eigenvalue multiplicity in quartic graphs abstract let g be a connected cubic graph of order n with as an eigenvalue of multiplicity k we show that i if 1 0 then k 1 2 n with equality if and only if 1 and g is the petersen graph ii if 1 then k 1 2 n 1 with equality if and only if g k 4 iii if 0 then k 1 2 n 1 with equality if and only if g 2 k 3

h measures a new approach for studying homogenisation oscillations and concentration effects in partial differential equations new mathematical objects called h measures are introduced for studying oscillations and concentration effects in partial differential equations applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach

periodic nonlinear schrodinger equation and invariant measures in this paper we continue some investigations on the periodic nlseiu u iu xx u u p 2 p 6 started in lrs we prove that the equation is globally wellposed for a set of data of full normalized gibbs measrue e beta h phi hd phi x h phi tfrac 1 2 int left phi right 2 tfrac 1 p int left phi right p after suitablel 2 truncation the set and the measure are invariant under the flow the proof of a similar result for the kdv and modified kdv equations is outlined the main ingredients used are some estimates from b1 on periodic nls and kdv type equations

on the initial boundary value problem for the bipolar hydrodynamic model for semiconductors abstract the global existence and zero relaxation limit results of weak solutions of the initial boundary value problem to the bipolar hydrodynamic model for semiconductors are established by the theory of compensated compactness the boundary conditions of weak solutions in the sense of traces are discussed

statistics of noise driven coupled nonlinear oscillators applications to systems with kerr nonlinearity we present exact analytical results for the statistics of nonlinear coupled oscillators under the influence of additive white noise we suggest a perturbative approach for analysing the statistics of such systems under the action of a deterministic perturbation based on the exact expressions for probability density functions for noise driven oscillators using our perturbation technique we show that our results can be applied to studying the optical signal propagation in noisy fibres at nearly zero dispersion as well as to weakly nonlinear lattice models with additive noise the approach proposed can account for a wide spectrum of physically meaningful perturbations and is applicable to the case of large noise strength

equivalence of the euler and lagrangian equations of gas dynamics for weak solutions abstract this paper demonstrates the equivalence of the euler and the lagrangian equations of gas dynamics in one space dimension for weak solutions which are bounded and measurable in eulerian coordinates the precise hypotheses include all known global solutions on r r in particular solutions containing vacuum states zero mass density are included furthermore there is a one to one correspondence between the convex extensions of the two systems and the corresponding admissibility criteria are equivalent in the presence of a vacuum the definition of weak solution for the lagrangian equations must be strengthened to admit test functions which are discontinuous at the vacuum as an application we translate a large data existence result of diperna for the euler equations for isentropic gas dynamics into a similar theorem for the lagrangian equations

free transportation cost inequalities via random matrix approximation by means of random matrix approximation procedure we re prove biane and voiculescu s free analog of talagrand s transportation cost inequality for measures on r in a more general setup furthermore we prove the free transportation cost inequality for measures on t as well by extending the method to special unitary random matrices

quasiconvexity and relaxation of nonconvex problems in the calculus of variations keywords quasiconvexity relaxation reference caa article 1982 002doi 10 1016 0022 1236 82 90046 5view record in web of science record created on 2008 11 25 modified on 2017 05 12

graphs with least eigenvalue 2 ten years on abstract the authors monograph spectral generalizations of line graphs was published in 2004 following the successful use of star complements to complete the classification of graphs with least eigenvalue 2 guided by citations of the book we survey progress in this area over the past decade some new observations are included

error bounds for finite difference approximations for a class of nonlinear parabolic systems in this paper we establish error bounds for a finite difference approximation to solutions of certain parabolic systems of the form t f po u we assume that the cauchy data is of class bv and we show that the sup norm of the error is bounded by o axiln axi at positive times

convergence of the viscosity method for isentropic gas dynamics a convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

effective velocity in compressible navier stokes equations with third order derivatives abstract a formulation of certain barotropic compressible navier stokes equations with third order derivatives as a viscous euler system is proposed by using an effective velocity variable the equations model for instance viscous korteweg or quantum navier stokes flows the formulation in the new variable allows for the derivation of an entropy identity which is known as the bd bresch desjardins entropy equation as a consequence of this estimate a new global in time existence result for the one dimensional quantum navier stokes equations with strictly positive particle densities is proved

