Application of Wavelet Shrinkage to Solving the Sideways Heat Equation

The sideways heat equation is considered in terms of the ill-posed operator equation Au = g, g R(A) L ²(), with a given noisy right hand side. For a reconstruction of the solution from indirect data the dual least square method generated by the family of Meyer wavelet subspaces is applied. An explicit relation between the truncation level of the wavelet expansion and the data error bound is found under which convergence results including error estimation are obtained. Next, a certain simple nonlinear modification of the method based on local refinements of the wavelet expansion of the noisy data is investigated. Moreover, it is shown that an accuracy of the solution reconstruction can be improved by adding some sufficiently large coefficients on the level higher than that indicated by the convergence theorem.This revised version was published online in October 2005 with corrections to the Cover Date.     

Generalized Solution of Linear Systems and Image Restoration

Let n, m be positive integers; we consider m×n real linear systems. We define regularized solutions of a linear system as the minimizers of an optimization problem. The objective function of this optimization problem can be seen as the Tikhonov functional when the p-norm is considered instead of the Euclidean norm. The cases p=1 and p= are studied. This analysis is used to restore defocused synthetic images and real images with encouraging results.     

Primitive and Poisson Spectra of Twists of Polynomial Rings

A family of flat deformations of a commutative polynomial ring S on n generators is considered, where each deformation B is a twist of S by a semisimple, linear automorphism of ^n–1, such that a Poisson bracket is induced on S. We show that if the symplectic leaves associated with this Poisson structure are algebraic, then they are the orbits of an algebraic group G determined by the Poisson bracket. In this case, we prove that if is 'generic enough', then there is a natural one-to-one correspondence between the primitive ideals of B and the symplectic leaves if and only if has a representative in GL( ⁿ ) which belongs to G. As an example, the results are applied to the coordinate ring of quantum 2 × 2 matrices which is not a twist of a polynomial ring, although it is a flat deformation of one; if q is not a root of unity, then there is a bijection between the primitive ideals of and the symplectic leaves.     

Evaluation ofS,T, U parameters, triple gauge boson vertices and some oblique corrections in extraU(1) superstring inspired model

Explicit evaluation of the following parameters has been carried out in the extraU (1) superstring inspired model: (i) As Mz₂ varies from 555 GeV to 620 GeV and (m _t) CDF = 175.6 ± 5.7 GeV (Table 1): (a) S^New varies from -0.100 ± 0.089 to -0.130 ± 0.090, (b) T^New varies from -0.098 ± 0.097 to -0.129 ± 0.098, (c) U^New varies from -0.229 ± 0.177 to -0.253 ± 0.206, (d) Τ_z varies from 2.487 ± 0.027 to 2.486 ± 0.027, (e) A_LR varies from 0.0125 ± 0.0003 to 0.0126 ± 0.0003, (f) A _FB ^b remains constant at 0.0080 ± 0.0007. Almost identical values are obtained for (m _t)D0 = 169 GeV (see table 2). (ii) Triple gauge boson vertices (TGV) contributions: AsMz ₂ varies from 555 GeV to 620 GeV and (m _t) CDF = 175.6 ±5.7 GeV. (a)√s = 500 GeV, asymptotic case: varies from -0.301 to -0.179; varies from -0.622 to -0.379; varies from +0.0061 to 0.0056; varies from -3.691 to -2.186. varies from +0.270 to +0.118; varies from +0.552 to 0.238; varies from +0.0004 to +0.0002; remains constant at -0.110. (b)√s = 700 GeV, asymptotic case: varies from -0.297 to -0.176; varies from -0.609 to -0.370; varies from -0.0082 to -0.0078; varies from -3.680 to -2.171.√s = 700 GeV, nonasymptotic case: varies from -0.173 to -0.299; varies from-0.343 to -0.591; varies from -0.005 to -0.011; remains constant at -0.110. The pattern of form factors values for√s = 1000, 1200 GeV is almost identical to that of√s= 700 GeV. Further the values of the form factors for (m _t)D0 (=169 GeV) follow identical pattern as that of (m _t) CDF form factors values (see tables 5, 6, 9, 10). We conclude that the values of all the form factors with the exception of these of , are comparable or larger than theS, T values and therefore the TGV contributions are important while deciding the use of extraU (1) model for doing physics beyond standard model.     

