graded c algebras c algebras graded by a semilattice appeared for the first time in the study of the quantum n body problem we show that these algebras are completely described by their homogeneous subalgebras and that they are invariant under several operations of c algebras such as tensor products we give the k theory of graded c algebras and we study some of their properties from the c algebra point of view

on the spectral analysis of quantum field hamiltonians abstract we define c algebras on a fock space such that the hamiltonians of quantum field models with positive mass are affiliated to them we describe the quotient of such algebras with respect to the ideal of compact operators and deduce consequences in the spectral theory of these hamiltonians we compute their essential spectrum and give a systematic procedure for proving the mourre estimate

critical properties of electron eigenstates in incommensurate systems this paper describes some properties of the eigenvalue equation n 1 n 1 2 cos 2 n n e n this is an example of the more general problem of a hermitian eigenvalue equation in the form of a difference equation with periodic coefficients these equations arise in solid state physics they occur in connection with tight binding models for electrons in one dimensional solids with an incommensurate modulation of the structure and in models for the energy bands of bloch electrons moving in a plane with a perpendicular magnetic field the model studied has a critical point when 1 following some earlier work by azbel azbel m ya phys rev lett 43 1954 1979 an approximate renormalization group transformation is derived this predicts that the spectrum and eigenstates have a remarkable recursive structure at the critical point which is dependent on the expansion of as a continued fraction also when is an irrational number there is a localization transition from extended states to localized states as increases through the critical point this localization transition which was previously discovered by aubry andre aubry s andre g ann israel phys soc 3 133 1979 using the thouless formula for the localization length is explained by the renormalization group transformation derived here

hamiltonians on discrete structures jumps of the integrated density of states and uniform convergence we study equivariant families of discrete hamiltonians on amenable geometries and their integrated density of states ids we prove that the eigenspace of a fixed energy is spanned by eigenfunctions with compact support the size of a jump of the ids is consequently given by the equivariant dimension of the subspace spanned by such eigenfunctions from this we deduce uniform convergence w r t the spectral parameter of the finite volume approximants of the ids our framework includes quasiperiodic operators on delone sets periodic and random operators on quasi transitive graphs and operators on percolation graphs

graded hilbert c modules the notion of hilbert c modules graded by a semilattice was first introduced in the study of many body systems we show the relationship between these hilbert c modules and their homogeneous subspaces we study the stability of graded hilbert c modules under tensor products and crossed products

spectral gaps for periodic schrodinger operators with strong magnetic fields we consider schrodinger operators h h ih d a ih d a with the periodic magnetic field b da on covering spaces of compact manifolds using methods of a paper by kordyukov mathai and shubin 14 we prove that under some assumptions on b there are in arbitrarily large number of gaps in the spectrum of these operators in the semiclassical limit of the strong magnetic field h 0

isometries fock spaces and spectral analysis of schrodinger operators on trees abstract we construct conjugate operators for the real part of a completely non unitary isometry and we give applications to the spectral and scattering theory of a class of operators on complete fock spaces natural generalizations of the schrodinger operators on trees we consider c algebras generated by such hamiltonians with certain types of anisotropy at infinity we compute their quotient with respect to the ideal of compact operators and give formulas for the essential spectrum of these hamiltonians

a characteristic finite element algorithm for time dependent 3 d advection dominated transport using unstructured grids abstract an algorithm based on operator splitting has been successfully implemented for solving unsteady advection dominated transport problems in 3 d specifically the general operator integration factor splitting method of maday et al is applied to the unsteady advection diffusion equation with source sink terms the algorithm incorporates a 3 d characteristic galerkin scheme to treat advection and a standard galerkin treatment of the diffusion and source sink terms up to third order operator splitting was implemented and validated against several analytical solutions the algorithm showed the expected error behaviour and good performance in modeling advection dominated transport problems the practical utility and effectiveness of the proposed numerical scheme was further demonstrated by solving the graetz nusselt problem i e high peclet number mass heat transport in a fully developed pipe flow

cantor and band spectra for periodic quantum graphs with magnetic fields we provide an exhaustive spectral analysis of the two dimensional periodic square graph lattice with a magnetic field we show that the spectrum consists of the dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant lyapunov function of a suitable kronig penney hamiltonian in particular between any two dirichlet eigenvalues the spectrum is a cantor set for an irrational flux and is absolutely continuous and has a band structure for a rational flux the dirichlet eigenvalues can be isolated or embedded subject to the choice of parameters conditions for both possibilities are given we show that generically there are infinitely many gaps in the spectrum and the bethe sommerfeld conjecture fails in this case

