Theory Of Generalized Functions Research Articles

The Weizsäcker functional: Some rigorous results

AbstractThe Weizsäcker functional TW is a necessary element to explain basic physical and chemical phenomena of atomic and molecular systems in the general density functional theory initiated by Hohenberg and Kohn. Here, rigorous inequalities which involve the functional TW and two arbitrary power‐type density functionals ωα ≔ ∫ ρα(r)dr are found by the successive applications of Sobolev and Hölder inequalities. Particular cases of these inequalities give lower bounds to the Weizsacker functional of an N‐electron system in terms of a fundamental and/or experimentally measurable quantity such as, e.g., the Thomas–Fermi kinetic energy T0, the Dirac–Slater exchange energy K0 and the average electronic density 〈ρ〉 in doing so, some known relationships appear. A numerical Hartree–Fock study of the accuracy of some resulting lower bounds is carried out. Finally, rigorous relationships between the Weizsäcker functional and the Boltzmann–Shannon information entropy of the system under consideration are given. © 1995 John Wiley & Sons, Inc.

On simple Igusa local zeta functions

The objective of this announcement is the statement of some recent results on the classification of generalized Igusa local zeta functions associated to irreducible matrix groups. The definition of a simple Igusa local zeta function will motivate a complete classification of certain generalized Igusa local zeta functions associated to simply connected simple Chevalley groups. In addition to the novelty of these results are the various methods used in their proof. These methods include use of the concept of canonical basis from quantum group theory and a formula expressing Serre’s canonical measure μc in terms of a suitably normalized Haar measure μ and density function Φ. The relevance of these results in the general theory of Igusa local zeta functions is also discussed.

Toward a Unified Theory of Generalized Functions: Convergence

Generalized functions are usually treated as bounded linear functionals on spaces of test functions. Here they are considered as objects which „convolute” with test functions and satisfy a certain associativity condition. In this setting simple definitions are given for convergence of sequences of test functions and generalized functions, definitions which do not depend on particular properties of the space of test functions. It is shown that for the most commonly used spaces, these general definitions reduce to the standard ones.

NEW GENERALIZED FUNCTIONSIN NONSTANDARD FRAMEWORK

In this paper we develope a theory of new generalized functions by using Nonstandard Analysis which is closely relevant to that of Colombeau's new generalized functions.

Operator-valued Bessel functions on Schrödinger-Fock spaces and Siegel domains of type II

In this paper, a general theory of operator-valued Bessel functions on mixed Schrödinger-Fock spaces and Siegel domains of type II is presented. In general, Bessel functions J λ are associated to the partial Fourier transform on a mixed Schrödinger-Fock space. J λ can be realized as a reduced Bessel function K π , which is a generalized inverse Laplace transform on a Siegel domain of type II. The application of this theory to representation theory will be studied in the next paper.

Multiplication and mixed differentiation in generalized Fock spaces

The classical distribution theory founded by Laurent Schwartz [26] has an intrinsic multiplication problem. He indicates this in his paper [27]. This problem is clearly demonstrated by selecting two particular distributions and attempting to define a multiplication that enjoys commutative, associative, and unit identity properties. This process leads to the contradiction, 1 = eP I [lo]. A new generalized function theory developed by J. Colombeau [4-91 provides a solution to the multiplication problem and also develops a general integration theory. T. Todorov reexamines the new generalized function theory in the spirit of non-denumerable sequences termed ultrapowers [29]. The L. Schwartz distribution theory was reexamined in the spirit of fundamental sequences [ 16,333. However, there are new considerations in the theory of new generalized function theory requiring an algebraic notion of a particular ideal contained in a specified test function space. A brief review of these constructions can be found in Section 7 and a comprehensive review can be found in Ref. [25]. A principle application for our development will be modeling a physical system consisting of an “infinite” number of particles. A system of n-particles termed bosoms or fermions has previously been modeled by n-dimensional symmetric or antisymmetric functions. When this system is enlarged to an infinite number of particles, the model for the state space vector has the form,

Construction of high-order deceptive functions using low-order Walsh coefficients

This paper constructs deceptive functions over bit strings of any length using low-order Walsh coefficients. Specifically, a partially deceptive construct is achieved with coefficients no higher than second order, and a fully deceptive construct is achieved with coefficients no higher than third order. Previous work on deception tacitly assumed that the very highest order Walsh coefficient, the order-l coefficient in a length-l problem, must be non-zero to achieve full deception, that condition where all schemata of orderl−1 or less containing the complement of the global optimum are superior to their competitors. The constructions of this paper show such assumptions to be false and lead to a more general theory of functions whose value only depends on the number of ones (or zeros) in the function's argument, so-calledfunctions of unitation. These results also help shed some light on Tanese's puzzling results on functions that contained large numbers of Walsh terms of fixed order.

