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

global strong solution for the korteweg system in dimension n 2 this work is devoted to prove the existence of global strong solution in dimension n 2 for a general isothermal model of capillary fluids derived by j e dunn and j serrin 1985 see 16 which can be used as a phase transition model in a first part we prove the existence of strong solution in finite time for large initial data with a precise bound by below on the life span t of existence of the solution this one depends on the norm of the initial data 0 v 0 the second part consists in proving the existence of global strong solution with particular choice on the capillary coefficient 2 and on the viscosity tensor which corresponds to the shallow water case 2 div du to do this we derivate different energy estimate on the density and the effective velocity v which ensures that the strong solution can be extended beyond t the main difficulty consists in controlling the vacuum or in other words to estimate the l norm of 1 the proof relies mostly on a method introduced by de giorgi 14 see also ladyzhenskaya et al in 30 for the parabolic case to obtain regularity results for elliptic equations with discontinuous diffusion coefficients and a suitable bootstrap argument

influence of the anisotropy parameter δ on the spectrum of the generalized q symmetrized harper equation the 1 generalization of the q symmetrized harper equation is discussed in terms of wavefunctions expressed by laurent series proceeding by recursion leads to a nontrivial dependent generalization of the characteristic energy polynomial with a special emphasis on a continuous dependence on the commensurability parameter the multiplicity parameter which is responsible for the amount of coprime realizations of the commensurability parameter is also accounted for the present energies have been derived so as to reproduce particular ones obtained before as limiting cases

gauss runge kutta time discretization of wave equations on evolving surfaces this paper is concerned with an error analysis for a full discretization of the linear wave equation on a moving surface the equation is discretized in space by the evolving surface finite element method discretization in time is done by gau runge kutta grk methods aiming for higher order accuracy in time and unconditional stability of the fully discrete scheme the latter is established in the natural time dependent norms by using the algebraic stability and the coercivity property of the grk methods together with the properties of the spatial semi discretization under sufficient regularity conditions optimal order error estimates for this class of fully discrete methods are shown numerical experiments are presented to confirm some of the theoretical results

spherical ornstein uhlenbeck processes the paper considers random motion of a point on the surface of a sphere in the case where the angular velocity is determined by an ornstein uhlenbeck process the solution is fully characterised by only one dimensionless number the persistence angle which is the typical angle of rotation during the correlation time of the angular velocity

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

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

a convergent finite volume scheme for diffusion on evolving surfaces a finite volume scheme for transport and diffusion problems on evolving hypersurfaces is discussed the underlying motion is assumed to be described by a fixed not necessarily normal velocity field the ingredients of the numerical method are an approximation of the family of surfaces by a family of interpolating simplicial meshes where grid vertices move on motion trajectories a consistent finite volume discretization of the induced transport on the simplices and a proper incorporation of a diffusive flux balance at simplicial faces the semi implicit scheme is derived via a discretization of the underlying conservation law and discrete counterparts of continuous a priori estimates in this geometric setup are proved the continuous solution on the continuous family of evolving surfaces is compared to the finite volume solution on the discrete sequence of simplicial surfaces and convergence of the family of discrete solutions on successively refined meshes is proved under suitable assumptions on the geometry and the discrete meshes furthermore numerical results show remarkable aspects of transport and diffusion phenomena on evolving surfaces and experimentally reflect the established convergence results finally we discuss how to combine the presented scheme with a corresponding finite volume scheme for advective transport on the surface via an operator splitting and present some applications

transport and diffusion of material quantities on propagating interfaces via level set methods we develop theory and numerical algorithms to apply level set methods to problems involving the transport and diffusion of material quantities in a level set framework level set methods are computational techniques for tracking moving interfaces they work by embedding the propagating interface as the zero level set of a higher dimensional function and then approximate the solution of the resulting initial value partial differential equation using upwind finite difference schemes the traditional level set method works in the trace space of the evolving interface and hence disregards any parameterization in the interface description consequently material quantities on the interface which themselves are transported under the interface motion are not easily handled in this framework we develop model equations and algorithmic techniques to extend the level set method to include these problems we demonstrate the accuracy of our approach through a series of test examples and convergence studies

