an almost sure large deviation principle for the hopfield model we prove a large deviation principle for the finite dimensional marginals of the gibbs distribution of the macroscopic overlap parameters in the hopfield model in the case where the number of random patterns m as a function of the system size n satisfies lim sup m n n 0 in this case the rate function is independent of the disorder for almost all realizations of the patterns

on a relation of pseudoanalytic function theory to the two dimensional stationary schrodinger equation and taylor series in formal powers for its solutions we consider the real stationary two dimensional schrodinger equation with the aid of any of its particular solutions we construct a vekua equation possessing the following special property the real parts of its solutions are solutions of the original schrodinger equation and the imaginary parts are solutions of an associated schrodinger equation with a potential having the form of a potential obtained after the darboux transformation using bers theory of taylor series for pseudoanalytic functions we obtain a locally complete system of solutions of the original schrodinger equation which can be constructed explicitly for an ample class of schrodinger equations for example it is possible when the potential is a function of one cartesian spherical parabolic or elliptic variable we give some examples of application of the proposed procedure for obtaining a locally complete system of solutions of the schrodinger equation the procedure is algorithmically simple and can be implemented with the aid of a computer system of symbolic or numerical calculation

cauchy integral formula for generalized analytic functions in hydrodynamics it is shown that for several classes of generalized analytic functions arising in linearized equations of hydrodynamics and magnetohydrodynamics the cauchy integral formulae follow from the one for generalized holomorphic vectors in a uniform fashion if hydrodynamic fields velocity pressure and vorticity admit representations in terms of corresponding generalized analytic functions those representations and the cauchy integral formulae form two essential parts of the generalized analytic function approach which readily yields either closed form solutions or boundary integral equations this approach is demonstrated for problems of axisymmetric and asymmetric stokes flows two phase axisymmetric stokes flows two dimensional and axisymmetric oseen flows

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

gibbs states of the hopfield model with extensively many patterns we consider the hopfield model withm n n patterns wheren is the number of neurons we show that if is sufficiently small and the temperature sufficiently low then there exist disjoint gibbs states for each of the stored patterns almost surely with respect to the distribution of the random patterns this solves a provlem left open in previous work the key new ingredient is a self averaging result on the free energy functional this result has considerable additional interest and some consequences are discussed a similar result for the free energy of the sherrington kirkpatrick model is also given

a nonlinearly preconditioned inexact newton algorithm for steady state lattice boltzmann equations most existing methods for calculating the steady state solution of the lattice boltzmann equations are based on pseudo time stepping which often requires a large number of time steps especially for high reynolds number problems to calculate the steady state solution directly without the time integration in this paper we propose and study a nonlinearly preconditioned inexact newton algorithm with a domain decomposition based linear solver for parallelization more precisely the proposed algorithmic framework involves an implicit second order discretization a two level inexact newton method and a nonlinear elimination preconditioner to accelerate the convergence of newton iteration a nonstandard pollution removing coarse space is introduced for the two level method numerical experiments are presented to demonstrate the robustness and efficiency of the algorithm especially for problems at a high reynolds number a comparison is also included to show the superiority of the proposed approach over o

open and traction boundary conditions for the incompressible navier stokes equations we present numerical schemes for the incompressible navier stokes equations nse with open and traction boundary conditions we use pressure poisson equation ppe formulation and propose new boundary conditions for the pressure on the open or traction boundaries after replacing the divergence free constraint by this pressure poisson equation we obtain an unconstrained nse for stokes equation with open boundary condition on a simple domain we prove unconditional stability of a first order semi implicit scheme where the pressure is treated explicitly and hence is decoupled from the computation of velocity using either boundary condition the schemes for the full nse that treat both convection and pressure terms explicitly work well with various spatial discretizations including spectral collocation and c 0 finite elements moreover when reynolds number is of o 1 and when the first order semi implicit time stepping is used time step size of o 1 is allowed in benchmark computations for the full nse besides standard stability and accuracy check various numerical results including flow over a backward facing step flow past a cylinder and flow in a bifurcated tube are reported numerically we have observed that using ppe formulation enables us to use the velocity pressure pairs that do not satisfy the standard inf sup compatibility condition our results extend that of johnston and liu h johnston j g liu accurate stable and efficient navier stokes solvers based on explicit treatment of the pressure term j comp phys 199 1 2004 221 259 which deals with no slip boundary conditions only

