violation of the fluctuation dissipation theorem in glassy systems basic notions and the numerical evidence this review reports on the research done during past years on violations of the fluctuation dissipation theorem fdt in glassy systems it is focused on the existence of a quasi fluctuation dissipation theorem qfdt in glassy systems and the current supporting knowledge gained from numerical simulation studies it covers a broad range of non stationary aging and stationary driven systems such as structural glasses spin glasses coarsening systems ferromagnetic models at criticality trap models models with entropy barriers kinetically constrained models sheared systems and granular media the review is divided into four main parts 1 an introductory section explaining basic notions related to the existence of the fdt in equilibrium and its possible extension to the glassy regime qfdt 2 a description of the basic analytical tools and results derived in the framework of some exactly solvable models 3 a detailed report of the current evidence in favour of the qfdt and 4 a brief digression on the experimental evidence in its favour this review is intended for inexpert readers who want to learn about the basic notions and concepts related to the existence of the qfdt as well as for the more expert readers who may be interested in more specific results

localization and diagonalization a review of functional integral techniques for low dimensional gauge theories and topological field theories localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low dimensional gauge theories are reviewed these are the functional integral counterparts of the mathai quillen formalism the duistermaat heckman theorem and the weyl integral formula respectively in each case the necessary mathematical background euler classes of vector bundles equivariant cohomology topology of lie groups is introduced and the finite dimensional integration formulae described then some applications to path integrals are discussed and an overview of the relevant literature is given the applications include supersymmetric quantum mechanics cohomological field theories phase space path integrals and two dimensional yang mills theory

oversampled filter banks as error correcting codes theory and impulse noise correction oversampled filter banks ofbs provide an overcomplete representation of their input signal this paper describes how ofbs can be considered as error correcting codes acting on real or complex sequences very much like classical binary convolutional codes act on binary sequences the structured redundancy introduced by ofbs in subband signals can be used to increase robustness to noise in this paper we define the notions of code subspace syndrome and parity check polynomial matrix for ofbs furthermore we derive generic expressions for projection based decoding suitable for the case when a simple second order model completely characterizes the noise incurred by subband signals we also develop a nonlinear hypotheses test based decoding algorithm for the case when the noise in subbands is constituted by a gaussian background noise and impulsive errors a model that adequately describes the action of both quantization noise and transmission errors simulation results show that the algorithm effectively removes the effect of impulsive errors occurring with a probability of 10 sup 3

a single saddle model for the β relaxation in supercooled liquids we analyse the relaxational dynamics of a system close to a saddle of the potential energy function within an harmonic approximation our main aim is to relate the topological properties of the saddle as encoded in its spectrum to the dynamical behaviour of the system in the context of the potential energy landscape approach this represents a first formal step to investigate the belief that the dynamical slowing down at tc is related to the vanishing of the number of negative modes found at the typical saddle point in our analysis we keep the description as general as possible using the spectrum of the saddle as an input we prove the existence of a timescale t which is uniquely determined by the spectrum but is not simply related to the fraction of negative eigenvalues the mean square displacement develops a plateau of length t such that a two step relaxation is obtained if t diverges at tc we analyse different spectral shapes and outline the conditions under which the mean square displacement exhibits a dynamical scaling identical to the relaxation regime of mode coupling theory with a power law approach to the plateau and power law divergence of t at tc

kramers equation and supersymmetry hamilton s equations with noise and friction possess a hidden supersymmetry valid for time independent as well as periodically time dependent systems it is used to derive topological properties of critical points and periodic trajectories in an elementary way from a more practical point of view the formalism provides new tools to study the reaction paths in systems with separated time scales a reduced current which contains the relevant part of the phase space probability current is introduced together with strategies for its computation

statistical mechanics of a single particle in a multiscale random potential parisi landscapes in finite dimensional euclidean spaces we construct an n dimensional gaussian landscape with multiscale translation invariant logarithmic correlations and investigate the statistical mechanics of a single particle in this environment in the limit of high dimension n the free energy of the system and overlap function are calculated exactly using the replica trick and parisi s hierarchical ansatz in the thermodynamic limit we recover the most general version of the derrida s generalized random energy model grem the low temperature behaviour depends essentially on the spectrum of length scales involved in the construction of the landscape if the latter consists of k discrete values the system is characterized by a k step replica symmetry breaking solution we argue that our construction is in fact valid in any finite spatial dimensions n 1 we discuss the implications of our results for the singularity spectrum describing multifractality of the associated boltzmann gibbs measure finally we discuss several generalizations and open problems such as the dynamics in such a landscape and the construction of a generalized multifractal random walk

an asymptotic solution of the first passage problem for singular diffusion in population biology the first passage problem is studied for a singular diffusion process arising in population biology with the deterministic part having a stable equilibrium point and small diffusion the laplace transform of the first passage time density satisfies an ordinary differential equation in a finite interval with two point boundary conditions a formal asymptotic solution to this problem is constructed when the singular boundary point is absorbing and the regular boundary point is either absorbing or reflecting the solution depends upon the initial value z of the process for z near the equilibrium point the density is approximated by the ornstein uhlenbeck equation for z near the singular boundary point it is approximated by the diffusion limit of the linear birth death process and elsewhere it is approximated by the wkb solution these different solutions are then connected together using these results the process is shown to exit with probability one in the limit of small diffusion the mean and varianc

