a hill series approach to wavefunction nodes the hill series method is shown to have several advantages for treating the schrodinger equation when the wavefunction obeys the dirichlet boundary condition r 0 a novel combination of the virial theorem the hill series method and a finite difference method is used to find the node positions of the wavefunction for several excited states of perturbed oscillator and coulombic systems and to find critical radii for the hydrogen atom

the affine polya szego principle equality cases and stability a brothers ziemer type theorem for the affine polya szego principle and a quantitative affine polya szego principle are established

harmonic crystal on the wall a microscopic approach a three dimensional winterbottom type construction in the regime of partial wetting is derived in a scaling limit of a gas of microscopic gaussian sos droplets under the fixed volume constraint the proof is based on a coarse graining of the random microscopic region wetted by the crystal random walk representations of various quantities related to free massless fields and a stability analysis of the torsional rigidity problem

a mass transportation approach to quantitative isoperimetric inequalities a sharp quantitative version of the anisotropic isoperimetric inequality is established corresponding to a stability estimate for the wulff shape of a given surface tension energy this is achieved by exploiting mass transportation theory especially gromov s proof of the isoperimetric inequality and the brenier mccann theorem a sharp quantitative version of the brunn minkowski inequality for convex sets is proved as a corollary

comment on the letter a new efficient method for calculating perturbation energies using functions which are not quadratically integrable by l skala and j cizek we suggest that the recent numerical integration perturbative approach to bound states as proposed by skala and cizek may be generalized via its renormalizations

finite difference order doubling in two dimensions an order doubling process previously used to obtain eighth order eigenvalues from the fourth order numerov method is applied to the perturbed oscillator in two dimensions a simple method of obtaining high order finite difference operators is reported and an odd parity boundary condition is found to be effective in facilitating the smooth operation of the order doubling process

characterization of sobolev and bv spaces abstract the main results of this paper are new characterizations of w 1 p 1 p and bv for r n an arbitrary open set using these results we answer some open questions of brezis 2002 10 and ponce 2004 25

existence and uniqueness for anisotropic and crystalline mean curvature flows an existence and uniqueness result up to fattening for crystalline mean curvature flows with forcing and arbitrary convex mobilities is proven this is achieved by introducing a new notion of solution to the corresponding level set formulation such a solution satisfies the comparison principle and a stability property with respect to the approximation by suitably regularized problems the results are valid in any dimension and for arbitrary possibly unbounded initial closed sets the approach accounts for the possible presence of a time dependent bounded forcing term with spatial lipschitz continuity as a by product of the analysis the problem of the convergence of the almgren taylor wang minimizing movements scheme to a unique up to fattening flat flow in the case of general possibly crystalline anisotropies is settled

blow up analysis of a finsler liouville equation in two dimensions abstract in this paper we establish a blow up analysis for the finsler liouville equation q u v x e u in two dimensions here the operator q is the so called finsler laplacian which is a generalized laplacian as a byproduct we obtain an existence result

the sharp quantitative isoperimetric inequality a quantitative sharp form of the classical isoperimetric inequality is proved thus giving a positive answer to a conjecture by hall

anisotropic singular perturbations the vectorial case we obtain the l 1 limit of the sequence where e e is the family of anisotropic perturbations of the nonconvex functional of vector valued functions the proof relies on the blow up argument introduced by fonseca and muller

nonlocal approximation of nonisotropic free discontinuity problems we prove that a class of free discontinuity problems with nonisotropic bulk and surface energy densities is approximated in the sense of gamma convergence by suitable families of nonlocal integral functionals some connections with the wulff set are also pointed out

boolean functions and finite difference eigenvalues finite difference calculations involving the expectation values of boolean functions are shown to yield detailed information about the properties of the energy levels and eigenfunctions for double well potentials

minimal energy for elastic inclusions we consider a variant of the isoperimetric problem with a non local term representing elastic energy more precisely our aim is to analyse the optimal energy of an inclusion of a fixed volume the energy of which is determined by surface and elastic energies this problem has been studied extensively in the physical metallurgical literature however the analysis has mainly been either i numerical or ii restricted to a specific set of inclusion shapes e g ellipsoids in this article we prove a lower bound for the energy with no a priori hypothesis on the shape or even number of the inclusions

