convergence and nonlinear stability of the lagrange galerkin method for the navier stokes equations the lagrange galerkin method is a numerical technique for solving convection dominated diffusion problems based on combining a special discretisation of the lagrangian material derivative along particle trajectories with a galerkin finite element method we present optimal error estimates for the lagrange galerkin mixed finite element approximation of the navier stokes equations in a velocity pressure formulation the method is shown to be nonlinearly stable

analytic representations with theta functions for systems on ℤ d and on an analytic representation with theta functions on a torus for systems with variables in d is considered another analytic representation with theta functions on a strip for systems with positions in a circle x1d54a and momenta in is also considered the reproducing kernel formalism for these two systems is studied wigner and weyl functions in this language are also studied

shannon entropy at avoided crossings in the quantum transition from order to chaos shannon entropy is studied for the series of avoided crossings that characterize the transition from order to chaos in quantum mechanics in order to be able to study jointly this entropy for discrete and continuous probability calculations have been performed on a quantized map the kicked harper map resulting in a different behavior as order chaos transition takes place for the discrete position representation and continuous coherent state representation cases this different behavior is analyzed in terms of the distribution of zeros of the husimi function

phase space lattices with threefold symmetry this paper presents a semiclassical analysis of the spectrum of the hamiltonian ir cos p e cos ci v 3 5 cos f p a 5 which is a model for bloch electrons in a magnetic field the energy levels are determined to a first approximation by bohr sommerfeld quantisation because of the translational symmetry in phase space this leads to a lattice of infinitely degenerate states this degeneracy is lifted by tunnelling effects which can be described by an effective hamiltonian of the same form as that above but with different values of 0 and h the calculation of the effective hamiltonian therefore defines a renormalisation group transformation and it predicts that the spectrum has a complex recursive structure which is confirmed by numerical experiments the results complement earlier work on lattices with fourfold symmetry in which a similar type of spectrum occurs the greater complexity of the problem in lattices with threefold symmetry necessitated developing simpler canonically invariant methods of analysis

an exact renormalisation group for bloch electrons in a magnetic field bloch electrons in a magnetic field can be modelled by a one dimensional hamiltonian h h x p which is periodic in x and p when beta the ratio of planck s constant h to the area of a unit cell is a rational number p q the spectrum is a set of q bloch bands if beta is perturbed away from this rational value the spectrum becomes a cantor set but it is still possible to define a set of generalised bloch waves which form a complete basis set for a given band projecting these states into the band and taking matrix elements of the hamiltonian leads to a new effective hamiltonian h1v x p and planck constant h v describing the spectrum of the vth band since degrees of freedom have been eliminated by projecting into a band this is a renormalisation group rg transformation the rg transformation can be iterated indefinitely if the measure of the spectrum is zero or if beta is a liouville number the measure of the spectrum vanishes if h x p has centres of threefold or fourfold symmetry in the phase plane and the rg transformation explains the hierarchical structure of the spectrum observed in these cases

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

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

on semiclassical dispersion relations of harper like operators we describe some semiclassical spectral properties of harper like operators i e of one dimensional quantum hamiltonians periodic in both momentum and position the spectral region corresponding to the separatrices of the classical hamiltonian is studied for the case of integer flux we derive asymptotic formulae for the dispersion relations the width of bands and gaps and show how geometric characteristics and the absence of symmetries of the hamiltonian influence the form of the energy bands

quantum variance and ergodicity for the baker s map we prove an egorov theorem or quantum classical correspondence for the quantised baker s map valid up to the ehrenfest time this yields a logarithmic upper bound for the decay of the quantum variance and as a corollary a quantum ergodic theorem for this map

universal fluctuations of zeros of chaotic wavefunctions wavefunctions of one and two dimensional quantum systems can be parametrized by a finite number of zeros lying in phase space we study correlations of these zeros for fully chaotic systems in terms of a statistical model based on random polynomials excellent agreement is found for the two point correlation function and nearest neighbour spacing distribution of this model and the results obtained for wavefunctions of dynamical systems we conjecture that these correlation functions are valid for any chaotic system after rescaling the phase space distances unfolding some consequences for the distribution of zeros due to time reversal symmetry are also discussed

equipartition of the eigenfunctions of quantized ergodic maps on the torus we give a simple proof of the equipartition of the eigenfunctions of a class of quantized ergodic area preserving maps on the torus examples are the irrational translations the skew translations the hyperbolic automorphisms and some of their perturbations

cyclic states berry phases and the schrodinger operator the evolution of a quantum mechanical system under a nonadiabatic external perturbation at a time interval 0 t is considered it is shown that all the cyclic states of the system mod psi t ei phi mod psi 0 are determined by the eigenvalues in quasi energies and associated eigenvectors mod phi in t of the schrodinger operator sq t h t i delta delta t acting on some hilbert space the set of linearly independent cyclic quasi energy states possess some properties similar to the properties of the stationary states of a closed system the berry phases of the states associated with eigenvectors of the discrete spectrum of sq t which are single valued functions of omega 2 pi t are supplied by the partial derivatives of the corresponding eigenvalues quasi energies with respect to omega the approach developed is illustrated by several applications to time dependent systems the system under an adiabatic perturbation the forced harmonic oscillator and the two level system even in the case of a system with a time independent hamiltonian there exist nonstationary quasi energy states that have nontrivial berry phases beta 2 pi not an integer the criterion of existence of such states is formulated in terms of the energies of the system and the corresponding expression for the berry phases is obtained some examples of nonstationary cyclic states of closed systems including the coherent states of a force free oscillator and the wannier states of electrons in the parabolic band are considered

