fully discrete entropy conservative schemes of arbitrary order we consider weak solutions of hyperbolic or hyperbolic elliptic systems of conservation laws in one space dimension and their approximation by finite difference schemes in conservative form the systems under consideration are endowed with an entropy entropy flux pair we introduce a general approach to construct second and third order accurate fully discrete in both space and time entropy conservative schemes in general these schemes are fully nonlinear implicit but in some important cases can be explicit or linear implicit furthermore semidiscrete entropy conservative schemes of arbitrary order are presented the entropy conservative schemes are used to construct a numerical method for the computation of weak solutions containing nonclassical regularization sensitive shock waves finally specific examples are investigated and tested numerically our approach extends the results and techniques by tadmor in numerical methods for compressible flows finite difference element and volume techniques asme new york 1986 pp 149 158 lefloch and rohde siam j numer anal 37 2000 pp 2023 2060

on isospectral periodic potentials in rn ii on considere un reseau a n dimensions l dans r n et un potentiel q satisfaisant q x d q x pour d l on considere spec k q l ensemble des avec multiplicites telles que u qu u a une solution satisfaisant u x d u x exp 2i k d d l

on the stability of dense point spectrum for self adjoint operators let a be a random self adjoint operator with fixed orthonormal eigenvectors but with independently distributed random eigenvalues typically for the eigenvalue distributions a is considered to have a dense point spectrum almost surely a s a class of perturbations b is exhibited such that a b has only point spectrum a s examples are also constructed including a rank one perturbation b such that a b has no eigenvalues for 0 a s despite a having dense point spectrum a s

on a decomposition lemma for positive semi definite block matrices abstract this short note in part of expository nature points out several new or recent consequences of a quite nice decomposition for positive semi definite matrices

the averaging method and ergodic theory for pseudo differential operators on compact hyperbolic surfaces let x g be a riemannian manifold with laplacian and let a be a bounded do on l2 x the average a of a is defined as m e it aeit where mf limt 12t ttf t dt weak topology formally the principal symbol of a is m a gt where gt is the geodesic flow when gt is periodic or with some modifications completely integrable a differs from m op a gt op m a gt by a compact operator we take up the opposite extreme where gt is ergodic when x g is a compact hyperbolic surface we may use representation theory of psl2 r to directly compare these averages we find that m op a gt op m a gt op a ais the space average a constant however auk uk m op a gt uk uk k where uk is an onb of laplace eigenfunctions the latter is known to be a at least along a generic sequence of uk s

some inequalities of majorization type abstract we show some majorization inequalities and apply them to derive norm eigenvalue singular value and trace inequalities of matrices we also present a generalization and a different proof of a recent result on majorization concerning partitioned hermitian positive semidefinite matrices

more extensions of a determinant inequality of hartfiel abstract hartfiel s determinant inequality first proved for two positive definite matrices has been recently extended to the case of sector matrices by lin hou and dong this paper contributes an improvement of lin s result with a simple argument we also present a complement of hou and dong s result

positive lyapunov exponents for schrodinger operators with quasi periodic potentials we present a new simple way to estimate the rate of exponential growth lyapunov exponent of solutions of the finite difference schrodinger equation h e psi n mathop limits def psi n 1 psi n 1 lambda f alpha n theta psi n heref is a non constant real analytic function of period 1 and is irrational for large we prove that the lyapunov exponent is positive for every energye in the spectrum ofh and a e in particular the absolutely continuous spectrum ofh is empty in the continuum we study the quasi periodic operator onl2 r h frac d 2 dx 2 k 2 cos x cos alpha x theta for largek and show that for wide intervals of low energies the lyapunov exponent is positive the main idea which originated from m herman s subharmonic argument 11 is to deform the phase to the complex plane this enables us to avoid small denominator problems by moving them off the axis making estimates much easier to perform we recover the information for real using an elementary extension of jensen s formula subharmonicity

band invariants for perturbations of the harmonic oscillator abstract we study the direct and inverse spectral problems for semiclassical operators of the form s s 0 2 v where s 0 1 2 2 r n x 2 is the harmonic oscillator and v r n r is a tempered smooth function we show that the spectrum of s forms eigenvalue clusters as tends to zero and compute the first two associated band invariants we derive several inverse spectral results for v under various assumptions in particular we prove that in two dimensions generic analytic potentials that are even with respect to each variable are spectrally determined up to a rotation