hilbert formulas for r analytic functions in the domain exterior to spindle hilbert formulas for an r analytic function defined by a generalized cauchy riemann system in the domain exterior to the contour of a spindle in the meridional cross section plane have been derived the derivation is based on the theory of riemann boundary value problems for analytic functions for numerical calculations fourier integrals with hilbert formulas representing the real and imaginary parts of the r analytic function have been reduced to the form of regular integrals the problem of the axially symmetric steady motion of a rigid spindle shaped body in a stokes fluid has been solved and the pressure in the fluid has been expressed analytically based on a hilbert formula as an illustration streamlines about the body vortex and pressure functions at the contour of the body and the drag force exerted on the body by the fluid have been calculated

absence of self averaging of the order parameter in the sherrington kirkpatrick model we prove that if n is the sherrington kirkpatrick sk hamiltonian and the quantity bar q n n 1 sum left langle s l right rangle h 2 converges in the variance to a nonrandom limit asn then the mean free energy of the model converges to the expression obtained by sk since this expression is known not to be correct in the low temperature region our result implies the non self averaging of the order parameter of the sk model this fact is an important ingredient of the parisi theory which is widely believed to be exact we also prove that the variance of the free energy of the sk model converges to zero asn i e the free energy has the self averaging property

the free energy of a class of hopfield models we show that in the limitp n 0 p n 0 the limit free energy of the hopfield model equals in probability the curie weiss free energy we prove also that the free energy of the hopfield model is self averaging for any finite

moderate deviations for the overlap parameter in the hopfield model we derive moderate deviation principles for the overlap parameter in the hopfield model of spin glasses and neural networks if the inverse temperature is different from the critical inverse temperature c 1 and the number of patterns m n satisfies m n n 0 the overlap parameter multiplied by n 1 2 1 obeys a moderate deviation principle with speed n1 2 and a quadratic rate function i e the gaussian limit for 1 2 remains visible on the moderate deviation scale at the critical temperature we need to multiply the overlap parameter by n 1 4 1 if then m n satisfies m n 6 log n m n 2n4 log n n 0 the rescaled overlap parameter obeys a moderate deviation principle with speed n1 4 and a rate function that is basically a fourth power the random term occurring in the central limit theorem for the overlap at c 1 is no longer present on a moderate deviation scale if the scaling is even closer to n1 4 e g if we multiply the overlap parameter by n1 4 log log n the moderate deviation principle breaks down the case of variable temperature converging to one is also considered if n converges to c fast enough i e faster than open image in new window the non gaussian rate function persists whereas for n converging to one slower than open image in new window the moderate deviations principle is given by the gaussian rate at the borderline the moderate deviation rate function is the one at criticality plus an additional gaussian term

on a factorization of second order elliptic operators and applications we show that given a nonvanishing particular solution of the equation the corresponding differential operator can be factorized into a product of two first order operators the factorization allows us to reduce the above equation to a first order equation which in a two dimensional case is a vekua equation of a special form under quite general conditions on the coefficients p and q we obtain an algorithm which allows us to construct in explicit form positive formal powers solutions of the vekua equation generalizing the usual powers z z0 n n 0 1 this result means that under quite general conditions one can construct an infinite system of exact solutions of the above equation explicitly and moreover at least when p and q are real valued this system will be complete in ker divp grad q in the sense that any solution of the above equation in a simply connected domain can be represented as an infinite series of obtained exact solutions which converges uniformly on any compact subset of finally we give a similar factorization of the operator divp grad q in a multidimensional case and obtain a natural generalization of the vekua equation which is related to second order operators in a similar way as its two dimensional prototype does

on the reduction of the multidimensional stationary schrodinger equation to a first order equation and its relation to the pseudoanalytic function theory given a particular solution of a one dimensional stationary schrodinger equation this equation of second order can be reduced to a first order linear ordinary differential equation this is done with the aid of an auxiliary riccati differential equation in the present work we show that the same fact is true in a multidimensional situation also for simplicity we consider the case of two or three independent variables one particular solution of the stationary schrodinger equation allows us to reduce this second order equation to a linear first order quaternionic differential equation as in the one dimensional case this is done with the aid of an auxiliary quaternionic riccati equation the resulting first order quaternionic equation is equivalent to the static maxwell system and is closely related to the dirac equation in the case of two independent variables it is the well known vekua equation from theory of pseudoanalytic or generalized analytic functions nevertheless we show that even in this case it is very useful to consider not only complex valued functions solutions of the vekua equation but complete quaternionic functions in this way the first order quaternionic equation represents two separate vekua equations one of which gives us solutions of the schrodinger equation and the other one can be considered as an auxiliary equation of a simpler structure moreover for the auxiliary equation we always have the corresponding bers generating pair f g the base of the bers theory of pseudoanalytic functions and what is very important the bers derivatives of solutions of the auxiliary equation give us solutions of the main vekua equation and as a consequence of the schrodinger equation based on this fact we obtain an analogue of the cauchy integral theorem for solutions of the stationary schrodinger equation other results from theory of pseudoanalytic functions can be written for solutions of the schrodinger equation moreover for an ample class of potentials in the schrodinger equation which includes for instance all radial potentials this new approach gives us a simple procedure allowing us to obtain an infinite sequence of solutions of the schrodinger equation from one known particular solution