chaos revealing multiplicative representation of quantum eigenstates the quantisation of the two dimensional toric and spherical phase spaces is considered in analytic coherent state representations every pure quantum state admits therein a finite multiplicative parametrisation by the zeros of its husimi function for eigenstates of quantised systems this description explicitly reflects the nature of the underlying classical dynamics in the semiclassical regime the distribution of the zeros in the phase space becomes one dimensional for integrable systems and highly spread out conceivably uniform for chaotic systems this multiplicative representation thereby acquires a special relevance for semiclassical analysis in chaotic systems

an exact effective hamiltonian for a perturbed landau level considers the effect of a scalar potential v x y on a landau level in two dimensions an exact effective hamiltonian is derived which describes the effect of the potential on a single landau level expressed as a power series in v ec where ec is the cyclotron energy the effective hamiltonian can be represented as a function h x p in a one dimensional phase space the function h x p resembles the potential v x y when the area of a flux quantum is much smaller than the square of the characteristic length scale of v then h approximately v also h x p retains the translational and rotational symmetries of v x y exactly but reflection symmetries are not retained beyond the lowest order of the perturbation expansion

time reversal symmetry and random polynomials we analyse the density of roots of random polynomials where each complex coefficient is constructed of a random modulus and a fixed deterministic phase the density of roots is shown to possess a singular component only in the case for which the phases increase linearly with the index of coefficients this means that contrary to earlier belief eigenvectors of a typical quantum chaotic system with some antiunitary symmetry will not display a clustering curve in the stellar representation moreover a class of time reverse invariant quantum systems is shown for which spectra display fluctuations characteristic of orthogonal ensemble while eigenvectors confer to predictions of unitary ensemble

general quantization of canonical maps on a two torus canonical maps on a two torus in phase space are quantized under most general conditions recent results by keating et al 1999 nonlinearity 12 579 are thus fully extended in two directions a the translational component of a general canonical map is included in the quantization b all values of planck s constant consistent with the toral boundary conditions bcs are considered generically these values are rational numbers whose numerator must satisfy a number theoretical condition besides the condition on planck s constant the quantization is possible only for particular allowed bcs on the torus the general equation determining these bcs is derived allowed bcs may not exist in some cases representative examples are the irrational skew translations and kronecker maps exact versions of egorov s theorem are shown to hold under some conditions composition and representation properties of the quantization scheme are studied

on the zeros of the husimi distribution the basic features of the zeros of the husimi phase space density of a quantum eigenstate are discussed it is demonstrated that some special properties of the harmonic oscillator related to the number of zeros and their location within the classical energy shell are not valid for anharmonic potentials as illustrated by numerical examples for the morse oscillator also discussed is the distribution of zeros in quantum poincare sections for systems with two degrees of freedom

on the transport diffusion algorithm and its applications to the navier stokes equations this paper deals with an algorithm for the solution of diffusion and or convection equations where we mixed the method of characteristics and the finite element method globally it looks like one does one step of transport plus one step of diffusion or projection but the mathematics show that it is also an implicit time discretization of thepde in lagrangian form we give an error bound h t h h t in the interesting case that holds also for the navier stokes equations even when the reynolds number is infinite euler equation

weak coherent states related to the multiplicative group mathbb c generalized coherent states which are associated with the multiplicative group of non zero complex numbers are introduced they have the property of temporal stability and they form a total set in some subspace of the full hilbert space they are highly non classical states in the sense that in general their bargmann functions have zeros which are related to regions where their wigner functions take negative values states with bargmann functions sin az and exp bz az k are examples of the general formalism and they are discussed in detail

random analytic chaotic eigenstates the statistical properties of random analytic functions z are investigated as a phase space model for eigenfunctions of fully chaotic systems we generalize to the plane and to the hyperbolic plane a theorem concerning the equidistribution of the zeros of z previously demonstrated for a spherical phase space su 2 polynomials for systems with time reversal symmetry the number of real roots is computed for the three geometries in the semiclassical regime the local correlation functions are shown to be universal independent of the system considered or the geometry of phase space in particular the autocorrelation function of is given by a gaussian function the connections between this model and the gaussian random function hypothesis as well as the random matrix theory are discussed

phase space approach to quantum dynamics replaces the schrodinger equation for the time propagation of states of a quantized 2d spherical phase space by the dynamics of a system of n particles lying in phase space this is done through factorization formulae of an analytic function theory arising in a coherent state representation the particles being the zeroes of the quantum state for linear hamiltonians like a spin in a uniform magnetic field the motion of the particles is classical however nonlinear terms induce interactions between the particles their time propagation is studied and it is shown that contrary to integrable systems for chaotic maps they tend to fill as their classical counterpart the whole phase space in a uniform way

