Arbitrary Terms Research Articles

On the equivalence of the background field method

An improved and generalized version of a recent proof of equivalence forS-matrices obtained in the background field method and in conventional quantization procedures is proposed. Furthermore, it is shown explicitly that theS-matrix is independent of the additional arbitrary gauge breaking term for the background field. In addition, the axial gauges as background field gauges are discussed briefly.

A certain class of [formula omitted]-series transformations

A large number of summation and transformation formulas for a certain class of double hypergeometric series are observed here to follow fairly readily from a single known result which, in turn, is a very special case of one of six general expansion formulas given in the literature. Generalizations and unifications of these expansion formulas involving series with essentially arbitrary terms are presented. It is also shown how the various series transformations considered in this paper admit themselves of q -extensions which are capable of unifying numerous scattered results in the theory of basic double hypergeometric functions.

Non-resonance for a strongly dissipative wave equation in higher dimensions

Under some regularity conditions, a non-resonance property is established for a semi-linear forced wave equation with a strong local damping term and Dirichlet boundary conditions in a bounded open domain. In dimension less than or equal to six, the damping term can grow at infinity like an arbitrarily large power of the velocity. If a viscosity term is added, in dimension less or equal to four a stronger result is obtained, and this property allows to construct almost periodic solutions for an arbitrary forcing term in a suitable regularity class.

Population dynamics of deer

We modelled a population of white-tailed deer and determined the optimal harvesting patterns. Since the deer naturallay fall into three distinct groups—fawns, does, and bucks—we used a three-dimensional system of difference equations to model the sizes of the three groups at yearly intervals. We include arbitrary harvesting terms in each equation which were to be chosen in order to maximize the total value of the harvest over a long period of time. We assumed that the mortality rates of each group were proportional to their sizes. The total yearly food supply remained constant. We used actual data to estimate realistic ranges on the parameters in the model. We defined the value function of the harvest as a weighted sum of the harvested deer and the live deer. We then used an iterative linear programming method to optimize this value function in order to determine an optimal harvesting pattern over a period of twenty years. Under normal conditions, we found that it is best to harvest mostly bucks, some does, and no fawns. We discovered that a “buck law”—a law which prohibits hunting anything but bucks—is not an optimal way of harvesting the deer. We examined the results for different values of the mortality parameters and found them to be the same. Therefore, our solution appears to be stable with respect to environmental conditions.

Higgs masses in the standard, multi-Higgs and supersymmetric models

Theoretical constraints and limits on the masses of Higgs scalars in the standard electroweak model, in electroweak models with additional Higgs doublets and in various supersymmetric models are presented. In the standard model, the lower limit on the Higgs mass, based on vacuum stability arguments, is reviewed in detail, as are “upper limits” based on perturbative constraints. In most grand unified and all supersymmetric models, however, at least two doublets are needed. The masses of the various Higgs scalars in the two-doublet model are discussed and constraints on their masses are found, including the generalization of the above limits. The results are then generalized to models with more than two doublets. Finally, recent attempts at constructing models with low-energy supersymmetry are reviewed and it is shown that in many models, fairly stringent tree-level mass relations among the Higgs scalars can be found. These relations are interesting in that they do not refer to the supersymmetric partners of ordinary particles, and they are most restrictive in models in which the supersymmetry is explicitly broken, i.e., via arbitrary mass terms.

A method for determination of higher-order magnetic anisotropy constants — Importance of the cubic K 3 and K 4 for certain energy levels models

A method for determination of magnetocrystalline anisotropy constants of arbitrary order is proposed. The method is based on a least squares fitting of a phenomenological anisotropy energy for a given symmetry truncated at an arbitrary order term to a theoretical anisotropy energy computed exactly for a given energy level model. Several applications of the method to cubic systems are considered. The study reveals that the widely used expressions in the literature for the cubic constants K 1 and K 2 in terms of free energy for the three symmetry direction are of rather limitedvalidity only. The higher-order cubic constants K 3, K 4 and K 5 are determined besides the usual K 1 and K 2 in temperature range 0 to 300 K. The importance of the higher- order terms with respect to the first term in the cubic anisotropy energy is discussed. The results show that the cubic constants K 3 and K 4 cannot be neglected for most of the energy level models studied at certain temperatures.

