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

strong limit point criteria for a class of singular discrete linear hamiltonian systems this paper is concerned with the limit point case for a class of singular discrete linear hamiltonian systems the limit point case is divided into the strong and the weak limit point cases several sufficient conditions for the strong limit point case are established in consequence two criteria of the strong limit point case for second order formally self adjoint vector difference equations are obtained

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

exact and approximation algorithms for the tactical fixed interval scheduling problem the tactical fixed interval scheduling problem tfisp is the problem of determining the minimum number of parallel nonidentical machines such that a feasible schedule exists for a given set of jobs in tfisp each job must belong to a specific job class and must be carried out in a prespecified time interval the problem is complicated by the restrictions that 1 each machine can handle only one job at a time 2 each machine can handle only jobs from a subset of the job classes and 3 preemption is not allowed in this paper we discuss the occurrence of tfisp in practice we analyze the computational complexity of tfisp and we present exact and approximation algorithms for solving tfisp the paper concludes with a computational study

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

a spectral analysis for self adjoint operators generated by a class of second order difference equations abstract a qualitative spectral analysis for a class of second order difference equations is given central to the analysis of equations in this class is the observation that real valued solutions exhibit a type of stable asymptotic behavior for certain real values of the spectral parameter this asymptotic behavior leads to the characterization of the limit point and limit circle nature of these equations and is used to show that a strong nonsubordinacy criterion is satisfied on subintervals of r for equations of limit point type these subintervals are part of the absolutely continuous spectrum of the self adjoint realization of these equations by other means the nature of the discrete spectrum for these self adjoint realizations is also discussed

eigenfunctions transfer matrices and absolutely continuous spectrum of one dimensional schrodinger operators on l r and its half line problem h on l2 0 with u 0 0 boundary conditions we will focus on a new approach to the absolutely continuous spectrum ac h and more generally ac h the essential support of the a c part of the spectral measures what is new in our approach is that it relies on estimates on the transfer matrix that is the 2 2 matrix te n m which takes u m 1 u m to u n 1 u n for solutions u of hu eu in the continuum case use u x u x instead of u x 1 u x we let te n te n 0 for example we will prove the following

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

continuous dependence of the n th eigenvalue of self adjoint discrete sturm liouville problems on the problem abstract this paper is concerned with continuous dependence of the n th eigenvalue of self adjoint discrete sturm liouville problems on the problem the n th eigenvalue is considered as a function on the space of the problems called the n th eigenvalue function a necessary and sufficient condition for all the n th eigenvalue functions to be continuous and several properties of the n th eigenvalue function on a subset of the space of the problems are given they play an important role in the study of continuous dependence of the n th eigenvalue function on the problem continuous dependence of the n th eigenvalue function on the equation and on the boundary condition is studied separately consequently the continuity and discontinuity of the n th eigenvalue function are completely characterized on the whole space of the problems especially asymptotic behavior of the n th eigenvalue function near each discontinuity point is given

eigenvalues of second order difference equations with periodic and antiperiodic boundary conditions this paper is concerned with periodic and antiperiodic boundary value problems for self adjoint second order difference equations existence of eigenvalues of these two different boundary value problems is proved numbers of their eigenvalues are calculated and their relationships are obtained in addition a representation of solutions of a nonhomogeneous linear equation with initial conditions is given

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

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

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

self adjoint extensions for second order symmetric linear difference equations abstract in this paper self adjoint extensions for second order symmetric linear difference equations with real coefficients are studied by applying the glazman krein naimark theory for hermitian subspaces both self adjoint subspace extensions and self adjoint operator extensions of the corresponding minimal subspaces are completely characterized in terms of boundary conditions where the two endpoints may be regular or singular