a quantitative sobolev inequality in bv a quantitative version of the standard sobolev inequality with sharp constant for functions u in w1 1 rn or bv rn is established in terms of a distance of u from the manifold of all multiples of characteristic functions of balls inequalities involving non euclidean norms of the gradient are discussed as well

equilibrium shapes of crystals attached to walls we discuss equilibrium shapes of crystals attached to walls optimal shapes for different configurations of walls are found and the minimality of the overall surface tension is proven with the help of a simple geometrical argument based on the isoperimetric inequality and monotonicity stability results in the form of bonnesen inequalities are obtained in the two dimensional case

existence and uniqueness for a crystalline mean curvature flow an existence and uniqueness result up to fattening for a class of crystalline mean curvature flows with natural mobility is proved the results are valid in any dimension and for arbitrary possibly unbounded initial closed sets the comparison principle is obtained by means of a suitable weak formulation of the flow while the existence of a global in time solution follows via a minimizing movement approach 2016 wiley periodicals inc

the hvt technique and the treatment of two basic inequalities the quantum mechanical hypervirial theorems hvt technique is used in treating two basic inequalities relating the ground state mean square radius of the orbit of a particle in a central potential and its kinetic energy respectively to the spacing of the two lowest energy levels e for quite a wide class of those potentials the parameters of which also lead to a sufficiently small dimensionless quantity s the difference between the two sides of each inequality is of order s3 and higher while e is of order s and higher and thus it is expected in general to be quite small

the bloch wave operator generalizations and applications part i the time independent case this is part 1 of a two part review on wave operator theory and methods the basic theory of the time independent wave operator is presented in terms of partitioned matrix theory for the benefit of general readers with a discussion of the links between the matrix and projection operator approaches the matrix approach is shown to lead to simple derivations of the wave operators and effective hamiltonians of lowdin bloch des cloizeaux and kato as well as to some associated variational forms the principal approach used throughout stresses the solution of the nonlinear equation for the reduced wave operator leading to the construction of the effective hamiltonians of bloch and of des cloizeaux several mathematical techniques which are useful in implementing this approach are explained some of them being relatively little known in the area of wave operator calculations the theoretical discussion is accompanied by several specimen numerical calculations which apply the described techniques to a selection of test matrices taken from the previous literature on wave operator methods the main emphasis throughout is on the use of numerical methods which use iterative or perturbation algorithms with simple pade approximant methods being found sufficient to deal with most of the cases of divergence which are encountered the use of damping factors and relaxation parameters is found to be effective in stabilizing calculations which use the energy dependent effective hamiltonian of lowdin in general the computations suggest that the numerical applications of the nonlinear equation for the reduced wave operator are best carried out with the equation split into a pair of equations in which the bloch effective hamiltonian appears as a separate entity the presentation of the theoretical and computational details throughout is accompanied by references to and discussion of many works which have used wave operator methods in physics chemistry and engineering some of the techniques described in this part 1 will be further extended and applied in part 2 of the review which deals with the changes which are required to extend wave operator theory to the case of a time dependent hamiltonian such as that which describes the interaction of a laser pulse with an atom or molecule

an accurate method for numerical calculations in quantum mechanics an accurate method for numerical calculations of matrix elements and for solving the eigenvalue problem in quantum mechanics is presented methods for numerical interpolation differentiation and integration provide 15 digit accuracy with double precision arithmetic operations a method for solution of the eigenvalue problem of an ordinary differential equation by using discretization and matrix eigenvalue methods provides 13 to 15 digit accuracy the efficiency of the proposed methods is demonstrated by the applications to bound states for the linear harmonic oscillator anharmonic oscillators the morse potential and the modified poschl teller potential

