characteristic cycles of perverse sheaves and milnor fibers we show that the milnor monodromy and the milnor numbers naturally appear in the characteristic cycles of irreducible perverse sheaves having the singularities of a complex hypersurface we find also a new numerical constraint on the milnor monodromy for general hypersurface singularities

half linear differential equations linearization technique and its application abstract oscillatory properties of the half linear second order differential equation r t x c t x 0 x x p 2 x p 1 are investigated where is viewed as a perturbation of an equation of the same form and these properties are compared with the oscillatory behavior of a certain associated linear second order differential equation as a corollary a conjecture posed by the first author in the paper o dosl perturbations of the half linear euler weber differential equations j math anal appl 323 2006 426 440 is proved and some new research problems are formulated

on the variety of a highest weight module 1 2 let u g be the enveloping algebra of g detailed information on the primitive spectrum prim u g of u g was summarized by conjecture 7 4 of 181 here a main question was to relate prim u g with the springer correspondence parts i ii of this conjecture were established in i ii and a partial solution to iii was given in 21 sufficient e g to treat g of type a meanwhile borho and brylinski 5 have established iii for induced ideals while barbasch and vogan 1 2 have established a version of iii in which gr i i e prim u g is replaced by a certain wavefront set known to be contained in y gr 1 this last work involves some case by case analysis and gives little indication as to why the springer correspondence should arise in the study of prim u g

a collection of problems for which gaussian elimination with partial pivoting is unstable a significant collection of two point boundary value problems is shown to give rise to linear systems of algebraic equations on which gaussian elimination with row partial pivoting is unstable when standard solution techniques are used

on perturbations of quasiperiodic schrodinger operators abstract using relative oscillation theory and the reducibility result of eliasson we study perturbations of quasiperiodic schrodinger operators in particular we derive relative oscillation criteria and eigenvalue asymptotics for critical potentials

oscillation constants for second order ordinary differential equations related to elliptic equations with p laplacian abstract in this paper we consider the second order nonlinear differential equation t 1 x t 1 p f x 0 x x p 2 x p 1 r with f satisfying x f x 0 x 0 we analyze the difference between the cases p p and p in each case we give a condition on the function f which guarantees that solutions of eq are non oscillatory the principal methods used in this paper are the riccati technique and its modifications the results of our paper complement and extend several previously obtained results on the subject

characteristic varieties and vanishing cycles on presente une etude systematique des varietes caracteristiques des systemes holonomes a singularites regulieres ce sont des sous varietes lagrangiennes coniques dans le fibre cotangent t x

differential operators on homogeneous spaces iii characteristic varieties of harish chandra modules and of primitive ideals letg be a semi simple complex algebraic group with lie algebra and flag varietyx g b for each primitive idealj with trivial central character in the enveloping algebrau we define a characteristic variety in the cotangent bundle ofx which projects under the springer resolution mapt x onto the closure of a nilpotent orbit we prove that this characteristic variety is theg saturation of the characteristic variety of a highest weight module with annihilatorj we conjecture that it is irreducible forg sl n our conjecture would provide a geometrical explanation for the classification of primitive ideals in terms of weyl group representations as achieved by a joseph the presentation of these ideas here is simultaneously used to some extent as an opportunity to continue our more general systematic discussion of differential operators on a complete homogeneous space and to study more generally characteristic varieties of harish chandra modules

half linear equations and characteristic properties of the principal solution abstract the characteristic properties of the principal solution for half linear differential equation a t x b t x 0 where the functions a b are positive and continuous for t 0 and u u p 2u p 1 are investigated in the linear case it is well known that the principal solution is the smallest one in a neighbourhood of infinity we show that this property continues to hold in the half linear case in addition it is proved that the principal solutions can be fully characterized by means of two different integral criteria which reduce to that one well known in the linear case

