the limit circle and limit point criteria for second order linear difference equations abstract this paper is concerned with the limit circle and limit point criteria of second order linear difference equations a sufficient and necessary condition a sufficient and necessary condition subject to a certain restriction and several sufficient conditions are established these results improve and extend some previous results

on periodic matrix valued weyl titchmarsh functions abstract we consider a certain class of herglotz nevanlinna matrix valued functions which can be realized as the weyl titchmarsh matrix valued function of some symmetric operator and its self adjoint extension new properties of weyl titchmarsh matrix valued functions as well as a new version of the functional model for such realizations are presented in the case of periodic herglotz nevanlinna matrix valued functions we provide a complete characterization of their realizations in terms of the corresponding functional model we also obtain properties of a symmetric operator and its self adjoint extension which generate a periodic weyl titchmarsh matrix valued function we study pairs of operators a symmetric operator and its self adjoint extension with constant weyl titchmarsh matrix valued functions and establish connections between such pairs of operators and representations of the canonical commutation relations for unitary groups of operators in weyl s form as a consequence of such an approach we obtain the stone von neumann theorem for two unitary groups of operators satisfying the commutation relations as well as some extension and refinement of the classical functional model for generators of those groups our examples include multiplication operators in weighted spaces first and second order differential operators as well as the schrodinger operator with linear potential and its perturbation by bounded periodic potential

a new approach to inverse spectral theory i fundamental formalism we present a new approach distinct from gelfand levitan to the the orem of borg marchenko that the m function equivalently spectral measure for a finite interval or half line schrodinger operator determines the poten tial our approach is an analog of the continued fraction approach for the moment problem we prove there is a representation for the m function m 2 r b

embeddings of spherical homogeneous spaces in characteristic p let g be a reductive group over an algebraically closed field of characteristic p 0 we study properties of embeddings of spherical homogeneous g spaces we look at frobenius splittings canonical or by a p 1 th power compatible with certain subvarieties we show the existence of rational g equivariant resolutions by toroidal embeddings and give results about cohomology vanishing and surjectivity of restriction maps of global sections of line bundles we show that the class of homogeneous spaces for which our results hold contains the symmetric homogeneous spaces in characteristic ne 2 and is closed under parabolic induction

titchmarsh sims weyl theory for complex hamiltonian systems on sturmian time scales abstract we study non self adjoint hamiltonian systems on sturmian time scales defining weyl sims sets which replace the classical weyl circles and a matrix valued m function on suitable cone shaped domains in the complex plane furthermore we characterize realizations of the corresponding dynamic operator and its adjoint and construct their resolvents even order scalar equations and the orr sommerfeld equation on time scales are given as examples illustrating the theory which are new even for difference equations these results unify previous discrete and continuous theories to dynamic equations on sturmian time scales

on the titchmarsh weyl m λ coefficient and spectral density for a dirac system for a dirac system on the line we study the jost solutions and the existence of half bound states a characterisation of this phenomenon is given in terms of the m function which supplements a well known spectral characterisation for the system as a corollary we deduce the spectral density behaviour at the spectral gap endpoints which is shown to depend on the possible existence of half bound states

weyl titchmarsh theory for symplectic difference systems in this work we establish weyl titchmarsh theory for symplectic difference systems this paper extends classical weyl titchmarsh theory and provides a foundation for studying spectral theory of symplectic difference systems

on the absolutely continuous spectrum of a vector matrix dirac system a dirac system is considered which has a matrix valued long range short range and oscillatory potentials the system has one singular endpoint at infinity additional conditions on the potential are given which guarantee particular asymptotic behaviour of an energy functional associated with a certain set of solutions this asymptotic behaviour guarantees the existence of a purely absolutely continuous spectrum outside a gap containing the origin

singular hamiltonian system with several spectral parameters ii odd order case abstract in this paper we deal with a singular hamiltonian system of odd order with several spectral parameters and we investigate the behavior of the solution of this system at singular point with the aid of the characteristic function theory moreover some results have been introduced for the weyl titchmarsh function for some special hamiltonian systems of odd order with several spectral parameters

the spectrum of differential operators with almost constant coefficients ii the absolutely continuous spectrum of differential operators of the form ly w 1 k 0n 1 k pky k k on l2 0 w is determined with pn x w x 0 the coefficients pk are assumed to satisfy pk x pk 2kw 1 x ck w pn 1 1 2n if the coefficients satisfy some additional smoothness and decay conditions the absolutely continuous part hac of any self adjoint extension of l is unitarily equivalent to the operator of multiplication by p x 0n ckx2k on l2 0 several extensions of this result as well as examples are shown