ground states of the 2d sticky disc model fine properties and n 3 4 law for the deviation from the asymptotic wulff shape we investigate ground state configurations for a general finite number n of parti cles of the heitmann radin sticky disc pair potential model in two dimensions exact energy minimizers are shown to exhibit large microscopic fluctuations about the asymptotic wulff shape which is a regular hexagon there are arbitrarily large n with ground state configura tions deviating from the nearest regular hexagon by a number of n 3 4 particles we also prove that for any n and any ground state configuration this deviation is bounded above by n 3 4 as a consequence we obtain an exact scaling law for the fluctuations about the asymptotic wulff shape in particular our results give a sharp rate of convergence to the limiting wulff shape

symmetrization with respect to the anisotropic perimeter and applications in this paper we introduce a new type of symmetrization which preserves the anisotropic perimeter of the level sets of a suitable concave smooth function in order to prove sharp comparison results for solutions of a class of homogeneous dirichlet fully nonlinear elliptic problems of second order and for suitable anisotropic hessian integrals

the jacobi eigenfunctions and the quantum mechanical hypervirial theorems method for bound state problems the jacobi eigenfunctions are suitably adjusted by means of the hypervirial theorems hvt scheme in order to obtain in many cases a very satisfactory analytic approximation of the s state eigenfunctions of the radial gaussian potential the results obtained show that the hvt scheme is a powerful method for obtaining analytically not only approximate eigenvalues but also the corresponding eigenfunctions

surface order large deviations for high density percolation we derive surface order large deviation estimates for the volume of the largest cluster and for the volume of the largest region surrounded by a cluster of a bernoulli percolation process restricted to a big finite box with sufficiently large parameter we also establish a useful version of the isoperimetric inequality which is the main tool of our proofs

comment on the asymptotic iteration method for polynomial potentials it is shown that polynomial potentials can be treated by a method which cuts out most of the complications of the popular aim approach several examples are given to show how expectation values and resonance energies can be calculated using the simplified approach

off diagonal matrix elements a modus tollens approach a numerical application of the modus tollens rule of classical logic is described and is shown to permit the calculation of off diagonal matrix elements by using a small basis set of implicit wavefunctions the success of the calculation is discussed in terms of the most simple form of the hohenberg kohn theorem

synchronization of phase oscillators with heterogeneous coupling a solvable case we consider an extension of kuramoto s model of coupled phase oscillators where oscillator pairs interact with different strengths when the coupling coefficient of each pair can be separated into two different factors each one associated to an oscillator kuramoto s theory for the transition to synchronization can be explicitly generalized and the effects of coupling heterogeneity on synchronized states can be analytically studied the two factors are respectively interpreted as the weight of the contribution of each oscillator to the mean field and the coupling of each oscillator to that field we explicitly analyze the effects of correlations between those weights and couplings and show that synchronization can be completely inhibited when they are strongly anti correlated numerical results validate the theory but suggest that finite size effect are relevant to the collective dynamics close to the synchronization transition where oscillators become entrained in synchronized frequency clusters

rigorous probabilistic analysis of equilibrium crystal shapes the rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade we review here the main results that have been obtained both in two and higher dimensions in particular we describe how the phenomenological wulff and winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases we focus on the main conceptual issues and describe the central ideas of the existing approaches

anisotropic elliptic problems involving hardy type potentials abstract in this paper we give existence uniqueness and regularity of the solutions of problems whose prototype is h u h o x 2 u f x in u 0 on where is a bounded open set of r n n 2 and 0 here h is a norm on r n h o is its polar and h u div h d u h d u is the anisotropic laplacian

comments on a new efficient method for calculating perturbation energies using functions which are not quadratically integrable the recently proposed method of calculating perturbation energies using a non normalizable wavefunction by skala and cizek is analysed and rigorously proved

a simple numerical method for singular potentials a one parameter family of coordinate transformations is shown to lead to a simple finite difference method which gives highly accurate energies and expectation values for the schrodinger equation in which the potential consists of a smooth term plus a perturbing term which is singular at the origin the method is effective down to very small values of the perturbation parameter and supplements the previously reported perturbation approach which is valuable for large values

