derivation of the gross pitaevskii hierarchy for the dynamics of bose einstein condensate consider a system of n bosons on the three dimensional unit torus interacting via a pair potential n2v n xi xj where x x1 xn denotes the positions of the particles suppose that the initial data n 0 satisfies the condition where hn is the hamiltonian of the bose system this condition is satisfied if n 0 wn n 0 where wn is an approximate ground state to hn and n 0 is regular let n t denote the solution to the schrodinger equation with hamiltonian hn gross and pitaevskii proposed to model the dynamics of such a system by a nonlinear schrodinger equation the gross pitaevskii gp equation the gp hierarchy is an infinite bbgky hierarchy of equations so that if ut solves the gp equation then the family of k particle density matrices k ut ut solves the gp hierarchy we prove that as n the limit points of the k particle density matrices of n t are solutions of the gp hierarchy our analysis requires that the n boson dynamics be described by a modified hamiltonian that cuts off the pair interactions whenever at least three particles come into a region with diameter much smaller than the typical interparticle distance our proof can be extended to a modified hamiltonian that only forbids at least n particles from coming close together for any fixed n 2006 wiley periodicals inc

unconditional uniqueness for the cubic gross pitaevskii hierarchy via quantum de finetti we present a new simpler proof of the unconditional uniqueness of solutions to the cubic gross pitaevskii hierarchy in one of the main tools in our analysis is the quantum de finetti theorem our uniqueness result is equivalent to the one established in the celebrated works of erd s schlein and yau 2015 wiley periodicals inc

face alignment by explicit shape regression we present a very efficient highly accurate explicit shape regression approach for face alignment unlike previous regression based approaches we directly learn a vectorial regression function to infer the whole facial shape a set of facial landmarks from the image and explicitly minimize the alignment errors over the training data the inherent shape constraint is naturally encoded into the regressor in a cascaded learning framework and applied from coarse to fine during the test without using a fixed parametric shape model as in most previous methods to make the regression more effective and efficient we design a two level boosted regression shape indexed features and a correlation based feature selection method this combination enables us to learn accurate models from large training data in a short time 20 min for 2 000 training images and run regression extremely fast in test 15 ms for a 87 landmarks shape experiments on challenging data show that our approach significantly outperforms the state of the art in terms of both accuracy and efficiency

on well posedness for general hierarchy equations of gross pitaevskii and hartree type gross pitaevskii and hartree hierarchies are infinite systems of coupled pdes emerging naturally from the mean field theory of bose gases their solutions are known to be related to initial value problems in particular the gross pitaevskii and hartree equations due to their physical and mathematical relevance the issues of well posedness and uniqueness for these hierarchies have recently been studied thoroughly using specific nonlinear and combinatorial techniques in this article we introduce a new approach for the study of such hierarchy equations by firstly establishing a duality between them and certain liouville equations and secondly solving the uniqueness and existence questions for the latter as an outcome we formulate a hierarchy equation starting from any initial value problem which is u 1 invariant and prove a general principle which can be stated formally as follows in particular several new well posedness results as well as a counterexample to uniqueness for the gross pitaevskii hierarchy equation are proved the novelty in our work lies in the aforementioned duality and the use of liouville equations with powerful transport techniques extended to infinite dimensional functional spaces

the quintic nls as the mean field limit of a boson gas with three body interactions we investigate the dynamics of a boson gas with three body interactions in dimensions d 1 2 we prove that in the limit of infinite particle number the bbgky hierarchy of k particle marginals converges to a limiting gross pitaevskii gp hierarchy for which we prove existence and uniqueness of solutions factorized solutions of the gp hierarchy are shown to be determined by solutions of a quintic nonlinear schrodinger equation our proof is based on and extends methods of erdos schlein yau klainerman machedon and kirkpatrick schlein staffilani