perturbations of the half linear euler weber type differential equation abstract we investigate oscillatory properties of the half linear second order differential equation r t x c t x 0 x x p 2 x p 1 viewed as a perturbation of another half linear differential equation of the same form r t x c t x 0 the obtained oscillation and nonoscillation criteria are formulated in terms of the integral c t c t h p t d t where h is a function which is close to the principal solution of in a certain sense a typical model of in applications is the half linear euler weber differential equation with the critical coefficients x p t p p t p log 2 t x 0 p p 1 p p p 1 2 p 1 p p 1 we establish oscillation and nonoscillation criteria for perturbations of this equation some open problems and perspectives of the further research along this line are also formulated

effective prufer angles and relative oscillation criteria abstract we present a streamlined approach to relative oscillation criteria based on effective prufer angles adapted to the use at the edges of the essential spectrum based on this we provided a new scale of oscillation criteria for general sturm liouville operators which answer the question whether a perturbation inserts a finite or an infinite number of eigenvalues into an essential spectral gap as a special case we recover and generalize the gesztesy unal criterion which works below the spectrum and contains classical criteria by kneser hartman hille and weber and the well known results by rofe beketov including the extensions by schmidt

relative oscillation theory for sturm liouville operators extended we extend relative oscillation theory to the case of sturm liouville operators hu r 1 pu qu with different p s we show that the weighted number of zeros of wronskians of certain solutions equals the value of krein s spectral shift function inside essential spectral gaps

nonoscillatory half linear differential equations and generalized karamata functions abstract we introduce a natural generalization of the concept of regularly varying functions in the sense of karamata and show that the class of generalized karamata functions is a well suited framework for the study of the asymptotic behavior of nonoscillatory solutions of the half linear differential equation of the type a p t y 1 y q t y 1 y 0

growth conditions for uniform asymptotic stability of damped oscillators abstract the present paper is devoted to an investigation on the uniform asymptotic stability for the linear differential equation with a damping term x h t x 2 x 0 and its generalization p x h t p x p p x 0 where 0 and p z z p 2 z with p 1 sufficient conditions are obtained for the equilibrium x x 0 0 to be uniformly asymptotically stable under the assumption that the damping coefficient h t is integrally positive the obtained condition for the damped linear differential equation is given by the form of a certain uniform growth condition on h t another representation which is equivalent to this uniform growth condition is also given our results assert that the equilibrium can be uniformly asymptotically stable even if h t is unbounded an example is attached to show this fact in addition easy to use conditions are given to guarantee that the uniform growth condition is satisfied moreover a sufficient condition expressed by an infinite series is presented the relation between the representation of an infinite series and the uniform growth condition is also clarified finally our results are extended to be able to apply to the above mentioned nonlinear differential equation

the transformation operator for schrodinger operators on almost periodic infinite gap backgrounds abstract we investigate the kernels of the transformation operators for one dimensional schrodinger operators with potentials which are asymptotically close to bohr almost periodic infinite gap potentials

asymptotic properties of kneser solutions to nonlinear second order odes with regularly varying coefficients in this work we investigate properties of a class of solutions to the second order ode p t u t q t f u t 0 on the interval a a 0 where p and q are functions regularly varying at infinity and f satisfies f l 0 f 0 f l 0 with l0 0

growth conditions for oscillation of nonlinear differential equations with p laplacian abstract in this paper oscillation criteria are established for all solutions of second order nonlinear differential equations of the form p x 1 t p g x 0 t 0 here p y is the one dimensional p laplacian operator and g x satisfies the signum condition x g x 0 if x 0 but is not assumed to be monotone the equation naturally includes the famous euler differential equation and half linear differential equations the main purpose is to examine the influence of certain growth conditions of the nonlinear term g x on the oscillation of solutions the conditions are shown to be sharp some of differential inequalities play important roles to prove our results a simple example is included to illustrate the main result a conjecture on the inverse problem is also given finally elliptic equations with p laplacian operator are discussed as an application to our results

power comparison theorems in half linear oscillation theory we compare oscillatory behavior of two half linear differential equations with different powers by the homogeneous functions appearing in these equations our results generalize and extend several previously proved power comparison theorems