on sectorial matrices abstract a square complex matrix a is said to be sectorial if the values of the corresponding quadratic form x ax belong to the sector z arg z of the complex plane with 0 2 we prove criterions for square matrix to be sectorial the case of sectorial block matrix is considered and the corresponding criterions by means of schur complements and quadratic forms are established we also consider square matrices of the class c connected with sectorial matrices via the cayley transform

pure point spectrum for discrete almost periodic schrodinger operators the finite difference schrodinger operator on m is considered hu j left sum limits v 1 m d v 2 right u j frac 1 varepsilon q j u j u in ell 2 mathbb z m where sum limits v 1 m d v 2 is the difference laplacian inm dimensions for sufficiently small almost periodic potentialsq j are constructed such that the operatorh has only pure point spectrum the method is an inverse spectral procedure which is a modification of the kolmogorov arnol d moser technique

band asymptotics in two dimensions let x be a rank one symmetric space e g s rp cp and let a be its laplace beltrami operator it is known that the spectrum of a is discrete and consists of a sequence of eigenvalues a k k a k 0 1 2 a 1 1 each occurring with multiplicity n b n l k o k 1 2 n being the dimension of x and a and b being positive integers which depend only on x let q be a zeroth order self adjoint pseudodifferential operator if we perturb a by adding q to it the eigenvalues 1 1 break up into bands

a fully discrete scheme for diffusive dispersive conservation laws we introduce a fully discrete in both space and time scheme for the numerical approximation of diffusive dispersive hyperbolic conservation laws in one space dimension this scheme extends an approach by lefloch and rohde 4 it satisfies a cell entropy inequality and as a consequence the space integral of the entropy is a decreasing function of time this is an important stability property shared by the continuous model as well following hayes and lefloch 2 we show that the limiting solutions generated by the scheme need not coincide with the classical oleinik kruzkov entropy solutions but contain nonclassical undercompressive shock waves investigating the properties of the scheme we stress various similarities and differences between the continuous model and the discrete scheme dynamics of nonclassical shocks nucleation etc

on isospectral periodic potentials on a discrete lattice i on considere une version discretisee et generalisee de l equation de schrodinger et on etudie le probleme aux valeurs propres

more results on singular value inequalities of matrices abstract the arithmetic geometric mean inequality for singular values due to bhatia and kittaneh says that 2 s j ab s j a a b b j 1 2 for any matrices a b we give a new equivalent form and some relevant generalizations of this inequality in particular we show that s j a 1 4 b 3 4 a 3 4 b 1 4 s j a b j 1 n for any n n positive semidefinite matrices a b which proves a special case of zhan s conjecture posed in 2000

fischer determinantal inequalities and highamʼs conjecture abstract we obtain a fischer type determinantal inequality for matrices with given angular numerical range we discuss the growth factor for gaussian elimination for linear systems in which the coefficient matrix has this form and give a proof of higham s conjecture

coupling of general lagrangian systems this work is devoted to the coupling of two fluid models such a s two euler systems in lagrangian coordinates at a fixed interface we define coupling conditi ons which can be expressed in terms of continuity of some well chosen variables and then solve the coupled riemann problem in the present setting where the interface is characteristic a particula r choice of dependent variables which are transmitted ensures the overall conservativity

sharp polynomial bounds on the number of scattering poles of radial potentials abstract it is shown that for scattering by a radially symmetric potential in r n n odd the number of poles in a disk of radius r satisfies an estimate n r c n r 1 n this bound is sharp as shown by the special case of potentials nonvanishing at the boundary where n r k n a n r n 1 o 1 a being the diameter of the support

on the phases of a complex matrix abstract in this paper we define the phases of a complex sectorial matrix to be its canonical angles which are uniquely determined from a sectorial decomposition of the matrix properties of matrix phases are studied including those of compressions schur complements matrix products and sums in particular by exploiting a notion known as the compound numerical range we establish a majorization relation between the phases of the eigenvalues of ab and the phases of a and b this is then applied to investigate the rank of i a b with phase information of a and b which plays a crucial role in feedback stability analysis a pair of problems banded sectorial matrix completion and decomposition is studied the phases of the kronecker and hadamard products are also discussed

