Necessary and sufficient conditions for the existence of a Lagrangian in field theory. III. Generalized analytic representations of tensorial field equations

In this paper we first study the equivalence transformations of class C², regular, tensorial, quasi-linear systems of field equations which (a) preserve the continuity, regularity, and quasi-linear structure of the systems; and (b) occur within a fixed system of Minkowski coordinates and field components. We identify, among the transformations of this class, those which either induce or preserve a self-adjoint structure of the field equations and we term them genotopic and isotopic transformations, respectively. We then give the necessary and sufficient conditions for an equivalence transformation of the above type to be either genotopic or isotopic. By using this methodology, we then extend the theorem on the necessary and sufficient condition for the existence of ordered direct analytic representations introduced in the preceding paper to the case of ordered indirect analytic representations in terms of the conventional Lagrange equations; we introduce a method for the construction of a Lagrangian, when it exists, in this broader context; and we explore some implications of the underlying methodology for the problem of the structure of the Lagrangian capable of representing interactions within the framework of the indirect analytic representations. Some of the several aspects which demand an inspection prior to the use of this analytic approach in actual models are pointed out. In particular, we indicate a possible deep impact in the symmetries and conservation laws of the system generated by the use of the concept of indirect analytic representation. As a preparatory step prior to the analysis of these problems, we study some methodological aspects which underlie the generalized Lagrange equations postulated in the first paper of this series for the case when they are regular, namely, when they are simple equivalence transformations of the conventional Lagrange equations. We first introduce a generalization of the action principle capable of inducing the generalized as well as the conventional equations. In this way we establish that the former equations are “bona fide” analytic equations. Finally, as our most general analytic framework for the case of unconstrained field equations, we work out the necessary and sufficient condition for the existence of ordered direct analytic representations of quasi-linear systems in terms of the generalized analytic equations and study their relationship to the conventional representations.     

On the nonequilibrium statistical mechanics of a binary mixture. I. The distribution functions

In this paper, we obtain the generalization of the BBGKY hierarchy for a binary mixture of chemically neutral particles. Using modified boundary conditions different from the ones proposed by Bogoliubov, we solve the hierarchy, and obtain explicitly the set of two-particle distribution functions for the several species of the mixture, up to first order in the density.     

Pseudo-Hermitian Reduction of a Generalized Heisenberg Ferromagnet Equation. II. Special Solutions

This paper is a continuation of our previous work in which we studied a sl (3, ?) Zakharov-Shabat type auxiliary linear problem with reductions of Mikhailov type and the corresponding integrable hierarchy of nonlinear evolution equations. Now, we shall demonstrate how one can construct special solutions over constant back- ground through Zakharov-Shabat’s dressing technique. That approach will be illustrated on the example of the generalized Heisenberg ferromagnet equation related to the linear problem for sl (3, ?). In doing this, we shall discuss the differences between the Hermitian and pseudo-Hermitian cases.     

Zum Beugungswirkungsgrad von Volumenhologrammen. Teil II. Reflexionshologramme

In this paper we give an exact solution of the coupled differential equation system as developed by Kogelnik [1] for the case of symmetrical recording without neglecting the second differentials for the reflection volume hologram. This solution is applicable to dielectric, absorption and so-called mixed volume holograms. There are some remarkable distinctions between our results and those got by Kogelnik for the case of absorption and mixed holograms. It is shown, that under special assumptions and approximations or for special combinations of the material parameters we get the same relations as Kogelnik.     

Chaotic capture of vortices by a moving body. II. Bound pair model

Previously, we have presented a simple model for the interaction of a fluid vortex structure with a moving bluff body, and demonstrated the existence of a trapping mechanism related to chaotic scattering. This single point vortex model required explicit perturbation to generate chaos and the subsequent complex dynamics. Here, we present a model which attempts to introduce internal degrees-of-freedom in the vortex structure in the simplest manner, by replacing the single vortex with a like-signed pair. We show that this model exhibits chaotic trapping without the need of explicit perturbation, however, the region of parameter space for which trapping occurs is exceedingly small due to the spatially dependent form of the perturbation. We claim that this result explains some the behavior observed in Navier-Stokes simulations of the same vortex-body system, where we find close correspondence between the dynamics of an extended vorticity distribution and the single vortex model. Finally, we generalize the model to unequal strength vortex pairs, and find more complex behavior which includes "partial" capture of the weaker vortex by the body. (c) 1994 American Institute of Physics.     