high order accurate solution of the incompressible navier stokes equations high order methods are of great interest in the study of turbulent flows in complex geometries by means of direct simulation with this goal in mind the incompressible navier stokes equations are discretized in space by a compact fourth order finite difference method on a staggered grid the equations are integrated in time by a second order semi implicit method stable boundary conditions are implemented and the grid is allowed to be curvilinear in two space dimensions the method is extended to three dimensions by a fourier expansion in every time step a system of linear equations is solved for the velocity and the pressure by an outer and an inner iteration with preconditioning the convergence properties of the iterative method are analyzed the order of accuracy of the method is demonstrated in numerical experiments the method is used to compute the flow in a channel the driven cavity and a constricted channel

on a transplant operator and explicit construction of cauchy type integral representations for p analytic functions we present a new technique for explicit construction of cauchy kernels and cauchy integral representations for a class of generalized analytic functions and p analytic functions

an efficient method for the incompressible navier stokes equations on irregular domains with no slip boundary conditions high order up to the boundary common efficient schemes for the incompressible navier stokes equations such as projection or fractional step methods have limited temporal accuracy as a result of matrix splitting errors or introduce errors near the domain boundaries which destroy uniform convergence to the solution in this paper we recast the incompressible constant density navier stokes equations with the velocity prescribed at the boundary as an equivalent system for the primary variables velocity and pressure equation for the pressure the key difference from the usual approaches occurs at the boundaries where we use boundary conditions that unequivocally allow the pressure to be recovered from knowledge of the velocity at any fixed time this avoids the common difficulty of an apparently over determined poisson problem since in this alternative formulation the pressure can be accurately and efficiently recovered from the velocity the recast equations are ideal for numerical marching methods the new system can be discretized using a variety of methods including semi implicit treatments of viscosity and in principle to any desired order of accuracy in this work we illustrate the approach with a 2 d second order finite difference scheme on a cartesian grid and devise an algorithm to solve the equations on domains with curved non conforming boundaries including a case with a non trivial topology a circular obstruction inside the domain this algorithm achieves second order accuracy in the l norm for both the velocity and the pressure the scheme has a natural extension to 3 d

a split step finite element method for incompressible navier stokes equations with high order accuracy up to the boundary abstract an efficient and accurate finite element algorithm is described for the numerical solution of the incompressible navier stokes ins equations the new algorithm that solves the ins equations in a velocity pressure reformulation is based on a split step scheme in conjunction with the standard finite element method the split step scheme employed for the temporal discretization of our algorithm completely separates the pressure updates from the solution of velocity variables when the pressure equation is formed explicitly the algorithm avoids solving a saddle point problem therefore our algorithm has more flexibility in choosing finite element spaces in contrast popular mixed finite element methods that solve the ins equations in the primitive variables or velocity divergence formulation lead to discrete saddle point problems whose solution depends on the choice of finite element spaces for velocity and pressure that is subject to the well known ladyzenskaja babuska brezzi lbb or inf sup condition for efficiency and robustness lagrange piecewise polynomial finite elements of equal order for both velocity and pressure are used accurate numerical boundary condition for the pressure equation is also investigated motivated by a post processing technique that calculates derivatives of a finite element solution with super convergent error estimates an alternative numerical boundary condition is proposed for the pressure equation at the discrete level the new numerical pressure boundary condition that can be regarded as a better implementation of the compatibility boundary condition improves the boundary layer errors of the pressure solution a normal mode analysis is performed using a simplified model problem on a uniform mesh to demonstrate the numerical properties of our methods convergence studies using p 1 elements support the analytical results and demonstrates that our algorithm with the new numerical boundary condition achieves the optimal second order accuracy for both velocity and pressure up to the boundary benchmark problems are also computed and carefully compared with existing studies finally as an example to illustrate that our approach can be easily adapted for higher order finite elements we solve the classical flow past a cylinder problem using p n finite elements with n 1