isophasal isopolar and isospectral schrodinger operators and elementary complex analysis we explicitly construct super exponentially decaying potentials on d so that the associated schrodinger operators are isophasal and isopolar we use similar techniques to construct isospectral schrodinger operators with periodic potentials on and isospectral schrodinger operators on

spectral problems of a class of non self adjoint one dimensional schrodinger operators abstract in this paper we investigate the one dimensional schrodinger operator l q with complex valued periodic potential q when q l 1 0 1 and q n 0 for n 0 1 2 where q n are the fourier coefficients of q with respect to the system e i 2 n x we prove that the bloch eigenvalues are 2 n t 2 for n z t c and find explicit formulas for the bloch functions then we consider the inverse problem for this operator

inequalities and equalities for the cartesian decomposition of complex matrices let a and b be hermitian matrices and let c a ib inequalities and equalities for the eigenvalues and singular values of the matrices a b and c are discussed known results on inequalities are surveyed new results on equality cases are proved and open problems are mentioned

a matrix subadditivity inequality for f a b and f a f b abstract in 1999 ando and zhan proved a subadditivity inequality for operator concave functions we extend it to all concave functions given positive semidefinite matrices a b and a non negative concave function f on 0 f a b f a f b for all symmetric norms in particular for all schatten p norms the case f t t is connected to some block matrix inequalities for instance the operator norm inequality a x x b max x b x for any partitioned hermitian matrix

the numerical viscosity of entropy stable schemes for systems of conservation laws i discrete approximations to hyperbolic systems of conservation laws are studied the amount of numerical viscosity present in such schemes is quantified and related to their entropy stability by means of comparison to this end conservative schemes which are also entropy conservative are constructed these entropy conservative schemes enjoy second order accuracy moreover they can be interpreted as piecewise linear finite element methods and hence can be formulated on various mesh configurations it is then shown that conservative schemes are entropy stable if and for three point schemes only they contain more viscosity than that present in the above mentioned entropy conservative ones

high order schemes entropy inequalities and nonclassical shocks we are concerned with the approximation of undercompressive regularization sensitive nonclassical solutions of hyperbolic systems of conservation laws by high order accurate conservative and semidiscrete finite difference schemes nonclassical shock waves can be generated by diffusive and dispersive terms kept in balance particular attention is given here to a class of systems of conservation laws including the scalar equations and the system of nonlinear elasticity and to linear diffusion and dispersion in either the conservative or the entropy variables first we investigate the existence and the properties of entropy conservative schemes a notion due to tadmor math comp 49 1987 pp 91 103 in particular we exhibit a new five point scheme which is third order accurate at least second we study a class of entropy stable and high order accurate schemes satisfying a single cell entropy inequality they are built from any high order entropy conservative scheme by adding to it a mesh independent numerical viscosity which preserves the order of accuracy of the base scheme these schemes can only converge to solutions of the system of conservation laws satisfying the entropy inequality these entropy stable schemes exhibit mild oscillations near shocks and interestingly may converge to classical or nonclassical entropy solutions depending on the sign of their dispersion coefficient then based on a third order entropy conservative scheme we propose a general scheme for the numerical computation of nonclassical shocks importantly our scheme satisfies a cell entropy inequality following hayes and lefloch siam j numer anal 35 1998 pp 2169 2194 we determine numerically the kinetic function which uniquely characterizes the dynamics of nonclassical shocks for each regularization of the conservation laws our results compare favorably with previous analytical and numerical results finally we prove that there exists no fully discrete and entropy conservative scheme and we investigate the entropy stability of a class of fully discrete lax wendroff type schemes

examples of discrete schrodinger operators with pure point spectrum we present a general approach for constructing potentials for the discrete schrodinger equation of arbitrary dimension having only pure point spectrum we give examples of limit periodic potentials of that kind such that the pure point spectrum is dense in an interval or a cantor set of measure zero