on the uniqueness of solutions to the periodic 3d gross pitaevskii hierarchy abstract in this paper we present a uniqueness result for solutions to the gross pitaevskii hierarchy on the three dimensional torus under the assumption of an a priori spacetime bound we show that this a priori bound is satisfied for factorized solutions to the hierarchy which come from solutions of the nonlinear schrodinger equation in this way we obtain a periodic analogue of the uniqueness result on r 3 previously proved by klainerman and machedon 75 except that in the periodic setting we need to assume additional regularity in particular we need to work in the sobolev class h for 1 by constructing a specific counterexample we show that on t 3 the existing techniques from the work of klainerman and machedon approach do not apply in the endpoint case 1 this is in contrast to the known results in the non periodic setting where these techniques are known to hold for all 1 in our analysis we give a detailed study of the crucial spacetime estimate associated to the free evolution operator in this step of the proof our methods rely on lattice point counting techniques based on the concept of the determinant of a lattice this method allows us to obtain improved bounds on the number of lattice points which lie in the intersection of a plane and a set of radius r depending on the number theoretic properties of the normal vector to the plane we are hence able to obtain a sharp range of admissible sobolev exponents for which the spacetime estimate holds

an adaptive support vector regression based on a new sequence of unified orthogonal polynomials in practical engineering small scale data sets are usually sparse and contaminated by noise in this paper we propose a new sequence of orthogonal polynomials varying with their coefficient unified chebyshev polynomials ucp which has two important properties namely orthogonality and adaptivity based on these new polynomials a new kernel function the unified chebyshev kernel uck is constructed which has been proven to be a valid svm kernel to find the optimal polynomial coefficient and the optimal kernel we propose an adaptive algorithm based on the evaluation criterion for adaptive ability of uck to evaluate the performance of the new method we applied it to learning some benchmark data sets for regression and compared it with other three algorithms the experiment results show that the proposed adaptive algorithm has excellent generalization performance and prediction accuracy and does not cost more time compared with other svms therefore this method is suitable for practical engineering application

randomization and the gross pitaevskii hierarchy we study the gross pitaevskii hierarchy on the spatial domain mathbb t 3 by using an appropriate randomization of the fourier coefficients in the collision operator we prove an averaged form of the main estimate which is used in order to contract the duhamel terms that occur in the study of the hierarchy in the averaged estimate we do not need to integrate in the time variable an averaged spacetime estimate for this range of regularity exponents then follows as a direct corollary the range of regularity exponents that we obtain is alpha frac 3 4 it was shown in our previous joint work with gressman j funct anal 266 7 4705 4764 2014 that the range alpha 1 is sharp in the corresponding deterministic spacetime estimate this is in contrast to the non periodic setting which was studied by klainerman and machedon commun math phys 279 1 169 185 2008 where the spacetime estimate is known to hold whenever alpha geq 1 the goal of our paper is to extend the range of in this class of estimates in a probabilistic sense we use the new estimate and the ideas from its proof in order to study randomized forms of the gross pitaevskii hierarchy more precisely we consider hierarchies similar to the gross pitaevskii hierarchy but in which the collision operator has been randomized for these hierarchies we show convergence to zero in low regularity sobolev spaces of duhamel expansions of fixed deterministic density matrices we believe that the study of the randomized collision operators could be the first step in the understanding of a nonlinear form of randomization

small delay approximation of stochastic delay differential equations delay differential equations evolve in an infinite dimensional phase space in this paper we consider the effect of external fluctuations noise on delay differential equations involving one variable thus leading to univariate stochastic delay differential equations sdde s for small delays a univariate nondelayed stochastic differential equation approximating such a sdde is presented another approximation complementary to the first is also obtained using an average of the sdde s drift term over the delayed dynamical variable which defines a conditional average drift this second approximation is characterized by the fact that the diffusion term is identical to that of the original sdde for small delays our approach yields a steady state probability density and a conditional average drift which are in close agreement with numerical simulations of the original sdde we illustrate this scheme with the delayed linear langevin equation and a stochastic version of the delayed logistic equation the technique can be used with any type of noise and is easily generalized to multiple delays s1063 651x 99 08304 x

influence of spatiotemporally correlated noise on structure formation in excitable media we discuss the influence of additive spatiotemporally correlated i e colored noise on pattern formation in a two dimensional network of excitable systems the signature of spatiotemporal stochastic resonance stsr is analyzed using cross correlation and information theoretic measures it is found that the stsr behavior is affected by both the spatial and temporal correlations of the noise due to an interplay with the length scales of the deterministic network increasing the spatiotemporal noise correlation shifts the occurrence of stsr to smaller values of the noise variance additionally if the spatial correlation of the noise exceeds that of the network the excitation patterns disappear in favor of cloudy structures directly rendering the underlying spatial noise field