finite element discretization of the stokes and navier stokes equations with boundary conditions on the pressure we consider the stokes and navier stokes equations with boundary conditions of dirichlet type on the velocity on one part of the boundary and involving the pressure on the rest of the boundary we write the variational formulations of such problems next we propose a finite element discretization of them and perform the a priori and a posteriori analysis of the discrete problem some numerical experiments are presented in order to justify our strategy

gibbs states of the hopfield model in the regime of perfect memory we study the thermodynamic properties of the hopfield model of an autoassociative memory ifn denotes the number of neurons andm n the number of stored patterns we prove the following results ifm n 0 asn then there exists an infinite number of infinite volume gibbs measures for all temperaturest 1 concentrated on spin configurations that have overlap with exactly one specific pattern moreover the measures induced on the overlap parameters are dirac measures concentrated on a single point and the gibbs measures on spin configurations are products of bernoulli measures ifm n asn for small enough we show that for temperaturest smaller than somet 1 the induced measures can have support only on a disjoint union of balls around the previous points but we cannot construct the infinite volume measures through convergent sequences of measures

boundary conditions for a divergence free velocity pressure formulation of the navier stokes equations new sets of boundary conditions for the velocity pressure formulation of the incompressible navier stokes equations are derived the boundary conditions have the same form on both inflow and outflow boundaries and lead to a divergence free solution moreover the specific form of the boundary condition makes it possible derive a symmetric positive definite equation system for the internal pressure numerical experiments support the theoretical conclusions

stability of pressure boundary conditions for stokes and navier stokes equations abstract the stability of a finite difference discretization of the time dependent incompressible navier stokes equations in velocity pressure formulation is studied in paticular we compare the stability for different pressure boundary conditions in a semiimplicit time integration scheme where only the viscous term is treated implicitly the stability is studied in three different ways by a normal mode analysis by numerical computation of the amplification factors and by direct numerical simulation of the governing equations all three approaches identify the same pressure boundary condition as the best alternative this condition implicitly enforces the normal derivative of the divergence to be zero on the boundary by coupling the normal derivative of the pressure to the normal component of the curl of the vorticity using this boundary condition we demonstrate that the time step is determined only by the convective term

an approximate numerical method for solving cauchy singular integral equations composed of multiple implicit parameter functions with unknown integral limits in contact mechanics abstract cauchy singular integral equations are widely used in physics and mathematics especially in solid contact mechanics the solution of cauchy singular integral equations composed of explicit functions has already been achieved in existing research however when dealing with the contact problem between two solid bodies with irregular surfaces described by implicit parametric functions difficulties arise when trying to solve the cauchy singular integral because it is composed of multiple implicit parameter functions moreover the integral limits are unknown and constrained by physical characteristics to solve this kind of problem an approximate calculation method with high accuracy will be provided in this paper specifically based on the quadrature method and taking the constraint function of the boundary as the convergence criterion both the integral limits satisfying the physical characteristics condition and the solution of the cauchy singular integral equations composed of multiple parameter functions can be derived by an iterative method finally five different examples are calculated using the new method and the absolute errors between the approximate values provided by the new method and the true values are analysed

a fourth order accurate method for the incompressible navier stokes equations on overlapping grids a method is described to solve the time dependent incompressible navier stokes equations with finite differences on curvilinear overlapping grids in two or three space dimensions the scheme is fourth order accurate in space and uses the momentum equations for the velocity coupled to a poisson equation for the pressure the boundary condition for the pressure is taken as u 0 extra numerical boundary conditions are chosen to make the scheme accurate and stable the velocity is advanced explicitly in time any standard time stepping scheme such as runge kutta can be used the poisson equation is solved using direct or iterative sparse matrix solvers or by the multigrid algorithm computational results in two and three space dimensions are given

an efficient numerical method for the equations of steady and unsteady flows of homogeneous incompressible newtonian fluid in the present paper we present some numerical methods to solve the equations of steady and unsteady flows such as those in the microcirculatory bed and large blood vessels arteries and veins respectively in the case of steady flows the method does not need neither any boundary conditions on pressure nor any small parameter and the main computation consists of solving some poisson equations in the case of unsteady flows the scheme uses a consistent neumann boundary condition for the pressure poisson equation at each time step a poisson and heat equation are solved for the pressure and each velocity component respectively the accuracy and efficiency of scheme are checked by a set of numerical tests