An Improved Method for Calculating the Quasi-Static EMP

A new time-independent numerical method is used to determine the late-time electromagnetic pulse (EMP) resulting from a surface nuclear burst. Completely arbitrary source terms are employed and the non-linear dependence of the air chemistry and air conductivity parameters on the electric field strength is incorporated into the solutions. Comparisons are made with previous analytic expressions. In addition, a parametric study is presented.

The exchange of massless spin-two particles

The theory of gravity is considered from the little group viewpoint. This leads to a theory with a constraint, which is equivalent to general relativity with an arbitrary cosmological term. With this framework (i) the cosmological constant cannot be put into the Lagrangian but it appears as an integration constant. (ii) The gravitational Lagrangian automatically takes the form of a finite polynomial of the metric. (iii) The so-called conformal factor is fixed, which removes an apparent difficulty in carrying out path integrals.

A clustered search algorithm incorporating arbitrary term dependencies

The documents in a database are organized into clusters, where each cluster contains similar documents and a representative of these documents. A user query is compared with all the representatives of the clusters, and on the basis of such comparisons, those clusters having many close neighbors with respect to the query are selected for searching. This paper presents an estimation of the number of close neighbors in a cluster in relation to the given query. The estimation takes into consideration the dependencies between terms. It is demonstrated by experiments that the estimate is accurate and the time to generate the estimate is small.

Long waves in world industrial production, energy consumption, innovations, inventions, and patents and their identification by spectral analysis

Long-term cycles have been much discussed in the literature since Kondratieff (1926). (“Cycles” is, of course, a quite arbitrary term for these time periods.) Spectral analysis was applied by the authors to long-time series of industrial production, energy consumption, inventions, innovations, and patents in order to reveal quantitative regularities in their behavior and or in their interdependence. An attempt was also made to identify logistics within those time series. In the long cycle of 50–53 years no significant autocorrelation could be detected. Logistics exist only in three special periods for innovations and inventions. Nondominant long cycles do appear in the interaction between innovations, production, patents, and energy consumption. The investigation shed light on the causal structure of the innovation system. In particular, it revealed a significant influence of industrial production on patents with a lag of 9 years.

Linear fluctuations in a plasma

The linear fluctuation spectrum of a plasma is calculated from the Boltzmann equation with an arbitrary linear collision term.

The mechanism of reference

Reference is what relates words to the world of objects on whose condition the truth of sentences hinges. It is natural to wonder what sorts of relations underlie the reference relation-to wonder, that is, what constitutes the mechanism of reference. This question comes naturally because reference has the appearance of a supervenient phenomenon: semantic facts, one feels, hold in virtue of facts of some other kind. To ask for a specification of the underlying facts is not (necessarily) to commit oneself to a reductionist view of reference; it may be that our best account of the reference relation, though substantial and illuminating, fails to reach the standards of genuine reduction-it takes the form of a 'picture' rather than a 'theory'.1 An account of reference, of the shape I envisage, would consist in a set of rules the rules of reference which determine a function from expressions to the items which are their referents. The rules would speak of relations, not themselves directly characterised in terms of the notion of reference, recognition and deployment of which is comprised in mastery of a language: a speaker has mastered the rules of reference for a language if and only if he appreciates (perhaps inchoately) what relations go to determine the referent of an arbitrary singular term. Specifying the rules of reference is thus part of an account of the principles in which a speaker's ability to interpret words as possessed of specific referents consists, and hence of his ability to come to know the truth conditions of sentences. Articulating the rules of reference is thus analogous to devising a theory of the structure of sentences: such a theory should likewise be construed as a representation of a speaker's ability to understand sentences by way of the meaning of their constituent parts. Recent discussion of the mechanism of reference has tended to focus on the case of proper names, as if these were the basic referential devices. This focus reflects the assumption that the mode of operation of names provides the model for an account of how

Solutions of the f-g field equations

The f-g field equations are solved for an arbitrary mixing term. Several techniques are given to generate solutions of f-g field equations with and without sources for a special mixing term. Some new solutions are also presented.

