Generalised splitting of spacetime

Normally, when a spacetime splitting is considered the ADM 3+1 split is brought to mind. In this paper, the idea of spacetime splitting is extended to include anm + n splitting of spacetime. The global spacetime has dimension (m + n) and the foliating spaces have dimensionm. There aren independent normals to each of these foliating spaces, thus givingn different extrinsic curvatures. The generalised Gauss-Weingarten and the generalised Gauss-Codazzi equations associated with this splitting are derived. These generalised equations reduce to the familar ADM equations when a 3+1 split is considered. The generalised equations are found to have a particularly elegant form when an orthogonal splitting of spacetime is examined.     

Integral equations for N-particle scattering

Integral equations are derived for the N-particle transition operators. The equations couple together only transition operators between two-body channels. The kernel of the equations becomes connected after a single iteration. Transition operators involving channels with three or more particles can be obtained by quadratures from the solution of the equations. It is also shown that the N-particle equations can be reduced to multichannel two-body equations by the use of the quasiparticle method.     

Bipartite Matching and Van der Waerden Conjecture

In this paper I show that the free energy F and the cost C associated to a bipartite matching problem can be explicitly estimated in term of the solution of a suitable system of equations (cavity equations in the following). The proof of these results relies on a well known result in combinatorics: the Van der Waerden conjecture (Egorychev–Falikman Theorem). Cavity equations, derived by a mean field argument by Mèzard and Parisi, can be considered as a smoothed form of the dual formulation for the bipartite matching problem. Moreover cavity equation are the Euler–Lagrange equations of a convex functional G parameterized by the temperature T. In term of their unique solution it is possible to define a free-energy-like function of the temperature g(T). g is a strictly decreasing concave function of T and C=g(0). The convexity of G allows to define an explicit algorithm to find the solution of the cavity equations at a given temperature T. Moreover, once the solution of the cavity equations at a given temperature T is known, the properties of g allow to find exact estimates from below and from above of the cost C.     

The twistor theory of equations of KdV type: I

This article is the first of two concerned with the development of the theory of equations of KdV type from the point of view of twistor theory and the self-dual Yang-Mills equations. A hierarchy on the self-dual Yang-Mills equations is introduced and it is shown that a certain reduction of this hierarchy is equivalent to then-generalized KdV-hierarchy. It also emerges that each flow of then-KdV hierarchy is a reduction of the self-dual Yang-Mills equations with gauge group SL _n . It is further shown that solutions of the self-dual Yang-Mills hierarchy and their reductions arise via a generalized Ward transform from holomorphic vector bundles over a twistor space. Explicit examples of such bundles are given and the Ward transform is implemented to yield a large class of explicit solutions of then-KdV equations. It is also shown that the construction of Segal and Wilson of solutions of then-KdV equations from loop groups is contained in our approach as an ansatz for the construction of a class of holomorphic bundles on twistor space.A summary of the results of the second part of this work appears in the Introduction.Most of this work was done while Darby Fellow of Mathematics at Lincoln College, Oxford     

TheN/D method off the mass shell for the Klein-Gordon equation and the inversion problem

The relativistic analogon of a procedure demonstrating the link between theS-wave off-the-mass- shellN/D equations (variables: the momentum, energy and radial coordinate of a scattered particle) and the Marchenko equations of the inversion problem is presented in the static scattering. For the Klein-Gordon formalism the transition from the former type of the equations to the latter requires: a decomposition of theN/D equation quantities into the components without theE-branch points, a suitable deformation of the integration path in theN/D equations and an assumption on the regular behaviour of the off-the-mass-shellN function discontinuities.     

A classical Proca particle

An elementary particle is described as a spherically symmetric solution of the Proca equations and the Einstein general relativity equations. The mass is found to be of the order of the Planck mass. If the motion of its center of mass is determined by the Dirac equations, it has a spin 1/2.This work is parallel to an earlier one involving the Klein- Gordon equation.     

N = 2 Supercomplexification of the Korteweg-de Vries,Sawada-Kotera and Kaup-Kupershmidt Equations

The supercomplexification is a special method of N = 2 supersymmetrization of the integrable equations in which the bosonic sector can be reduced to the complex version of these equations. The N = 2 supercomplex Korteweg-de Vries, Sawada-Kotera and Kaup-Kupershmidt equations are defined and investigated. The common attribute of the supercomplex equations is appearance of the odd Hamiltonian structures and superfermionic conservation laws. The odd bi-Hamiltonian structure, Lax representation and superfermionic conserved currents for new N = 2 supersymmetric Korteweg-de Vries equation and for Sawada-Kotera one, are given.     

