Equidistribution for meromorphic transforms and the ddc-method

In this paper we give an introduction to the notion of meromorphic transform. We describe some equidistribution problems and their solution, using the dd^c-method. In particular, we give some statistical properties of the equilibrium measure for meromorphic maps on compact Kähler manifolds: K-mixing, exponential decay of correlations and central limit theorem.

     

Equidistribution for meromorphic transforms and the dd<Superscript>c</Superscript>-method

In this paper we give an introduction to the notion of meromorphic transform. We describe some equidistribution problems and their solution, using the dd^c-method. In particular, we give some statistical properties of the equilibrium measure for meromorphic maps on compact Kähler manifolds: K-mixing, exponential decay of correlations and central limit theorem.     

A nearly optimal preconditioner for the Navier–Stokes equations

We present a preconditioner for the linearized Navier–Stokes equations which is based on the combination of a fast transform approximation of an advection diffusion problem together with the recently introduced ‘BFBT^T’ preconditioner of Elman (SIAM Journal of Scientific Computing, 1999; 20 :1299–1316). The resulting preconditioner when combined with an appropriate Krylov subspace iteration method yields the solution in a number of iterations which appears to be independent of the Reynolds number provided a mesh Péclet number restriction holds, and depends only mildly on the mesh size. The preconditioner is particularly appropriate for problems involving a primary flow direction. Copyright © 2001 John Wiley & Sons, Ltd.     

Solvability of a model problem for the Stokes equations in an infinite angle

We consider a model problem for the Stokes equations in the half-plane ? ₊ ² (x₂>0) with different boundary conditions on the semiaxes (x₂=0, x₁<0) and (x₂=0, x₁>0), which plays an important role in the studies of some free boundary problems, such as problem of filling or drying a capillary. The proof of the solvability of the problem in weighted Sobolev and Hölder spaces is presented, and estimates for the solution as well as the asymptotic formula for the solution in the vicinity of the singular point x=0 are obtained. The proof is based on an explicit formula for the solution in terms of its Mellin transform, which makes it possible to obtain the estimates uniform with respect to one of the parameters of the problem (in the problem of filling a capillary it is proportional to the velocity of filling). Bibliography: 9 titles.     

Exterior penalty in optimal control problem with state-control constraints

We consider an abstract optimal control problem with additional equality and inequality state and control constraints, we use the exterior penalty function to transform the constrained optimal control problem into a sequence of unconstrained optimal control problems, under conditions in control lie in L ₁, the sequence of the solution to the unconstrained problem contains a subsequence converging of the solution of constrained problem, this convergence is strong when the problemis non convex, and is weak if the problemis convex in control. This generalizes the results of P.Nepomiastcthy [4] where he considered the control in the Hilbert space L ²(I,? ^m ).     

Nonlinear Fourier Transforms and the mKdV Equation in the Quarter Plane

The unified transform method introduced by Fokas can be used to analyze initial‐boundary value problems for integrable evolution equations. The method involves several steps, including the definition of spectral functions via nonlinear Fourier transforms and the formulation of a Riemann‐Hilbert problem. We provide a rigorous implementation of these steps in the case of the mKdV equation in the quarter plane under limited regularity and decay assumptions. We give detailed estimates for the relevant nonlinear Fourier transforms. Using the theory of L²‐RH problems, we consider the construction of quarter plane solutions which are C¹ in time and C³ in space.     

SUPERCRITICAL SEMILINEAR WAVE EQUATION WITH NON-NEGATIVE POTENTIAL

We prove a weighted L ^∞ estimate for the solution to the linear wave equation with a smooth positive time independent potential. The proof is based on application of generalized Fourier transform for the perturbed Laplace operator and a finite dependence domain argument. We apply this estimate to prove the existence of global small data solution to supercritical semilinear wave equations with potential.     

A regularization procedure for the auto‐correlation equation

The paper deals with the auto‐correlation equation and its regularization by means of a Lavrent'ev regularization procedure in L². The solution of this quadratic integral equation of the first kind and of the regularized equation of the second kind are obtained by reduction to a boundary value problem for the Fourier transform of the solution. We prove convergence of the approximate solution to the exact solution and derive a stability estimate for the error. Copyright © John Wiley & Sons, Ltd.     

Numerische Behandlung von Verzweigungsproblemen bei gewöhnlichen Differentialgleichungen

Summary We present a new method for the numerical solution of bifurcation problems for ordinary differential equations. It is based on a modification of the classical Ljapunov-Schmidt-theory. We transform the problem of determining the nontrivial branch bifurcating from the trivial solution into the problem of solving regular nonlinear boundary value problems, which can be treated numerically by standard methods (multiple shooting, difference methods).