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

on some inequalities for unitarily invariant norms and singular values abstract in this note we improve and generalize some existing inequalities for unitarily invariant norms and singular values and present some other inequalities for unitarily invariant norms

determinantal and eigenvalue inequalities for matrices with numerical ranges in a sector abstract let a a 11 a 12 a 21 a 22 m n where a 11 m m with m n 2 be such that the numerical range of a lies in the set e i z c z z tan for some 0 2 and 0 2 we obtain the optimal containment region for the generalized eigenvalue satisfying a 11 0 0 a 22 x 0 a 12 a 21 0 x for some nonzero x c n and the optimal eigenvalue containment region of the matrix i m a 11 1 a 12 a 22 1 a 21 in case a 11 and a 22 are invertible from this result one can show det a sec 2 m det a 11 det a 22 in particular if a is an accretive dissipative matrix then det a 2 m det a 11 det a 22 these affirm some conjectures of drury and lin

complex hill s equation and the complex periodic korteweg de vries equations on etudie la classe spectrale des operateurs differentiels l d 2 dx 2 q avec un potentiel q de periode un on traite des potentiels complexes et des operateurs auto adjoints

numerical approximation of a macroscopic model of pedestrian flows this paper is concerned with the numerical approximation of the solutions of a macroscopic model for the description of the flow of pedestrians solutions of the associated riemann problem are known to be possibly nonclassical in the sense that the underlying discontinuities may well violate the lax inequalities which makes their numerical approximation very sensitive this study proposes to extend the transport equilibrium strategy proposed in c chalons c r acad sci paris ser i 342 2006 pp 623 626 and c chalons transport equilibrium schemes for computing nonclassical shocks i scalar conservation laws submitted for computing the nonclassical solutions of scalar conservation laws with either a concave convex or a convex concave flux function and supplemented with an invertible kinetic function these strong properties are not fulfilled in the present setting since the flow function admits several inflection points and the kinetic function is not invertible we nevertheless succeed in

bernstein inequalities and applications to analytic geometry and differential equations we introduce new methods of complex analysis inequalities of bernstein type to study projections of analytic sets these techniques are applied to problems of bifurcations of periodic orbits of differential equations such as the local hilbert s 16th problem

almost periodic jacobi matrices associated with julia sets for polynomials let be the jacobi matrix associated with polynomialt z of degreen 2 the spectrum of is the julia set associated witht z which in many cases is a cantor set let 1 denote the result of omitting the first row and column ofj then it is shown that the spectrum of 1 may be purely discrete

franco american meeting on the mathematics of random and almost periodic potentials we present the abstracts of papers given at a conference held at the ecole polytechnique in june 1983 on the mathematics of random and almost periodic potentials

factorizing complex symmetric matrices with positive definite real and imaginary parts complex symmetric matrices whose real and imaginary parts are positive definite are shown to have a growth factor bounded by 2 for lu factorization this result adds to the classes of matrix for which it is known to be safe not to pivot in lu factorization block mathrm ldl t factorization with the pivoting strategy of bunch and kaufman is also considered and it is shown that for such matrices only 1 times 1 pivots are used and the same growth factor bound of 2 holds but that interchanges that destroy band structure may be made the latter results hold whether the pivoting strategy uses the usual absolute value or the modification employed in linpack and lapack

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

spectral statistics on zoll surfaces the subject of this article is a family of partly conjectural spectral invariants of a laplace operator with a discrete spectrum j known as its k level correlation functions k and level spacings distribution the 2 level or pair correlation function 2 for instance measures the asymptotic number of spacings i j between normalized eigenvalues i as the eigenvalues tend to infinity the normalization is so that the mean level spacing i 1 i is one these invariants have been studied by many physicists including wigner dyson and berry and are conjectured to have certain universal forms depending on the qualitative behaviour of the corresponding classical geodesic flow

the singular values of a b and a ib we obtain several inequalities relating the singular values of a b and those of a ib when a and b are hermitian and when one or both of them are positive semidefinite

band asymptotics with non smooth potentials let s denote the n dimensional sphere with its canonical riemannian metric and a its laplace beltrami operator if q is a real valued function on s we can consider the self adjoint operator s a q for the case when q is smooth weinstein proved in w the following very beautiful theorem on the asymptotic behavior of the spectrum of s let dk k k n 1 be the kth eigenvalue of a and let d be its multiplicity by the minimax principle the eigenvalues of s can be written in the form ak p i l dk where ipclk il is bounded by a constant independent of k and i let 0 denote the space of oriented closed geodesics on s and let dp be the unique probability measure on 0 which is rotationally invariant by integrating q along closed geodesics we obtain a smooth function s on 0 which we call the x ray transform of q then weinstein s theorem is

