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Thesis advisor, Lawrence J. Ziomek

We present an improved version of our earlier work on summing the planar graphs in phi^3 field theory. The present treatment is also based on our world sheet formalism and the mean field approximation, but it makes use of no further approximations. We derive a set of equations between the expectation values of the world sheet fields, and we investigate them in certain limits. We show that the equations can give rise to (metastable) string forming solutions.

This paper deals with the set of $\alpha\in{\mathbb{R}}$ such that $\alpha \zeta^{n} \bmod 1$ tends to $0$ for a fixed $\zeta\in{\mathbb{R}}$, which we call $\mathscr{M}_{\zeta}$. Predominately the case of Pisot numbers $\zeta$ is studied. In this case the inclusions $\mathcal{O}_{\mathbb{Q}(\zeta)}\subset\mathscr{M}_{\zeta}\subset\mathbb{Q}(\zeta)$ are known. We will show the properties of $\mathscr{M}_{\zeta}$ are connected to the module structure of the ring of integers $\mathcal{O}_{\mathbb{Q}(\zeta)}$. We will describe the module structure of $\mathscr{M}_{\zeta}$ and how much $\mathscr{M}_{\zeta}$ differs from $\mathcal{O}_{\mathbb{Q}(\zeta)}$. The results besides allow to give some information on the shape of integral bases of real number fields.

The convergence theory for the set of simultaneously $\psi$-approximable points lying on a planar curve is established. Our results complement the divergence theory developed in `Diophantine approximation on planar curves and the distribution of rational points' (by V. Beresnevich, H. Dickinson, and S. Velani, with an Appendix: `Sums of two squares near perfect squares' by R.C. Vaughan, To appear: Annals of Math., Pre-print: arkiv:math.NT/0401148, (2004), 1-52.) and thereby completes the general metric theory for planar curves.

The convergence theory for the set of simultaneously $\psi$-approximable points lying on a planar curve is established. Our results complement the divergence theory developed in `Diophantine approximation on planar curves and the distribution of rational points' (by V. Beresnevich, H. Dickinson, and S. Velani, with an Appendix: `Sums of two squares near perfect squares' by R.C. Vaughan, To appear: Annals of Math., Pre-print: arkiv:math.NT/0401148, (2004), 1-52.) and thereby completes the general metric theory for planar curves.

This paper goes back to a famous problem of Mahler in metrical Diophantine approximation. The problem has been settled by Sprindzuk and subsequently improved by Alan Baker and Vasili Bernik. In particular, Bernik's result establishes a convergence Khintchine type theorem for Diophantine approximation by polynomials, that is it allows arbitrary monotonic error of approximation. In the present paper the monotonicity assumption is completely removed.

For an interval $E=[a,b]$ on the real line, let $\mu$ be either the equilibrium measure, or the normalized Lebesgue measure of $E$, and let $V^{\mu}$ denote the associated logarithmic potential. In the present paper, we construct a function $f$ which is analytic on $E$ and possesses four branch points of second order outside of $E$ such that the family of the admissible compacta of $f$ has no minimizing elements with regard to the extremal theoretic-potential problem, in the external field equals $V^{-\mu}$.

We consider the variant of stochastic homogenization theory introduced in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Serie I 2006 and Journal de Mathematiques Pures et Appliquees 2007]. The equation under consideration is a standard linear elliptic equation in divergence form, where the highly oscillatory coefficient is the composition of a periodic matrix with a stochastic diffeomorphism. The homogenized limit of this problem has been identified in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Serie I 2006]. We first establish, in the one-dimensional case, a convergence result (with an explicit rate) on the residual process, defined as the difference between the solution to the highly oscillatory problem and the solution to the homogenized problem. We next return to the multidimensional situation. As often in random homogenization, the homogenized matrix is defined from a so-called corrector function, which is the solution to a problem set on the entire space. We describe and prove the almost sure convergence of an approximation strategy based on truncated versions of the corrector problem.

In this paper, I generalize a previous one about hilbertian kernels and approximation theory

In this paper, I generalize a previous one about hilbertian kernels and approximation theory

In this paper, I generalize a previous one about hilbertian kernels and approximation theory

We discuss about the Hermitian treatment of Dyson-type boson expansion theory. We show that the basic assumption of the conventional treatment does not hold in general and the method is only approximately valid. We also show that the approximation is the same order as that of truncation of the expansion usually done in the Hermitian type boson expansion theory.

We discuss about the Hermitian treatment of Dyson-type boson expansion theory. We show that the basic assumption of the conventional treatment does not hold in general and the method is only approximately valid. We also show that the approximation is the same order as that of truncation of the expansion usually done in the Hermitian type boson expansion theory.

