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

frequency domain analysis of exponentially damped linear systems abstract in this paper frequency domain analysis of linear exponentially damped systems subjected to arbitrary initial conditions is studied the standard discrete fourier transform dft procedure combined with fast fourier transform fft algorithms is applied to compute dynamic responses after transforming the original motion equations subjected to non zero initial conditions into new ones with completely null initial conditions by using the pseudo force approach a multi degree of freedom mdof system and a continuum system with exponential damping are considered as the illustrative problems the accuracy of the proposed method is evaluated by comparing the results with those obtained from simulink based on the state space method

initial conditions in frequency domain analysis the fem applied to the scalar wave equation the present paper describes a procedure to consider initial conditions in the frequency domain analysis of continuous media discretized by the fem the frequency domain formulation presented here is based on a standard dft procedure the fft algorithm being employed to transform from time to the frequency domain and vice versa the standard galerkin finite element method displacement model is used to replace the original differential governing equation by an integral equation amenable to numerical solution the scalar wave equation one and two dimensional is used to illustrate the proposed approach at the end of the paper examples of wave propagation for a taut string a one dimensional rod and a membrane are presented to illustrate the robustness of the formulation presented here

dynamic analysis of the linear discrete dynamic systems subjected to the initial conditions by using an fft based spectral analysis method abstract in this paper a fast fourier transforms fft based spectral analysis method is introduced for the dynamic analysis of the linear discrete dynamic systems subjected to the non zero initial conditions the fft based spectral analysis method is developed first for the one degree of freedom dof system and then extended to the multi dof systems to evaluate the accuracy and convergence of the fft based spectral analysis method the forced vibration of a viscously damped 3 dof system is considered as an illustrative problem the accuracy of the proposed fft based spectral analysis method is evaluated by comparing the forced vibration responses obtained by using the fft based spectral analysis method with the exact analytical solutions as well as with the numerical results obtained by using the runge kutta method

rigorous solution of the gardner problem we prove rigorously the well known result of gardner about the typical fractional volume of interactions between n spins which solve the problem of storing a given set of p random patterns the gardner formula for this volume in the limit n p p n is proven for all values of besides we prove a useful criterion for the factorisation of all correlation functions for a class of models of classical statistical mechanics

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

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

clifford algebras of hyperbolic involutions for as a central simple algebra of even degree with hyperbolic orthogonal involution we describe the canonically induced involution underline sigma on the even clifford algebra c 0 a sigma underline sigma of a sigma when deg a equiv 0 mod 8 a cong m 2 b and the interesting part of underline sigma is isomorphic to the canonical involution on an exterior power algebra of b as a corollary we get some properties of the involution on the exterior power algebra

a variety associated to an algebra with involution abstract a variety iv a is canonically associated to a central simple algebra a with orthogonal involution of the first kind the properties of this variety are the main topic of this paper the paper is divided into two parts we first define this variety to be the subvariety of the grassmannian of the n dimensional subspaces of the n2 dimensional vector space a consisting of all the n dimensional right ideals i of a such that i i 0 the geometric properties of iv a are investigated in particular the generic property of this variety is identified further the kernel of the scalar extension map br f br k where k is the function field of iv a is also computed lastly a criterion on the embeddability of the field k into the function field of the brauer severi variety of a is given in the second part the quillen k theory of iv a is computed the k theory result then leads to an index reduction formula for the function field k

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

tight bounds on the capacity of binary input random cdma systems in this paper we consider code division multiple access cdma communication over a binary input additive white gaussian noise awgn channel using random spreading for a general class of symmetric distributions for spreading sequences in the limit of a large number of users we prove an upper bound to the capacity the bound matches the formula obtained by tanaka using the replica method we also show concentration of various relevant quantities including mutual information and free energy the mathematical methods are quite general and allow us to discuss extensions to other multiuser scenarios

nonlinear neural networks ii information processing information processing in nonlinear neural networks with a finite numberq of stored patterns is studied each network is characterized completely by its synaptic kernelq at low temperatures the nonlinearity typically results in 2q 2 q metastable pure states in addition to theq retrieval states that are associated with theq stored patterns these spurious states start appearing at a temperature tilde t q which depends onq we give sufficient conditions to guarantee that the retrieval states bifurcate first at a critical temperaturetc and that tilde t q tc 0 asq hence there is a large temperature range whereonly the retrieval states and certain symmetric mixtures thereof exist the latter are unstable as they appear attc for clipped synapses the bifurcation and stability structure is analyzed in detail and shown to approach that of the linear hopfield model asq we also investigate memories that forget and indicate how forgetfulness can be explained in terms of the eigenvalue spectrum of the synaptic kernelq

