigorous proof is given of the path integral representation of correlation functions and the reduction of wave functions by the measurement.

A one-dimensional free boundary problem of hyperbolic type will be analyzed. This problem arises from the physical model "Peel a thin film from a domain". The local solutions have been already given for our problem. However, it is not sure that the solutions can be extended globally. In this note, by applying an iteration method, it will be shown that time-global solutions exist.

This paper is concerned with the mathematical modeling of second harmonic generation of nonlinear optics in periodic structures. The governing equation is a system of nonlinear Maxwell's equations. The well-posedness of the model is studied. Preliminary results on L_p regularity of solutions for the model problems are presented. The results are expected to be useful in our current effort of solving the underlying optimal design problem. Some future directions are also highlighted.

The interaction between diffusion and absorption gives several interesting phenomena in the flow through porous media with some kind of evaporation effect. In particular, it can be observed from numerical computations that the effect of strong absorption causes support splitting or non-splitting phenomena. The aim of this paper is to find some sufficient conditions under which such phenomena appear.

The author gave a general lecture based on the paper entitled PRINCIPLE OF TELETHOSCOPE which will be published in the Proceedings of the Graz Workshop "Functional-analytic and complex methods, their interactions and applications to Partial Differential Equations". In this Proceedings, we shall cite the main materials of the paper. We shall present a principe of telethoscope which is stating that many solutions of linear partial differential equations of parabolic and hyperbolic types are represented by their local data of space and time of the solutions.

Bifurcation of stationary solutions to a reaction-diffusion system with simple nonlocal unilateral boundary conditions described by variational inequalities is studied with diffusion coefficients as a two-dimensional parameter. Using former results about a destabilizing effect of unilateral conditions, the existence of bifurcation points is obtained in the domain of parameters where a bifurcation for the corresponding classical boundary conditions is excluded. Our new results concerning smooth bifurcation branches for variational inequalities are applied to these bifurcation points.

Local bifurcation of positive solutions from the line of trivial solutions is considered for a nonlinear elliptic boundary value problem arising in population dynamics, having nonlinear boundary conditions. The bifurcation theory based on the Lyapunov and Schmidt procedure and the super-sub-solution method are used.

The oblique derivative problem for the Laplace equation is studied on a plain domain. For a domain with Ljapunov boundary and Hölderian boundary condition a classical solution is studied. For a domain with Lipschitz boundary and L^p boundary condition a solution in the sense of the nontangential limit is studied. For a domain with bounded cyclic variation and a real measure as a boundary condition a solution in the sense of distribution is studied.

The problem for harmonic functions outside a cut of an arbitrary shape in a plane is studied. The Dirichlet condition is given on one side of the cut and Neumann condition is specified on the other side. Basing on the method of potentials the problem is reduced to the uniquely solvable Fredholm integral equation of the second kind. The behavior of a solution gradient at the tips of the cut is studied.

We consider an approach to the classical center-focus problem. For the wide class of systems with analytical right hand sides, the necessary and sufficient conditions for the center are given, which are expressed through the variational equations of higher order. The necessary conditions for the center are obtained along with the asymptotics of the Poincaré mapping in case of the focus. The effectiveness of the suggested method is demonstrated on some examples.

A significant role in mathematical modeling and algorithms for applications to processing of genetic data (for example, Gene expression data) is played by infinite-dimensional P-spaces. Connected with them are infinite-dimensional P-groups and P-algebras. In particular, it's the group of all local analytical homeomorphisms of the real line into itself with a fixed point and the group of all formal series with real coefficients with formal fixed point. Both groups are P-groups. We prove here that these groups have the classical "canonical coordinates of first and second kinds". In genetic data processing the utilization of the topological-algebraic properties of P-spaces, P-groups and P-algebras permit to find "Genes functionality".

A stationary diffraction problem of acoustic wave by a thin elastic cylindrical shell is considered. The wave field (the potential of the speeds of the medium particles) has to satisfy the Helmholtz equation as well as the radiation conditions at infinity and the condition for equality of normal components of the speeds of the particles of the medium and the shell. The last one and the equations of the theory of the thin elastic shell (Kirhgoff-Ljav equations) allows us to derive the boundary condition for the wave field. It contains the tangential derivatives up to the sixth order. The problem of diffraction on a non-momentum circular shell is considered as a model. The solution of the mode! problem clarifies the type of asymptotic expansion (ansatz) for the wave field in the problem of diffraction by a convex shell of an arbitrary form. The wave field scattered in the bright region is represented by uniform asymptotic expansion in inverse powers of frequency. It contains 3 terms corresponding to the direct, reflected and head waves. The last one is closely connected with the lengthwise oscillations of the shell.

