the units of a ring spectrum and a logarithmic cohomology operation recall that if r is a commutative ring then the set r r of invertible elements of r is naturally an abelian group under multiplication this construction is a functor from commutative rings to abelian groups in general there is no obvious relation between the additive group of a ring r and the multiplicative group of units r however under certain circumstances one can define a homomorphism from a subgroup of r to a suitable completion of r e g the natural logarithm q 0 r or the p adic logarithm 1 pzp zp the logarithmic cohomology operation is a homotopy theoretic analogue of the above where r is a commutative s algebra and completion is bousfield localization with respect to a morava k theory the purpose of this paper is to give a formula for the logarithmic operation in certain contexts in terms of power operations before giving our results we briefly explain some of the concepts involved

existence and quasilinearization in banach spaces abstract we establish some existence results for the nonlinear problem a u f in a reflexive banach space v without and with upper and lower solutions we then consider the application of the quasilinearization method to the above mentioned problem under fairly general assumptions on the nonlinear operator a and the banach space v we show that this problem has a solution that can be obtained as the strong limit of two quadratically convergent monotone sequences of solutions of certain related linear equations

localization and periodicity in unstable homotopy theory in this paper we develop a hierarchy of natural localizations of spaces called the vn periodizations for n 0 which may be used to expose and study periodic phenomena in unstable homotopy theory these vn periodizations act less radically than the corresponding homological localizations 2 and respect fibrations to a very considerable extent a major part of this paper is devoted to developing the general theory of periodizations of spaces thereby providing a foundation for the study of the vn periodization and many others some of this general theory has been developed independently by dror farjoun 13 14 and we refer the reader to his work for an alternative approach with other interesting general results during the past decade remarkable progress was made by ravenel hopkins devinatz and smith 12 16 17 33 toward a global understanding of stable periodic phenomena and we hope that the present paper will help to prepare the way for a similar understanding of unstable periodic phenomena an excellent exposition of localization and periodicity in stable homotopy theory is now available in ravenel s book 35 major features of stable homotopy are understood chromatically as manifestations of more basic periodic phenomena these phenomena belong to a hierarchy starting with those detected rationally followed by those detected in classical k theory and in the successive morava k theories most fundamentally each finite cw spectrum has an intrinsic periodicity given by a vn self map which becomes a selfequivalence after suitable localization for simplicity we describe our results in the pointed homotopy category hoo of connected cw complexes for spaces w y e hoo we call y w periodic or w local when the pointed mapping space map w y is contractible or equivalently when xt w y for t 0 as shown more generally in 3 corollary 7 2 10 or 13 there is a natural initial example x pwx of a map from x to a w periodic space pwx in hoo and we call this the w periodization or w localization of x our notation is derived from the classical example where w sn l and pwx is the n th postnikov section

generalized quasilinearization method for mixed boundary value problems a generalized quasilinearization method for a nonlinear mixed bvp is developed and a sequence of approximate solutions converging monotonically and quadratically to a solution of the given problem is presented

a numerical approach to nonlinear two point boundary value problems for odes in this paper we propose a numerical approach to solve some problems connected with the implementation of the newton type methods for the resolution of the nonlinear system of equations related to the discretization of a nonlinear two point bvps for odes with mixed linear boundary conditions by using the finite difference method

low energy properties of the random displacement model abstract we study low energy properties of the random displacement model a random schrodinger operator describing an electron in a randomly deformed lattice all periodic displacement configurations which minimize the bottom of the spectrum are characterized while this configuration is essentially unique for dimension greater than one there are infinitely many different minimizing configurations in the one dimensional case the latter leads to unusual low energy asymptotics for the integrated density of states of the one dimensional random displacement model for symmetric bernoulli distributed displacements it has a 1 log 2 singularity at the bottom of the spectrum in particular it is not holder continuous