on the hill determinant method explicit examples are given of the false eigenvalues of the hill determinant method the existence of which has been suspected by workers using analytic methods a simple criterion is discovered for distinguishing between false and physical eigenvalues

a new efficient method for calculating perturbative energies using functions which are not square integrable regularization and justification the method recently proposed by skala and cizek for calculating perturbation energies in a strict sense is ambiguous because it is expressed as a ratio of two quantities which are separately divergent even though this ratio comes out finite and gives the correct perturbation energies the calculational process must be regularized to be justified we examine one possible method of regularization and show that the proposed method gives traditional quantum mechanical results

minimizers of anisotropic surface tensions under gravity higher dimensions via symmetrization we consider a variational model describing the shape of liquid drops and crystals under the influence of gravity resting on a horizontal surface making use of anisotropic symmetrization techniques we establish the existence convexity and symmetry of minimizers for a class of surface tensions admissible to the symmetrization procedure in the case of smooth surface tensions we obtain the uniqueness of minimizers via an ode characterization

shooting methods for the schrodinger equation a simple stable shooting method for the schrodinger equation is described and is shown to work with various integrators including the numerov one quantum mechanical techniques are extended to apply to problems for which e is not a traditional eigenvalue and are shown to permit the extraction of useful information even from an unstable shooting method

phase diagram for the kuramoto model with van hemmen interactions we consider a kuramoto model of coupled oscillators that includes quenched random interactions of the type used by van hemmen in his model of spin glasses the phase diagram is obtained analytically for the case of zero noise and a lorentzian distribution of the oscillators natural frequencies depending on the size of the attractive and random coupling terms the system displays four states complete incoherence partial synchronization partial antiphase synchronization and a mix of antiphase and ordinary synchronization

direct expectation value calculations a new application of the hellmann feynman theorem is shown to allow accurate calculation of expectation values along with energies for the schrodinger equation for variable boundary position and boundary conditions power series and finite difference versions of the method are developed and applied to perturbed oscillator and perturbed hydrogen atom problems

the low temperature expansion of the wulff crystal in the 3d ising model we compute the expansion of the surface tension of the 3d random cluster model for q 1 in the limit where p goes to 1 we also compute the asymptotic shape of a plane partition of n as n goes to this same shape determines the wulff crystal to order o in the 3d ising model and more generally in the 3d random cluster model for q 1 at temperature

relaxation of multiple integrals in the space bv ω rp a characterisation of the surface energy density for the relaxation in v r p of the functional is obtained a lemma of de giorgi is used to modify a sequence near the boundary without increasing its total energy

a finite difference approach for the calculation of perturbed oscillator energies a simple numerical method for calculating eigenvalues and corresponding eigenvectors of the schrodinger equation for a perturbed oscillator is described the derived results are compared with previously derived numerical data and with available exact values

anisotropic isoperimetric inequalities involving boundary momentum perimeter and volume abstract we consider a scale invariant functional involving the anisotropic p momentum the anisotropic perimeter and the volume we show that the wulff shape associated with the finsler norm f considered and centered at the origin is the unique minimizer of the anisotropic functional taken into consideration among all bounded convex sets

on the wulff crystal in the ising model we study the phase separation phenomenon in the ising model in dimensions d 3 to this end we work in a large box with plus boundary conditions and we condition the system to have an excess amount of negative spins so that the empirical magnetization is smaller than the spontaneous magnetization m we confirm the prediction of the phenomenological theory by proving that with high probability a single droplet of the minus phase emerges surrounded by the plus phase moreover the rescaled droplet is asymptotically close to a definite deterministic shape the wulff crystal which minimizes the surface free energy in the course of the proof we establish a surface order large deviation principle for the magnetization our results are valid for temperatures t below a limit of slab thresholds t c conjectured to agree with the critical point t c moreover t should be such that there exist only two extremal translation invariant gibbs states at that temperature a property which can fail for at most countably many values and which is conjectured to be true for every t the proofs are based on the fortuin kasteleyn representation of the ising model along with coarse graining techniques to handle the emerging macroscopic objects we employ tools from geometric measure theory which provide an adequate framework for the large deviation analysis finally we propose a heuristic picture that for subcritical temperatures close enough to t c the dominant minus spin cluster of the wulff droplet permeates the entire box and has a strictly positive local density everywhere