The boundary value problem for the equation of gravity-inertial waves outside several cuts in a plane is studied. The jump of the unknown function and the jump of the analogue of the Neumann operator acting on this function are specified on the cuts. The problem is studied under different conditions at infinity, which lead to different uniqueness and existence theorems. The solution of this problem is constructed in the explicit form by means of single layer potential and angular potential.

A high-order accurate algorithm for computation of stress fields inside polygonal domains with cracks, holes and V-notches is presented. The algorithm is based on Fredholm second kind integral equations, which ensures numerical stability. Special basis functions in the domain corners and careful numerical implementation of integral operators gives high order of accuracy. Numerical results of the highest quality are presented. A relative error on the order of 10^-5 is obtained using less than two hundred discretization points and less than ten seconds on a workstation. Results from large scale computations are also presented.

Finite volume schemes are widely used in the numerical solution of conservation laws such as those occuring in C.F.D. Finite element approximations are naturally well suited to second-order operators like those modelling diffusion terms. Then an attractive approach for solving time-dependent convection-dominated diffusion problems consists in combining finite volume for convection and finite element for diffusion. In this paper, we propose to provide a review and a numerical comparison of these approaches in the framework of triangular meshes in two-dimension. New methods for combining finite volumes and finite elements are also defined.

In this paper we obtain a boundary integral method in order to determine the oscillatory Stokes flow due to the translational oscillations of two bodies in an unbounded viscous incompressible fluid. As an application of this boundary integral method, we treat both cases of small- and high-frequency oscillations.

The paper describes a program complex for evaluation of Macdonald's function of an arbitrary real argument and of a real and some complex values of order. Approximation method, quadrature formulas, decomposition on Chebyshev polynomials, recurrent relations and others are used for computations. The programs of complex are used in the numerical solution of mixed boundary value problems by means of the method of integral transforms. The using of different quadrature formulas for evaluation K_1/2+iβ(x) are considering in details (Philon's quadrature formulas, Gauss-Legendre quadrature formulas, quadrature formulas of trapezoidal kind). The author's latest papers are observed. It is shown that the best accuracy and speed are achieved on using of quadrature formulas of trapezoidal kind in computation of modified Bessel function K_1/2+iβ(x). The special procedure for "optimal" step's choice is used based on asymptotic decompositions of these functions. The suggested method allows to decrease substantially the machine time which is important in computation of special functions.

A new analytical solution of the heat conduction equation inside a spherical droplet is suggested. The droplet is assumed to be heated by convection from surrounding hot air — a situation typical in many engineering applications. The well known 'rapid mixing limit' is obtained from the general solution in the limiting case of infinitely large thermal conductivity of droplets. Results are applied to a typical problem of fuel droplet heating in a diesel engine. In agreement with the previously reported results based on numerical analysis, it is shown that the infinite conductivity model can substantially overpredict the heating-up time of droplets' surfaces.

It is shown how various formulas for integral transforms and intertwining operators in classical transmutation theory can be adapted to the quantum group context and q-analysis. Other material on integrable systems, tau functions, and group theory is sketched.

Random surface reflection problems are most importantly characterized by ξ₀, the rms surface displacement, Λ, proportional to the correlation length, and λ, the radiation wavelength (wave numbers: K = 2π/Λ, k = 2πλ). For small λ, higher frequencies (HF), geometric optics/acoustics is useful. Indeed for HF and small-sloped (ξ₀/Λ small), Gaussian surfaces, the angular distribution is itself Gaussian. For larger λ perturbation theory is usual. It will be seen that the present expansion has substantial advantages in such a (commonly occurring) regime. The stochastic expansion [also called the Wiener-Hermite (WH)] is used to represent field functions reflected from random surfaces which surfaces have Gaussian statistics. The special problem considered is for a two-dimensional conducting surface with an incident, polarized, electromagnetic wave. This is the same boundary condition as that for the reflection of sound from the ocean surface, incident from the water side. The WH expansion is in terms of polynomials of a basic element. Here we use the presumed Gaussian surface as that basis. The polynomials are constructed so as to be mutually, statistically orthogonal. The expansion of exp(aξ) (ξ the Gaussian surface) is known to all WH orders, a decided advantage since this functional is central to reflection problems. The expansions have been used extensively and successfully for turbulence problems. In this work, we include but the first three terms; multiple reflections are clearly visible. Such reflections greatly enhance the expected back scatter at low angles of incidence. The reflection approaches, for larger surface slopes, a Lambert's law: the reflected field following the square of the cosine of the angle, regardless of the incident direction as is the case if the reflected field is fully diffuse. The result is shown to be greatly superior to perturbation theory, as judged by energy conservation. The method is readily extended to more general problems. Results will be presented for a variety of surface correlation lengths, for various incident angles and for several frequencies. It is emphasized that the expansion is not a perturbation one for it contains a swath composed of all perturbation orders.