The results for the total multi-photon decay rates of the 3p and 4s levels of hydrogen, presented by D. Solovyev and L. Labzowsky within the cascade approximation, are revisited. The corrected results for certain decay channels differ from original ones of those authors sometimes by order of magnitude. Some aspects with respect to the cosmological recombination process are clarified.

A general, modular, pulse propagation model for underwater acoustics that is based on linear systems theory for sound-speed profiles as a function of depth is presented. The development and computer implementation of the model, together with results from preliminary computer simulation studies involving the transmission of CW and LFM pulses from a planar array of complex weighted point sources is reported. The studies examined free-space propagation problems (i.e., no boundaries) in homogeneous and inhomogeneous media using a transfer function of the ocean medium based on the WKB approximation. The two main outputs from the model are the predicted complex acoustic field as a function of frequency and spatial location and the time-domain output electrical signal from each element in a receive planar array.

Solving large-scale optimization on-the-fly is often a difficult task for real-time computer graphics applications. To tackle this challenge, model reduction is a well-adopted technique. Despite its usefulness, model reduction often requires a handcrafted subspace that spans a domain that hypothetically embodies desirable solutions. For many applications, obtaining such subspaces case-by-case either is impossible or requires extensive human labors, hence does not readily have a scalable solution for growing number of tasks. We propose linear variational subspace design for large-scale constrained quadratic programming, which can be computed automatically without any human interventions. We provide meaningful approximation error bound that substantiates the quality of calculated subspace, and demonstrate its empirical success in interactive deformable modeling for triangular and tetrahedral meshes.

The invariant integration method for Chern-Simons theory for gauge group SU(2) and manifold \Gamma\H^3 is verified in the semiclassical approximation. The semiclassical limit for the partition function associated with a connected sum of hyperbolic 3-manifolds is presented. We discuss briefly L^2 - analytical and topological torsions of a manifold with boundary.

In this paper we develop the convergence theory of simultaneous, inhomogeneous Diophantine approximation on manifolds. A consequence of our main result is that if the manifold $M \subset \mathbb{R}^n$ is of dimension strictly greater than $(n+1)/2$ and satisfies a natural non-degeneracy condition, then $M$ is of Khintchine type for convergence. The key lies in obtaining essentially the best possible upper bound regarding the distribution of rational points near manifolds.

In this paper we develop the convergence theory of simultaneous, inhomogeneous Diophantine approximation on manifolds. A consequence of our main result is that if the manifold $M \subset \mathbb{R}^n$ is of dimension strictly greater than $(n+1)/2$ and satisfies a natural non-degeneracy condition, then $M$ is of Khintchine type for convergence. The key lies in obtaining essentially the best possible upper bound regarding the distribution of rational points near manifolds.

We establish the critical line of the one-component $\Phi^4$ (or Landau-Ginzburg) model on the simple cubic lattice in three dimensions. Our study is performed in the framework of the non-perturbative renormalization group in the local potential approximation. Soft as well as ultra-sharp infra-red regulators are both considered. While the latter gives poor results, the critical line given by the soft cut-off compares well with the Monte Carlo simulations data of Hasenbusch (J. Phys. A : Math. Gen. 32 (1999) 4851) with a relative error of, at worst, $\sim 3. 10^{-3}$ on published points (critical parameters) of this line.

In the framework of analytic approach to QCD, which has been recently intensively developed, the dependence of nonperturbative contributions in a running coupling of strong interaction on initial perturbative approximation to 3-loop order is studied. The nonperturbative contributions are obtained in an explicit form. In the ultraviolet region they are shown to be represented in the form of the expansion in the inverse powers of Euclidean momentum squared. The expansion coefficients are calculated for different numbers of active quark flavors $n_f$ and for different numbers of loops taken into account. For all $n_f$ of interest it is shown that 2-loop order and 3-loop order corrections result in partial compensation of 1-loop order leading in the ultraviolet region nonperturbative contribution.

Ten years ago, Beresnevich-Dickinson-Velani initiated a project that develops the general Hausdorff measure theory of dual approximation on non-degenerate manifolds. In particular, they established the divergence part of the theory based on their general ubiquity framework. However, the convergence counterpart of the project remains wide open and represents a major challenging question in the subject. Until recently, it was not even known for any single non-degenerate manifold. In this paper, we settle this problem for all curves in $\mathbb{R}^2$, which represents the first complete theory of its kind for a general class of manifolds.