a proof of the local borg marchenko theorem in 2 5 a local version of a basic uniqueness theorem in inverse spectral theory is given this paper gives a simple proof of this theorem

self adjoint extensions for linear hamiltonian systems with two singular endpoints abstract this paper is concerned with self adjoint extensions for a linear hamiltonian system with two singular endpoints the domain of the closure of the corresponding minimal hamiltonian operator h 0 is described by properties of its elements at the endpoints of the discussed interval decompositions of the domains of the corresponding left and right maximal hamiltonian operators are provided and expressions of the defect indices of h 0 in terms of those of the left and right minimal operators are given based on them characterizations of all the self adjoint extensions for a hamiltonian system are obtained in terms of square integrable solutions as a consequence the characterizations of all the self adjoint extensions are given for systems in several special cases

uber bewertungen welche unter einer reduktiven gruppe invariant sind letg be a reductive group defined over an algebraically closed fieldk and letx be ag variety in this paper we studyg invariant valuationsv of the fieldk of rational functions onx these objects are fundamental for the theory of equivariant completions ofx letb be a borel subgroup andu the unipotent radical ofb it is proved thatv is uniquely determined by its restriction toku then we study the set of invariant valuations having some fixed restrictionv0 tokb ifv0 is geometric i e induced by a prime divisor then this set is a polyhedron in some vector space in characteristic zero we prove that this polyhedron is a simplicial cone and in fact the fundamental domain of finite reflection groupwx thus the classification of invariant valuations is almost reduced to the classification of valuations ofkb

borg type theorems for matrix valued schrodinger operators abstract a borg type uniqueness theorem for matrix valued schrodinger operators is proved more precisely assuming a reflectionless potential matrix and its spectrum a half line 0 we derive the triviality of the potential matrix our approach is based on trace formulas and matrix valued herglotz representation theorems as a by product of our techniques we obtain an extension of borg s classical result from the class of periodic scalar potentials to the class of reflectionless matrix valued potentials

a stochastic calculus model of continuous trading optimal portfolios the problem of choosing a portfolio of securities so as to maximize the expected utility of wealth at a terminal planning horizon is solved via stochastic calculus and convex analysis this problem is decomposed into two subproblems with security prices modeled as semimartingales and trading strategies modeled as predictable processes the set of terminal wealths is identified as a subspace in a space of integrable random variables the first subproblem is to find the terminal wealth that maximizes expected utility convex analysis is used to derive necessary and sufficient conditions for optimality and an existence result the second subproblem of finding the admissible trading strategy that generates the optimal terminal wealth is a martingale representation problem the primary advantage of this approach is that explicit formulas can readily be derived for the optimal terminal wealth and the corresponding expected utility as is shown for the case of an exponential utility function and a risky security modeled as geometric brownian motion

frobenius splitting and cohomology vanishing for schubert varieties let k be an algebraically closed field of characteristic p 0 and x be a projective variety over k we then have the absolute frobenius f x x and an injection ox f 6x given by f fp f e o9x if this makes ox a direct summand in f ox as an x module we call x a frobenius split variety for such a variety the vanishing theorem for ample line bundles follows trivially from serre s vanishing theorem for tensoring ox f ox by an ample line bundle l and noting that l f x f f l f lp projection formula we get that the map in cohomology h x l h x f lp h x lp is an injection iterating this we see that h x l injects into h x lp for every p but for large v the latter is zero thus frobenius split varieties have quite pleasant properties it also turns out that using duality for the frobenius morphism or equivalently the cartier operator one can give a manageable criterion for frobenius splitting the point here is the local nature of duality the relevance of the compatibility of local and global duality was suggested to us by grothendieck s proof of h x y 0 for a noncomplete variety x of dimension n 4 theorem 6 9 and by kempf s paper 10 by this criterion and the bott samelson demazure desingularisation of schubert varieties it follows very easily that schubert varieties in characteristic p are frobenius split the following vanishing theorem is then an immediate consequence let g be a reductive group over the field k of arbitrary characteristic zero or positive let q be a parabolic subgroup and x c g q a schubert variety let l be an ample line bundle on g q then hi x l 0 for i 0 and the restriction map h0 g q l h0 x l is surjective if char k 0 this is a consequence of the compatible frobenius splitting of x in g q and the char k 0 case is handled by semicontinuity the above result for special schubert varieties has been proved by seshadrimusili lakshmibai 12 13 and by kempf 9 by using characteristic free methods for x g q andersen 1 and haboush 5 have given simple proofs using characteristic p methods