d module structure of local cohomology modules of toric algebras let s be a toric algebra over a field k of characteristic 0 and let i be a monomial ideal of s we show that the local cohomology modules h i i s are of finite length over the ring of differential operators d s k generalizing the classical case of a polynomial algebra s as an application we compute the characteristic cycles of some local cohomology modules

hamiltonian reduction and nearby cycles for mirabolic d modules abstract we study holonomic d modules on sl n c c n called mirabolic modules analogous to lusztig s character sheaves we describe the supports of simple mirabolic modules we show that a mirabolic module is killed by the functor of hamiltonian reduction from the category of mirabolic modules to the category of representations of the trigonometric cherednik algebra if and only if the characteristic variety of the module is contained in the unstable locus we introduce an analogue of verdier s specialization functor for representations of cherednik algebras which agrees on category o with the restriction functor of bezrukavnikov and etingof in type a we also consider a verdier specialization functor on mirabolic d modules we show that hamiltonian reduction intertwines specialization functors on mirabolic d modules with the corresponding functors on representations of the cherednik algebra this allows us to apply known hodge theoretic purity results for nearby cycles in the setting considered by bezrukavnikov and etingof

existence of regularly varying solutions with nonzero indices of half linear differential equations with retarded arguments sharp conditions are established for the existence of a pair of regularly varying solutions with nonzero indices of half linear functional differential equations of the type x t asgnx t q t x g t asgnx g t a 0 q t 0 g t

global asymptotic stability for damped half linear oscillators abstract a necessary and sufficient condition is established for the equilibrium of the oscillator of half linear type with a damping term p x h t p x p x 0 to be globally asymptotically stable the obtained criterion is given by the form of a certain growth condition of the damping coefficient h t and it can be applied to not only the cases of large damping and small damping but also the case of fluctuating damping the presented result is new even in the linear cases p 2 it is also discussed whether a solution of the half linear differential equation r t p x c t p x 0 that converges to a non zero value exists or not some suitable examples are included to illustrate the results in the present paper

relative oscillation theory for dirac operators we develop relative oscillation theory for one dimensional dirac operators which rather than measuring the spectrum of one single operator measures the difference between the spectra of two different operators this is done by replacing zeros of solutions of one operator by weighted zeros of wronskians of solutions of two different operators in particular we show that a sturm type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the number of eigenvalues in essential spectral gaps furthermore the connection with krein s spectral shift function is established as an application we extend a result by k m schmidt on the finiteness infiniteness of the number of eigenvalues in essential spectral gaps of perturbed periodic dirac operators

hypertoric category o abstract we study the representation theory of the invariant subalgebra of the weyl algebra under a torus action which we call a hypertoric enveloping algebra we define an analogue of bgg category o for this algebra and identify it with a certain category of sheaves on a hypertoric variety we prove that a regular block of this category is highest weight and koszul identify its koszul dual compute its center and study its cell structure we also consider a collection of derived auto equivalences analogous to the shuffling and twisting functors for bgg category o

lagrangian construction for representations of hecke algebras 1 let k be a non archimedean local field with the residue field consisting of q elements according to langlands philosophy irreducible complex representations of a reductive k group g should roughly speaking correspond to homomorphisms of the weil deligne group into the socalled l group let zc g be the iwahori subgroup the category of admissible gkmodules generated by z fixed vectors is known to be equivalent to the category of finite dimensional representations of the hecke algebra h z g z langlands philosophy suggests via that equivalence the following classification of irreducible hecke algebra modules stated in purely complex terms and not involving any local held

multiplier ideals v filtrations and transversal sections we show that the restriction to a smooth transversal section commutes to the computation of multiplier ideals and v filtrations as an application we prove the constancy of the jumping numbers and the spectrum along any stratum of a whitney regular stratification

extensions of modules over weyl algebras in this paper we calculate some ext groups of singular modules over the complex weyl algebra an in particular we determine conditions under which ext is an infinite dimensional vector space when n 2 or 3