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

an su 2 analogue of the azbel hofstadter hamiltonian motivated by the recent findings of wiegmann zabrodin and faddeev kashaev concerning the appearance of the quantum symmetry in the problem of a bloch electron on a two dimensional magnetic lattice we introduce a modification of the tight binding azbel hofstadter hamiltonian that is a specific spin s euler top that can be considered as its classical analogue the eigenvalue problem for the proposed model in the coherent state representation is described by the s gap lame equation and thus is completely solvable we observe a striking similarity between the shapes of the spectra of the two models for various values of the spin s

on the multifractal spectrum of the fibonacci chain we consider the multifractal properties of a quasiperiodic tight binding hamiltonian where the hopping elements are arranged according to the fibonacci chain by using the trace map approach and an assumption relating the cycles of the trace map to the integrated density of states it is shown that the maximal scaling of the spectrum does not always occur at the edges or the centre of the spectrum which is confirmed by numerical simulations a good description of the multifractal spectrum is obtained by applying the 3z model which was recently developed for the harper model

localization problem in one dimension mapping and escape a one dimensional schr odinger equation in a discontinuous quasiperiodic potential is reduced to a recursion relation for transfer matrices and then to one for traces of these matrices when the potential is periodic the bandwidth goes to zero as an algebraic function of the period with a critical index which depends upon the potential strength this critical index is also evaluated as the solution to an escape rate problem for the recursion relations

a grid based particle method for solving partial differential equations on evolving surfaces and modeling high order geometrical motion we develop numerical methods for solving partial differential equations pde defined on an evolving interface represented by the grid based particle method gbpm recently proposed in s leung h k zhao a grid based particle method for moving interface problems j comput phys 228 2009 7706 7728 in particular we develop implicit time discretization methods for the advection diffusion equation where the time step is restricted solely by the advection part of the equation we also generalize the gbpm to solve high order geometrical flows including surface diffusion and willmore type flows the resulting algorithm can be easily implemented since the method is based on meshless particles quasi uniformly sampled on the interface furthermore without any computational mesh or triangulation defined on the interface we do not require remeshing or reparametrization in the case of highly distorted motion or when there are topological changes as an interesting application we study locally inextensible flows governed by energy minimization we introduce tension force via a lagrange multiplier determined by the solution to a helmholtz equation defined on the evolving interface extensive numerical examples are also given to demonstrate the efficiency of the proposed approach

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

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

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

existence of global strong solutions in critical spaces for barotropic viscous fluids this paper is dedicated to the study of viscous compressible barotropic fluids in dimension n 2 we address the question of the global existence of strong solutions for initial data close to a constant state having critical besov regularity first this article shows the recent results of charve and danchin arch ration mech anal 198 1 233 271 2010 and chen et al commun pure appl math 63 1173 1224 2010 with a new proof our result relies on a new a priori estimate for the velocity that we derive via the intermediary of the effective velocity which allows us to cancel out the coupling between the density and the velocity as in haspot well posedness in critical spaces for barotropic viscous fluids 2009 second we improve the results of charve and danchin 2010 and chen et al 2010 by adding as in charve and danchin 2010 some regularity on the initial data in low frequencies in this case we obtain global strong solutions for a class of large initial data which rely on the results of hoff arch rational mech anal 139 303 354 1997 hoff commun pure appl math 55 11 1365 1407 2002 and hoff j math fluid mech 7 3 315 338 2005 and those of charve and danchin 2010 and chen et al 2010 we conclude by generalizing these results for general viscosity coefficients

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

global existence in critical spaces for flows of compressible viscous and heat conductive gases we are concerned with global existence and uniqueness of strong solutions for a general model of viscous and heat conductive gases the initial data are supposed to be close to a stable equilibrium with constant density and temperature using uniform estimates for the linearized system with a convection term we get global well posedness in a functional setting invariant with respect to the scaling of the associated equations in space dimension n 3 we also show a smoothing effect on the velocity and the temperature and a decay on the difference between the density and the constant reference state these results extend a previous paper devoted to the barotropic case see 5

relative entropy in multi phase models of 1d elastodynamics convergence of a non local to a local model abstract in this paper we study a local and a non local regularization of the system of nonlinear elastodynamics with a non convex energy we show that solutions of the non local model converge to those of the local model in a certain regime the arguments are based on the relative entropy framework and provide an example of how local and non local regularizations may compensate for non convexity of the energy and enable the use of the relative entropy stability theory even if the energy is not quasi or poly convex

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

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

l 2 estimates for the evolving surface finite element method in this paper we consider the evolving surface nite element method for the advection and di usion of a conserved scalar quantity on a moving surface in an earlier paper using a suitable variational formulation in time dependent sobolev space we proposed and analysed a nite element method using surface nite elements on evolving triangulated surfaces an optimal order h error bound was proved for linear nite elements in this work we prove the optimal error bound in l t uniformly in time

on the total bandwidth for the rational harper s equation in the last years several contributions have been done around the total bandwidth of the spectrum for the harper s operator in particular an interesting conjecture has been proposed by thouless which gives also strong convincing arguments for the proof in special cases on the other hand in the study of the cantor structure of the spectrum b helffer and j sjostrand have justified an heuristic semiclassical approach proposed by m wilkinson the aim of this article is to explain how one can use the first step of this approach to give a rigorous semi classical proof of the thouless formula in some of the simplest cases we shall also indicate how one can hope with more effort to prove rigorously recent results of last and wilkinson on the same conjecture