on the m function and borg marchenko theorems for vector valued sturm liouville equations we will consider a vector valued sturm liouville equation of the form r u pu qu lambdawu xis an element of 0 b with p 1 w qis an element ofl loc 1 0 b mxm being hermitian and under some additional conditions on p 1 and w we give an elementary deduction of the leading order term asymptotics for the titchmarsh weyl m function corresponding to this equation in the special case of p w i qis an element ofl 1 0 infinity mxm and the neumann boundary conditions at 0 we will also prove that m 1 root lambda i r i r 1 where r lim n infinity r n sigma n 1 infinity q n for recursively defined sequences r n and q n if qis an element ofl loc 1 0 b mxm 0 bless than or equal toinfinity the same formula is valid with an exponentially small error for large lambda it is clear that expansions of this type are helpful in finding representatives of the kdv invariants for p w i we prove that the spectral measure corresponding to the equation r u lambdau uniquely determines q as well as b and the boundary conditions at 0 and b we finally give a new proof of a local form of the borg marchenko theorem cf gesztesy and simon on local borg marchenko uniqueness results commun math phys 211 273 287 2000 chap 3 a theorem which is due to simon see simon a new approach to inverse spectral theory i fundamental formalism ann math 150 1 29 1999 in the scalar case for applications to physics it is worth mentioning that vector valued sturm liouville equations appear in some problems in magneto hydro dynamics

fuzzy stochastic linear programming survey and future research directions in many real life problems one has to base decision on information which is both fuzzily imprecise and probabilistically uncertain although consistency indexes providing a union nexus between possibilistic and probabilistic representation of uncertainty exist there are no reliable transformations between them this calls for new paradigms for incorporating the two kinds of uncertainty into mathematical models fuzzy stochastic linear programming is an attempt to fulfill this need it deals with modelling and problem solving issues related to situations where randomness and fuzziness co occur in a linear programming framework in this paper we provide a survey of the essential elements methods and algorithms for this class of linear programming problems along with promising research directions being a survey the paper includes many references to both give due credit to results in the field and to help readers obtain more detailed information on issues of interest

weyl titchmarsh m function asymptotics local uniqueness results trace formulas and borg type theorems for dirac operators we explicitly determine the high energy asymptotics for weyl titchmarsh matrices associated with general dirac type operators on half lines and on r we also prove new local uniqueness results for dirac type operators in terms of exponentially small differences of weyl titchmarsh matrices as concrete applications of the asymptotic high energy expansion we derive a trace formula for dirac operators and use it to prove a borg type theorem

the xi function despite the fact that spectral and inverse spectral properties of one dimensional schrsdinger operators h d2 dx2 have been extensively studied for seventy five years there remain large areas where our knowledge is limited for example while the inverse theory for operators on l2 co co is well understood in case v is periodic 12 24 25 35 39 42 49 it is not understood in case limjx o v x co and h has discrete spectrum our goal here is to introduce a special function x a on r associated to h which we believe will be a valuable tool in the spectral and inverse spectral theory in a sense we will make precise it complements the weyl m functions m x a a main application of which we will make here concerns a generalization of the trace formula for schrsdinger operators to general v s recall the well known trace formula for periodic potentials let v x v x t 1 then by floquet theory see e g 10 37 44

on local borg marchenko uniqueness results we provide a new short proof of the following fact first proved by one of us in 1998 if two weyl titchmarsh m functions m j z of two schrodinger operators h j d 2 dx 2 q j j 1 2 in l 2 o r o r are exponentially close that is m 1 z m 2 z z o e 2 im z 1 2 a o r then q 1 q 2 a e on o the result applies to any boundary conditions at x o and x r and should be considered a local version of the celebrated borg marchenko uniqueness result which is quickly recovered as a corollary to our proof moreover we extend the local uniqueness result to matrix valued schrodinger operators

spectral theory of hamiltonian systems with almost constant coefficients abstract we derive the spectral theory for general linear hamiltonian systems the coefficients are assumed to be asymptotically constant and satisfy certain smoothness and decay conditions these latter constraints preclude the appearance of singular continuous spectra the results are thus far reaching extensions of earlier theorems of the authors two three and four dimensional systems are studied in greater detail the results also apply to the case of the dirichlet index and dirichlet spectrum