asymptotic behavior of positive solutions of sublinear differential equations of emden fowler type this paper is concerned with asymptotic analysis of positive solutions of the second order nonlinear differential equation a x t q t f x t 0 where q a 0 is a continuous function which is regularly varying and f 0 0 is a continuous increasing function which is regularly varying of index c 0 1 an application of the theory of regular variation gives the possibility of determining precise information about the asymptotic behavior at infinity of intermediate solutions of eq a

global existence and nonexistence of solutions for second order nonlinear differential equations abstract this paper deals with the global existence and nonexistence of solutions of the second order nonlinear differential equation x x 0 satisfying x 0 x 0 and x 0 x 1 where is a positive parameter and with 0 and 0 is strictly increasing odd bijective and continuous on necessary and sufficient conditions are obtained for the initial value problem to have a unique global solution which is oscillatory and periodic examples are given to illustrate our main result finally a nonexistence result for the equation with a damping term is discussed as an application to our result

an implementation of the look ahead lanczos algorithm for non hermitian matrices the nonsymmetric lanczos method can be used to compute eigenvalues of large sparse non hermitian matrices or to solve large sparse non hermitian linear systems however the original lanczos algorithm is susceptible to possible breakdowns and potential instabilities an implementation of a look ahead version of the lanczos algorithm is presented that except for the very special situation of an incurable breakdown overcomes these problems by skipping over those steps in which a breakdown or near breakdown would occur in the standard process the proposed algorithm can handle look ahead steps of any length and requires the same number of matrix vector products and inner products as the standard lanczos process without look ahead

asymptotic formulae for solutions of half linear differential equations we establish asymptotic formulae for regularly varying solutions of the half linear differential equation r t y 1 sgn y p t y 1 sgn y where r p are positive continuous functions on a and 1 the results can be understood in several ways some open problems posed in the literature are solved results for linear differential equations are generalized some of the observations are new even in the linear case a refinement on information about behavior of solutions in standard asymptotic classes is provided a precise description of regularly varying solutions which are known to exist is given regular variation of all positive solutions is proved

oscillation and non oscillation of euler type half linear differential equations abstract we investigate oscillatory properties of second order euler type half linear differential equations whose coefficients are given by periodic functions and functions having mean values we prove the conditional oscillation of these equations in addition we prove that the known oscillation constants for the corresponding equations with only periodic coefficients do not change in the studied more general case the presented results are new for linear equations as well

asymptotic formulas for non oscillatory solutions of perturbed half linear euler equation abstract we establish asymptotic formulas for non oscillatory solutions of the half linear second order differential equation x t p x c t x 0 where this equation is viewed as a perturbation of the half linear euler equation

eigenvalue excluding for perturbed periodic one dimensional schrodinger operators subject of investigation in this paper is a one dimensional schrodinger equation where the potential is a sum of a periodic function and a perturbation decaying at graphic 1 it is well known that the essential spectrum consists of spectral bands and that there may or may not be additional eigenvalues below the lowest band or in the gaps between the bands while enclosures for gap eigenvalues can comparatively easily be obtained from numerical approximations e g by d weinstein s bounds there seems to be no method available so far which is able to exclude eigenvalues in spectral gaps i e which identifies subregions of a gap which contain no eigenvalues here we propose such a method it makes heavy use of computer assistance nevertheless the results are completely rigorous in the strict mathematical sense because all computational errors are taken into account 1 embed inline graphic 1 gif

oscillation constants for euler type differential equations involving the p t laplacian abstract this paper deals with the oscillation problem for the nonlinear differential equation x p t 2 x t p t x p t 2 x 0 where is a positive parameter and p t 1 is a nondecreasing and smooth function some non oscillation criteria are given for this equation the obtained results are the best possible in a certain sense we use the riccati technique and the function sequence technique moreover we propose a conjecture concerning the oscillation problem for this equation in the case when p t tends to infinity as t

