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.     

Solution of the Bethe-Goldstone equation for the reaction matrix in finite nuclei

A method for solving the BG equation for the reaction matrixt in finite nuclei is presented. The application of this method is demonstrated for a one-dimensional case, which is similar to the problem where the internucleon potential acts only in the relatives-state. The single particle potential has a harmonic oscillator form and the phenomenological internucleon potentialv(r) contains a hard core and an attractive part of the Yukawa type. By taking the exclusion principle into account exactly an infinite system of integral equations is obtained. It is proved that the solution of the corresponding finite system converges to the exact solution. An iteration method for solving such a finite system with an arbitrary number of equations is developed. Its main feature consists in the exclusion of the dependence on the hard core part ofv(r) (which is treated as the limit case of a rectangular repulsive potential with a variable heightv ₀). This exclusion transforms the original system to a system of integral equations depending only on the attractive part ofv(r) and to a linear algebraic system. Both these systems can be solved by iteration for all values ofv ₀ as well as for v₀= +. The numerical results confirm the rapid convergence of the proposed iteration method and demonstrate that the solution of the finite system with a sufficiently large number of equations approximates the exact solution very precisely.     

Is there a tachyon in the Nambu-Goto string with massive ends?

The dependence of Casimir energy on quark mass is investigated in the model of relativistic strings with massive quarks attached to the ends. The quark dynamics are treated in the nonrelativistic approximation, and the equations of motion and boundary conditions are linearized. The Casimir energyE as a function of quark massm is found by two methods (numerically and analytically). Different subtraction procedures for both approaches result in different functional dependences ofE onm. But both cases have values ofm for which the Casimir energy is definitely positive. The sign of this energy is known to coincide with the sign of the squared mass of the ground state in the string spectrum. Hence, the obtained result indicates that it is possible at least in principle to solve the tachyon problem in the model of relativistic strings with massive ends.     

The gravitational field in the wave zone I. The radiating terms of the metric tensor

We consider an asymptotically flat space-time generated by a perfect fluid source of compact spatial support. Using the de Donder gauge conditions, the Einstein equations are reduced to a new form of Poisson-type equations. A formal iterative scheme is set up to solve these equations by expanding the components of the metric tensor in powers ofc ^–1. The coefficient of each power ofc ^–1 depends on the asymptotically retarded timeu andx, y, z and satisfies a Poisson-type equation. Assuming asymptotic flatness the solution is carried out in the first orders. The results are explicit expressions of the metric up to orderc ^–4 in terms of the source functions. These expressions hold over all space-time. A further expansion in powers ofr ^–1 gives the first terms of the metric that contribute to gravitational radiation.     

A cosmological solution of a 5D space-time-mass gravity in vacuum

A cosmological solution is obtained in a 5D space-time-mass cosmological model by applying a homogeneous and isotropic metric, whose tensorsg _ij (i, j =0–4) are functions of both timet and massm, to the field equations in vacuum. It shows that in this theoryg ₄₄of the fifth coordinate is contracted not only by special functional form oft as in the Kaluza-Klein theory but also by that ofm. This procedure realizes the universe which evolves into a conventional 4D radiation-dominated era from a 5D vacuum era.     

ArbitraryN-vortex solutions to the first order Ginzburg-Landau equations

We prove that a set ofN not necessarily distinct points in the plane determine a unique, real analytic solution to the first order Ginzburg-Landau equations with vortex numberN. This solution has the property that the Higgs field vanishes only at the points in the set and the order of vanishing at a given point is determined by the multiplicity of that point in the set. We prove further that these are the onlyC solutions to the first order Ginzburg-Landau equations.This work is supported in part through funds provided under Contract PHY 77-18762     

Simplex equations and their solutions

We investigate ann-simplex generalization of the classical and quantum Yang-Baxter equation. For the case ofsl(2) we find the most general solution of the classicaln-simplex equation for alln. These classical solutions can be quantized (in the sense of quantum group theory) forn=2,3 and we exhibit a quantum solution to the tetrahedron equations (n=3). The classical nondegenerate solutions cannot be quantized forn=4.     

Analysis of the static spherically symmetricSU(n)-Einstein-Yang-Mills equations

The singular boundary value problem that arises for the static spherically symmetricSU(n)-Einstein-Yang-Mills equations in the so-called magnetic case is analyzed. Among the possible actions ofSU(2) on aSU(n)-principal bundles over space-time there is one which appears to be the most natural. If one assumes that no electrostatic type component is present in the Yang-Mills fields and the gauge is suitably fixed a set ofn-1 second order and two first order differential equations is obtained forn-1 gauge potentials and two metric components as functions of the radial distance. This system generalizes the one for the casen=2 that leads to the discrete series of the Bartnick-Mckinnon and the corresponding black hole solutions. It is highly nonlinear and singular atr= and atr=0 or at the black hole horizon but it is known to admit at least one series of discrete solutions which are scaled versions of then=2 case. In this paper local existence and uniqueness of solutions near these singular points is established which turns out to be a nontrivial problem for generaln. Moreover, a number of new numerical soliton (i.e. globally regular) numerical solutions of theSU(3)-EYM equations are found that are not scaledn=2 solutions.     