stability and convergence of efficient navier stokes solvers via a commutator estimate for strong solutions of the incompressible navier stokes equations in bounded domains with velocity specified at the boundary we establish the unconditional stability and convergence of discretization schemes that decouple the updates of pressure and velocity through explicit time stepping for pressure these schemes require no solution of stationary stokes systems nor any compatibility between velocity and pressure spaces to ensure an inf sup condition and are representative of a class of highly efficient computational methods that have recently emerged the proofs are simple based upon a new sharp estimate for the commutator of the laplacian and helmholtz projection operators this allows us to treat an unconstrained formulation of the navier stokes equations as a perturbed diffusion equation c 2006 wiley periodicals inc

a composite grid solver for conjugate heat transfer in fluid structure systems we describe a numerical method for modeling temperature dependent fluid flow coupled to heat transfer in solids this approach to conjugate heat transfer can be used to compute transient and steady state solutions to a wide range of fluid solid systems in complex two and three dimensional geometry fluids are modeled with the temperature dependent incompressible navier stokes equations using the boussinesq approximation solids with heat transfer are modeled with the heat equation appropriate interface equations are applied to couple the solutions across different domains the computational region is divided into a number of sub domains corresponding to fluid domains and solid domains there may be multiple fluid domains and multiple solid domains each fluid or solid sub domain is discretized with an overlapping grid the entire region is associated with a composite grid which is the union of the overlapping grids for the sub domains a different physics solver fluid solver or solid solver is associated with each sub domain a higher level multi domain solver manages the entire solution process we propose and analyze some centered discrete approximations to the interface equations that have some desirable stability properties the coupled interface equations may be solved directly when using explicit time stepping methods in the sub domains resulting in a strongly coupled approach the stability of the interface treatment in this case is independent of the relative sizes of the material properties in the two domains with the time step only depending on the usual von neumann conditions for each sub domain for implicit time stepping methods we solve the interface equations in a weakly coupled fashion to avoid forming a coupled implicit system across all sub domains the convergence of this approach does depend on the relative sizes of the thermal conductivities and diffusivities we analyze different iteration strategies for solving these implicit equations including the use of mixed robin approximations at the interface numerical results are presented to illustrate the method the accuracy of the technique is verified using the method of analytic solutions and by computing the solution to some heat exchanger problems where the exact solution is known the technique is also applied to the modeling of an inertial confinement fusion hohlraum target and the flow of coolant past an hexagonal array of heated fuel rods the multi domain solver runs in parallel on distributed memory computers and some parallel results are provided

an analysis on neural dynamics with saturated sigmoidal functions we propose a unified approach to study the relation between the set of saturated attractors and the set of system parameters of the hopfield model linsker s model and the dynamic link network dln which use saturated sigmoidal functions in its dynamics of the state or weight the key point for this approach is to rigorously derive a necessary and sufficient condition to test whether a given saturated state in the hopfield model or weight vector in linsker s model and the dln is stable or not for any given set of system parameters and used this to determine the complete regime in the parameter space over which the given state or weight is stable our approach allows us to give an exact characterization between the parameters and the capacity in the hopfield model to generalize our previous results on linsker s network and the dln to have a better understanding of the underlying mechanism among these models the method reported here could be adopted to analyze a variety of models in the field of the neural networks

explicit solutions of generalized cauchy riemann systems using the transplant operator abstract in kravchenko 2008 8 it was shown that the tool introduced there and called the transplant operator transforms solutions of one vekua equation into solutions of another vekua equation related to the first via a schrodinger equation in this paper we prove a fundamental property of this operator it preserves the order of zeros and poles of generalized analytic functions and transforms formal powers of the first vekua equation into formal powers of the same order for the second vekua equation this property allows us to obtain positive formal powers and a generating sequence of a complicated vekua equation from positive formal powers and a generating sequence of a simpler vekua equation similar results are obtained regarding the construction of cauchy kernels elliptic and hyperbolic pseudoanalytic function theories are considered and examples are given to illustrate the procedure

on the two dimensional stationary schrodinger equation with a singular potential abstract we study the equation u x y x y u x y 0 when the potential has the following expression r a r 2 where r are the polar coordinates in r 2 and a is a 2 periodic function of class c k with k 1 we obtain series representations for solutions in a full neighborhood of the singular point and we also give representations in terms of pseudoanalytic formal powers in sectors having the singular point as a vertex the results are obtained with the aid of the reduction of the stationary schrodinger equation to a vekua equation of a special form and by using recent developments in pseudoanalytic function theory

