An automatic integration procedure for infinite range integrals involving oscillatory kernels

Let the real functionsK(x) andL(x) be such thatM(x)=K(x)+iL(x)=e^ix g(x), whereg(x) is infinitely differentiable for all largex and is non-oscillatory at infinity. We develop an efficient automatic quadrature procedure for numerically computing the integrals _a K(t)f(t) and _a L(t)f(t)dt, where the functionf(t) is smooth and nonoscillatory at infinity. One such example for which we also provide numerical results is that for whichK(x)=J (x) andL(x)=Y (x), whereJ (x) andY (x) are the Bessel functions of order . The procedure involves the use of an automatic scheme for Fourier integrals and the modified W-transformation which is used for computing oscillatory infinite integrals.     

Fast Preconditioned Iterative Methods for Convolution-Type Integral Equations

We consider solving the Fredholm integral equation of the second kind with the piecewise smooth displacement kernel x(t) + _j=1 ^m µ_j x(t – t _j) + ₀ k(t – s)x(s) ds = g(t), 0 t , where t _j (–, ), for 1 j m. The direct application of the quadrature rule to the above integral equation leads to a non-Toeplitz and an underdetermined matrix system. The aim of this paper is to propose a numerical scheme to approximate the integral equation such that the discretization matrix system is the sum of a Toeplitz matrix and a matrix of rank two. We apply the preconditioned conjugate gradient method with Toeplitz-like matrices as preconditioners to solve the resulting discretization system. Numerical examples are given to illustrate the fast convergence of the PCG method and the accuracy of the computed solutions.     

Finite-dimensional attainable sets for stochastic control systems

Letx _t ^u () be a stochastic control system on the probability space (, ,P) intoR ⁿ. We say that the pointxR ⁿ is (, ) attainable at timet if there exists an admissible controlu such thatP _xo{x _t ^u ()S (x)}, wherex ₀()=x ₀, 0, 10, andS (x) is the closed Euclidean -ball inR ⁿ centered atx. We define the attainable setA (t) to be the set of all pointsxR ⁿ which are (, ) attainable at timet. For a large class of stochastic control systems, it is shown thatA (t) is compact for eacht and continuous as a function oft in an appropriate metric. From this, the existence of stochastic time-optional controls is established for a large class of nonlinear stochastic differential equations.This research was supported by the National Research Council of Canada, Grant No. A-9072.     

Existence of quantum diffusions

Summary A quantum diffusion (A, A, j) comprises of unital *-algebras A and A and a family of identity preserving *-homomorphisms j=(j _t : t0) from A into A. Also j satisfies a system of quantum stochastic differential equations dj _t (x ₀=j _t( _j ⁱ (x ₀))dM _i ⁱ , j ₀(x ₀)=x ₀I for all x ₀A where _j ⁱ , 1i, jN are maps from A to itself and are known as the structure maps. In this paper an existence proof is given for a class of quantum diffusions, for which the structure maps are bounded in the operator norm sense. A solution to the system of quantum stochastic differential equations is first produced using a variation of the Picard iteration method. Another application of this method shows that the solution is a quantum diffusion.     

A Stability Estimate for a Solution to the Problem of Determination of Two Coefficients of a Hyperbolic Equation

We consider the problem of determination of two coefficients (x) and q(x) in a hyperbolic equation. Here (x) is the coefficient of the first derivative with respect to t and q(x) is the coefficient of the solution itself. We suppose that these coefficients are small in some norm and supported in a disk D. Oscillations are excited by the impulse function (t)(x·) supported on the straight line t=0, x·=0. Here is a unit vector playing the role of a parameter of the problem. The acoustic field generated by this source lying outside D is measured at the points of the boundary of D together with the normal derivative on some time interval of a fixed length T for two different values of the parameter . We prove that, for a sufficiently large T, the given information determines the sought coefficients uniquely. We obtain a stability estimate for a solution to the problem.     

