boundedness in the higher dimensional attractionrepulsion chemotaxis growth system this paper deals with an attractionrepulsion chemotaxis system with logistic source ut u uv uw f u x t 0 vt v1v 1u x t 0 wt w2w 2u x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain rn n3 with nonnegative initial data u0 v0 w0 w1 w2 w2 where 0 0 i 0 i 0 i 1 2 and f u auu2 with a0 and 0 based on the maximal sobolev regularity and semigroup technique it is proved that the system admits a unique globally bounded classical solution provided that n3 1 2 and there exists 0 0 such that 1 2 0 the main aim of this paper is to solve the higher dimensional boundedness question addressed by xie and xiang in ima j appl math 81 2016 165198

decorrelation estimates for random discrete schrodinger operators in dimension one and applications to spectral statistics the purpose of the present work is to establish decorrelation estimates for some random discrete schrodinger operator in dimension one we prove that the minami estimates are consequences of the wegner estimates and localization we also prove decorrelation estimates at distinct energies for the random hopping model and schrodinger operators with alloy type potentials these results are used to give a description of the spectral statistics

large time behavior of an attraction repulsion chemotaxis system abstract in this study we consider an attraction repulsion chemotaxis system u t u u v u w x t 0 v t v u v x t 0 w t w u w x t 0 with homogeneous neumann boundary conditions where and are positive parameters under the critical condition that 0 we prove that the system possesses a unique global solution which is uniformly bounded in the physical domain r n n 2 3 in addition we assert that we can find 0 0 such that for all of the initial data u 0 that satisfy u 0 0 the solution of the system approaches the steady state u 0 u 0 u 0 exponentially as time tends to infinity where u 0 1 u 0

an optimal wegner estimate and its application to the global continuity of the integrated density of states for random schrodinger operators we prove that the integrated density of states ids of random schrodinger operators with anderson type potentials on l 2 r d for d geq1 is locally holder continuous at all energies with the same holder exponent 0 alpha leq1 as the conditional probability measure for the single site random variable as a special case we prove that if the probability distribution is absolutely continuous with respect to lebesgue measure with a bounded density then the ids is lipschitz continuous at all energies the single site potential u in l 0 infty r d must be nonnegative and compactly supported the unperturbed hamiltonian must be periodic and satisfy a unique continuation principle we also prove analogous continuity results for the ids of random anderson type perturbations of the landau hamiltonian in two dimensions all of these results follow from a new wegner estimate for local random hamiltonians with rather general probability measures

dynamical localization for the random dimer schrodinger operator we study the one dimensional random dimer model with hamiltonian h v where for all x mathbb z v 2x v 2x 1 and where the v 2x are i i d bernoulli random variables taking the values v v 0 we show that for all values of vand with probability one in the spectrum of his pure point if v 1 and v 1 sqrt 2 the lyapunov exponent vanishes only at the two critical energies given by e v for the particular value v 1 sqrt 2 respectively v sqrt 2 we show the existence of new additional critical energies at e 3 sqrt 2 respectively e 0 on any compact interval inot containing the critical energies the eigenfunctions are then shown to be semi uniformly exponentially localized and this implies dynamical localization for all q 0 and for all ell 2 mathbb z with sufficiently rapid decrease mathop sup limits t r psi i left q right kern 1pt left t right mathop sup limits t left langle p i left h omega right psi t left x right q p i left h omega right psi t right rangle infty here psi t e ih omega t psi and pi h is the spectral projector of h onto the interval i in particular if v 1 and v sqrt 2 these results hold on the entire spectrum so that one can take i h

aggregation vs global diffusive behavior in the higher dimensional keller segel model abstract we consider the classical parabolic parabolic keller segel system u t u u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r n it is proved that in space dimension n 3 for each q n 2 and p n one can find e 0 0 such that if the initial data u 0 v 0 satisfy u 0 l q e and v 0 l p e then the solution is global in time and bounded and asymptotically behaves like the solution of a discoupled system of linear parabolic equations in particular u v approaches the steady state m m as t where m is the total mass m u 0 of the population moreover we shall show that if is a ball then for arbitrary prescribed m 0 there exist unbounded solutions emanating from initial data u 0 v 0 having total mass u 0 m