singular second order operators the maximal and minimal operators and selfadjoint operators in between differential operators arising from the differential equation py qy lambda wy are put in the hilbert space setting l 2 a b w a new and simpler characterization of the minimal operator is exhibited selfadjoint operators which lie in between the minimal and maximal operator are easily described in terms of conditions on boundary coefficients which look like and indeed are the same as those imposed on regular problems examples drawn from mathematical physics include the legendre laguerre hermite and bessel problems

on the existence of optimal controls the optimal control problem where the state is governed by an ito stochastic differential equation possibly just an ordinary differential equation is formulated in martingale terms under a coercivity condition which is weaker than compactness of the control set a convexity condition and mild continuity hypotheses on the data it is shown by the direct method that optimal controls exist hard and soft constraints are allowed in the absence of soft constraints it is shown that there exists an optimal control that is a function only of the present time and state i e the synthesis problem has a solution the main tool here is krylov s markovian selection theorem

applicable stochastic control from theory to practice stochastic control has come of age it is currently applied in a wide variety of fields spanning cancer research and chemotherapy economics and finance engineering management and many other areas of scientific and social investigation both theoretical models formulated to elucidate some characteristics of a problem under study and practical applications which have sought to provide specific answers to well stated problems have been the subjects of intense research for example economists have simultaneously groped with the underlying theoretical constructs justifying the adoption of particular econometric models while at the same time attempting to devise algorithms and optimization approaches to provide economic policy guidelines based on these models in the same spirit t remendous interest in options management which are based on stochastic dynamic models has stimulated much research and numerical analyses not only in universities but also in the back rooms of trading firms the usefulness of stochastic control was recognized once it became clear that it can be used as a unifying framework for treating multi stage or multi period optimization problems subject to stochastic disturbances however just as it is convenient as a formalism to formulate such problems it is becoming increasingly clear that it is practically difficult to solve them there are in fact only few problems one can solve completely and obtain optimal solutions for this reason there is today an increased awareness that a broad range of techniques including stochastic approximations perturbation techniques markov decision processes etc are an essential part of stochastic control kits it is in such a spirit that computer software such as pandore of inria the adaptive control algorithms for econometric models developed by kendrick and many others has been devised providing thereby computer aided expertise for managing stochastic control problems this software is essential even after the great effort applied to solving such problems over the last three decades ever since bellman s principle of optimality there are very few non trivial problems we can solve theoretically these include the well known linear quadratic stochastic control scp problem some version of the multi armed bandit problem and some specific linear optimization problems for example even the adaptive control problems studied over the last few years by kendrick and his co workers and presumed to be convex are believed today to include some unspecified nonconvexities private communication by d kendrick and hans amman in this spirit the intent in this paper is to introduce salient aspects of stochastic control which are useful for applications in management a review of applications can be found in neck 1984 for example as well as in books such as tapiero 1977 1988 bensoussan hurst and naslund 1974 bensoussan kleindorfer and tapiero 1978 1980 sethi and thompson 1981 kamien and schwarz 1981 malliaris and brock 1982 and whittle 1982 further due to an awesome amount of research published in this area appearing under various connotations such a task has become unwieldy rather

on the adjoint process for optimal control of diffusion processes for the markovian control problem begin gathered mathop min limits u e left int 0 t l t x u dt c x t right hfill dx f t x u dt sigma t x dw hfill end gathered it is shown that the adjoint process appearing in the maximum principle has the form p t x e tx left frac partial c partial x x t phi t t int t t frac partial l partial x s x s hat u s x s phi s t ds right where hat u is the optimal feedback control e tx denotes conditional expectation and phi is the fundamental matrix solution of dy frac partial f partial x t x t hat u t x t ydt sum k frac partial sigma k partial x t x y dw k here sigma k is the kth column of sigma and w k is thekth component of the brownian motion w it is also shown that p t x t satisfies denotes transpose begin gathered dp left frac partial f partial x t x t hat u t x t p

computation of the m matrix for fourth order problems this paper is concerned with the numerical computation of the titchmarsh weyl m matrix we show how an algorithm may be developed which relates solutions of an initial value problem to an approximation of the m matrix in some special cases we compare the results from our algorithm with results obtained using special functions

spectral properties of a differential system the spectrum of the differential operators associated with the differential system u r d d p u dy y qy on l2 0 is determined for this the coefficients are assumed to satisfy rather general properties which combine smoothness and decay with this the asymptotics of the eigenfunctions can be determined this in turn leads to properties of the spectra with the aid of the m matrix

