abrikosov lattices in finite domains in 1957 abrikosov published his work on periodic solutions to the linearized ginzburg landau equations abrikosov s analysis assumes periodic boundary conditions which are very different from the natural boundary conditions the minimizer of the ginzburg landau energy functional should satisfy in the present work we prove that the global minimizer of the fully non linear functional can be approximated in every rectangular subset of the domain by one of the periodic solution to the linearized ginzburg landau equations in the plane furthermore we prove that the energy of this solution is close to the minimum of the energy over all abrikosov s solutions in that rectangle

computations of the first eigenpairs for the schrodinger operator with magnetic field this paper is devoted to computations of eigenvalues and eigenvectors for the schrodinger operator with constant magnetic field in a domain with corners as the semi classical parameter h tends to 0 the eigenvectors corresponding to the smallest eigenvalues concentrate in the corners they have a two scale structure consisting of a corner layer at scale sqrt h and an oscillatory term at scale h the high frequency oscillations make the numerical computations particularly delicate we propose a high order finite element method to overcome this difficulty relying on such a discretization we illustrate theoretical results on plane sectors squares and other straight or curved polygons we conclude by discussing convergence issues

magnetic wkb constructions this paper is devoted to the semiclassical magnetic laplacian until now wkb expansions for the eigenfunctions were only established in the presence of a non zero electric potential here we tackle the pure magnetic case thanks to feynman hellmann type formulas and coherent states decomposition we develop here a magnetic born oppenheimer theory exploiting the multiple scales of the problem we are led to solve an effective eikonal equation in pure magnetic cases and to obtain wkb expansions we also investigate explicit examples for which we can improve our general theorem global wkb expansions quasi optimal agmon estimates and upper bound of the tunelling effect in symmetric cases we also apply our strategy to get more accurate descriptions of the eigenvalues and eigenfunctions in a wide range of situations analyzed in the last two decades

peak power in the 3d magnetic schrodinger equation this paper is devoted to the spectral analysis of the magnetic neumann laplacian on an infinite cone of aperture alpha when the magnetic field is constant and parallel to the revolution axis and when the aperture goes to zero we prove that the first n eigenvalues exist and admit asymptotic expansions in powers of alpha 2

lifting inequalities for polytopes we present a method of lifting linear inequalities for the flag f vector of polytopes to higher dimensions known inequalities that can be lifted using this technique are the non negativity of the toric g vector and that the simplex minimizes the cd index we obtain new inequalities for six dimensional polytopes in the last section we present the currently best known inequalities for dimensions 5 through 8

on simple polytopes letp be a simpled polytope ind dimensional euclidean space mathbb e d and let p be the subalgebra of the polytope algebra generated by the classes of summands ofp it is shown that the dimensions of the weight spaces r p of p are theh numbers ofp which describe the dehn sommerville equations between the numbers of faces ofp and reflect the duality between r p and d r p moreover p admits a lefschetz decomposition under multiplication by the element of 1 p corresponding top itself which yields a proof of the necessity of mcmullen s conditions in theg theorem on thef vectors of simple polytopes the lefschetz decomposition is closely connected with the new hodge riemann minkowski quadratic inequalities between mixed volumes which generalize minkowski s second inequality also proved are analogous generalizations of the aleksandrov fenchel inequalities a striking feature is that these are obtained without using brunn minkowski theory indeed the brunn minkowski theorem without characterization of the cases of equality can be deduced from them the connexion found between p and the face ring of the dual simplicial polytopep enables this ring to be looked at in two ways and a conjectured formulation of theg theorem in terms of a gale diagram ofp is also established

peak quasisymmetric functions and eulerian enumeration abstract via duality of hopf algebras there is a direct association between peak quasisymmetric functions and enumeration of chains in eulerian posets we study this association explicitly showing that the notion of cd index long studied in the context of convex polytopes and eulerian posets arises as the dual basis to a natural basis of peak quasisymmetric functions introduced by stembridge thus eulerian posets having a nonnegative cd index for example face lattices of convex polytopes correspond to peak quasisymmetric functions having a nonnegative representation in terms of this basis we diagonalize the operator that associates the basis of descent sets for all quasisymmetric functions to that of peak sets for the algebra of peak functions and study the g polynomial for eulerian posets as an algebra homomorphism