Geometry of the submanifoldsESX n . III. Parallelism inESX n and its submanifolds

A connection which is both Einstein and semisymmetric is called anES connection. And a generalizedn-dimensional Riemannian manifold on which the differential geometric structure is imposed byg through anES connection, is called ann-dimensionalES manifold and denoted byESX _n . This paper is the third part of a systematic study of the submanifoldsX _m ofESX _n . In the first part, we introduced a new concept of theC-nonholonomic frame of reference inESX _n at points ofX _m and dealt with its consequences. In the second part, the generalized fundamental equations on a hypersubmanifold ofESX _n were derived as an application of theC-nonholonomic frame of reference. The purpose of the present paper is to study parallelism inESX _n and in its submanifoldX _m , using theC-nonholonomic frame of reference and the new concept ofES _i curves.     

Structure of Lanczos-Lovelock Lagrangians in critical dimensions

The Lanczos-Lovelock models of gravity constitute the most general theories of gravity in D-dimensions which satisfy (a) the principle of of equivalence, (b) the principle of general covariance, and (c) have field equations involving derivatives of the metric tensor only up to second order. The mth order Lanczos-Lovelock Lagrangian is a polynomial of degree m in the curvature tensor. The field equations resulting from it become trivial in the critical dimension D = 2m and the action itself can be written as the integral of an exterior derivative of an expression involving the vierbeins, in the differential form language. While these results are well known, there is some controversy in the literature as to whether the Lanczos-Lovelock Lagrangian itself can be expressed as a total divergence of quantities built only from the metric and its derivatives (without using the vierbeins) in D = 2m. We settle this issue by showing that this is indeed possible and provide an algorithm for its construction. In particular, we demonstrate that, in two dimensions, = _j R^j{R \sqrt{-g} = \partial_j R^j} for a doublet of functions R ^j  = (R ⁰, R ¹) which depends only on the metric and its first derivatives. We explicitly construct families of such R ^j -s in two dimensions. We also address related questions regarding the Gauss-Bonnet Lagrangian in D = 4. Finally, we demonstrate the relation between the Chern-Simons form and the mth order Lanczos-Lovelock Lagrangian.     

The estimates of surface heterogeneity by series expansions

Conclusion In the foregoing treatment, Cerofolini's method for the estimate of the surface heterogeneity, based on the relation (1), has been analyzed from the standpoint of numerical mathematics. The limited amount of observed data, their experimental error and the properties of regressional procedures led to notable modifications of the original method. The main result is that the series expansion does not give straightforward solution of the problem, tedious iterations are necessary even for simple analytical forms off(U). Serious difficulties are introduced by the consideration of the incipient second-layer coverages, by existence of a number of maxima inf(U) and by existence of homogeneous regions. Deleterious impairment can be caused by differences in coefficients for the expansions of observed data (N, p) and of(p, T, U). As in other methods, there is no possibility to estimate the absolute value of the least adsorption energyU ₀, only the relative quantitiesq are extracted from a single adsorption isotherm. In spite of the large amount of needed computations, the method does not give any possibility for estimates of the reliability of the calculated distribution. This contrasts with the recently developed method [8], in which the observed data are correlated by the sum (10). By means of this method, continuous and discontinuous distributions can be dealt with equally well and it can be easily tested whether any supposed distribution fits the data within experimental error. When isotherms at several temperatures and/or calorimetric data are at hand, analysis of the thermodynamic properties of different adsorption sites can be performed. The chief difficulty of the method, the solution of a system of non-linear equations, is much less tedious than the treatment analyzed in this paper.     

Theory of melting,vacancy model

A theory of melting based on vacancy model is formulated. The polymer solution theory is used for derivation of the melting equation for a two-species model of melting solid. Under simplifying assumptions the analysis leads to a simple correlation betweenT _m and 〈v〉, the average energy of interaction between the vibrating atoms. Pseudopotential method is used for calculating 〈v〉 for the alkali metals lithium, sodium, potassium and rubidium at temperatureT _m. The calculated values ofT _m〈v〉 are in accord with those expected from our model. Application to the high pressure melting curves of solids is also discussed.     

Canonical Structure and Symmetries of the Schlesinger Equations

The Schlesinger equations S _(n,m) describe monodromy preserving deformations of order m Fuchsian systems with n + 1 poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of n copies of m × m matrix algebras equipped with the standard linear Poisson bracket. In this paper we present a new canonical Hamiltonian formulation of the general Schlesinger equations S _(n,m) for all n, m and we compute the action of the symmetries of the Schlesinger equations in these coordinates.     

A minimal energy tracking method for non-radially symmetric solutions of coupled nonlinear Schrödinger equations

We aim at developing methods to track minimal energy solutions of time-independent m-component coupled discrete nonlinear Schrödinger (DNLS) equations. We first propose a method to find energy minimizers of the 1-component DNLS equation and use it as the initial point of the m-component DNLS equations in a continuation scheme. We then show that the change of local optimality occurs only at the bifurcation points. The fact leads to a minimal energy tracking method that guides the choice of bifurcation branch corresponding to the minimal energy solution curve. By combining all these techniques with a parameter-switching scheme, we successfully compute a non-radially symmetric energy minimizer that can not be computed by existing numerical schemes straightforwardly.     