Feynman amplitude circuit breakers

It is shown that point splitting may be used to regularize arbitrary Feynman amplitudes and Taylor subtraction terms of the type found in the Zimmermann formulation (BPHZ) of renormalized perturbation theory. Point splitting refers to the Fourier transformation of the Feynman integrand with respect to the internal (loop) momenta. It is shown that this has the effect in configuration space of breaking open the circuits of the diagram. The regularization may be removed, interchangeably, before or after the propagator epsilon is taken to zero in the sense of distribution theory. This result is proven elsewhere but is mentioned here for its importance. The result may make the method useful in establishing the unitarity of BPHZ. Two appendices contain a review of circuit-based graph theory and a demonstration that an arbitrary circuit based graph may sometimes have no realization as a graph.

Effects of Indoor Residence on Radiation Doses from Routine Releases of Radionuclides to the Atmosphere

Dose reduction factors from indoor residence during routine releases of radionuclides to the atmosphere were studied using models that are suitable for application to arbitrary source terms. Dose reduction factors for internal exposure to inhaled radionuclides account for air ventilation and deposition on inside building surfaces. Estimated internal dose reduction factors are approx. 0.2 to 0.8 for particulates and 0.07 to 0.4 for radioiodine. Dose reduction factors for external photon exposure from airborne and surface-deposited sources are based on the point-kernel integration method. Values for source terms from a fuel reprocessing plant and a hypothetical reactor accident are within a factor of 2 of the value 0.5 adopted by the US Nuclear Regulatory Commission (NRC) for population dose assessments. For the release at Three Mile Island nuclear station, however, the external dose reduction factor may be an order of magnitude less than the value adopted by the NRC.

Equations for the propagation of finite local disturbances in vector fields of physics and mechanics

The purpose of this theoretical investigation is to determine the propagation velocities of local disturbances that may occur in the curl and divergence of a vector field v that satisfies the equation ∂v/∂t+f=0, where f is a general vector field. Equations of this form frequently occur in physics and mechanics. A disturbance in the curl v (div v) field is defined by ∂curl v/∂t≠0 (∂div v/∂t≠0). Such a disturbance is termed a local disturbance if it is surrounded by a quiescent layer of finite thickness. Equations are derived for the velocity U(r,t) of curl v lines and the velocity W(r,t) of div v elements. These velocities conserve, respectively, the circulation of v about arbitrary circuits and the efflux of v through arbitrary closed surfaces. These conservation conditions are not sufficient to completely specify the two velocity fields, and each contains an arbitrary term. More convenient for three-dimensional applications than the circulation is the gyration G (spatial integral of curl v); and U is found to also conserve G in regions T where the curl v lines retain their individual shapes and orientations as they move, and our consideration of curl v disturbances is confined to such regions. The centroid of efflux is defined for arbitrary volumes, and the centroid of gyration for volumes in T. The movement of these centroids is linked to the velocities U and W by the conservation relations. Contributions from the arbitrary term in U (W) are suppressed by integration over a local curl v (div v) disturbance and by construction of a scalar potential field that satisfies Laplace’s equation in the interior of the disturbance with Dirichlet (Neumann) conditions related to f on the closed boundary. Formulas for the velocities of the centroids are thus obtained for local curl v and div v disturbances as functions of f, and these velocities are interpreted as propagation velocities of the disturbances. Only one type of disturbance need be localized for its formula to apply. These formulas enable us to calculate how such local disturbances will propagate without integrating forward in time; and in certain cases, only the fields outside a disturbance are required for this calculation. Applications to Maxwell’s electrodynamic equations are presented as examples.