A Stability Estimate for a Solution to a Two-Dimensional Inverse Problem of Electrodynamics

We consider the problem of finding the three coefficients c(x), (x), and q(x) in a hyperbolic equation. Here c(x) is the coefficient at the Laplace operator, (x) is the coefficient of the first time derivative, and q(x) is the coefficient of the lower-order term. The problem results from the inverse electrodynamic problem of finding the electrodynamic parameters of an isotropic medium under the assumption that the properties of the medium and the exterior current are independent of one coordinate. We suppose that the coefficients c(x)-1, (x), and q(x) are small in some norm and their supports are contained in some disk B. This is equivalent to the assumption that the electrodynamic parameters of the medium are close to constants. We suppose that the source initiating oscillations has the form of the impulse function (t)(x·) localized on the set t=0, x·=0. Here is a unit vector playing the role of a parameter of the problem. The electromagnetic field excited by this source applied outside B is measured at points of the boundary of the domain B on some time interval of a fixed length T counted from the moment of arrival of the signal from the source for three different values of the parameter . It is proven that, for a sufficiently large T, these data determine the sought coefficients uniquely. We obtain a conditional stability estimate for a solution to the problem.     

A Strongly Semismooth Integral Function and Its Application

As shown by an example, the integral function f : ⁿ , defined by f(x) = _a ^b[B(x, t)]₊ g(t) dt, may not be a strongly semismooth function, even if g(t) 1 and B is a quadratic polynomial with respect to t and infinitely many times smooth with respect to x. We show that f is a strongly semismooth function if g is continuous and B is affine with respect to t and strongly semismooth with respect to x, i.e., B(x, t) = u(x)t + v(x), where u and v are two strongly semismooth functions in ⁿ . We also show that f is not a piecewise smooth function if u and v are two linearly independent linear functions, g is continuous and g 0 in [a, b], and n 2. We apply the first result to the edge convex minimum norm network interpolation problem, which is a two-dimensional interpolation problem.     

Density functions for prime and relatively prime numbers

Letr ^*(x) denote the maximum number of pairwiserelatively prime integers which can exist in an interval (y,y+x] of lengthx, and let ^*(x) denote the maximum number ofprime integers in any interval (y,y+x] whereyx. Throughout this paper we assume the primek-tuples hypothesis. (This hypothesis could be avoided by using an alternative sievetheoretic definition of ^*(x); cf. the beginning of Section 1.) We investigate the differencer ^*(x)—^*(x): that is we ask how many more relatively prime integers can exist on an interval of lengthx than the maximum possible number of prime integers. As a lower bound we obtainr ^*(x)—^*(x)<x ^c for somec>0 (whenx). This improves the previous lower bound of logx. As an upper bound we getr ^*(x)—^*(x)=o[x/(logx)²]. It is known that ^*(x)—(x)>const.[x/(logx)²];.; thus the difference betweenr ^*(x) and ^*(x) is negligible compared to ^*(x)—(x). The results mentioned so far involve the upper bound or maximizing sieve. In Section 2, similar comparisons are made between two types of minimum sieves. One of these is the erasing sieve, which completely eliminates an interval of lengthx; and the other, introduced by Erdös and Selfridge [1], involves a kind of minimax for sets of pairwise relatively prime numbers. Again these two sieving methods produce functions which are found to be closely related.     

Wave operators in a multistratified strip

One investigates the scattering theory for the positive self-adjoint operatorH=–· acting in with = × and a bounded open set in ^n–1,n2. The real-valued function belongs toL (), is bounded from below byc>0 and there exist real-valued functions ₁ and ₂ inL () such that – _j ,j=1,2 is a short range perturbation of _j when (–1) ^j x _n +. One assumes _j = ^(j) 1_R,j=1,2, with ^(j) L bounded from below byc>0. One proves the existence and completeness of the generalized wave operators _j ^± =s – _j e ^–itHj ,j=1,2, withH _j =–· _j and _j : equal to 1 if (–1) ^j x _n >0 and to 0 if (–1) ^j x _n <0. The ranges ofW _j ^± :=( _j ^± )^* are characterized so that W ₁ ^± =Ran and . The scattering operator can then be defined.     

Plane Domains Which Are Spectrally Determined

This paper studies the trace of the heat kernelZ(t) _j=1 exp ( _j t), where{ _j } are the eigenvalues of atwo-dimensional Dirichlet or Neumann Laplace operator. FromZ(t), a sequence of invariants (geometrical invariants)such as area, boundary measure, Euler characteristics, etc., can bedetermined. Using these invariants, the existence of the nondisk domainswhich are determined from the information of Dirichlet and Neumannspectrum, can be shown. In addition, we prove that the number of suchdomains is infinite (uncountable) and these domains are not similar eachother.     

