the complex geometry of weak piecewise smooth solutions of integrable nonlinear pde s of shallow water and dym type an extension of the algebraic geometric method for nonlinear integrable pde s is shown to lead to new piecewise smooth weak solutions of a class of n component systems of nonlinear evolution equations this class includes among others equations from the dym and shallow water equation hierarchies the main goal of the paper is to give explicit theta functional expressions for piecewise smooth weak solutions of these nonlinear pde s which are associated to nonlinear subvarieties of hyperelliptic jacobians

the jacobi rosochatius problem on an ellipsoid the lax representations and billiards in this article we construct the lax representations of the geodesic flow the jacobi rosochatius problem and its perturbations by means of separable polynomial potentials on an ellipsoid we prove complete integrability in the case of a generic symmetric ellipsoid and describe analogous systems on complex projective spaces also we consider billiards within an ellipsoid under the influence of the hook and rosochatius potentials between the impacts a geometric interpretation of the integrability analogous to the classical chasles and poncelet theorems is given

oscillation of self adjoint second order vector difference equations to the parameter abstract this paper is concerned with oscillation of self adjoint second order vector difference equations with respect to a parameter properties of zeros and monotonicity of matrix valued solutions are studied the oscillation of two consecutive polynomials for vector valued solutions is discussed a separation theorem for matrix valued solutions is also obtained

negative order mkdv hierarchy and a new integrable neumann like system abstract the purpose of this paper is to develop the negative order mkdv hierarchy and to present a new related integrable neumann like hamiltonian flow from the view point of inverse recursion operator and constraint method the whole mkdv hierarchy both positive and negative is generated by the kernel elements of lenard s operators pair and recursion operator through solving a key operator equation the whole mkdv hierarchy is shown to have the lax representation in particular some new integrable equation together with the liouville equations the sine gordon equation and the sinh gordon equation are derived from the negative order mkdv hierarchy it is very interesting that the restricted flow corresponding to the negative order mkdv hierarchy is just a new kind of neumann like system this new neumann like system is obtained through restricting the mkdv spectral problem onto a symplectic submanifold and is proven to be completely integrable under the dirac poisson bracket which we define on the symplectic submanifold finally with the help of the constraint between the neumann like system and the negative order mkdv hierarchy all equations in the hierarchy are proven to have the parametric representations of solutions in particular we obtain the parametric solutions of the sine gordon equation and the sinh gordon equation

eigenvalue comparisons for second order difference equations with neumann boundary conditions abstract the boundary value problem for second order difference equation r i 1 y i 1 b i y i a i y i 0 1 i n y 0 y 1 y n 1 y n 0 with 0 1 and 2 was recently discussed in ji and yang 2007 j ji b yang eigenvalue comparisons for a class of boundary value problems of second order difference equations linear algebra appl 420 1 2007 218 227 in this paper we extend our earlier results to the second order difference equations with neumann boundary conditions the case of 1 as in ji and yang 2007 mentioned above we will also focus on the structure of its eigenvalues and comparisons of all eigenvalues as the coefficients a i b i and r i change

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

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

extension groups between simple mackey functors in order to better understand the structure of indecomposable projective mackey functors we study extension groups of degree 1 between simple mackey functors we explicitly determine these groups between simple functors indexed by distinct normal subgroups we next study the conditions under which it is possible to restrict ourselves to that case and we give methods for calculating extension groups between simple mackey functors which are not indexed by normal subgroups we then focus on the case where the simple mackey functors are indexed by the same subgroup in this case the corresponding extension group can be embedded in an extension group between modules over a group algebra and we describe the image of this embedding in particular we determine all extension groups between simple mackey functors for a p group and for a group that has a normal p sylow subgroup finally we compute higher extension groups between simple mackey functors for a group that has a p sylow subgroup of order p

spectral theory of discrete linear hamiltonian systems this paper is concerned with spectral problems for a class of discrete linear hamiltonian systems with self adjoint boundary conditions where the existence and uniqueness of solutions of initial value problems may not hold a suitable admissible function space and a difference operator are constructed so that the operator is self adjoint in the space then a series of spectral results are obtained the reality of eigenvalues the completeness of the orthogonal normalized eigenfunction system rayleigh s principle the minimax theorem and the dual orthogonality especially the number of eigenvalues including multiplicities and the number of linearly independent eigenfunctions are calculated