topological properties of quantum periodic hamiltonians we consider periodic quantum hamiltonians on the torus phase space harper like hamiltonians we calculate the topological chern index which characterizes each spectral band in the generic case this calculation is made by a semiclassical approach with the use of quasi modes as a result the chern index is equal to the homotopy of the path of these quasi modes on phase space as the floquet parameter of the band is varied it is quite interesting that the chern indices defined as topological quantum numbers can be expressed from simple properties of the classical trajectories

phase space flow in the husimi representation we derive a continuity equation for the husimi function evolving under a general non hermitian hamiltonian and identify the phase space flow associated with it for the case of unitary evolution we obtain explicit formulas for the quantum flow which can be written as a classical term plus quantum corrections the quantum terms can be expanded in powers of planck s constant providing a series of semiclassical corrections to the classical flow we test the exact and semiclassical formulas for a particle in a double well potential and find numerical evidence that the zeros of the husimi function are always saddle points of the flow merging or splitting of stagnation points reported for the wigner flow was not observed

chaotic analytic zero points exact statistics for those of a random spin state a natural statistical ensemble of 2j points on the unit sphere can be associated via the majorana representation with a random quantum state of spin j and an exact expression is obtained here for the general k point correlation function in this ensemble the pair correlation in the large j limit takes the relatively simple form where and is the angular separation of the pair of points on the sphere it appears from the numerical work of others that in this limit these statistics are typical of the zero points of analytic functions associated with chaotic quantum dynamical systems

renyi wehrl entropies as measures of localization in phase space we generalize the concept of the wehrl entropy of quantum states which gives a basis independent measure of their localization in phase space we discuss the minimal values and the typical values of these r enyi wehrl entropies for pure states for spin systems according to lieb s conjecture the minimal values are provided by the spin coherent states though lieb s conjecture remains unproven we give new proofs of partial results that may be generalized for other systems we also investigate random pure states and calculate the mean r enyi wehrl entropies averaged over the natural measure in the space of pure quantum states

eigenfunction statistics of complex systems a common mathematical formulation we derive a common mathematical formulation for the eigenfunction statistics of hermitian operators represented by a multiparametric probability density the system information in the formulation enters through two parameters only namely system size and the complexity parameter a function of all system parameters including size the behavior is contrary to the eigenvalue statistics which is sensitive to the complexity parameter only and shows a single parametric scaling the existence of a mathematical formulation of both eigenfunctions and eigenvalues common to a wide range of complex systems indicates the possibility of a similar formulation for many physical properties this also suggests the possibility to classify them in various universality classes defined by the complexity parameter

a completely explicit scheme of cauchy problem in bslm for solving the navier stokes equations abstract this paper presents a backward semi lagrangian method bslm with third order convergence in both time and space for solving incompressible navier stokes equations the third order backward differentiation formula for the total time derivative and the projection method for the steady state stokes equation are used a fourth order difference scheme together with a local bi cubic interpolation is used to solve the resulted two governing equations for the velocity and pressure this paper mainly focuses on the development of an efficient scheme for solving the nonlinear cauchy problem of the characteristic curve we employ a modified linear multi step method of the implicit type based on the error correction strategy a novel contribution of this paper is the design of a completely explicit formula for the three foot points the proposed method is superior to existing methods in terms of computational costs and accuracy allowing the use of a large time step size the fully explicit formula for the foot points significantly improves the performance of a common bslm for a wide range of practical applications

bounds on the abc spectral radius and abc energy of graphs abstract in this paper we obtain some bounds for the abc spectral radius of general graphs in continuing we get some properties of the abc eigenvalues of graphs and then we establish some new bounds for the abc energy e a b c finally we set up the correlation between the abc energy and other classes of graph energies

extended cahill glauber formalism for finite dimensional spaces i fundamentals the cahill glauber approach for quantum mechanics on phase space is extended to the finite dimensional case through the use of discrete coherent states all properties and features of the continuous formalism are appropriately generalized the continuum results are promptly recovered as a limiting case the jacobi theta functions are shown to have a prominent role in the context

generalized wannier function and renormalization of harper s equation harper s equation a model for bloch electrons in a magnetic field has a band spectrum when the dimensionless magnetic field beta is a rational number p q this paper considers the definition of generalized wannier functions which can be used to represent the bloch bands of the spectrum of the rational harper equation by means of a von neumann lattice one representation of these bloch bands can be extended to the irrational case and taking matrix elements in this basis leads to a renormalization group transformation acting on the hamiltonian the results in the present paper considerably extend a previous analysis of this renormalization group transformation in that the formalism is suitable for systematic calculations of the renormalized hamiltonian and that the transformation preserves the rotational symmetry of the harper hamiltonian in phase space