Boundary integral approach to the scattering of nonplanar acoustic waves by rigid bodies

The acoustic scattering of an incident wave by a rigid body can be described by a singular Fredholm integral equation of the second kind. This equation is derived by solving the wave equation using generalized function theory, Green's function for the wave equation in unbounded space, and the acoustic boundary condition for a perfectly rigid body. This paper will discuss the derivation of the wave equation, its reformulation as a boundary integral equation, and the solution of the integral equation by the Galerkin method. The accuracy of the Galerkin method can be assessed by applying the technique outlined in the paper to reproduce the known pressure fields that are due to various point sources. From the analysis of these simpler cases, the accuracy of the Galerkin solution can be inferred for the scattered pressure field caused by the incidence of a dipole field on a rigid sphere. The solution by the Galerkin technique can then be applied to such problems as a dipole model of a propeller whose pressure field is incident on a rigid cylinder. This is the groundwork for modeling the scattering of rotating blade noise by airplane fuselages.

The Fourier transform of the unit step function

The purpose of this paper is to highlight difficulties which can arise when the Fourier transform of the unit step function is introduced to non‐mathematics specialists without reference to generalized function theory.

Sound radiation from a vibrating body in relative motion with its surrounding medium

As an extension of the authors' previous work [J. Acoust. Soc. Am. Suppl. 1 83, S77 (1988)], this paper considers sound radiation from a vibrating body in relative motion with its surrounding medium, namely, either a vibrating body radiating sound into a homogeneous inviscid fluid in uniform motion or a moving vibrating body radiating sound into a stationary inviscid fluid. The resulting sound fields in both cases can be expressed in terms of volume integrals and surface integrals based on the exact theory of Lighthill's acoustic analogy and generalized function theory. However, quantities involved in the volume integrals are not known a priori. Determination of these quantities is equivalent to solving an inhomogeneous wave equation governing the entire flow field, which is virtually impossible for most problems of interest. This paper shows that for a vibrating body in constant and rectilinear relative motion with its surrounding medium, the volume integrals can be transformed into surface integrals. Consequently, the acoustic field can be completely determined given the boundary conditions on the vibrating surface. The total radiated power is calculated and shown to be greatly enhanced by the Mach number. [Work supported in part by ONR, NSF, and Institute for Manufacturing Research at Wayne State University.]

Correlation functions of hot electrons in semiconductors.

We present a general theory of the correlation functions for a steady state which is valid for arbitrary strengths of an applied electric field, as obtained in high-field transport in semiconductors. When limiting to the first two moments of the distribution function, we find a closed set of equations coupling energy and velocity which are amenable to an analytical solution. Thus, the theory provides an interpolation formula which gives smooth, analytical expressions for the correlation functions. These expressions can be made to fit the computer simulation by using these simulations to estimate the various parameters. The theory is found to be in excellent agreement with numerical calculations for a simple model semiconductor performed with an ensemble Monte Carlo technique.

Frontal lobes and language

Numerous theories discuss the neuropsychological functions of the frontal lobes, most based on some concept of supramodality, and an extensive literature presents the phenomenology and semiology of language and communication deficits after focal brain lesions involving the frontal lobes. Despite this, few attempts have been made to link the clinical phenomenology to a theory. This paper presents (1) a general theory of frontal functions; (2) a brief summary of experimental and anatomical literatures in support of defined frontal functional systems; (3) clinical observations that delineate these functional systems for the specific modalities of language and communication; (4) a review of the available literature supporting the idea of specific modal and supramodal language and communication capacities; (5) hypotheses about the distributed anatomy of these functional systems; and (6) implications for traditional clinical notions of aphasia, particularly in relation to a general theory of frontal lobe functions.

Central Fourier analysis for Lorentz spaces on compact Lie groups

We determine the critical indexes for theLp,q uniform boundedness of the characters of a compact connected semisimple Lie group. These results show a basic difference between the simple and the semisimple case, a situation which does not appear when dealing withLp spaces. We provide applications to the general theory of central functions on compact Lie groups.