Semi-infinite homology and 2D gravity. I

In [33], we studied the constraint problem for two-dimensional quantum gravity in the conformal gauge. In this gauge, we proposed an ansatz for the gravitational sector. Using this ansatz, we established a striking connection between the matrix models and continuum 2D gravity. We also announced several results on semi-infinite homology of the Virasoro algebra with coefficients in a suitable class of positive energy modules. In this article, we will provide details of the proof of the announced results.Supported in part by the Alfred P. Sloan FoundationSupported by NSF Grant DMS-8703581     

Interpretation of S-state ion E.P.R. spectra

Although a considerable body of data exists on the parametrization of the ground-state splittings of S-state ions in crystals, relatively little progress has been made in obtaining a quantitative understanding of the mechanisms which determine these parameters. In the course of summarizing our present understanding, we emphasize the need for making planned experiments explicitly aimed at testing theoretical models, such as those proposed in this article. The variable frequency E.P.R. technique is described in some detail, as it has proved to be particularly useful in this respect.     

Microscopic modes in a Fermi superfluid. I. Linearized kinetic equations

This is the first of two papers in which microscopic expressions for the amplitudes and dispersion relations for hydrodynamic modes in an isotropic Fermi superfluid are derived. In this first paper we derive closed, decoupled, linearized kinetic equations for the bogolon spin density and total density in a Fermi superfluid with fluctuating superfluid velocity, and we discuss the form of the hydrodynamic equations that result from these equations.     

The Influence of Neutral Particle Sources on the Dynamics of Tokamak Plasmas. II. Particle Dynamics and Induced Electromagnetic Fields

In part I of our paper [1] a kinetic theory of the influence of neutral particle sources on the dynamics of tokamak plasmas has been developed and the general solution of this problem has been given for an arbitrary source function. By the use of these results we calculate in this part II the particle's distribution function for different simple source functions. In the case of recycling perpendicular to the limiter surface we derive analytical expressions for the charge density, the induced electric potential and the anomaleous electron density flux transverse to the magnetic field.     

Stochastic dynamics of parametrically excited two d.o.f. systems with symmetry

In this paper we investigate the non-linear dynamics of a two-degree-of-freedom system with symmetries subject to random parametric excitation. The study of this non-linear near-Hamiltonian system is simplified by using the symmetry and separation of scales present in the problem. To this end, we study the equations as a random perturbation of a four-dimensional weakly dissipative Hamiltonian system. We achieve the model-reduction through stochastic averaging and the reduced process is simply a Markov process on a line. Examination of the reduced Markov process on the line yields many important results, namely, probability density functions, and stochastic bifurcations. The steady state dynamics is computed explicitly. Phenomenological and dynamical bifurcations are investigated. The approach adopted in this paper can in principle, be applied to any four-dimensional integrable system.     

Temperature dependent self-diffusion coefficient measurements of glycerol by the pulsed N.M.R. technique

A review of the coupled Hartree-Fock method for calculation of second-order one-electron properties is given with special reference to force constants and static electric and magnetic properties. A comprehensive analysis of the computational problems is attempted and we present the logic of a computer programme. A number of results obtained for the H₂O molecule are reported in part II of this work.     

Yukawa potentials in systems with partial periodic boundary conditions. II. Lekner sums for quasi-two-dimensional systems

Yukawa potentials may be long-ranged when the Debye screening length is large. In computer simulations, such long-ranged potentials have to be taken into account with convenient algorithms to avoid systematic bias in the sampling of the phase space. Recently, we provided Ewald sums for quasi-two-dimensional systems with Yukawa interaction potentials [J. Chem. Phys. 126, 056101 (2007); Molec. Phys. paper I of this series]. Sometimes, Lekner sums are used as an alternative to Ewald sums for Coulomb systems. In the present work, we derive the Lekner sums for quasi-two-dimensional systems with Yukawa interaction potentials and we give numerical tests for practical implementations. The main result of this paper is to emphasize that Lekner sums cannot be considered as an alternative to Ewald sums for Yukawa potentials. As a conclusion to this work: Lekner sums should not be used for quasi-two-dimensional systems with Yukawa interaction potentials.     

Scaling features of noncoding DNA.