global bounded weak solutions to a degenerate quasilinear attractionrepulsion chemotaxis system with rotation we consider the following quasilinear attractionrepulsion chemotaxis system with rotation ut um us1 u v w x v us2 u v w x w x t 0 vt v uv x t 0 wt w uw x t 0 umus1v us2w v w 0 x t 0 where r2 is a bounded domain with smooth boundary and are positive constants here s1 sij 22 and s2 sij 22 are chemosensitivity tensors with sij sijc2 0 3 which are assumed to satisfy s1 cs1 and s2 cs2 with some positive constants cs1 cs2 for all u v w x 0 3 it is shown that whenever m 1 for any sufficiently smooth non negative initial data the system possesses at least one global bounded weak solution

modeling conception as an aggregated bernoulli outcome with latent variables via the em algorithm we describe a general method for analyzing aggregated bernoulli outcomes the research is motivated by an epidemiological reproductive study where the outcome was whether pregnancy was detected in a woman s particular menstrual cycle and the bernoulli trials corresponded to days with intercourse during the cycle each cycle is either viable or not i e is or is not susceptible to conception we develop an em algorithm approach to maximizing the observed data pseudo likelihood based on a set of unobservable latent outcomes linked to the specific days with intercourse this method is flexible in that it allows one to model effect of covariates on the susceptibility factor and on the latent outcomes application of the method to fertility studies enables one to investigate covariates with a long term or transient effect on the daily conception probability a complication is that most couples contribute more than one cycle in a prospective study a generalized estimating equation approach adjusts for the dependency among outcomes within individual couples the method can be applied in any setting where dependency among bernoulli trials is induced by a susceptibility factor and the trial outcomes are only observable in the aggregate

global existence and boundedness in a keller segel stokes system involving a tensor valued sensitivity with saturation the 3d case abstract in this paper we continue to deal with the initial boundary value problem for the coupled keller segel stokes system n t u n n n s x n c c x t 0 t c t u c c c n x t 0 t u t p u n x t 0 t u 0 x t 0 t where r d is a bounded domain with smooth boundary and the chemotactic sensitivity s is not a scalar function but rather attains values in r d d and satisfies s x n c c s 1 n with some c s 0 and 0 when d 2 our previous work j differential equations 2015 has established the existence of global bounded classical solutions under the subcritical assumption 0 which is consistent with the corresponding results of the fluid free system but the method seems to be invalid in the three dimensional setting in this paper for the case d 3 we develop a new method to establish the existence and boundedness of global classical solutions for arbitrarily large initial data under the assumption 1 2 which is slightly stronger than the corresponding subcritical assumption 1 3 on the fluid free system where such an assumption is essentially necessary and sufficient for the existence of global bounded solutions the key idea here is to establish the general l p regularity of u from a rather low l p regularity of n which will be obtained by a new combinational functional

a parabolic elliptic elliptic attraction repulsion chemotaxis system with logistic source abstract this paper deals with the parabolic elliptic elliptic attraction repulsion chemotaxis system with logistic source u t d u u u v u w r u u 2 x t 0 0 v u v x t 0 0 w u w x t 0 under no flux boundary conditions in bounded domain with smooth boundary where r and are assumed to be positive when r 3 d u is assumed to satisfy d 0 0 d u c d u m 1 with m 1 and c d 0 it is proved that if 0 and 1 3 then for any given u 0 w 1 the system possesses a global and bounded classical solution for the case where d u 1 and n 3 the convergence rate of the solution is established when the random motion of the chemotactic species is neglected i e d u 0 and r n n 2 is a convex domain boundedness and the finite time blow up of the solution are investigated