on random matrices from the compact classical groups if m is a matrix taken randomly with respect to normalized haar measure on u n o n or sp n then the real and imaginary parts of the random variables tr mk k 1 converge to independent normal random variables with mean zero and variance k 2 as the size n of the matrix goes to infinity for the unitary group this is a direct consequence of the strong szeg6 theorem for toeplitz determinants we will prove a conjecture of diaconis

l1 convergence rates to the barenblatt solution for the damped compressible euler equations abstract in our previous work 17 it is shown that any l weak entropy solution of damped compressible euler equation converges to the barrenblatt s solution with finite mass in l 1 norm with a convergence rate for 1 3 in this paper by introducing an elaborate iterative method and using the intensive entropy analysis we prove the l 1 convergence to the barrenblatt s solution for any large 3 furthermore we get better l 1 convergence rate for 2 3

oscillations and concentrations in weak solutions of the incompressible fluid equations the authors introduce a new concept of measure valued solution for the 3 d incompressible euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations one application of the concepts developed here is the following important result a sequence of leray hopf weak solutions of the navier stokes equations converges in the high reynolds number limit to a measure valued solution of 3 d euler defined for all positive times the authors present several explicit examples of solution sequences for 3 d incompressible euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations an extensions of the concept of young measure is developed to incorporate these complex phenomena in the measure valued solutions constructed here

an extension of the star complement technique for regular graphs abstract we extend the means by which a regular graph can sometimes be identified from a star complement in two applications we determine i the regular graphs with a path as a star complement for the eigenvalue 1 ii the extremal regular graphs which have another type of tree as a star complement for an eigenvalue 1 0

mass transportation proofs of free functional inequalities and free poincare inequalities this work is devoted to direct mass transportation proofs of families of functional inequalities in the context of one dimensional free probability avoiding random matrix approximation the inequalities include the free form of the transportation log sobolev hwi interpolation and brunn minkowski inequalities for strictly convex potentials sharp constants and some extended versions are put forward the paper also addresses two versions of free poincare inequalities and their interpretation in terms of spectral properties of jacobi operators the last part establishes the corresponding inequalities for measures on r with the reference example of the marcenko pastur distribution

almost sure weak convergence for the generalized orthogonal ensemble the generalized orthogonal ensemble satisfies isoperimetric inequalities analogous to the gaussian isoperimetric inequality and an analogue of wigner s law let v be a continuous and even real function such that v x tracev x n defines a uniformly p convex function on the real symmetric n n matrices x for some p 2 then dx e v x dx z satisfies deviation and transportation inequalities analogous to those satisfied by gaussian measure 6 27 but for the schatten cp norm the map that associates to each x msn its ordered eigenvalue sequence induces from a measure which satisfies similar inequalities it follows from such concentration inequalities that the empirical distribution of eigenvalues converges weakly almost surely to some non random compactly supported probability distribution as n

a relaxation theorem and its application to the equilibrium of gases keywords relaxed problem lower convex envelope gas equilibrium van der waal s equation reference caa article 1981 001doi 10 1007 bf00280643view record in web of science record created on 2008 11 25 modified on 2017 05 12

free functional inequalities on the circle abstract in this paper we deal with free functional inequalities on the circle there are some interesting changes from their classical counterparts for example the free poincare inequality has a slight change which seems to account for the lack of invariance under rotations of the base measure another instance is the modified wasserstein distance on the circle which provides the tools for analyzing transportation log sobolev and hwi inequalities these new phenomena also indicate that they have classical counterparts which does not seem to have been investigated thus far

statistical mechanics of the nonlinear schrodinger equation we investigate the statistical mechanics of a complex fieldo whose dynamics is governed by the nonlinear schrodinger equation such fields describe in suitable idealizations langmuir waves in a plasma a propagating laser field in a nonlinear medium and other phenomena their hamiltonian h phi int omega frac 1 2 nabla phi 2 1 p phi p dx is unbounded below and the system will under certain conditions develop self focusing singularities in a finite time we show that when is the circle and thel 2 norm of the field which is conserved by the dynamics is bounded byn the gibbs measure obtained is absolutely continuous with respect to wiener measure and normalizable if and only ifp andn are such that classical solutions exist for all time no collapse of the solitons this measure is essentially the same as that of a one dimensional version of the more realisitc zakharov model of coupled langmuir and ion acoustic waves in a plasma we also obtain some properties of the gibbs state by both analytic and numerical methods asn and the temperature are varied