von neumann lattices of wannier functions for bloch electrons in a magnetic field the problem of bloch electrons in a magnetic field in two dimensions can be reduced to a one dimensional problem with a hamiltonian h that is a periodic function of x and p wannier functions can be defined for the sub bands of the spectrum of this effective hamiltonian when the chern class quantized hall conductance integer of the sub band is zero the weyl wigner formalism can be used to represent these wannier functions by a von neumann lattice it is shown how this von neumann lattice of wannier functions can be defined for irrational as well as rational magnetic fields an important benefit from using the weyl wigner formalism is that symmetries of the periodic potential are reflected by symmetries of the effective hamiltonian in phase space it is shown how the wannier functions can be defined so that their wigner functions have the point symmetries of the effective hamiltonian an example of how these results can prove useful is given if we take matrix elements of the hamiltonian between the wannier states of a sub band we derive a new effective hamiltonian describing this sub band which is again a periodic function of coordinate and momentum operators since by projecting onto a sub band we have also reduced the number of degrees of freedom this operation is a renormalization group transformation it is shown that the symmetry of the new effective hamiltonian in phase space is the same as that of the original one this preservation of symmetry helps to explain some unusual properties of the spectrum when the hamiltonian has fourfold symmetry

on abc eigenvalues and abc energy abstract let g be a graph with n vertices and let d i be the degree of its i th vertex the abc matrix of g is the square matrix of order n whose i j entry is equal to d i d j 2 d i d j if the i th vertex and the j th vertex of g are adjacent and 0 otherwise this matrix related closely to the atom bond connectivity abbreviated as abc index was recently introduced by estrada as a matrix representation of the probability of visiting a nearest neighbor edge from one side or the other of a given edge in a graph which in the context of molecular graphs can be related to the polarizing capacity of the bond considered the abc eigenvalues of g are the eigenvalues of its abc matrix and the abc energy of g is the sum of the absolute values of its abc eigenvalues in this paper some new results for the abc eigenvalues and the abc energy of a graph are presented

numerical analysis of convection diffusion reaction problems with higher order characteristics finite elements part ii fully discretized scheme and quadrature formulas in this paper a higher order lagrange galerkin discretization method is analyzed when applied to a variable coefficient convection possibly degenerated diffusion reaction equation with mixed dirichlet robin boundary conditions in a previous paper a bermu dez m r nogueiras and c va zquez siam j numer anal to appear the proposed second order time discretization scheme has been rigorously introduced for exact and approximated characteristics moreover the l infty l 2 stability property and l infty l 2 error estimates of order o delta t 2 have been obtained as a continuation of that work consistency error estimates of order o delta t 2 h k are obtained for the fully discretized lagrange galerkin scheme moreover adequate quadrature formulas are proposed for the practical implementation of the method with particular finite element spaces finally some numerical tests illustrate the theoretical results and the performance of the combination of second order lagrange galerkin schemes with quadrature formulas

hofstadter rules and generalized dimensions of the spectrum of harper s equation we consider the harper model which describes two dimensional bloch electrons in a magnetic field for irrational flux through the unit cell the corresponding energy spectrum is known to be a cantor set with multifractal properties in order to relate the maximal and minimal fractal dimension of the spectrum of harper s equation to the irrational number involved we combine a refined version of the hofstadter rules with results from semiclassical analysis and tunnelling in phase space for quadratic irrationals with continued fraction expansion the maximal fractal dimension exhibits oscillatory behaviour as a function of n which can be explained by the structure of the renormalization flow the asymptotic behaviour of the minimal fractal dimension is given by as the generalized dimensions can be related to the anomalous diffusion exponents of an initially localized wavepacket our results imply that the time evolution of high order moments is sensible to the parity of n

singular bohr sommerfeld rules the object of this paper is to present a geometrical formulation of the singular bohr sommerfeld rules given in 8 in a more general context including characteristic manifolds with several saddle points several examples are detailed and numerical checkings of the results are provided

on the normalized laplacian energy and general randic index r 1 of graphs in this paper we consider the energy of a simple graph with respect to its normalized laplacian eigenvalues which we call the l energy over graphs of order n that contain no isolated vertices we characterize the graphs with minimal l energy of 2 and maximal l energy of 2 n 2 we provide upper and lower bounds for l energy based on its general randic index r 1 g we highlight known results for r 1 g most of which assume g is a tree we extend an upper bound of r 1 g known for trees to connected graphs we provide bounds on the l energy in terms of other parameters one of which is the energy with respect to the adjacency matrix finally we discuss the maximum change of l energy and r 1 g upon edge deletion

on the abc spectra radius of unicyclic graphs abstract the abc matrix of a graph g is a square matrix where the i j entry a b c i j d i d j 2 d i d j if there is an edge between vertices i and j in g and a b c i j 0 otherwise the abc spectral radius of g denoted by 1 g is the largest eigenvalue of the abc matrix of g in this paper we prove that for a unicyclic graph g of order n 4 2 1 c n 1 g 1 s n e with equality if and only if g c n for the lower bound and if and only if g s n e for the upper bound this solves a conjecture by ghorbani et al

generalized quantum baker maps as perturbations of a simple kernel we present a broad family of quantum baker maps that generalize the proposal of schack and caves to any even hilbert space with arbitrary boundary conditions we identify a structure common to all maps consisting of a simple kernel perturbed by diffraction effects this essential baker s map has a different semiclassical limit and can be diagonalized analytically for hilbert spaces spanned by qubits in all cases this kernel provides an accurate approximation to the spectral properties char22 eigenvalues and eigenfunctions char22 of all the different quantizations