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

boundedness and finite time collapse in a chemotaxis system with volume filling effect abstract we consider the elliptic parabolic pde system u t u u u v x t 0 0 v m u x t 0 with nonnegative initial data u 0 having mean value m under homogeneous neumann boundary conditions in a smooth bounded domain r n the nonlinearities and are supposed to generalize the prototypes u u 1 p u u u 1 q 1 with p 0 and q r problems of this type arise as simplified models in the theoretical description of chemotaxis phenomena under the influence of the volume filling effect as introduced by painter and hillen k j painter t hillen volume filling and quorum sensing in models for chemosensitive movement can appl math q 10 2002 501 543 it is proved that if p q 2 n then all solutions are global in time and bounded whereas if p q 2 n q 0 and is a ball then there exist solutions that become unbounded in finite time the former result is consistent with the aggregation inhibiting effect of the volume filling mechanism the latter however is shown to imply that if the space dimension is at least three then chemotactic collapse may occur despite the presence of some nonlinearities that supposedly model a volume filling effect in the sense of painter and hillen

potential for bias in estimating human fecundability parameters a comparison of statistical models fecundability studies where couples attempting pregnancy subject to failure conception one time in each menstrual cycle present a natural discrete failure time scenario because the biologic capacity to conceive varies among couples in the population a complication arises in choosing a method of analysis related to the fact that the maximum follow up time can vary from study to study and follow up time could potentially have different effects on parameters based on different approaches to modelling traditional development in fertility studies has been based on an implicit assumption that binary outcomes for different menstrual cycles are mutually independent we contrast traditional models to a random effects model where cycle viability is modelled as subject specific we clarify the interpretations for different parameters from different models we show that the traditional approach yields some regression parameters that depend on follow up time limiting the generalizability of inferences based on this analytic approach by contrast the subject specific model consistently estimates parameters of interest if the underlying distribution is properly specified data from a fecundability study carried out in north carolina serves to illustrate these points

finite time blowup and global in time unbounded solutions to a parabolic parabolic quasilinear keller segel system in higher dimensions abstract in this paper we consider quasilinear keller segel type systems of two kinds in higher dimensions in the case of a nonlinear diffusion system we prove an optimal with respect to possible nonlinear diffusions generating explosion in finite time of solutions finite time blowup result in the case of a cross diffusion system we give results which are optimal provided one assumes some proper non decay of a nonlinear chemical sensitivity moreover we show that once we do not assume the above mentioned non decay our result cannot be as strong as in the case of nonlinear diffusion without nonlinear cross diffusion terms to this end we provide an example interesting by itself of global in time unbounded solutions to the nonlinear cross diffusion keller segel system with chemical sensitivity decaying fast enough in a range of parameters in which there is a finite time blowup result in a corresponding case without nonlinear cross diffusion

modeling human fertility in the presence of measurement error the probability of conception in a given menstrual cycle is closely related to the timing of intercourse relative to ovulation although commonly used markers of time of ovulation are known to be error prone most fertility models assume the day of ovulation is measured without error we develop a mixture model that allows the day to be misspecified we assume that the measurement errors are i i d across menstrual cycles heterogeneity among couples in the per cycle likelihood of conception is accounted for using a beta mixture model bayesian estimation is straightforward using markov chain monte carlo techniques the methods are applied to a prospective study of couples at risk of pregnancy in the absence of validation data or multiple independent markers of ovulation the identifiability of the measurement error distribution depends on the assumed model thus the results of studies relating the timing of intercourse to the probability of conception should be interpreted cautiously

prevention of blow up by fast diffusion in chemotaxis abstract in this paper we study a strongly coupled parabolic system with cross diffusion term which models chemotaxis the diffusion coefficient goes to infinity when cell density tends to an allowable maximum value such fast diffusion leads to global existence of solutions in bounded domains for any given initial data irrespective of the spatial dimension which is usually the goal of many modifications to the classical keller segel model the key estimates that make this possible have been obtained by a technique that uses ideas from moser s iterations