the spectrum of differential operators of order 2n with almost constant coefficients abstract we discuss the spectral properties of higher order ordinary differential operators if the coefficients differ from constants by small perturbations then the spectral properties are preserved in this context small perturbations are either short range i e integrable or long range but slowly varying this generalizes classical results on second order operators our approach relies on an analysis of the associated differential equations with the help of uniform asymptotic integration techniques

a general stochastic maximum principle for optimal control problems the maximum principle for nonlinear stochastic optimal control problems in the general case is proved the control domain need not be convex and the diffusion coefficient can contain a control variable

trace formulas and borg type theorems for matrix valued jacobi and dirac finite difference operators abstract borg type uniqueness theorems for matrix valued jacobi operators h and supersymmetric dirac difference operators d are proved more precisely assuming reflectionless matrix coefficients a b in the self adjoint jacobi operator h as a s b with s the right left shift operators on the lattice z and the spectrum of h to be a compact interval e e e e we prove that a and b are certain multiples of the identity matrix an analogous result which however displays a certain novel nonuniqueness feature is proved for supersymmetric self adjoint dirac difference operators d with spectrum given by e 1 2 e 1 2 e 1 2 e 1 2 0 e e our approach is based on trace formulas and matrix valued exponential herglotz representation theorems as a by product of our techniques we obtain the extension of flaschka s borg type result for periodic scalar jacobi operators to the class of reflectionless matrix valued jacobi operators

perturbations of orthogonal polynomials with periodic recursion coefficients the results of denisov rakhmanov szeg shohat nevai and killip simon are extended from asymptotically constant orthogonal polynomials on the real line oprl and unit circle opuc to asymptotically periodic oprl and opuc the key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations

m matrix asymptotics for sturm liouville problems on graphs we consider a system formulation for sturm liouville operators with formally self adjoint boundary conditions on a graph an m matrix associated with the boundary value problem is defined and related to the matrix prufer angle associated with the system boundary value problem and consequently with the boundary value problem on the graph asymptotics for the m matrix are obtained as the eigenparameter tends to negative infinity we show that the boundary conditions may be recovered up to a unitary equivalence from the m matrix and that the m matrix is a herglotz function this is the first in a series of papers devoted to the reconstruction of the sturm liouville problem on a graph from its m matrix

on the optimal control of stochastic systems with an exponential of integral performance index abstract in this paper a theory of optimal control is developed for stochastic systems whose performance is measured by the exponential of an integral form such a formulation of the cost function is shown to be not only general and useful but also analytically tractable starting with very general classes of stochastic systems optimality conditions are obtained which exploit the multiplicative decomposability of the exponential of integral form specializing to partially observed systems of stochastic differential equations with brownian motion disturbances optimality conditions are obtained which parallel those for systems with integral costs also treated are the special cases of linear systems with exponential of quadratic costs for which explicit optimal controls are obtainable in addition several general results of independent interest are obtained which concern optimality of stochastic systems

stochastic control theory and operational research abstract the paper gives an overview of stochastic optimal control theory and its applications to operational research stochastic control theory deals with the intertemporal optimization of dynamic systems under uncertainty after a short review of deterministic optimal control theory and the theory of stochastic dynamic systems a general stochastic optimal control problem is formulated both in discrete and continuous time among the method for solving such problems particularly stochastic dynamic programming and the stochastic maximum principle are introduced examples from the areas of finance advertising production inventory models and information policy illustrate the applicability of stochastic control theory to operational research problems moreover its actual and potential impact on or is discussed from a methodological point of view

on weyl titchmarsh theory for singular finite difference hamiltonian systems we develop the basic theory of matrix valued weyl titchmarsh m functions and the associated green s matrices for whole line and half line self adjoint hamiltonian finite difference systems with separated boundary conditions

singular hamiltonian system with several spectral parameters abstract in this paper the weyl titchmarsh theory has been constructed for the singular 2n dimensional even order hamiltonian system with several spectral parameters in particular we consider that the left end point of the interval is regular and the right end point of the interval is singular for the hamiltonian system with several parameters using the nested circles approach we prove that at least n linearly independent solutions are squarly integrable with respect to some matrix functions