on 2d discrete schrodinger operators associated with multiple orthogonal polynomials a class of cross shaped difference operators on a two dimensional 2d lattice is introduced the main feature of the operators in this class is that their formal eigenvectors consist of multiple orthogonal polynomials in other words this scheme generalizes the classical connection between jacobi matrices and orthogonal polynomials to the case of operators on lattices furthermore we also show how to obtain 2d discrete schrodinger operators out of this construction and give a number of explicit examples based on known families of multiple orthogonal polynomials

the coexistence problem for the discrete mathieu operator we solve the coexistence problem for the periodic discrete mathieu operator in all parametric cases the main tool in the proof will be bezout s theorem for projective plane curves as an additional result we obtain the gap opening and gap growth powers for this operator

purely absolutely continuous spectrum for almost mathieu operators using a recent result of sinai we prove that the almost mathieu operators acting onl2 ly n l 1 l cos n n have a purely absolutely continuous spectrum for almost all a provided that is a good irrational and is sufficiently small furthermore the generalized eigen functions are quasiperiodic

natural boundaries and spectral theory we present and exploit an analogy between lack of absolutely continuous spectrum for schrodinger operators and nat ural boundaries for power series among our new results are gen eralizations of hecke s example and natural boundary examples for random power series where independence is not assumed

quantum graphs ii some spectral properties of quantum and combinatorial graphs the paper deals with some spectral properties of mostly infinite quantum and combinatorial graphs quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics nanotechnology optics and other areas a schnol type theorem is proven that allows one to detect that a point belongs to the spectrum when a generalized eigenfunction with an sub exponential growth integral estimate is available a theorem on spectral gap opening for decorated quantum graphs is established its analogue is known for the combinatorial case it is also shown that if a periodic combinatorial or quantum graph has a point spectrum it is generated by compactly supported eigenfunctions scars

approximating spectral invariants of harper operators on graphs abstract we study harper operators and the closely related discrete magnetic laplacians dml on a graph with a free action of a discrete group as defined by sunada sun a main result in this paper is that the spectral density function of dmls associated to rational weight functions on graphs with a free action of an amenable discrete group can be approximated by the average spectral density function of the dmls on a regular exhaustion with either dirichlet or neumann boundary conditions this then gives a criterion for the existence of gaps in the spectrum of the dml as well as other interesting spectral properties of such dmls the technique used incorporates some results of algebraic number theory

toeplitz algebras and spectral results for the one dimensional heisenberg model we determine the structure of the spectrum and obtain nonpropagation estimates for a class of toeplitz operators acting on a subset of the lattice zn this class contains the hamiltonian of the one dimensional heisenberg model

spectral and scattering theory of spatially cut off p ϕ 2 hamiltonians we study spatially cut off p 2 hamiltonians we show the local finiteness of the pure point spectrum outside of thresholds the limiting absorption principle and asymptotic completeness of scattering for such hamiltonians our results imply the absence of singular continuous spectrum

an algorithmic regularity lemma for hypergraphs in this paper we will consider the problem of designing an efficient algorithm that finds an epsilon regular partition of an l uniform hypergraph

essential spectra of difference operators on periodic graphs let x be a discrete metric space we suppose that the group acts freely on x and that the number of orbits of x with respect to this action is finite then we call x a periodic discrete metric space we examine the fredholm property and essential spectra of band dominated operators on lp x when 1 p our approach is based on the theory of band dominated operators on and their limit operators in the case where x is the set of vertices of a combinatorial graph the graph structure defines a schrodinger operator on lp x in a natural way we illustrate our approach by determining the essential spectra of schrodinger operators with slowly oscillating potential both on zig zag and on hexagonal graphs the latter being related to nano structures

1d quasiperiodic operators latent symmetries a large class of discrete quasiperiodic operators is shown to be decomposed into orbits ofsl 2 z action with equal densities of states moreover under some natural assumptions all nontrivial representatives of the mentioned action transform operators with pure point spectrum into those with absolutely continuous spectrum some applications of these results are presented

effective masses for laplacians on periodic graphs abstract we consider laplacians on both periodic discrete and periodic metric equilateral graphs their spectrum consists of an absolutely continuous part which is a union of non degenerate spectral bands and flat bands i e eigenvalues of infinite multiplicity we estimate effective masses associated with the ends of each spectral band in terms of geometric parameters of the graphs moreover in the case of the bottom of the spectrum we determine two sided estimates on the effective mass in terms of geometric parameters of the graphs the proof is based on floquet theory factorization of fiber operators perturbation theory and the relation between effective masses for laplacians on discrete and metric graphs obtained in our paper