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

on subordinacy and analysis of the spectrum of schrodinger operators with two singular endpoints the theory of subordinacy is extended to all one dimensional schrodinger operatorsfor which the corresponding differential expression l d 2 dr 2 v r is in the limit point case at both ends of an interval a b with v r locally integrable this enables a detailed classification of the absolutely continuous and singular spectra to be established in terms of the relative asymptotic behaviour of solutions of lu xu x e as r a and r b the result provides a rigorous but straightforward method of direct spectral analysis which has very general application and somefurther properties of the spectrum are deduced from the underlying theory

the absolutely continuous spectrum of one dimensional schrodinger operators with decaying potentials we investigate one dimensional schrodinger operators with asymptotically small potentials it will follow from our results that if with then is an essential support of the absolutely continuous part of the spectral measure we also prove that if then the spectrum is purely absolutely continuous on these results are optimal

spectral theory of second order vector difference equations abstract this paper is concerned with spectral problems of second order vector difference equations with two point boundary value conditions where the matrix valued coefficient of the leading term may be singular a concept of self adjointness of the boundary value conditions is introduced the self adjointness of the corresponding difference operator is discussed on a suitable admissible function space and fundamental spectral results are obtained the dual orthogonality of eigenfunctions is shown in a special case rayleigh s principles and the minimax theorems in two linear spaces are given as an application a comparison theorem for eigenvalues of two sturm liouville problems is presented

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

embeddings of hecke algebras in group algebras let g be a finite group and let b be a subgroup of g the endomorphism algebra of the induced permutation representation indg 1 is called the hee but the multiplying elements are different from those mentioned above even in characteristic zero since their definition involves only division by 1 h where h n w b in particular if r is a field of characteristic p the duals of our elements h turn out to be words in the steinberg idempotents of parabolic subgroups of g our results provide explicit formulae for an arbitrary product of these in rg see theorem 1 10 this modular case is discussed in section 3 together with related questions on the representation theory of g the modular case was partially treated by kuhn 6 and landrock 7 and we thank particularly the former for sharing his ideas with us 159 0021 8693 87 3 00

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

on decomposition of modular representations from cohen macaulay geometries abstract chain spaces for a cohen macaulay complex split into subspaces corresponding to cycles for truncations of the complex precisely when the cycles are non degenerate in the natural bilinear form application to the tits building provides a simple geometric approach to the well known splitting of ind g b 1 for a lie type group g in natural characteristic an application to a sporadic geometry is also indicated

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

regular approximations of spectra of singular discrete linear hamiltonian systems with one singular endpoint abstract this paper is concerned with regular approximations of spectra of singular discrete linear hamiltonian systems with one singular endpoint for any given self adjoint subspace extension sse of the corresponding minimal subspace its spectrum can be approximated by eigenvalues of a sequence of induced regular sses generated by the same difference expression on smaller finite intervals it is shown that every sse of the minimal subspace has a pure discrete spectrum and the k th eigenvalue of any given sse is exactly the limit of the k th eigenvalues of the induced regular sses that is spectral exactness holds in the limit circle case furthermore error estimates for the approximations of eigenvalues are given in this case in addition in the limit point and intermediate cases spectral inclusive holds

discrete sturm liouville problems jacobi matrices and lagrange interpolation series abstract the close relationship between discrete sturm liouville problems belonging to the so called limit circle case the indeterminate hamburger moment problem and the search of self adjoint extensions of the associated semi infinite jacobi matrix is well known in this paper all these important topics are also related with associated sampling expansions involving analytic lagrange type interpolation series

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

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

on subordinacy and analysis of the spectrum of one dimensional schrodinger operators abstract by considering the behaviour as n of the ratio of l2 0 n norms of solutions of d 2 dr 2 v r u xu 0 r x r a characterisation of the absolutely continuous and singular spectra of one dimensional schrodinger operators is deduced the analysis is applicable to all operators for which l d 2 dr 2 v r is regular at 0 and in the limit point case at infinity with v r locally integrable

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

asymptotics of orthogonal polynomials with slowly oscillating recurrence coefficients abstract we study solutions of three term recurrence relations whose n step transfer matrices belong to the uniform stolz class in particular we derive the first order of their uniform asymptotics for orthonormal polynomials we show more namely we find the constructive formula for the density of their orthogonality measure in terms of turan determinants and we determine their exact asymptotic behavior we treat both bounded and unbounded cases in a unified way

