On the Wl q-Regularity of Solutions to Systems of Differential Equations in the Case When the Equations Are Constructed from Discontinuous Functions

Some solution, final in a sense from the standpoint of the theory of Sobolev spaces, is obtained to the problem of regularity of solutions to a system of (generally) nonlinear partial differential equations in the case when the system is locally close to elliptic systems of linear equations with constant coefficients. The main consequences of this result are Theorems 5 and 8. According to the first of them, the higher derivatives of an elliptic C ^l -smooth solution to a system of lth-order nonlinear partial differential equations constructed from C ^l -smooth functions meet the local Hoelder condition with every exponent , 0<<1. Theorem 8 claims that if a system of linear partial differential equations of order l with measurable coefficients and right-hand sides is uniformly elliptic then, under the hypothesis of a (sufficiently) slow variation of its leading coefficients, the degree of local integrability of lth-order partial derivatives of every W ^l _q,loc-solution, q>1, to the system coincides with the degree of local integrability of lower coefficients and right-hand sides.     

On the deficiency index of a certain class of differential operators

This paper studies problems on the behavior of solutions for a certain class of differential equations of the form l _n y = λy as x → +0, where l _n is a scalar, formally self-adjoint, quasidifferential expression of an arbitrary order with complex coefficients. The results obtained allow us to find the defficiency index of the operator generated by the expression l _n . __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 36, Suzdal Conference-2004, Part 2, 2005.     

Compression and denoising using <Emphasis Type="Italic">l</Emphasis><Subscript>0</Subscript>-norm

In this paper, we deal with l ₀-norm data fitting and total variation regularization for image compression and denoising. The l ₀-norm data fitting is used for measuring the number of non-zero wavelet coefficients to be employed to represent an image. The regularization term given by the total variation is to recover image edges. Due to intensive numerical computation of using l ₀-norm, it is usually approximated by other functions such as the l ₁-norm in many image processing applications. The main goal of this paper is to develop a fast and effective algorithm to solve the l ₀-norm data fitting and total variation minimization problem. Our idea is to apply an alternating minimization technique to solve this problem, and employ a graph-cuts algorithm to solve the subproblem related to the total variation minimization. Numerical examples in image compression and denoising are given to demonstrate the effectiveness of the proposed algorithm.     

Block Coordinate Relaxation Methods for Nonparametric Wavelet Denoising

Abstract

An important class of nonparametric signal processing methods entails forming a set of predictors from an overcomplete set of basis functions associated with a fast transform (e.g., wavelet packets). In these methods, the number of basis functions can far exceed the number of sample values in the signal, leading to an ill-posed prediction problem. The “basis pursuit” denoising method of Chen, Donoho, and Saunders regularizes the prediction problem by adding an l ₁ penalty term on the coefficients for the basis functions. Use of an l ₁ penalty instead of l ₂ has significant benefits, including higher resolution of signals close in time/frequency and a more parsimonious representation. The l ₁ penalty, however, poses a challenging optimization problem that was solved by Chen, Donoho and Saunders using a novel application of interior-point algorithms (IP). This article investigates an alternative optimization approach based on block coordinate relaxation (BCR) for sets of basis functions that are the finite union of sets of orthonormal basis functions (e.g., wavelet packets). We show that the BCR algorithm is globally convergent, and empirically, the BCR algorithm is faster than the IP algorithm for a variety of signal denoising problems.     

Third power moments of the error terms corresponding to certain arithmetic functions

For positive integersn, letd(l ₁,M ₁;l ₂,M ₂;n) denote the number of factorizationsn=n ₁ n ₂ where each of the factorsn∈ℕ belongs to a prescribed congruence classl _i moduloM _i (i=1,2). In this article an asymptotic result is derived for the third power moment of the error term in the formula for the Dirichlet summmatory function ofd(l ₁,M ₁;l ₂,M ₂;n). This extends a recent result of [17] for the classic “unrestricted” case ofd(n)=d(1,1;1,1; n). Moreover, the technique developed is applied to the analogous problem related to Fourier coefficients of cusp forms. In memory of Professor Karl Prachar This article is part of a research project supported by theAustrian Science Foundation (Nr. P 9298-PHY)     

Variable Selection via A Combination of the L0 and L1 Penalties

Variable selection is an important aspect of high-dimensional statistical modeling, particularly in regression and classification. In the regularization framework, various penalty functions are used to perform variable selection by putting relatively large penalties on small coefficients. The L₁ penalty is a popular choice because of its convexity, but it produces biased estimates for the large coefficients. The L₀ penalty is attractive for variable selection because it directly penalizes the number of non zero coefficients. However, the optimization involved is discontinuous and non convex, and therefore it is very challenging to implement. Moreover, its solution may not be stable. In this article, we propose a new penalty that combines the L₀ and L₁ penalties. We implement this new penalty by developing a global optimization algorithm using mixed integer programming (MIP). We compare this combined penalty with several other penalties via simulated examples as well as real applications. The results show that the new penalty outperforms both the L₀ and L₁ penalties in terms of variable selection while maintaining good prediction accuracy.     