modified prufer angle and conditional oscillation of perturbed linear and half linear differential equations abstract the research and results described in this paper belong to the qualitative theory of differential equations more precisely the partial differential equations with the one dimensional p laplacian using a method whose core is formed by the prufer technique we identify a borderline case between oscillatory and non oscillatory equations moreover we are able to decide whether the studied equations are oscillatory or not even in the so called critical i e the borderline case the advantage of our approach is the fact that we obtain new and strong results for linear and half linear equations i e the equations with the one dimensional p laplacian at the same time in addition we are able to work with equations whose coefficients are non constant and non periodic the novelty of our results is documented by examples and corollaries

half linear oscillation criteria perturbation in term involving derivative abstract we consider the nonoscillatory half linear differential equation r t x c t x 0 x x p 2 x p 1 and we study oscillatory properties of its perturbation r t r t x c t c t x 0 we use the riccati technique and relationship between and a certain associated linear equation the results are applied to the perturbed euler type equation

convergence of solutions of nonlinear systems with integrable forcing term and its applications to a biological model a non autonomous nonlinear system with a time variational forcing term is considered for such a system the global attraction of the origin is discussed whose result is suggestive to an ecological problem as well because the system is reduced to a lotka volterra predator prey model with prey receiving an environmental time variation by an appropriate transformation a numerical simulation is also shown to suggest a new mathematical insight beyond our intuition

global attractivity for half linear differential systems with periodic coefficients abstract the system to be considered in this paper is x y a t x y b t p x p y here a t is a 2 2 diagonal matrix and b t is a 2 2 anti diagonal matrix and q z z q 2 z with q 1 the coefficients of b t are assumed to be periodic but the coefficients of a t are not necessarily periodic the system is of half linear type sufficient conditions are given for the zero solution of the half linear system to be globally asymptotically stable the zero solution of the system x y b t p x p y is stable but is not attractive our concern is to clarify a positive effect of the diagonal part a t x y on the global asymptotic stability for the half linear system some simple examples are included to illustrate the main result

oscillation and non oscillation criteria for linear and half linear difference equations abstract we find very general classes of linear and half linear difference equations which are conditionally oscillatory we identify the critical oscillation constant whose value implies the oscillation or non oscillation of studied equations our results are divided into oscillatory and non oscillatory theorems which determine the critical oscillation constant for coefficients given by sequences having mean values in addition our approach enables to analyse also the oscillatory properties of equations whose coefficients are not given by sequences with mean values we point out that the obtained results are new even for linear equations with periodic coefficients such consequences are formulated at the end of this paper

a nonoscillation theorem for half linear differential equations with periodic coefficients abstract the half linear differential equation p x a t p x b t p x 0 is considered under the assumption that the coefficient a t and an indefinite integral b t of b t are periodic functions with period t 0 it is proved that p 1 p b t a t b t 0 0 t t is sufficient for all nontrivial solutions to be nonoscillatory here p 1 and q y y q 2 y for q p or q p p p 1 the proof is given by means of riccati technique the condition is shown to be sharp sufficient conditions are also presented for all nontrivial solutions are oscillatory in the linear case p 2 some examples and simulations are included to illustrate our results

critical coupling constants and eigenvalue asymptotics of perturbed periodic sturm liouville operators perturbations of asymptotic decay c r 2 arise in the partial wave analysis of rotationally symmetric partial differential operators we show that for each end point 0 of the spectral bands of a perturbed periodic sturm liouville operator there is a critical coupling constant c crit such that eigenvalues in the spectral gap accumulate at 0 if and only if c c crit 1 the oscillation theoretic method used in the proof also yields the asymptotic distribution of the eigenvalues near 0

positive solutions of fourth order emden fowler type differential equations in the framework of regular variation abstract the fourth order nonlinear differential equations a x 4 q t x sgn x 0 0 1 with regularly varying coefficient q t are studied in the framework of regular variation it is shown that thorough and complete information can be acquired about the existence of all possible regularly varying solutions of a and their accurate asymptotic behavior at infinity