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

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

eigenvalues of second order difference equations with coupled boundary conditions abstract this paper is concerned with coupled boundary value problems for self adjoint second order difference equations existence of eigenvalues is proved numbers of eigenvalues are calculated and relationships between the eigenvalues of a self adjoint second order difference equation with three different coupled boundary conditions are established these results extend the relevant existing results of periodic and antiperiodic boundary value problems

spectral properties of self adjoint subspaces abstract this paper focuses on spectral properties of self adjoint subspaces in the product space of a hilbert space several classifications of the spectrum of a self adjoint subspace are introduced including discrete spectrum essential spectrum continuous spectrum singular continuous spectrum absolutely continuous spectrum and singular spectrum their relationships between the subspace and its operator part are established respectively it is shown that all self adjoint subspace extensions of a closed hermitian subspace have the same essential spectrum as a simple application it is shown that every self adjoint subspace extension of the corresponding minimal subspace to a singular second order symmetric linear difference equation has a bounded spectrum under some conditions this shows that there is an essential difference between spectral properties of singular symmetric difference operators and classical regular or singular symmetric differential operators

discrete spectra criteria for certain classes of singular differential and difference operators in this paper we consider singular differential and difference operators of the form m y 1 n w t r t y n n and b y k n 1 n w k delta n r k delta n y k as well as their full term analogies we establish criteria for discreteness and boudnedness bellow of their spectrum property bd difference operators are studied as the discrete counterparts of the continuous ones

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

regular approximations of spectra of singular second order symmetric linear difference equations abstract this paper is concerned with regular approximations of spectra of singular second order symmetric linear difference equations for any given self adjoint subspace extension of the corresponding minimal subspace its spectrum can be approximated by eigenvalues of a sequence of regular self adjoint subspace extensions generated by the same difference expression on smaller finite intervals

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 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

algorithms for preemptive scheduling of different classes of processors to do jobs with fixed times abstract we consider scheduling problems that involve processors of different classes and jobs with fixed start and finish times we allow preemption of jobs but impose restrictions on the assignment of processors to jobs we present polynomial time algorithms for finding the minimal cost preemptive solutions for two special cases in which the number of job classes is at most one more than the number of processor classes we then discuss a generalized version in which a processor can be assigned only to a subset of the jobs and a job can be done only by a subset of the processors we prove that for this case the problem of finding a preemptive feasible schedule with a given combination of processors can be solved in polynomial time by transforming it into a transportation network problem next we extend these results to cyclic scheduling problems further we discuss some transformations among these class scheduling problems and provide simpler proofs of np completeness for the nonpreemptive versions of the first two cases

limit point and limit circle criteria for singular second order linear difference equations with complex coefficients this paper is concerned with limit point and limit circle criteria of singular second order linear difference equations with complex coefficients formally self adjoint second order linear difference operators are formulated several sufficient conditions and sufficient and necessary conditions for the limit point and limit circle cases are established these results extend some relevant existing results

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

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

error estimate of eigenvalues of perturbed second order discrete sturm liouville problems abstract this paper is concerned with eigenvalues of perturbed second order vector discrete sturm liouville problems by some variational properties of eigenvalues of discrete sturm liouville problems error estimates of eigenvalues of perturbed problems sufficiently close to a given sturm liouville problem are given under a certain non singularity condition perturbations of the coefficient functions of the difference equation the weight function and the coefficients of the boundary condition are all considered this together with higher dimension involved results in a certain complexity of the problem and difficulty of study as a direct consequence continuous dependence of eigenvalues on problems is obtained under the non singularity condition in addition an example is presented to illustrate the necessity of the non singularity condition

on the computational complexity of maximum class scheduling in this paper we consider several generalizations of the fixed job scheduling problem fsp which appear in a natural way in the aircraft maintenance process at an airport a number of jobs have to be carried out where the main attributes of a job are a fixed start time a fixed finish time a value representing the job s priority and a job class for carrying out these jobs a number of machines are available these machines can be split up into a number of disjoint machine classes for each combination of a job class and a machine class it is known whether or not it is allowed to assign a job in the job class to a machine in the machine class furthermore the jobs must be carried out in a non preemptive way and each machine can be carrying out at most one job at the same time within this setting one can ask for a feasible schedule for all jobs or if such a schedule does not exist for a feasible schedule for a subset of the jobs of maximum total value in this paper we present a complete classification of the computational complexity of two classes of combinatorial problems related this operational job scheduling problem

absolute continuity for unbounded jacobi matrices with constant row sums we investigate the spectral properties of a class of jacobi matrices in which the subdiagonal entries are quadratics and the row sums are constants

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