a spectral lagrange galerkin method for convection dominated diffusion problems abstract the spectral lagrange galerkin method is a numerical technique for time dependent convection diffusion problems based on combining a lagrangian formulation of the equations with the spectral method the resulting scheme can be shown to be unconditionally stable for linear advection diffusion equations and optimal convergence rates may be obtained both the spectral accuracy and the efficiency of the scheme are examined theoretically and by various numerical experiments and extensions of the method to give high accuracy in time are presented

quantum chaotic dynamics and random polynomials we investigate the distribution of roots of polynomials of high degree with random coefficients which among others appear naturally in the context of quantum chaotic dynamics it is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane in order to further increase this tendency we study in detail the particular case of self inversive random polynomials and show that for them a finite portion of all roots lies exactly on the unit circle correlation functions of these roots are also computed analytically and compared to the correlations of eigenvalues of random matrices the problem of ergodicity of chaotic wavefunctions is also considered for that purpose we introduce a family of random polynomials whose roots spread uniformly over phase space while these results are consistent with random matrix theory predictions they provide a new and different insight into the problem of quantum ergodicity special attention is devoted to the role of symmetries in the distribution of roots of random polynomials

an iteration free backward semi lagrangian scheme for solving incompressible navier stokes equations a backward semi lagrangian method based on the error correction method is designed to solve incompressible navier stokes equations the time derivative of the stokes equation is discretized with the second order backward differentiation formula for the induced steady stokes equation a projection method is used to split it into velocity and pressure fourth order finite differences for partial derivatives are used to the boundary value problems for the velocity and the pressure also finite linear systems for poisson equations and helmholtz equations are solved with a matrix diagonalization technique for characteristic curves satisfying highly nonlinear self consistent initial value problems the departure points are solved with an error correction strategy having a temporal convergence of order two the constructed algorithm turns out to be completely iteration free in particular the suggested algorithm possesses a good behavior of the total energy conservation compared to existing methods to assess the effectiveness of the method two dimensional lid driven cavity problems with large different reynolds numbers are solved the doubly periodic shear layer flows are also used to assess the efficiency of the algorithm

the classical limit for a class of quantum baker s maps we show that the class of quantum baker s maps defined by schack and caves have the proper classical limit provided the number of momentum bits approaches infinity this is done by deriving a semi classical approximation to the coherent state propagator

the chaotic analytic function in contrast to a stationary gaussian random function of a real variable which is free to have any correlation function the closest analogous analytic random function in the complex plane has no true freedom it is statistically unique since it has arisen only recently as an apparently universal feature in the physical context of quantum chaos i refer to it here as the chaotic analytic function i note that it is implied by the assumption that a quantum chaotic wavefunction has gaussian randomness and has a constant value for the average of its wigner function in phase space interpreted literally this shows that the chaotic analytic function is the bargmann function of a pure white noise wavefunction more physically if constant is replaced by smooth on the scale of a planck area these assumptions are the semiclassical ones made by berry for chaotic eigenstates the analysis shows that the chaotic analytic function is still obtained semiclassically

computer search for trees with minimal abc index the abc index is a degree based molecular structure descriptor that found chemical applications finding the connected graph s of a given order whose abc index is minimal is a hitherto unsolved problem but it is known that these must be trees in this paper by combining mathematical arguments and computer based modeling we establish the basic structural features of the minimum abc trees

geometrical approach to the distribution of the zeros for the husimi function we construct a semiclassical expression for the husimi function of autonomous systems in one degree of freedom by smoothing with a gaussian function an expression that captures the essential features of the wigner function in the semiclassical limit our approximation reveals the centre and chord structure that the husimi function inherits from the wigner function down to very shallow valleys where the husimi zeros lie this explanation for the distribution of zeros along curves relies on the geometry of the classical torus rather than the complex analytic properties of the wkb method in the bargmann representation we evaluate the zeros for several examples

the semiclassical small ℏ limit of loci of roots of subdominant solutions for polynomial potentials in this paper a description of the small limit of loci of zeros of subdominant solutions for polynomial potentials is given called as fundamental solutions in our earlier papers the considered potentials are those which provide us with the simple turning points only three types of stokes graphs sg s associated with the potentials are investigated the general non critical ones the general critical ones but with only single internal stokes line sl and the stokes graphs corresponding to arbitrary multiple well real even degree polynomial potentials with internal stokes lines distributed on the real axis only all these cases are considered in their both versions of the quantized and not quantized in particular due to the fact that the small limit is semiclassical it is shown that loci of roots of subdominant solutions in the cases considered are collected along stokes lines there are infinitely many roots of subdominant solutions on such lines escaping to infinity and a finite number of th

the minimum abc energy of trees abstract let g be a graph of order n and d i be the degree of its i th vertex the abc matrix of g is the matrix of order n whose i j entry is equal to d i d j 2 d i d j if the i th vertex and the j th vertex of g are adjacent and 0 otherwise the abc eigenvalues of g are the eigenvalues of its abc matrix and the abc energy of g is the sum of the absolute values of its abc eigenvalues in chen 2018 9 the author conjectured that the star has the minimum abc energy among all trees in this paper we prove the conjecture is true

