a group theoretic consequence of the donald flanigan conjecture abstract the donald flanigan conjecture asserts that for any finite group g and prime p dividing its order g the group algebra f pg can be deformed into a semisimple and hence rigid algebra we show that this implies that there is some element g g whose centralizer cg g has a normal subgroup of index p the method is to observe that the donald flanigan deformation must be a jump whence from the deformation theory h1 f pg f pg 0 using a standard result linking hochschild and group cohomology one sees that some h1 cg g f p must be non zero giving the result our corollary to the d f conjecture has recently been verified by fleischmann janiszczak and lempken using the classification of finite simple groups

localization phenomenon in gaps of the spectrum of random lattice operators we consider a class of random lattice operators including schrodinger operators of the formh w gv wherew x is a real valued periodic function g is a positive constant andv x x d are independent identically distributed real random variables we prove that if the operator w has gaps in the spectrum andg is sufficiently small then the operatorh develops pure point spectrum with exponentially decaying eigenfunctions in a vicinity of the gaps

convergence of a stochastic approximation version of the em algorithm the expectation maximization em algorithm is a powerful computational technique for locating maxima of functions it is widely used in statistics for maximum likelihood or maximum a posteriori estimation in incomplete data models in certain situations however this method is not applicable because the expectation step cannot be performed in closed form to deal with these problems a novel method is introduced the stochastic approximation em saem which replaces the expectation step of the em algorithm by one iteration of a stochastic approximation procedure the convergence of the saem algorithm is established under conditions that are applicable to many practical situations moreover it is proved that under mild additional conditions the attractive stationary points of the saem algorithm correspond to the local maxima of the function presented to support our findings

a characterization of the anderson metal insulator transport transition we investigate the anderson metal insulator transition for random schrodinger operators we define the strong insulator region to be the part of the spectrum where the random operator exhibits strong dynamical localization in the hilbert schmidt norm we introduce a local transport exponent e and set the weak metallic transport region to be the part of the spectrum with nontrivial transport i e e 0 we prove that these insulator and metallic regions are complementary sets in the spectrum of the random operator and that the local transport exponent e provides a characterization of the metal insulator transport transition moreover we show that if there is such a transition then e has to be discontinuous at a transport mobility edge more precisely we show that if the transport is nontrivial then e 1 2d where d is the space dimension these results follow from a proof that slow transport of quantum waves in random media implies the starting hypothesis for the authors bootstrap multiscale analysis we also conclude that the strong insulator region coincides with the part of the spectrum where we can perform a bootstrap multiscale analysis proving that the multiscale analysis is valid all the way up to a transport mobility edge

a new proof of localization in the anderson tight binding model we give a new proof of exponential localization in the anderson tight binding model which uses many ideas of the frohlich martinelli scoppola and spencer proof but is technically simpler particularly the probabilistic estimates

localization for random and quasiperiodic potentials a survey is made of some recent mathematical results and techniques for schrodinger operators with random and quasiperiodic potentials a new proof of localization for random potentials established in collaboration with h von dreifus is sketched

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

sampling and equidistribution theorems for elliptic second order operators lifting of eigenvalues and applications abstract we consider elliptic second order partial differential operators with lipschitz continuous leading order coefficients on finite cubes and the whole euclidean space we prove quantitative sampling and equidistribution theorems for eigenfunctions the estimates are scale free in the sense that for a sequence of growing cubes we obtain uniform estimates these results are applied to prove lifting of eigenvalues as well as the infimum of the essential spectrum and an uncertainty relation aka spectral inequality for short energy interval spectral projectors several applications including random operators are discussed in the proof we have to overcome several challenges posed by the variable coefficients of the leading term

on guided waves in 3 d photonic crystals the propagation of guided waves in 3 d photonic crystals is studied in this paper we suppose that the spectrum of reference medium a perfectly 3 d photonic crystal has band gaps if a line defect is introduced along one of the lattice vectors we prove that spectrum can be created inside the gaps of the reference spectrum if the line defect is sufficiently strong furthermore we also prove that the generalized eigenfunctions corresponding to the generalized eigenvalues created by line defect are exponentially decaying away from the line defect

