Search

A procedure that can establish the identity of apparently different graphs is given. When certain instructions are followed, any adjacency matrix can be brought by permutations of rows and columns to a prescribed standard form, which is defined by a unique numbering of vertices. Identity of two graphs may then be established by a direct comparison of the transformed adjacency matrices. This approach is simple and practical. Furthermore, the associated unique numbering of atoms in a molecule may in itself be of significance for chemical nomenclature. Several molecular graphs describing polycyclic hydrocarbons illustrate the numbering procedure. The systematic generation of all structures having any prescribed number of vertices and edges is possible by adopting the present scheme of numbering of graphs. This is illustrated for systems having four and five vertices (atoms).

THE PAPER is primarily a survey concerned with method; it describes a scheme and technique for deriving and justifying equations in slow time (averaged equations) from the point of view of the stroboscopic method. Typical classes of problems drawn mainly from monographs, are discussed, in which the averaged equations are obtained by the classical Krylov-Bogolyubov procedure.

The set of trajectories isolated from the k-th 1.s.a. will be denoted by {Tk}. We will describe a trajectory by a series of numbers rP,kr,,A . . . rvjk=RlrA, where /J is the ordinal number of a trajectory isolated from the k-th l.s.a., j is the ordinal number of a fragment of the p-th trajectory, not including logical conditions and contained between logical conditions, r is the number of operators {AR} in this fragment.

A recently proposed degenerate-kernel scheme for solving Fredholm integral equations of the second kind is extended to singular equations of the Lippmann-Schwinger type. Numerical calculations of on- and off-shell t-matrix elements are carried out at positive and negative energies, for the 1S0 Reid soft-core nucleon-nucleon potential. Satisfactory convergence is achieved, even with the simplest version of the s

In connection with the determination of the bilinear equal-time commutation relations between distinct fields and a new classification scheme of elementary particles a correspondence scheme between fields and particles and a schematic representation of the allowed transitions between different isomultiplets are shown, and the vacuum expectation value is evaluated. (JFP)

A study of four techniques is examined to determine which method has the greater overall utility. An iterative least-squares error minimization method provides the greatest accuracy, while a linear algebraic method is accurate only for two-exponential data with small error. Consequently, a three-method sequential identification scheme is devised utilizing a graphical method, a transform method, and an iterative method. This scheme is successfully applied to physiological data containing three and four exponents.

The Adler treatment of the off-mass-shell corrections to pion-nucleon forward amplitudes has been re-examined in the light of the new experimental data. This treatment is found to be consistent both with the experimental knowledge on |gA| and with the chiral-symmetry-breaking scheme of Gell-Mann, Oakes and Renner, in whose framework it had previously been employed to evaluate the kaon-nucleon sigma-term.

It is argued here that the concept of dynamical origin of charge as formulated in a previous paper requires the quantisation of space-time. Indeed, in this scheme, it is pointed out that the quantisation of electric charge in unit ofe is a direct consequence of this space-time quantisation.

This paper describes a technique for the reconstruction of an arbitrary straight-edge, wire-frame object from two or more central projections. The planes and centers of projections must be known. To reconstruct the object, matches must be established among the vertices in the different projections. A necessary (but not sufficient) condition for creating a match is given, as well as a scheme to eliminate wrong matches. In the projection of an object, false vertices may be present and some real vertices may be absent. A procedure is described for handling these two special situations. It is shown that for any particular configuration of projection planes (and centers), it is possible to specify an object which cannot be reconstructed. Conversely, given the number of edges of an object, it is shown that there is a maximum number of projections that will permit a complete reconstruction.

We consider the evaluation of the average electronic spectrum in a disordered binary alloy. The single-site approximation is reformulated in terms of restricted averages of the total scattering operator and equations relevant to the average-$t$-matrix (ATA) and coherent-potential (CPA) approximations are obtained. Specializing to the muffin-tin model of the alloy, we derive expressions for the average total and component densities of states. The results for the ATA are new and the implied numerical calculations appear to be tractable. We find that the Gyorffy-Stocks version of the muffin-tin CPA is equivalent to the earlier approach of Soven and also, that their mixed ATA-CPA scheme can be viewed as an approximation to just the average-$t$-matrix equations.

Various numerical methods applied to Schwarzschild's equation have been found to produce solutions that exhibit pronounced two-grid-interval oscillations. Though such solutions remain bounded as the time integration proceeds, the short-wave oscillations eventually obliterate the desired components of the solution. An analysis is presented that explains this oscillatory behavior, and a new finite-difference scheme is proposed that produces a smooth solution.