existence of positive solutions of bvps for third order discrete nonlinear difference systems in this paper we consider the following boundary value problem of discrete email protected 3u 1 k f 1 k u 1 k u 2 k 0 email protected 0 t d 3u 2 k f 2 k u 1 k u 2 k 0 email protected 0 t with the dirichlet boundary conditionu 1 0 u 1 1 u 1 t 3 0 u 2 0 u 2 1 u 2 t 3 some new results of the existence nonexistence and multiplicity are obtained by using krasnosel skii s fixed point theorem in a cone in particular it is proved that the above boundary value problem has n positive solutions under suitable conditions where n is an arbitrary positive integer

on the linear and nonlinear discrete second order neumann boundary value problems abstract by construct the spectrum of the linear eigenvalue problem we are concerned with the global structure of the set of sign changing solutions to the discrete second order neumann boundary value problem p t 1 u t 1 q t u t ra t f u t t 1 t z u 0 u t 0 furthermore we obtain the existence of the sign changing solutions of the above nonlinear problem

hierarchy of lower korteweg de vries equations and supersymmetry structure of miura transformations the infinite hierarchy of lower korteweg de vries kdv equations which corresponds to negative degrees of olver s operator is constructed basic forms of these equations are found the supersymmetry structure of the miura s transformations is considered

eigenvalue comparisons for a class of boundary value problems of second order difference equations abstract we consider the boundary value problem for second order difference equation r i 1 y i 1 b i y i a i y i 0 1 i n y 0 y 1 y n 1 y n 0 in this study we do not require the positiveness of a k k 1 n we focus on the structure of eigenvalues of this problem and comparisons of all eigenvalues as the coefficients a i i 1 n b i i 1 n r i i 0 n and the parameters change

multiple positive solutions for nonlinear dynamical systems on a measure chain in this paper we consider the following dynamical system on a measure chain u 1 t f1 t u1 t u2 t 0 t a b u 2 t f2 t u1 t u2 t 0 t a b with the sturm liouville boundary value conditions ui a u i a 0 ui b ui b 0 for i 1 2 some results are obtained for the existence of three positive solutions of the above problem by using leggett williams fixed point theorem

an alternative approach to solve the mixed akns equations abstract the algebraic geometric solutions of the mixed akns equations are investigated through a finite dimensional lie poisson hamiltonian system which is generated by the nonlinearization of the adjoint equation related to the akns spectral problem first each mixed akns equation can be decomposed into two compatible lie poisson hamiltonian flows then the separated variables on the coadjoint orbit are introduced to study these lie poisson hamiltonian systems further based on the hamilton jacobi theory the relationship between the action angle coordinates and the jacobi inversion problem is established in the end using riemann jacobi inversion the algebraic geometric solutions of the first three mixed akns equations are obtained

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

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

standard stratifications of ei categories and alperin s weight conjecture abstract we characterize the finite ei categories whose representations are standardly stratified with respect to the natural preorder on the simple representations the orbit category of a finite group with respect to any set of subgroups is always such a category taking the subgroups to be the p subgroups of the group we reformulate alperin s weight conjecture in terms of the standard and proper costandard representations of the orbit category we do this using the properties of the ringel dual construction and a theorem of dlab which have elsewhere been described for standardly stratified algebras where there is a partial order on the simple modules we indicate that these results hold in the generality of an algebra whose simple modules are preordered rather than partially ordered

positive solutions of bvps for third order nonlinear difference systems abstract in this paper we consider the following discrete system boundary value problems 3 u 1 k f 1 k u 1 k u 2 k 0 k o t 3 u 2 k f 2 k u 1 k u 2 k 0 k o t with the dirichlet boundary condition u 1 0 u 1 1 u 1 t 3 0 u 2 0 u 2 1 u 2 t 3 some sufficient conditions are obtained for the existence of one or two positive solutions to the above system by using a nonlinear alternative of leray schauder type and krasnosel skii s fixed point theorem in a cone

some remarks on g functors and the brauer morphism keywords finite group g functor restriction maps transfer maps mackey s formula frobenius reciprocity brauer morphism defect base nilpotent ideal burnside ring idempotent formula induction formula symmetric g functor character ring green ring reference ctg article 1988 003 record created on 2008 12 16 modified on 2017 05 12

