Non-singular Part Research Articles

Spectra of positive and negative energies in the linearized NLS problem

AbstractWe study the spectrum of the linearized NLS equation in three dimensions in association with the energy spectrum. We prove that unstable eigenvalues of the linearized NLS problem are related to negative eigenvalues of the energy spectrum, while neutrally stable eigenvalues may have both positive and negative energies. The nonsingular part of the neutrally stable essential spectrum is always related to the positive energy spectrum. We derive bounds on the number of unstable eigenvalues of the linearized NLS problem and study bifurcations of embedded eigenvalues of positive and negative energies. We develop the L2‐scattering theory for the linearized NLS operators and recover results of Grillakis [5] with a Fermi golden rule. © 2004 Wiley Periodicals, Inc.

Time correlation functions of hard sphere and soft sphere fluids.

We explore the transition between soft particle fluids of increasing steepness to the hard sphere limit. We analyze the analytic forms of the time correlation functions used in determining transport coefficients in Green-Kubo formulas for fluids composed of particles interacting through a repulsive r(-n) potential. We focus on the steeply repulsive n--> infinity limit where the potential tends to the hard sphere interaction. Dufty [Mol. Phys. 100, 2331 (2002)] developed a theoretical framework that can be used to characterize the transition from a steeply repulsive continuous potential toward the hard sphere potential for the shear stress time correlation function. This function was shown to consist of a rapidly decaying contribution (which is singular in the steeply repulsive limit) and a slowly decaying nonsingular part which can be reasonably well represented by Enskog's prediction on times of order and in excess of the mean collision time. We extend this treatment to the bulk viscosity and thermal conductivity. We focus on the bulk viscosity (pressure) correlation function as it is purely singular for hard spheres, and has no kinetic or cross term contributions in this limit. There is no relaxation of this correlation function on the mean collision or Enskog time scale for hard spheres. We show that it is not possible to represent the steeply repulsive behavior of this function entirely in terms of a sech function, i.e., C(B)(t)=sech(a(n)t/tau(n)), where a(n) is a numerical factor, t is time, and tau(n) is a relaxation time proportional to n(-1). An additional singular function, which we call w(t), is required to obtain the correct short-time behavior of C(B)(t) and the Enskog value for the bulk viscosity. With this additional function, the value of a(n) in the n--> infinity limit is a(n)=square root of 2 which is consistent with the second moment of the time expansion of the time correlation function. We compute this function for large n and extrapolate it to n--> infinity, determining one possible analytic form. The shear stress correlation function also gives a(n)=square root of 2 in the hard sphere limit for the singular part when the sech and w functions are used. This function has a nonsingular component, even in the hard sphere limit. We explore various forms for the crossover function X(t/tau(n)) introduced by Dufty, which weights the limiting singular and nonsingular contributions to C(S)(t) particularly at intermediate times. The qualitative behavior for the heat flux time correlation function (used to obtain the thermal conductivity) is much the same as the shear case. The w(t) derived by several self-consistent extrapolations appears, within the simulation statistics, to be the same for the bulk and shear viscosity, and for the thermal conductivity cases.

A B-Spline Higher-Order Panel Method Applied to Two-Dimensional Lifting Problem

A higher-order panel method based on B-spline representation for both the geometry and the solution is developed for the solution of the flow around two-dimensional lifting bodies. The influence functions due to the normal dipole and the source are separated into the singular and nonsingular parts; then the former is integrated analytically, whereas the latter is integrated using Gaussian quadrature. Through a desingularization process, the accuracy of the present method can be increased without limit to any order by selecting a proper numerical quadrature. A null pressure jump Kutta condition at the trailing edge is found to be effective in stabilizing the solution process and in predicting the correct solution. Numerical experiments indicate that the present method is robust and predicts the pressure distribution around lifting foils with far fewer panels than existing low-order panel methods.

Multiplicity of Functions On Singular Varieties

Let V be a local complete intersection in a complex manifold W. For a function g on W, we set f = g|V and f′ = g|V′, where V′ denotes the non-singular part of V. For each compact connected component S of the union of the singular set of V and the critical set of f′, we define the virtual multiplicity [Formula: see text] of f at S as the residue of the localization by df′ of the Chern class of the virtual cotangent bundle of V. The multiplicity m(f, S) of f at S is then defined by [Formula: see text], where μ(V, S) is the (generalized) Milnor number of [2]. If S = {p} is an isolated point and if g is holomorphic, we give an explicit expression of [Formula: see text] as a Grothendieck residue on V. In the global situation, where we have a holomorphic map of V onto a Riemann surface, we prove a singular version of a formule of B. Iversen [13].