A straightforward attempt to estimate the influence of the first vertex correction for a homogeneous electron gas is made. The approximation scheme is also tested on a solvable model of a core electron coupled to an electron gas.

A generalization of the Korn inequality which permits reduction of the proof of solvability of the problem of total shell energy minimization in some class of admissible displacements to the verification of some algebraic condition which the strains must satisfy, and to the proof of existence theorems for the solution (or to the verification of the equilibrium conditions). Existence theorems are proved by the scheme mentioned in the Novozhilov-Bolabukh shell theory [1] and in the Reissner theory [2, 3].

Some established results concerning best $L_p $-approximations are introduced, and it is shown that a minimax approximation can generally be found as the limit of a sequence of such approximations. The values of p required, however, are too large to be practicable, and the main part of this paper is devoted to the development of an extrapolation scheme which enables minimax approximations to be predicted from best $L_p $-approximations for comparatively small values of p. Numerical evidence is presented to show that the scheme works well in practice.

Some analytic properties of the coherent potential approximation (CPA) are derived. As shown recently by Muller-Hartmann (1973), the CPA is analytic: this result is rederived by using a slightly different method. It is then shown that the conditional densities of states are always positive and that the iteration scheme, starting from the averaged t-matrix approximation, is always convergent to the CPA. Finally all these results are extended to the molecular CPA first proposed by Tsukada (1969).

Given the iterative scheme x i+1 = BTx i + r where B, T are fixed n ×: n real matrices, r a fixed real n -vector and x i a real n -vector we investigate the convergence and monotonicity of schemes of the type v i+1 w i+1 = B O O B S 11 −S 12 −S 21 S 22 v i w i + r r , where S ij are n ×: n real matrices related to T . The n -vector iterates v i , w i bracket in a certain sense solutions x of x = BTx + r . We also give necessary and sufficient conditions for the monotonicity of the original iterative scheme itself x i+1 = BTx i + r . This leads to monotonici results for classical iterative schemes such as the Jacobi, Gauss-Seidel, and successive overrelaxation methods.

It is shown that Hubbard's solution of the Hubbard model with inclusion of the scattering and also the broadening corrections can be written as the solution of the single-site CPA self-consistency condition with a continuous distribution function for the stochastic variables. The distribution function is determined and discussed. Es wird gezeigt, das Hubbards Losung fur das Hubbardmodell bei Berucksichtigung der Scattering- und auch der Broadening-Korrekturen geschrieben werden kann als Losung der Selbstkonsistenzgleichung der Einzentren-CPA mit einer kontinuierlichen Verteilungsfunktion fur die stochastischen Variablen. Die Verteilungsfunktion wird bestimmt und diskutiert.

In [7] D. G. Moursund examined the problem of approximating xliz over an interval [a, b] (a > 0) by applying the Newton-Raphson iteration scheme to classes of polynomial and rational approximants. G. Meinardus and G. D. Taylor [6] have observed that the above problem may be posed in a more general setting and the results obtained by Moursund extended to the approximation of additional functions. Specifically, given a compact subset X of [a, b], let C(X) denote the space of all continuous real-valued functions defined on X normed by llfll = max(I w(x)f(x)i: x E X}, where w E C(X) and w > 0 on X. Let K be a convex subset of C(X) and @ a continuous mapping of K into C(X). The problem of interest was then to approximate g E @b(K) by elements of Q(M) where M is a subset of K consisting of members of a Haar subspace of C[a, b]. By imposing certain restrictions on CD, K and Al, Meinardus and Taylor were able to develop a theory for the above nonlinear approximation problem which is analogous to the classical Chebyshev theory. Moreover they were able to obtain results similar to those of P. H. Sterbenz and C. T. Fike [13] and R. F. King and D. L. Phillips [2] in the more general setting. Since the theory obtained has application to many iterative processes that can be used to approximate functions such as eX, In x and xlkil, IZ = 2, 3 ,... the following definition is made

Starting from an exact three-body formulation of nuclear rearrangement, inclastic and break-up processes, we develop an operator reduction procedure of the three-body equations, by which the components from implicit channels—channels different from the initial and final ones—are formally eliminated. This procedure allows us to introduce in a simple way a distorted-wave approximation scheme involving generalized transition potentials, which take account also of contributions from the implicit channels. In order to facilitate the comparison between this approximation scheme and the DWBA, we recast the former one in an alternative form, in which the DWBA contributions appear explicitly separated from the additional ones. The importance of these latter contributions is discussed. As far as rearrangement processes are concerned, it is shown that the polar- and triangular-diagram mechanism contributions—contained in the generalized transition potential—sum up to give, in interesting practical cases, the DWBA.