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

a well posed and stable stochastic galerkin formulation of the incompressible navier stokes equations with random data we present a well posed stochastic galerkin formulation of the incompressible navier stokes equations with uncertainty in model parameters or the initial and boundary conditions the stochastic galerkin method involves representation of the solution through generalized polynomial chaos expansion and projection of the governing equations onto stochastic basis functions resulting in an extended system of equations a relatively low order generalized polynomial chaos expansion is sufficient to capture the stochastic solution for the problem considered we derive boundary conditions for the continuous form of the stochastic galerkin formulation of the velocity and pressure equations the resulting problem formulation leads to an energy estimate for the divergence with suitable boundary data on the pressure and velocity the energy estimate implies zero divergence of the velocity field based on the analysis of the continuous equations we present a semi discretized system where the spatial derivatives are approximated using finite difference operators with a summation by parts property with a suitable choice of dissipative boundary conditions imposed weakly through penalty terms the semi discrete scheme is shown to be stable numerical experiments in the laminar flow regime corroborate the theoretical results and we obtain high order accurate results for the solution variables and the velocity divergence converges to zero as the mesh is refined

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

ordered overlaps in disordered mean field models let m n be a sequence of integers with m as n and m o n for bounded i i d r v i k and bounded i i d r v i we study the large deviation of the family of ordered scalar products x k n 1 i 1 n i i k k m under the distribution conditioned on the i k s to get a full large deviation principle it is necessary to specify also the total norm k m x k 2 1 2 which turns to be associated with some extra gaussian distribution our results apply to disordered mean field systems including generalized hopfield models in the regime of a sublinear number of patterns we build also a class of examples where this norm is the crucial order parameter

a stable partitioned fsi algorithm for rigid bodies and incompressible flow part ii general formulation abstract a stable partitioned algorithm is developed for fluid structure interaction fsi problems involving viscous incompressible flow and rigid bodies this added mass partitioned amp algorithm remains stable without sub iterations for light and even zero mass rigid bodies when added mass and viscous added damping effects are large the scheme is based on a generalized robin interface condition for the fluid pressure that includes terms involving the linear acceleration and angular acceleration of the rigid body added mass effects are handled in the robin condition by inclusion of a boundary integral term that depends on the pressure added damping effects due to the viscous shear forces on the body are treated by inclusion of added damping tensors that are derived through a linearization of the integrals defining the force and torque added damping effects may be important at low reynolds number or for example in the case of a rotating cylinder or rotating sphere when the rotational moments of inertia are small in this second part of a two part series the general formulation of the amp scheme is presented including the form of the amp interface conditions and added damping tensors for general geometries a fully second order accurate implementation of the amp scheme is developed in two dimensions based on a fractional step method for the incompressible navier stokes equations using finite difference methods and overlapping grids to handle the moving geometry the numerical scheme is verified on a number of difficult benchmark problems

an isoparametric spectral element method for solution of the navier stokes equations in complex geometry abstract an isoparametric spectral element technique is presented for solution of the two dimensional navier stokes equations in arbitrary curvy geometries the spatial discretization is described and the method is illustrated by solution of poisson s equation in a cylindrical annulus a fractional step navier stokes solver based on a discretely consistent poisson equation for the pressure is then presented and exponential convergence in space is demonstrated for the case of separated flow between eccentric rotating cylinders the technique is made efficient by the use of explicit collocation rather than galerkin operators for the convective terms and use of static condensation to solve the elliptic equations resulting from the stokes problem the algorithm is amenable to a high degree of parallelism in both multiprocessor and vector pipeline environments

self averaging in a class of generalized hopfield models we prove the almost sure convergence to zero of the fluctuations of the free energy in a class of disordered mean field spin systems that generalize the hopfield model in two ways i multi spin interactions are permitted and ii the random variables xi imu describing the patterns can have arbitrary distributions with mean zero and finite 4 epsilon th moments the number of patterns m is allowed to be an arbitrary multiple of the system size this generalizes a previous result of bovier gayrard and picco for the standard hopfield model and improves a result of feng and tirozzi that required m to be a finite constant note that the convergence of the mean of the free energy is not proven

free energy and some sample path properties of a random walk with random potential we study the asymptotic behavior of the free energy for a model defined by sinai of one dimensional random walk with random potential in particular we obtain a central limit theorem and a strong law of large numbers for this free energy we use some results on the free energy to study some sample path properties of this random walk which are related respectively to its recurrence and localization some exponents describing the recurrence and localization are found