about the ergodic regime in the analogical hopfield neural networks moments of the partition function in this paper we introduce and exploit the real replica approach for a minimal generalization of the hopfield model by assuming the learned patterns to be distributed according to a standard unit gaussian we consider the high storage case when the number of patterns linearly diverges with the number of neurons we study the infinite volume behavior of the normalized momenta of the partition function we find a region in the parameter space where the free energy density in the infinite volume limit self averages around its annealed approximation as well as the entropy and the internal energy density moreover we evaluate the corrections to their extensive counterparts with respect to their annealed expressions the fluctuations of properly introduced overlaps which act as order parameters are also discussed

the legendre transform of two replicas of the sherrington kirkpatrick spin glass model we prove a variational inequality linking the values of the free energy per site at different temperatures this inequality is based on the legendre transform of the free energy of two replicas of the system we prove that equality holds when 1 sqrt 2 and fails when 1 sqrt 2 1 sqrt 2 we also use our inequality numerically within the low temperature region to improve slightly the best previously known lower bounds for the free energy and the ground state energy per site

fft based spectral element analysis for the linear continuum dynamic systems subjected to arbitrary initial conditions by using the pseudo force method in this paper a fast fourier transform fft based spectral element method sem is developed for the linear continuum dynamic systems subjected to arbitrary non null initial conditions in the fft based sem the original equations of motion subjected to arbitrary initial conditions are transformed into a new set of equations of motion subjected to completely null initial conditions by using the pseudo force method so that the conventional spectral element analysis can be applied to obtain desired dynamic responses a simply supported beam and a cantilevered beam are considered as the illustrative problems to evaluate the fft based sem the dynamic responses obtained by using the fft based sem are shown to be in good agreement with the analytical solutions obtained by using the mode superposition method copyright 2007 john wiley sons ltd

the thermodynamic limit in mean field spin glass models we present a simple strategy in order to show the existence and uniqueness of the infinite volume limit of thermodynamic quantities for a large class of mean field disordered models as for example the sherrington kirkpatrick model and the derrida p spin model the main argument is based on a smooth interpolation between a large system made of n spin sites and two similar but independent subsystems made of n 1 and n 2 sites respectively with n 1 n 2 n the quenched average of the free energy turns out to be subadditive with respect to the size of the system this gives immediately convergence of the free energy per site in the infinite volume limit moreover a simple argument based on concentration of measure gives the almost sure convergence with respect to the external noise similar results hold also for the ground state energy per site

notes on the polynomial identities in random overlap structures in these notes we review first in some detail the concept of random overlap structure rost applied to fully connected and diluted spin glasses we then sketch how to write down the general term of the expansion of the energy part from the boltzmann rost for the sherrington kirkpatrick model and the corresponding term from the ramost which is the diluted extension suitable for the viana bray model

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

the sherrington kirkpatrick model of spin glasses and stochastic calculus the high temperature case we study the fluctuations of free energy energy and entropy in the high temperature regime for the sherrington kirkpatrick model of spin glasses we introduce here a new dynamical method with the help of brownian motions and continuous martingales indexed by the square root of the inverse temperature as parameter thus formulating the thermodynamic formalism in terms of random processes the well established technique of stochastic calculus leads us naturally to prove that these fluctuations are simple gaussian processes with independent increments a generalization of a result proved by aizenman lebowitz and ruelle 1

on the free energy of the hopfield model the general theory of inhomogeneous mean field systems of raggio and werner provides a variational expression for the almost sure limiting free energy density of the hopfield model h n p xi s frac 1 2n sum limits i j 1 n sum limits mu 1 n xi i mu xi j mu s i s j for ising spinssi andp random patterns 1 2 n under the assumption that mathop lim limits n to gamma n 1 sum limits i 1 n delta xi i lambda xi i xi i 1 xi i 2 xi i p exists almost surely in the space of probability measures overp copies of 1 1 including an external field term p h i 1 n i si we give a number of general properties of the free energy density and compute it for a p 2 in general and b p arbitrary when is uniform and at most the two componentsh 1 andh 2 are nonzero obtaining the almost sure formula f beta h tfrac 1 2 f ew beta h mu 1 h mu 2 tfrac 1 2 f ew beta h mu 1 h mu 2 for the free energy wherefcw denotes the limiting free energy density of the curie weiss model with unit interaction constant in both cases we obtain explicit formulas for the limiting almost sure values of the so called overlap parameters m n mu beta h n 1 sum limits i 1 n xi i mu left langle s i right rangle in terms of the curie weiss magnetizations for the general i i d case with prob i 1 1 2 we obtain the lower bound 1 4 2 p 1 for the temperaturetc separating the trivial free regime where the overlap vector is zero from the nontrivial regime where it is nonzero this lower bound is exact forp 2 or e 0 or e 1 2 forp 2 we identify an intermediate temperature region between t 1 4 2 and tc 1 4 2 where the overlap vector is homogeneous i e all its components are equal and nonzero t marks the transition to the nonhomogeneous regime where the components of the overlap vector are distinct we conjecture that the homogeneous nonzero regime exists forp 3 and that t max 1 4 2 p 1 0