integrals invariant manifolds and degeneracy for central force problems in rn since the notion of angular momentum is defined in any dimension by using the exterior product in rn one would guess that central force problems in any dimension are completely integrable as it is known for n 2 or 3 this is proved explicitly in this paper by constructing n first integrals independent and involution the energy and some combinations of the angular momentum components it is shown that these problems are always reduced to a two dimensional plane and the invariant manifolds are topologically the cartesian product of those of the reduced problem times n 2 circle factors it is also proved that for n 2 these problems are degenerate in the sense that their hamiltonians do not satisfy one of the hypotheses of the kolmogorov arnold moser theorem for persistence of invariant tori when one considers small perturbations

nonlinearizing linear equations to integrable systems including new hierarchies with nonholonomic deformations we propose a scheme for nonlinearizing linear equations to generate integrable nonlinear systems of both the ablowitz kaup newell segur akns and the kaup newell classes based on the simple idea of dimensional analysis and detecting the building blocks of the lax pair along with the well known equations we discover a novel integrable hierarchy of higher order nonholonomic deformations for the akns family e g the korteweg de vries kdv the modified kdv the nonlinear schrodinger and the sine gordon equation showing thus a twofold universality of the recently found deformation for the kdv equation a karsu et al j math phys 49 073516 2008

hierarchy of negative order equation and its lax pair a simple method is presented to construct the hierarchy of a negative order equations from a given eigenvalue problem the hierarchy of the negative order equation we obtained is the natural continuation of a corresponding hierarchy of a higher order equation although most of the negative order equations are complicated we give their lax representations three examples are discussed to illustrate the effectiveness of our method

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

integrable rosochatius deformations of the restricted soliton flows a method to construct integrable rosochatius deformations of the restricted soliton flows in the setup of lax formulation is presented the integrable rosochatius deformations of the restricted soliton flows such as the restricted ablowitz kaup newell segur flow the restricted tu meng flow the restricted tu flow with neumann type constraints and the restricted modified korteweg de vries flow together with their lax representations are presented in addition a lax representation of the jacobi rosochatius system is obtained

soliton solutions for a negative order non isospectral akns equation abstract in this paper the bilinear form of a negative order non isospectral akns equation is given the soliton solutions are obtained through hirota s direct method and the wronskian technique respectively

spectral theory of higher order discrete vector sturm liouville problems abstract this paper is concerned with spectral problems of higher order vector difference equations with self adjoint boundary conditions where the coefficient of the leading term may be singular a suitable admissible function space is constructed so that the corresponding difference operator is self adjoint in it and the fundamental spectral results are obtained rayleigh s principles and minimax theorems in two special linear spaces are given as an application comparison theorems for eigenvalues of two sturm liouville problems are presented especially the dual orthogonality and multiplicity of eigenvalues are discussed

multiple positive solutions of bvps for third order discrete difference systems this paper is concerned with the following email protected 3u 1 k f 1 k u 1 k u 2 k 0 email protected 0 t d 3u 2 k f 2 k u 1 k u 2 k 0 email protected 0 t with the dirichlet boundary conditionu 1 0 u 1 1 u 1 t 3 0 u 2 0 u 2 1 u 2 t 3 by using leggett williams fixed point theorem sufficient conditions are obtained for the existence of three positive solutions to the above system

soliton scattering with amplitude changes of a negative order akns equation the paper investigates a nonlinear model which provides soliton scattering with amplitude changes this is a nonlinear equation in negative order akns hierarchy we derive its n soliton solutions through hirota method for dynamics 1 soliton is characterized by an asymmetric shape and time dependent amplitude then by means of asymptotic analysis it is found that two soliton scattering provides not only phase shifts but also amplitude changes for each soliton which differs from classical soliton interactions where only phase shifts appear appendix a gives a tentative discussion on lax integrability and formal hamiltonian structures of the negative order hierarchy and appendix b lists reductions of equations and solutions

eigenvalues of discrete linear second order periodic and antiperiodic eigenvalue problems with sign changing weight abstract by virtue of the eigenvalues of discrete linear second order neumann eigenvalue problems we study the eigenvalues of discrete linear second order periodic and antiperiodic eigenvalue problems with sign changing weight we find that these two problems have t real eigenvalues including the multiplicity respectively furthermore the numbers of positive eigenvalues are equal to the numbers of positive elements in the weight function and the numbers of negative eigenvalues are equal to the numbers of negative elements in the weight function furthermore these eigenvalues including the eigenvalues of neumann problem satisfy the order relation