multiple critical behavior of probabilistic limit theorems in the neighborhood of a tricritical point we derive probabilistic limit theorems that reveal the intricate structure of the phase transitions in a mean field version of the blume emery griffiths model phys rev a4 1971 1071 1077 these probabilistic limit theorems consist of scaling limits for the total spin and moderate deviation principles mdps for the total spin the model under study is defined by a probability distribution that depends on the parameters n and k which represent respectively the number of spins the inverse temperature and the interaction strength the intricate structure of the phase transitions is revealed by the existence of 18 scaling limits and 18 mdps for the total spin these limit results are obtained as k converges along appropriate sequences n kn to points belonging to various subsets of the phase diagram which include a curve of second order points and a tricritical point the forms of the limiting densities in the scaling limits and of the rate functions in the mdps reflect the influence of one or more sets that lie in neighborhoods of the critical points and the tricritical point of all the scaling limits the structure of those near the tricritical point is by far the most complex exhibiting new types of critical behavior when observed in a limit theorem phase diagram in the space of the two parameters that parametrize the scaling limits

the distribution of the partition function of the hopfield model with finite number of patterns we derive the leading term in the large n asymptotic expansion of the partition function of the hopfield model with finite number of patterns we show that this leading order term is deterministic in the high temperature region in the low temperature region and at the critical point it is random with the distribution governed by normal or iterated exponential distributions

a free energy bound for the hopfield model the author gives a simple upper and lower bound on the free energy density of the hopfield model of size n with p stored patterns in the limit where n and p tend to infinity with p n to alpha 1 the two bounds coincide for alpha 0

large deviation principles for the hopfield model and the kac hopfield model we study the kac version of the hopfield model and prove a lebowitz penrose theorem for the distribution of the overlap parameters at the same time we prove a large deviation principle for the standard hopfield model with infinitely many patterns

thermodynamic limit of the q state potts hopfield model with infinitely many patterns we prove the almost sure convergence of the free energy and of the overlap order parameters in aq state version of the hopfield neural network model we compute explicitly these limits for all temperatures different from some critical value the number of stored patterns is allowed to grow with the size of the systemn like lnq lnn we study the limiting behavior of the extremal states of the model that are the measures induced on the gibbs measures by the overlap parameters

a stable partitioned fsi algorithm for rigid bodies and incompressible flow part i model problem analysis a stable partitioned algorithm is developed for fluid structure interaction fsi problems involving viscous incompressible flow and rigid bodies this added mass partitioned amp algorithm remains stable without sub iterations for light and even zero mass rigid bodies when added mass and viscous added damping effects are large the scheme is based on a generalized robin interface condition for the fluid pressure that includes terms involving the linear acceleration and angular acceleration of the rigid body added mass effects are handled in the robin condition by inclusion of a boundary integral term that depends on the pressure added damping effects due to the viscous shear forces on the body are treated by inclusion of added damping tensors that are derived through a linearization of the integrals defining the force and torque added damping effects may be important at low reynolds number or for example in the case of a rotating cylinder or rotating sphere when the rotational moments of inertia are small in this first part of a two part series the properties of the amp scheme are motivated and evaluated through the development and analysis of some model problems the analysis shows when and why the traditional partitioned scheme becomes unstable due to either added mass or added damping effects the analysis also identifies the proper form of the added damping which depends on the discrete time step and the grid spacing normal to the rigid body the results of the analysis are confirmed with numerical simulations that also demonstrate a second order accurate implementation of the amp scheme

composite overlapping meshes for the solution of partial differential equations abstract we discuss the generation of curvilinear composite overlapping grids and the numerical solution of partial differential equations on them a composite overlapping grid consists of a set of curvilinear component grids that cover a region and overlap where they meet continuity conditions interpolation are imposed at the overlapping boundaries the principal advantage of composite grids is in the generation of grids for regions with complicated geometries the grid construction program cmpgrd is used to create composite grids with any number of component grids we describe some techniques for the solution of elliptic and time dependent pdes on composite meshes applications to the solution of the compressible navier stokes equations are presented

some rigorous results on the hopfield neural network model we analyze the thermal equilibrium distribution of 2p mean field variables for the hopfield model withp stored patterns in the case where 2p is small compared to the number of spins in particular we give a full description of the free energy density in the thermodynamic limit and of the so called symmetric solutions for the mean field equations