local fluctuation of the spectrum of a multidimensional anderson tight binding model we consider the anderson tight binding modelh v acting inl 2 z d and its restrictionh to finite hypercubes z d herev v x x z d is a random potential consisting of independent identically distributed random variables let e j j be the eigenvalues ofh and let j e e j e j 1 be its rescaled eigenvalues then assuming that the exponential decay of the fractional moment of the green function holds for complex energies neare and that the density of statesn e exists ate we shall prove that the random sequence j e j considered as a point process onr 1 converges weakly to the stationary poisson point process with intensity measuren e dx as gets large thus extending the result of molchanov proved for a one dimensional continuum random schrodinger operator on the other hand the exponential decay of the fractional moment of the green function was established recently by aizenman molchanov and graf as a technical lemma for proving anderson localization at large disorder or at extreme energy thus our result in this paper can be summarized as follows near the energye where anderson localization is expected there is no correlation between eigenvalues ofh if is large

boundedness blowup and critical mass phenomenon in competing chemotaxis abstract we consider the following attraction repulsion keller segel system u t u u v u w x t 0 v t v u v x t 0 0 w u w x t 0 u x 0 u 0 x v x 0 v 0 x x with homogeneous neumann boundary conditions in a bounded domain r 2 with smooth boundary the system models the chemotactic interactions between one species denoted by u and two competing chemicals denoted by v and w which has important applications in alzheimer s disease here all parameters and are positive by constructing a lyapunov functional we establish the global existence of uniformly in time bounded classical solutions with large initial data if the repulsion dominates or cancels attraction i e if the attraction dominates i e a critical mass phenomenon is found specifically speaking we find a critical mass m 4 such that the solution exists globally with uniform in time bound if m m and blows up if m m and m 4 m m n where n denotes the set of positive integers and m u 0 d x the initial cell mass

anderson localization for bernoulli and other singular potentials we prove exponential localization in the anderson model under very weak assumptions on the potential distribution in one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment in particular this solves the longstanding problem of localization for bernoulli potentials i e potentials that take only two values in dimensions greater than one we prove localization at high disorder for potentials with holder continuous distributions and for bounded potentials whose distribution is a convex combination of a holder continuous distribution with high disorder and an arbitrary distribution these include potentials with singular distributions we also show that for certain bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component

global existence vs blowup in a fully parabolic quasilinear 1d keller segel system abstract we show that the one dimensional fully parabolic keller segel system with nonlinear diffusion possesses global in time solutions provided the nonlinear diffusion is equal to 1 1 u 1 independently on the volume of the initial data we also show that in the critical case i e for 1 the same result holds for initial masses smaller than a prescribed constant additionally we prove the existence of initial data for which a solution blows up in a finite time for any nonlinear diffusion integrable at infinity however in the parabolic elliptic case the above mentioned integrability condition on nonlinear diffusion sharply distinguishes between global existence and blowup cases we are unable to recover the entire global existence counterpart of this result in a fully parabolic case

global dynamics for an attraction repulsion chemotaxis model with logistic source abstract in this paper we investigate attraction repulsion chemotaxis system with logistic source under homogeneous neumann boundary conditions in a bounded domain with smooth boundary under appropriate regularity assumptions on the initial data by some l p estimate techniques using different methods we show that the system possesses global classical solution or at least one global weak solution in addition to we also consider large time behavior of solutions and steady solutions to an attraction repulsion chemotaxis system these results generalize and improve previously known ones and partially results are new

dynamical localization ii with an application to the almost mathieu operator several recent works have established dynamical localization for schrodinger operators starting from control on the localization length of their eigenfunctions in terms of their centers of localization we provide an alternative way to obtain dynamical localization without resorting to such a strong condition on the exponential decay of the eigenfunctions furthermore we illustrate our purpose with the almost mathieu operator h cos 2 x 15 and with good diophantine properties more precisely for almost all for all q 0 and for all functions l2 mathbb z of compact support we show that mathop sup limits t left langle e ith theta lambda omega psi left x right q e ith theta lambda omega psi right rangle c psi the proof applies equally well to discrete and continuous random hamiltonians in all cases it uses as input a repulsion principle of singular boxes supplied in the random case by the multi scale analysis