on a sharp transition from area law to perimeter law in a system of random surfaces we introduce and study a phase transition which is associated with the spontaneous formation of infinite surface sheets in a bernoulli system of random plaquettes the transition is manifested by a change in the asymptotic behavior of the probability of the formation of a surface spanning a prescribed loop as such this transition offers a generalization of the bond percolation phenomenon at low plaquette densities the probability for large loops is shown to decay exponentially with the loops area whereas for high densities the decay is by a perimeter law furthermore we show that the two phases of the three dimensional plaquette system are in a precise correspondence with the two phases of the dual system of random bonds thus if a natural conjecture about the phase structure of the bond percolation model is true then there is a sharp transition in the asymptotic behavior of the surface events our analysis incorporates block variables in terms of which a non critical system is transformed into one which is close to a trivial high or low density fixed point stochastic geometric effects like those discussed here play an important role in lattice gauge theories

uniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets given an elliptic integrand of class mathscr c 2 alpha we prove that finite unions of disjoint open wulff shapes with equal radii are the only volume constrained critical points of the anisotropic surface energy among all sets with finite perimeter and reduced boundary almost equal to its closure

a generator of high order embedded p stable methods for the numerical solution of the schro dinger equation a generator of new embedded p stable methods of order 2n 2 where n is the number of layers used by the embedded methods for the approximate numerical integration of the one dimensional schrodinger equation is developed in this paper these new methods are called embedded methods because of a simple natural error control mechanism numerical results obtained for one dimensional differential equations of the schrodinger type show the validity of the developed theory

on stable parametric finite element methods for the stefan problem and the mullins sekerka problem with applications to dendritic growth we introduce a parametric finite element approximation for the stefan problem with the gibbs thomson law and kinetic undercooling which mimics the underlying energy structure of the problem the proposed method is also applicable to certain quasi stationary variants such as the mullins sekerka problem in addition fully anisotropic energies are easily handled the approximation has good mesh properties leading to a well conditioned discretization even in three space dimensions several numerical computations including for dendritic growth and for snow crystal growth are presented

equilibrium shapes for planar crystals in an external field the equilibrium shape of a two dimensional crystal in a convex background potential g x is analyzed for g 0 the shape of minimum energy may be deduced from surface tension via the wulff construction but if g is not constant little is known beyond the case of a crystal sitting in a uniform field only an unpublished result of okikiolu shows each connected component of the equilibrium crystal to be convex here it will be shown that any such component minimizes energy uniquely among convex sets of its area if the wulff shape and g x are symmetric under x x it follows that the equilibrium crystal is unique convex and connected this last result leads to a new proof that convex crystals away from equilibrium remain convex as they evolve by curvature driven flow subsequent work with felix otto shows without assuming symmetry that no equilibrium crystal has more than two convex components

polynomial type eigenfunctions the differential equation for the harmonic oscillator is generalised to include an interaction potential containing a positive definite quadratic denominator conditions are developed under which certain eigenfunctions take the form of an exponential function multiplied by a polynomial the problem reduces to that of finding the eigenvalues of a certain matrix that is tridiagonal in form properties of the eigenvalues of this matrix are investigated since they are functions of a parameter occurring in the positive definite quadratic form the asymptotic forms of these eigenvalues are developed together with computed results expressed as curves showing the variation of the eigenvalues with respect to this parameter

low lying states of two dimensional double well potentials wavefunctions energies and selected expectation values of the low lying stationary states of two dimensional double well potentials are obtained from the long time solutions of the corresponding time dependent schrodinger equation the numerical method consists of transformation to a diffusion like equation which is then solved by an alternating direction implicit finite difference method the method is tested for cases in which the energies have been obtained by other methods then the dependence of the energies and other properties on the potential parameters is discussed on the basis of results for four states of 35 different sets of potential parameters including some that lead to pseudo degeneracies