A crossing-symmetric model for low-energy πK scattering in investigated. The model satisfies elastic unitarity to a good approximation, on and off the mass shell. With some minimal additional input from current algebra,s-wave phase shifts andp-wave scattering lengths are determined. A comparison is made with further current algebra theorems including those arising from the (3*, 3)+(3, 3*) symmetry-breaking scheme.

The mechanism of fine structure transitions in atom-atom collisions is discussed, within the framework of the molecular representation, in terms of the evolution of the colliding system from one Hund's coupling case to another. Simple models of the process can be constructed. The results are compared in the case of Na(3p)-He collisions with the results of exact calculations. Choosing the molecular potentials given by Nikitin or by Bottcher and taking into account trajectory effects, the calculated cross sections are in close agreement with experiment. The discrepancy with the quantal results of Reid, which is about 20% for the fine structure transition, is primarily due to the use of different potential and to trajectory effects.

It is shown that a phase-locked loop is a rather close approximation to a real junction. A lot of phenomena observed in Josephson junctions are easily identified in the « equivalent circuit ». It is further shown that the inclusion of an upper cut-off frequency in the loop gain causes a conductivity term similar to the cos ϕ-term experimentally observed in plasma resonance experiments.

A self-consistent and “spin-conserving” approximation scheme in the theory of triplet pairing superfluidity in3He is developed in terms of normal and anomalous Gor'kov Green's functions and the antisymmetrized interaction function. Both the normal and anomalous self-energy parts must be derived from the same diagrammatic approximation to a certain functional Φ, which in turn can be obtained from an approximation to the thermodynamic potential. With the help of this prescription the particle-holeT-matrix approximation to the normal system is generalized by taking into account that in the superfluid state the spin fluctuations are propagated also by pairs of anomalous Green's functions. The resulting equations for the components (with respect to the Pauli matrices) of the normal and anomalous self-energy parts and theT-matrix are treated in a self-consistent way. Explicit expressions for the effective interaction kernels occurring in the gap equations are given in terms of normal and anomalous “bare particle-hole susceptibilities.” The results are discussed in connection with the original theory of Layzer and Fay and the recent one of Anderson and Brinkman.

A general class of evaluation schemes for polynomials in one or several variables is discussed. By the same concept, error bounds are obtained for various methods, for instance Horner's scheme and Clenshaw's method, which are strict in some cases with a loss of a factor logn at best. For multivariable polynomials, a new family of evaluation schemes is suggested which generalizes a modification of Clenshaw's method and is therefore expected to have a favorable stability behavior with respect to round-off.

A new scheme is proposed for computation of the real and imaginary parts of the spectral functions of solids, including the density of states and wavenumber dependent susceptibility. Improvements over previous schemes result from analytic integration over small spherical cells with quadratic expansion of energy denominators.

A ring R is called a qc-ring if each cyclic R-module is quasi-injective. For various properties of these rings we refer to Ahsan (1) and Koehler (15). In this paper we shall obtain some additional results related to qc-rings. The scheme of the paper is as follows. Section 2 contains various preliminary definitions and results. In Section 3, we shall prove that every commutative hypercyclic ring is a qc-ring. In this section, we shall also show that a qc-ring which satisfies the ascending chain condition on its annihilators has nilpotent Jacobson-radical. Finally, in Section 4, we shall study rings all of whose proper factor rings are qc. Such rings will be called “ restricted qc ”.

Spin-projected one-particle density and spin density matrices are presented as polynomials of suitable unprojected quantities with generalized Sasaki-Ohno coefficients. Thus an explicit form of Harriman's theorems is given. For the two-particle spatial density matrix an expansion in direct products of powers of unprojected residual electron and spin density matrices is given. For these basic matrices of the scheme the variational spin-extended equations are formulated.

An iteration scheme for determining nonstationary heat flux is constructed for the linear case. The scheme is based on the two gradient slope methods.

In this work we generalise some previously obtained results concerning the quantisation of classical finite models according to the symmetric (Fermi-like) scheme of quantisation. We consider models whose dynamics is defined through some non-singular Lie bracket and show that we can make the dynamics with any prescribed bracket relations, as defined by a certain type ofnon-singular symmetric brackets, coexist. The quantisation scheme established is: (a) defined up to an arbitrary factor and, (b) sensitive to the addition of total time derivatives to the corresponding Lagrangian. Both unconstrained and constrained models are considered.