there is no theory of everything inside e 8 we analyze certain subgroups of real and complex forms of the lie group e8 and deduce that any theory of everything obtained by embedding the gauge groups of gravity and the standard model into a real or complex form of e8 lacks certain representation theoretic properties required by physical reality the arguments themselves amount to representation theory of lie algebras in the spirit of dynkin s classic papers and are written for mathematicians

broken replica symmetry bounds in the mean field spin glass model by using a simple interpolation argument in previous work we have proven the existence of the thermodynamic limit for mean field disordered models including the sherrington kirkpatrick model and the derrida p spin model here we extend this argument in order to compare the limiting free energy with the expression given by the parisi ansatz and including full spontaneous replica symmetry breaking our main result is that the quenched average of the free energy is bounded from below by the value given in the parisi ansatz uniformly in the size of the system moreover the difference between the two expressions is given in the form of a sum rule extending our previous work on the comparison between the true free energy and its replica symmetric sherrington kirkpatrick approximation we give also a variational bound for the infinite volume limit of the ground state energy per site

vector precoding for gaussian mimo broadcast channels impact of replica symmetry breaking the replica method of statistical physics is employed for the large system analysis of vector precoding for the gaussian multiple input multiple output broadcast channel the transmitter comprises a linear front end combined with nonlinear precoding minimizing transmit energy by means of input alphabet relaxation for the common discrete lattice based relaxation the problem violates replica symmetry and a replica symmetry breaking rsb ansatz is taken the limiting empirical distribution of the precoder s output and the limiting transmit energy are derived for one step rsb particularizing to a zero forcing zf linear front end a decoupling result is derived for discrete lattice based relaxations the impact of rsb is demonstrated for the transmit energy the spectral efficiencies of the aforementioned precoding methods are compared to linear zf and tomlinson harashima precoding thp focusing on quaternary phase shift keying qpsk significant performance gains of both lattice and convex relaxations are revealed for medium to high signal to noise ratios snrs when compared to linear zf precoding thp is shown to be outperformed as well comparing certain lattice based relaxations for qpsk against a convex counterpart the latter is found to be superior for low and high snrs but slightly inferior for medium snrs in terms of spectral efficiency

on an achievable rate of large rayleigh block fading mimo channels with no csi training based transmission over rayleigh block fading multiple input multiple output mimo channels is investigated as a training method a combination of a pilot assisted scheme and a biased signaling scheme is considered the achievable rates of successive decoding sd receivers based on the linear minimum mean squared error lmmse channel estimation are analyzed in the large system limit by using the replica method under the assumption of replica symmetry it is shown that negligible pilot information is best in terms of the achievable rates of the sd receivers in the large system limit the obtained analytical formulas of the achievable rates can improve the existing lower bound on the capacity of the mimo channel with no channel state information csi derived by hassibi and hochwald for all snrs the comparison between the obtained bound and a high snr approximation of the channel capacity derived by zheng and tse implies that the high snr approximation is unreliable unless quite high snr is considered energy efficiency in the low snr regime is also investigated in terms of the power per information bit required for reliable communication the required minimum power is shown to be achieved at a positive rate for the sd receiver with no csi whereas it is achieved in the zero rate limit for the case of perfect csi available at the receiver moreover numerical simulations imply that the presented large system analysis can provide a good approximation for not so large systems the results in this paper imply that sd schemes can provide a significant performance gain in the low to moderate snr regimes compared to conventional receivers based on one shot channel estimation

rigorous results for the free energy in the hopfield model we prove that the free energy of the hopfield model with a finite number of patterns can be represented in terms of an asymptotic series expansion in inverse powers of the neurons number the series is borel summable for large temperatures we also establish mathematically some other interesting properties partly used before in a seminal paper by amit gutfreund and sompolinsky

on the distribution of overlaps in the sherrington kirkpatrick spin glass model this paper describes some of the analytic tools developed recently by ghirlanda and guerra in the investigation of the distribution of overlaps in the sherrington kirkpatrick spin glass model and of parisi s ultrametricity in particular we introduce to this task a simplified but also generalized model on which the gaussian analysis is made easier moments of the hamiltonian and derivatives of the free energy are expressed as polynomials of the overlaps under the essential tool of self averaging we describe with full rigour various overlap identities and replica independence that actually hold in a rather large generality the results are presented in a language accessible to probabilists and analysts