the wulff theorem revisited the parametrized indicator measures and the brunn minkowski inequality are used to prove that the wulff set w is a minimizer for the surface energy where the density is the support function of w the support of the indicator measures associated to minimizing sequences is characterized it is shown that if w is polyhedral then minimizing sequences cannot oscillate

the schrodinger equation with position dependent mass it is shown that the traditional hartree second order shooting method for eigenvalues can be modified to give accurate energy levels for a schrodinger equation involving a position dependent mass some numerical applications are given to demonstrate the effectiveness of the method

a variational principle for domino tilings 1 1 description of results a domino is a 1 x 2 or 2 x 1 rectangle and a tiling of a region by dominos is a way of covering that region with dominos so that there are no gaps or overlaps in 1961 kasteleyn kal found a formula for the number of domino tilings of an m x n rectangle with mn even as shown in figure 1 for rm n 68 temperley and fisher tf used a differenit method and arrived at the same result at almost exactly the same time both lines of calculation showed that the logarithm of number of tilings divided by the number of dominos in a tiling that is mn 2 converges to 2g 7r 0 58 here g is catalan s constant on the other hand in 1992 elkies et al eklp studied domino tilings of regions they called aztec diamonds figure 2 shows an aztec diamond of order 48 and showed that the logarithm of the number of tilings divided by the number of dominos converges to the smaller number log 2 2 0 35 thus even though the region in figure 1 has slightly smaller area than the region in figure 2 the former has far more domino tilings for regions with other shapes neither of these asymptotic formulas may apply in the present paper we consider simply connected regions of arbitrary shape we give an exact formula for the limiting value of the logarithm of the number of tilings per unit area as a function of the shape of the boundary of the region as the size of the region goes to infinity in particular we show that computation of this limit is intimately linked with an understanding of long range variations in the local statistics of random domino tilings such variations can be seen by comparing figures 1 and 2 each of the two tilings is random in the sense that the algorithm pwi that was used to create it generates each of the possible tilings of the region being tiled with the same probability hence one can expect each tiling to be qualitatively typical of the overwhelming majority of tilings of the region in

on the improvement of convergence of hill determinants we propose a simple approach to analytically determine a parameter of a convergence factor exp gx2 4 in the hill determinant method it provides much more rapid convergence of the method than the other analytical approaches currently used for the g determination

parametric approximation of isotropic and anisotropic elastic flow for closed and open curves deckelnick and dziuk math comput 78 266 645 671 2009 proved a stability bound for a continuous in time semidiscrete parametric finite element approximation of the elastic flow of closed curves in mathbb r d d geq2 we extend these ideas in considering an alternative finite element approximation of the same flow that retains some of the features of the formulations in barrett et al j comput phys 222 1 441 462 2007 siam j sci comput 31 1 225 253 2008 ima j numer anal 30 1 4 60 2010 in particular an equidistribution mesh property for this new approximation we obtain also a stability bound for a continuous in time semidiscrete scheme apart from the isotropic situation we also consider the case of an anisotropic elastic energy in addition to the evolution of closed curves we also consider the isotropic and anisotropic elastic flow of a single open curve in the plane and in higher codimension that satisfies various boundary conditions

a uniqueness proof for the wulff theorem the wulff problem is a generalisation of the isoperimetric problem and is relevant for the equilibrium of small elastic crystals it consists in minimising the generally anisotropic surface energy among sets of given volume a solution of this problem is given by a geometric construction due to wulff in the class of sets of finite perimeter this was first shown by j e taylor who using methods of geometric measure theory also proved uniqueness here a more analytic uniqueness proof is presented the main ingredient is a sharpened version of the brunn minkowski inequality

convex symmetrization and polya szego inequality abstract we prove a polya szego inequality involving a convex symmetrization of functions and we investigate the equality case