the comparison of the riemann solutions in gas dynamics in this paper we compare two riemann solution constructions of isentropic and nonisentropic gas dynamic equations with the same initial values we obtain that when rarefaction waves occur in the non isentropic solution the same family rarefaction waves occur in the isentropic solution however when advancing or reflected shock waves occur in the isentropic case the same kind shock waves occur in the nonisentropic case moreover by numerical calculation it can be found that the wave intensities of the two solutions are quite close our conclusion reflects that almost non isentropic gas dynamic process can be simulated by isentropic gas dynamic process in practice c 2015 elsevier inc all rights reserved

uniqueness and stability of riemann solutions with large oscillation in gas dynamics we prove the uniqueness of riemann solutions in the class of entropy solutions in with arbitrarily large oscillation for the 3 3 system of euler equations in gas dynamics the proof for solutions with large oscillation is based on a detailed analysis of the global behavior of shock curves in the phase space and the singularity of centered rarefaction waves near the center in the physical plane the uniqueness of riemann solutions yields their inviscid large time stability under arbitrarily large perturbation of the riemann initial data as long as the corresponding solutions are in l and have local bounded total variation satisfying a natural condition on its growth with time no specific reference to any particular method for constructing the entropy solutions is needed the uniqueness result for riemann solutions can easily be extended to entropy solutions u x t piecewise lipschitz in x for any t 0 with arbitrarily large oscillation

statistical mechanics of the periodic benjamin ono equation the periodic benjamin ono equation is an autonomous hamiltonian system with a gibbs measure on l2 t this paper shows that the gibbs measures on bounded balls of l2 satisfy some logarithmic sobolev inequalities the space of n soliton solutions of the periodic benjamin ono equation as discovered by case is a hamiltonian system with an invariant gibbs measure as n these gibbs measures exhibit a concentration of measure phenomenon case introduced soliton solutions that are parameterized by atomic measures in the complex plane the limiting distributions of these measures give the density of a compressible gas that satisfies the isentropic euler equations the periodic benjamin ono equation is an autonomous hamiltonian system with a gibbs measure on l2 t this paper shows that the gibbs measures on bounded balls of l2 satisfy some logarithmic sobolev inequalities the space of n soliton solutions of the periodic benjamin ono equation as discovered by case is a hamiltonian system with an invariant gibbs measure as n these gibbs measures exhibit a concentration of measure phenomenon case introduced soliton solutions that are parameterized by atomic measures in the complex plane the limiting distributions of these measures give the density of a compressible gas that satisfies the isentropic euler equations

hypercontractivity and spectral gap of symmetric diffusions with applications to the stochastic ising models abstract we describe some results relating the spectral gap and logarithmic sobolev constants we work initially in the context of symmetric diffusions on a finite dimensional manifold and later apply our results to the analysis of certain infinite systems of mildly interacting diffusions

existence of the entropy solution for a viscoelastic model in this paper we consider the cauchy problem for a viscoelastic model with relaxationut x 0 f u t 1 f u 0with discontinuous large initial data where 0 0 are constants when the system is nonstrictly hyperbolic under the additional assumptionv0x l the system is reduced to an inhomogeneous scalar balance law by employing the special form of the system itself after introducing a definition of entropy solutions to the system we prove the existence uniqueness and continuous dependence of the global entropy solution for the system when the system is strictly hyperbolic some special entropy pairs of the lax type are constructed in which the progression terms are functions of a single variable and the necessary estimates for the major terms are obtained by using the theory of singular perturbation of the ordinary differential equations the special entropy pairs are used to prove the existence of the global entropy solutions for the corresponding cauchy problem by applying the method of compensated compactness

eigenvalue multiplicity in triangle free graphs abstract let g be a connected triangle free graph of order n 5 with 1 0 as an eigenvalue of multiplicity k 1 we show that if d is the maximum degree in g then k n d 1 moreover if k n d 1 then either a g is non bipartite and k 2 3 1 2 2 1 with equality only if g is strongly regular or b g is bipartite and k d 1 with equality only if g is a bipolar cone in each case we discuss the extremal graphs that arise