numerical analysis of a second order pure lagrange galerkin method for convection diffusion problems part i time discretization we propose and analyze a second order pure lagrangian method for variable coefficient convection possibly degenerate diffusion equations with mixed dirichlet robin boundary conditions first the method is rigorously introduced for exact and approximate characteristics next l infty h 1 stability is proved and l infty h 1 error estimates of order o delta t 2 are obtained moreover l infty l 2 stability and l infty l 2 error estimates of order o delta t 2 with constants bounded in the hyperbolic limit are shown for the particular case of dirichlet boundary conditions diffusion tensor a epsilon i and right hand side f 0 the l infty h 1 stability estimate is independent of epsilon moreover for incompressible flows the constants in the stability inequalities are independent of the final time in a second part of this work the pure lagrangian scheme will be combined with galerkin discretization using finite elements spaces and numerical examples will be presented

eigenfunctions of non integrable systems in generalised phase spaces the authors use the generalised coherent state distribution to study in phase space the eigenfunctions of non integrable hamiltonians the methods available in the weyl group the husimi distribution are extended to the more complicated phase spaces that arise from other lie groups they demonstrate the feasibility of numerical calculations in two models with w2 and su 3 symmetries emphasising the similarities and showing the emergence of classical invariant structures in the semiclassical limit

wigner function statistics in classically chaotic systems we have studied statistical properties of the values of the wigner function w x of 1d quantum maps on compact 2d phase space of finite area v for this purpose we have defined a wigner function probability distribution p w 1 v w w x dx which has by definition fixed first and second moments in particular we concentrate on relaxation of time evolving quantum states in terms of w x starting from a coherent state we have shown that for a classically chaotic quantum counterpart the distribution p w in the semiclassical limit becomes a gaussian distribution that is fully determined by the first two moments numerical simulations have been performed for the quantum sawtooth map and the quantized kicked top in a quantum system with hilbert space dimension n 1 the transition of p w to a gaussian distribution was observed at times t log n in addition it has been shown that the statistics of wigner functions of propagator eigenstates is gaussian as well in the classically fully chaotic regime we have also studied the structure of the nodal cells of the wigner function in particular the distribution of intersection points between the zero manifold and arbitrary straight lines

classical limit of the quantized hyperbolic toral automorphisms the canonical quantization of any hyperbolic symplectomorphisma of the 2 torus yields a periodic unitary operator on an dimenional hilbert space n 1 h we prove that this quantum system becomes ergodic and mixing at the classical limit n n prime which can be interchanged with the time average limit the recovery of the stochastic behaviour out of a periodic one is based on the same mechanism under which the uniform distribution of the classical periodic orbits reproduces the lebesgue measure the wigner functions of the eigenstates supported on the classical periodic orbits are indeed proved to become uniformly speread in phase space

a high order characteristics method for the incompressible navier stokes equations abstract we analyze a high order characteristics method for the navier stokes equations we focus on the cases of the first second and third order in time schemes with finite element spatial discretization a numerical comparison between the first and second order schemes is done for steady or transient states flows

unconditional stability and error estimates of modified characteristics fems for the navier stokes equations the paper is concerned with the unconditional stability and convergence of characteristics type methods for the time dependent navier stokes equations we present optimal error estimates in l 2 l2 and h 1 h1 norms for a typical modified characteristics finite element method unconditionally while all previous works require certain time step restrictions the analysis is based on an iterated characteristic time discrete system with which the error function is split into a temporal error and a spatial error with a rigorous analysis to the characteristic time discrete system we prove that the difference between the numerical solution and the solution of the time discrete system is tau independent where tau denotes the time stepsize thus numerical solution in w 1 infty w1 is bounded and optimal error estimates can be obtained in a traditional way numerical results confirm our analysis and show clearly the unconditional stability and convergence of the modified characteristics finite element method for the time dependent navier stokes equations the approach used in this paper can be easily extended to many other characteristics based methods

first order correction to the renormalization of the phase space lattice hamiltonian we develop a renormalization group analysis for a hamiltonian which is a periodic function of and this is a model for bloch electrons in a magnetic field and includes harper s model as a special case the eigenfunctions are bloch waves when the ratio of to the area of the unit cell is a rational number p q the renormalization procedure produces an effective hamiltonian for a subset of the spectrum which can be expanded in the zeroth order term has been obtained in earlier papers here we obtain the first order term of this expansion in terms of the q dimensional vectors which define the rational bloch states k and are bloch wavevectors the effective hamiltonian is not invariant under gauge transformations multiplying the vector we show that the infinitesimal gauge transformation is equivalent to evolving under a gauge hamiltonian obtained to lowest order in by quantizing