fast global oscillations in networks of integrate and fire neurons with low firing rates we study analytically the dynamics of a network of sparsely connected inhibitory integrate and fire neurons in a regime where individual neurons emit spikes irregularly and at a low rate in the limit when the number of neurons n 1 the network exhibits a sharp transition between a stationary and an oscillatory global activity regime where neurons are weakly synchronized the activity becomes oscillatory when the inhibitory feedback is strong enough the period of the global oscillation is found to be mainly controlled by synaptic times but depends also on the characteristics of the external input in large but finite networks the analysis shows that global oscillations of finite coherence time generically exist both above and below the critical inhibition threshold their characteristics are determined as functions of systems parameters in these two different regimes the results are found to be in good agreement with numerical simulations

derivation of hartree s theory for generic mean field bose systems in this paper we provide a novel strategy to prove the validity of hartree s theory for the ground state energy of bosonic quantum systems in the mean field regime for the well known case of trapped bose gases this can be shown using the strong quantum de finetti theorem which gives the structure of infinite hierarchies of k particles density matrices here we deal with the case where some particles are allowed to escape to infinity leading to a lack of compactness our approach is based on two ingredients 1 a weak version of the quantum de finetti theorem and 2 geometric techniques for many body systems our strategy does not rely on any special property of the interaction between the particles in particular our results cover those of benguria lieb and lieb yau for respectively bosonic atoms and boson stars

on the uniqueness of solutions to the gross pitaevskii hierarchy the purpose of this note is to give a new proof of uniqueness of the gross pitaevskii hierarchy first established in 1 in a different space based on space time estimates similar in spirit to those of 2

dynamics of encoding in neuron populations some general mathematical features the use of a population dynamics approach promises efficient simulation of large assemblages of neurons depending on the issues addressed and the degree of realism incorporated in the simulated neurons a wide range of different population dynamics formulations can be appropriate here we present a common mathematical structure that these various formulations share and that implies dynamical behaviors that they have in common this underlying structure serves as a guide toward efficient means of simulation as an example we derive the general population firing rate frequency response and show how it may be used effectively to address a broad range of interacting population response and stability problems a few specific cases will be worked out a summary of this work appears at the end before the appendix

the quintic nonlinear schrodinger equation on three dimensional zoll manifolds let m g be a three dimensional smooth compact riemannian manifold such that all geodesics are simple and closed with a common minimal period such as the 3 sphere bbb s 3 with canonical metric in this work the global well posedness problem for the quintic nonlinear schr odinger equation i partial t u delta u pm u 4u u t 0 u 0 is solved for small initial data u 0 in the energy space h 1 m which is the scaling critical space further local well posedness for large data as well as persistence of higher initial sobolev regularity is obtained this extends previous results of burq g erard tzvetkov to the endpoint case

derivation of the gross pitaevskii equation for the dynamics of bose einstein condensate consider a system of n bosons in three dimensions interacting via a repulsive short range pair potential n 2 v n xi xj where x x1 xn denotes the positions of the particles let hn denote the hamiltonian of the system and let n t be the solution to the schrodinger equation suppose that the initial data n 0 satisfies the energy condition h n 0 h k n n 0i c k n k for k 1 2 we also assume that the k particle density matrices of the initial state are asymptotically factorized as n we prove that the k particle density matrices of n t are also asymptotically factorized and the one particle orbital wave function solves the gross pitaevskii equation a cubic non linear schrodinger equation with the coupling constant given by the scattering length of the potential v we also prove the same conclusion if the energy condition holds only for k 1 but the factorization of n 0 is assumed in a stronger sense ams subject classification number 81v70 81t18 35q55