Theory of interfaces between discrete and continuous media — Application to transition metal surface states

A general theory of response functions for partly discrete and partly continuous composite systems is given. It is then used for the study of the influence of work function and tunneling on transition metal surface states. It is shown on a simple model that when account is taken of the work function, localized surface states may appear above the bulk band. A composite system formed out of a thin vacuum slab sandwiched between two identical transition metal surface (tunnel junction) is then considered. It is found that each surface state is split in two, having slightly different energies.

Extension of Kirchhoff's formula to radiation from moving surfaces

Kirchhoff's formula for radiation from a closed surface has been used recently for prediction of the noise of high speed rotors and propellers. Because the closed surface on which the boundary data are prescribed in these cases is in motion, an extension of Kirchhoff's formula to this condition is required. In this paper such a formula, obtained originally by Morgans for the interior problem, is derived for regions exterior to surfaces moving at speeds below the wave propagation speed by making use of some results of generalized function theory. It is shown that the usual Kirchhoff formula is a special case of the main result of the paper. The general result applies to a deformable surface. However, the special form it assumes for a rigid surface in motion is also noted. In addition, Morgans' result is further extended by showing that edge line integrals appear in the formula when applied to a surface that is piecewise smooth. Some possible areas of application of the formula to problems of current interest in aeroacoustics are discussed.

Application of generalized function theory on the complete solutions of plates and shells

In this paper general solutions of partial differential equations with constant coefficients, especially in the theory of plates and shells, are discussed and obtained in closed or integral forms by a Fourier transform method based on the theory of generalized functions. Some of these general solutions can be directly used to solve mechanics problems.

A Conversation with George A. Barnard

A Conversation with George A. Barnard

On class numbers of hyperelliptic function fields with Hasse-Witt-invariant zero

with a l e ~, a o = 1, a2o_ i = qg i al for i = 0 , 1 . . . . . g and the class number h is given by h = L (1) (the general theory of algebraic functions of one variable is found in Eichler [3]). Especially, if a 1 = a 2 -= . . . ag ~ 0 (rood p), then it is said that the Hasse-Wit t invar iant of A is zero (cf. Stichtenoth [8]). Our purpose of this note is to determine the L-function and the class number of some kind of hyperell iptic function field having Hasse-Wit t invar iant zero. F o r a e K (a + 0) we will denote by ord (a) the order of a in the cyclic group K* = K {0} and then, for n = 1, 2 , . . . , we will put I (n , a) ~(q" l ) /ord (a); clearly, I(n, a) is the index of a with respect to some suitable generator of the cyclic group K~*, where K n = GF (q") GF (pnr). Then our main results are stated as follows.

General properties of electromagnetic response functions

The present state of the general theory of response functions of material media, describing the universal properties of these functions that characterize all types of medium, is surveyed. Topics covered include recent results on the range of admissible values of static permeability, the form of the Landau functional for the electrodynamic ordering of a medium, and the electrodynamics of a medium interacting with a magnetic monopole. Applications to high-temperature superconductivity, anomalous diamagnetism, and monopole detection are discussed.

Completeness of transfinite evaluation in an extension of the lambda calculus

The lambda calculus was conceived by Church, and to some extent by Curry (see [1] and [2]), as a general theory of functions, more precisely as a formalization of two basic functional concepts: application and abstraction. This claim to universality was impossible to maintain, for it soon became apparent that functions in the calculus are required to have properties which are by no means universal. Self-application is certainly a very special feature, but more relevant from our point of view is the existence of fixed points, which is definitely a nonuniversal property of functions. This situation was not ignored, but was considered rather as a form of inconsistency very much related to the paradoxes of set theory (for example, see the discussion of the paradoxical combinator in [2]).More recently, the lambda calculus has been studied by people in computer science, and as a result the algorithmic nature of the system has been more explicitly recognized. Eventually, the right approach was taken by D. Scott (see [4] and [5]) who restricted the objects of the calculus to continuous functionals over complete lattices, which can be combined in structures where application and abstraction are properly interpreted. It soon became clear that complete lattices were not necessary, and more general structures were introduced. On the other hand the requirement of continuity was essential for the construction, and has remained a basic feature. It is one of the purposes of this paper to show that in the general situation continuity can be replaced by monotonicity.The work of Scott was extended by Wadsworth [6], who gave a precise formulation of the operational semantics, and proved the equivalence to Scott's interpretation.