distinguishability of particles in glass forming systems abstract the distinguishability of particles has important implications for calculating the partition function in statistical mechanics while there are standard formulations for systems of identical particles that are either fully distinguishable or fully indistinguishable many realistic systems do not fall into either of these limiting cases in particular the glass transition involves a continuous transition from an ergodic liquid system of indistinguishable particles to a nonergodic glassy system where the particles become distinguishable while the question of partial distinguishability of microstates has been treated previously in quantum information theory this issue has not yet been addressed for a system of classical particles in this paper we present a general theoretical formalism for quantifying particle distinguishability in classical systems this formalism is based on a classical definition of relative entropy such as applied in quantum information theory example calculations for a simple glass forming system demonstrate the continuous onset of distinguishability as temperature is lowered we also examine the loss of distinguishability in the limit of long observation time coinciding with the restoration of ergodicity we discuss some of the general implications of our work including the direct connection to topological constraint theory of glass we also discuss qualitative features of distinguishability as they relate to the second and third laws of thermodynamics

random walks in asymmetric random environments we consider random walks on zd with transition ratesp x y given by a random matrix ifp is a small random perturbation of the simple random walk we show that the walk remains diffusive for almost all environmentsp ifd 2 the result also holds for a continuous time markov process with a random drift the corresponding path space measures converge weakly in the scaling limit to the wiener process for almost everyp

joint source channel coding using real bch codes for robust image transmission in this paper a new still image coding scheme is presented in contrast with standard tandem coding schemes where the redundancy is introduced after source coding it is introduced before source coding using real bch codes a joint channel model is first presented the model corresponds to a memoryless mixture of gaussian and bernoulli gaussian noise it may represent the source coder the channel coder the physical channel and their corresponding decoder decoding algorithms are derived from this channel model and compared to a state of art real bch decoding scheme a further comparison with two reference tandem coding schemes and the proposed joint coding scheme for the robust transmission of still images has been presented when the tandem scheme is not accurately tuned the joint coding scheme outperforms the tandem scheme in all situations compared to a tandem scheme well tuned for a given channel situation the joint coding scheme shows an increased robustness as the channel conditions worsen the soft performance degradation observed when the channel worsens gives an additional advantage to the joint source channel coding scheme for fading channels since a reconstruction with moderate quality may be still possible even if the channel is in a deep fade

an asymptotic solution to a two dimensional exit problem arising in population dynamics a study is made of a two dimensional stochastic system with small stochastic fluctuations arising in population biology at the boundary of the state space the diffusion matrix becomes singular by an asymptotic analysis expressions are derived that determine the probability of exit at each of the two boundaries and the expectation and variance of the exit time these expressions contain constants that can be computed numerically

determination of the number of errors in dft codes subject to low level quantization noise this paper analyzes the effects of quantization or other low level noise on the error correcting capability of a popular class of real number bose chaudhuri hocquenghem bch codes known as discrete fourier transform dft codes in the absence of low level noise a modified version of the peterson gorenstein zierler pgz algorithm allows the correction of up to corrupted entries in the real valued code vector of an dft code in this paper we analyze the performance of this modified pgz algorithm in the presence of low level quantization or other noise that might affect each entry of the code vector and not simply of them we focus on the part of the algorithm that determines the number of errors that have corrupted the real number codeword our approach for determining the number of errors is more effective than existing systematic approaches in the literature and results in an explicit lower bound on the precision needed to guarantee the correct determination of the number of errors our simulations suggest that this bound can be tight finally we prove that the optimal bit allocation for dft codes in terms of correctly determining the number of errors is the uniform one

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

aging and relaxation dynamics in free energy landscapes with multiple minima abstract we consider the stochastic dynamics of a system thermally relaxing in a free energy landscape with multiple attractors and show that lack of translational homogeneity in this landscape leads to aging effects e g to the dependence of the susceptibilities on the time elapsed from a thermal quench to the imposition of the probing field we then prove an inequality between response and correlation which generalizes the fluctuation dissipation theorem to a situation far from thermodynamical equilibrium as an application and a check we specialize our formalism in a way which we suggest is appropriate for spin glass systems we assume a hierarchical organization of the landscape and find aging behavior in the response curves in good agreement with relevant experimental data we finally conclude with a summary and a brief discussion of different approaches to slow relaxation in complex systems

correcting wavelet codes using interleaving and trapping a rate k n wavelet code a form of real convolutional code that encodes k numbers into n numerical code symbols using the present and past input numbers may be used to transport numerical data with redundancy factor rm n rm k such codes offer detection and correction within roundoff tolerances wavelet codes can be processed using two different algebraic representations vector matrix or polyphase polynomial when an interleaver intersperses the code symbols any burst of errors within the interelaver s frame length appears in only one component of the polyphase format a special extended polyphase parity check matrix exposes the error bursts directly for redundancies greater than 1 or a deconvolver operation achieves the same when the redundancy factor is 1 the trapped errors correct the symbols by subtraction simulations verify the effectiveness of this trapping technique

dynamics and geometric properties of the k trigonometric model we analyse the dynamics and the geometric properties of the potential energy surfaces pes of the k trigonometric model ktm defined by a fully connected k body interaction this model has no thermodynamic transition for k 1 a second order one for k 2 and a first order one for k 2 in this paper we i show that the single particle dynamics can be traced back to an effective dynamical system with only one degree of freedom ii compute the diffusion constant analytically iii determine analytically several properties of the self correlation functions apart from the relaxation times which we calculate numerically iv relate the collective correlation functions to those of the effective degree of freedom using an exact dyson like equation v using two analytical methods calculate the saddles of the pes that are visited by the system evolving at fixed temperature on the one hand we minimize v 2 as usually done in the numerical study of supercooled liquids and on the other hand we compute the saddles with minimum distance in configuration space from initial equilibrium configurations we find the same result from the two calculations and we speculate that the coincidence might go beyond the specific model investigated here