non gaussian error probability in optical soliton transmission we find the probability distribution of the fluctuating parameters of a soliton propagating through a medium with additive noise our method is a modification of the instanton formalism method of optimal fluctuation based on a saddle point approximation in the path integral we first solve consistently a fundamental problem of soliton propagation within the framework of noisy nonlinear schrodinger equation we then consider model modifications due to in line filtering amplitude and phase modulation control it is examined how control elements change the error probability in optical soliton transmission even though a weak noise is considered we are interested here in probabilities of error causing large fluctuations which are beyond perturbation theory we describe in detail a new phenomenon of soliton collapse that occurs under the combined action of noise filtering and amplitude modulation 2004 elsevier b v all rights reserved

viscous limits for piecewise smooth solutions to systems of conservation laws in this paper we study the zero dissipation problem for a general system of conservation laws with positive viscosity it is shown that if the solution of the problem with zero viscosity is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition then there exist solutions to the corresponding system with viscosity that converge to the solutions of the system without viscosity away from shock discontinuities at a rate of order e as the viscosity coefficient e goes to zero the proof uses a matched asymptotic analysis and an energy estimate related to the stability theory for viscous shock profiles

weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift diffusion equation we investigate the relaxation problem for the hydrodynamic isentropic euler poisson system when the momentum relaxation time tends to zero very sharp estimates on the solutions independent of the relaxation time are obtained and used to establish compactness

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

first order asymptotics of matrix integrals a rigorous approach towards the understanding of matrix models we investigate the large n limit of spectral measures of matrices which relate to the gibbs measures of a number of statistical mechanical systems on random graphs these include the ising and potts models on random graphs for most of these models we prove that the spectral measures converge almost surely and describe their limit via solutions to an euler equation for isentropic flow with negative pressure p 3 1 2 3

sharp uniform convexity and smoothness inequalities for trace norms we prove several sharp inequalities specifying the uniform convexity and uniform smoothness properties of the schatten trace ideals c p which are the analogs of the lebesgue spaces l p in non commutative integration the inequalities are all precise analogs of results which had been known in l p but were only known in c p for special values of p in the course of our treatment of uniform convexity and smoothness inequalities for c p we obtain new and simple proofs of the known inequalities for l p

multiscale young measures in homogenization of continuous stationary processes in compact spaces and applications abstract we introduce a framework for the study of nonlinear homogenization problems in the setting of stationary continuous processes in compact spaces the latter are functions f t r n q q with f t x f t x where q is a compact hausdorff topological space f c q and t x q q x r n is an n dimensional continuous dynamical system endowed with an invariant radon probability measure it can be easily shown that for almost all q the realization f t x belongs to an algebra with mean value that is an algebra of functions in buc r n containing all translates of its elements and such that each of its elements possesses a mean value this notion was introduced by zhikov and krivenko v v zhikov e v krivenko homogenization of singularly perturbed elliptic operators mat zametki 33 1983 571 582 english transl in math notes 33 1983 294 300 we then establish the existence of multiscale young measures in the setting of algebras with mean value where the compactifications of r n provided by such algebras plays an important role these parametrized measures are useful in connection with the existence of correctors in homogenization problems we apply this framework to the homogenization of a porous medium type equation in r n with a stationary continuous process as a stiff oscillatory external source this application seems to be new even in the classical context of periodic homogenization

eulerian droplet model delta shock waves and solution of the riemann problem abstract we study an eulerian droplet model which can be seen as the pressureless gas system with a source term a subsystem of this model and the inviscid burgers equation with source term the condition for loss of regularity of a solution to burgers equation with source term is established the same condition applies to the eulerian droplet model and its subsystem the riemann problem for the eulerian droplet model is constructively solved by going through the solution of the riemann problems for the inviscid burgers equation with a source term and the subsystem respectively under suitable generalized rankine hugoniot relations and entropy condition the existence of delta shock solution is established the existence of a solution to the generalized rankine hugoniot conditions is proven some numerical illustrations are presented