almost mathieu operators and rotation c algebras a concept of eigenstates is discussed for almost mathieu operators relating these operators to the rotation c algebras they are associated with and to the integrated density of states for irrational rotations the eigenstates give rise to a notion of multiplicity for all points in the spectrum of an almost mathieu operator an integer which is either 1 or 2 it is shown how the occurrence of point spectrum relates to the multiplicity of the corresponding eigenvalues

on the continuity of spectra for families of magnetic pseudodifferential operators for families of magnetic pseudodifierential operators deflned by symbols and magnetic flelds depending continuously on a real parameter we show that the corresponding family of spectra also varies continuously with 1

localization for a class of one dimensional quasi periodic schrodinger operators we prove for small and satisfying a certain diophantine condition the operator h varepsilon 2 delta frac 1 2 pi cos 2 pi j alpha theta j in mathbb z has pure point spectrum for almost all a similar result is established at low energy for h frac d 2 dx 2 k 2 cos 2 pi x cos 2 pi alpha x theta providedk is sufficiently large

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 the spectral analysis of many body systems we describe the essential spectrum and prove the mourre estimate for quantum particle systems interacting through k body forces and creation annihilation processes which do not preserve the number of particles for this we compute the hamiltonian algebra of the system i e the c algebra c generated by the hamiltonians we want to study and show that as in the n body case it is graded by a semilattice hilbert c modules graded by semilattices are involved in the construction of c for example if we start with an n body system whose hamiltonian algebra is cn and then we add field type couplings between subsystems then the many body hamiltonian algebra c is the imprimitivity algebra of a graded hilbert cn module

on the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators the article is devoted to the following question consider a periodic self adjoint difference differential operator on a graph quantum graph g with a co compact free action of the integer lattice mathbb z n it is known that a local perturbation of the operator might embed an eigenvalue into the continuous spectrum a feature uncommon for periodic elliptic operators of second order in all known constructions of such examples the corresponding eigenfunction is compactly supported one wonders whether this must always be the case the paper answers this question affirmatively what is more surprising one can estimate that the eigenmode must be localized not far away from the perturbation in a neighborhood of the perturbation s support the width of the neighborhood dependent upon the unperturbed operator only the validity of this result requires the condition of irreducibility of the fermi floquet surface of the periodic operator which is known in some cases and is expected to be satisfied for periodic schrodinger operators

a fast approximation algorithm for computing thefrequencies of subgraphs in a given graph in this paper we give an algorithm which given a labeled graph on n vertices and a list of all labeled graphs on k vertices provides for each graph h of this list an approximation to the number of induced copies of h in g with total error small this algorithm has running time o n 1 over log log n cdot m n where m n is the time needed to square a n by n matrix with 0 1 entries over the integers the main tool in designing this algorithm is a variant of the regularity lemma of szemeredi

anderson localization for the almost mathieu equation a nonperturbative proof we prove that for any diophantine rotation angle and a e phase the almost mathieu operator h n n 1 n 1 cos 2 n n has pure point spectrum with exponentially decaying eigenfunctions for 15 we also prove the existence of some pure point spectrum for any 5 4

absence of localisation in the almost mathieu equation the author considers the discrete schrodinger operator on z with the potential lambda cos 2 pi alpha n theta this one dimensional model occurs in the study of an electron in a two dimensional periodic potential with a uniform magnetic field first it is proved that for every alpha and for lambda 2 this operator has no eigenvalue i e localised state furthermore at lambda 2 the eigenvalues if they exist belong to the set where the lyapunov exponent vanishes and the associated eigenvectors at in l2 z but are not summable

difference equations and pseudo differential operators on zn abstract in this paper we develop the calculus of pseudo differential operators on the lattice z n which we can call pseudo difference operators an interesting feature of this calculus is that the global frequency space t n is compact so the symbol classes are defined in terms of the behaviour with respect to the lattice variable we establish formulae for composition adjoint transpose and for parametrix for the elliptic operators we also give conditions for the l 2 weighted l 2 and l p boundedness of operators and for their compactness on l p we describe a link to the toroidal quantization on the torus t n and apply it to give conditions for the membership in schatten classes on l 2 z n furthermore we discuss a version of fourier integral operators on the lattice and give conditions for their l 2 boundedness the results are applied to give estimates for solutions to difference equations on the lattice z n moreover we establish garding and sharp garding inequalities with an application to the unique solvability of parabolic equations on the lattice z n

unbounded elements affiliated with c algebras and non compact quantum groups the affiliation relation that allows to include unbounded elements operators into thec algebra framework is introduced investigated and applied to the quantum group theory the quantum deformation of the two fold covering of the group of motions of euclidean plane is constructed a remarkable radius quantization is discovered it is also shown that the quantumsu 1 1 group does not exist on thec algebra level for real value of the deformation parameter