c filtered modules and proper costratifying systems abstract in this paper we define and study the notion of a proper costratifying system which is a generalization of the so called proper costandard modules to the context of stratifying systems the proper costandard modules were defined by v dlab in his study of quasi hereditary algebras see dlab 1996 d1

mixed hierarchy of soliton equations the mixed hierarchy of soliton equations in 1 1 dimensions is introduced it contains nonholonomic deformations of soliton equations such the kdv6 equation and the kupershmidt deformations of soliton equations as special members based on the commutator representation method a recipe for constructing zero curvature representations of mixed hierarchy is proposed as applications we obtain the mixed hierarchies and their zero curvature representations for the korteweg de vries hierarchy the ablowitz kaup newell segur hierarchy the modified korteweg de vries hierarchy the toda lattice hierarchy and the volterra lattice hierarchy

an involutive system and integrable c neumann system associated with the modified korteweg de vries hierarchy in this article a system of finite dimensional involutive functions is presented and proven to be integrable in the liouville sense by using the nonlinearization method the c neumann system associated with the modified korteweg de vries mkdv hierarchy is obtained thus the c neumann system is shown to be completely integrable via a gauge transformation between it and an integrable hamiltonian system finally the solution of a stationary mkdv equation and the involutive solutions of the mkdv hierarchy are secured as two examples the involutive solutions are given for the mkdv equation vt 1 4vxxx 3 2v2vx 0 and the 5th mkdv equation vt 1 16vxxxxx 5 8v2vxxx 5 2vvxvxx 5 8v3x 3 40v4vx 0

two new hierarchies containing the sine gordon and sinh gordon equation and their lax representations abstract an approach for obtaining a new lax pair from a given old lax pair is proposed and two hierarchies of nonlinear evolution equations which include the well known sine gordon and sinh gordon equation are presented in this article by the use of the general structure of commutator representations the lax representations of the two new hierarchies are obtained particularly the new lax pairs of the sine gordon and sinh gordon equation are given all of these lax representations are of operator form

negative powers of olver recursion operators olver s concept of a recursion operator for symmetries of an evolution equation is extended to include negative powers of the operator some negative order kdv equations are derived in this way in particular the inverse image of the trivial zero symmetry generator has an elegant formulation in terms of an eigenfunction of the associated schrodinger equation used in the inverse scattering solution of the kdv equation and this formulation is used to show that this new equation is the miura transform of a sinh gordon equation

existence of positive solutions of bvps for second order nonlinear difference systems this paper is concerned with the following system d 2u i k f i k u 1 k u 2 k u n k 0 k 0 t i 1 2 nwith the dirichlet boundary conditionu i 0 u i t 2 0 i 1 2 n some new results are obtained for the existence and multiplicity of positive solutions to the above system by using nonlinear alternative of leray schauder type krasnosel skii s fixed point theorem in a cone and leggett williams fixed point theorem in particular it proves that the above system has n positive solutions under suitable conditions where n is an arbitrary integer

two fold integrable hierarchy of nonholonomic deformation of the derivative nonlinear schrodinger and the lenells fokas equation the concept of the nonholonomic deformation formulated recently for the ablowitz kaup newell segur family is extended to the kaup newell class applying this construction we discover a novel integrable mixed twofold hierarchy related to the deformed derivative nonlinear schrodinger dnls equation and found the exact soliton solutions exhibiting unusual accelerating motion for both its field and the perturbing functions extending the idea of deformation the integrable perturbation of the gauge related chen lee liu dnls equation is constructed together with its soliton solution we show that the recently proposed lenells fokas lf equation falls in the deformed dnls hierarchy sharing the accelerating soliton and other unusual features higher order integrable deformations of the lf and the dnls equations are proposed

negative volterra flows taking the standard zero curvature approach we derive an infinite set of integrable equations which taken together form the negative volterra hierarchy the resulting equations turn out to be nonlocal which is usual for the negative flows however in some cases the nonlocality can be eliminated studying the combined action of both positive classical and negative volterra flows i e considering the differential consequences of equations of the extended volterra hierarchy we deduce local equations which seem to be promising from the viewpoint of applications the presented results give answers to some questions related to the classification of integrable differential difference equations we also obtain dark solitons of the negative volterra hierarchy using an elementary approach