<Emphasis Type="Italic">N</Emphasis> = 1, 2 supersymmetric non-local gas equation

In this paper we study the supersymmetrization of the N = 1 and N = 2 nonlocal gas equation. We show that this system is bi-Hamiltonian. While the N = 1 supersymmetrization allows the hierarchy of equations to be extended to negative orders (local equations), we argue that this is not the case for the N = 2 supersymmetrization. In the bosonic limit, however, the N = 2 system of equations lead to a new coupled integrable system of equations. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

Theory of Gravitational-Inertial Field of Universe. I. Gravitational-Inertial Field Equations

In the series of present articles the original proposition is a generalization of the real world tensor by the introduction of a inertial field tensor. From this generalization it follows, particularly, that ?_ig_lm ? g_lm;i ≠ 0. This allows to use as a Lagrangian density of the field the expression A_g = k₁ g_lm;ig^lm _;kg^ik. On the basis of variational equations a system of more general covariant equations of the gravitational-inertial field is obtained. In the Einstein approximation these equations reduce to the field equations of Einstein. The solution of fundamental problems in the general theory of relativity by means of the new equations gives the same results as the solution by means of Einstein's equations. However, application of these equations to the cosmologic problem gives a result different from that obtained by Friedmann's theory. In particular, the solution gives the Hubble law as the law of motion of a free body in the inertial field - in contrast to Galileo-Newton's law.     

A note on the solution of a spinor equation

An equation of spinor algebra, which is specified by two positive integers,M andN, is solved by relating it to the problem of integrating a two-dimensional Hamiltonian homogeneous polynomial system of ordinary differential equations, whose degree isN}-1. The case in whichN=1 reduces to a well-known result of spinor algebra. The caseM=N=4 is of relevance in the study of symmetry operators of Maxwell's equations on a curved space-time. It is also shown, using spinor notation, that the first integral for a general two-dimensional Hamiltonian system of ordinary differential equations (whether polynomial or analytic) is determinable in a purely algebraic manner, i.e., by using no integration.     

New Concepts from the Evans Unified Field Theory. Part One: The Evolution of Curvature,Oscillatory Universe Without Singularity,Causal Quantum Mechanics,and General Force and Field Equations

No Heading The Evans field equation is solved to give the equations governing the evolution of scalar curvature R and contracted energy-momentum T. These equations show that R and T are always analytical, oscillatory, functions without singularity and apply to all radiated and matter fields from the sub-atomic to the cosmological level. One of the implications is that all radiated and matter fields are both causal and quantized, contrary to the Heisenberg uncertainty principle. The wave equations governing this quantization are deduced from the Evans field equation. Another is that the universe is oscillatory without singularity, contrary to contemporary opinion based on singularity theorems. The Evans field equation is more fundamental than, and leads to, the Einstein field equation as a particular example, and so modifies and generalizes the contemporary Big Bang model. The general force and conservation equations of radiated and matter fields are deduced systematically from the Evans field equation. These include the field equations of electrodynamics, dark matter, and the unified or hybrid field.     

On fluctuations in liquids and gases

The Lagrangian formalism is used to derive a system of nonlinear inhomogeneous dissipative differential equations describing the nonlinear dynamics of interrelated fluctuations of density, δρ, and temperature, δT, in a medium. With these equations, the unstable (with respect to initial conditions) phase trajectory describing parameter fluctuations in the ρ-T plane was obtained. By numerically solving the equations, we show that δρ and δT oscillate in time almost periodically, which is typical of fluctuations.     

The neutrino energy-momentum tensor and the Weyl equations in curved space-time

In general, a first order Lagrangian gives rise to second order Euler-Lagrange equations. However, there are important examples where the associated Euler-Lagrange equations are of first order only, the Weyl neutrino equations being of this type. In this paper we therefore consider first order spinor Lagrangians which give rise to firstorder Euler-Lagrange equations. Specifically, the most general first order spinor field equations of rank one in curved space-time which are derivable from a first order Lagrangian of the same type are explicitly constructed. Subject to a certain restriction, the Weyl neutrino equation is the only possibility. Furthermore, if the spinor field satisfies the Weyl neutrino equation, then the associated energy momentum tensor is the conventional neutrino energymomentum tensor.     