A test is made of the factorisation scheme proposed by Miettinen for the inclusive reaction π±p → Λ + X at 18.5 GeV/c and K−p→ Λ + X at 14.3 GeV/c. The test is applied to the lambda differential cross section and polarisation via a sum rule derived within the concepts of duality and exchange degeneracy, and SU3. The original scheme is found not to agree with the experimental data. A possible explanation of the disagreement is discussed.

We propose a scheme of approximation where strong-interaction processes with a purely longitudinal configuration of external momenta are described by fields which depend on longitudinal spacetime variables only, but retain their full four-dimensional spin structure. The evolution of these fields is governed by an effective Lagrangian, which can be obtained from the four-dimensional Lagrangian using a variational method. We discuss briefly the main features of the effective Lagrangians for one-dimensional and two-dimensional configurations of the external momenta.

The results that follow are intended to be understood as informal counterparts to formal theorems of Zermelo-Fraenkel set theory with choice. Basic notation not explained here can usually be found in [5]. It will also be necessary to assume a knowledge of the fundamentals of boolean and generic extensions, in the style of Jech's monograph [3]. Consistency results will be stated as assertions about the existence of certain complete boolean algebras, B, C, etc., either outright or in the sense of a countable standard transitive model M of ZFC augmented by hypotheses about the existence of various large cardinals. Proofs will usually be phrased in terms of the forcing relation ⊩ over such an M, especially when they make heavy use of genericity. They are then assertions about Shoenfield-style P-generic extensions M(G), in which the ‘names’ are required without loss of generality to be elements of MB = (VB)M, B is the boolean completion of P in M (cf. [3, p. 50]: the notation there is RO(P)), the generic G is named by Ĝ ∈ MB such that (⟦p ∈ Ĝ⟧B = p and (cf. [11, p. 361] and [3, pp. 58–59]), and for p ∈ P and c1, …, cn ∈ MB, p ⊩ φ(c1, …, cn) iff ⟦φ(c1, …, cn)⟧B ≥ p (cf. [3, pp. 61–62]).Some prior acquaintance with large cardinal theory is also needed. At this writing no comprehensive introductory survey is yet in print, though [1], [10], [12]and [13] provide partial coverage. The scheme of definitions which follows is intended to fix notation and serve as a glossary for reference, and it is followed in turn by a description of the results of the paper. We adopt the convention that κ, λ, μ, ν, ρ and σ vary over infinite cardinals, and all other lower case Greek letters (except χ, φ, ψ, ϵ) over arbitrary ordinals.

A prescription is given for the calculation of n -body repulsive potentials within a Heitler-London scheme generalized to many-molecule systems. The expression obtained for the potentials are correct to first order in the electron overlap integrals. In this approximation the n -body potential can be defined in diagrammatic terminology as the sum of all linked diagrams depending on n molecular coordinates and consisting of certain matrix elements of single electron and electron pair energy operators and of electron overlap integrals. This definition leads directly to a closed form expression for the total energy, including all pairwise additive and non-additive interactions, of a lattice system of molecules.

A scheme for representing vectors and matrices as functions of a certain abstract symbol λ is set up: though λ has no numerical significance, it is found to behave as if it were an eigenvalue of a certain singular matrix A. The resulting ‘eigensymbol’ theory is developed and applied to the quantum theory of Fermi systems. It is shown how a Feynman principle for such systems may be formulated in analogy with the familiar Feynman principle for a system with canonical p and q . The results are illustrated by the case of a simple Fermi oscillator.

This article considers a model for the spread of an infection of the type proposed by K. Cooke. The model involves a threshold for becoming infective that leads to functional, rather than ordinary, differential equations. Three types of result are presented. The principal result is to show that the model yields a mathematically well-posed problem. We show existence, uniqueness, and continuous dependence on initial data. Then an approximate solution is obtained for small values of a threshold parameter. Finally, a numerical scheme is proposed for one case. A number of graphs illustrate the effect of varying certain parameters in the model in this case.

An iterative scheme for calculation of eigenvalues is defined in terms of successive partitionings. The major part of this investigation is devoted to tridiagonal matrices, and a method for calculation of all eigenvalues of such a matrix is defined.

AN implicit iterative difference scheme for the solution of a quasilinear heat conduction equation is considered. An example of the divergence of the iterations is presented.

The use of a scheme based on the Galerkin—Petrov method for quantum chemical calculations is suggested.The results of its application to the calculation of the ground and excited states of the quartic oscillator are presented.

A class of discrete noiseless channels having upper and lower bounds on the separation between adjacent nonzero input symbols is considered. Recursion relations are derived for determining the number of input sequences which satisfy the constraints for all block lengths, and the asymptotic information rate is calculated. Applications to compaction and synchronization are discussed. An optimal algebraic block coding scheme for such channels is developed.