theory of oscillatory firing induced by spatially correlated noise and delayed inhibitory feedback a network of leaky integrate and fire neurons with global inhibitory feedback and under the influence of spatially correlated noise is studied we calculate the spectral statistics of the network power spectrum of the population activity cross spectrum between spike trains of different neurons as well as of a single neuron power spectrum of spike train cross spectrum between external noise and spike train within the network as shown by comparison with numerical simulations our theory works well for arbitrary network size if the feedback is weak and the amount of external noise does not exceed that of the internal noise by means of our analytical results we discuss the quality of the correlation induced oscillation in a large network as a function of the transmission delay and the internal noise intensity it is shown that the strongest oscillation is obtained in a system with zero internal noise and adiabatically long delay i e the delay period is longer than any other time scale in the system for a neuron with a strong intrinsic frequency the oscillation becomes strongly anharmonic in the case of a long delay time we also discuss briefly the kind of synchrony introduced by the feedback induced oscillation

quantum fluctuations and rate of convergence towards mean field dynamics the nonlinear hartree equation describes the macroscopic dynamics of initially factorized n boson states in the limit of large n in this paper we provide estimates on the rate of convergence of the microscopic quantum mechanical evolution towards the limiting hartree dynamics more precisely we prove bounds on the difference between the one particle density associated with the solution of the n body schrodinger equation and the orthogonal projection onto the solution of the hartree equation

integrate and fire neurons with threshold noise a tractable model of how interspike interval correlations affect neuronal signal transmission many neurons exhibit interval correlations in the absence of input signals we study the influence of these intrinsic interval correlations of model neurons on their signal transmission properties for this purpose we employ two simple firing models one of which generates a renewal process while the other leads to a nonrenewal process with negative interval correlations different methods to solve for spectral statistics in the presence of a weak stimulus spike train power spectra cross spectra and coherence functions are presented and their range of validity is discussed using these analytical results we explore a lower bound on the mutual information rate between output spike train and input stimulus as a function of the system s parameters we demonstrate that negative correlations in the baseline activity can lead to enhanced information transfer of a weak signal by means of noise shaping of the background noise spectrum we also show that an enhancement is not compulsory char22 for a stimulus with power exclusively at high frequencies the renewal model can transfer more information than the nonrenewal model does we discuss the application of our analytical results to other problems in neuroscience our results are also relevant to the general problem of how a signal affects the power spectrum of a nonlinear stochastic system

the derivation of the mathbb t 3 t 3 energy critical nls from quantum many body dynamics we derive the 3d energy critical quintic nls from quantum many body dynamics with 3 body interaction in the mathbb t 3 periodic setting due to the known complexity of the energy critical setting previous progress was limited in comparison to the 2 body interaction case yielding energy subcritical cubic nls previously the only result for the 3d energy critical case was hong et al j funct anal 270 1 34 67 2016 which proved the uniqueness part of the argument in the case of small solutions in the main part of this paper we develop methods to prove the convergence of the bbgky hierarchy to the infinite gross pitaevskii gp hierarchy and separately the uniqueness of large gp solutions since the trace estimate used in the previous proofs of convergence is the false endpoint trace estimate in our setting we instead introduce a new frequency interaction analysis and apply the finite dimensional quantum de finetti theorem for the large solution uniqueness argument we discover the new hierarchical uniform frequency localization hufl property for the gp hierarchy and use it to prove a new type of uniqueness theorem the hufl property reduces to a new statement even for nls with the help of colliander et al ann math 2 167 3 767 865 2008 and ionescu and pausader duke math j 161 1581 1612 2012 which proved the global well posedness for the quintic nls this new uniqueness theorem establishes global uniqueness

rigorous derivation of the gross pitaevskii equation with a large interaction potential consider a system of n bosons in three dimensions interacting via a repulsive short range pair potential n 2 v n xi xj where x x1 xn denotes the positions of the particles let hn denote the hamiltonian of the system and let n t be the solution to the schrodinger equation suppose that the initial data n 0 satisfies the energy condition h n 0 hn n 0i cn and that the one particle density matrix converges to a projection as n then we prove that the k particle density matrices of n t factorize in the limit n moreover the one particle orbital wave function solves the time dependent gross pitaevskii equation a cubic non linear schrodinger equation with the coupling constant proportional to the scattering length of the potential v in 12 we proved the same statement under the condition that the interaction potential v is sufficiently small in the present work we develop a new approach that requires no restriction on the size of the potential