spectral behavior of quasi periodic potentials spectral properties of schrodinger operators of the typeh v where is the laplacian v a quasiperiodic potential and a coupling constant are developed v is taken to be finite sum of exponentials with generic frequencies for small a strong stability is shown on the other hand examples in the finite diffeence case are given for which a transition in the type of spectrum occurs as is increased

schrodinger operators with complex valued potentials and no resonances in dimension d 3 we give examples of nontrivial compactly supported complex valued potentials such that the associated schrodinger operators have neither resonances nor eigenvalues if d 2 we show that there are potentials with no resonances or eigenvalues away from the origin these schrodinger operators are isophasal and have the same scattering phase as the laplacian on rd in odd dimensions d 3 we study the fundamental solution of the wave equation perturbed by such a potential if the space variables are held fixed it is superexponentially decaying in time

a schauder and riesz basis criterion for non self adjoint schrodinger operators with periodic and antiperiodic boundary conditions under the assumption that v l2 0 dx we derive necessary and sufficient conditions in terms of spectral data for non self adjoint schrodinger operators d2 dx2 v in l2 0 dx with periodic and antiperiodic boundary conditions to possess a riesz basis of root vectors i e eigenvectors and generalized eigenvectors spanning the range of the riesz projection associated with the corresponding periodic and antiperiodic eigenvalues we also discuss the case of a schauder basis for periodic and antiperiodic schrodinger operators d2 dx2 v in lp 0 dx p 1

inverse spectral results on two dimensional tori is much more complicated than the corresponding problem on surfaces of genus 2 on one hand for certain classes of potentials one can reconstruct v from the spectrum of 1 1 on the other hand there are potentials that have lots of isospectral deformations in their analysis of 1 1 an important element is the decomposition of the fourier series of v into primitive summands and we begin here by saying a few words about this decomposition since it will also be important in the results to be described an element of a lattice is said to be primitive if it is not a positive integer multiple of another element of the lattice now let a e27i w x ct e r

fischer type determinantal inequalities for accretive dissipative matrices abstract let a a 11 a 12 a 21 a 22 be an n n accretive dissipative matrix k and l be the orders of a 11 and a 22 respectively and let m min k l then det a a det a 11 det a 22 where a 2 3 m 2 if m n 3 2 n 2 if n 3 m n 2 this improves a result of ikramov

nonclassical shocks and kinetic relations finite difference schemes we consider hyperbolic systems of conservation laws that are not genuinely nonlinear the solutions generated by diffusive dispersive regularizations may include nonclassical n c shock waves that do not satisfy the classical liu entropy criterion we investigate the numerical approximation of n c shocks via conservative difference schemes constrained only by a single entropy inequality the schemes are designed by comparing their equivalent equations with the continuous model and include discretizations of the diffusive and dispersive terms limits of these schemes are characterized via the kinetic relation introduced earlier by the authors we determine the kinetic function numerically for several examples of systems and schemes this study demonstrates that the kinetic relation is a suitable tool for the selection of unique n c solutions and for the study of their sensitive dependence on the critical parameters the ratios of diffusion dispersion and diffusion mesh size the shock strength and the order of discretization of the flux

diophantine tori and weyl laws for non selfadjoint operators in dimension two we study the distribution of eigenvalues for non selfadjoint perturbations of selfadjoint semiclassical analytic pseudodifferential operators in dimension two assuming that the classical flow of the unperturbed part is completely integrable an asymptotic formula of weyl type for the number of eigenvalues in a spectral band bounded from above and from below by levels corresponding to diophantine invariant lagrangian tori is established the weyl law is given in terms of the long time averages of the leading non selfadjoint perturbation along the classical flow of the unperturbed part

anderson localization for time quasi periodic random schrodinger and wave equations we prove that at large disorder with large probability and for a corresponding set of diophantine frequencies of large measure anderson localization in d is stable under localized time quasi periodic perturbations by proving that the associated quasi energy operator has pure point spectrum the main tools are the frohlich spencer mechanism for the random component and the bourgain goldstein schlag mechanism for the quasi periodic component this paper paves the way for the construction of time quasi periodic kam type of solutions of non linear random schrodinger equations in bw