a new approach to inverse spectral theory ii general real potentials and the connection to the spectral measure we continue the study of the a amplitude associated to a half line schrodinger operator d 2 dx 2 q in l 2 0 b b a is related to the weyl titchmarsh m function via m k 2 k a 0 a e 2 k d o e 2 k for all 0 we discuss five issues here first we extend the theory to general q in l 1 0 for all a including q s which are limit circle at infinity second we prove the following relation between the a amplitude and the spectral measure p a 2 1 2 sin 2 dp since the integral is divergent this formula has to be properly interpreted third we provide a laplace transform representation for m without error term in the case b fourth we discuss m functions associated to other boundary conditions than the dirichlet boundary conditions associated to the principal weyl titchmarsh m function finally we discuss some examples where one can compute a exactly

ergodic properties and rotation number for linear hamiltonian systems abstract this paper is concerned with the dynamical behavior of the solutions of a class of linear hamiltonian systems including those to which kotani s theory applies we first present a symplectic l 2 perron transformation which takes these systems into skew symmetric form this allows us to study the average of the trajectories and the fourier coefficients of the solutions in addition from the construction of two invariant complex lagrange planes the differentiability of the rotation number is analyzed

on the rank of the matrix radius of the limiting set for a singular linear hamiltonian system this paper is concerned with the rank of the matrix radius of the limiting set for a singular hamiltonian system with one singular end point the exact relationship between the rank of the matrix radius and the number of square integrable solutions is obtained and then the defect index of the corresponding minimal operator can be represented in terms of the rank of the matrix radius so two results obtained by allan m krall siam j math anal 20 1989 664 are improved in addition it is discussed that the rank of the matrix radius is independent of the spectral parameter and a certain matrix especially the classification of singular linear hamiltonian systems is present and several sufficient and necessary conditions for the limit point and limit circle cases are established

self adjoint extensions for discrete linear hamiltonian systems abstract this paper is concerned with self adjoint extensions for a class of discrete linear hamiltonian systems by applying the generalized von neumann theory and the gkn theory for hermitian subspaces a complete characterization of all the self adjoint subspace extensions for the systems are obtained in terms of boundary conditions via linear independent square summable solutions as a consequence characterizations of all the self adjoint subspace extensions are given in the limit point and limit circle cases in addition some sufficient conditions for the corresponding minimal subspace to be an operator or a densely defined operator are given and consequently characterizations of all the self adjoint operator extensions for the system are obtained in particular even order formally self adjoint vector difference equations are discussed as a special case

hamiltonian systems of limit point or limit circle type with both endpoints singular abstract a linear hamiltonian system j y a b y is considered on an open interval a b where both a and b are singular the system is assumed to be of limit point or limit circle type at the endpoints a theory of boundary problems for such systems is developed explicit boundary conditions are given resolvent operators constructed and unique solutions established the results given extend to hamiltonian systems a theory of singular boundary value problems due to m h stone and k kodaira

trace identities in the inverse scattering transform method associated with matrix schrodinger operators trace identities arising in the scattering theory of one dimensional matrix schrodinger operators are deduced they derive from the properties of an asymptotic expansion of the trace of the resolvent kernel in inverse powers of the spectral parameter applications of these trace identities for characterizing infinite families of conservation laws for nonlinear evolution equations are given

absolutely continuous spectra for matrix dirac systems this paper investigates the matrix dirac systems under some conditions on the potential matrices it is shown that the spectrum of the dirac operator is purely absolutely continuous outside the gaps containing the eigenvalues of q

the glazman krein naimark theory for a class of discrete hamiltonian systems abstract in this paper the glazman krein naimark theory for a class of discrete hamiltonian systems is developed a minimal and a maximal operators gkn sets and a boundary space for the system are introduced algebraic characterizations of the domains of self adjoint extensions of the minimal operator are given a close relationship between the domains of self adjoint extensions and the gkn sets is established it is shown that there exist one to one correspondences among the set of all the self adjoint extensions the set of all the d dimensional lagrangian subspaces of the boundary space and the set of all the complete lagrangian subspaces of the boundary space