A comprehensive approach to an equation of state for hard spheres and Lennard-Jones fluids

We present a simple method of obtaining various equations of state for hard sphere fluid in a simple unifying way.We will guess equations of state by using suitable axiomatic functional forms (n=1,2,3,4,5) for surface tension S n m (r),r ≥ d/2 with intermolecular separation r as a variable,where m is an arbitrary real number (pole).Among the equations of state obtained in this way are Percus-Yevick,scaled particle theory and Carnahan-Starling equations of state.In addition,we have found a simple equation of state for the hard sphere fluid in the region that represents the simulation data accurately.It is found that for both hard sphere fluids as well as Lennard-Jones fluids,with m=3/4 the derived equation of state (EOS) gives results which are in good agreement with computer simulation results.Furthermore,this equation of state gives the Percus-Yevick (pressure) EOS for the m=0,the Carnahan-Starling EOS for m=4/5,while for the value of m=1 it corresponds to a scaled particle theory EOS.     

On the hierarchy equations of the wave-operator for open-shell systems

Starting with the open-shell analogue of the Gell Mann-Low theorem of many-body perturbation theory, a non-perturbative linear operator equation is derived for the linked part of the wave-operatorW for open-shell systems. It is shown that, for a proper treatment of the linked nature of the wave-operator, a separation into its connected and disconnected components has to be made, and this leads to a hierarchy of equations for the various connected components. It is proved that the set of equations can be cast into a form equivalent to the non-perturbative equations of the wave-operator recently derived by Mukherjee and others in a coupled-cluster or exp(T) type formalism if a consistent use is made of a ‘core-valence separability’ condition introduced earlier. A comparison of the coupled-cluster representation ofW with the perturbative representation reveals that various alternative forms ofW in the coupled-cluster representation are possible and these reflect alternative ways of realising the core-valence expansion of the wave-operator. In particular it is emphasised how the use of Mandelstam block-ordering simplifies the coupled-cluster theories to a considerable extent and a comparison is made with coupled-cluster methods for open-shells put forward very recently by Ey and Lindgren. Finally, it is shown how difference energies of interest may be derived in a compact manner using the Mandelstam block-ordering of the wave-operator.     

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.     

Mass-transport models with multiple-chipping processes

We study mass-transport models with multiple-chipping processes. The rates of these processes are dependent on the chip size and mass of the fragmenting site. In this context, we consider k-chip moves (where k = 1, 2, 3, ...); and combinations of 1-chip, 2-chip and 3-chip moves. The corresponding mean-field (MF) equations are solved to obtain the steady-state probability distributions, P(m) vs. m. We also undertake Monte Carlo (MC) simulations of these models. The MC results are in excellent agreement with the corresponding MF results, demonstrating that MF theory is exact for these models.     

Asymptotics for the Fredholm determinant of the sine kernel on a union of intervals

In the bulk scaling limit of the Gaussian Unitary Ensemble of hermitian matrices the probability that an interval of lengths contains no eigenvalues is the Fredholm determinant of the sine kernel over this interval. A formal asymptotic expansion for the determinant ass tends to infinity was obtained by Dyson. In this paper we replace a single interval of lengths bysJ, whereJ is a union ofm intervals and present a proof of the asymptotics up to second order. The logarithmic derivative with respect tos of the determinant equals a constant (expressible in terms of hyperelliptic integrals) timess, plus a bounded oscillatory function ofs (zero ifm=1, periodic ifm=2, and in general expressible in terms of the solution of a Jacobi inversion problem), pluso(1). Also determined are the asymptotics of the trace of the resolvent operator, which is the ratio in the same model of the probability that the set contains exactly one eigenvalue to the probability that it contains none. The proofs use ideas from orthogonal polynomial theory.Research supported by National Science Foundation grant DMS-9216203.     

Teleparallelism as a universal connection on null hypersurfaces in general relativity

We show that there exists a close relationship between inner geometry of a null hypersurfaceN ₃ and the Newman-Penrose (NP) spin coefficient formalism. Projecting the null complexNP tetrad ontoN ₃ we get two triads of basis vectors inN ₃. Inner geometry ofN ₃ is based on the assumption that these vectors are parallelly transported along the surface; this gives rise to the teleparallel connection as a metric nonsymmetric affine connection. The gauge freedom for the choice of the basis triads is given by the isotropy subgroup of the local Lorentz group leaving invariant the direction of the null generators ofN ₃, and teleparallelism is determined by the equivalence class of the basis triads with respect to the global gauge group. Nine of the twelve NP coefficients are identified as the triad components of the torsion and the second fundamental form ofN ₃. The resulting generalized Gauss-Codazzi equations are identical to 9 of the NP equations, i.e., to the half of the Ricci identities. This result gives a geometrical meaning to the entire formalism. Finally we present a general proof of Penrose's theorem that the shear of the null generators ofN ₃ is the only initial null datum for a gravitational field onN ₃.