sonic hyperbolic phase transitions and chapman jouguet detonations abstract we prove that the cauchy problem for an n n system of strictly hyperbolic conservation laws in one space dimension admits a weak global solution also in presence of sonic phase boundaries applications to chapman jouguet detonations liquid vapor transitions and elastodynamics are considered

global bv entropy solutions and uniqueness for hyperbolic systems of balance laws we consider the cauchy problem for n n strictly hyperbolic systems of nonresonant balance laws

a bound on the total variation of the conserved quantities for solutions of a general resonant nonlinear balance law we introduce a new potential interaction functional and use it to define a new glimm type functional that bounds the total variation of the conserved quantities at time t 0 by the total variation at time t 0 in glimm approximate solutions of a general resonant nonlinear balance law

an extension of glimm s method to inhomogeneous strictly hyperbolic systems of conservation laws by weaker than weak solutions of the riemann problem abstract we construct a generalized solution of the riemann problem for strictly hyperbolic systems of conservation laws with source terms and we use this to show that glimm s method can be used directly to establish the existence of solutions of the cauchy problem the source terms are taken to be of the form a g and this enables us to extend the method introduced by lax to construct general solutions of the riemann problem our generalized solution of the riemann problem is weaker than weak in the sense that it is weaker than a distributional solution thus we prove that a weak solution of the cauchy problem is the limit of a sequence of glimm scheme approximate solutions that are based on weaker than weak solutions of the riemann problem by establishing the convergence of glimm s method it follows that all of the results on time asymptotics and uniqueness for glimm s method in the presence of a linearly degenerate field now apply unchanged to inhomogeneous systems

spectral monodromy of non self adjoint operators in the present paper we build a combinatorial invariant called the spectral monodromy from the spectrum of a single non self adjoint h pseudodifferential operator with two degrees of freedom in the semi classical limit our inspiration comes from the quantum monodromy defined for the joint spectrum of an integrable system of n commuting self adjoint h pseudodifferential operators given by s vu ngoc quantum monodromy in integrable systems commun math phys 203 2 465 479 1999 the first simple case that we treat in this work is a normal operator in this case the discrete spectrum can be identified with the joint spectrum of an integrable quantum system the second more complex case we propose is a small perturbation of a self adjoint operator with a classical integrability property we show that the discrete spectrum in a small band around the real axis also has a combinatorial monodromy the main difficulty in this case is that we do not know the description of the spectrum everywher

smooth and discontinuous junctions in the p system abstract consider the p system describing the subsonic flow of a fluid in a pipe with section a a x we prove that the resulting cauchy problem generates a lipschitz semigroup provided the total variation of the initial datum and the oscillation of a are small an explicit estimate on the bound of the total variation of a is provided showing that at lower fluid speeds higher total variations of a are acceptable an example shows that the bound on tv a is mandatory for otherwise the total variation of the solution may grow arbitrarily

semiclassical eigenvalues and shape problems on surfaces of revolution mark kac can one hear the shape of drums am math monthly 73 1 23 1966 asked if the shape of a region rn could be determined from its sound spectrum of the laplacian he proved the conjecture for special classes of domains including polygons and balls in rn similar problems could be raised in other geometric contexts including shape of metric for laplacians on manifolds or shape of potential for schrodinger operators hv v the latter two problems for surfaces of revolution are addressed an explicit reconstruction procedure will be outlined that leads from the joint spectrum of h or v and the angular momentum algebra so n to the shape of and v respectively the metric result applies to generic surfaces of revolution while the schrodinger result allows all zonal axisymmetric potentials v on so the work extends the kac s n ball result as well as the n sphere zonal schrodinger theory d gurarie a averaging methods in spectral theory of schr

extending morphisms from finite to algebraic groups keywords extending morphisms finite groups of lie type algebraic groups indecomposable representations maximal subgroups reference ctg article 1990 001doi 10 1016 0021 8693 90 90195 t record created on 2008 12 16 modified on 2017 05 12