Hyper-Kähler Hierarchies and Their Twistor Theory

A twistor construction of the hierarchy associated with the hyper-K?hler equations on a metric (the anti-self-dual Einstein vacuum equations, ASDVE, in four dimensions) is given. The recursion operator R is constructed and used to build an infinite-dimensional symmetry algebra and in particular higher flows for the hyper-K?hler equations. It is shown that R acts on the twistor data by multiplication with a rational function. The structures are illustrated by the example of the Sparling–Tod (Eguchi–Hansen) solution. An extended space-time ? is constructed whose extra dimensions correspond to higher flows of the hierarchy. It is shown that ? is a moduli space of rational curves with normal bundle ?(n)⊕?(n) in twistor space and is canonically equipped with a Lax distribution for ASDVE hierarchies. The space ? is shown to be foliated by four dimensional hyper-K?hler slices. The Lagrangian, Hamiltonian and bi-Hamiltonian formulations of the ASDVE in the form of the heavenly equations are given. The symplectic form on the moduli space of solutions to heavenly equations is derived, and is shown to be compatible with the recursion operator. Received: 27 January 2000 / Accepted: 20 March 2000     

Triangular Newton Equations with Maximal Number of Integrals of Motion

Abstract

We study two-dimensional triangular systems of Newton equations (acceleration = velocity-independent force) admitting three functionally independent quadratic integrals of motion. The main idea is to exploit the fact that the first component M ₁(q ₁) of a triangular force depends on one variable only. By using the existence of extra integrals of motion we reduce the problem to solving a simultaneous system of three linear ordinary differential equations with nonconstant coefficients for M ₁(q ₁). With the help of computer algebra we have found and solved these ordinary differential equations in all cases. A complete list of superintegrable triangular equations in two dimensions is been given. Most of these equations were not known before.     

Equivalence of the Regge and Einstein equations using Cartan's moment of rotation

An explicit derivation of the Einstein tensor via Cartan's moment of rotation on an infinitesimal lattice is presented. With the standard form of the Einstein equations assumed, the equivalence of the Regge equations with matter to the Einstein equations is demonstrated in detail using a spherically symmetric example with proper time slicing. Such an example has been numerically evolved to withinr=2M using null struts. These results make Regge calculus more readily applicable and provide a justification for its use.     

Selection rules,causality, and unitarity in statistical and quantum physics

The integrodifferential equations satisfied by the statistical frequency functions for physical systems undergoing stochastic transitions are derived by application of a causality principle and selection rules to the Markov chain equations. The result equations can be viewed as generalizations of the diffusion equation, but, unlike the latter, they have a direct bearing onactive transport problems in biophysics andcondensation aggregation problems of astrophysics and phase transition theory. Simple specific examples of the effects of severe selection rules, such as the relaxational Boltzmann transport equation and the diffusion equation, are also given. Finally, partial differential equations for the probability amplitudes of quantum mechanics are derived, usingunitarity instead of causality, and a selection rule is applied directly to obtain ageneralization of the Dirac equation in which infinite transitions between states arenot allowed.     

Fermion Fields,Boson Fields,and Gravitational Field

If a tetrad theory is derivable from a variational principle with a Lagrangian ?? of the form ?? = ??_F+??_M 6 tetrad components will be defined by the vacuum equations if the energy momentum tensor is symmetric. Therefore, we look for a realisation of a programme proposed in a little different way by TREDER according to which the 16 tetrad field equations should degenerate to 10 equations for the Riemannian metric if boson fields are the only source of the gravitational field.     

Trilocal structures. V. Wave equations

In addition to the usual centroid-time wave equation, a trilocal structure will need to satisfy two relative-time wave equations. When the trilocal wave function is expanded in tree functions, each of the three wave equations becomes an infinite matrix equation, but when the four auxiliary conditions (defined in earlier articles in this series) are introduced, each wave equation reduces to a set of 16 linear homogeneous equations in 16 unknown expansion coefficients (the first 16 coefficients in the tree expansion). The 48 linear equations, in the 16 unknownC _j , are given explicitly. Every 16-by-16 determinant, formed from any 16 of these 48 linear homogeneous equations, must vanish if the trilocal structure is to be an acceptable solution; this requirement will be used in later calculations.     

Integrable Evolution Equations on Spaces of Tensor Densities and Their Peakon Solutions

We study a family of equations defined on the space of tensor densities of weight λ on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. We present their Lax pair formulations and describe their bihamiltonian structures. We prove local wellposedness of the corresponding Cauchy problem and include results on blow-up as well as global existence of solutions. Moreover, we construct “peakon” and “multi-peakon” solutions for all λ ≠ 0, 1, and “shock-peakons” for λ = 3. We argue that there is a natural geometric framework for these equations that includes other well-known integrable equations and which is based on V. Arnold’s approach to Euler equations on Lie groups.