existence of compactly supported global minimisers for the interaction energy the existence of compactly supported global minimisers for continuum models of particles interacting through a potential is shown under almost optimal hypotheses the main assumption on the potential is that it is catastrophic or not h stable which is the complementary assumption to that in classical results on thermodynamic limits in statistical mechanics the proof is based on a uniform control on the local mass around each point of the support of a global minimiser together with an estimate on the size of the gaps it may have the class of potentials for which we prove the existence of global minimisers includes power law potentials and for some range of parameters morse potentials widely used in applications we also show that the support of local minimisers is compact under suitable assumptions

quantum averaging ii kolmogorov s algorithm a complete analogue of kolmogorov s perturbation algorithm in classical mechanics is presented for perturbations of self adjoint operators the resulting perturbation theory is different from the usual rayleigh schr dinger or kato rellich perturbation theory

hill determinants and 1 n theory a computationally efficient form of the hill determinant method is devised giving both e and x2 values and is used to test the accuracy of that method in the light of recent criticisms l 1 and n 1 asymptotic theories are used to check the numerical results

injectivity and self contact in nonlinear elasticity let be a bounded open connected subset of 3 with a sufficiently smooth boundary the additional condition det dx vol is imposed on the admissible deformations of a hyperelastic body whose reference configuration is we show that the associated minimization problem provides a mathematical model for matter to come into frictionless contact with itself but not interpenetrate we also extend j ball s theorems on existence to this case by establishing the existence of a minimizer of the energy in the space w1 p 3 p 3 that is injective almost everywhere

on the sign of the second order energy shift in the rayleigh schrodinger perturbation theory for a highly excited state the second order energy shift in the rayleigh schrodinger perturbation theory is most likely to be positive for large quantum numbers i e for excited states within the wilson sommerfeld approximation good for large quantum numbers we show on the contrary that the sufficient condition for the second order shift to be negative is that the total potential be symmetric and both the unperturbed and perturbed potentials be monotonically increasing

reply to comments on a new efficient method for calculating perturbation energies using functions which are not quadratically integrable in a previous paper sk la and c zek 1996 a new approach to calculating perturbation energies of bound states based on the use of functions which are not quadratically integrable was suggested this method has been analysed in a comment by guardiola and ros 1996 the present comment is the reply to this previous comment the aim is to give a simple and straightforward proof of the method and clarify some other points important for its application

a hill series method for resonances a simple algorithm is given to treat perturbed oscillator bound states and resonances the method is applied to a bound state problem for which the energy is a non analytic function of a perturbation parameter and also to two resonance problems one of which has a spectrum with an unusual dual nature

rayleigh schrodinger perturbation theory with a strong perturbation anharmonic oscillators the bound state solutions of schrodinger s equation for the anharmonic oscillator potentials v x2 lambda x2k k 2 3 have been investigated using elementary techniques of low order variational perturbation theory for the quartic oscillator k 2 a scaled harmonic potential provides a remarkably accurate model for all lambda although this model is slightly less satisfactory for higher order anharmonicities k or 3 the perturbation procedures remain effective and can be applied successfully provided that higher order terms are calculated

theory of collective firing induced by noise or diversity in excitable media large variety of physical chemical and biological systems show excitable behavior characterized by a nonlinear response under external perturbations only perturbations exceeding a threshold induce a full system response firing it has been reported that in coupled excitable identical systems noise may induce the simultaneous firing of a macroscopic fraction of units however a comprehensive understanding of the role of noise and that of natural diversity present in realistic systems is still lacking here we develop a theory for the emergence of collective firings in nonidentical excitable systems subject to noise three different dynamical regimes arise subthreshold motion where all elements remain confined near the fixed point coherent pulsations where a macroscopic fraction fire simultaneously and incoherent pulsations where units fire in a disordered fashion we also show that the mechanism for collective firing is generic it arises from degradation of entrainment originated either by noise or by diversity