surface superconductivity in 3 dimensions we study the ginzburg landau system for a superconductor occupying a 3 dimensional bounded domain and improve the estimate of the upper critical field h c3 obtained by k lu and x pan in j diff eqns 168 2000 386 452 we also analyze the behavior of the order parameters we show that under an applied magnetic field lying below and not far from h c3 order parameters concentrate in a vicinity of a sheath of the surface that is tangential to the applied field and exponentially decay both in the normal and tangential directions away from the sheath in the l 2 sense as the applied field decreases further but keeps in between and away from h c2 and h c3 the superconducting sheath expands but does not cover the entire surface and superconductivity at the surface portion orthogonal to the applied field is always very weak this phenomenon is significantly different to the surface superconductivity on a cylinder of infinite height studied by x pan in comm math phys 228 2002 327 370 where under an axial applied field lying in between h c2 and h c3 the entire surface is in the superconducting state

analytic description of layer undulations in smectic a liquid crystals we investigate the layer undulations that appear in smectic a liquid crystals when a magnetic field is applied in the direction parallel to the smectic layers in an earlier work garcia cervera and joo in j comput theor nanosci 7 795 801 2010 the authors characterized the critical field using the landau de gennes model for smectic a liquid crystals in this paper we obtain an asymptotic expression of the unstable modes using convergence theory and a sharp estimate of the critical field under the assumption that the layers are fixed at the boundaries the maximum layer undulation occurs in the middle of the cell and the displacement amplitude decreases near the boundaries our estimate of the critical field is consistent with the helfrich hurault theory when natural boundary conditions are considered the displacement amplitude does not diminish near the boundary in sharp contrast with the dirichlet case and the critical field is reduced compared to the one calculated in the classical theory this is consistent with the experiments carried out by ishikawa and lavrentovich phys rev e 63 030501 r 2001 furthermore we prove the existence and stability of the solution to the nonlinear system of the landau de gennes model using bifurcation theory numerical simulations are used to illustrate the predictions of the analysis

non linear surface superconductivity for type ii superconductors in the large domain limit the ginzburg landau model for superconductivity is considered in two dimensions we show for smooth bounded domains that superconductivity remains concentrated near the surface when the applied magnetic field is decreased below hc3 as long as it is greater than hc2 we demonstrate this result in the large domain limit i e when the domain s size tends to infinity additionally we prove that for applied fields greater than hc2 the only solution in 2 satisfying normal state conditions at infinity is the normal state the above results have been proved in the past for the linear case here we prove them for non linear problems

twisted incidence algebras and kazhdan lusztig stanley functions abstract we introduce a new multiplication in the incidence algebra of a partially ordered set and study the resulting algebra as an application of the properties of this algebra we obtain a combinatorial formula for the kazhdan lusztig stanley functions of a poset as special cases this yields new combinatorial formulas for the parabolic and inverse parabolic kazhdan lusztig polynomials for the generalized toric h vector of an eulerian poset and for the lusztig vogan polynomials

optimal prefix codes with fewer distinct codeword lengths are faster to construct abstract a new method for constructing minimum redundancy binary prefix codes is described our method does not explicitly build a huffman tree instead it uses a property of optimal prefix codes to compute the codeword lengths corresponding to the input weights let n be the number of weights and k be the number of distinct codeword lengths as produced by the algorithm for the optimal code the running time of our algorithm is o k n following our previous work no algorithm can possibly construct optimal prefix codes in o k n time if the given weights are presorted our algorithm performs o 9 k log 2 k n comparisons

on the third critical field in ginzburg landau theory using recent results by the authors on the spectral asymptotics of the neumann laplacian with magnetic field we give precise estimates on the critical field h c 3 describing the appearance of superconductivity in superconductors of type ii furthermore we prove that the local and global definitions of this field coincide near h c 3 only a small part near the boundary points where the curvature is maximal of the sample carries superconductivity we give precise estimates on the size of this zone and decay estimates in both the normal to the boundary and parallel variables

the phase transitions from chiral nematic toward smectic liquid crystals a chen lubensky energy is used to investigate phase transitions from chiral nematic to smectic c and smectic a liquid crystal phases we consider a liquid crystalline material confined between two parallel plates where the dimensions of are assumed to be large relative both to the width of a smectic layer and the material s chiral pitch we take boundary conditions so that the smectic phase melts at the plates surfaces and prove the existence of energy minimizers in an admissible set consisting of order parameters psi in mathcal h 2 0 omega and molecular directors mathbf n in mathbf w 1 2 omega mathbb s 2 then under the physically observed assumption that the frank elasticity constants become large near a phase transition we establish estimates for the transition region separating phases in particular we derive analytic estimates proving that chirality lowers the transition temperature regime above which minimizers are nematic and below which minimizers are in a smectic phase