Specific heat and energy for the three-dimensional O(2) model

We investigate the three-dimensional O(2) model on lattices of size 8 3 to 160 3 close to the critical point at zero magnetic field. We confirm explicitly the value of the critical coupling J c found by Ballesteros et al. and estimate the universal values g r and ξ/ L. At criticality we study the finite size dependencies of the energy density ϵ and the specific heat C. We find that the nonsingular part of the specific heat C ns is linearly dependent on 1/α. From the critical behaviour of the specific heat results at T ≠ T c on the largest lattices we determine the universal amplitude ratio A +/ A −. The α-dependence of this ratio is close to the phenomenological relation A +/ A − = 1 − 4α.

A meshless, integration-free, and boundary-only RBF technique

Based on the radial basis function (RBF), nonsingular general solution, and dual reciprocity method (DRM), this paper presents an inherently meshless, integration-free, boundary-only RBF collocation technique for numerical solution of various partial differential equation systems. The basic ideas behind this methodology are very mathematically simple. In this study, the RBFs are employed to approximate the inhomogeneous terms via the DRM, while nonsingular general solution leads to a boundary-only RBF formulation for homogenous solution. The present scheme is named as the boundary knot method (BKM) to differentiate it from the other numerical techniques. In particular, due to the use of nonsingular general solutions rather than singular fundamental solutions, the BKM is different from the method of fundamental solution in that the former does not require the artificial boundary and results in the symmetric system equations under certain conditions. The efficiency and utility of this new technique are validated through a number of typical numerical examples. Completeness concern of the BKM due to the sole use of the nonsingular part of complete fundamental solution is also discussed.

Resonant equations and special functions

Differential equations of the form Lf= g, where L is a differential operator, are termed Resonant if g satisfies Lg=0. In the case when L represents a linear harmonic oscillator, resonance occurs when the forcing term g has the same frequency as that of the unperturbed system. Resonance is associated with a transition from boundedness to unboundedness of the solution. We study the cases where L is the Legendre or Hermite operator. The first case arose in the context of supersymmetric Casimir operators for the di-spin algebra, and has solutions expressible in terms of singular functions, Legendre functions and polylogarithms. The non-singular polynomial parts of a certain class of solutions exhibit interesting properties. The non-resonant Hermite equation supports the theory of the quantum mechanical harmonic oscillator. A standard technique for its solution involves a Darboux/Infeld-Hull factorization of the Hamiltonian as a product of two first-order linear operators. The algebra of these operators can also be used to study the solutions of the resonant Hermite equation. A lowest order solution is found by elementary means, and then higher order solutions are generated by the repeated application of a ladder operator.

Height pairings on Shimura curves and p-adic uniformization

In a recent paper [16] of one of us it was shown that there is a close connection between the value of the height pairing of certain arithmetic 0-cycles on Shimura curves and the values at the center of their symmetry of the derivatives of certain metaplectic Eisenstein series of genus 2. On the one hand, the height pairing can be written as a sum of local height pairings. For example, if the 0-cycles have disjoint support on the generic fiber, then their height pairing is a sum of an archimedean contribution and a contribution from each of the (finitely many) finite primes p for which the cycles meet in the fiber at p. On the other hand, it turns out that the non-singular part of the Fourier expansion of the central derivative of the metaplectic Eisenstein series also has a decomposition into a sum of contributions indexed by the places of Q. Then, one would like to compare the height pairing and the Fourier coefficients by proving an identity of local contributions place by place. In loc. cit. the identity for the archimedean place was proved, and it was shown that the identity at a non-archimedean place of good reduction is a consequence of results of Gross and Keating, [10], (for the algebraicgeometric side) and of Kitaoka, [12], (for the analytic side). It then remains to consider the finite primes p where the Shimura curve has bad reduction. These are of three sorts: (i) the primes p at which the quaternion algebra defining the Shimura curve is split, but which divide the level, (ii) the primes p at which the quaternion algebra remains division, but no level structure is imposed, and (iii) the primes p at which the quaternion algebra remains division and where a higher level structure is imposed at p. So far, little is known concerning cases (i) and (iii).