     

Novel numerical techniques based on Fokas transforms,for the solution of initial boundary value problems

The unified transform method of A. S. Fokas has led to important new developments, regarding the analysis and solution of various types of linear and nonlinear PDE problems. In this work we use these developments and obtain the solution of time-dependent problems in a straightforward manner and with such high accuracy that cannot be reached within reasonable time by use of the existing numerical methods. More specifically, an integral representation of the solution is obtained by use of the A. S. Fokas approach, which provides the value of the solution at any point, without requiring the solution of linear systems or any other calculation at intermediate time levels and without raising any stability problems. For instance, the solution of the initial boundary value problem with the non-homogeneous heat equation is obtained with accuracy 10⁻¹⁵, while the well-established Crank–Nicholson scheme requires 2048 time steps in order to reach a 10⁻⁸ accuracy.     

On the Solution of Linear Recurrence Equations

In this article, we present a general solution for linear divide-and-conquer recurrences of the formu_n=^k_i=1a_i u[ⁿ _bi]+g(n)Our approach handles more cases than the Master method does [1]. We achieve this advantage by defining a new transform – the Order transform – which has useful properties for providing asymptotic answers (compared to other transforms which supply exact answers). This transform helps in mapping the sequence under consideration to the two-dimensional plane where the solution becomes easier to obtain. We demonstrate the power of the final results by solving many difficult examples.     

Search for maximum points of a vector criterion based on decomposition properties

We address the problem of optimal reconstruction of the values of a linear operator on ℝ ^d or ℤ ^d from approximate values of other operators. Each operator acts as the multiplication of the Fourier transform by a certain function. As an application, we present explicit expressions for optimal methods of reconstructing the solution of the heat equation (for continuous and difference models) at a given instant of time from inaccurate measurements of this solution at other time instants.     

On CCZ-equivalence of addition mod 2 n

We show that addition mod 2 ⁿ is CCZ-equivalent to a quadratic vectorial Boolean function. We use this to reduce the solution of systems of differential equations of addition to the solution of an equivalent system of linear equations and to derive a fully explicit formula for the correlation coefficients, which leads to enhanced results about the Walsh transform of addition mod 2 ⁿ . The results have direct applications in the cryptanalysis of cryptographic primitives which use addition mod 2 ⁿ .     

The Wirtinger-Steklov inequality between the norm of a periodic function and the norm of the positive cutoff of its derivative

We prove that there is no single uniform tight frame in Euclidean (unitary) space such that a solution of the ℓ ₁-norm minimization problem for the frame representation is attained on the frame coefficients. Then we find an exact solution of the ℓ ₁-minimization problem for the Mercedes-Benz frame in ℝ ^N . We also give some examples of connections between optimization problems of various types.     

On the Local Uniqueness of Solutions of Variational Inequalities Under H-Differentiability

In this paper, we give some sufficient conditions for the local uniqueness of solutions to nonsmooth variational inequalities where the underlying functions are H-differentiable and the underlying set is a closed convex set/polyhedral set/box/polyhedral cone. We show how the solution of a linearized variational inequality is related to the solution of the variational inequality. These results extend/unify various similar results proved for C ¹ and locally Lipschitzian variational inequality problems. When specialized to the nonlinear complementarity problem, our results extend/unify those of C ² and C ¹ nonlinear complementarity problems.     

A regularization method for approximating the inverse Laplace transform

A new method for approximating the inerse Laplace transform is presented. We first change our Laplace transform equation into a convolution type integral equation, where Tikhonov regularization techniques and the Fourier transformation are easily applied. We finally obtain a regularized approximation to the inverse Laplace transform as finite sum     

Error estimates for mixed finite element methods for nonlinear parabolic problems

We consider a class of mixed finite element methods for nonlinear parabolic problems over a plane domain. The finite element spaces taken are Raviart-Thomas spaces of index k, k ? 0. We obtain optimal order L²- and almost optimal order L^∞-error estimates for the finite element solution and order optimal L²-error estimates for its gradient. We also derive the error estimates for the time derivatives of the solution. Our results extend those previously obtained by Johnson and Thomée for the corresponding linear problems with k ? 1.     

On the condition number of the antireflective transform

Deconvolution problems with a finite observation window require appropriate models of the unknown signal in order to guarantee uniqueness of the solution. For this purpose it has recently been suggested to impose some kind of antireflectivity of the signal. With this constraint, the deconvolution problem can be solved with an appropriate modification of the fast sine transform, provided that the convolution kernel is symmetric. The corresponding transformation is called the antireflective transform. In this work we determine the condition number of the antireflective transform to first order, and use this to show that the so-called reblurring variant of Tikhonov regularization for deconvolution problems is a regularization method. Moreover, we establish upper bounds for the regularization error of the reblurring strategy that hold uniformly with respect to the size n of the algebraic system, even though the condition number of the antireflective transform grows with n. We briefly sketch how our results extend to higher space dimensions.     

A singular value decomposition for the radon transform in n-dimensional euclidean space

We construct the singular value decomposition of the Radon transform when the Radon transform is restricted to functions which are either square integrable on the unit disc in IR ⁿ with respect to one of the weights (1-r ²)^n/2-λ: or square integrable on IR ⁿ with respect to exp(r ²). An application to calculating mollifiers for approximate inversion of the sampled Radon transform is discussed.