2 dimensional localization of acoustic waves in random perturbation of periodic media abstract in this work we study internal band edges localization of acoustic waves in 2 dimensional space obtained by random perturbation of some periodic media our results rely on the study of lifshitz tails for the integrated density of states for random acoustic operators of the form a 1 localization is then deduced by the standard multiscale argument

the space clamped hodgkin huxley system with random synaptic input inhibition of spiking by weak noise and analysis with moment equations we consider a classical space clamped hodgkin huxley model neuron stimulated by synaptic excitation and inhibition with conductances represented by ornstein uhlenbeck processes using numerical solutions of the stochastic model system obtained by an euler method it is found that with excitation only there is a critical value of the steady state excitatory conductance for repetitive spiking without noise and for values of the conductance near the critical value small noise has a powerfully inhibitory effect for a given level of inhibition there is also a critical value of the steady state excitatory conductance for repetitive firing and it is demonstrated that noise in either the excitatory or inhibitory processes or both can powerfully inhibit spiking furthermore near the critical value inverse stochastic resonance was observed when noise was present only in the inhibitory input process the system of deterministic differential equations for the approximate first and second order moments of the model is derived they are solved using runge kutta methods and the solutions are compared with the results obtained by simulation for various sets of parameters including some with conductances obtained by experiment on pyramidal cells of rat prefrontal cortex the mean and variance obtained from simulation are in good agreement when there is spiking induced by strong stimulation and relatively small noise or when the voltage is fluctuating at subthreshold levels in the occasional spike mode sometimes exhibited by spinal motoneurons and cortical pyramidal cells the assumptions underlying the moment equation approach are not satisfied the simulation results show that noisy synaptic input of either an excitatory or inhibitory character or both may lead to the suppression of firing in neurons operating near a critical point and this has possible implications for cortical networks although suppression of firing is corroborated for the system of moment equations there seem to be substantial differences between the dynamical properties of the original system of stochastic differential equations and the much larger system of moment equations

constructive proof of localization in the anderson tight binding model we prove that for large disorder or near the band tails the spectrum of the anderson tight binding hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues the corresponding eigenfunctions are exponentially well localized these results hold in arbitrary dimension and with probability one in one dimension we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder our techniques extend to other physical systems which exhibit localization phenomena such as infinite systems of coupled harmonic oscillators or random schrodinger operators in the continuum

hypoelliptic diffusions filtering and inference from complete and partial observations the statistical problem of parameter estimation in partially observed hypoelliptic diffusion processes is naturally occurring in many applications however because of the noise structure where the noise components of the different co ordinates of the multi dimensional process operate on different timescales standard inference tools are ill conditioned we propose to use a higher order scheme to approximate the likelihood such that the different timescales are appropriately accounted for we show consistency and asymptotic normality with non typical convergence rates when only partial observations are available we embed the approximation in a filtering algorithm for the unobserved co ordinates and use this as a building block in a stochastic approximation expectation maximization algorithm we illustrate on simulated data from three models the harmonic oscillator the fitzhugh nagumo model used to model membrane potential evolution in neuroscience and the synaptic inhibition and excitation model used for determination of neuronal synaptic input

localization of classical waves ii electromagnetic waves we consider electromagnetic waves in a medium described by a position dependent dielectric constant we assume that is a random perturbation of a periodic function and that the periodic maxwell operator has a gap in the spectrum where we prove the existence of localized waves i e finite energy solutions of maxwell s equations with the property that almost all of the wave s energy remains in a fixed bounded region of space at all times localization of electromagnetic waves is a consequence of anderson localization for the self adjoint operators we prove that in the random medium described by the random operator exhibits anderson localization inside the gap in the spectrum of this is shown even in situations when the gap is totally filled by the spectrum of the random operator we can prescribe random environments that ensure localization in almost the whole gap

eigenvalue branches of the perturbed maxwell operator m λd in a gap of σ m the propagation of guided waves in photonic crystal fibers pcfs is studied a pcf can be regarded as a perfectly two dimensional photonic crystal with a line defect along the axial direction this problem can be treated as an eigenvalue problem for a family of noncompact self adjoint operators under the assumption that the background spectrum has a gap we prove that a line defect can create an eigenvalue of any given fixed value in the gap provided that the defect is strong enough based on a decoupling of regions in r2 by means of dirichlet and neumann boundaries then using the trace ideal estimates we study asymptotic distribution of eigenvalues and bounds on the number of eigenvalue branches in particular we show that if the defect is weak enough no eigenvalues can be created inside the gap

the law of the euler scheme for stochastic differential equations i convergence rate of the distribution function we study the approximation problem ofef x t byef x t n where x t is the solution of a stochastic differential equation x t n is defined by the euler discretization scheme with stept n andf is a given function for smoothf s talay and tubaro have shown that the erroref x t f x t n can be expanded in powers of 1 n which permits to construct romberg extrapolation precedures to accelerate the convergence rate here we prove that the expansion exists also whenf is only supposed measurable and bounded under an additional nondegeneracy condition of hormander type for the infinitesimal generator of x t to obtain this result we use the stochastic variations calculus in the second part of this work we will consider the density of the law ofx t n and compare it to the density of the law ofx t