total bandwidth for the harper equation iii corrections to scaling earlier work on the spectral measure for the harper equation discrete mathieu equation showed that if there is a large common period p for the lattice and the sinusoidal potential the spectral measure scaled as c p where c is a constant independent of the number of oscillations q of the potential in p lattice spacings the corrections to this scaling law are found for q 1 and q 2 for q 1 it is shown that the corrections to scaling are logarithmic functions of p while for q 2 the leading corrections are reduced by a factor p 2 analytical and numerical support is given for the assertion that this difference between odd and even values of q persists for higher values of q provided they are substantially less than p

a lagrangian method for the shallow water equations based on a voronoi mesh one dimensional results abstract a lagrangian method for the shallow water equations in one dimension is presented the method is based on the following idea place n points on a line to represent the fluid each point represents all the fluid that is closer to this point than to any other this partitions the line into intervals the principle of least action yields a set of difference equations which can be easily solved at each time step to give new values for the height and velocity of the fluid particles in each interval the points are then moved to their new locations and the intervals reconstructed in the event that two points get too close to each other consider the points to have collided the collision is treated as an inelastic collision and the two points are merged into one according to certain conservation laws it is then possible to handle flows with shocks by modelling a shock as a collision between two particles that is a shock is thought of as one fluid particle overtaking another one and colliding with it the results of this method on the dam breaking problem are presented the solution is compared to the exact solution and to a solution by the random choice method

crossed products of c algebras and spectral analysis of quantum hamiltonians we study spectral properties of a hamiltonian by analyzing the structure of certain c algebras to which it is affiliated the main tool we use for the construction of these algebras is the crossed product of abelian c algebras generated by the classical potentials by actions of groups we show how to compute the quotient of such a crossed product with respect to the ideal of compact operators and how to use the resulting information in order to get spectral properties of the hamiltonians this scheme provides a unified approach to the study of hamiltonians of anisotropic and many body systems including quantum fields

mesh duality and legendre duality we describe a sense in which mesh duality is equivalent to legendre duality that is a general pair of meshes which satisfy a definition of duality for meshes are shown to be the projection of a pair of piecewise linear functions that are dual to each other in the sense of a legendre dual transformation in applications the latter functions can be a tangent plane approximation to a smoother function and a chordal plane approximation to its legendre dual convex examples include one from meteorology and also the relation between the delaunay mesh and the voronoi tessellation the latter are shown to be the projections of tangent plane and chordal approximations to the same paraboloid

random schrodinger operators arising from lattice gauge fields i existence and examples we consider new models of ergodic schrodinger operators whose existence relies on a cohomological theorem of feldman and moore in ergodic theory these operators generalize the harper operator which describes the case of a constant magnetic field an example is the case when the magnetic field is given by independent random variables attached to the lattice plaquettes a generalization of the feldman moore theorem by lind to non abelian groups also allows us to consider schrodinger operators obtained from non abelian lattice gauge fields the existence result extends to more general graphs like to operators on tilings and to higher dimensions we compute some moment expansions for the density of states for example for independent identically and uniformly distributed magnetic fields a model which has been studied at least since 1970 and whose existence can also be seen without involving the above mentioned existence theorem we show that the nth moment is the number of closed paths in the two dimensional lattice starting at the origin for which the winding number vanishes at each plaquette point this goes beyond the brinkman rice self retracing path approximation other examples are a higher dimensional example a one dimensional anderson model which can be treated in this framework as well as the hofstadter model with constant magnetic field where one averages over all possible magnetic fields we also reprove a result of jitomirskaya mandelshtam stating that the deterministic aharonov bohm model is a compact perturbation of the free laplacian

cantor spectrum for the almost mathieu equation for a dense g of pairs in r2 we prove that the operator hu n u n 1 u n 1 cos 2 n u n has a nowhere dense spectrum along the way we prove several interesting results about the case pq of which we mention a if q is not an integral multiple of then all gaps are open and b if q is even and is chosen suitably then the middle gap is closed for all

the positivity of the lyapunov exponent and the absence of the absolutely continuous spectrum for the almost mathieu equation this paper contains the rigorous proof of the formulated by andre and aubry following statement the cauchy solutions of the discrete schrodinger equation with the potential qn g cos 2 n grow exponentially for every irrational g 1 and almost every e 0 2 according to known this fact implies the absence of the absolutely continuous component of the spectrum for the corresponding operator