emergence of the triangular lattice near a flat wall the bifurcation of periodic solutions near a flat wall for applied magnetic fields which are slightly weaker than hc2 is considered for a reduced ginzburg landau model obtained in the large limit we formally demonstrate that following the bifurcation of the first mode when the applied magnetic field is further decreased there is a second bifurcation after which the solution develops continuously into the well known triangular lattice

design and analysis of dynamic huffman codes a new one pass algorithm for constructing dynamic huffman codes is introduced and analyzed we also analyze the one pass algorithm due to faller gallager and knuth in each algorithm both the sender and the receiver maintain equivalent dynamically varying huffman trees and the coding is done in real time we show that the number of bits used by the new algorithm to encode a message containing t letters is t bits more than that used by the conventional two pass huffman scheme independent of the alphabet size this is best possible in the worst case for any one pass huffman method tight upper and lower bounds are derived empirical tests show that the encodings produced by the new algorithm are shorter than those of the other one pass algorithm and except for long messages are shorter than those of the two pass method the new algorithm is well suited for on line encoding decoding in data networks and for tile compression

verification of minimum redundancy prefix codes we show that verifying a given prefix code for optimality requires spl omega nlogn time indicating that the verification problem is not asymptotically easier than the construction problem alternatively we give linear time verification algorithms for several special cases that are either typical in practice or theoretically interesting

the onset of superconductivity in semi infinite strips the existence of a superconducting branch bifurcating from the normal state is proved in semi infinite strips it is proved that the critical magnetic field at which bifurcation takes place or the onset field for a semi infinite strip is greater than the onset field for an infinite strip with the same width in addition we find the loci of the vortices far away from the corners and show convergence of the bifurcating modes in long rectangles to those in the semi infinite strip with the same width

numerical approximations of the ginzburg landau models for superconductivity in this paper we review various methods for the numerical approximations of the ginzburg landau models of superconductivity particular attention is given to the different treatment of gauge invariance in both the finite element finite difference and finite volume settings representative theoretical results typical numerical simulations and computational challenges are presented generalizations to other relevant models are also discussed

remarks on the spectrum of the neumann problem with magnetic field in the half space we consider a schrodinger operator with a constant magnetic field in a one half three dimensional space with neumann type boundary conditions it is known from the works by lu pan and helffer morame that the lower bound of its spectrum is less than b the intensity of the magnetic field provided that the magnetic field is not normal to the boundary we prove that the spectrum under b is a finite set of eigenvalues each of infinite multiplicity in the case when the angle between the magnetic field and the boundary is small we give a sharp asymptotic expansion of the number of these eigenvalues

radial solutions for the ginzburg landau equation with applied magnetic field abstract the purpose of this work is a systematic study of symmetric vortices for the ginzburg landau model of superconductivity along a cylinder with applied magnetic field parallel to its axis the ginzburg landau constant of the material and the degree d of the vortex are fixed for any given parameters r the radius of the cylinder and h the intensity of the applied magnetic field one can find a symmetric vortex a which satisfies the boundary conditions b r e i d and curl a b r he 3 it is then shown that symmetric vortices form a family depending continuously on two real parameters and c which describe the behaviour at the center of the vortex as the boundary conditions depend smoothly on those parameters one can distinguish two main connected domains of vortices the first one defined by the boundary conditions b r e i d and ra r b r is a zone of stability where remains increasing the second one defined by the boundary conditions b r e i d and ra r b r d is a zone where some instability can occur attention is focused on the smooth branch of vortices which separates those two domains it is indexed by the parameter running from some l 0 to and the limit as decreases to l corresponds to a berger and chen type vortex

stable nucleation for the ginzburg landau system with an applied magnetic field we consider the ginzburg landau system with an applied magnetic field and analyze the behavior of solutions when the domain is a cylinder of radius overline r and the applied field is parallel to the axis it is shown that there is an upper critical value overline h such that if the modulus of the applied field is greater than overline h the normal nonsuperconducting state in which the order parameter is identically zero is stable and if the modulus of the applied field is slightly below oh the normal state is unstable in addition it is shown that there is a positive lower critical value underline h leq overline h such that the normal state is unstable if the modulus of the applied field is less than underline h and stable if the modulus is slightly above underline h in the case of type ii materials for whic h the ginzburg landau constant is large it is shown that there is a discrete set of radii such that if overline r notin cal b kappa and kappa overline r is sufficiently large then for each applied field of modulus slightly less than overline h or slightly more than underline h there is precisely one small superconducting solution up to a gauge transformation which is stable moreover for this solution the complex valued order parameter is zero only on the axis of the cylinder and its winding number is proportional to the product of 2 and the cross sectional area of the cylinder in addition the solution exhibits surface superconductivity as predicted by the physicists de gennes and st james