Coefficients in some Young Derived Sequences

Letη^q={ηrqn}_n≥0denote theS_ncharacter sequence obtained fromηby aq-column dilation of the Young diagrams, and lety^l(η^q)denote itslth Young derived sequence [7]. We prove a formula expressing the coefficients iny^l(η^q)in terms of those inηand of semistandard tableaux. The asymptotics of these sequences yield new extensions of the Mehta integral.     

Time optimal boundary controllability of a simple linear viscoelastic liquid

A viscoelastic liquid confined between two parallel plates is considered under the action of L₂ (O, T)-boundary controls. Applying Laplace-Transform-techniques it is shown that L₂ (O, l)-states are exactly controllable in finite time, depending on the speed of propagation of singularities. Finally the existence of time-optimal controls respecting a given norm bound is shown.     

On the use of AGE algorithm with a high accuracy Numerov type variable mesh discretization for 1D non-linear parabolic equations

In this paper, the use of N-AGE and Newton-N-AGE iterative methods on a variable mesh for the solution of one dimensional parabolic initial boundary value problems is considered. Using three spatial grid points, a two level implicit formula based on Numerov type discretization is discussed. The local truncation error of the method is of O(k²h_l^-1 +kh_l +h_l³)O({k^2h_l^{-1} +kh_l +h_l^3}), where h _l  > 0 and k > 0 are the step lengths in space and time directions, respectively. We use a special technique to handle singular parabolic equations. The advantage of using these algorithms is highlighted computationally.     

Controller design to optimize a composite performance measure

This paper studies a mixed objective problem of minimizing a composite measure of thel ₁, ₂, andl -norms together with thel -norm of the step response of the closed loop. This performance index can be used to generate Pareto-optimal solutions with respect to the individual measures. The problem is analyzed for discrete-time, single-input single-output (SISO), linear time-invariant systems. It is shown via Lagrange duality theory that the problem can be reduced to a convex optimization problem with a priori known dimension. In addition, continuity of the unique optimal solution with respect to changes in the coefficients of the linear combination that defines the performance measure is estabilished.This research was supported by the National Science Foundation under Grants No. ECS-92-04309, ECS-92-16690 and ECS-93-08481.     

Weak solutionsu(x,t) to parabolic partial differential equations with coefficients that depend uponu(yl, σ, (t,u(x,t))), 1 = 1, … k

The existence of weak solutionsu(x, t) to parabolic partial differential equations with coefficients that depend onu(y_l, σ_l(t, u(x, t))), l = 1,… k, is demonstrated using a retardation of the time arguments in the coefficients along with regularity and compactness results for solutions of linear parabolic partial differential equations.     

An algorithm for Gaussian quadrature given modified moments

Summary An algebraic algorithm, the long quotient- modified difference (LQMD) algorithm, is described for the Gaussian quadrature of the one-dimensional product integral f(x)w(x)dx when the weight function (x) is known through modified momentsv _l =; theP _l (x) are any polynomials of degreel satisfying 3-term recurrence relations with known coefficients. The algorithm serves to establish the co-diagonal matrix, the eigenvalues of which are the Gaussian abscissas. Applied to ordinary moments it requires far fewer divisions than the quotient-difference algorithm; if theP _l (x) are themselves orthogonal with a kernelw ₀ 03F0;, there is no instability due to rounding errors. For smooth kernels (x) it is safe to use secondorder interpolation in determining the eigenvalues by Givens' method. The Christoffel weights can be expressed as ratios of two terms which are most easily calculated in a Sturm sequence beginning with the highest value ofl. A formula for the Christoffel weights applicable for rational versions of theQR algorithm is also derived. Convergence and the propagation of rounding errors are illustrated by several examples, and anAlgol procedure is given.Based in part on a project report presented by A. F. Donovan in partial fulfilment of the requirements for the degree of B. Sc. (Honours), University of Salford (1969).     

Cubic harmonics as linear combinations of spherical harmonics

Orthonormal functions belonging to different irreducible representations of the cubic point groupO _h are calculated by a simple numerical procedure. The coefficients of the expansion in spherical harmonics are tabulated forA ₁,A ₂ andE up tol=30.

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Zusammenfassung Es wird ein einfaches numerisches Verfahren beschrieben zur Berechnung eines Orthonormalsystems von Funktionen, die wie die verschiedenen irreduziblen Darstellungen der kubischen GruppeO _h transformieren. Die Koeffizienten der Entwicklung nach Kugelfunktionen sind bisl=30 fürA ₁,A ₂ undE tabelliert.