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A basic model in financial mathematics was introduced by Black, Scholes and Merton in 1973 (BSM model). A classical discrete approximation in distribution is the binomial model given by Cox, Ross and Rubinstein in 1979 (CRR model). The BSM and the CRR models have been used for example to price European call and put options. Our aim in this work is to give a strong (almost sure, pathwise) discrete approximation of the BSM model using a suitable nested sequence of simple, symmetric random walks. The approximation extends to the stock price process, the value process, the replicating portfolio, and the greeks. An important tool in the approximation is a discrete version of the Feynman--Kac formula as well. It is hoped that such a discrete pathwise approximation can be useful for example when teaching students whose mathematical background is limited, e.g. does not contain measure theory or stochastic analysis.

We report many-body calculations of the self-energy and lifetime of Shockley and image states on the (100) and (111) surfaces of Cu that go beyond the $GW$ approximation of many-body theory. The self-energy is computed in the framework of the GW\Gamma approximation by including short-range exchange-correlation (XC) effects both in the screened interaction W (beyond the random-phase approximation) and in the expansion of the self-energy in terms of W (beyond the GW approximation). Exchange-correlation effects are described within time-dependent density-functional theory from the knowledge of an adiabatic nonlocal XC kernel that goes beyond the local-density approximation.

Several questions of approximation theory are discussed: 1) can one approximate stably in $L^\infty$ norm $f^\prime$ given approximation $f_\delta, \parallel f_\delta - f \parallel_{L^\infty} < \delta$, of an unknown smooth function $f(x)$, such that $\parallel f^\prime (x) \parallel_{L^\infty} \leq m_1$? 2) can one approximate an arbitrary $f \in L^2(D), D \subset \R^n, n \geq 3$, is a bounded domain, by linear combinations of the products $u_1 u_2$, where $u_m \in N(L_m), m=1,2,$ $L_m$ is a formal linear partial differential operator and $N(L_m)$ is the null-space of $L_m$ in $D$, $3) can one approximate an arbitrary $L^2(D)$ function by an entire function of exponential type whose Fourier transform has support in an arbitrary small open set? Is there an analytic formula for such an approximation? N(L_m) := \{w: L_m w=0 \hbox{in\} D\}$?

We investigate the question of how well points on a nondegenerate $k$-dimensional submanifold $M \subseteq \mathbb R^d$ can be approximated by rationals also lying on $M$, establishing an upper bound on the "intrinsic Dirichlet exponent" for $M$. We show that relative to this exponent, the set of badly intrinsically approximable points is of full dimension and the set of very well intrinsically approximable points is of zero measure. Our bound on the intrinsic Dirichlet exponent is phrased in terms of an explicit function of $k$ and $d$ which does not seem to have appeared in the literature previously. It is shown to be optimal for several particular cases. The requirement that the rationals lie on $M$ distinguishes this question from the more common context of (ambient) Diophantine approximation on manifolds, and necessitates the development of new techniques. Our main tool is an analogue of the Simplex Lemma for rationals lying on $M$ which provides new insights on the local distribution of rational points on nondegenerate manifolds.

The inhomogeneous Groshev type theory for dual Diophantine approximation on manifolds is developed. In particular, the notion of nice manifolds is introduced and the divergence part of the theory is established for all such manifolds. Our results naturally incorporate and generalize the homogeneous measure and dimension theorems for non-degenerate manifolds established to date. The generality of the inhomogeneous aspect considered within enables us to make a new contribution even to the classical theory in R^n. Furthermore, the multivariable aspect considered within has natural applications beyond the standard inhomogeneous theory such as to Diophantine problems related to approximation by algebraic integers.

It is shown that in the weak field approximation solutions of Brans-Dicke equations are simply related to the solutions of General Relativity equations for the same matter distribution. A simple method is developed which permits to obtain Brans-Dicke solutions from Einstein solutions when both theories are considered in their linearized forms. To illustrate the method some examples found in the literature are discussed.

We determine upper asymptotic estimates of Kolmogorov and linear $n$-widths of unit balls in Sobolev and Besov norms in $L_{p}$-spaces on compact Riemannian manifolds. The proofs rely on estimates for the near-diagonal localization of the kernels of elliptic operators. We also summarize some of our previous results about approximations by eigenfunctions of elliptic operators on compact homogeneous manifolds.

We determine upper asymptotic estimates of Kolmogorov and linear $n$-widths of unit balls in Sobolev and Besov norms in $L_{p}$-spaces on compact Riemannian manifolds. The proofs rely on estimates for the near-diagonal localization of the kernels of elliptic operators. We also summarize some of our previous results about approximations by eigenfunctions of elliptic operators on compact homogeneous manifolds.