quenched large deviation principle for the overlap of a p spins system in this note we prove a quenched large deviation result for the overlap of a p spins interaction system at high temperature the rate function of the large deviation principle is proved to be deterministic and some of its basic properties are studied our result is based on a pure state result for a multidimensional p spins system combined with a careful application of the gartner ellis theorem

rigorous bounds on the storage capacity of the dilute hopfield model we study a neural network model consisting ofn neurons where a dendritic connection between each pair of neurons exists with probabilityp and is absent with probability 1 p for the hopfield hamiltonian on such a network we prove that ifp c lnn n 1 2 the model can store at leastm cpn patterns where c 0 027 ifc 3 and decreases proportional to 1 lnc forc small this generalizes the results of newman for the standard hopfield model

discriminant and clifford algebras the centralizer of a square central skew symmetric unit in a central simple algebra with orthogonal involution carries a unitary involution the discriminant algebra of this unitary involution is shown to be an orthogonal summand in one of the components of the clifford algebra of the orthogonal involution as an application structure theorems for orthogonal involutions on central simple algebras of degree 8 are obtained

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

nonlinear neural networks i general theory a neural network is called nonlinear if the introduction of new data into the synaptic efficacies has to be performed through anonlinear operation the original hopfield model is linear whereas for instance clipped synapses constitute a nonlinear model here a general theory is presented to obtain the statistical mechanics of a neural network with finitely many patterns and arbitrary symmetric nonlinearity the problem is reduced to minimizing a free energy functional over all solutions of a fixed point equation with synaptic kernelq the case of clipped synapses with bimodal and gaussian probability distribution is analyzed in detail to this end a simple theory is developed that gives the spectrum ofq and thereby all the solutions that bifurcate from the high temperature phase

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

rigorous results for mean field models for spin glasses in this paper i explain why i believe that it is important to prove rigorous results about mean field models for spin glasses and why i think it is difficult i also describe at a high level what mathematicians have been able to prove

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

fluctuation of the free energy in the hopfield model we prove that in the ergodic region t j2 1 r the deviation of the total free energy of the hopfield neural network converges in distribution asn to a shifted gaussian variable moreover the free energy per site converges in probability to lim 1 n ln n

a new look at independence the concentration of measure phenomenon in product spaces is a farreaching abstract generalization of the classical exponential inequalities for sums of independent random variables we attempt to explain in the simplest possible terms the basic concepts underlying this phenomenon the basic method to prove concentration inequalities and the meaning of several of the most useful inequalities

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

random site spin glass models a simple general method is presented for solving mean field spin glass models where the bond randomness is expressible in terms of an underlying site randomness both separable and non separable models can be solved

the adaptive interpolation method a simple scheme to prove replica formulas in bayesian inference in recent years important progress has been achieved towards proving the validity of the replica predictions for the asymptotic mutual information or free energy in bayesian inference problems the proof techniques that have emerged appear to be quite general despite they have been worked out on a case by case basis unfortunately a common point between all these schemes is their relatively high level of technicality we present a new proof scheme that is quite straightforward with respect to the previous ones we call it the adaptive interpolation method because it can be seen as an extension of the interpolation method developped by guerra and toninelli in the context of spin glasses with an interpolation path that is adaptive in order to illustrate our method we show how to prove the replica formula for three non trivial inference problems the first one is symmetric rank one matrix estimation or factorisation which is the simplest problem considered here and the one for which the method is presented in full details then we generalize to symmetric tensor estimation and random linear estimation we believe that the present method has a much wider range of applicability and also sheds new insights on the reasons for the validity of replica formulas in bayesian inference

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

on the use of pseudo forces to consider initial conditions in 3d time and frequency domain acoustic analysis this work is mainly concerned with a general strategy based on well known concepts of classical mechanics for taking into account initial conditions in frequency domain fd and time domain td analyses a general approach extended here to three dimensional applications is presented special problems associated with analyses through discrete fourier transform dft algorithms as those occurring in consequence of a non correct choice of extended period or those connected with aliasing phenomenon are also discussed furthermore an alternative starting procedure for time marching schemes in td analyses is proposed at the end of the paper to validate the proposed techniques and to demonstrate their generality two and three dimensional problems with non homogeneous initial conditions are solved through frequency and time domain approaches by employing the finite element method fem numerical results are compared with existing analytical solutions

large time asymptotics for the fluctuation spde in the kuramoto synchronization model abstract we investigate the long time asymptotics of the fluctuation spde in the kuramoto synchronization model we establish the linear behavior for large time and weak disorder of the quenched limit fluctuations of the empirical measure of the particles around its mckean vlasov limit this is carried out through a spectral analysis of the underlying unbounded evolution operator using arguments of perturbation of self adjoint operators and analytic semigroups we state in particular a jordan decomposition of the evolution operator which is the key point in order to show that the fluctuations of the disordered kuramoto model are not self averaging

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