the generalized buckley leverett and the regularized buckley leverett equations this paper studies solutions of the generalized buckley leverett bl equation with variable porosity and solutions of the regularized bl equation with the burgers benjamin bona mahony type regularization we construct global in time weak solutions to the cauchy problem and to the initial and boundary value problem for the generalized bl equation and global classical solutions for the regularized bl equation solutions of the regularized bl equation are shown to converge to the corresponding solution of the generalized bl equation when the coefficient of the bbm term and the coefficient of the burgers term obey o 2

on the front tracking algorithm in this paper we present an improved version of the front tracking algorithm for systems of conservation laws the formulation and the theoretical analysis are here somewhat simpler than in previous algorithms at the same time our version leads to a more efficient numerical scheme

globally lipschitz continuous solutions to a class of quasilinear wave equations abstract this work investigates the existence of globally lipschitz continuous solutions to a class of cauchy problem of quasilinear wave equations applying lax s method and generalized glimm s method we construct the approximate solutions of the corresponding perturbed riemann problem and establish the global existence for the derivatives of solutions then the existence of global lipschitz continuous solutions can be carried out by showing the weak convergence of residuals for the source term of equation

homogeneous quantization and multiplicities of group representations let b be a compact manifold a cone over b is a principal r bundle x with base b let a x a x be the mapping associated with the action of a r on x x is called a symplectic cone if it possesses a symplectic form such that a a a compact lie group g is said to act in a homogeneous fashion on x if it acts on x in such a way that both and the principal bundle structure are preserved it is known that to such an action one can associate in a fairly canonical way a representation of g on a hilbert space h see 3 in this paper we propose a symplectic recipe for the multiplicities with which h decomposes into g irreducibles and show that this recipe is correct generically

on linearized coupling conditions for a class of isentropic multiphase drift flux models at pipe to pipe intersections in this paper a general drift flux model describing a subsonic and isentropic multi phase fluid in connected pipes is considered each phase is assumed to be isentropic with its own sonic speed the components are gamma law gases with 1 for such a computational challenge at a junction is the computation of rarefaction waves which do not have a readily available analytical form firstly the well posedness of the riemann problem at the junction is discussed it is suggested that rarefaction waves should be linearized in order to obtain a more efficient numerical method for coupling such multi component flow some computational results on the dynamics of the multi phase gas in the pipes demonstrate the qualitative behavior of this approach

semitoric integrable systems on symplectic 4 manifolds let m be a symplectic 4 manifold a semitoric integrable system on m is a pair of smooth functions j h c m r for which j generates a hamiltonian s 1 action and the poisson brackets j h vanish we shall introduce new global symplec tic invariants for these systems some of these invariants encode topological or geometric aspects while others encode analytical information about the singularities and how they stand with respect to the system our goal is to prove that a semitoric system is completely determined by the invariants we introduce

semiclassical inverse spectral theory for singularities of focus focus type we prove assuming that the bohr sommerfeld rules hold that the joint spectrum near a focus focus singular value of a quantum integrable system determines the classical lagrangian foliation around the full focus focus leaf the result applies for instance to pseudodifferential operators on cotangent bundles and berezin toeplitz operators on prequantizable compact symplectic manifolds

generators relations and coverings of algebraic groups ii theorem 1 1 let g be the group of rational points of a simply connected simple afine algebraic group defined and split over a fmite j eld of q elements then the schur multiplier m of g is trivial with the following exceptions a zf g is a 4 a 2 a 3 c 2 f 2 or g 4 then h4 is z cyclic of order 2 if g is a 9 b 3 or g 3 it is iz b zf g is a 4 resp d 2 then m is the direct product of z resp z with itself

constructing integrable systems of semitoric type let m be a connected symplectic 4 manifold a semitoric integrable system on m essentially consists of a pair of independent real valued smooth functions j and h on m for which j generates a hamiltonian circle action under which h is invariant in this paper we give a general method to construct starting from a collection of five ingredients a symplectic 4 manifold equipped a semitoric integrable system then we show that every semitoric integrable system on a symplectic 4 manifold is obtained in this fashion in conjunction with the uniqueness theorem proved recently by the authors this gives a classification of semitoric integrable systems on 4 manifolds in terms of five invariants