the proof of the l 2 decoupling conjecture we prove the l 2 decoupling conjecture for compact hypersurfaces with positive denite second fundamental form and also for the cone this has a wide range of important consequences one of them is the validity of the discrete restriction conjecture which implies the full range of expected l p strichartz estimates for both the rational and up to n losses the irrational torus another one is an improvement in the range for the discrete restriction theorem for lattice points on the sphere various applications to additive combinatorics incidence geometry and number theory are also discussed our argument relies on the interplay between linear and multilinear restriction theory

multi variable regression methods using modified chebyshev polynomials of class 2 abstract so far many regression works have been implemented by using linear regression methods although more accurate predictions results could be obtained polynomial regression is not used as much as compared to linear regression in real applications due to occurrence of coefficient explosion to overcome this problem two regression algorithms using chebyshev polynomials of class 2 based on cascade regression and feature selection are proposed in this paper in the experimental part three separate experiments including function interpolation and real case regression were conducted on three datasets to test the proposed algorithms as shown by the experimental results the proposed algorithms performed better than other regression methods in terms of both accuracy and processing time

global well posedness of the energy critical nonlinear schrodinger equation with small initial data in h1 t3 a refined trilinear strichartz estimate for solutions to the schrodinger equation on the flat rational torus t3 is derived by a suitable modification of critical function space theory this is applied to prove a small data global well posedness result for the quintic nonlinear schrodinger equation in hs t3 for all s 1 this is the first energy critical global well posedness result in the setting of compact manifolds

local existence of solutions to randomized gross pitaevskii hierarchies in this paper we study the local in time existence of solutions to randomized forms of the gross pitaevskii hierarchy on periodic domains in particular we study the independently randomized gross pitaevskii hierarchy and the dependently randomized gross pitaevskii hierarchy which were first introduced in the author s joint work with staffilani 2013 for these hierarchies we construct local in time low regularity solutions in spaces which contain a random component the constructed density matrices will solve the full randomized hierarchies thus extending the results from the author and staffilani s joint work where solutions solving arbitrarily long subhierarchies were given our analysis will be based on the truncation argument which was first used in the deterministic setting in the work of t chen and pavlovic 2013 the presence of randomization in the problem adds additional difficulties most notably to estimating the duhamel expansions that are crucial in the truncation argument these difficulties are overcome by a detailed analysis of the duhamel expansions in the independently randomized case we need to keep track of which randomization parameters appear in the duhamel terms whereas in the dependently randomized case we express the duhamel terms directly in terms of the initial data in both cases we can obtain stronger results with respect to the time variable if we assume additional regularity on the initial data

population coding in neuronal systems with correlated noise neuronal representations of external events are often distributed across large populations of cells we study the effect of correlated noise on the accuracy of these neuronal population codes our main question is whether the inherent error in the population code can be suppressed by increasing the size of the population n in the presence of correlated noise we address this issue using a model of a population of neurons that are broadly tuned to an angular variable in two dimensions the fluctuations in the neuronal activities are modeled as gaussian noises with pairwise correlations that decay exponentially with the difference between the preferred angles of the correlated cells we assume that the system is broadly tuned which means that both the correlation length and the width of the tuning curves of the mean responses span a substantial fraction of the entire system length the performance of the system is measured by the fisher information fi which bounds its estimation error by calculating the fi in the limit of a large n we show that positive correlations decrease the estimation capability of the network relative to the uncorrelated population the information capacity saturates to a finite value as the number of cells in the population grows in contrast negative correlations substantially increase the information capacity of the neuronal population these results are supplemented by the effect of correlations on the mutual information of the system our analysis provides an estimate of the effective number of statistically independent degrees of freedom denoted n mathrm eff that a large correlated system can have according to our theory n mathrm eff remains finite in the limit of a large n estimating the parameters of the correlations and tuning curves from experimental data in some cortical areas that code for angles we predict that the number of effective degrees of freedom embedded in localized populations in these areas is less than or of the order of ensuremath approx 10 2