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

on a lagrangian method for the convergence from a non local to a local korteweg capillary fluid model in the present article we are interested in further investigations for the barotropic compressible navier stokes system endowed with a non local capillarity we studied in 7 thanks to an accurate study of the associated linear system using a lagrangian change of coordinates we provide more precise energy estimates in terms of hybrid besov spaces naturally depending on a threshold frequency which is determined in function of the physical parameter distinguishing the low and the high regimes it allows us in particular to prove the convergence of the solutions from the non local to the local korteweg system another mathematical interest of this article is the study of the effect of the lagrangian change on the non local capillary term

symmetry properties of exact energy solutions to the harper equation and related q normalizations proofs are given that exact solutions to the wavefunctions of the harper equation can be established in an explicit manner by resorting to three term recurrence relations implied by a q calculus approach proposed previously the q normalization of wavefunctions resulting in the appearance of peaks is discussed the exact q 6 energy solution has also been derived and analysed in some more detail there are reasons to say that fractal structures concerning band and gap distributions have to be accounted for if q 6

energy spectra for one dimensional quasiperiodic potentials bandwidth scaling mapping and relation with local isomorphism energy spectra for one dimensional tight binding models with two types of quasiperiodic potentials are studied for which the incommensurability is characterised by quadratic irrationals one is the step potential model for which the structure is a generalised fibonacci chain for special structures scaling properties of the spectrum are found numerically a critical index delta for the total bandwidth is determined after deriving recursion relations for the trace of transfer matrices it is shown that generalised fibonacci chains have the same energy spectrum if and only if they are locally isomorphic the other potential is sinusoidal for which the critical index delta is determined at the critical point

a global existence result for the compressible navier stokes equations in the critical l p framework the present paper is dedicated to the global well posedness issue for the barotropic compressible navier stokes system in the whole space mathbb r d with d 2 we aim at extending the work by danchin inventiones mathematicae 141 3 579 614 2000 to a critical framework which is not related to the energy space for small perturbations of a stable equilibrium state in the sense of suitable lp type besov norms we establish the global existence as a consequence like for incompressible flows one may exhibit a class of large highly oscillating initial velocity fields for which global existence and uniqueness holds true in passing we obtain new estimates for the linearized and the paralinearized systems which may be of interest for future works on compressible flows

lie algebra of diffat2 and bloch electrons in a constant uniform magnetic field magnetic translation operators that generate symmetries of a bloch electron in a constant uniform magnetic field are shown to span the sine bracket algebra which is the unique lie algebra deformation of the area preserving diffeomorphisms of a two torus the deformation parameter is identified with the number of magnetic flux quanta through a unit cell of the substrate lattice the symmetries of a spin 1 2 bloch electron are shown to realize the supersymmetric extension of the sine bracket algebra

local in time results for local and non local capillary navier stokes systems with large data abstract in this article we study three capillary compressible models the classical local navier stokes korteweg system and two non local models for large initial data bounded away from zero and with a reference pressure state which is not necessarily stable p can be non positive we prove that these systems have a unique local in time solution and we study the convergence rate of the solutions of the non local models towards the local korteweg model the results are given for constant viscous coefficients and we explain how to extend them for density dependant coefficients

existence of global strong solution for korteweg system with large infinite energy initial data this work is devoted to the study of the initial boundary value problem for a general isothermal model of capillary fluids derived by j e dunn and j serrin 1985 see cite fds which can be used as a phase transition model we will prove the existence of local and global under a condition of smallness on the initial data strong solutions with discontinuous initial density where ln rho 0 belongs to the besov space b n 2 infty r n in passing our result allow to consider initial data with discontinuous interfaces up our knowledge in the literature the results of existence of strong solutions require generally continuous initial density it implies also that we can work with initial data space of infinite energy our result relies on the fact that the density can be written as the sum of the solution rho l associated to linear system and a remainder density bar rho which is more regular than rho l by taking into account the regularizing effects induced on the bilinear convection term the main difficulty concerns the proof of new estimate of maximum principle type for the linear system associated to the korteweg system the proof is based on a characterization of the besov space in terms of the semi group associated to this linear system let also mention that we prove the existence of global strong solution with a smallness hypothesis which is subcritical in terms of the scaling of the equations it allows us to exhibit a family of large energy initial data for the scaling of the equations providing global strong solution in particular for the first time up our knowledge we show the existence of global strong solution for some large energy initial data when n 2 we finish this paper by introducing the notion of quasi solutions for the korteweg s system a tool which has been developed in the framework of the compressible navier stokes equations cite arxiv arxiv1 hal cras1 cras2 which enables us to improve the previous result and by obtaining the existence of global strong solution with large initial velocity in b n 1 2 infty in particular when n 2 it implies the existence of global strong solution with large energy data as a corollary we get global existence and uniqueness for highly compressiblekorteweg system when n geq2 it means that for any large initial data under an irrotational condition on the initial velocity we have the existence of global strong solution provided that the pressure is sufficiently highly compressible