the beta geometric distribution applied to comparative fecundability studies a convenient measure of fecundability is time number of menstrual cycles required to achieve pregnancy couples attempting pregnancy are heterogeneous in their per cycle probability of success if success probabilities vary among couples according to a beta distribution then cycles to pregnancy will have a beta geometric distribution the likelihood ratio test can thus be employed in studies of exposures that may impair fecundability covariates are incorporated in a natural way the model is illustrated by applying it to data on cycles to pregnancy in smokers and nonsmokers with adjustment for covariates summary in fre excerpt

on a quasilinear parabolic elliptic chemotaxis system with logistic source abstract this paper deals with a quasilinear parabolic elliptic chemotaxis system with logistic source under homogeneous neumann boundary conditions in a smooth bounded domain for the case of positive diffusion function it is shown that the corresponding initial boundary value problem possesses a unique global classical solution which is uniformly bounded moreover if the diffusion function is zero at some point or a positive diffusion function and the logistic damping effect is rather mild we proved that the weak solutions are global existence finally it is asserted that the solutions approach constant equilibria in the large time for a specific case of the logistic source

boundedness in a quasilinear parabolic parabolic keller segel system with subcritical sensitivity abstract we consider the quasilinear parabolic parabolic keller segel system u t d u u s u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a convex smooth bounded domain r n with n 1 it is proved that if s u d u c u with 2 n and some constant c 0 for all u 1 then the classical solutions to the above system are uniformly in time bounded provided that d u satisfies some technical conditions such as algebraic upper and lower growth resp decay estimates as u this boundedness result is optimal according to a recent result by the second author winkler 2010 27 which says that if s u d u c u for u 1 with c 0 and some 2 n n 2 then for each mass m 0 there exist blow up solutions with mass u 0 m in addition this paper also proves a general boundedness result for quasilinear non uniformly parabolic equations by modifying the iterative technique of moser alikakos alikakos 1979 1

boundedness and blow up in the higher dimensional attraction repulsion chemotaxis system with nonlinear diffusion abstract this paper deals with the following quasilinear attraction repulsion chemotaxis system u t d u u u v u w x t 0 v t v u v x t 0 0 w u w x t 0 in a bounded domain r n n 2 under homogeneous neumann boundary conditions the parameters and are positive and the diffusion d u is supposed to generalize the prototype d u d 0 u with d 0 0 and r under the assumption 2 n 1 it is proved that the corresponding initial boundary problem possesses a nonnegative globally bounded solution moreover by constructing a lyapunov functional and by assuming that 2 n 1 and n 3 it is proved that there exist radially symmetric solutions which blow up in finite time in b r 0 with r 0 finally it is also shown that blow up may occur if 2 n 1 and 0 when is a ball in r n with n 3

a quasilinear attraction repulsion chemotaxis system of parabolic elliptic type with logistic source abstract this study considers the following quasilinear attraction repulsion chemotaxis system of parabolic elliptic type with logistic source u t d u u u v u w f u x t 0 0 v u v x t 0 0 w u w x t 0 under homogeneous neumann boundary conditions in a bounded domain r n n 2 with smooth boundary where d u c d u m 1 with m 1 0 and c d 0 and f u a b u with a 0 b 0 and 1 in the case of non degenerate diffusion i e 0 we show that the system admits a unique global bounded classical solution provided that the repulsion prevails over the attraction in the sense that 0 or the logistic dampening is sufficiently strong or the diffusion is sufficiently strong while in the case of degenerate diffusion i e 0 we show that the system admits a global bounded weak solution at least under the same assumptions finally we obtain the large time behavior of solutions for a specific logistic source

boundedness vs blow up in a chemotaxis system abstract we determine the critical blow up exponent for a keller segel type chemotaxis model where the chemotactic sensitivity equals some nonlinear function of the particle density assuming some growth conditions for the chemotactic sensitivity function we establish an a priori estimate for the solution of the problem considered and conclude the global existence and boundedness of the solution furthermore we prove the existence of solutions that become unbounded in finite or infinite time in that situation where this a priori estimate fails