variational technique and principal solution in half linear oscillation criteria we investigate oscillatory properties of half linear differential equations the investigated half linear equation is viewed as a perturbation of another nonoscillatory equation of the same form and perturbation is allowed both in the absolute term and in the derivative term first the sturmian type theorem for solutions of the associated riccati equation is established in the second part of the paper we prove new half linear oscillation criteria using the variational technique

convergence and instability in pcg methods for bordered systems abstract bordered almost block diagonal systems arise from discretizing a linearized first order system of n ordinary differential equations in a two point boundary value problem with nonseparated boundary conditions the discretization may use spline collocation finite differences or multiple shooting after internal condensation if necessary the bordered almost block diagonal system reduces to a standard finite difference structure which can be solved using a preconditioned conjugate gradient method based on a simple matrix splitting technique this preconditioned conjugate gradient method is guaranteed to converge in at most 2n 1 iterations we exhibit a significant collection of two point boundary value problems for which this preconditioned conjugate gradient method is unstable and hence convergence is not achieved

characteristic cycles for the loop grassmannian and nilpotent orbits x which is a linear combination of closures of conormal bundles to submanifolds of x intuitively the microlocal multiplicities c me asurethesingularity of at in settings related to representation theory a group g acts on x is g equivariant and the microlocal multiplicities play a significant but only partially understood role in representation theory see ro sv abv and kasa e g in this paper we compute microlocal multiplicities for certain cases of interest in representation theory let g be a connected reductive group with loop group lg and let p bethesubgroup of lg consisting of loops with positive fourier coefficients then p orbits on theloop grassmannian lg p correspond to the irreducible representations l of thedual re ductivegroup g for dominant weights and of a torus of

renormalized oscillation theory for hamiltonian systems abstract we extend a result on renormalized oscillation theory originally derived for sturm liouville and dirac type operators on arbitrary intervals in the context of scalar coefficients to the case of general hamiltonian systems with block matrix coefficients in particular this contains the cases of general sturm liouville and dirac type operators with block matrix valued coefficients as special cases the principal feature of these renormalized oscillation theory results consists in the fact that by replacing solutions by appropriate wronskians of solutions oscillation theory now applies to intervals in essential spectral gaps where traditional oscillation theory typically fails

de haan type increasing solutions of half linear differential equations abstract we study asymptotic behavior of eventually positive increasing solutions to the half linear equation r t y 1 sgn y p t y 1 sgn y where r p are positive continuous functions and 1 we give conditions which guarantee that any such a solution is in the class in the de haan sense we also discuss regularly varying solutions and connections with a generalized regular variation and other related concepts the results can be viewed as a half linear extension of existing statements for linear equations but in some aspects they are new also in the linear case

a necessary and sufficient condition for global asymptotic stability of time varying lotka volterra predator prey systems abstract the purpose of this paper is to present a necessary and sufficient condition which guarantees that an interior equilibrium of a certain predator prey system is globally asymptotically stable this ecological system is a model of lotka volterra type whose prey population receives time variation of the environment we assume that the time varying coefficient is weakly integrally positive and has a weaker property than uniformly continuous our necessary and sufficient condition is expressed by an improper double integral on the time varying coefficient our work is inspired by the study of the stability theory for damped linear oscillators

the characteristic cycle and the singular support of a constructible sheaf we define the characteristic cycle of an etale sheaf as a cycle on the cotangent bundle of a smooth variety in positive characteristic using the singular support recently defined by beilinson we prove a formula a la milnor for the total dimension of the space of vanishing cycles and an index formula computing the euler poincare characteristic generalizing the grothendieck ogg shafarevich formula to higher dimension an essential ingredient of the construction and the proof is a partial generalization to higher dimension of the semi continuity of the swan conductor due to deligne laumon we prove the index formula by establishing certain functorial properties of characteristic cycles

half linear ode and modified riccati equation comparison theorems integral characterization of principal solution abstract in this paper we study the half linear differential equation with one dimensional p laplacian r t p x c t p x 0 where p x x p 2 x and p 1 using a suitable modification of the so called linearization technique we derive new results which allow to compare the solutions of two equations with different p and provide new integral characterization of the principal solution