mean matrices and infinite divisibility abstract we consider matrices m with entries mij m i j where 1 n are positive numbers and m is a binary mean dominated by the geometric mean and matrices w with entries wij 1 m i j where m is a binary mean that dominates the geometric mean we show that these matrices are infinitely divisible for several much studied classes of means

norm inequalities for cartesian decompositions abstract let the cartesian decomposition of a complex n n matrix t be t a ib with a b hermitian let j and j be the eigenvalues of a and b respectively ordered so that 1 n and 1 n we prove that diag 1 i 1 n i n 2 t for every unitarily invariant norm this settles affirmatively a conjecture of ando and bhatia t ando r bhatia eigenvalue inequalities associated with the cartesian decomposition linear and multilinear algebra 22 1987 133

optimal measures for toeplitz matrices abstract we extend and refine a result of r mceachin concerning 2 2 toeplitz matrices even in the complex case the schur norm of such a matrix is equal to the total variation of an appropriately chosen representing measure the measure may be supported on two or three points from this phenomenon we derive formulas for the schur norms of the matrices we point out a connection with the extreme points of the unit ball in a certain function space

an overview of periodic elliptic operators the article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas close attention is paid to studying analytic properties of bloch and fermi varieties which influence significantly most properties of such operators the approaches described are applicable not only to the standard model example of schrodinger operator with periodic electric potential v x but to a wide variety of elliptic periodic equations and systems equations on graphs operator and other operators on abelian coverings of compact bases many important applications are mentioned however due to the size restrictions they are not dealt with in details

determinants of laplacians and isopolar metrics on surfaces of infinite area we construct a determinant of the laplacian for infinite area surfaces which are hyperbolic near infinity and without cusps in the case of a convex co compact hyperbolic metric the determinant can be related to the selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the laplacian in the hyperbolic near infinity case the determinant is analyzed through the zeta regularized relative determinant for a conformal metric perturbation we establish that this relative determinant is a ratio of entire functions of order two with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics these results are applied to the problem of compactness in the smooth topology for the class of metrics with a given set of eigenvalues and resonances

rigid motions action angles relative cohomology and polynomials with roots on the unit circle revisiting canonical integration of the classical solid near a hyperbolic or elliptic uniform rotation normal canonical coordinates p q are constructed so that the hamiltonian becomes a function normal form of x pq or of x p2 q2 the two cases are treated simultaneously distinguishing them respectively by a label a in terms of various power series with coefficients which are shown to be polynomials in a variable ra2 depending on the inertia moments the normal forms are derived via the analysis of a relative cohomology problem and shown to be obtainable without reference to the construction of the normal coordinates via elliptic integrals unlike the derivation of the normal coordinates p q results and conjectures also emerge about the properties of the above polynomials and the location of their roots in particular a class of polynomials with all roots on the unit circle arises

norm inequalities for accretive dissipative operator matrices abstract an operator on a hilbert space is accretive dissipative if its real and imaginary parts are both positive in this paper we consider norm inequalities for an accretive dissipative operator matrix the inequalities obtained extend some known results

lieb functions and sectorial matrices abstract in this article we present a special treatment of sectorial matrices under lieb functions this treatment results from a new characterization of sectorial matrices using blocks which then implies a new set of inequalities governing such matrices where singular values determinants numerical radius and unitarily invariant norms inequalities are retrieved in a general form that implies many well known results in the literature

a metal insulator transition for the almost mathieu model we study the spectrum of the almost mathieu hamiltonian left h x psi right left n right psi left n 1 right psi left n 1 right 2 mu cos left x n theta right psi left n right n in mathbb z where is an irrational number andx is in the circle mathbb t for a small enough coupling constant and anyx there is a closed energy set of non zero measure in the absolutely continuous spectrum ofh for sufficiently high and almost allx we prove the existence of a set of eigenvalues whose closure has positive measure the two results are obtained for a subset of irrational numbers of full lebesgue measure