negative even grade mkdv hierarchy and its soliton solutions in this paper we provide an algebraic construction for the negative even mkdv hierarchy which gives rise to time evolutions associated with even graded lie algebraic structures we propose a modification of the dressing method in order to incorporate a non trivial vacuum configuration and construct a deformed vertex operator for sl 2 that enable us to obtain explicit and systematic solutions for the whole negative even grade equations

spectral theory of left definite difference operators this paper is concerned with the spectral theory for the second order left definite difference boundary value problems existence of eigenvalues of boundary value problems is proved numbers of their eigenvalues are calculated and fundamental spectral results are obtained

multiple positive solutions of a discrete difference system this paper is concerned with the following system d 2u 1 k f 1 k u 1 k u 2 k 0 k 0 t d 2u 2 k f 2 k u 1 k u 2 k 0 k 0 t u 1 0 u 1 t 2 0 u 2 0 u 2 t 2 by using leggett williams fixed point theorem sufficient conditions are obtained for the existence of three positive solutions to the above system

the generalized kupershmidt deformation for the fifth order coupled kdv equations hierarchy abstract based on the kupershmidt deformation we propose the generalized kupershmidt deformation gkd to construct new systems from integrable bi hamiltonian system as applications the generalized kupershmidt deformation of the fifth order coupled kdv equations hierarchy with self consistent sources and its lax representation are presented

neumann type integrable reduction for nonlinear evolution equations in 1 1 and 2 1 dimensions a family of new neumann type systems is given in view of the nonlinearization technique realizing the variable separation of the modified jaulent miodek hierarchy and a new coupled modified kadomtsev petviashvili equation on the symplectic submanifold tsn 1 by two casimir functions and a special solution of the lenard eigenvalue equation we deduce the lax moser matrix of the neumann type systems that yields integrals of motion and the constrained hamiltonians whose vector fields are tangent to tsn 1 based on the dirac poisson bracket and a lax equation on tsn 1 a new systematic way is proposed to prove the liouville integrability of a family of neumann type systems synchronously the dirac poisson bracket and the generating function are used to reveal the explicit relation between infinite dimensional integrable systems and neumann type systems and then we point out that compatible solutions of neumann type systems yield the finite parametric solutions of 1 1 and 2 1 dimensional integrable nonlinear

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 theory of sturm liouville difference operators abstract we present several classes of explicit self adjoint sturm liouville difference operators with either a non hermitian leading coefficient function or a non hermitian potential function or a non definite weight function or a non self adjoint boundary condition these examples are obtained using a general procedure for constructing difference operators realizing discrete sturm liouville problems and the minimum conditions for such difference operators to be self adjoint with respect to a natural quadratic form it is shown that a discrete sturm liouville problem admits a difference operator realization if and only if it does not have all complex numbers as eigenvalues spectral properties of self adjoint sturm liouville difference operators are studied in particular several eigenvalue comparison results are proved

arithmetic of del pezzo surfaces of degree 4 and vertical brauer groups we show that brauer classes of a locally solvable degree 4 del pezzo surface x are vertical for some projection away from a plane g x p 1 i e that every brauer class is obtained by pullback from an element of br k p 1 as a consequence we prove that a brauer class obstructs the existence of a k rational point if and only if all k fibers of g fail to be locally solvable or in other words if and only if x is covered by curves that each have no adelic points using work of wittenberg we deduce that for certain quartic del pezzo surfaces with nontrivial brauer group the algebraic brauer manin obstruction is sufficient to explain all failures of the hasse principle conditional on schinzel s hypothesis and the finiteness of tate shafarevich groups the proof of the main theorem is constructive and gives a simple and practical algorithm distinct from that in 5 for computing all classes in the brauer group of x modulo constant algebras c 2014 elsevier inc all rights reserved