random matrices and complexity of spin glasses cern y abstract we give an asymptotic evaluation of the complexity of spherical p spin spin glass models via random matrix theory this study enables us to obtain detailed infor mation about the bottom of the energy landscape including the absolute minimum the ground state the other local minima and describe an interesting layered structure of the low critical values for the hamiltonians of these models we also show that our ap proach allows us to compute the related tap complexity and extend the results known in the physics literature as an independent tool we prove a ldp for the k th largest eigenvalue of the goe extending the results of bdg01 how many critical values of given index and below a given level does a typical random morse function have on a high dimensional manifold our work addresses this question in a very special case we look at certain natural random gaussian functions on the n dimensional sphere known as p spin spherical spin glass models we cannot yet answer the question above about the typical number but we can study thoroughly the mean number which we show is exponentially large in n we introduce a new identity based on the classical kac rice formula relating random matrix theory and the problem of counting these critical values using this identity and tools from random matrix theory we give an asymptotic evaluation of the complexity of these spherical spin glass models the complexity mentioned here is defined as the mean number of critical points of given index whose value is below or above a given level this includes the important question of counting the mean number of local minima below a given level and in particular the question of finding the ground state energy the minimal value of the hamiltonian we show that this question is directly related to the study of the edge of the spectrum of the gaussian orthogonal ensemble goe the question of computing the complexity of mean field spin glass models has recently been thoroughly studied in the physics literature see for example clr03 and the refer ences therein mainly for a different measure of the complexity i e the mean number of solutions to the thouless anderson palmer equations or tap complexity our approach to the complexity enables us to recover known results in the physics literature about tap complexity to compute the ground state energy when p is even and to describe an interesting layered structure of the low energy levels of the hamiltonians of these models which might prove useful for the study of the metastability of langevin dynamics for these models in longer time scales than those studied in bdg01 the paper is organised as follows in section 2 we give our main results in section 3 we prove two main formulas theorem 2 1 and 2 2 relating random matrix theory specif ically the goe and spherical spin glasses these formulas are direct consequences of the kac rice formula we learned the version needed here in the book at07 for another modern account see aw09 the main new ingredient is the fact that for spherical spin glass models the hessian of the hamiltonian at a critical point conditioned on the value of the hamiltonian is a symmetric gaussian random matrix with independent entries up to symmetry plus a diagonal matrix this implies in particular that it is possible to

a role for a brachistochrone in the broken ergodicity scenario abstract we show that the time evolution of ergodic decomposition is outlined by a variational principle for a rather universal class of complex systems with a quasicontinuous spectrum of intrinsic relaxation timescales in turn this variational approach enables us to single out a logarithmic scaling law for the barriers on the hierarchical level such a logarithmic dependence has already been shown to hold valid for certain complex systems moreover it represents a key factor to explain the experimentally ubiquitous debye kohlrausch relaxation law and plays a central role on different theoretical model descriptions within this field

complex field coding for ofdm over fading wireless channels orthogonal frequency division multiplexing ofdm converts a time dispersive channel into parallel subchannels and thus facilitates equalization and de coding but when the channel has nulls close to or on the fast fourier transform fft grid uncoded ofdm faces serious symbol recovery problems as an alternative to various error control coding techniques that have been proposed to ameliorate the problem we perform complex field coding cfc before the symbols are multiplexed we quantify the maximum achievable diversity order for independent and identically distributed i i d or correlated rayleigh fading channels and also provide design rules for achieving the maximum diversity order the maximum coding gain is given and the encoder enabling the maximum coding gain is also found simulated performance comparisons of cfc ofdm with existing block and convolutionally coded ofdm alternatives favor cfc ofdm for the code rates used in a hiperlan2 experiment

generalized ensemble and tempering simulations a unified view from the underlying master equations we derive one dimensional stochastic processes that describe generalized ensemble simulations as well as tempering simulated and parallel simulations the representations obtained are either in the form of a one dimensional fokker planck equation or a hopping process on a one dimensional chain in particular we discuss the conditions under which these representations are valid approximate markovian descriptions of the random walk in order parameter or control parameter space they allow a unified discussion of the stationary distribution on as well as of the stationary flow across each space we demonstrate that optimizing the flow is equivalent to minimizing the first passage time for crossing the space and discuss the consequences of our results for optimizing simulations finally we point out the limitations of these representations under conditions of broken ergodicity

diffusion across characteristic boundaries with critical points we consider the problems of the effect of small white noise perturbations on a deterministic dynamical system in the plane with i an asymptotically stable equilibrium point or limit cycle and ii an equilibrium point surrounded by closed trajectories the mean exit time and the distribution of exit points for each problem is determined by solving singularly perturbed elliptic boundary value problems in domains with closed characteristic boundaries with critical points uniformly valid asymptotic solutions are constructed for each of the problems for the asymptotically stable equilibrium point the method of matched asymptotic expansions with the integral condition of matkowsky and schuss is employed a method of averaging combined with boundary layer analysis is used for the problem of an equilibrium point surrounded by closed trajectories the influence on the solutions of the critical points on the boundary is exhibited and explained an application to the physical pendulum is given finally our r

generalized algorithm based fault tolerance error correction via kalman estimation an extension to algorithm based fault tolerance abft methodologies shows how parity values dictated by a real convolutional code can be employed by kalman estimation techniques to perform real number correction for protecting linear processing systems intermittent failures appearing in the output samples are detected and corrected using only the syndromes normally generated in abft schemes the algebraic structure of a real convolutional code provides separation needed by recursive kalman state estimators to affect mean square error correction state and parity measurement equations model fault and computational noise in both the linear processing and parity generation subassemblies and in a departure from previous models the noise sources are considered time varying the kalman one step estimator which makes decisions on all parity values up to the present point is determined and it separates naturally into detection and correction operations permitting corrective action only when the detection levels exceed thresholds based on roundoff noise energy the detector corrector uses efficient multirate block processing techniques as determined by the real convolutional code a smoothed fixed lag kalman estimator which uses parity values for a fixed amount beyond the point of interest is needed to complete the correction it employs one step estimator quantities and implementation simplifications are possible examples showing the correction behavior and mean square error performance are presented and the size of overhead calculations for detection and correction is estimated a protected processing system is constructed by introducing additional subassemblies mostly comparators with the detection and correction parts already described under the usual assumptions of at most a single subassembly failure no improperly detected or corrected data leave the overall protected configuration

subspace based error and erasure correction with dft codes for wireless channels this paper deals with the decoding of lowpass discrete fourier transform dft codes in the presence of both errors and erasures we propose a subspace based approach for the error localization that is similar to the subspace approaches followed in the array signal processing for direction of arrival doa estimation the basic idea is to divide a vector space into two orthogonal subspaces of which one is spanned by the error locator vectors the locations of the errors are estimated from the spanning eigenvectors of the complement subspace however unlike the subspace approach in doa estimation which is similar to estimating the subspaces from the syndrome covariance matrix after a projection in the proposed approach the subspaces are estimated from the modified syndrome covariance matrix after a whitening transform simulation results with a gauss markov source reveal that the proposed algorithm is more efficient than the coding theoretic approach on impulsive channels as well as the subspace approach with projection on lossy channels

relations between invasion percolation and critical percolation in two dimensions we study invasion percolation in two dimensions we compare connectivity properties of the origin s invaded region to those of a the critical percolation cluster of the origin and b the incipient infinite cluster to exhibit similarities we show that for any k 1 the k point function of the first socalled pond has the same asymptotic behavior as the probability that k points are in the critical cluster of the origin more prominent though are the differences we show that there are infinitely many ponds that contain many large disjoint pc open clusters further for k 1 we compute the exact decay rate of the distribution of the radius of the kth pond and see that it differs from that of the radius of the critical cluster of the origin we finish by showing that the invasion percolation measure and the incipient infinite cluster measure are mutually singular

matched uniform approximations for a singular boundary point and an interior turning point the singular perturbation problem for a second order ode with singular boundary point and interior turning point is studied two uniform asymptotic approximations are separately constructed around singular point and turning point the restricted uniform approximation about the turning point corresponds to o malley s uniform reduction while that about the singular point is found by a similar method and is entirely new higher order matching of both dominant and recessive terms of the two approximations leads to a complete asymptotic approximation of the general solution a subsequent paper applies these results to higher order resonance criteria

invasion percolation a new form of percolation theory a new kind of percolation problem is described which differs from ordinary percolation theory in that it automatically finds the critical points of the system the model is motivated by the problem of one fluid displacing another from a porous medium under the action of capillary forces but in principle it may be applied to any kind of invasion process which proceeds along a path at least resistance the name invasion percolation is proposed for this new process similarities to and differences from ordinary percolation theory are discussed

information capacity of a perceptron the authors study the information storage capacity of a simple perceptron in the error regime for random unbiased patterns the geometrical analysis gives a logarithmic dependence for the information content in the asymptotic limit in this case the statistical physics approach when used at the simplest level of replica theory does not give satisfactory results however for perceptrons with finite stability the information content can be simply calculated with statistical physics methods in a region above the critical storage level for biased as well as for unbiased patterns

asymptotic expansion of the witten deformation of the analytic torsion given a compact riemannian manifold md g a finite dimensional representation 1 m gl v of the fundamental group 1 m on a vector spacevof dimensionland a hermitian structure on the flat vector bundle formula associated to ray singer rs have introduced the analytic torsiont t m g 0 witten s deformationdq t of the exterior derivativedq dq t e htdqeht withh m ra smooth morse function can be used to define a deformationt h t 0 of the analytic torsiontwitht h 0 t the main results of this paper are to provide assuming that grad ghis morse smale an asymptotic expansion for log t h t fort of the form formula and to present two different formulae fora0 as an application we obtain a shorter derivation of results due to ray singer rs cheeger ch muller mu1 2 which in increasing generality concern the equality for odd dimensional manifolds of the analytic torsion with the average of the reidemeister torsion corresponding to the triangulation t h g and the dual triangulation td d h g

random cellular froths in spaces of any dimension and curvature froth is a random partition of a d dimensional space by cells this assembly of cells obeys two fundamental laws euler s relation and the condition of maximum vertex figure imposed by geometry and by topological stability respectively these two conditions generate a set of relations between the variables that fully characterize the system topologically the number of degrees of freedom of the system and a set of useful independent variables the even valences have been found the influence of the space dimension and curvature on the range of variability of these valences is discussed and up to d 5 the regions in valence space corresponding to differently curved froths are calculated explicitly

subspace algorithms for error localization with quantized dft codes recently a class of real number bose chaudhuri hocquengem codes known as discrete fourier transform dft codes have been considered as joint source and channel codes for providing robustness to erasures and errors over wireless networks we propose three subspace algorithms for error localization with quantized dft codes the algorithms are similar to the music the minimum norm and the esprit algorithms used in array signal processing for direction of arrival estimation they provide different but related formulations of the error localizations by first partitioning a vector space into the channel error subspace and its orthogonal complement the noise subspace the locations of the errors are determined from either the error subspace eigenvectors or the noise subspace eigenvectors we also present a brief performance analysis of the localization error in terms of the perturbation of the error subspace due to quantization simulation results show that their localization performances are similar and they perform better than the coding theoretic approach over a broad range of channel error to quantization noise ratios

roughness and scaling in cellular patterns analysis of a simple model a foam can be decomposed into successive layers of cells at the same topological distance to an origin which is either an arbitrary cell or a basal plane the shape of these layers profile and thickening indicates the degree of randomness of the cellular pattern to support this idea we analyse the layer shapes in 2d rectangular models of foam as confirmed by numerical simulations the fluctuations in the direction normal to the layers are self affine on a significant range of scales with values of the exponents compatible with the kpz universality class 0 5 for the roughness exponent and z 1 5 for the dynamic exponent measuring the increase of the intralayer correlation length these fluctuations are not sufficient however to affect the dominant behaviour of the number of cells per layer found to saturate in cylindrical geometry and to increase linearly in concentric geometry

impulsive noise cancellation in multicarrier transmission a parallel between reed solomon codes in the complex field and multicarrier transmission using orthogonal frequency division multiplexing ofdm is first presented this shows that when the signal is sent over some channel composed of gaussian plus impulsive noise the impulsive noise can be removed by a procedure similar to channel decoding using information carried by the syndrome this result is first derived in a simple situation oversampled discrete multitone additive channel which is merely of theoretical interest in any case consecutive zeros in the output of the ofdm modulator do not correspond to real subcarriers pilot tones are transmitted for synchronization or channel estimation purposes these pilot tones are generally scattered among the information ones our approach is to use these pilot tones as syndromes in order to correct impulsive noise we show that the correction capacity is conditioned by the position of these pilot tones in the transmitted sequence a protection subsystem based on hypotheses tests is introduced after the decoding operation in order to detect malfunctions of this decoder the efficiency of this technique is corroborated with simulations in the slightly modified hiperlan2 context other extensions are then provided in order to increase the practical usefulness of the method

signal processing applications of oblique projection operators oblique projection operators are used to project measurements onto a low rank subspace along a direction that is oblique to the subspace they may be used to enhance signals while nulling interferences in the paper the authors give several basic results for oblique projections including formulas for constructing oblique projections with desired range and null space they analyze the algebra and geometry of oblique projections in order to understand their properties they then show how oblique projections can be used to separate signals from structured noise such as impulse noise damped or undamped interfering sinusoids such as power line interference and narrow band noise in some of the problems addressed the oblique projection provides an alternative way to implement an already known solution expressing these solutions as oblique projections brings geometrical insight to the study of the solution the geometry of oblique projections enables one to compute performance in terms of angles between signal and noise subspaces as a special case of removing impulse noise the authors can use oblique projections to interpolate missing data samples in array processing oblique projections can be used to simultaneously steer beams and nulls in communications oblique projections can be used to remove intersymbol interference

oxidation of self duality to 12 dimensions and beyond using partial curvature flows and the transitive action of subgroups of rm o d mathbb z on the indices 1 ldots d of the components of the yang mills curvature in an orthonormal basis we obtain a nested system of equations in successively higher dimensions d each implying the yang mills equations on d dimensional riemannian manifolds possessing special geometric structures this matryoshka of self duality equations contains the familiar self duality equations on riemannian fourfold as well as their generalisations on complex kahler threefold and on seven and eight dimensional manifolds with g2 and spin 7 holonomy the matryoshka allows enlargement oxidation to a remarkable system in 12 dimensions invariant under sp 3 there are hints that the underlying geometry is related to the sextonions a six dimensional algebra between the quaternions and octonions

fast fault tolerant digital convolution using a polynomial residue number system a fault tolerant convolution algorithm that is an extension of residue number system fault tolerance schemes applied to polynomial rings is described the algorithm is suitable for implementation on multiprocessor systems and is able to concurrently mask processor failures a fast algorithm based on long division for detecting and correcting multiple processor failures is presented moduli polynomials that yield an efficient and robust fast fourier transform fft based algorithm are selected for this implementation a single fault detection and correction is studied and a generalized likelihood ratio test is applied to optimally detect system failures in the presence of computational noise the coding scheme is capable of protecting over 90 of the computation involved in convolution parts not covered by the scheme are assumed to be protected via triple modular redundancy this hybrid approach can detect and correct any single system failure with as little as 70 overhead compared with 200 needed for a system fully protected via modular redundancy

phase space geometry and slow dynamics we describe a non arrhenius mechanism for the slowing down of dynamics that is inherent to the high dimensionality of the phase space we show that such a mechanism is at work both in a family of mean field spin glass models without any domain structure and in the case of ferromagnetic domain growth the marginality of spin glass dynamics as well as the existence of a quasi equilibrium regime can be understood within this scenario we discuss the question of ergodicity in an out of equilibrium situation

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

dynamical partitions of space in any dimension topologically stable cellular partitions of d dimensional spaces are studied a complete statistical description of the average structural properties of such partitions is given in terms of a sequence of or variables for d even or odd these variables are the average coordination numbers of the 2k dimensional polytopes which make up the cellular structure a procedure to produce d dimensional space partitions through cell division and cell coalescence transformations is presented classes of structures which are invariant under these transformations are found and the average properties of such structures are illustrated homogeneous partitions are constructed and compared with the known structures obtained by voronoi partitions and sphere packings in high dimensions

discrete cosine transform in error control coding the authors define a new class of real number linear block codes using the discrete cosine transform dct they also show that a subclass with a bch like structure can be defined and therefore encoding decoding algorithms for bch codes can be applied a 16 10 dct code is given as an example

exponential growth of ponds in invasion percolation on regular trees in invasion percolation the edges of successively maximal weight the outlets divide the invasion cluster into a chain of ponds separated by outlets on the regular tree the ponds are shown to grow exponentially with law of large numbers central limit theorem and large deviation results the tail asymptotics for a fixed pond are also studied and are shown to be related to the asymptotics of a critical percolation cluster with a logarithmic correction