log holder continuity of the integrated density of states for stochastic jacobi matrices we consider the integrated density of states k e of a general operator on l2 v of the formh h0 v where h 0 u n sum limits left i right 1 u n i and vu n v n u n wherev is a general bounded ergodic stationary process on v we show that k e k e c log e e 1 when e e 1 2 the key is a thouless formula for the strip

anderson localization for the almost mathieu equation ii point spectrum for lambda 2 we prove that for any 2 and a e the pure point spectrum of the almost mathieu operator h n n 1 n 1 cos 2 n n contains the essential closure hat sigma of the spectrum corresponding eigenfunctions decay exponentially the singular continuous component if it exists is concentrated on a set of zero measure which is nowhere dense in hat sigma

perturbations of orthogonal polynomials with periodic recursion coefficients the results of denisov rakhmanov szeg shohat nevai and killip simon are extended from asymptotically constant orthogonal polynomials on the real line oprl and unit circle opuc to asymptotically periodic oprl and opuc the key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations

schrodinger operators on periodic discrete graphs abstract we consider schrodinger operators with periodic potentials on periodic discrete graphs the spectrum of the schrodinger operator consists of an absolutely continuous part a union of a finite number of non degenerated bands plus a finite number of flat bands i e eigenvalues of infinite multiplicity we obtain estimates of the lebesgue measure of the spectrum in terms of geometric parameters of the graph and show that they become identities for some class of graphs moreover we obtain stability estimates and show the existence and positions of large number of flat bands for specific graphs the proof is based on the floquet theory and the precise representation of fiber schrodinger operators constructed in the paper

approximate hypergraph partitioning and applications szemeredi s regularity lemma is a cornerstone result in extremal combinatorics it roughly asserts that any dense graph is composed of a finite number of pseudorandom graphs the regularity lemma has found many applications in theoretical computer science and thus a lot of attention was given to designing algorithmic versions of this lemma our main results in this paper are the following i we introduce a new approach to the problem of constructing regular partitions of graphs which results in a surprisingly simple o n time algorithmic version of the regularity lemma thus improving over the previous o n 2 time algorithms furthermore unlike all the previous approaches for this problem see n alon and a naor siam j comput 35 2006 pp 787 803 r a duke h lefmann and v rodl siam j comput 24 1995 pp 598 620 a frieze and r kannan electron j combin 6 1999 article 17 a frieze and r kannan the regularity lemma and approximation schemes for dense problems in proceedings of the 37th annual symposium on foundations of computer science burlington vt 1996 ieee computer society press los alamitos ca 1996 pp 12 20 and y kohayakawa v rodl and l thoma siam j comput 32 2003 pp 1210 1235 which only guaranteed to find tower size partitions our algorithm will find a small regular partition if one exists in the graph ii for any constant r geq3 we give an o n time randomized algorithm for constructing regular partitions of r uniform hypergraphs thus improving the previous o n 2r 1 time deterministic algorithms a czygrinow and v rodl siam j comput 30 2000 pp 1041 1066 a frieze and r kannan the regularity lemma and approximation schemes for dense problems in proceedings of the 37th annual symposium on foundations of computer science burlington vt 1996 ieee computer society press los alamitos ca 1996 pp 12 20 these two results are obtained as an application of an efficient algorithm for approximating partition problems of hypergraphs which we obtain here given a directed hypergraph with bounded edge arities a set of constraints on the set sizes and densities of a possible partition of its vertex set and an approximation parameter we provide in o n time a partition approximating the constraints if a partition satisfying them exists we can also test in o 1 time for the existence of such a partition given the approximation parameter this algorithm extends the result of goldreich goldwasser and ron for graph partition problems o goldreich s goldwasser and d ron j acm 45 1998 pp 653 750 and encompasses more recent hypergraph related results such as the maximal constraint satisfaction approximation of g andersson and l engebretsen random structures algorithms 21 2002 pp 14 32

gauss polynomials and the rotation algebra newton s binomial theorem is extended to an interesting noncommutative setting as follows if in a ring ba ab with commuting witha andb then the generalized binomial coefficient left begin array 20 c n k end array right r arising in the expansion left a b right n sum limits k 0 n left begin array 20 c n k end array right gamma a n k b k resulting from these relations is equal to the value at of the gaussian polynomial left begin array 20 c n k end array right frac left n right left k right left n k right where m 1 x m 1 x m 1 1 x this is of course known in the case 1 from this it is deduced that in the universal c algebraa gq generated by unitariesu andv such thatvu e 2 i uv the spectrum of the self adjoint element u v u v has all the gaps that have been predicted to exist provided that either is rational or is a liouville number in the latter case the gaps are labelled in the natural way viak theory by the set of all non zero integers and the spectrum is a cantor set