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

finite gap solutions of 2 1 dimensional integrable nonlinear evolution equations generated by the neumann systems each soliton equation in the korteweg de vries kdv hierarchy the 2 1 dimensional breaking soliton equation and the 2 1 dimensional caudrey dodd gibbon kotera sawada cdgks equation are reduced to two or three neumann systems on the tangent bundle tsn 1 of the unit sphere sn 1 the lax moser matrix for the neumann systems of degree n 1 is deduced in view of the mckean trubowitz identity and a bilinear generating function whose favorite characteristic accounts for the problem of the genus of riemann surface matching to the number of elliptic variables from the lax moser matrix the constrained hamiltonians in the sense of dirac poisson bracket for all the neumann systems are written down in a uniform recursively determined by integrals of motion the involution of integrals of motion and constrained hamiltonians is completed on tsn 1 by using a lax equation and their functional independence is displayed over a dense open subset of tsn 1 by a direct calculation which contribute to the liouville integ

socles and radicals of mackey functors we study the socle and the radical of a mackey functor m for a finite group g over a field k usually of characteristic p 0 for a subgroup h of g we construct bijections between some classes of the simple subfunctors of m and some classes of the simple kn g h submodules of m h we relate the multiplicity of a simple mackey functor sh vg in the socle of m to the multiplicity of v in the socle of a certain kn g h submodule of m h we also obtain similar results for the maximal subfunctors of m we then apply these general results to some special mackey functors for g including the functors obtained by inducing or restricting a simple mackey functor mackey functors for a p group the fixed point functor and the burnside functor bkg for instance we find the first four top factors of the radical series of bkg for a p group g and assuming further that g is an abelian p group we find the radical series of bkg

the socle of a projective mackey functor for a p group we study the simple subfunctors of indecomposable projective mackey functors for a p group p unlike the case of group algebras the mackey algebra is not in general self injective thus the socle of an indecomposable projective functor is not in general simple we first show that the simple subfunctors of a projective functor p h k where h p are indexed by the normalizer in h of a subgroup of h we then study the socle of a specific projective mackey functor namely the burnside functor b p and we focus on the case where p is abelian in particular our study enables us to determine the socle of an indecomposable projective mackey functor indexed by a cyclic p group an abelian p group of rank 2 and an elementary abelian p group of rank 3

existence of positive solutions of boundary value problem for a discrete difference system in this paper we consider the following boundary value problem of discrete system d 2u 1 k f 1 k u 1 k u 2 k 0 k 0 t d 2u 2 k f 2 k u 1 k u 2 k 0 k 0 t with the dirichlet boundary value conditionu 1 0 u 1 t 2 0 u 2 0 u 2 t 2 some sufficient conditions are obtained for the existence of one or two positive solutions to the above system by using nonlinear alternative of leray schauder type and krasnosel skii s fixed point theorem in a cone

the quasi periodic solutions of mixed kdv equations the quasi periodic solutions of mixed kdv equations are constructed to derive the solutions we introduce the mixed finite dimensional integrable hamiltonian systems fdihs which are linearized on the jacobi variety of the associated hyperelliptic curve a relation between the fdihs and the mixed kdv equations is revealed based on this relation we construct the quasi periodic solutions for the mixed kdv equations with aid of the riemann theta function the quasi periodic solutions for the kdv6 equation and the negative order kdv equation are obtained as special members of the mixed kdv equations

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

sine gordon and modified korteweg de vries charges the sine gordon and the integrated modified korteweg de vries equations are shown to conserve the same infinite set of charges the charges are determined by a recursion relation as a consequence the solutions of all the equations generated by the charges have in common all time independent properties

eigenvalues of second order symmetric equations on time scales with periodic and antiperiodic boundary conditions this paper studies periodic and antiperiodic boundary value problems for second order symmetric linear equations on time scales by properties of eigenvalues of the dirichlet boundary value problem and some oscillation results existence of eigenvalues of these two different boundary value problems is proved numbers of their eigenvalues are calculated and their relationships are obtained these results not only unify the existing ones of periodic and antiperiodic boundary value problems for second order symmetric differential and difference equations but also contain more complicated time scales

mackey functors with chain conditions abstract letabe a green functor for a finite groupgand letmbe a lefta module mis callednoetherian artinian if the lattice of all submodules ofmsatisfies acc respectively dcc mis calledtotally noetherian totally artinian ifmxis noetherian respectively artinian for every finiteg setx in this paper it is shown that the totally noetherian totally artinian lefta modules behave in the same way as the classical noetherian artinian modules the left noetherian left artinian green functors are less well behaved but some properties from classical algebra still hold it is shown that cohen s theorem holds in this setting our main result shows that if a commutative green functorais artinian then the ringsa h jac a h are semisimple artinian for all subgroupshofg