A variational approach to perturbed soliton equations

A general perturbation method is presented for soliton equations modified by small but arbitrary perturbation terms. Equations for the adiabatic time evolution of the soliton parameters are derived from an approximate variational principle. The method is applied to the Korteweg-de Vries and nonlinear Schrödinger equations modified by various dissipative perturbation terms.

A general theory of linear finite-difference equations with a few applications

Recently a new formalism has been developed giving the solutions in explicit form of multiterm linear homogeneous recursion relations with nonconstant coefficients. The basic idea was to relate this problem to the one of partitioning an interval into parts of given lengths. This idea is extensively used here to obtain the solutions of linear inhomogeneous difference equations. The resulting method has the advantage of being general: It does not rely on any special device and does not assume any special form for the recursion relation. Applications of these techniques to physical problems are presented elsewhere. Here we show how the method works for first and second order equations. The three-term Legendre polynomial recursion relation with an arbitrary inhomogeneous term is discussed in detail. The Legendre polynomials are then viewed as special cases of combinatorics functions, P (j,m;z), based on the partitions of an interval (j,m), 0⩽j⩽m, into parts of lengths 1 and 2. P (j,m;z) reduces to the Legendre polynomial, Pm(z), for j=0. It is also interesting to note that P (j,m;z) are not orthogonal in general. However, a new set of orthogonal functions can be constructed as the solutions of the inhomogeneous Legendre recursion relation with a suitable choice of the inhomogeneous term.

Effects of a neutral current on the neutrino bremsstrahlung

The energy-loss rate due to the neutrino bremsstrahlung from relativistic electrons in a dense matter such as a star is calculated in the framework of Weinberg's unified theory of weak and electromagnetic interactions by using the method of neutrino-energy expansion. The momentum-transfer integration, though complicated, can be carried out analytically, but the subsequent two electron integrals, called the distribution integrals, are helpless and do not allow any further analytic calculation. In order to avoid this difficulty, we set up a set of variables whose integration is simple enough to be expanded in series. It turns out that this series becomes a power series in the Fermi energy inverse after the distribution integrals. This enables us to keep only the leading terms, greatly simplifying the final result. Thus in principle we will be able to calculate arbitrary higher-order terms, even though most of them require prohibitively tedious labor. We carry out the nexthigher-order calculations to find that the correction to the lowest-order contribution is within 5%. This confirms the earlier calculations of Festa and Ruderman, even though the two approaches are quite different: the neutrino-bremsstrahlung energy-loss rate is proportional to ${T}^{6}$ and independent of the density at extremely relativistic degenerate regions. We find that the asymptotic expression for the energy-loss rate is of the form $A(\ensuremath{\beta})+{(\frac{\ensuremath{\rho}}{{\ensuremath{\mu}}_{e}})}^{\ensuremath{-}\frac{1}{3}}kTB(\ensuremath{\beta})$. The most striking effect of the neutral current is the muon-neutrino emission. The ratio of the energy-loss rate in Weinberg's theory to that in the conventional theory is $\frac{1}{2}+4{{sin}^{4}\ensuremath{\theta}}_{W}$ at the extreme relativistic degenerate and nondegenerate nonrelativistic regions.

Hyperinflation and the Supply of Money

Hyperinflation is a somewhat arbitrary term that is used to describe the monetary experience of six European countries after World War I and Hungary and China after World War II. All eight countries experienced enormous growth of the nominal money stock, mainly because governments resorted to the printing press to finance expenditures. The primary focus of economic studies of hyperinflation has been to test quantity theory propositions concerning the stability of the demand for money during hyperinflation. Cagan's [5] study presented the first formal model whose empirical results appeared to confirm quantity theorists' beliefs concerning hyperinflation; i.e., that hyperinflation was due to growth of the nominal money stock acting against a stable demand for real money balances. More recently, models of hyperinflation have been derived by Allais [1] and Barro [2]. Both of these models are developed within the same theoretical framework employed by Cagan. Only the functional forms of the behavioral equations differ from Cagan's model. The Barro and Allais models appear to provide even stronger support for a stable demand for