     

Convergence of double Fourier series

Summary Let a2π-periodic function f(x, y) be continuous in some neighbourhood of the point (x, y) except possibly along finitely many lines l ₁ , l ₂ , ..., l_k terminating at (x, y). The problem of convergence of the Fourier series of f(x, y) at the point (x, y) is examined in some detail. It is established that under certain restrictions on the variation of f(x, y), and also on the lines l ₁ , l ₂ , ..., l_k, the fourier series converges to a value bounded above by the limit superior, and below by the limit inferior of f(x+u, y+v), u, v →0, this value depending on the manner in which the series is summed. The preparation of this paper was financed, in part, by a Canadian Mathematical Congress Summer Research Grant (1968). Entrata in Redazione il 3 febbraio 1969.     

On the solution of a nonlinear programming problem by decomposition

A decomposition method, used in least-weight plastic design, is extended to solve problems with nonlinearity arising from variable structure geometry. The problem considered is that of finding vectorsx ₁,x ₂, andq that minimize [l max{|x ₁|, |x ₂|}], subject toAx ₁=b ₁ andAx ₂=b ₂, where both the vectorl and the matrixA are nonlinear functions ofq.     

Complexity bound for the absolute factorization of parametric polynomials

An algorithm is constructed for the absolute factorization of polynomials with algebraically independent parametric coefficients. It divides the parameter space into pairwise disjoint pieces such that the absolute factorization of polynomials with coefficients in each piece is given uniformly. Namely, for each piece there exist a positive integer l ≤ d, l variables C₁, ..., C_l algebraically independent over the ground field F, and rational functions b_J,j of the parameters and of the variables C₁, ..., C_l such that for any parametric polynomial f with coefficients in this piece, there exist c₁, ..., with f = Π _j G _j where G _j = Σ_|J| B _J,j Z ^J is absolutely irreducible. Here Z = (Z₀, ..., Z_n) are the variables of f, each B_J,j is the value of b_J,j at the coefficients of f and c₁, ..., c_l, and denotes the algebraic closure of F. The number of pieces does not exceed (2d²+1)^2n+3d+5, and the algorithm performs arithmetic operations in F (thus the number of operations is exponential in the number r = ( _n+1 ^n+1+d ) of coefficients of f), and its binary complexity is bounded by if F = ℚ and by if , where d is an upper bound on the degrees of polynomials. The techniques used include the Hensel lemma and the quantifier elimination in the theory of algebraically closed fields. Bibliography: 20 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 316, 2004, pp. 5–29.     

Automatic Smoothing With Wavelets for a Wide Class of Distributions

Wavelet-based denoising techniques are well suited to estimate spatially inhomogeneous signals. Waveshrink (Donoho and Johnstone) assumes independent Gaussian errors and equispaced sampling of the signal. Various articles have relaxed some of these assumptions, but a systematic generalization to distributions such as Poisson, binomial, or Bernoulli is missing. We consider a unifying l₁-penalized likelihood approach to regularize the maximum likelihood estimation by adding an l₁ penalty of the wavelet coefficients. Our approach works for all types of wavelets and for a range of noise distributions. We develop both an algorithm to solve the estimation problem and rules to select the smoothing parameter automatically. In particular, using results from Poisson processes, we give an explicit formula for the universal smoothing parameter to denoise Poisson measurements. Simulations show that the procedure is an improvement over other methods. An astronomy example is given.     

Heat kernel expansions on the integers and the Toda lattice hierarchy

We consider the heat equation u _t = Lu where L is a second-order difference operator in a discrete variable n. The fundamental solution has an expansion in terms of the Bessel functions of imaginary argument. The coefficients α _k (n, m) in this expansion are analogs of Hadamard’s coefficients for the (continuous) Schr?dinger operator. We derive an explicit formula for α _k in terms of the wave and the adjoint wave functions of the Toda lattice hierarchy. As a first application of this result, we prove that the values of these coefficients on the diagonals n = m and n = m + 1 define a hierarchy of differential-difference equations which is equivalent to the Toda lattice hierarchy. Using this fact and the correspondence between commutative rings of difference operators and algebraic curves we show that the fundamental solution can be summed up, giving a finite formula involving only two Bessel functions with polynomial coefficients in the time variable t, if and only if the operator L belongs to the family of bispectral operators constructed in [18].      

Automorphisms of elementary adjoint Chevalley groups of types Al, Dl, and El over local rings with 1/2

It is proved that every automorphism of an elementary adjoint Chevalley group of type A _l , D _l , or E _l over a local commutative ring with 1/2 is a composition of a ring automorphism and conjugation by some matrix from the normalizer of that Chevalley group in GL(V) (V is an adjoint representation space).     

Special classes of <Emphasis Type="Italic">l</Emphasis>-rings

We study a special class of lattice-ordered rings and a special radical. We prove that a special radical of an l-ring is equal to the intersection of the right l-prime l-ideals for each of which the following condition holds: the quotient l-ring by the maximal l-ideal contained in a given right l-ideal belongs to the special class. The prime radical of an l-ring is equal to the intersection of the right l-semiprime l-ideals. We introduce the notion of a completely l-prime l-ideal. We prove that N ₃(R) is equal to the intersection of the completely l-prime, right l-ideals of an l-ring R, where N ₃(R) is the special radical of the l-ring R defined by the class of l-rings without positive divisors of zero.