the brunn minkowski inequality in 1978 osserman 124 wrote an extensive survey on the isoperimetric inequality the brunn minkowski inequality can be proved in a page yet quickly yields the classical isoperimetric inequality for important classes of subsets of rn and deserves to be better known this guide explains the relationship between the brunn minkowski inequality and other inequalities in geometry and analysis and some applications

lower semicontinuity of surface energies using the theory of indicator measures a lower semicontinuity result for quasiconvex functions in w 1 1 and assuming only l 1 convergence is obtained

embedded methods for the numerical solution of the schrodinger equation abstract new methods for the approximate numerical integration of the one dimensional schrodinger equation are developed in this paper complete phase lag analysis of the new methods is included these new methods are called embedded methods because of a simple natural error control mechanism numerical results obtained for a one dimensional schrodinger equation show the validity of the developed theory

equilibria and dislocations in epitaxial growth abstract an overview of recent analytical developments in epitaxial growth is presented the energy release via the onset of dislocations is addressed regularity of quasistatic equilibria is studied both in the absence and in the presence of dislocations morphological evolution of anisotropic epitaxially strained films is considered under the assumption that the surface evolves by surface diffusion under the influence of a chemical potential

the exponential cosine screened potential the concept of forward nested multiplication is used to develop a unified power series hill determinant method for the calculation of schrodinger equation energy levels a numerical application is made to the exponential cosine screened potential

clustering coding switching hierarchical ordering and control in a network of chaotic elements abstract a network of chaotic elements is investigated with the use of globally coupled maps a simple coding of many attractors with clustering is shown through the coding the attractors are organized so that their change exhibits bifurcation like phenomena a precision dependent tree is constructed which leads to the similarity of our attractor with those of spin glasses hierarchical dynamics is constructed on the tree which leads to the dynamical change of trees and the temporal change of effective degrees of freedom by a simple input on a site we can switch among attractors and tune the strength of chaos a threshold on a cluster size is found beyond which a peculiar posi nega switch occurs possible application to biological information processing is discussed with the emphasis on the fuzzy switch chaotic search and hierarchical code categorization

a new finite difference scheme with minimal phase lag for the numerical solution of the schrodinger equation a new finite difference approach for the numerical solution of the eigenvalue problem of the schrodinger equations is developed in this paper the new approach is based on the maximization of the algebraic order and the minimization of the phase lag we investigate two cases i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wavefunctions tend to zero for x ii the general case for positive and negative eigenvalues and for the well known cases of the morse potential and woods saxon or optical potential numerical and theoretical results show that this new approach is more efficient when compared with previously derived methods

the wulff construction and asymptotics of the finite cluster distribution for two dimensional bernoulli percolation we consider two dimensional bernoulli percolation at densityp p c and establish the following results

the stochastic random cluster process and the uniqueness of random cluster measures the random cluster model is a generalization of percolation and ferromagnetic potts models due to fortuin and kasteleyn not only is the random cluster model a worthwhile topic for study in its own right but also it provides much information about phase transitions in the associated physical models this paper serves two functions first we introduce and survey random cluster measures from the probabilist s point of view giving clear statements of some of the many open problems second we present new results for such measures as follows we discuss the relationship between weak limits of random cluster measures and measures satisfying a suitable dlr condition using an argument based on the convexity of pressure we prove the uniqueness of random cluster measures for all but at most countably many values of the parameter p related results concerning phase transition in two or more dimensions are included together with various stimulating conjectures the uniqueness of the infinite cluster is employed in an intrinsic way in part of these arguments in the second part of this paper is constructed a markov process whose level sets are reversible markov processes with random cluster measures as unique equilibrium measures this construction enables a coupling of random cluster measures for all values of p furthermore it leads to a proof of the semicontinuity of the percolation probability and provides a heuristic probabilistic justification for the widely held belief that there is a first order phase transition if and only if the cluster weighting factor q is sufficiently large

geometric variational problems of statistical mechanics and of combinatorics we present the geometric solutions of the various extremal problems of statistical mechanics and combinatorics together with the wulff construction which predicts the shape of the crystals we discuss the construction which exhibits the shape of a typical young diagram and of a typical skyscraper