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

analytic representation of finite quantum systems a transform between functions in r and functions in zd is used to define the analogue of number and coherent states in the context of finite d dimensional quantum systems the coherent states are used to define an analytic representation in terms of theta functions all states are represented by entire functions with growth of order 2 which have exactly d zeros in each cell the analytic function of a state is constructed from its zeros results about the completeness of finite sets of coherent states within a cell are derived

penalty finite element method for the navier stokes equations abstract we present an analysis of a penalty formulation of the stationary navier stokes equations for an incompressible fluid subject to restrictions on the viscosity and prescribed body force it is shown that there exists a unique solution to this penalty problem the solution to the penalty problem is shown to converge to the solution of the navier stokes problem as o e where e 0 is the penalty parameter existence uniqueness and stability properties for the approximate problem are then developed and we derive estimates for finite element approximation of the penalized navier stokes problem presented here numerical studies are conducted to examine rates of convergence and sample numerical results presented for test cases

on the distribution of zeros of chaotic wavefunctions wavefunctions in a phase space representation can be expressed as entire functions of their zeros if the phase space is compact these zeros seem to hide a lot of relevant and explicit information about the underlying classical dynamics besides an understanding of their statistical properties may prove to be useful in the analytical calculations of the wavefunctions in quantum chaotic systems this motivates us to pursue the present study which by a numerical statistical analysis shows that both long range correlations as well as short range correlations exist between zeros while the latter turn out to be universal and parametric independent the former seem to be system dependent and are significantly affected by various parameters i e symmetry localization etc furthermore for the delocalized quantum dynamics the distribution of these zeros seem to mimic that of the zeros of the random functions as well as random polynomials

periodic orbit basis for the quantum baker map a set of quantum states dynamically related to the classical periodic orbits of a chaotic map is used as a basis in which the description of the eigenstates of its quantum version is greatly simplified this set can be improved with the inclusion of short time propagation along the stable and unstable manifolds of the periodic orbits resulting in a construction similar to the scar functions of vergini j phys a 33 4709 2000 the average participation ratio is used to quantify the quality of the basis

diabolical points in the spectra of triangles accidental degeneracies between energy levels e j and e j 1 of a real hamiltonian can occur generically in a family of hamiltonians labelled by at least two parameters x y energy level surfaces in e x y space have locally a double cone diabolo connection and we refer to the degeneracies themselves as diabolical points we studied the family of systems in which a particle moves freely within hard walled triangles vibrations of triangular membranes with x and y labelling two of the angles using an efficient green function technique to compute the levels we found several diabolical points for low lying levels as well as some symmetry degeneracies the lowest diabolical point occurred for levels 5 and 6 of the triangle 130 57 30 73 18 70 the conical structure was confirmed by noting that the normal derivative u of the wavefunction at a boundary point changed sign during a small circuit of the diabolical point the form of the variation of u around a circuit and the changing pattern of nodal lines of agreed with theoretical expectations an estimate of the total number of degeneracies n d j involving levels 1 through j based on the energy scaling of cone angles and the level spacing distribution gave n d j j 2 5 and our limited data support this prediction analytical theory confirmed that for thin triangles where our computational method is slow there are no degeneracies in the energy range studied

vortex structure and characterization of quasiperiodic functions quasiperiodic functions qpfs are characterized by their full vortex structure in one unit cell this characterization is much finer and more sensitive than the topological one given by the total vorticity per unit cell the chern index it is shown that qpfs with an arbitrarily prescribed vortex structure exist by constructing explicitly such a standard qpf two qpfs with the same vortex structure are equivalent in the sense that their ratio is a function which is strictly periodic nonvanishing and at least continuous a general qpf can then be approximately reconstructed from its vortex structure on the basis of the standard qpf and the equivalence concept as another application of this concept a simple method is proposed for calculating the quasiperiodic eigenvectors of periodic matrices possible applications to the quantum chaos problem on a phase space torus are briefly discussed

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

on extremality of abc spectral radius of a tree abstract the abc matrix of a graph g recently introduced by estrada is the square matrix of order g whose i j entry is equal to d i d j 2 d i d j if the i th vertex and the j th vertex of g are adjacent and 0 otherwise where d i is the degree of the i th vertex of g the abc spectral radius of g is the largest eigenvalue of the abc matrix of g which is denoted by g in this paper we prove that for any tree t of order n 3 2 cos n 1 t n 2 with equality in the left resp right inequality if and only if t is isomorphic to the path p n resp the star k 1 n 1

multistep characteristic method for incompressible flow in porous media in this paper we present a multistep difference scheme for the problem of miscible displacement of incompressible fluid flow in porous media the discretization involves a three level time scheme based on the characteristic method and a five point finite difference scheme for space discretization we prove that the convergence is of order o h 2 dt 2 which is in contrast to the convergence of order o h dt proved for a singlestep characteristic with the same space discretization numerical experiments demonstrate the stability and second order convergence of the scheme