global solutions for initial boundary value problem of quasilinear wave equations abstract this work investigates the existence of globally lipschitz continuous solutions to a class of initial boundary value problem of quasilinear wave equations applying the lax s method and generalized glimm s method we construct the approximate solutions of initial boundary riemann problem near the boundary layer and perturbed riemann problem away from the boundary layer by showing the weak convergence of residuals for the approximate solutions we establish the global existence for the derivatives of solutions and obtain the existence of global lipschitz continuous solutions of the problem

bohr sommerfeld conditions for integrable systems with critical manifolds of focus focus type we present a detailed study in the semi classical regime h of microlocal properties of systems of two commuting h pseudo dierential operators ph ph such that the joint principal symbol p p p has a special kind of singularity called a focusfocus singularity typical examples include the quantum spherical pendulum or the quantum champagne bottle in the spirit of colin de verdiere and parisse we show that such systems have a universal behavior described by singular quantization conditions of bohr sommerfeld type these conditions are used to give a precise description of the joint spectrum of such systems including the phenomenon of quantum monodromy and dierent formulations of the counting function for the joint eigenvalues close to the singularity in which a logarithm of the semi classical constant h appears thanks to numerical computations done by ms child for the case of the champagne bottle we are able to accurately illustrate our statements

concentration of source terms in generalized glimm scheme for global entropy solutions of nonlinear hyperbolic balance laws abstract in this paper we investigate the cauchy problem for a nonlinear hyperbolic system of balance laws with sources a x g and a t h to get the approximate solutions of our problem we consider a version of generalized riemann problem that concentrates the variation of a on a thin t shaped region of each grid a new version of glimm scheme is introduced to construct the approximate solutions and its stability is proved by considering two types of conditions on a finally we verify the consistency of the scheme and the entropy inequality to establish the global existence of entropy solutions

spectral asymptotics of semiclassical unitary operators we introduce axiomatically a semiclassical quantization for non necessarily self adjoint operators then we focus on unitary operators and prove that in the semiclassical limit the convex hull of the joint spectrum of a finite commuting family of semiclassical unitary operators converges to the convex hull of the joint image of the principal symbols which can be shown to be a subset of a d torus mathbb s 1 d this result covers in particular hbar pseudodifferential and berezin toeplitz operators part of the paper is devoted to the definition of this notion of convex hull for subsets of tori the proof of our result builds on recent results for semiclassical self adjoint operators and involves the inverse cayley transform for unitary operators

semiclassical quantization and spectral limits of ħ pseudodifferential and berezin toeplitz operators we introduce a minimalistic notion of semiclassical quantization and use it to prove that the convex hull of the semiclassical spectrum of a quantum system given by a collection of commuting operators converges to the convex hull of the spectrum of the associated classical system this gives a quick alternative solution to the isospectrality problem for quantum toric systems if the operators are uniformly bounded the convergence is uniform analogous results hold for non commuting operators

on the congruence subgroup problem determination of the metaplectic kernel let f be a global field i e either a number field or a function field in one variable over a finite field and let oo be the set of its archimedean places let g be a finite non empty set of places of f containing oo let o o denote the ring of g integers of f let a denote the ring of g adeles i e the restricted direct product of the completions f v for r eg let f be an absolutely simple simply connected subgroup of sl defined over f recall that a subgroup f of c f is an g arithmetic subgroup if fnsl n o has finite index in f as well as in o fy f c sl n o an g arithmetic subgroup f is a congruence subgroup if there exists a non zero ideal a in o o g such that

convergence of the 2 2 godunov method for a general resonant nonlinear balance law we introduce a generalized solution of the riemann problem for a general resonant nonlinear balance law and we prove the convergence of the 2 x 2 godunov numerical method based on these solutions in particular we obtain generic conditions that guarantee a canonical structure for the elementary waves in the solution of the riemann problem and an interesting multiplicity of time asymptotic wave patterns is observed and characterized

inverse spectral theory for semiclassical jaynes cummings systems quantum semitoric systems form a large class of quantum hamiltonian integrable systems with circular symmetry which has received great attention in the past decade they include systems of high interest to physicists and mathematicians such as the jaynes cummings model 1963 which describes a two level atom interacting with a quantized mode of an optical cavity and more generally the so called systems of jaynes cummings type in this paper we consider the joint spectrum of a pair of commuting semiclassical operators forming a quantum integrable system of jaynes cummings type we prove assuming the bohr sommerfeld rules hold that if the joint spectrum of two of these systems coincide up to mathcal o hbar 2 then the systems are isomorphic