weyl titchmarsh m function asymptotics for matrix valued schrodinger operators we explicitly determine the high energy asymptotics for weyl titchmarsh matrices corresponding to matrix valued schr odinger operators associated with general self adjoint m times m matrix potentials q in l 1 text loc x 0 infty m times m where m in mathbb n more precisely assume that for some n in mathbb n and x 0 in mathbb r q n 1 in l 1 x 0 c m times m for all c x 0 and that x geq x 0 is a right lebesgue point of q n 1 in addition denote by i m the m times m identity matrix and by c varepsilon the open sector in the complex plane with vertex at zero symmetry axis along the positive imaginary axis and opening angle varepsilon with 0 varepsilon frac12 pi then we prove the following asymptotic expansion for any point m z x of the unique limit point or a point of the limit disk associated with the differential expression i m frac d 2 dx 2 q x in l 2 x 0 infty m and a dirichlet boundary condition at x x 0 begin equation m z x underset z to infty z in c varepsilon i i m z 1 2 sum k 1 n m k x z k 2 o z n 2 quad text where n in mathbb n nonumber end equation the expansion is uniform with respect to arg z for z to infty in c varepsilon and uniform in x as long as x varies in compact subsets of mathbb r intersected with the right lebesgue set of q n 1 moreover the m times m expansion coefficients m k x can be computed recursively analogous results hold for matrix valued schr odinger operators on the real line 2000 mathematics subject classification 34e05 34b20 34l40 34a55

weyl titchmarsh theory for a class of discrete linear hamiltonian systems abstract this paper is concerned with establishing the weyl titchmarsh theory for a class of discrete linear hamiltonian systems over a half line fundamental properties of solutions regular spectral problems and the corresponding maximal and minimal operators are first studied matrix disks are constructed and proved to be nested and converge to a limiting set some precise relationships among the rank of the matrix radius of the limiting set the number of linearly independent square summable solutions and the defect indices of the minimal operator are established based on the above results a classification of singular discrete linear hamiltonian systems is given in terms of the defect indices of the minimal operator and several equivalent conditions on the cases of limit point and limit circle are obtained respectively especially several problems in the limit point case are more carefully investigated including fundamental properties of square summable solutions properties of the weyl function which is the unique element in the limiting set in this case and inhomogeneous boundary problems self adjointness of the corresponding hamiltonian operator relationship between the spectrum of the hamiltonian operator and the analyticity of the weyl function as well as the dependence of the spectrum on the boundary data in which some interesting separation results for the spectrum are obtained finally another set of four equivalent conditions on the limit point case are established

on essential spectra of singular linear hamiltonian systems abstract the paper is concerned with essential spectra of singular linear hamiltonian systems of arbitrary order with arbitrary equal defect indices several sufficient conditions for the essential spectral points are given in terms of the number of linearly independent square integrable solutions of the corresponding hamiltonian system and a sufficient and necessary condition for the essential spectral points is obtained for hamiltonian systems of even order with one singular endpoint an advantage of these results is that one can determine the essential spectral points of hamiltonian systems by the information of solutions obtained by numerous tools available in the fundamental theory of differential equations in addition two illustrative examples are provided to show how to get some information about the essential spectrum by our results

classification of sturm liouville differential equations with complex coefficients and operator realizations in this paper a new classification of sturm liouville differential equations with complex coefficients is given compared with the corresponding result of brown et al this classification reveals the great effects of rotation angle and it is independent of the rotation angles moreover the asymptotic behaviours of functions in the maximal domain are presented and j self adjoint extensions associated with the differential equations are characterized

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

on the optimal reward function of the continuous time multiarmed bandit problem the optimal reward function associated with the so called multiarmed bandit problem for general markov feller processes is considered it is shown that this optimal reward function has a simple expression product form in terms of individual stopping problems without any smoothness properties of the optimal reward function neither for the global problem nor for the individual stopping problems some results relative to a related problem with switching cost are obtained

spectral theory for systems of ordinary differential equations with distributional coefficients abstract we study the spectral theory for the first order system j u q u w f of differential equations on the real interval a b when j is a constant invertible skew hermitian matrix and q and w are matrices whose entries are distributions of order zero with q hermitian and w non negative also we do not pose the definiteness condition often required for the coefficients of the equation specifically we construct minimal and maximal relations and study self adjoint restrictions of the maximal relation for these we determine green s function and prove the existence of a spectral or generalized fourier transformation we have a closer look at the special cases when the endpoints of the interval a b are regular as well as the case of a 2 2 system two appendices provide necessary details on distributions of order zero and the abstract spectral theory for relations

on a weyl titchmarsh theory for discrete symplectic systems on a half line recently bohner and sun introduced basic elements of a weyl titchmarsh theory into the study of discrete symplectic systems we extend this development through the introduction of weyl titchmarsh functions together with a preliminary study of their properties a limit point criterion is described and characterized green s function for the half line is introduced as a limit of such functions in the regular case and half line solutions obtained are seen to satisfy lambda dependent boundary conditions at infinity