a multi objective evolutionary algorithm for feature selection based on mutual information with a new redundancy measure a new feature redundancy measurement based on mutual information was proposed a multi objective evolutionary algorithm for feature selection was presented pareto optimality was used to evaluate candidate feature subsets and find compact feature subsets experiments showed that our algorithm could select compact and discriminative feature subsets feature selection is an important task in data mining and pattern recognition especially for high dimensional data it aims to select a compact feature subset with the maximal discriminative capability the discriminability of a feature subset requires that selected features have a high relevance to class labels whereas the compactness demands a low redundancy within the selected feature subset this paper defines a new feature redundancy measurement capable of accurately estimating mutual information between features with respect to the target class mifs cr based on a relevance measure and this new redundancy measure a multi objective evolutionary algorithm with class dependent redundancy for feature selection mecy fs is presented the mecy fs algorithm employs the pareto optimality to evaluate candidate feature subsets and finds compact feature subsets with both the maximal relevance and the minimal redundancy experiments on benchmark datasets are conducted to validate the effectiveness of the new redundancy measure and the mecy fs algorithm is verified to be able to generate compact feature subsets with a high predictive capability

correlations without synchrony peaks in spike train correlograms are usually taken as indicative of spike timing synchronization between neurons strictly speaking however a peak merely indicates that the two spike trains were not independent two biologically plausible ways of departing from independence that are capable of generating peaks very similar to spike timing peaks are described here covariations over trials in response latency and covariations over trials in neuronal excitability since peaks due to these interactions can be similar to spike timing peaks interpreting a correlogram may be a problem with ambiguous solutions what peak shapes do latency or excitability interactions generate when are they similar to spike timing peaks when can they be ruled out from having caused an observed correlogram peak these are the questions addressed here the previous article in this issue proposes quantitative methods to tell cases apart when latency or excitability covariations cannot be ruled out

derivation of the cubic non linear schrodinger equation from quantum dynamics of many body systems we prove rigorously that the one particle density matrix of three dimensional interacting bose systems with a short scale repulsive pair interaction converges to the solution of the cubic non linear schrodinger equation in a suitable scaling limit the result is extended to k particle density matrices for all positive integer k

time structure of the activity in neural network models several neural network models in continuous time are reconsidered in the framework of a general mean field theory which is exact in the limit of a large and fully connected network the theory assumes pointlike spikes which are generated by a renewal process the effect of spikes on a receiving neuron is described by a linear response kernel which is the dominant term in a weak coupling expansion it is shown that the resulting spike response model is the most general renewal model with linear inputs the standard integrate and fire model forms a special case in a network structure with several pools of identical spiking neurons the global states and the dynamic evolution are determined by a nonlinear integral equation which describes the effective interaction within and between different pools we derive explicit stability criteria for stationary incoherent and oscillatory coherent solutions it is shown that the stationary state of noiseless systems is almost always unstable noise suppresses fast oscillations and stabilizes the system furthermore collective oscillations are stable only if the firing occurs while the synaptic potential is increasing in particular collective oscillations in a network with delayless excitatory interaction are at most semistable inhibitory interactions with short delays or excitatory interactions with long delays lead to stable oscillations our general results allow a straightforward application to different network models with spiking neurons furthermore the theory allows an estimation of the errors introduced in firing rate or graded response models

mean field dynamics singular potentials and rate of convergence we consider the time evolution of a system of n identical bosons whose interaction potential is rescaled by n 1 we choose the initial wave function to describe a condensate in which all particles are in the same one particle state it is well known that in the mean field limit n the quantum n body dynamics is governed by the nonlinear hartree equation using a nonperturbative method we extend previous results on the mean field limit in two directions first we allow a large class of singular interaction potentials as well as strong possibly time dependent external potentials second we derive bounds on the rate of convergence of the quantum n body dynamics to the hartree dynamics

bilinear eigenfunction estimates and the nonlinear schrodinger equation on surfaces we study the cubic non linear schrodinger equation nls on compact surfaces on the sphere mathbb s 2 and more generally on zoll surfaces we prove that for s 1 4 nls is uniformly well posed in hs which is sharp on the sphere the main ingredient in our proof is a sharp bilinear estimate for laplace spectral projectors on compact surfaces