This paper is concerned with the existence, uniqueness and uniform decay for the solutions of the coupled Klein-Gordor-Schrödinger damped equations

where Ω is a bounded domain of ℝⁿ, n < 3, F : ℝ² → ℝ is a C¹–function; γ, β, θ are constants such that γ, β > 0 and 1 ≤ 2θ ≤ 2.

This talk overviews some recent and new results about phase transitions in hyperbolic conservation laws. Various applications are considered: liquid ↔ vapor phase transitions, a combustion model and some equations describing traffic flows. These examples lead to different mathematical problems having the same underlining framework.

We consider the nonlinear model of the wave equation

subject to the following nonlinear boundary conditions We show existence of solutions by means of Faedo-Galerkin method and the uniform decay is obtained by using of the multiplier technique.

In ¹, H.O. Cordes mainly gave some a priori estimates of classical solutions of the Dirichlet problem for quasilinear elliptic equations of second order

In ², Yu.A. Alkhutov and I.T. Mamedov discussed the solvability of Dirichlet problem for linear nondivergence uniformly parabolic equations with measurable coefficients: where the coefficients satisfy the conditions: In this paper, we try to discuss the solvability of mixed boundary value problem for nondivergence uniformly elliptic systems of second order equations with measurable coefficients in higher dimensional domains.

This contribution is concerned with a phenomenological model of isotropic finite elasto-plasticity valid for small elastic strains applied to polycrystalline material. We prove a local in time existence and uniqueness result making use of an extended Korn's first inequality. To the best of our knowledge this is the first rigorous result concerning classical solutions in geometrically nonlinear finite visco-plasticity.

In this paper the nonlinear inverse scattering problem for the determination of the mass density of an elastic inclusion is recast as an optimization problem. Three fully nonlinear iterative algorithms which have been proven to be effective for the electromagnetic and acoustic case, are applied to a more complicated elastodynamic wave inversion. Finally with the use of synthetic examples advantages and disadvantages of these three iterative inversion methods are presented.

We describe a multipole theory of photonic crystal or microstructured optical fibers (MOF). We present general results obtained with this method concerning practical aspects of MOFs, such as dispersion, losses and number of guided modes. This article is intended to introduce the study of MOFs with the multipole method and therefore concentrates on a simple presentation of the theory and on the most important results.

In this study we propose a multiplicative regularization scheme to deal with the problem of detecting and imaging of homogeneous dielectric objects (the so-called binary objects). By considering the binary regularizer as a multiplicative constraint for the Contrast Source Inversion (CSI) method we are able to avoid the necessity of determining the regularization parameter before the inversion process has been started. Numerical results show that the binary CSI method is able to obtain a reasonable reconstruction result even when a wrong estimate of the material parameter is used.

We consider "furtivity" and "masking" problems in time dependent acoustic obstacle scattering. Roughly speaking a "furtivity" ("masking") problem consists in making "undetectable" ("unrecognizable") an object immersed in a medium where an acoustic wave that scatters on the object is propagating. The detection (recognition) of the obstacle must be made through the knowledge of the acoustic field scattered by the object when hit by the propagating wave. These problems are interesting in several application fields. We formulate a mathematical model for the "furtivity" and "masking" problems considered consisting in optimal control problems for the wave equation. Using the Pontryagin maximum principle we show that the solution of these control problems can be characterized as the solution of a suitable exterior problem for a system of two coupled wave equations. The numerical solution of these systems involving partial differential equations in four (space, time) independent variables is a critical issue when reliable and efficient procedures to solve the furtivity or masking problem are required. High performance parallel algorithms are desirable to solve these systems. We suggest a computational method well suited for parallel computing and based on an adapted version of the operator expansion method originally introduced in ¹ and developed by the authors and some co-authors (see ², ³, ⁴, ⁵ and the references therein). Some numerical results in the form of computer animations can be found in the website http://www.econ.unian.it/recchioni/w8. Finally we make some comments on the possible extension of this work to electromagnetic obstacle scattering.