Square-root information filtering and fixed-interval smoothing with singularities

The square-root information filter algorithm and the related covariance smoother algorithm have been generalized to handle singular state-transition matrices and perfect measurements. This allows the use of SRIF techniques for problems with delays and state constraints. The generalized algorithms use complete QR factorization to isolate deterministically known parts of the state and nonsingular parts of the state transition and disturbance influence matrices. These factorizations and the corresponding changes of coordinates are used to solve the recursive least-squares problems that are basic to the SRIF technique. Numerical stability, computation time, and storage requirements are comparable to the traditional SRIF algorithms.

, , , and from 3D Ising energy and specific heat

We analyse Monte Carlo data for the energy and specific heat at and close to the critical point of the 3D cubic Ising model. From the finite-size scaling of the energy E and the specific heat C at criticality we obtain the estimate . Furthermore, one obtains precise estimates for the `backgrounds' (nonsingular parts) and . Fitting solely off-critical energy estimates to a scaling law, we find, depending on the choice of the reduced temperature, either and , or and . Including information from the data at , we obtain the estimate . We also determine the universal combination in both phases.

IGB-offset for plane curves—loop removal by scanning of interval sequences

The generation of NC-tool paths for 2 1 2 - axes machining is based on offset curves to arbitrary simple piecewise smooth G 0 generator curves. Global and local properties that are of interest for the offset generation from the NC and the differential geometric viewpoint are discussed and described in this paper. These properties lead to an approach for generating global offset curves in a more direct and technological way. The key point of this approach is the restriction of the offset image to nonsingular parts of the generator curve with respect to the given offset distance. In general, this restricted offset image is discontinuous, but the restriction guarantees a minimal number of self-intersection points. The correspondence of self-intersections to parameter intervals is used to describe uniquely a set of relevant trim intervals. The result after the trimming of these intervals is a continuous offset curve that is, up to some technological constraints, equal to the offset curve of the right- or left-sided rolling ball blend of the generator curve. This offset carries the Gauss-Bonnet values of the generator as invariants and is, therefore, called IGB-offset. Global loops of the IGB-offset are classified by the superposition of remaining self-intersection intervals and are described by the union of elements with even index of finite interval sequences in the parameter space of the IGB-offset.

Generalized Hestenes’ Lemma and extension of functions

Suppose we have an m-jet field on V C Rn which is a Whitney field on the nonsingular part M of V We show that, under certain hypotheses about the relationship between geodesic and euclidean distance on V, if the field is flat enough at the singular part S then it is a Whitney field on V (the order of flatness required depends on the coefficients in the hypotheses). These hypotheses are satisfied when V is subanalytic. In Section TI) we show that a C2 function f on M can be extended to one on V if the diSerential dy goes to 0 faster than the order of divergence of the principal curvatures of M and if the first covariant derivative of df is sufficiently flat. For the general case of cm functions with m > 2, we give a similar result for codim M = 1 in Section II

Crack growth criteria incorporating non-singular stresses: size effect in apparent fracture toughness

Simple criteria accounting for the non-singular stresses at the crack tip are considered. They are based on the comparison of the local stress concentration with the material microstrength. The local stress concentration is estimated either as the magnitude of the conventional elastic stress ahead of the process zone, or by its averaging over the process zone length. When a criterion of this type is used to find the critical load and then the conventional fracture toughness, the latter will be dependent of the crack length. This size effect in fracture toughness manifests itself as an increase (if the non-singular part of the near-tip stress field is positive) or decrease (if the non-singular part is negative) in the apparent fracture toughness as the crack length increases. The obtained dependence is compared with available experimental data. It is also shown that when the load can be resolved into a superposition of elementary loads, the size effect can asymptotically (for long cracks) be presented as a weighted sum of the elementary size effects (i.e. the size effects associated with the elementary loads) with the weights equal to the relative contributions of the elementary loads into the total stress intensity factor.