mathematical analysis of the guided waves in photonic crystal fibers abstract the propagation of guided waves in photonic crystal fibers pcfs is studied the structure of a pcf can be regarded as a perfect two dimensional photonic crystal with a line defect along the invariant direction this problem can be treated as an eigenvalue problem for a family of noncompact self adjoint operators we prove that line defects do not change the essential spectrum of the associated background medium this result plays a key role for studying the influence of line defects on the background spectrum a modified combes thomas estimate is also formulated

nonlinear maxwell equations in inhomogeneous media we study the basic properties of the maxwell equations for nonlinear inhomogeneous media assuming the classical nonlinear optics representation for the nonlinear polarization as a power series we show that the solution exists and is unique in an appropriate space if the excitation current is not too large the solution to the nonlinear maxwell equations is represented as a power series in terms of the solution of the corresponding linear maxwell equations this representation holds at least for the time period inversely proportional to the appropriate norm of the solution to the linear maxwell equation we derive recursive formulas for the terms of the power series for the solution including an explicit formula for the first significant term attributed to the nonlinearity

parameter estimation for partially observed hypoelliptic diffusions hypoelliptic diffusion processes can be used to model a variety of phenomena in applications ranging from molecular dynamics to audio signal analysis we study parameter estimation for such processes in situations where we observe some components of the solution at discrete times since exact likelihoods for the transition densities are typically not known approximations are used that are expected to work well in the limit of small intersample times t and large total observation times n t hypoellipticity together with partial observation leads to ill conditioning requiring a judicious combination of approximate likelihoods for the various parameters to be estimated we combine these in a deterministic scan gibbs sampler alternating between missing data in the unobserved solution components and parameters numerical experiments illustrate asymptotic consistency of the method when applied to simulated data the paper concludes with an application of the gibbs sampler to molecular dynamics data

localization for some continuous random hamiltonians in d dimensions abstract we prove the existence with probability one of an interval of pure point spectrum for some families of continuous random schrodinger operators in d dimensions for anderson like models with positive short range single site potentials we also prove that the corresponding eigenfunctions decay exponentially and that the integrated density of states is lipschitz continuous for the other families of random potentials that we study we show that the corresponding eigenfunctions decay faster than an inverse power of x which depends upon the decay rate of the single site potential to obtain these results we develop an extension of the classical aronszajn donoghue theory for a class of relatively compact perturbations and a spectral averaging method which extends kotani s trick to these more general families of operators

the absolute continuity of the spectrum of maxwell operator in a periodic media we prove that the spectrum of maxwell operator b e rot e a x rot b x b with non negative and periodic functions a x and b x is absolutely continuous

localization of classical waves i acoustic waves we consider classical acoustic waves in a medium described by a position dependent mass density x we assume that x is a reandom perturbation of a periodic function 0 x and that the periodic acoustic operator a 0 nabla cdot tfrac 1 varrho 0 x nabla has a gap in the spectrum we prove the existence of localized waves i e finite energy solutions of the acoustic equations with the property that almost all of the wave s energy remains in a fixed bounded region of space at all times with probability one localization of acoustic waves is a consequence of anderson localization for the self adjoint operators a nabla cdot tfrac 1 varrho 0 x nabla onl2 d we prove that in the random medium described by x the random operatora exhibits anderson localization inside the gap in the spectrum ofa0 this is shown even in situations when the gap is totally filled by the spectrum of the random opertor we can prescribe random environments that ensure localization in almost the whole gap

on the spectrum of schrodinger operators with a random potential we investigate the spectrum of schrodinger operatorsh of the type h qi f x xi i qi and i independent identically distributed random variables i d we establish a strong connection between the spectrum ofh and the spectra of deterministic periodic schrodinger operators from this we derive a condition for the existence of forbidden zones in the spectrum ofh for random one and three dimensional kronig penney potentials the spectrum is given explicitly

on absence of diffusion near the bottom of the spectrum for a random schrodinger operator on l 2 bf r nu we consider a random schrodinger operator onl2 v of the form h omega delta v omega v omega x sigma chi c i x q i omega ci being a covering of v with unit cubes around the sites of v and qi i i d random variables with values in 0 1 we assume that theqi s are continuously distributed with bounded densityf q and that 0 p q0 1 2 1 then we show that an ergodic mean of the quantity dx x 2 exp ith x 2 t 1 vanishes provided ge h where is well localized around the origin andge is a positivec function with support in 0 e e e f our estimate ofe f is such that the set x v v x e f may contain with probability one an infinite cluster of cubes ci which are nearest neighbours the proof is based on the technique introduced by frohlich and spencer for the analysis of the anderson model