homotopical localizations of spaces for a map f of spaces dror farjoun and the author have constructed an f localization functor where a space y is called f local when map f y is an equivalence this very general construction gives all known idempotent homotopy functors of spaces the main theorem of this paper shows that f localization functors always preserve fiber sequences of connected h spaces up to small error terms for instance the localization with respect to the n th morava k theory preserves such fiber sequences up to error terms with at most three nontrivial homotopy groups this implies for example that a k 1 homology equivalence of h spaces must induce an isomorphism of v 1 periodic homotopy groups results are also obtained on the a nullification or a periodization functors which are just the f localization functors for the maps f from spaces a to points two spaces are said to have the same nullity when they give the same nullification functors and it is shown that arbitrary sets of nullity classes have both least upper bounds and greatest lower bounds the a nullifications of nilpotent postnikov spaces are completely determined

existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti periodic boundary conditions this paper describes the method of quasilinearization for first order nonlinear impulsive functional differential equations with anti periodic boundary conditions a monotone iterative technique coupled with lower and upper solutions is employed to obtain sequences of approximate solutions converging monotonically and quadratically to the unique solution of the problem at hand 2007 elsevier ltd all rights reserved

on the telescopic homotopy theory of spaces in telescopic homotopy theory a space or spectrum x is approximated by a tower of localizations lnx n 0 taking account of vn periodic homotopy groups for progressively higher n for each n 1 we construct a telescopic kuhn functor n carrying a space to a spectrum with the same vn periodic homotopy groups and we construct a new functor n left adjoint to n using these functors we show that the nth stable monocular homotopy category comprising the nth fibers of stable telescopic towers embeds as a retract of the nth unstable monocular homotopy category in two ways one giving infinite loop spaces and the other giving infinite ln suspension spaces we deduce that ravenel s stable telescope conjectures are equivalent to unstable telescope conjectures in particular we show that the failure of ravenel s nth stable telescope conjecture implies the existence of highly connected infinite loop spaces with trivial johnson wilson e n homology but nontrivial vn periodic homotopy groups showing a fundamental difference between the unstable chromatic and telescopic theories as a stable chromatic application we show that each spectrum is k n equivalent to a suspension spectrum as an unstable chromatic application we determine the e n localizations and k n localizations of infinite loop spaces in terms of e n localizations of spectra under suitable conditions we also determine the e n localizations and k n localizations of arbitrary postnikov h spaces

parabolic problems for the anderson model i intermittency and related topics the objective of this paper is a mathematically rigorous investigation of intermittency and related questions intensively studied in different areas of physics in particular in hydrodynamics on a qualitative level intermittent random fields are distinguished by the appearance of sparsely distributed sharp peaks which give the main contribution to the formation of the statistical moments the paper deals with the cauchy problem t u t x hu t x u 0 x t 0 x 0 t x d for the anderson hamiltonianh x x d where is a generally unbounded spatially homogeneous random potential this first part is devoted to some basic problems using percolation arguments a complete answer to the question of existence and uniqueness for the cauchy problem in the class of all nonnegative solutions is given in the case of i i d random variables necessary and sufficient conditions for intermittency of the fieldsu t ast are found in spectral terms ofh rough asymptotic formulas for the statistical moments and the almost sure behavior ofu t x ast are also derived

brownian survival and lifshitz tail in perturbed lattice disorder abstract we consider the annealed asymptotics for the survival probability of brownian motion among randomly distributed traps the configuration of the traps is given by independent displacements of the lattice points we determine the long time asymptotics of the logarithm of the survival probability up to a multiplicative constant as applications we show the lifshitz tail effect of the density of states of the associated random schrodinger operator and derive a quantitative estimate for the strength of intermittency in the parabolic anderson problem

existence and quasilinearization methods in hilbert spaces abstract in this paper we discuss some existence results and the application of quasilinearization methods developed so far for differential equations to the solution of the abstract problem l u f u in a hilbert space h under fairly general assumptions on l f and h we show that this problem has a solution that can be obtained as the limit of a quadratically convergent nondecreasing sequence of approximate solutions if the assumptions are strengthened we show that the abstract problem has a solution which is quadratically bracketed between two monotone sequences of approximate solutions of certain related linear equations