well posedness for a model derived from an attraction repulsion chemotaxis system abstract in this paper we are interested in a model derived from an attraction repulsion chemotaxis system in high dimensions t u u 1 u v 2 u w x r n t 0 1 v v u x r n t 0 2 w w u x r n t 0 u x 0 u 0 x x r n with the parameters 1 0 2 0 1 0 2 0 and nonnegative initial data u 0 x l 1 r n l r n we prove that a global bounded solution exists when the repulsion prevails over the attraction in the sense of 1 1 moreover we give the smoothness of the solution and obtain its decay rates in w s p r n which coincide with the ones of the classical heat equation conversely when n 2 1 2 we prove that the finite time blow up may occur

global existence of solutions to a parabolic parabolic system for chemotaxis with weak degradation abstract we study the global existence of solutions to a parabolic parabolic system for chemotaxis with a logistic source in a two dimensional domain where the degradation order of the logistic source is weaker than quadratic we introduce nonlinear production of a chemoattractant and show the global existence of solutions under certain relations between the degradation and production orders

global existence and asymptotic behavior of classical solutions to a parabolic elliptic chemotaxis system with logistic source on rn in the current paper we consider the following parabolic elliptic semilinear keller segel model on rnrn ut u u v au bu2 x rn t 0 0 i v u x rn t 0 where 0 a 0 b 0 are constant real numbers and n is a positive integer we first prove the local existence and uniqueness of classical solutions u x t u0 v x t u0 u x t u0 v x t u0 with u x 0 u0 u0 x u x 0 u0 u0 x for various initial functions u0 x u0 x next under some conditions on the constants a b a b and the dimension n we prove the global existence and boundedness of classical solutions u x t u0 v x t u0 u x t u0 v x t u0 for given initial functions u0 x u0 x finally we investigate the asymptotic behavior of the global solutions with strictly positive initial functions or nonnegative compactly supported initial functions under some conditions on the constants a b a b and the dimension n we show that for every strictly positive initial function u0 u0 limt supx rn u x t u0 ab v x t u0 ab 0 and that for every nonnegative initial function u0 u0 with non empty and compact support supp u0 supp u0 there are 0 clow u0 cup u0 such that limt sup x ct u x t u0 ab v x t u0 ab 0 0 c clow u0 and limt sup x ct u x t u0 v x t u0 0 c cup u0

decorrelation estimates for some continuous and discrete random schrodinger operators in dimension one and applications to spectral statistics abstract the purpose of the present work is to establish decorrelation estimates at distinct energies for some random schrodinger operator in dimension one in particular we establish the result for some random operators on the continuum with alloy type potential these results are used to give a description of the spectral statistics

correlation estimates in the anderson model we give a new proof of correlation estimates for arbitrary moments of the resolvent of random schrodinger operators on the lattice that generalizes and extends the correlation estimate of minami for the second moment we apply this moment bound to obtain a new n level wegner type estimate that measures eigenvalue correlations through an upper bound on the probability that a local hamiltonian has at least n eigenvalues in a given energy interval another consequence of the correlation estimates is that the results on the poisson statistics of energy level spacing and the simplicity of the eigenvalues in the strong localization regime hold for a wide class of translation invariant selfadjoint lattice operators with decaying off diagonal terms and random potentials

localization for some continuous random schrodinger operators we study the spectrum of random schrodinger operators acting onl2 r d of the following type h delta w sum x in mathbb z d t x v x the t x x in mathbb z d are i i d random variables under weak assumptions onv we prove exponential localization forh at the lower edge of its spectrum in order to do this we give a new proof of the wegner estimate that works without sign assumptions onv

generalized eigenvalue counting estimates for the anderson model we generalize minami s estimate for the anderson model and its extensions to n eigenvalues allowing for n arbitrary intervals and arbitrary single site probability measures with no atoms as an application we derive new results about the multiplicity of eigenvalues and mott s formula for the ac conductivity when the single site probability distribution is holder continuous