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

indicators of quantum chaos based on eigenvector statistics expansion coefficients of the coherent states in the basis of hamiltonian eigenstates contain information about the local character of motion of a quantum system the author analyses three quantities the minimal number of relevant eigenvectors the sum of moduli of coefficients and the shannon entropy of a coherent state and show that all of them might be used as indicators of quantum chaos they also allow one to distinguish between three known universality classes results obtained are confirmed by a numerical study of kicked tops

a six fold rotation operator for the wannier functions of the phase space lattice hamiltonian we consider a set of localized wannier functions which can be used to describe the bloch bands of one dimensional hamiltonians whose associated weyl function is periodic under a hexagonal lattice of translations in phase space these hamiltonians can be used to describe electrons on two dimensional hexagonal lattices penetrated by a magnetic field the effect on the wannier functions of a rotation of the phase plane is considered the resulting transformation is found to be surprisingly complicated when the quantized hall conductance integer for the bloch band is non zero

numerical analysis of convection diffusion reaction problems with higher order characteristics finite elements part i time discretization in this paper a higher order characteristics time discretization scheme is analyzed for a variable coefficient convection possibly degenerate diffusion reaction equation with mixed dirichlet robin boundary conditions first the proposed second order time discretization scheme is rigorously introduced for exact and approximate characteristics next under not very restrictive hypotheses on the data the l infty l 2 stability is proved and l infty l 2 error estimates of order o delta t 2 are obtained lagrange galerkin schemes using different finite elements spaces will be analyzed in the second part of this work to appear in siam j numer anal where quadrature formulas are proposed for practical implementation

the monge metric on the sphere and geometry of quantum states topological and geometrical properties of the set of mixed quantum states in the n dimensional hilbert space are analysed assuming that the corresponding classical dynamics takes place on the sphere we use the vector su 2 coherent states and the generalized husimi distributions to define the monge distance between two arbitrary density matrices the monge metric has a simple semiclassical interpretation and induces a non trivial geometry among all pure states the distance from the maximally mixed state proportional to the identity matrix admits the largest value for the coherent states while the delocalized chaotic states are close to this contrasts the geometry induced by the standard trace hilbert schmidt or bures metrics for which the distance from is the same for all pure states we discuss possible physical consequences including unitary time evolution and the process of decoherence we introduce also a simplified monge metric defined in the space of pure quantum states and more suitable for numerical computation

phase retrieval problem in quantum chaos and its relation to the origin of irreversibility i abstract the purpose of this paper are i to study characteristics features of phase in quantum dynamics corresponding to chaos and ii to discuss implication of these features for the origin of irreversibility the first is done by focusing attention on topological defects of phase in the husimi representation we find that the process of generating topological defects corresponds to the folding of morse smale horseshoe dynamics moreover the distribution of topological defects changes from regular to random under the time evolution this change is a distinctive feature of quantum dynamics corresponding to chaos the second is done in terms of phase retrieval problem we present numerical results which strongly suggest the following the randomness in the distribution of topological defects is the cause of difficulty in retrieving phase based on these results we present the following hypothesis the loss of phase information in statistical mechanics can be understood in terms of the randomness of phase in quantum dynamics corresponding to chaos

semiclassical wavefunctions of nonintegrable systems and localization on periodic orbits we review semiclassical approximations for wavefunctions including the ebk approximation for integrable systems and the recent work of bogomolny and berry for nonintegrable systems stemming from the periodic orbit theory of gutzwiller and balian and block in particular we focus on the localization around periodic orbits scarring first appreciated by heller and the description of this scarring in both coordinate and phase space we examine individual wavefunctions of a schematic shell model in phase space and find that few are ergodic we also find that the degree of localization depends both on the degree of chaos in the classical limit and on the nearness of the eigenvalue to an energy that quantizes the scarring periodic orbit

analysis of the spectrum of a particle on a triangular lattice with two magnetic fluxes by algebraic and numerical methods the authors consider an electron on a triangular lattice with two magnetic fluxes using a c algebra formalism a perturbation theory around rational fluxes is developed this semiclassical type of analysis leads to the wilkinson rammal formula for the energy eigenvalues the analytic result is verified numerically with high accuracy up to second order crossing of eigenvalues near band edges is shown analytically as well as numerically

intrinsic ordering of quasienergy states for mixed regular chaotic quantum systems zeros of the husimi distribution a method is proposed which allows an intrinsic ordering of the quasienergy states for a time periodic quantum system according to the number of essential zeros of the husimi distribution in phase space the corresponding classical system shows a typical mixed dynamics with coexisting regular and chaotic regions in phase space the ordering of states allows a straightforward comparison with semiclassical quantities moreover a direct connection of the organization of zeros with the underlying classical phase space dynamics is demonstrated

chaotic eigenfunctions in phase space we study individual eigenstates of quantized area preserving maps on the 2 torus which are classically chaotic in order to analyze their semiclassical behavior we use the bargmann husimi representations for quantum states as well as their stellar parametrization which encodes states through a minimal set of points in phase space the constellation of zeros of the husimi density we rigorously prove that a semiclassical uniform distribution of husimi densities on the torus entails a similar equidistribution for the corresponding constellations we deduce from this property a universal behavior for the phase patterns of chaotic bargmann eigenfunctions which is reminiscent of the wkb approximation for eigenstates of integrable systems though in a weaker sense in order to obtain more precise information on chaotic eigenconstellations we then model their properties by ensembles of random states generalizing former results on the 2 sphere to the torus geometry this approach yields statistical predictions for the constellations which fit quite well the chaotic data we finally observe that specific dynamical information e g the presence of high peaks like scars in husimi densities can be recovered from the knowledge of a few long wavelength fourier coefficients which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions