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

on symmetric cauchy riemann manifolds the riemannian symmetric spaces play an important role in different branches of mathematics by definition a connected riemannian manifold m is called symmetric if to every a m there exists an involutory isometric diffeomorphism sa m m having a as isolated fixed point in m or equivalently if the differential dasa is the negative identity on the the tangent space ta tam of m at a in case such a transformation sa exists for a m it is uniquely determined and is the geodesic reflection of m about the point a as a consequence for every riemannian symmetric space m the group g gm generated by all symmetries sa a m is a lie group acting transitively on m in particular m can be identified with the homogeneous space g k for some compact subgroup k g using the elaborate theory of lie groups and lie algebras e cartan classified all riemannian symmetric spaces

geometrical aspects of stability theory for hill s equations where a and b are real parameters apparently the first stability diagram was drawn in the classical paper by b van der pol m j o sti utt 15 since then many papers and textbooks have appeared on the subject we mention j j stoker 13 j meixner f w sch fke 12 d m levy 8 j b keller 11 m i weinstein j b keller 16 17 j poschel e trubowltz 9 and v i arnold 3 5 6 this literature contains extensive estimates on the order of tangency of resonance tongues and the behavior of stability boundaries at infinity a full understanding of the stability diagrams is still lacking for instance the appearance of instability pockets in some cases see figure 1 cf 15 calls for a geometric explanation it should be noted that this phenomenon does not occur in the classical mathieu case cf 12 for a preliminary study near resonances using singularity theory we refer to afsharnejad 1 related bifurcational aspects of nonlinear perturbations of mathieu s equation near resonances were studied by broer vegter 7 the global geometry of the symplectic group was used for a stability proof of a hill equation in levi 10 we introduce the main object of study of this paper

asymptotic analysis of the lyapunov exponent and rotation number of the random oscillator and application we construct asymptotic expansions for the exponential growth rate lyapunov exponent and rotation number of the random oscillator when the noise is large small rapidly varying or slowly varying we then apply our results to problems in the stability of the random oscillator the spectrum of the one dimensional random schr6dinger operator and wave propagation in a one dimensional random medium

holomorphic extension and decomposition from a totally real manifold this paper is to develop an elementary cohomological approach for decomposing a function into boundary values of holomorphic functions and for discussing the corresponding microlocal analysis and hyperfunction theory

effect of non gaussian noise sources in a noise induced transition here we study a noise induced transition when the system is driven by a noise source taken as colored and non gaussian we show using both a theoretical approximation and numerical simulations that there is a shift of the transition as the noise departs from the gaussian behavior also we confirm the reentrance effect found for colored gaussian noise and show the behavior of the transition line in the phase like diagram as the noise departs from gaussianity in the large correlation time limit

colored noise induced parametric resonance abstract the phenomenon of parametric resonance for harmonic oscillator is studied in the presence of a parametric colored gaussian noise together with the main resonance the higher order ones are observed for the mean square of displacement the influence of noise color on the shape of parametric resonance regions is investigated

cr structures with group action and extendability of cr functions part i presents results on local embedding of cr structures we consider an abstract cr manifold whose structure is invariant under a transversal lie group action we show that such a manifold can always be locally embedded in complex space as a generic submanifold the proof is based on selection of canonical coordinates and repeated use of the newlander nirenberg theorem 13 when the lie group is abelian the embedding can be given a particularly simple form let l 1 be the codimension of our submanifold called m throughout the paper it is then convenient to denote by n i the dimension of the ambient complex space and by zl z w 1 w t the complex coordinates we shall systematically write

short time propagators for nonlinear fokker planck equations path integral solutions to time evolution equations in statistical physics have recently aroused great interest the main problem in applying these methods is to find a valid propagator in the short time regime of evolution a new method is proposed to obtain a set of accurate short time propagators by the construction of a simple auxiliary fokker planck equation this equation takes into account the full relevant functional dependence of the original drift and diffusion terms by using a suitable decomposition of the drift and diffusion coefficients it is possible to derive a new representation of the dirac function from this representation the short time behaviour of the solutions is given not only for the infinitesimal time interval but also for a discrete finite one which has a more practical numerical sense this picture leads to accurate short time propagators which include the prescribed boundary conditions

constructions and complexity of secondary polytopes abstract the secondary polytope a of a configuration a of n points in affine d 1 space is an n d polytope whose vertices correspond to regular triangulations of conv a in this article we present three constructions of a and apply them to study various geometric combinatorial and computational properties of secondary polytopes the first construction is due to gel fand kapranov and zelevinsky who used it to describe the face lattice of a we introduce the universal polytope u a d r n a combinatorial object depending only on the oriented matroid of a the secondary a can be obtained as the image of u a under a canonical linear map onto r n the third construction is based upon gale transforms or oriented matroid duality it is used to analyze the complexity of computing a and to give bounds in terms of n and d for the number of faces of a

hill s equation with random forcing terms motivated by a class of orbit problems in astrophysics this paper considers solutions to hill s equation with forcing strength parameters that vary from cycle to cycle the results are generalized to include period variations from cycle to cycle the development of the solutions to the differential equation is governed by a discrete map for the general case of hill s equation in the unstable limit we consider separately the cases of purely positive matrix elements and those with mixed signs we then find exact expressions bounds and estimates for the growth rates we also find exact expressions estimates and bounds for the infinite products of several 2 times 2 matrices with random variables in the matrix elements in the limit of sharply spiked forcing terms the delta function limit we find analytic solutions for each cycle and for the discrete map that matches solutions from cycle to cycle for this case we find the growth rates and the condition for instability in the limit of large forcin

cross correlated bounded noises induced the population extinction and enhancement of stability in a population growth model abstract the bounded noises as a new typed non gaussian noise have important practical significance in this paper the steady state and transient dynamical properties of the population growth model subjected to the cross correlated sine wiener bounded noise are investigated based on the approximate fokker planck equation the analytical expressions of stationary probability distribution function and extinction time of the population growth model are obtained the results show that the bounded noise intensity and the cross correlated bounded noise intensity can cause the population extinction in particular the population density are sensitive to the environmental fluctuation multiplicative noise and the self correlated time on the other hand the effects of noise intensities on the extinction time are discussed the results indicate that the extinction time can be decreased with the intrinsic stochasticity additive noise and the self correlated time increasing for the positive correlated case the result is similar to the additive noise and the self correlated time however for the negative correlated case the extinction time can be increased with the intensity increasing in particular the environmental fluctuation multiplicative noise causes noise enhanced stability for the negative correlated case and the extinction time firstly decreases for the small environmental fluctuation and then increases for the large environmental fluctuation for the positive correlated case the numerical results are in basic agreement with the theoretical predictions

hill s equation with small fluctuations cycle to cycle variations and stochastic processes hill s equations arise in a wide variety of physical problems and are specified by a natural frequency a periodic forcing function and a forcing strength parameter this classic problem is generalized here in two ways a to random hill s equations which allow the forcing strength qk the oscillation frequency k and the period k of the forcing function to vary from cycle to cycle and b to stochastic hill s equations which contain at least one additional term that is a stochastic process this paper considers both random and stochastic hill s equations with small parameter variations so that pk qk qk lk k k and are all o e where e 1 we show that random hill s equations and stochastic hill s equations have the same growth rates when the parameter variations pk and lk obey certain constraints given in terms of the moments of for random hill s equations the growth rates for the solutions are given by the growth rates of a matrix transformation under matrix multiplicat

on the rotation distance between binary trees abstract we develop combinatorial methods for establishing lower bounds on the rotation distance between binary trees i e equivalently on the flip distance between triangulations of a polygon these methods lead to sharp estimates for certain particular pairs of trees as an application we prove that for each n there exist size n trees at distance 2 n o n i e the diameter of the nth associahedron has at least this value

enhancement of stochastic resonance the role of non gaussian noises abstract we have analyzed the phenomenon of stochastic resonance in a double well potential driven by a colored non gaussian noise using a path integral approach we have obtained a consistent markovian approximation that enables us to get through the two state theory analytical expressions for the signal to noise ratio finding an enhancement of this quantity when the system departs from gaussian behavior this finding is supported by extensive numerical simulations we discuss the relation of these results to some experiments in sensory systems

transitions induced by bounded noise abstract we study noise induced transitions when the system is driven separately by a gaussian colored noise and a non gaussian bounded noise with the same mean and stationary correlation clear differences in the nature of such transitions for these two cases are observed these transitions are shown to be enhanced essentially by the bounded noisy driving the results are obtained through a new approach based on numerical analysis of hyperbolic systems

transitions induced by cross correlated bounded noises and time delay in a genotype selection model abstract we present a numerical investigation of occurrence of transitions in a genotype selection model with time delay where two bounded noises are cross correlated stationary probability distribution spd function is obtained it is found that the multiplicative bounded noise can facilitate the gene separation and it plays a constructive role in the genetic selection progress while the additive bounded noise suppresses the gene separation the strong correlation between noises gives a big chance to one type haploid out of the group besides what is more interesting is that the correlation time can induce a new transitions i e the curve of the spd changes from unimodal to bimodal and then to four peaks as the correlation time increases

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

spatiotemporal bounded noises and transitions induced by them in solutions of the real ginzburg landau model in this work we introduce two spatiotemporal colored bounded noises based on the zero dimensional cai lin and tsallis borland noises then we study and characterize the dependence of the defined stochastic processes on both a temporal correlation parameter ensuremath tau and a spatial coupling parameter ensuremath lambda in particular we found that varying ensuremath lambda may induce a transition of the distribution of the noise from bimodality to unimodality with the aim of investigating the role played by bounded noises in nonlinear dynamical systems we analyze the behavior of the real ginzburg landau time varying model additively perturbed by such noises the observed phase transition phenomenology is quite different from that observed when the perturbations are unbounded in particular we observed an inverse order to disorder transition and a reentrant transition with dependence on the specific type of bounded noise

comparison of some approaches for analysis of stochastic systems with short correlation time noise abstract we give a strict comparison of the diffusion bourret and van kampen approximations in the analysis of stochastic equations with short correlation time disturbances through the example of a random harmonic oscillator we show that the diffusion approximation does not always gives satisfactory results

the diameter of associahedra abstract it is proven here that the diameter of the d dimensional associahedron is 2 d 4 when d is greater than 9 two maximally distant vertices of this polytope are explicitly described as triangulations of a convex polygon and their distance is obtained using combinatorial arguments this settles two problems posed about twenty five years ago by daniel sleator robert tarjan and william thurston

effective markovian approximation for non gaussian noises a path integral approach abstract we have analyzed diffusion in a double well potential driven by a colored non gaussian noise using a path integral approach we have obtained a consistent markovian approximation to the initially non markovian problem such an approximation allows us to get analytical expressions for the mean first passage time that has been tested against extensive numerical simulations

average first passage time of a quasi hamiltonian mathieu oscillator with parametric and forcing excitations abstract a linear oscillator simultaneously subjected to stochastic forcing and parametric excitation is considered the time required for this system to evolve from a low initial energy level until a higher energy state for the first time is a random variable its expectation satisfies the pontryagin equation of the problem which is solved with the asymptotic expansion method developed by khasminskii this allowed deriving closed form expressions for the expected first passage time a comprehensive parameter analysis of these solutions is performed beside identifying the important dimensionless groups governing the problem it also highlights three important regimes which are called incubation multiplicative and additive because of their specific features those three regimes are discussed with the parameters of the problem

bounded noise induced transitions in a tumor immune system interplay by studying a recent biophysical model of tumor growth in the presence of the immune system here we propose that the phenomenon of evasion of tumors from immune control at a temporal mesoscale might in some cases be due to random fluctuations in the levels of the immune system bounded noises are considered but the gaussian approach is also used for analytical reference after showing that in the case of bounded noises there may be multiple attractors in the space of probability densities we numerically show that the velocity of convergence toward asymptotic density is very slow and that a transitory analysis is needed then by simulations using the sine wiener and the tsallis noises we show that if the level of the noise is sufficiently large then there may be the onset of noise induced transitions in the transitory density evaluated at realistic times namely the transitions are from unimodal density centered at low values of tumor burden to bimodal densities that have a second maximum centered at higher values however those transitions depend on the distribution of the noise

effect of non gaussian noises on the stochastic resonance like phenomenon in gated traps we exploit a simple one dimensional trapping model introduced before prompted by the problem of ion current across a biological membrane the voltage sensitive channels are open or closed depending on the value taken by an external potential that has two contributions a deterministic periodic and a stochastic one here we assume that the noise source is colored and non gaussian with a q dependent probability distribution where q is a parameter indicating the departure from gaussianity we analyze the behavior of the oscillation amplitude as a function of both q and the noise correlation time the main result is that in addition to the resonant like maximum as a function of the noise intensity there is a new resonant maximum as a function of the parameter q

stochastic resonance in a tumor immune system subject to bounded noises and time delay abstract immunotherapy is one of the most recent approaches in cancer therapy a mathematical model of tumor immune interaction subject to a periodic immunotherapy treatment imitated by a periodic signal correlative and bounded stochastic fluctuations and time delays is investigated by numerical simulations for its signal power amplification spa within the tailored parameter regime the synchronous response of tumor growth to the immunotherapy stochastic resonance sr versus both the noises and delays is obtained the details are as follows i the peak values of spa versus the noise intensity a in the proliferation term of tumor cells decrease as the frequency of periodic signal increases i e an increase of the frequency restrains the sr ii an increase of the amplitude of periodic signal restrains the sr versus a but boosts up the sr versus the noise intensity b in the immune term iii there is an optimum cross correlated degree between the two bounded noises at which the system exhibits the strongest sr versus the delay time the reaction time of tumor cell population to their surrounding environment constraints iv upon increasing the delay time double sr versus the delay time the time taken by both the tumor antigen identification and tumor stimulated proliferation of effectors emerges these results may be helpful for an immunotherapy treatment for the sufferer

the stochastic mathieu s equation in this manuscript we consider generalizations of the classical mathieu s equation to stochastic systems unlike previous works we focus on internal frequencies that vary continuously between periodic and stochastic variables by numerically integrating the system of equations using a symplectic method we determine the lyapunov exponents for a wide range of parameters to quantify how the growth rates vary in parameter space in the nearly periodic limit we recover the same growth rates as the classical mathieu s equation as the stochasticity increases the maximum growth decreases and the unstable region broadens which signifies that the stochasticity can both stabilize and destabilize a system in relation to the deterministic limit in addition to the classical parametric modes there is also an unstable stochastic mode that arises only for moderate stochasticity the power spectrum of the solutions shows that an increase in stochasticity tends to narrow the width of the subharmonic peak and increase the decay away from this peak

transitions induced by time delays and cross correlated sine wiener noises in a tumor immune system interplay abstract a model of tumor immune system interplay with time delays and cross correlated sine wiener noises is investigated by numerical simulations for the stationary probability distribution spd and stationary mean value x s t of tumor cell population results show that i the structure of the spd transfers from bimodal to unimodal as the reaction time of the cell population to environmental constraints increases ii conversely as the time of immune system to develop specific immune competence and the correlation time of noises increase the structure of the spd transfers from unimodal to bimodal iii as the cross correlated intensity between noises increases the transitions induced by are suppressed

a property of the functions and distributions annihilated by a locally integrable system of complex vector fields page introduction 387 1 basic notation and ingredients 389 2 approximation and constancy on the fibers of the classical solutions of the homogeneous equations 393 3 first consequences and interpretation of the fundamental theorem 398 4 distribution solutions of the homogeneous equations 402 5 regularity of the solutions of the homogeneous equations 409 6 example of a system that is analytic hypo elliptic but not co hypo elliptic 413 references 420

colored noise a case study of the cumulant method abstract the time ordered operator cumulant expansion r kubo r f fox is one of the standard tools for treating classical or quantum systems influenced by colored noise because of its poor convergency properties partial summations become imperative if the noise is not weak a recently developed resummed version of the cumulant expansion is shown in the present paper to yield in a conventional way correct physical results in situations where the cumulant expansion extended to the fourth cumulant drastically fails the theory amounts to a renormalization procedure of the bare physical parameters occurring in the second cumulant expression

stochastic equations of the langevin type under a weakly dependent perturbation asymptotic expansions for the probability density of the solution of a stochastic differential equation under a weakly dependent perturbation are proposed in particular linear partial differential equations for the first two terms of the correlation time expansion are derived it is shown that in these expansions the boundary layer part appears and non gaussianity of the perturbation is important for the fokker planck approximation correction

a general scheme for studying the stochastic dynamics of a parametric oscillator driven by coloured noise the stochastic dynamics of a classical parametric oscillator driven by coloured noise with exponential correlation is studied both by analytical and numerical methods the critical strength of the parametric modulation at which the motion of the oscillator becomes unbounded and the dependence of this critical strength on the magnitude and the correlation length of the noise are investigated the results show that the critical strength does not depend on either the strength or the correlation length of the coloured noise but that the squeezing induced by the parametric oscillation is enhanced by increasing the correlation length of the noise

cellular polarization interaction between extrinsic bounded noises and the wave pinning mechanism cell polarization cued or uncued is a fundamental mechanism in cell biology as an alternative to the classical turing bifurcation it has been proposed that the onset of cell polarity might arise by means of the well known phenomenon of wave pinning gamba et al proc natl acad sci usa 102 16927 2005 a particularly simple and elegant deterministic model of cell polarization based on the wave pinning mechanism has been proposed by edelstein keshet and coworkers biophys j 94 3684 2008 this model consists of a small biomolecular network where an active membrane bound factor interconverts into its inactive form that freely diffuses in the cell cytosol however biomolecular networks do communicate with other networks as well as with the external world thus their dynamics must be considered as perturbed by extrinsic noises these noises may have both a spatial and a temporal correlation and in any case they must be bounded to preserve the biological meaningfulness of the perturbed parameters here we numerically show that the inclusion of external spatiotemporal bounded parametric perturbations in the above wave pinning based model of cellular polarization may sometimes destroy the polarized state the polarization loss depends on both the extent of temporal and spatial correlations and on the kind of noise employed for example an increase of the spatial correlation of the noise induces an increase of the probability of cell polarization however if the noise is spatially homogeneous then the polarization is lost in the majority of cases these phenomena are independent of the type of noise conversely an increase of the temporal autocorrelation of the noise induces an effect that depends on the model of noise

stochastic resonance induced by bounded noise and periodic signal in an asymmetric bistable system abstract stochastic resonance sr induced by the bounded noise in an asymmetric bistable system is investigated based on stochastic simulation the signal power amplification s p a is derived for the case of the additive modulated signal the simulation results indicate that 1 the sr phenomenon is weakened by the asymmetry of the system 2 increasing the frequency restricts the sr of the system 3 there is an optimum value of amplitude a matching the strongest sr phenomenon