inverse tunnelingestimates and applications to the study of spectral statistics ofrandom operators on the real line we present a proof of a minami type estimate for one dimensional random schrodinger operators valid at all energies in the localization regime provided a wegner estimate is known to hold the minami type estimate is then applied to two models to obtain results on their spectral statistics the heuristics underlying our proof of minami type estimates is that close by eigenvalues of a one dimensional schrodinger operator correspond either to eigenfunctions that live far away from each other in space or they come from some tunneling phenomena in the second case one can undo the tunneling and thus construct quasi modes that live far away from each other in space resume nous demontrons une inegalite de type minami pour des operateurs de schrodinger aleatoires uni dimensionnel dans toute la region localisee si une estimee de wegner est connue cette estimee de type de minami est alors appliquee pour obtenir les statistiques spectrales pour deux modeles l heuristique qui guide ce travail est que des valeurs propres proches pour un operateur de schrodinger sur un intervalle sont soient localisees loin l une de l autre soit sont la consequence d un phenomene d effet tunnel dans le second cas on peut en defaisant cet effet tunnel construire des quasi modes qui sont localises loin l un de l autre 0 introduction consider the following two random operators on the real line the anderson model 0 1 h d2 dx2 w

models for longitudinal data a generalized estimating equation approach this article discusses extensions of generalized linear models for the analysis of longitudinal data two approaches are considered subject specific ss models in which heterogeneity in regression parameters is explicitly modelled and population averaged pa models in which the aggregate response for the population is the focus we use a generalized estimating equation approach to fit both classes of models for discrete and continuous outcomes when the subject specific parameters are assumed to follow a gaussian distribution simple relationships between the pa and ss parameters are available the methods are illustrated with an analysis of data on mother s smoking and children s respiratory disease

boundedness in quasilinear keller segel systems of parabolic parabolic type on non convex bounded domains abstract this paper deals with the quasilinear fully parabolic keller segel system u t d u u s u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a bounded domain r n with smooth boundary n n the diffusivity d u is assumed to satisfy some further technical conditions such as algebraic growth and d 0 0 which says that the diffusion is allowed to be not only non degenerate but also degenerate the global in time existence and uniform in time boundedness of solutions are established under the subcritical condition that s u d u k u e for u 0 with 2 n k 0 and e 0 when d 0 0 this paper represents an improvement of tao and winkler 17 because the domain does not necessarily need to be convex in this paper in the case r n and d 0 0 uniform in time boundedness is an open problem left in a previous paper 7 this paper also gives an answer to it in bounded domains

reflectionless modes in chains with large size homogeneous impurities the authors study the spectra and localization properties of tight binding chains with randomly distributed binary blocks of arbitrary size m one having site energies epsilon a and the other epsilon b respectively they demonstrate that for block size m greater than one and delta mod epsilon a epsilon b less than a critical value delta c m perfect transmission resonance modes exist in the band their number is proportional to m and they occur via dominant 1 e2 divergencies of the localization length no transmission is found for delta delta c m the results are understood by solving exactly the scattering problem from a single homogeneous impurity block of arbitrary size m in the limit of hard impurities m to infinity transmission stops only when delta delta c infinity 4v v being the intersite matrix element that is when the pure a and b bands become detached

bootstrap multiscale analysis and localization in random media we introduce an enhanced multiscale analysis that yields subexponentially decaying probabilities for bad events for quantum and classical waves in random media we obtain exponential decay for the resolvent of the corresponding random operators in boxes of side l with probability higher than 1 e l for any 0 1 the starting hypothesis for the enhanced multiscale analysis only requires the verification of polynomial decay of the finite volume resolvent at some sufficiently large scale with probability bigger than 1 d is the dimension note that from the same starting hypothesis we get conclusions that are valid for any 0 1 this is achieved by the repeated use of a bootstrap argument as an application we use a generalized eigenfunction expansion to obtain strong dynamical localization of any order in the hilbert schmidt norm and better estimates on the behavior of the eigenfunctions

boundedness of the attraction repulsion keller segel system a b s t r a c t this paper considers the initial boundary value problem of the attraction repulsion keller segel model describing aggregation of microglia in the central nervous system in alzheimer s disease due to the interaction of chemoattractant and chemorepellent if repulsion dominates over attraction we show the global existence of classical solution in two dimensions and weak solution in three dimensions with large initial data u x 0 u0 x v x 0 v 0 x w x 0 w 0 x x