Integral averages and oscillation of second order sublinear differential equations

New oscillation criteria are given for the second order sublinear differential equation

On the Asymptotic Behaviour of the Solutions of a System of Functional Equations

In this paper we study the asymptotic behaviour of the solutions of the functional equation

Finite extinction time for some perturbed Hamilton-Jacobi equations

In this paper we study initial value problems likeu _t–R¦u¦^m+u^q=0 in ⁿ× ₊, u(·,0⁺)=u_o(·) in ^N, whereR > 0, 0 <q < 1,m 1, andu _o is a positive uniformly continuous function verifying –R¦u _o¦^m+u ₀ ^q 0 in ^N . We show the existence of the minimum nonnegative continuous viscosity solutionu, as well as the existence of the function t(·) defined byu(x, t) > 0 if 0<t<t (x) andu(x, t)=0 ift t (x). Regularity, extinction rate, and asymptotic behavior of t(x) are also studied. Moreover, form=1 we obtain the representation formulau(x, t)=max{([(u _o(x – t))^1–q –(1–q)t]₊)^1/(1–q): ¦¦R}, (x, t) ₊ ^N+1 .Partially supported by the DGICYT No. 86/0405 project.     

A central limit theorem for solutions of the porous medium equation

We study the large–time behavior of the second moment (energy) for the flow of a gas in a N-dimensional porous medium with initial density v₀(x) 0. The density v(x, t) satisfies the nonlinear degenerate parabolic equation v_t = v^m where m > 1 is a physical constant. Assuming that for some > 0, we prove that E(t) behaves asymptotically, as t , like the energy E_B(t) of the Barenblatt-Pattle solution B(|x|, t). This is shown by proving that E(t)/E_B(t) converges to 1 at the (optimal) rate t^{–2/(N(m-1)+2)}. A simple corollary of this result is a central limit theorem for the scaled solution E(t)^N/2v(E(t)^1/2x, t).     

Dual-orthogonal polynomials

Let (x, ) and (x,) be two functions,x[a, b] and { _j } _j=1 and { _j } _j=1 be two sequences where _{i j} and _{i j} whenij. We define the vector spacesU _k =span{(x, _j )} _j=1 ^k andV _k =span{(x, _j )} _j=1 ^k where we assume thatdim(U _k )=dim(V _k )=k,k1. We then look for the generalized polynomialsp _m xU _m+1\U _m so that _a ^b p _m (x)(x, _j )d(x)=0,j=1,2,...,m. If such generalized polynomials exist for allm1 we say that {p _m } _m=1 is a dual-orthogonal polynomial sequence from {(x, _j )} _j=1 to {(x, _j )} _j=1 with respect to the distribution (x),x[a, b]. In this article we present existence theorems for dual-orthogonal polynomials, explicit formulae forp _m(x), theorems about the zeros ofp _m(x), and, in the end, a Gauss-type quadrature formula for dual-orthogonal polynomials.     

Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R 2

For a bounded regular Jordan domain in R ², we introduce and study a new class of functions K() related on its Green function G. We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular nonlinear elliptic equation u+(x,u)=0, in D(), with u=0 on and uC(), where is a nonnegative Borel measurable function in ×(0,) that belongs to a convex cone which contains, in particular, all functions (x,t)=q(x)t ^–,>0 with nonnegative functions qK(). Some estimates on the solution are also given.     

The Most Visited Sites of Certain Lévy Processes

Let X be a symmetric Lévy process with

On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence

Let M _f(r) and _f(r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let be a continuously differentiable function convex on (–, +) and such that x = o((x)) as x +. We establish that, in order that the equality be true for any entire function f, it is necessary and sufficient that ln (x) = o((x)) as x +.     

Large deviations for squares of Bessel and Ornstein-Uhlenbeck processes

Let (X_t⁽⁾,t0) be the BESQ process starting at x. We are interested in large deviations as for the family {^–1X_t⁽⁾,tT}, – or, more generally, for the family of squared radial OU process. The main properties of this family allow us to develop three different approaches: an exponential martingale method, a Cramér–type theorem, thanks to a remarkable additivity property, and a Wentzell–Freidlin method, with the help of McKean results on the controlled equation. We also derive large deviations for Bessel bridges.Mathematics Subject Classification (2000): 60F10, 60J60     

Controllability of Nonlinear Evolution Systems with Preassigned Responses

In this paper, we study the approximate controllability with preassigned responses of the nonlinear delay systems x(t)=A(t)x(t)+f(t, x(t), x((t)), u(t)) and L(x(t), x(t))=A(t)x(t)+f(t, x(t), x((t)), u(t)). The controllability is not governed by an associated linear system, but by conditions on f or A involving the domain of A(t). No compactness assumptions are imposed in the main results.