Renormalization and Dimensional Regularization for a Scalar Field with Gauss–Bonnet-Type Coupling to Curvature

We consider a scalar field with a Gauss–Bonnet-type coupling to the curvature in a curved space–time. For such a quadratic coupling to the curvature, the metric energy–momentum tensor does not contain derivatives of the metric of orders greater than two. We obtain the metric energy–momentum tensor and find the geometric structure of the first three counterterms to the vacuum averages of the energy–momentum tensors for an arbitrary background metric of an N-dimensional space–time. In a homogeneous isotropic space, we obtain the first three counterterms of the n-wave procedure, which allow calculating the renormalized values of the vacuum averages of the energy–momentum tensors in the dimensions N=4,5. Using dimensional regularization, we establish that the geometric structures of the counterterms in the n-wave procedure coincide with those in the effective action method.     

Cohomology of symplectic reductions of generic coadjoint orbits

Let be a generic coadjoint orbit of a compact semi-simple Lie group . Weight varieties are the symplectic reductions of by the maximal torus in . We use a theorem of Tolman and Weitsman to compute the cohomology ring of these varieties. Our formula relies on a Schubert basis of the equivariant cohomology of , and it makes explicit the dependence on and a parameter in .

     

Regularized Total Least Squares Based on Quadratic Eigenvalue Problem Solvers

This paper presents a new computational approach for solving the Regularized Total Least Squares problem. The problem is formulated by adding a quadratic constraint to the Total Least Square minimization problem. Starting from the fact that a quadratically constrained Least Squares problem can be solved via a quadratic eigenvalue problem, an iterative procedure for solving the regularized Total Least Squares problem based on quadratic eigenvalue problems is presented. Discrete ill-posed problems are used as simulation examples in order to numerically validate the method. AMS subject classification (2000) 65F20, 65F30.Received March 2003. Revised November 2003. Accepted January 2004. Communicated by Per Christian Hansen.     

Adiabatic Invariance and Applications: From Molecular Dynamics to Numerical Weather Prediction

A wide class of Hamiltonian systems exhibit a mixture of slow motion with superimposed fast oscillations. Under the assumption of scale separation, these systems can be investigated using the principle of adiabatic invariance. In this paper, we start with a review of some of the main theoretical and numerical findings. We then briefly summarize a few important implications for molecular dynamics (MD) before we provide a more extensive discussion of numerical weather prediction (NWP). In particular, the conservative Hamiltonian particle-mesh (HPM) method is extended to Euler's equation and the fundamental concepts of geostrophic and hydrostatic balance are illustrated on the level of fluid blobs. We also demonstrate numerically that symplectic time-stepping methods are able to maintain hydrostatic balance to high accuracy.     

On Manifolds Whose Tangent Bundle is Big and 1-Ample

A line bundle over a complex projective variety is called bigand 1-ample if a large multiple of it is generated by globalsections and a morphism induced by the evaluation of the spanningsections is generically finite and has at most 1-dimensionalfibers. A vector bundle is called big and 1-ample if the relativehyperplane line bundle over its projectivisation is big and1-ample. The main theorem of the present paper asserts that any complexprojective manifold of dimension 4 or more, whose tangent bundleis big and 1-ample, is equal either to a projective space orto a smooth quadric. Since big and 1-ample bundles are ‘almost’ample, the present result is yet another extension of the celebratedMori paper ‘Projective manifolds with ample tangent bundles’(Ann. of Math. 110 (1979) 593–606). The proof of the theorem applies results about contractionsof complex symplectic manifolds and of manifolds whose tangentbundles are numerically effective. In the appendix we re-provethese results. 2000 Mathematics Subject Classification 14E30,14J40, 14J45, 14J50.     

An application of anisotropic regularization to the existence of weak Pareto minimal points

Given a function f on Rⁿ, we introduce the concept of anisotropic regularization as a generalization of Tikhonov regularization f_ε(x)=f(x)+εx. When f is a continuous -function on Rⁿ and K is a box in Rⁿ, we study the properties of and the limiting behavior of solutions of a regularized box variational inequality problem , with emphasis on the existence of weak Pareto minimal points with respect to K. This work generalizes results of Sznajder and Gowda (1998) proved in the setting of nonlinear complementarity problems.