topology and phase transitions the case of the short range spherical model we characterize the topology of the phase space of the berlin kac spherical model in the context of the so called topological hypothesis for spins lying in hypercubic lattices of dimension d for zero external field we are able to characterize the topology exactly up to homology we find that even though there is a continuum of changes in the topology of the corresponding manifolds for d 3 there are abrupt discontinuities in some topological functions that could be good candidates to associate with the phase transitions that occur at the thermodynamic level we show however that these changes do not coincide with the phase transitions and conversely that no topological discontinuity can be associated to the points where the phase transitions take place at variance with what happens in the mean field version of this same model we show that these abrupt topological changes are accessible thermodynamically we conclude that even in short range systems the topological mechanism does not seem to be responsible for the triggering of a phase transition we also analyze the case of spins connected to a macroscopic number of but not all neighbors and find that similar to the results found for the fully connected version in this case the topological hypothesis seems to hold the phase transition coincides with an accumulation point of the topological changes present in configuration space the question of the ensemble equivalence in the short range spherical model is also considered

stochastic equivariant cohomologies and cyclic cohomology we give two stochastic diffeologies on the free loop space which allow us to define stochastic equivariant cohomology theories in the chen souriau sense and to establish a link with cyclic cohomology with the second one we can establish a stochastic fixed point theorem

linear processes in high dimensions phase space and critical properties in this work we investigate the generic properties of a stochastic linear model in the regime of high dimensionality we consider in particular the vector autoregressive var model and the multivariate hawkes process we analyze both deterministic and random versions of these models showing the existence of a stable phase and an unstable phase we find that along the transition region separating the two regimes the correlations of the process decay slowly and we characterize the conditions under which these slow correlations are expected to become power laws we check our findings with numerical simulations showing remarkable agreement with our predictions we finally argue that real systems with a strong degree of self interaction are naturally characterized by this type of slow relaxation of the correlations

frame theoretic analysis of dft codes with erasures discrete fourier transform dft codes have been used as joint source channel codes in order to provide robustness against channel errors and erasures this paper presents a frame theoretic analysis of lowpass dft codes with erasures first it is shown that such codes make a special class of frames then the message reconstructions by the frame theoretic signal space projection and coding theoretic syndrome decoding of erasures are analyzed and their equivalence is proved the error performance of lowpass dft codes is analyzed using the frame theory the analysis helps in classifying the subframes of a dft frame into several classes this classification leads to the development of packetization criteria in order to guarantee minimum mean square reconstruction error and thus to minimize the effect of the quantization error simulation results substantiate the presented theory and the proposed packetization schemes

orientability for gauge theories on calabi yau manifolds abstract we study orientability issues of moduli spaces from gauge theories on calabi yau manifolds our results generalize and strengthen those for donaldson thomas theory on calabi yau manifolds of dimensions 3 and 4 we also prove a corresponding result in the relative situation which is relevant to the gluing formula in dt theory

ising model and n 2 supersymmetric theories we establish a direct link between massive ising model and arbitrary massiven 2 supersymmetric qft s in two dimensions this explains why the equations which appear in the computation of spin correlations in the non critical ising model are the same as those describing the geometry of vacua inn 2 theories the tau function appearing in the ising model i e the spin correlation function is reinterpreted in then 2 context as a new index in special cases this new index is related to the ray singer analytic torsion and can be viewed as a generalization of that to the loop space of kahler manifolds

class of majority decodable real number codes a majority decoding algorithm for a class of real number codes is presented majority decoding has been a relatively simple and fast decoding technique for codes over finite fields when applied to decode real number codes the robustness of the majority decoding to the presence of background noise which is usually an annoying problem for existing decoding algorithms for real number codes is its most prominent property the presented class of real number codes has generator matrices similar to those of the binary reed muller codes and is decoded by similar majority logic

role of saddles in mean field dynamics above the glass transition recent numerical developments in the study of glassy systems have shown that it is possible to give a purely geometric interpretation of the dynamic glass transition by considering the properties of unstable saddle points of the energy here we further develop this approach in the context of a mean field model by analytically studying the properties of the closest saddle point to an equilibrium configuration of the system we prove that when the glass transition is approached the energy of the closest saddle goes to the threshold energy defined as the energy level below which the degree of instability of the typical stationary points vanishes moreover we show that the distance between a typical equilibrium configuration and the closest saddle is always very small and that surprisingly it is almost independent of the temperature

random walk in random environment a counterexample we describe a family of random walks in random environments which have exponentially decaying correlations nearest neighbor transition probabilities which are bounded away from 0 and yet are subdiffusive in any dimensiond

variational principle for the pareto power law a mechanism is proposed for the appearance of power law distributions in various complex systems it is shown that in a conservative mechanical system composed of subsystems with different numbers of degrees of freedom a robust power law tail can appear in the equilibrium distribution of energy as a result of certain superpositions of the canonical equilibrium energy densities of the subsystems the derivation only uses a variational principle based on the boltzmann entropy without assumptions outside the framework of canonical equilibrium statistical mechanics two examples are discussed free diffusion on a complex network and a kinetic model of wealth exchange the mechanism is illustrated in the general case through an exactly solvable mechanical model of a dimensionally heterogeneous system

landscape approach for pinned elastic interfaces we discuss the large scale effective free energy landscape for elastic objects pinned by a random potential in the static approach converging analytical results show that this landscape consists in a succession of parabolic wells of random depth matching on singular points where the effective force is discontinuous we discuss the consequences for the dynamics of these pinned interfaces

spectral properties of unimodular lattice triangulations random unimodular lattice triangulations have been recently used as an embedded random graph model which exhibit a crossover behavior between an ordered large world and a disordered small world behavior using the ergodic pachner flips that transform such triangulations into another and an energy functional that corresponds to the degree distribution variance markov chain monte carlo simulations can be applied to study these graphs here we consider the spectra of the adjacency and the laplacian matrix as well as the algebraic connectivity and the spectral radius power law dependencies on the system size can clearly be identified and compared to analytical solutions for periodic ground states for random triangulations we find a qualitative agreement of the spectral properties with well known random graph models in the microcanonical ensemble analytical approximations agree with numerical simulations in the canonical ensemble a crossover behavior can be found for the algebraic connectivity and the spectral radius thus combining large world and small world behavior in one model the considered spectral properties can be applied to transport problems on triangulation graphs and the crossover behavior allows a tuning of important transport quantities