Several theoretical aspects of utilization of open waveguide resonators (OWR) in circular waveguides for the non-destructive control of dielectric parameters of various media (dielectrics, magnetic, chiral materials etc) that are widely used in practice ^1,2, are discussed in the paper. The parameter control by means of OWR relies on the solution of relevant boundary value and eigen boundary value problems for corresponding OWR and on definition of functional relation between electromagnetic characteristics (Q-factor, resonant frequency, amplitudes of scattered field) and parameters of the material under investigation (permittivity and permeability, chirality factor).

This paper presents new results of imaging floating objects from underwater by the generalized dual space indicator method with an automatic filtering technique. It is assumed that there is an unknown object floating in the surface of a shallow water waveguide. Sound signals are generated from an array of transducers, and the scattered fields are received by a hydrophone array below the floating object. The imaging of the unknown object is formulated as a dual space indicator. Mathematical analysis and computational results are presented.

It is shown that the use of: 1) the physical optics approximation (POA) to simulate the response (i.e., predictor), on a line above the open cylindrical boundary in its cross-section plane, to two probe plane waves of different frequencies, 2) the intersecting canonical boundary approximation (ICBA) of the wave-boundary interaction for a trial boundary (i.e., estimator), and 3) an analysis of the comparison equation and/or cost function, employing the ICBA as estimator and either the ICBA or POA as predictor, enables a non-ambiguous identification of the scattering boundary.

We consider the inverse problem of recovering a 2D periodic structure from scattered waves measured above the structure. Following an approach by Kirsch and Kress, the inverse problem is reformulated as a nonlinear optimization problem. The resulting Tikhonov regularized least squares problem is then solved iteratively by the Levenberg-Marquardt algorithm. Numerical results for synthetic data demonstrate the practicability of the inversion algorithm. We also present some convergence results for the Tikhonov regularization of the reconstruction problem and for the optimization method.

We propose a strategy for a priori choice of regularizing parameters and discretization mesh sizes in the Tikhonov regularization for guaranteeing a convergence rate. Our strategy is based on the conditional stability estimate of the inverse problem under consideration.

As relevant in mineral exploration of the Earth using inductive electromagnetic means, (diffusive) scattering by a perfectly conducting ellipsoidal anomaly placed in a homogeneous conductive medium and illuminated by a time-harmonic magnetic dipole, is considered by means of a low frequency expansion in positive integral powers of (jk), k complex wavenumber of the exterior medium. The aim is to construct a simple yet robust model of the fields to identify the anomaly when magnetic fields are collected nearby. The approach is illustrated by field calculations and inversions of both synthetic and experimental data.

The goal of this paper is to present recent progress on several aspects of an inverse medium problem. A recursive linearization method for solving the nonlinear inverse problem is introduced. Convergence of the method is studied. We address issues on regularity and stability for the scattering map which maps the scatterer to the scattered field. Preliminary computational results are also presented.

The operative algorithm of unbalance evaluation of a rotor based on the solution of the inverse problem of recognition is suggested. The vibrations of the rotor supports (accelerations or velocities or displacements) in two mutually perpendicular directions during the work for a few rotor rotations are used as the initial information. The Tikhonov regularization method is applied for solution of this ill-posed (unstable) problem taking into consideration the inaccuracy of mathematical model. The method of choice of the special mathematical model, which leads to increasing of exactness of approximate solution, is proposed. The numerical calculations of examples are given to illustrate this algorithm.

A directional Newton method is proposed for solving systems of m equations in n unknowns. The method does not use the inverse, or generalized inverse, of the Jacobian, and applies to systems of arbitrary m, n. Quadratic convergence is established under typical assumptions (first derivative "not too small", second derivative "not too large"). The method is stable under singularities in the Jacobian.

We present here a charge simulation method for problems of two-dimensional flows past obstacles in a two-dimensional periodic array. Our problem is mathematically a boundary value problem of the Stokes equation with periodic boundary conditions, to which it is difficult to give a good approximation by the ordinary charge simulation method. We propose a charge simulation method for our problem based on the periodic fundamental solution obtained by Hasimoto. In our method, the solution is approximated by using a linear combination of the periodic fundamental solutions.

We prove existence and uniqueness of the solution to nonlinear polar non-Lipschitz Volterra integral equation in weighted Colombeau spaces of generalized functions when the driving term is in (resp. W^-∞,p). We prove the coherence with classical Lipschitz cases.

To classify high energy physics events, we propose to use linear and non linear discriminant functions. Global event shape and connected new morphological variables are considered. Three kinds of non linear discriminant functions are designed. All these have proven to be more efficient classifier than the linear functions. The efficiencies and purities achieved with the non linear classifiers are in average 1 to 7.