We review evidence supporting the idea that the DNA sequence in genes containing noncoding regions is correlated, and that the correlation is remarkably long range--indeed, base pairs thousands of base pairs distant are correlated. We do not find such a long-range correlation in the coding regions of the gene, and utilize this fact to build a Coding Sequence Finder Algorithm, which uses statistical ideas to locate the coding regions of an unknown DNA sequence. Finally, we describe briefly some recent work adapting to DNA the Zipf approach to analyzing linguistic texts, and the Shannon approach to quantifying the "redundancy" of a linguistic text in terms of a measurable entropy function, and reporting that noncoding regions in eukaryotes display a larger redundancy than coding regions. Specifically, we consider the possibility that this result is solely a consequence of nucleotide concentration differences as first noted by Bonhoeffer and his collaborators. We find that cytosine-guanine (CG) concentration does have a strong "background" effect on redundancy. However, we find that for the purine-pyrimidine binary mapping rule, which is not affected by the difference in CG concentration, the Shannon redundancy for the set of analyzed sequences is larger for noncoding regions compared to coding regions.     

Microscopic modes in a Fermi superfluid. II. Dispersion relations

This is the second of two papers in which microscopic expressions for the amplitudes and dispersion relations for hydrodynamic modes in an isotropic Fermi superfluid are derived. In this paper we obtain approximate solutions to the linearized kinetic equations for the bogolon spin density and total density for the case of long-wavelength disturbances after long times when a fluctuating superfluid velocity is present. In so doing, we obtain microscopic expressions for the amplitude and dispersion relations for the spin diffusion mode, the two shear modes, and the four longitudinal modes (two first-sound modes and two second-sound modes).     

Theory of q-deformed forms. I. q-deformed alternating tensor and q-deformed wedge product

In this paper we deal with the q-deformed alternating tensor and prove the associativity of the q-deformed wedge product. Moreover, we construct the theory of q-deformed homology in order to prove the q-deformed Stokes theorem. Lastly we prove the q-deformed Poincaré lemma.     

Theory of strong interactions. I. Kinematics

This paper is the first of a series devoted to strong interaction theory. As a general introduction to the series, we give a critical survey of theories and models presently in force.In a previous paper, we studied the kinematics of a single unstable hadron, the state of which was described as an incoherent superposition of states with different masses. To such a state there corresponds a function on the Poincaré group, called a characteristic function. Here we take up these old results anew, we work them out, and we go more thoroughly into their physical foundations. Then we describe the kinematics of several particles.The theory of strong interactions must take into account two opposed features of unstable hadrons: their identity and their difference with stable hadrons. The identity implies that an unstable hadron is in a certain state, described by a density operator; the difference is that this state has a mass spectrum, whose width cannot be neglected.We make the further assumption that the state is an incoherent superposition of components with different masses. This assumption is compared to the statistics experiment of Baton and Laurens, in which the components with different masses have been effectively separated.We define the characteristic function of such a state, and we point out its analogy with a usual characteristic function in probability theory. The physical meaning of characteristic functions is studied on the example of a spin, then in the general case.Then we study the characteristic function of several hadrons, and we define two notions: the global particle and the inclusive characteristic function.     

The Weil-B.R.S. algebra of a Lie algebra and the anomalous terms in gauge theory

In this paper, we explain the computation we made in collaboration with M. Talon and C.M. Viallet of anomalous terms in gauge theory [1], [2], [3]. We relate our constructions to standard mathematical constructions. The paper is self-contained in the sense that all mathematical concepts and results we use are explained.     

Connections among quantum logics. Part 1. Quantum propositional logics

In this paper, we propose a theory of quantum logics which is general enough to enable us to reexamine previous work on quantum logics in the context of this theory. It is then easy to assess the differences between the different systems studied. The quantum logical systems which we incorporate are divided into two groups which we call quantum propositional logics and quantum event logics. We include the work of Kochen and Specker (partial Boolean algebras), Greechie and Gudder (orthomodular partially ordered sets), Domotar (quantum mechanical systems), and Foulis and Randall (operational logics) in quantum propositional logics; and Abbott (semi-Boolean algebras) and Foulis and Randall (manuals) in quantum event logics. In this part of the paper, we develop an axiom system for quantum propositional logics and examine the above structures in the context of this system.     

Theory of q-deformed forms. II. q-deformed differential forms and q-deformed Hamilton equation

In this paper we introduce the q-deformed differential forms and quantum-algebra-valued q-deformed forms. We use these to obtain the q-inner derivative and investigate its properties. As a physical application we discuss the q-deformed Hamilton equation.