information transmission using the nonlinear fourier transform part iii spectrum modulation motivated by the looming capacity crunch in fiber optic networks information transmission over such systems is revisited among numerous distortions interchannel interference in multiuser wavelength division multiplexing wdm is identified as the seemingly intractable factor limiting the achievable rate at high launch power however this distortion and similar ones arising from nonlinearity are primarily due to the use of methods suited for linear systems namely wdm and linear pulse train transmission for the nonlinear optical channel exploiting the integrability of the nonlinear schrodinger nls equation a nonlinear frequency division multiplexing nfdm scheme is presented which directly modulates noninteracting signal degrees of freedom under nls propagation the main distinction between this and previous methods is that nfdm is able to cope with the nonlinearity and thus as the signal power or transmission distance is increased the new method does not suffer from the deterministic crosstalk between signal components which has degraded the performance of previous approaches in this paper emphasis is placed on modulation of the discrete component of the nonlinear fourier transform of the signal and some simple examples of achievable spectral efficiencies are provided

construction of a gibbs measure associated to the periodic benjamin ono equation we define a finite borel measure of gibbs type supported by the sobolev spaces of negative indexes on the circle the measure can be seen as a limit of finite dimensional measures these finite dimensional measures are invariant by the ode s which correspond to the projection of the benjamin ono equation posed on the circle on the first n n 1 modes in the trigonometric bases

invariant measures and long time behaviour for the benjamin ono equation iii we complete the program developed in our previous works aiming to construct an infinite sequence of invariant measures of gaussian type associated with the conservation laws of the benjamin ono equation

a note on the eigenvalue density of random matrices the distribution of eigenvalues of n n random matrices in the limit n is the solution to a variational principle that determines the ground state energy of a confined fluid of classical unit charges this fact is a consequence of a more general theorem proven here in the statistical mechanics of unstable interactions our result establishes the eigenvalue density of some ensembles of random matrices which were not covered by previous theorems

on the statistical mechanics approach in the random matrix theory integrated density of states we consider the ensemble of random symmetricn n matrices specified by an orthogonal invariant probability distribution we treat this distribution as a gibbs measure of a mean field type model this allows us to show that the normalized eigenvalue counting function of this ensemble converges in probability to a nonrandom limit asn and that this limiting distribution is the solution of a certain self consistent equation

viscous limits for piecewise smooth solutions of the p system abstract we study the zero dissipation problem for a one dimensional model system for the isentropic flow of a compressible viscous gas the so called p system with viscosity when the solution of the inviscid problem is piecewise smooth and having finitely many noninteracting shocks satisfying the entropy condition there exists unique solution to the viscous problem which converges to the given inviscid solution away from shock discontinuities at a rate of order e as the viscosity coefficient e goes to zero the proof is given by a matched asymptotic analysis and an elementary energy method and we do not need the smallness condition on the shock strength

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

kinetic formulation of the isentropic gas dynamics and p systems we consider the 2 x 2 hyperbolic system of isentropic gas dynamics in both eulerian or lagrangian variables also called the p system we show that they can be reformulated as a kinetic equation using an additional kinetic variable such a formulation was first obtained by the authors in the case of multidimensio nal scalar conservation laws a new phenomenon occurs here namely that the advection velocity is now a combination of the macroscopic and kinetic velocities various applications are given we recover the invariant regions deduce new l estimates using moments lemma and prove l w stability for 7 3

convergence of the viscosity method for a nonstrictly hyperbolic conservation law a convergence theorem for the method of artificial viscosity applied to the nonstrictly hyperbolic systemvt vu x 0 u t left frac 1 2 u 2 mathop smallint limits v s s delta r 3 ds right x 0 0 r 3 is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

global solutions to the compressible euler equations with geometrical structure we prove the existence of global solutions to the euler equations of compressible isentropic gas dynamics with geometrical structure including transonic nozzle flow and spherically symmetric flow due to the presence of the geometrical source terms the existence results themselves are new especially as they pertain to radial flow in an unbounded region vec x geqq 1 and to transonic nozzle flow arbitrary data withl bounds are allowed in these results a shock capturing numerical scheme is introduced to compute such flows and to construct approximate solutions the convergence and consistency of the approximate solutions generated from this scheme to the global solutions are proved with the aid of a compensated compactness framework