schrodinger operators with non degenerately vanishing magnetic fields in bounded domains we establish an asymptotic estimate of the lowest eigenvalue bf of the schrodinger operator 2 bf with a magnetic field in a bounded 2 dimensional domain where curl f vanishes non degenerately and b is a large parameter our study is based on an analysis on an eigenvalue variation problem for the sturm liouville problem using the estimate we determine the value of the upper critical field for superconductors subject to non homogeneous applied magnetic fields and localize the nucleation of superconductivity

the ultimate planar convex hull algorithm we present a new planar convex hull algorithm with worst case time complexity o n log h where n is the size of the input set and h is the size of the output set i e the number of vertices found to be on the hull we also show that this algorithm is asymptotically worst case optimal on a rather realistic model of computation even if the complexity of the problem is measured in terms of input as well as output size the algorithm relies on a variation of the divide and conquer paradigm which we call the marriage before conquest principle and which appears to be interesting in its own right

the zero set of the order parameter in two dimensions we survey the possible zero sets of the order parameter wavefunction in two dimensional domains certain structures are feasible including the well known abrikosov lattice the giant vortex and the line cut in multiply connected domains other proposed structures such as the vortex ring and the noninteger vortex point are shown to be impossible

three space economical algorithms for calculating minimum redundancy prefix codes the minimum redundancy prefix code problem is to determine for a given list w spl omega sub 1 spl omega sub n of n positive symbol weights a list l l sub 1 l sub n of n corresponding integer codeword lengths such that spl sigma sub i 1 sup n 2 sup li spl les 1 and spl sigma sub i 1 sup n spl omega sub i l sub i is minimized let us consider the case where w is already sorted in this case the output list l can be represented by a list m m sub 1 m sub h where m sub l for l 1 h denotes the multiplicity of the codeword length l in l and h is the length of the greatest codeword fortunately h is proved to be o min log 1 p sub 1 n where p sub 1 is the smallest symbol probability given by spl omega sub 1 spl sigma sub i 1 sup n spl omega sub i we present the fast lazyhuff f lazyhuff the economical lazyhuff e lazyhuff and the best lazyhuff b lazyhuff algorithms f lazyhuff runs in o n time but requires o min h sup 2 n additional space on the other hand e lazyhuff runs in o n nlog n h time requiring only o h additional space finally b lazyhuff asymptotically overcomes the previous algorithms requiring only o n time and o h additional space moreover our three algorithms have the advantage of not writing over the input buffer during code calculation a feature that is very useful in some applications

magnetic bottles in connection with superconductivity abstract motivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension two a lot of papers devoted to the analysis in a semi classical regime of the lowest eigenvalue of the schrodinger operator with magnetic field have appeared recently here we mention the works by bernoff sternberg lu pan and del pino felmer sternberg this partially recovers questions analyzed in a different context by the authors around the question of the so called magnetic bottles our aim is to analyze the former results to treat them in a more systematic way and to improve them by giving sharper estimates of the remainder in particular we improve significantly the lower bounds and as a by product we solve a conjecture proposed by bernoff sternberg concerning the localization of the ground state inside the boundary in the case with constant magnetic fields

hard lefschetz theorem for nonrational polytopes the hard lefschetz theorem is known to hold for the intersection cohomology of the toric variety associated to a rational convex polytope one can construct the intersection cohomology combinatorially from the polytope hence it is well defined even for nonrational polytopes when there is no variety associated to it we prove the hard lefschetz theorem for the intersection cohomology of a general polytope

boundary concentration for eigenvalue problems related to the onset of superconductivity we examine the asymptotic behavior of the eigenvalue h and corresponding eigenfunction associated with the variational problem in the regime h 1 here a is any vector field with curl equal to 1 the problem arises within the ginzburg landau model for superconductivity with the function h yielding the relationship between the critical temperature vs applied magnetic field strength in the transition from normal to superconducting state in a thin mesoscopic sample with cross section omega subset r 2 we first carry out a rigorous analysis of the associated problem on a half plane and then rigorously justify some of the formal arguments of bs obtaining an expansion for while also proving that the first eigenfunction decays to zero somewhere along the sample boundary when is not a disc for interior decay we demonstrate that the rate is exponential

eigenvalue problems of ginzburg landau operator in bounded domains in this paper we study the eigenvalue problems for the ginzburg landau operator with a large parameter in bounded domains in r2 under gauge invariant boundary conditions the estimates for the eigenvalues are obtained and the asymptotic behavior of the associated eigenfunctions is discussed these results play a key role in estimating the critical magnetic field in the mathematical theory of superconductivity