decomposing the essential spectrum abstract we use c algebra theory to provide a new method of decomposing the essential spectra of self adjoint and non self adjoint schrodinger operators in one or more space dimensions

laplacians on periodic graphs with guides abstract we consider laplace operators on periodic discrete graphs perturbed by guides i e graphs which are periodic in some directions and finite in other ones the spectrum of the laplacian on the unperturbed graph is a union of a finite number of non degenerate bands and eigenvalues of infinite multiplicity we show that the spectrum of the perturbed laplacian consists of the unperturbed one plus the additional so called guided spectrum which is a union of a finite number of bands we estimate the position of the guided bands and their length in terms of geometric parameters of the graph we also determine the asymptotics of the guided bands for guides with large multiplicity of edges moreover we show that the possible number of guided bands their length and position can be rather arbitrary for some specific periodic graphs with guides

scaling localization and bandwidths for equations with competing periods abstract the finite size scaling theory of the measure of the spectrum of harper s equation is reexamined for sequences of fractions tending to a rational limit a simple criterion is derived which determines whether the corrections to scaling behave as log p 2 or p 2 as the denominator p is increased the question of how special are the properties of the harper equation is studied it is shown that if the pure cosine term in the diagonal term is replaced by a distorted periodic function the different subbands undergo a transition from localized to extended at a value of the strength of the off diagonal term that depends on energy in contrast to the harper equation where the transition is energy independent this has a crucial effect on the measure of the spectrum

scaling for the discrete mathieu equation the scaling of the total width of the band for the discrete mathieu equation is studied in the critical region near the transition between localized and extended states for the special case in which there is one period of the modulation forp lattice spacings a general expression for the bandwidthw in the critical region is found at the critical point an analytic expression forpw is found which agrees to one part in 108 with the result deduced from numerical work

metal insulator transition for the almost mathieu operator we prove that for diophantine and almost every the almost mathieu operator h n n 1 ni 1 cos 2 n n exhibits localization for 2 and purely absolutely continuous spectrum for 2

numerical treatment of nonconservation forms of the equations of shallow water theory abstract nonconservation forms of the equations of shallow water theory are solved by an extension of the random choice method the nonconservation riemann problem is analyzed in terms of two nonlinearly interacting waves and is solved by means of a numerical integration along characteristic curves numerical results for the dam failure problem in cartesian and cylindrical coordinates are presented the results are shown to be satisfactory the random choice method has the advantage of describing the breakdown of discontinuities in shallow water theory without need of ad hoc techniques and without introducing numerical diffusion and dispersion

wkb analysis of energy band structure of modulated systems a discrete wkb approximation is applied to harper s equation describing the motion of an electron in a square lattice subject to a magnetic field the treatment is valid when the modulation parameter proportional to the magnetic field is a rational number m n in the limit of large n a uniform asymptotic expression is derived for the floquet wavenumber which characterizes the dynamics and spectral properties the case of energies near the classical separatrix crucial to the properties of the spectrum near the transition between localized and extended states is included an application is a detailed calculation of the total width of the bands in the critical region for m 1 and 2 the results are consistent with thouless conjecture that the bandwidth obeys a universal scaling form independent of m

magnetic eigenmaps for community detection in directed networks communities in directed networks have often been characterized as regions with a high density of links or as sets of nodes with certain patterns of connection our approach for community detection combines the optimization of a quality function and a spectral clustering of a deformation of the combinatorial laplacian the so called magnetic laplacian the eigenfunctions of the magnetic laplacian which we call magnetic eigenmaps incorporate structural information hence using the magnetic eigenmaps dense communities including directed cycles can be revealed as well as role communities in networks with a running flow usually discovered thanks to mixture models furthermore in the spirit of the markov stability method an approach for studying communities at different energy levels in the network is put forward based on a quantum mechanical system at finite temperature

computational aspects of the random choice method for shallow water equations abstract a numerical solution of the shallow water equations is presented the construction of the solution is based on the random choice method consisting in the solution of riemann problems and sampling techniques as an illustration the method is applied to the problem of the breaking of a dam the main advantages of the method described lie in its ability to simulate discontinuities and in its lack of numerical dissipation

essential spectrum of the discrete laplacian on a perturbed periodic graph abstract we address the laplacian on a perturbed periodic graph which might not be a periodic graph we give a criterion for the essential spectrum of the laplacian on the perturbed graph to include that on the unperturbed graph this criterion is applicable to a wide class of graphs obtained by a non compact perturbation such as adding or removing infinitely many vertices and edges using this criterion we demonstrate how to determine the spectra of cone like graphs the upper half plane and graphs obtained from z 2 by randomly adding vertices