norm inequalities for sector block matrices abstract we study unitarily invariant norm inequalities for sector block matrices and obtain new inequalities which extend some recent results of jabbarzadeh and kaleibary 2020 and generalize a norm inequality due to lin and zhou 2013 at the same time we refine an inequality raised by zhang 2015

extensions of hartfiel s inequality to multiple matrices abstract hartfiel s determinant inequality which refines haynsworth s determinant inequality is a remarkable refinement of the fundamental determinant inequality det a b det a det b where a b are positive definite matrices in this paper we first extend hartfiel s determinant inequality to multiple positive definite matrices and then we further extend the result to a larger class of matrices namely matrices whose numerical ranges are contained in a sector

cartesian decompositions and schatten norms abstract let t be a hilbert space operator with t a i b where a and b are hermitian we prove sharp inequalities comparing the norms t p with a 2 b 2 1 2 p and a 2 p b 2 p 1 2

semiclassical approximation for schrodinger operators on a two sphere at high energy let h s v be a schrodinger operator acting in l2 s with s the two dimensional unit sphere s the spherical laplacian and v a smooth potential approximate eigenfunctions and eigenvalues for h are obtained involving expansions in inverse powers of the classical angular momentum variables provided that these variables are in a region of phase space where the corresponding classical hamiltonian is nearly integrable the analysis is carried out in a bargmann representation where s becomes a quadratic expression in the sum of two quantum harmonic oscillator hamiltonians and v becomes a pseudodifferential operator

spectral properties of a certain class of complex potentials in this paper we discuss spectral properties of the schroedinger operator a q on compact homogeneous spaces for certain complex valued potentials q we show for instance that for these potentials the spectrum of a q is identical with the spectrum of a introduction let g be a compact lie group and a the bi invariant laplacebeltrami operator on g fegan observed in f that there exist functions q on g for which the schroedinger operator q possesses an infinite sequence of eigenvalues al 2 such that each a is also an eigenvalue of a in this paper we will show that a potentials with this property exist not just on compact lie groups but on arbitrary compact homogeneous spaces as well and b for such potentials the entire spectrum of a q is identical with the spectrum of a in fegan s examples as well as in ours the potentials q are complex valued this means that the operator a q is not selfadjoint fortunately however there exists a rather nice spectral theory for such operators for instance their generalized eigenspaces are finite dimensional and span l2 see 2 below in view of this we were surprised to find relatively little material in the literature on such operators a notable exception is peter sarnack s beautiful paper on schroedinger operators with quasiperiodic q s see s we will describe and sketch the proof of our main result let g be a compact connected lie group h a closed subgroup and x the homogeneous space g h for simplicity we will assume for the moment that g is semisimple though this assumption is not essential see 4 for the case g s let g be the unitary dual of g we can order g lexicographically by choosing a system of positive weights consider the hilbert space direct sum 1 1 l2 x ha a eg ha being the g invariant subspace of l2 x on which g acts according to the representation a each irreducible subspace of ha possesses a unique maximal weight vector let h be the subspace of ha spanned by these vectors and let 1 2 l2 x ha received by the editors august 18 1982 and in revised form november 19 1982 1980 mathematics subject classification primary 58g25 secondary 35p99 research supported by the national university of mexico c1983 american mathematical society 0002 9947 83 1 00 25 per page

localization in v dimensional incommensurate structures we exhibit a class of quasi periodic unbounded potential in thev dimensional discrete schrodinger equation for which the spectrum is only pure point with exponentially localized states and a dense set of eigenvalues in

extension of matic s results matic 2014 6 proved two inequalities regarding the ratio det a d det a where a is positive definite and d is a positive definite block diagonal matrix in this paper we extend these results to a larger class of matrices namely matrices whose numerical ranges are contained in a sector

norm inequalities involving accretive dissipative 2 2 block matrices abstract let t 11 t 12 t 21 and t 22 be n n complex matrices and let t t 11 t 12 t 21 t 22 be accretive dissipative it is shown that if f is an increasing convex function on 0 such that f 0 0 then f t 12 2 f t 21 2 f t 2 for every unitarily invariant norm moreover if f is an increasing concave function on 0 such that f 0 0 then f t 12 2 f t 21 2 4 f t 2 4 for every unitarily invariant norm among other inequalities for the schatten p norms it is shown that t 12 p p t 21 p p 2 p 1 t 11 p p 2 t 22 p p 2 for p 2