replacing the finite difference methods for nonlinear two point boundary value problems by successive application of the linear shooting method abstract this paper studies numerical solution of nonlinear two point boundary value problems for second order ordinary differential equations first it establishes a connection between the finite difference method and the quasi linearization method we prove that using finite differences to discretize the sequence of linear differential equations arising from quasi linearization newton method on operator level leads to the usual iteration formula of the newton finite difference method from the provided derivation it can easily be inferred that such a relation holds also for the picard and the constant slope methods based on this result we propose a way of replacing the newton picard and constant slope finite difference methods by respective successive application of the linear shooting method this approach has a number of advantages it removes the necessity of solving systems of algebraic equations hence working with matrices altogether compared to the usual finite difference method with general solver it reduces the number of computational operations from o n 3 where n is the number of mesh points to only o n

minimizing the ground state energy of an electron in a randomly deformed lattice we provide a characterization of the spectral minimum for a random schrodinger operator of the form h delta sum i in mathbb z d q x i omega i in l 2 mathbb r d where the single site potential q is reflection symmetric compactly supported in the unit cube centered at 0 and the displacement parameters i are restricted so that adjacent single site potentials do not overlap in particular we show that a minimizing configuration of the displacements is given by a periodic pattern of densest possible 2 d clusters of single site potentials

the image of j in the ehp sequence the ehp sequence is based on a result of james j who showed that there is h a map h such that s q sn s2n 1 is a fibration when localized at 2 the map sn sn is usually labeled e the boundary homomorphism in the homotopy sequence is usually labeled p for our purposes it will be most convenient to combine all the ehp sequences into one system this gives the following filtration of o201 s q s

a generalized quasilinearization method for second order nonlinear differential equations with nonlinear boundary conditions we consider the differential equation l u f u where l is a formally self adjoint second order differential expression and f is nonlinear with nonlinear boundary conditions under appropriate assumptions on l f and the boundary conditions existence of solutions is established using the method of lower and upper solutions a generalized quasilinearization method is then developed for this problem and we obtain two monotonic sequences of approximate solutions converging quadratically to a solution of the equation

moment asymptotics for the parabolic anderson problem with a perturbed lattice potential abstract the parabolic anderson problem with a random potential obtained by attaching a long tailed potential around a randomly perturbed lattice is studied the moment asymptotics of the total mass of the solution is derived the results show that the total mass of the solution concentrates on a small set in the space of configuration

solving boundary value problems of ordinary differential equations with non separated boundary conditions abstract this paper focuses on solving the two point boundary value problem in which boundary conditions are systems of nonlinear equations the shooting method was used together with a combination of newton s method and broyden s method to update the initial values of the differential equations the experiments showed that the proposed method performed well in the sense that the overall amount of work was less than that of the newton shooting method

constructive proof for existence of nonlinear two point boundary value problems owing to the importance of differential equations in physics the existence of solutions for differential equations has been paid much attention in this paper the existence of solution are obtained for the nonlinear second order two point boundary value problem in the reproducing kernel space under certain assumptions on right hand side we propose constructive proof for the existence result and a method is presented to obtain the exact solution expressed by the form of series this paper is a extension of previous paper wei jiang minggen cui the exact solution and stability analysis for integral equation of third or first kind with singular kernel appl math comput 202 2 2008 666 674 which extends a method of solving linear problems to present method for solving nonlinear problems

almost sure asymptotics for the continuous parabolic anderson model we consider the parabolic anderson problem t u u x u on d with initial condition u 0 x 1 here 0 is a diffusion constant and is a random homogeneous potential we concentrate on the two important cases of a gaussian potential and a shot noise poisson potential under some mild regularity assumptions we derive the second order term of the almost sure asymptotics of u t 0 as t