equivariant quantum cohomology of cotangent bundle of g p abstract let g denote a complex semisimple linear algebraic group p a parabolic subgroup of g and p g p we identify the quantum multiplication by divisors in t p in terms of stable basis which is introduced in 9 using this and the restriction formula for stable basis 17 we show that the g c equivariant quantum multiplication formula in t p is conjugate to the formula conjectured by braverman

relative oscillation theory weighted zeros of the wronskian and the spectral shift function we develop an analog of classical oscillation theory for sturm liouville operators which rather than measuring the spectrum of one single operator measures the difference between the spectra of two different operators

nonprincipal solutions of half linear second order differential equations abstract we consider the nonoscillatory half linear second order differential equation r t x c t x 0 x x p 2 x p 1 and we suppose that we know its principal solution x we establish an asymptotic formula for nonprincipal solutions of and we use this asymptotic formula to obtain an oscillation criterion for

a taxonomy for conjugate gradient methods the conjugate gradient method of hestenes and stiefel is an effective method for solving large sparse hermitian positive definite hpd systems of linear equations ax b generalizations to non hpd matrices have long been sought the recent theory of faber and manteuffel gives necessary and sufficient conditions for the existence of a cg method this paper uses these conditions to develop and organize such methods it is shown that any cg method for ax b is characterized by an hpd inner product matrix b and a left preconditioning matrix c at each step the method minimizes the b norm of the error over a krylov subspace this characterization is then used to classify known and new methods finally it is shown how eigenvalue estimates may be obtained from the iteration parameters generalizing the well known connection between cg and lanczos such estimates allow implementation of a stopping criterion based more nearly on the true error

uniform global asymptotic stability for oscillators with superlinear damping abstract the present paper deals with the damped superlinear oscillator x h t q x 2 x 0 where 0 and q z z q 2 z with q 2 the origin x x 0 0 is the only equilibrium of this oscillator we herein establish a sufficient condition for the equilibrium to be uniformly globally asymptotically stable we conclude that under the assumption that the damping coefficient h t is integrally positive if the integral from to t of a particular solution of the first order nonlinear differential equation u h t q u 1 0 diverges to negative infinity uniformly with respect to then the equilibrium is uniformly globally asymptotically stable the above mentioned result is expressed by an implicit condition we examine when the implicit condition is satisfied and when it is not satisfied we also give explicit sufficient conditions which assure that the equilibrium is uniformly globally asymptotically stable using the obtained result we present an example of which the equilibrium is uniformly globally asymptotically stable even if h t is unbounded

scattering theory for schrodinger operators on steplike almost periodic infinite gap backgrounds abstract we develop direct scattering theory for one dimensional schrodinger operators with steplike potentials which are asymptotically close to different bohr almost periodic infinite gap potentials on different half axes

generalized prufer variables for perturbations of jacobi and cmv matrices abstract prufer variables are a standard tool in spectral theory developed originally for perturbations of the free schrodinger operator we adapt the generalized prufer variables introduced by kiselev remling simon for perturbations of an arbitrary schrodinger operator to the setting of jacobi and szeg recursions we present an application to random l 2 perturbations of jacobi and cmv matrices and an application to decaying oscillatory perturbations of periodic jacobi and cmv matrices

a little microlocal morse theory if a complex analytic function f has a stratified isolated critical point then it is known that the cohomology of the milnor fibre of f has a direct sum decomposition in terms of the normal morse data to the strata we use microlocal morse theory to obtain the same result under the weakened hypothesis that the vanishing cycles along f have isolated support we also investigate an index theoretic proof of this fact

on irreducible representations of compact p adic analytic groups we prove that the canonical dimension of a coadmissible representation of a semisimple p adic lie group in a p adic banach space is either zero or at least half the dimension of a nonzero coadjoint orbit to do this we establish analogues for p adically completed enveloping algebras of bernstein s inequality for modules over weyl algebras the beilinson bernstein localisation theorem and quillen s lemma about the endomorphism ring of a simple module over an enveloping algebra