direct numerical simulation of particulate flows with an overset grid method abstract we evaluate an efficient overset grid method for two dimensional and three dimensional particulate flows for small numbers of particles at finite reynolds number the rigid particles are discretised using moving overset grids overlaid on a cartesian background grid this allows for strongly enforced boundary conditions and local grid refinement at particle surfaces thereby accurately capturing the viscous boundary layer at modest computational cost the incompressible navier stokes equations are solved with a fractional step scheme which is second order accurate in space and time while the fluid solid coupling is achieved with a partitioned approach including multiple sub iterations to increase stability for light rigid bodies through a series of benchmark studies we demonstrate the accuracy and efficiency of this approach compared to other boundary conformal and static grid methods in the literature in particular we find that fully resolving boundary layers at particle surfaces is crucial to obtain accurate solutions to many common test cases with our approach we are able to compute accurate solutions using as little as one third the number of grid points as uniform grid computations in the literature a detailed convergence study shows a 13 fold decrease in cpu time over a uniform grid test case whilst maintaining comparable solution accuracy

the replica symmetric solution without replica trick for the hopfield model we derive the saddle point equations for the order parameters of the hopfield model in the case of replica symmetry without using the replica trick but assuming that the edwards anderson parameter is a self averaging quantity

new applications of pseudoanalytic function theory to the dirac equation in the present work we establish a simple relation between the dirac equation with a scalar and an electromagnetic potential in a two dimensional case and a pair of decoupled vekua equations in general these vekua equations are bicomplex however we show that the whole theory of pseudoanalytic functions without modifications can be applied to these equations under a certain nonrestrictive condition as an example we formulate the similarity principle which is the central reason why a pseudoanalytic function and as a consequence a spinor field depending on two space variables share many of the properties of analytic functions one of the surprising consequences of the established relation with pseudoanalytic functions consists in the following result consider the dirac equation with a scalar potential depending on one variable with fixed energy and mass in general this equation cannot be solved explicitly even if one looks for wavefunctions of one variable nevertheless for such dirac equation we obtain an algorithmically simple procedure for constructing in explicit form a complete system of exact solutions depending on two variables these solutions generalize the system of powers 1 z z2 in complex analysis and are called formal powers with their aid any regular solution of the dirac equation can be represented by its taylor series in formal powers

a collocated method for the incompressible navier stokes equations inspired by the box scheme we present a new finite difference numerical method to solve the incompressible navier stokes equations using a collocated discretization in space on a logically cartesian grid the method shares some common aspects with and it was inspired by the box scheme it uses centered second order accurate finite difference approximations for the spatial derivatives combined with semi implicit time integration the proposed method is constructed to ensure discrete conservation of mass and momentum by discretizing the primitive velocity pressure form of the equations the continuity equation is enforced exactly to machine accuracy at the collocated locations whereas the momentum equations are evaluated in a staggered manner this formulation preempts the appearance of spurious pressure modes in the embedded elliptic problem associated with the pressure the method shows uniform order of accuracy both in space and time for velocity and pressure in addition the skew symmetric form of the non linear advection term of the navier stokes equations improves discrete conservation of kinetic energy in the inviscid limit to within the order of the truncation error of the time integrator the method has been formulated to accommodate different types of boundary conditions fully periodic periodic channel inflow outflow and lid driven cavity always ensuring global mass conservation a novel aspect of this finite difference formulation is the derivation of the discretization near boundaries using the weak form of the equations as in the finite element method the method of manufactured solutions is utilized to perform accuracy analysis and verification of the solver to assess the applicability of the new method presented in this paper four realistic flow problems have been simulated and results are compared with those in the literature these cases include a lid driven cavity backward facing step kovasznay flow and fully developed turbulent channel

rigorous results on the thermodynamics of the dilute hopfield model we study the hopfield model of an autoassociative memory on a random graph onn vertices where the probability of two vertices being joined by a link isp n assuming thatp n goes to zero more slowly thano 1 n we prove the following results 1 if the number of stored patternsm n is small enough such thatm n np n 0 asn then the free energy of this model converges upon proper rescaling to that of the standard curie weiss model for almost all choices of the random graph and the random patterns 2 if in additionm n ln n ln 2 we prove that there exists fort 1 a gibbs measure associated to each original pattern whereas for higher temperatures the gibbs measure is unique the basic technical result in the proofs is a uniform bound on the difference between the hamiltonian on a random graph and its mean value