spectral and propagation results for magnetic schrodinger operators a c algebraic framework abstract we study generalized magnetic schrodinger operators of the form h h a v h a v where h is an elliptic symbol a i a with a a vector potential defining a variable magnetic field b and v is a scalar potential we are mainly interested in anisotropic functions b and v the first step is to show that these operators are affiliated to suitable c algebras of magnetic pseudodifferential operators a study of the quotient of these c algebras by the ideal of compact operators leads to formulae for the essential spectrum of h h a v expressed as a union of spectra of some asymptotic operators supported by the quasi orbits of a suitable dynamical system the quotient of the same c algebras by other ideals give localization results on the functional calculus of the operators h h a v which can be interpreted as non propagation properties of their unitary groups

a sum rule for the dispersion relations of the rational harper equation the authors derive an exact sum rule satisfied by the dispersion relations of the commensurable case of harper s equation 1955 they use this result to derive a lower bound for the total bandwidth of the spectrum and to provide a stronger analytical justification for a result due to thouless 1990 concerning the total bandwidth when the commensurability is a high order rational

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

quantum hall effect on the hyperbolic plane in this paper we study both the continuous model and the discrete model of the quantum hall effect qhe on the hyperbolic plane the hall conductivity is identified as a geometric invariant associated to an imprimitivity algebra of observables we define a twisted analogue of the kasparov map which enables us to use the pairing between k theory and cyclic cohomology theory to identify this geometric invariant with a topological index thereby proving the integrality of the hall conductivity in this case

absence of cantor spectrum for a class of schrodinger operators it is shown that the complete localization of eigenvectors for the almost mathieu operator entails the absence of cantor spectrum for this operator

on the structure of the essential spectrum of elliptic operators on metric spaces abstract we give a description of the essential spectrum of a large class of operators on metric measure spaces in terms of their localizations at infinity these operators are analogues of the elliptic operators on euclidean spaces and our main result concerns the ideal structure of the c algebra generated by them

the essential spectrum of schrodinger operators on lattices the paper is devoted to the study of the essential spectrum of discrete schr dinger operators on the lattice by means of the limit operators method this method has been applied by one of the authors to describe the essential spectrum of continuous electromagnetic schr dinger operators square root klein gordon operators and dirac operators under quite weak assumptions on the behaviour of the magnetic and electric potential at infinity the present paper aims at illustrating the applicability and efficiency of the limit operators method to discrete problems as well we consider the following classes of the discrete schr dinger operators 1 operators with slowly oscillating at infinity potentials 2 operators with periodic and semi periodic potentials 3 schr dinger operators which are discrete quantum analogues of the acoustic propagators for waveguides 4 operators with potentials having an infinite set of discontinuities and 5 three particle schr dinger operators which describe the motion of two particles around a heavy nuclei on the lattice

the hypergraph regularity method and its applications szemeredi s regularity lemma asserts that every graph can be decomposed into relatively few random like subgraphs this random like behavior enables one to find and enumerate subgraphs of a given isomorphism type yielding the so called counting lemma for graphs the combined application of these two lemmas is known as the regularity method for graphs and has proved useful in graph theory combinatorial geometry combinatorial number theory and theoretical computer science here we report on recent advances in the regularity method for k uniform hypergraphs for arbitrary k 2 this method purely combinatorial in nature gives alternative proofs of density theorems originally due to e szemeredi h furstenberg and y katznelson further results in extremal combinatorics also have been obtained with this approach the two main components of the regularity method for k uniform hypergraphs the regularity lemma and the counting lemma have been obtained recently rodl and skokan based on earlier work of frankl and rodl generalized szemeredi s regularity lemma to k uniform hypergraphs and nagle rodl and schacht succeeded in proving a counting lemma accompanying the rodl skokan hypergraph regularity lemma the counting lemma is proved by reducing the counting problem to a simpler one previously investigated by kohayakawa rodl and skokan similar results were obtained independently by w t gowers following a different approach

remark on the continuity of the density of states of ergodic finite difference operators we give an elementary proof that for a large class ofd dimensional finite difference operators including tight binding models for electron propagation and models for harmonic phonons with random masses or couplings the integrated density of states is a continuous function of the energy

discrete magnetic laplacian we consider a 2 dimensional discrete operator which we call the discrete magnetic laplacian dml it is an analogue of the magnetic schrodinger operator it follows from well known arguments that dml has the same spectrum as a subset inr as the almost mathieu operator am they also have the same integrated density of states ids which is known to be continuous we show that dml is an element in a ii1 factor and its ids can be expressed through the trace in the ii1 factor it follows that dml never has anyl2 eigenfunctions i e has no point spectrum then we formulate a natural algebraic conjecture which implies that the spectrum of dml hence the spectrum of am is a cantor set