delays and oscillations in networks of spiking neurons a two timescale analysis oscillations are a ubiquitous feature of many neural systems spanning many orders of magnitude in frequency one of the most prominent oscillatory patterns with possible functional implications is that occurring in the mammalian thalamocortical system during sleep this system is characterized by relatively long delays reaching up to 40 msec and gives rise to low frequency oscillatory waves motivated by these phenomena we study networks of excitatory and inhibitory integrate and fire neurons within a fokker planck delay partial differential equation formalism and establish explicit conditions for the emergence of oscillatory solutions and for the amplitude and period of the ensuing oscillations for relatively large values of the delays when a two timescale analysis is employed the full partial differential equation is replaced in this limit by a discrete time iterative map leading to a relatively simple dynamic interpretation this asymptotic result is shown numerically to hold to a good approximation over a wide range of parameter values leading to an accurate characterization of the behavior in terms of the underlying physical parameters our results provide a simple mechanistic explanation for one type of slow oscillation based on delayed inhibition which may play an important role in the slow spindle oscillations occurring during sleep moreover they are consistent with experimental findings related to human motor behavior with visual feedback

derivation of the two dimensional nonlinear schrodinger equation from many body quantum dynamics we derive rigorously for both r 2 and l l 2 the cubic nonlinear schrodinger equation in a suitable scaling limit from the two dimensional many body bose systems with short scale repulsive pair interactions we first prove convergence of the solution of the bbgky hierarchy corresponding to the many body systems to a solution of the infinite gross pitaevskii hierarchy corresponding to the cubic nls and then we prove uniqueness for the infinite hierarchy which requires number theoretical techniques in the periodic case

the gross pitaevskii hierarchy on general rectangular tori in this work we study the gross pitaevskii hierarchy on general rational and irrational rectangular tori of dimensions two and three this is a system of infinitely many linear partial differential equations which arises in the rigorous derivation of the nonlinear schrodinger equation we prove a conditional uniqueness result for the hierarchy in two dimensions this result allows us to obtain a rigorous derivation of the defocusing cubic nonlinear schrodinger equation from the dynamics of many body quantum systems on irrational tori this question was posed as an open problem in the previous work of kirkpatrick schlein and staffilani

asynchronous states in networks of pulse coupled oscillators we use a mean field approach to analyze the stability of the asynchronous state in a population of all to all pulse coupled nonlinear oscillators we determine the conditions that must be satisfied by the time constants and phase dependence characterizing the coupling between the oscillators in order for the asynchronous state to be stable we also consider the effects of noise this work complements results on synchronous states in similar models and allows us to study the validity of firing rate models commonly used for neural networks

stationary solutions of linear stochastic delay differential equations applications to biological systems recently k uchler and mensch stochastics stochastics rep 40 23 1992 derived exact stationary probability densities for linear stochastic delay differential equations this paper presents an alternative derivation of these solutions by means of the fokker planck approach introduced by guillouzic phys rev e 59 3970 1999 61 4906 2000 applications of this approach which is argued to have greater generality are discussed in the context of stochastic models for population growth and tracking movements

gross pitaevskii equation as the mean field limit of weakly coupled bosons we consider the dynamics of n boson systems interacting through a pair potential n 1va xi xj where va x a 3v x a we denote the solution to the n particle schrodinger equation by n t recall that the gross pitaevskii gp equation is a nonlinear schrodinger equation and the gp hierarchy is an infinite bbgky hierarchy of equations so that if ut solves the gp equation then the family of k particle density matrices open image in new window solves the gp hierarchy under the assumption that a n for 0 3 5 we prove that as n the limit points of the k particle density matrices of n t are solutions of the gp hierarchy with the coupling constant in the nonlinear term of the gp equation given by v x dx the uniqueness of the solutions of this hierarchy remains an open question

strichartz estimates for schrodinger equations on irrational tori in this paper we prove new strichartz estimates for linear schrodinger equations posed on d dimensional irrational tori then we use these estimates to prove subcritical and critical local well posedness results for nonlinear schrodinger equations on irrational tori

population dynamics of spiking neurons fast transients asynchronous states and locking an integral equation describing the time evolution of the population activity in a homogeneous pool of spiking neurons of the integrate and fire type is discussed it is analytically shown that transients from a state of incoherent firing can be immediate the stability of incoherent firing is analyzed in terms of the noise level and transmission delay and a bifurcation diagram is derived the response of a population of noisy integrate and fire neurons to an input current of small amplitude is calculated and characterized by a linear filter l the stability of perfectly synchronized locked solutions is analyzed