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

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

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

total bandwidth for harper equation correction to renormalization analysis this paper studies the correction to the effective hamiltonian of the harper equation in respect of the total bandwidth the effective hamiltonian describes the further splitting from the spectrum of a lower order fraction when the spectrum of a higher order fraction is considered the total bandwidth for a split sub band is given by the thouless scaling law modified by the curvature of the effective hamiltonian at its saddle points in this paper it is shown that the leading term of the effective hamiltonian is exactly solvable moreover the correction to the curvature of the effective hamiltonian is obtained numerically and the explicit analytical expression is also found this gives the second sum rule which is of importance in deriving the thouless scaling law for a generic fraction

lumped finite elements for reaction cross diffusion systems on stationary surfaces all the authors am is cv mf thank the isaac newton institute for mathematical sciences for its hospitality during the programme coupling geometric pdes with physics for cell morphology motility and pattern formation epsrc ep k032208 1 this work am has received funding from the european union s horizon 2020 research and innovation programme under the marie sklodowska curie grant agreement no 642866 am and cv acknowledge support from the engineering and physical sciences research council ep j016780 1 on modelling analysis and simulation of spatial patterning on evolving biological surfaces and the leverhulme trust research project grant rpg 2014 149 on unravelling new mathematics for 3d cell migration am was partially supported by a fellowship from the simons foundation am is a royal society wolfson research merit award holder generously funded by the wolfson foundation

global strong solution for the korteweg system with quantum pressure in dimension n ge 2 this work is devoted to prove the existence of global strong solution in dimension n ge 2 for a isothermal model of capillary fluids derived by dunn and serrin see arch ration mech anal 88 2 95 133 1985 which can be used as a phase transition model we will restrict us to the case of the so called compressible navier stokes system with quantum pressure which corresponds to the korteweg system with capillary coefficient such that kappa rho frac kappa 1 rho with kappa 1 0 and rho the density in a first part we prove the existence of strong solution in finite time for large initial data in critical besov spaces with precise estimate on the life span t the second part consists in proving the existence of global strong solution with large initial data for a specific choice on the capillary coefficient kappa 1 mu 2 with mu the viscosity coefficient to do this we derive different energy estimates on the density and the effective velocity v which allow us to extend the strong solution beyond t the main difficulty consists in estimating the l infty norm of frac 1 rho the proof relies mostly on a method introduced by de giorgi mem accad sci torino cl sci fis mat nat 3 3 25 43 1957 see also ladyzhenskaya et al linear and quasilinear equations of parabolic type ams translations providence 1968 for the parabolic case to obtain regularity results for elliptic equations with discontinuous diffusion coefficients and a suitable bootstrap argument

global existence in critical spaces for compressible navier stokes equations we investigate global strong solutions for isentropic compressible fluids with initial data close to a stable equilibrium we obtain the existence and uniqueness of a solution in a functional setting invariant by the scaling of the associated equations more precisely the initial velocity has the same critical regularity index as for the incompressible homogeneous navier stokes equations and one more derivative is needed for the density we point out a smoothing effect on the velocity and a l 1 decay on the difference between the density and the constant reference state the proof lies on uniform estimates for a mixed hyperbolic parabolic linear system with a convection term

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

level statistics in a quasiperiodic system the authors examine the level statistics of harper s equation a simple model of a one dimensional quasiperiodic system through the distribution of the spacings between adjacent eigenvalues each normalised by the average local density of states they present exact results for the distribution for both small and large amplitude alpha of the incommensurate potential showing it has a simple form in each of these limits they show numerically that these distributions are preserved as alpha approaches its critical value 2 from each side and also that at the critical point the distribution of the normalised spacings has a distinct but simple form

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