inequalities between f a b and f a f b abstract the conjecture posed by aujla and silva j s aujla f c silva weak majorization inequalities and convex functions linear algebra appl 369 2003 217 233 is proved it is shown that for any m tuple of positive semidefinite n n complex matrices a j and for any non negative convex function f on 0 with f 0 0 the inequality f a 1 f a 2 f a m f a 1 a 2 a m holds for any unitarily invariant norm it is also proved that f a 1 f a 2 f a m f a 1 a 2 a m where f is a non negative concave function on 0 and is normalized

spectra of non selfadjoint hill s operators and a class of riemann surfaces i am grateful to a boutet de monvel for the invitation to laboratory of mathematical physics and geometry university paris 7 where the initial version of this paper was written it gives me a special pleasure to thank h mckean for his permanent interest and support of my study of nonselfadjoint periodic operators i am thankful to the referees of this paper whose criticism helped me to improve the exposition the final version of the paper was written at the ben gurion university of the negev the financial support of the ministry of science of israel is gratefully appreciated

non classical shocks and kinetic relations scalar conservation laws this paper analyzes the non classical shock waves which arise as limits of certain diffusive dispersive approximations to hyperbolic conservation laws such shocks occur for non convex fluxes and connect regions of different convexity they have negative entropy dissipation for a single convex entropy function but not all convex entropies and do not obey the classical oleinik entropy criterion we derive necessary conditions for the existence of non classical shock waves and construct them as limits of traveling wave solutions for several diffusive dispersive approximations

high order entropy conservative schemes and kinetic relations for van der waals fluids abstract we consider the mixed hyperbolic elliptic system of two conservation laws modeling the dynamics of van der waals fluids viscosity and capillarity effects are taken into account we introduce a new class of semidiscrete high order schemes which are entropy conservative in the sense of tadmor when the viscosity is neglected and otherwise dissipate the associated mathematical entropy our numerical schemes generalize the works by e tadmor 1987 math comput 49 91 and p g lefloch and c rohde 2000 siam j numer anal 37 2023 who proposed second order and third order entropy conservative schemes respectively following b t hayes and p g lefloch 1998 siam j numer anal 35 2169 we demonstrate numerically that balanced viscosity and capillarity terms in van der waals fluids may generate nonclassical shock waves or subsonic propagating phase transitions such waves are undercompressive and do not satisfy standard entropy criteria they must be characterized by a kinetic function which we determine numerically in this paper from vanishing viscosity and capillarity the kinetic relation is an efficient tool to discuss the interplay among the viscosity capillarity and discretization parameters in van der waals fluids

singularities of the complex korteweg de vries flows on etudie le probleme isospectral de l equation de hill complexe etant donne 2g 1 valeurs propres simples dans le plan complexe trouver la classe spectrale correspondante q on considere la caracterisation de la classe isospectrale et ses sections singulieres on construit q a l aide des flots non lineaires de korteweg de vries dans c 1 les fonctions regulieres de periode 1

asymptotics for a resonance counting function for potential scattering on cylinders abstract we study certain resonance counting functions for potential scattering on infinite cylinders or half cylinders under certain conditions on the potential we obtain asymptotics of the counting functions with an explicit formula for the constant appearing in the leading term

weak majorization inequalities and convex functions abstract let f be a convex function defined on an interval i 0 1 and a b n n complex hermitian matrices with spectrum in i we prove that the eigenvalues of f a 1 b are weakly majorized by the eigenvalues of f a 1 f b further if f is log convex we prove that the eigenvalues of f a 1 b are weakly majorized by the eigenvalues of f a f b 1 as applications we obtain generalizations of the famous golden thomson trace inequality a representation theorem and a harmonic geometric mean inequality some related inequalities are discussed

circular symmetry and the trace formula let m be a compact n dimensional manifold without boundary and let p be a positive first order elliptic self adjoint classical pseudodifferential operator on lz m the hilbert space of half densities on m the spectrum of p is discrete and accumulates only at we denote the eigenvalues with multiplicities by 2o 21 2 recall that the singularities of the inverse fourier transform of the spectral measure