An ad hoc method to solve boundary value problems which are posed on infinite intervals is to reduce the infinite interval to a finite but large one and to impose additional boundary conditions at the far end. These boundary conditions should be posed in a way so that they express the asymptotic behaviour of the actual solution well. In this paper a rigorous theory is derived which defines classes of appropriate additional boundary conditions. Appropriate is to be understood in the sense that the solutions of the approximate problems converge to the actual solution of the 'infinite' problem as the length of the finite interval tends to infinity. Moreover boundary conditions which produce convergence with the largest expectable order are devised. (Author)

The small slope approximation is widely used to model the incoherent scattering cross-section per unit area from stochastic rough interfaces. It consists of an integral multiplied by a prefactor. The prefactor involves a closed-form algebraic expression, but evaluation of the integral is nontrivial. This paper develops tractable methods for evaluating the integral. The rough-surface scenarios considered generally assume spectra that have tails that decrease according to a single specified power law. This assumption is typically valid for the air-sea interface and ocean bottom at frequencies below 10 and 40 kHz respectively. The scenarios involve various tradeoffs, but the most significant tradeoff involves tractability vs broad applicability. In the most tractable scenario, it is assumed that only the isotropic tail of the spectrum is relevant. Alternately, information about the spectral peak and the power-law tail can be used to generate an isotropic difference spectrum formed by taking the difference between two power laws. The difference spectrum peak can be sharpened either by grafting it to a decaying exponential at low-wavenumber, or by replacing it entirely with a Pierson-Moskowitz (isotroptic) spectrum modified by substituting its exponential factor exp(-a/k2) with exp(-a/k). The latter spectrum is tractable, and while incorporating very sharp spectral peaks, it is rather inflexible. A typical azimuthal dependence is also considered.

The 8th Texas International Conference on Approximation Theory, sponsored by NSF and the U.S. Army Research Office, was hosted by Texas A&M University in College Station,"January 8-- 12, 1995. The organizers of this conference were Charles K. Chui, Larry L. Schumaker, Joseph D. Ward, Ewald Quak, and N. Sivakumar. Previous conferences in this series were held in 1973, 1976, 1980, and 1992 in Austin, and 1983, 1986, and 1989 in College Station. The conference was attended by 210 mathematicians from 21 different countries. In addition to a large number of contributed talks, there were three special sessions on interdisciplinary topics of current research interest, arranged by Akram Aldroubi (on Wavelets), Chandrajit Bajaj (on Scientific Visualization), and Joseph Warren (on Computer Geometric Aided Design).Two volumes of the conference proceedings, carefully edited by C. K. Chui and L. L. Schumaker, will be published by World Scientific Publishing. The central theme of Volume I is the core of approximation theory. It includes such important areas as qualitative approximations, interpolation theory, rational approximations, radial-'basis' functions, and splines. The second volume' focuses on topics related to wavelet analysis, including multiresolution and multi-level approximation. sub division schemes in CAGD. and applications.

The workshop Stochastic Systems and Their Applications was held in Newport, RI on April 15 & 16, 1994. The main topics of the conference were asymptotic methods in stochastic systems theory, and related applications. The goal of the conference was to review recent advances in asymptotic methods and expose some important new application areas where those methods might be useful. (AN)

The nonlinear damping in a strongly turbulent convecting plasma computed by Kraichnan's modified direct inteaction approximation and the power spectrum are rederived in a physically transparent form using Kolmogoroff's theory of turbulence. (Author)

Recently J. R. Rice ('Approximation of Functions', Elsevier Pub. Co. 1965, p. 111-133) initiated a geometrical study of non-linear approximations. This paper offers a small contribution to certain analytical aspects of mean- square non-linear approximations. One result is to exhibit a family of non- linear topological subspaces of the space C(a,b) with the mean-square metric which has a local unique best approximation property.

The effects of transverse shear deformations upon the classical buckling load of fiber-reinforced cylindrical shells under uniform axial compression have been analyzed. A stability determinant which upon evaluation yields an expression for the buckling load has been developed using a modified form of the Reissner variational principle. The buckling loads predicted by the stability determinant, with transverse shear effects neglected, have been calculated and shown to agree with previously published results for boron-epoxy and glass-epoxy cylinders. Inclusion of the transverse shear effects in the two cases investigated shows little reduction in the classical buckling loads. For general application, charts are presented to give stability criteria for fiber- reinforced composites as a function of the geometric and mechanical properties. For the determination of natural frequencies of fiber-reinforced shells, with transverse shear effects included, a frequency equation in determinant form is presented.