the fixed point theorem in equivariant cohomology in this paper we study the s equivariant de rham cohomology of infinite dimensional s manifolds our main example is the free loop space lx where x is a finite dimensional manifold with the circle acting by rotating loops we construct a new form of equivariant cohomology h which agrees with the usual periodic equivariant cohomology in finite dimensions and we prove a suitable analogue of the classical fixed point theorem which is valid for loop spaces lx this gives a cohomological framework for studying differential forms on loop spaces and we apply these methods to various questions which arise from the work of witten 16 atiyah 2 and bismut 5 in particular we show following atiyah in 2 that the a polynomial of x arises as an equivariant characteristic class in the theory h of the normal bundle to x considered as the space of constant loops in lx

inhomogeneous percolation problems and incipient infinite clusters the authors consider inhomogeneous percolation models with density pc f x and examine the forms of f x which produce incipient structures taking f x approximately mod x mod lambda and assuming the existence of a correlation length exponent v for the homogeneous percolation model they prove that in d 2 the borderline value of lambda is lambda b 1 v if lambda 1 v then with probability one there is no infinite cluster while if lambda 2 the models studied suggest what sort of incipient objects should be examined in random surface models

systematic dft frames principle eigenvalues structure and applications motivated by a host of recent applications requiring some amount of redundancy frames are becoming a standard tool in the signal processing toolbox in this paper we study a specific class of frames known as discrete fourier transform dft codes and introduce the notion of systematic frames for this class this is encouraged by a new application of frames namely distributed source coding that uses dft codes for compression studying their extreme eigenvalues we show that unlike dft frames systematic dft frames are not necessarily tight then we come up with conditions for which these frames can be tight in either case the best and worst systematic frames are established in the minimum mean squared reconstruction error sense eigenvalues of dft frames and their subframes play a pivotal role in this work particularly we derive some bounds on the extreme eigenvalues dft subframes which are used to prove most of the results these bounds are valuable independently

zero temperature relaxation of three dimensional ising ferromagnets we investigate the properties of the ising glauber model on a periodic cubic lattice of linear dimension l after a quench to zero temperature the resulting evolution is extremely slow with long periods of wandering on constant energy plateaus punctuated by occasional energy decreasing spin flip events the characteristic time scale for this relaxation grows exponentially with the system size we provide a heuristic and numerical evidence that exp l 2 for all but the smallest size systems the long time state is almost never static instead the system contains a small number of blinker spins that continue to flip forever with no energy cost thus the system wanders ad infinitum on a connected set of equal energy blinker states these states are composed of two topologically complex interwoven domains of opposite phases the average genus g l of the domains scales as l with 1 7 thus domains typically have many holes leading to a plumber s nightmare geometry

monte carlo simulation on aging processes within one pure state of the sk spin glass model monte carlo simulations on the sk model have been done to investigate aging processes after a rapid quench from to the spin glass phase taking care of time ranges of simulation we examine time evolutions of energy of the system parisi s overlap distribution function auto correlation and clones correlation functions distribution functions of the two correlations and magnetization induced by the field applied after a certain waiting time the data simulated exhibit a rich variety of aging phenomena most of them can be interpreted in a unified way though qualitatively by the scenario of growth of quasi equilibrium domains which we have recently introduced the results are consistent qualitatively with asymptotic behaviours of some of the basic assumptions and their results in recent analytical theory on the same sk model so long as the limiting procedures in finite systems are taken properly also they suggest that a basin of attraction of one dominant pure state spans almost an entire phase space of the system with a common time reversal symmetry

efficient algorithms for burst error recovery using fft and other transform kernels we show that the problem of signal reconstruction from missing samples can be handled by using reconstruction algorithms similar to the reed solomon rs decoding techniques usually the rs algorithm is used for error detection and correction of samples in finite fields for the case of missing samples of a speech signal we work with samples in the field of real or complex numbers and we can use fft or some new transforms in the reconstruction algorithm dsp implementation and simulation results show that the proposed methods are better than the ones previously published in terms of the quality of recovered speech signal for a given complexity the burst error recovery method using the fft kernel is sensitive to quantization and additive noise like the other techniques however other proposed transform kernels are very robust in correcting bursts of errors with the presence of quantization and additive noise

distributed source channel coding based on real field bch codes we use real number codes to compress statistically dependent sources and establish a new framework for distributed lossy source coding in which we compress sources before rather than after quantization this change in the order of binning and quantization blocks makes it possible to model the correlation between continuous valued sources more realistically and compensate for the quantization error partially we then focus on the asymmetric case i e lossy source coding with side information at the decoder the encoding and decoding procedures are described in detail for a class of real number codes called discrete fourier transform dft codes both for the syndrome and parity based approaches we leverage subspace based decoding to improve the decoding and by extending it we are able to perform distributed source coding in a rate adaptive fashion to further improve the decoding performance when the statistical dependency between sources is unknown we also extend the parity based approach to the case where the transmission channel is noisy and thus we perform distributed joint source channel coding in this context the proposed system is well suited for low delay communications as the mean squared reconstruction error mse is shown to be reasonably low for very short block length

redundancy the discrete fourier transform and impulse noise cancellation the relationship between the discrete fourier transform and error control codes is examined under certain conditions we show that discrete time sequences carry redundant information which then allow for the detection and correction of errors an application of this technique to impulse noise cancellation for pulse amplitude modulation transmission is described