spectral theory of extended harper s model and a question by erdős and szekeres the extended harper s model proposed by d j thouless in 1983 generalizes the famous almost mathieu operator allowing for a wider range of lattice geometries parametrized by three coupling parameters by permitting 2d electrons to hop to both nearest and next nearest neighboring nnn lattice sites while still exhibiting its characteristic symmetry aubry andre duality previous understanding of the spectral theory of this model was restricted to two dual regions of the parameter space one of which is characterized by the positivity of the lyapunov exponent in this paper we complete the picture with a description of the spectral measures over the entire remaining self dual region for all irrational values of the frequency parameter the magnetic flux in the model most notably we prove that in the entire interior of this regime the model exhibits a collapse from purely ac spectrum to purely sc spectrum when the nnn interaction becomes symmetric in physics literature extensive numerical analysis had indicated such spectral collapse however so far not even a heuristic argument for this phenomenon could be provided on the other hand in the remaining part of the self dual region the spectral measures are singular continuous irrespective of such symmetry the analysis requires some rather delicate number theoretic estimates which ultimately depend on the solution of a problem posed by erd s and szekeres on the product prod k 1 n 1 z a k publ de l institut mathematique paris 1950

an efficient characteristic galerkin scheme for the advection equation in 3 d conventional galerkin based finite element algorithms perform poorly when modeling advective transport here we develop and test a characteristic galerkin scheme to solve the unsteady three dimensional advective transport equation the algorithm uses tracking of the gaussian quadrature points to project the information from the eulerian background grid to the lagrangian grid numerical experiments were carried out to investigate the performance of this scheme despite speculations to the contrary in the literature this scheme does not suffer from instability problems further it is conservative and shows good phase characteristics with slight numerical diffusion we conclude that the characteristic galerkin method is a viable and efficient scheme for solving advection problems in three dimensions

spectral analysis of the acoustic propagator in a multistratified domain abstract the aim of this paper is to study the acoustic propagatorhin a perturbed multistratified domain with the conjugate operator method hdenotes the self adjoint operator c x inl2 with a dirichlet condition at the boundary herecdenotes the propagation velocity depending on the nature of the stratification of we shall give some spectral properties ofhand we establish the limiting absorption principle in an optimal scale of besov spaces

on the measure of the spectrum for the almost mathieu operator we obtain partial results on the conjecture that for the almost mathieu operator at irrational frequency the measure of the spectrum s 4 2 for 2 we show that if n is rational and alpha n to alpha irrational then s alpha n lambda theta to 4 2 lambda

on the essential spectrum of elliptic differential operators abstract let a be a c algebra of bounded uniformly continuous functions on a finite dimensional real vector space x such that a is stable under translations and contains the continuous functions that have a limit at infinity denote a the boundary of x in the character space of a then to each operator a in the crossed product a x one may naturally associate a family of bounded operators a on l 2 x indexed by the characters a we show that the essential spectrum of a is the union of the spectra of the operators a the applications cover very general classes of singular elliptic operators

multi dimensional conservative semi lagrangian method of characteristics cip for the shallow water equations a new characteristic approach that guarantees conservative property is proposed and is applied to the shallow water equations cip csl constrained interpolation profile conservative semi lagrangian interpolation is applied to the cip method of characteristics in order to enhance the mass conservation of the numerical result although the characteristic formulation is originally derived from non conservative form present scheme achieves complete mass conservation by solving mass conservation simultaneously and reflecting conserving mass in interpolation profile present method has less height error compared to the cip method of characteristics by several orders of magnitude by the enhanced conservation property present scheme is applicable to nonlinear problem such as shock furthermore application to two dimensions including the coriolis term is straightforward with directional splitting technique

the application of shape preserving splines for the solution of differential equations this paper investigates the use of shape preserving interpolants based on bernstein polynomials for the numerical solution of differential equations a simple and accurate algorithm is presented for the integration of the initial value ordinary differential equation as well as for the partial differential equation of the burger type

on the construction of the voronoi mesh on a sphere abstract a new construction of the voronoi mesh on the sphere is presented the main feature is that the algorithm adds points one at a time until the final voronoi mesh is built up by adding one point to an existing voronoi mesh of k points only local changes are needed to construct a voronoi mesh of k 1 points this construction is particularly well suited to time dependent problems since using information from the voronoi mesh at the previous time step allows us to reduce the construction to o n operations when the two configurations are close while the algorithm does not break down when they are far apart numerical experiments are presented to substantiate the o n operation count for a typical case