toeplitz operators and hamiltonian torus actions this paper is devoted to semi classical aspects of symplectic reduction consider a compact prequantizable kahler manifold m with a hamiltonian torus action in the seminal paper v guillemin s sternberg geometric quantization and multiplicities of group representations invent math 67 3 1982 515 538 guillemin and sternberg introduced an isomorphism between the invariant part of the quantum space associated to m and the quantum space associated to the symplectic quotient of m provided this quotient is non singular we prove that this isomorphism is a fourier integral operator and that the toeplitz operators of m descend to toeplitz operators of the reduced phase space we also extend these results to the case where the symplectic quotient is an orbifold and estimate the spectral density of a reduced toeplitz operator a result related to the riemann roch kawasaki theorem

berezin toeplitz operators a semi classical approach this article is devoted to the toeplitz operators 4 in the context of the geometric quantization 11 15 we propose an ansatz for their schwartz kernel from this we deduce the main known properties of the principal symbol of these operators and obtain new results we define their covariant and contravariant symbols which are full symbols and compute the product of these symbols in terms of the kahler metric this gives canonical star products on the kahlerian manifolds this ansatz is also useful to introduce the notion of microsupport

tame component of the schur multipliers of finite groups of lie type let f be a finite field of characteristic p let be an almost f simple simply connected algebraic group defined over f for any f subgroup a of y we shall denote by r f the finite group of f rational points of z and shall consider q z as an r f module with trivial x f action let g f f let j be the p primary component of q z and j be the sum of all primeto p primary components of q z then q z jo j and the schur multiplier h2 g q z is the direct sum of h g j and h g j the subgroup h2 g j is the p primary component of the schur multiplier of g we shall call it the wild component of the schur multiplier and h2 g j which is the sum of prime to p primary components of the schur multiplier the tame component it is a classical result of schur 41 that the tame component of the schur multiplier of sl 2 5 is trivial for any finite field 5 griess has proved 3 1 that the schur multiplier of su 3 5 is trivial we shall use these two results and the tits building of y f to give a new proof of the following theorem of steinberg 6 71 and griess 31

on the generalized buckley leverett equation in this paper we study the generalized buckley leverett equation with nonlocal regularizing terms one of these regularizing terms is diffusive while the other one is conservative we prove that if the regularizing terms have order higher than one combined there exists a global strong solution for arbitrarily large initial data in the case where the regularizing terms have combined order one we prove the global existence of solution under some size restriction for the initial data moreover in the case where the conservative regularizing term vanishes regardless of the order of the diffusion and under a certain hypothesis on the initial data we also prove the global existence of a strong solution and we obtain some new entropy balances finally we provide numerics suggesting that if the order of the diffusion is 0 1 a finite time blow up of the solution is possible

moment polytopes for symplectic manifolds with monodromy abstract a natural way of generalising hamiltonian toric manifolds is to permit the presence of generic isolated singularities for the moment map for a class of such almost toric 4 manifolds which admits a hamiltonian s 1 action we show that one can associate a group of convex polygons that generalise the celebrated moment polytopes of atiyah guillemin sternberg as an application we derive a duistermaat heckman formula demonstrating a strong effect of the possible monodromy of the underlying integrable system

semi classical properties of geometric quantization with metaplectic correction the geometric quantization of a symplectic manifold endowed with a prequantum bundle and a metaplectic structure is defined by means of an integrable complex structure we prove that its semi classical limit does not depend on the choice of the complex structure we show this in two ways first by introducing unitary identifications between the quantum spaces associated to the various complex polarizations and second by defining an asymptotically flat connection in the bundle of quantum spaces over the space of complex structures furthermore berezin toeplitz operators are intertwined by these identifications and have principal and subprincipal symbols defined independently of the complex structure the relation with the schrodinger equation and the group of prequantum bundle automorphisms is considered as well