instantons and intermittency we describe the method for finding the non gaussian tails of the probability distribution function pdf for solutions of a stochastic differential equation such as the convection equation for a passive scalar the random driven navier stokes equation etc the existence of such tails is generally regarded as a manifestation of the intermittency phenomenon our formalism is based on the wkb approximation in the functional integral for the conditional probability of large fluctuation we argue that the main contribution to the functional integral is given by a coupled field force configuration char22 the instanton as an example we examine the correlation functions of the passive scalar u advected by a large scale velocity field ensuremath delta correlated in time we find the instanton determining the tails of the generating functional and show that it is different from the instanton that determines the probability distribution function of high powers of u we discuss the simplest instantons for the navier stokes equation textcopyright 1996 the american physical society

displacement convexity for the generalized orthogonal ensemble the generalized orthogonal ensemble of n n real symmetric matrices x has probability measure nu n left dx right z n 1 exp left ntracev left x right right where dx is the product of lebesgue measure on the matrix entries and v left x right geqslant left 2 delta right log left x right with 0 the eigenvalue distribution is concentrated on left a mathord left vphantom a 2 right kern nulldelimiterspace 2 a mathord left vphantom a 2 right kern nulldelimiterspace 2 right for some a 0 or if the variance of the trace is o 1 n2 then the empirical distribution of eigenvalues converges weakly almost surely to some non random probability measure on a 2 a 2 as o left sqrt log n mathord left vphantom n n 2 right kern nulldelimiterspace n 2 right these conditions are satisfied for certain polynomial potentials the logarithmic energy is displacement convex as a functional on charge distributions with fixed mean along the real line when the trace distribution satisfies a logarithmic sobolev inequality or equivalently a quadratic transportation inequality the joint eigenvalue distributions and the limiting equilibrium measure likewise satisfy quadratic transportation inequalities in the sense of talagrand 24

kinetic formulation for systems of two conservation laws and elastodynamics for scalar conservation laws the kinetic formulation makes it possible to generate all the entropies from a simple kernel we show how this concept replaces and simplifies greatly the concept of young measures avoiding the difficulties encountered when working in lp the general construction of the two kinetic functions that generate the entropies of 2 2 strictly hyperbolic systems is also developed here we show that it amounts to building a universal entropy i e one that can be truncated by a kinetic value along riemann invariants for elastodynamics this construction can be completed and specialized using the additional galilean invariance this allows a full characterization of convex entropies it yields a kinetic formulation consisting of two semi kinetic equations which as usual are equivalent to the infinite family of all the entropy inequalities

global well posedness in the energy space for the benjamin ono equation on the circle we prove that the benjamin ono equation is locally well posed in h 1 2 mathbb t this leads to a global well posedeness result in h 1 2 mathbb t thanks to the energy conservation

global solutions to one dimensional shallow water magnetohydrodynamic equations abstract in this paper we study the cauchy problem for the one dimensional shallow water magnetohydrodynamic equations the main difficulty is the case of zero depth h 0 since the nonlinear flux function p h is singular and the definition of solution is not clear near h 0 first assuming that h has a positive and lower bound we establish the pointwise convergence of the viscosity solutions by using the div curl lemma from the compensated compactness theory to special pairs of functions c f e and obtain a global weak entropy solution second under some technical conditions on the initial data such that the riemann invariants w z are monotonic and increasing we introduce a variant of the vanishing artificial viscosity to select a weak solution finally we extend the results to two special cases where p h is for the polytropic gas or for the chaplygin gas

isometric immersions of surfaces with two classes of metrics and negative gauss curvature the isometric immersion of two dimensional riemannian manifolds or surfaces with negative gauss curvature into the three dimensional euclidean space is studied in this paper the global weak solutions to the gauss codazzi equations with large data in l infty are obtained through the vanishing viscosity method and the compensated compactness framework the l infty uniform estimate and h 1 compactness are established through a transformation of state variables and construction of proper invariant regions for two types of given metrics including the catenoid type and the helicoid type the global weak solutions in l infty to the gauss codazzi equations yield the c1 1 isometric immersions of surfaces with the given metrics

statistics of soliton bearing systems with additive noise we present a consistent method to calculate the probability distribution of soliton parameters in systems with additive noise even though the noise is weak we are interested in probabilities of large fluctuations generally non gaussian which are beyond perturbation theory our method is a development of the instanton formalism method of optimal fluctuation based on a saddle point approximation in the path integral we first