a new hamiltonian system abstract this paper aims to share a new first order differential equation that contains the continuous analogous of the orthogonal polynomials on the unit circle we introduce some basic results on the system and solutions of the system using nested circle approach we introduce the possible number of square integrable solutions of the system at the end of the paper we share a limit point criteria for the two dimensional system of equations

limit point and limit circle classification for symplectic systems on time scales abstract in this paper we study the limit point and limit circle classification for symplectic systems on time scales which depend linearly on the spectral parameter in a broader context we develop a unified weyl titchmarsh theory for continuous and discrete linear hamiltonian and symplectic systems both separated and coupled boundary conditions are allowed our results include the study of the weyl disks and circles and their limiting behavior as well as a precise analysis of the number of linearly independent square integrable solutions we also prove an analogue of the famous weyl alternative we connect and unify many known results in the weyl titchmarsh theory for continuous discrete and special time scales systems and explain the differences between them some of our statements in particular those connected with coupled endpoints or the weyl alternative are new even in the continuous time setting

weighted cohomology of arithmetic groups let g be a semisimple algebraic group defined over the rational numbers k a maximal compact subgroup ofg g r and g q a neat arithmetic subgroup letx g k be the locally symmetric space associated to and e the local system on x constructed out of a finite dimensional irreducible algebraic representation e of g fix a maximally q split torus s in g s is assumed to be nontrivial so that x is noncompact let a s r 0 and by a slight abuse of notation let x a denote the group of rational characters of s choose a minimal rational parabolic subgroup p0 s the choice determines a notion of positivity in x a and a set of positive roots among the roots of s in g let 0 be the half sum of the positive roots weighted cohomology is an invariant of introduced by m goresky g harder and r macpherson 12 in the study 15 16 12 14 of the trace of hecke operators in the cohomology of for each p x a q there are weight profiles p and p upper and lower p see 1 5 and groups w ph i e and w h i e the notion of profile here differs from but is equivalent to that of 12 see remark 1 2 below if p q 2 0 then the profiles p and q are dual and the corresponding groups are in poincare duality for very positive p one gets the compactly supported cohomology h i c x e and dually for very negative p the full cohomology h i x e the definition of the weighted cohomology groups is somewhat involved so i will attempt to give some idea of the motivation behind it detailed definitions are recalled in section 1 associated to p and p are complexes of sheaves wpc e and wpc e on the reductive borel serre compactification x of x let p be a rational parabolic subgroup np its unipotent radical sp the split centre of p np and xp the p boundary component in x the idea behind the construction in 12 is as follows the stalk cohomology of the direct image ri e by i x x at x xp is

the asymptotic form of the spectral functions associated with a class of sturm liouville equations on utilise une forme modifiee de la formule d inversion de stieltjes avec des resultats recents sur la forme asymptotique des m fonctions de titchmarch weyl pour obtenir une expression asymptotique pour les fonctions spectrales t associees a l equation differentielle y q x y y o x b

orbits invariants and representations associated to involutions of reductive groups by an involution of an algebraic group or lie algebra we mean an automorphism 0 such that 0 2 is the identity automorphism let 0 be an involution of a complex reductive lie algebra q and let p denote the 1 eigenspace of 0 on q get g be the adjoint group of q and let k g glog go kostant and rallis 10 have made a detailed study of the action of k on p in particular they obtain very precise information concerning the orbits of k on p and the representation of k on c p the algebra of polynomial functions on p in this paper we consider a global analogue of the problem studied by kostant and rallis let g be a reductive affine algebraic group over an algebraically closed field k of characteristic 2 let 0 be an involution of g and let k gegi0 g g if g is semisimple and the base field is c then g k is the complexification of a riemannian symmetric space we study the action of k and of groups closely related to k by left translation on the coset space g k for technical reasons it is convenient to study a slightly different but equivalent problem let r g g be defined by r g g0 g i then z g p is a closed subvariety of g furthermore z is constant on left cosets of g modulo k and induces an isomorphism o g k p we have o k g k k o g k for k e k and geg thus p is stable under conjugation by k and a is a k isomorphism of affine k varieties we consider the action of k on p by conjugation let x e g have jordan decomposition x x x then we show that x e p if and only if both x and x are in p thus we have a jordan decomposition for elements of p this will play a key role in our results a torus s of g is o anisotropic if o s s 1 for every ses it is known that any two maximal 0 anisotropic tori of g are k let a be a maximal 0 anisotropic torus of g the following result characterizes semisimple elements of p