Nonsingular Greens function method for crystals

By use of simple mathematical transformations, an integral equation for the Greens function method leads to a homogeneous equation which contains only the nonsingular part of the Greens function. No limitations are imposed on the crystal potential, so the solutions may be found for any unit cell. The method is applicable to any crystal structure. Using bcc iron as an example we present various calculations which make it possible to predict ahead of time which atomic levels will form bands and which will not.

On the computation of Fourier transforms of singular functions

In this paper we consider a method for the computation of finite Fourier transforms of functions having endpoint singularities of algebraic or logarithmic type. This method may be considered as a modification of Clenshaw-Curtis quadrature. The nonoscillatory and nonsingular part of the integrand is replaced by a truncated Chebyshev series approximation and the integral is then evaluated using recurrence relations. The numerical stability of these recurrence relations is investigated.

Exact Gaussian quadrature methods for near-singular integrals in the boundary element method

The boundary element method requires accurate calculation of line and surface integrals in which a factor such as 1n(1/ r), 1/ r or 1/ r 2 is ill-behaved when r is nearly zero over a small portion of the domain. This paper shows that the underlying theory of Gaussian integration can be applied to get a quadrature formula for any particular distance from the element. An accurate integral can then be calculated with very few evaluation points by (a) computing the distance to the element; (b) interpolating in a table to obtain quadrature points and weights; and (c) summing the weighted values of the non-singular part of the integrand. This provides a fast, accurate value of the integral with no uncertainty as to the number of evaluation points that are required. Line integrals with near-singular 1/ r, 1/ r 2 and 1n(1/ r) terms are used as examples. Accuracy of the formulas is verified experimentally for a wide range of distances. Interpolation in tabulated data is shown to intriduce only minor errors. The line integral formulas are applied to a triangular region, yielding exact formulas of any desired degree for the case where a 1/ r singularity is at a vertex of the triangle.

The interaction representation in the self-focusing theory

The general scheme of dividing the solution of the nonlinear Schrödinger equation (2D case) into the focused (singular) part of the beam called the “peak” and the rest (nonsingular) part, the “background”, is presented. The peak is compared with a manifold of self-similar modes whose dynamics is analysed. It is shown that if the beam power is less than critical, the self-similar beam is strictly localized. If the beam power is more than critical the self-similar solution is a quasi-localized mode with tunnel emission of power from the paraxial region and the width of the peak tends to zero. The parametric peak-background interaction analysis proves that the limiting self-focusing law a(t)∼[ (t 0−t) ln|ln(t 0−t)| ] 1 2 is determined only by the peak dynamics.

On the evaluation of the real parts of forward pp amplitudes from the data at small t

The author discusses the problem of the evaluation of the real parts of forward pp amplitudes directly from the data at small t. The expressions for d sigma /dt and for a number of polarisation parameters are derived in the limit of small t by means of an expansion of the non-singular parts of the amplitudes in powers of t. The analysis of the data at small t available in the energy range 330-800 MeV is performed and the real parts of forward mixed amplitudes are determined. A method suitable for calculation of the Bethe phases at low and medium energies is proposed, and the real parts of forward hadronic amplitudes are calculated. Comparison of these 'measured' real parts with those predicted by the forward dispersion relations reveals a rather satisfactory agreement for the real parts of all three forward hadronic amplitudes.

The boundary value problem in fermion systems

The half-space boundary value problem for fermions near zero temperature in plane geometry is solved for diffuse boundary scattering by numerically constructing the spatial propagator in terms of the eigenfunctions of a generalized eigenvalue problem for the linearized Uehling-Uhlenbeck collision integral. The slip length is calculated for several interparticle scattering laws and compared with a relaxation time ansatz result and the experimental values for normal fluid3He. It is shown that the nonsingular part of the collision operator is relatively compact to the singular part.

Strongly convergent Green's function expansions for rectangularly shielded microstrip lines

The exact analytical treatment of the quasi-TEM mode in various cross-sectional configurations of microstrip lines may be used on Carleman-type singular integral equations (SIEs). Their kernel is a Laplacian Green's function G with source point limited on the interface separating the dielectric media. Strongly convergent expansion for G, particularly suited for the subsequent solution of the SIE and for exact field-point evaluations in rectangularly shielded microstrip configurations, are developed. Extraction of the singular logarithmic term leads to rapidly converging series expansions for the nonsingular part. The convergence of certain of these series is further improved when the field point lies also on the interface or when the source point approaches the shielding boundaries. >