broken ergodicity and the geometry of rugged landscapes we study the nature and mechanisms of broken ergodicity be in specific random walk models corresponding to diffusion on random potential surfaces in both one dimension and high dimension using both rigorous results and nonrigorous methods we confirm several aspects of the standard be picture and show that others apply in one dimension but need to be modified in higher dimensions these latter aspects include the notions that a fixed temperature confining barriers increase logarithmically with time that components are necessarily bounded regions of state space which depend on the observational time scale and that the system continually revisits previously traversed regions of state space we examine our results in the context of several experiments and discuss some implications of our results for the dynamics of disordered and or complex systems

new topological field theories in two dimensions it is shown that two 1 1 dimensional 2d free abelian and self interacting non abelian gauge theories without any interaction with matter fields belong to a new class of topological field theories tfts these new theories capture together some of the key features of witten and schwarz types of tft because they are endowed with symmetries that are reminiscent of the schwarz type theories but their lagrangian density has the appearance of the witten type theories the topological invariants for these theories are computed on a 2d compact manifold and their recursion relations are obtained these new theories are shown to provide a class of tractable field theoretical models for the hodge theory in two dimensions of flat minkowski spacetime where there are no propagating degrees of freedom associated with the 2d gauge boson

improved coding theoretic and subspace based decoding algorithms for a wider class of dct and dst codes the decoding of a class of discrete cosine transform dct and discrete sine transform dst codes that are maximum distance separable codes mds is considered in this paper these class of codes are considered for error correction over real fields all the existing algebraic decoding algorithms are capable of decoding only a subclass of these codes which can be characterized into the bose chaudhuri hocquenghem bch form and fails to decode the remaining even though they are mds in this paper we propose a new generic algorithm along the lines of coding theoretic and subspace methods to decode the entire class of mds dct and dst codes the proposed subspace approaches are similar to popular esprit and music algorithms the proposed algorithms also perform significantly better than the existing algorithms on the bch like subclass a perturbation analysis is also presented to study the effect of various parameters on the error localization due to the quantization noise simulation results are presented to demonstrate the capability of proposed algorithms to decode the entire class and to perform significantly better on the bch like subclass than the existing algorithm under the influence of quantization noise

zero temperature freezing in the three dimensional kinetic ising model we investigate the relaxation of the ising glauber model on a periodic cubic lattice after a quench to zero temperature in contrast to the conventional picture from phase ordering kinetics we find the following i domains at long time are highly interpenetrating and topologically complex with average genus growing algebraically with system size ii the long time state is almost never static but rather contains blinker spins that can flip ad infinitum with no energy cost iii the energy relaxation is extremely slow with a characteristic time that grows exponentially with system size

an investigation of the hidden structure of states in a mean field spin glass model we study the geometrical structure of the states in the low temperature phase of a mean field model for generalized spin glasses the p spin spherical model this structure cannot be revealed by the standard methods mainly due to the presence of an exponentially high number of states each one having a vanishing weight in the thermodynamic limit performing a purely entropic computation based on the tap equations for this model we define a constrained complexity which gives the overlap distribution of the states we find that this distribution is continuous non random and highly dependent on the energy range of the considered states furthermore we show which is the geometrical shape of the threshold landscape giving some insight into the role played by threshold states in the dynamical behaviour of the system

the incipient infinite cluster in two dimensional percolation letpp be the probability measure on the configurations of occupied and vacant vertices of a two dimensional graphg under which all vertices are independently occupied respectively vacant with probabilityp respectively 1 p letph be the critical probability for this system andw the occupied cluster of some fixed vertexw0 we show that for many graphsg such as mathbb z 2 or its covering graph which corresponds to bond percolation on mathbb z 2 the following two conditional probability measures converge and have the same limit v say

localization formulas superconnections and the index theorem for families in this paper we give a new proof of the localization formulas of berline and vergne 9 and duistermaat and heckman 18 when interpreted in the framework of atiyah 2 the probabilistic heat equation proof of the index theorem given in our paper 12 appears as the rigorous infinite dimensional version of this new proof of the localization formulas in finite dimensions the results of quillen 25 on superconnections are briefly presented the heat equation proofs 15 of the index theorem for families are described it is shown that in this framework the superconnections formalism is the operator theoretic description of integration along the fiber in the loop space

heat equation derivative formulas for vector bundles we use martingale methods to give bismut type derivative formulas for differentials and co differentials of heat semigroups on forms and more generally for sections of vector bundles the formulas are mainly in terms of weitzenbock curvature terms in most cases derivatives of the curvature are not involved in particular our results improve b k driver s formula in 1997 j math pures appl 9 76 703 737 for logarithmic derivatives of the heat kernel measure on a riemannian manifold our formulas also include the formulas of k d elworthy and x m li 1998 c r acad sci paris ser i math 327 87 92

limit theorems for 2d invasion percolation we prove limit theorems and variance estimates for quantities related to ponds and outlets for 2d invasion percolation we first exhibit several properties of a sequence o n of outlet variables the nth of which gives the number of outlets in the box centered at the origin of side length 2n the most important of these properties describes the sequence s renewal structure and exponentially fast mixing behavior we use these to prove a central limit theorem and strong law of large numbers for o n we then show consequences of these limit theorems for the pond radii and outlet weights

outlets of 2d invasion percolation and multiple armed incipient infinite clusters we study invasion percolation in two dimensions focusing on properties of the outlets of the invasion and their relation to critical percolation and to incipient infinite clusters iic s first we compute the exact decay rate of the distribution of both the weight of the k th outlet and the volume of the k th pond next we prove bounds for all moments of the distribution of the number of outlets in an annulus this result leads to almost sure bounds for the number of outlets in a box b 2 n and for the decay rate of the weight of the k th outlet to pc we then prove existence of multiplearmed iic measures for any number of arms and for any color sequence which is alternating or monochromatic we use these measures to study the invaded region near outlets and near edges in the invasion backbone far from the origin