the friedrichs extension of singular differential operators abstract the friedrichs extension of singular ordinary differential operators of order 2 n is characterized in terms of singular boundary conditions these are given in terms of principal solutions

fourth order canonical forms of singular self adjoint boundary conditions abstract canonical forms of self adjoint boundary conditions are well known in the second order sturm liouville case for both regular and singular problems these are critical for the theoretical investigation of the eigenvalues as well as their numerical computation recently canonical forms have been found for fourth order regular problems these are much more complicated than the second order ones here we find canonical forms for fourth order singular problems with one or both endpoints of the domain interval singular and with arbitrary deficiency index d in the regular fourth order case d 4 in the singular case d can assume any value between 0 and 4 and moreover depends on the nature of the singularities at the two endpoints these different values of d and their dependence on the endpoints introduces serious additional complications

symplectic geometry and dissipative differential operators abstract everitt and markus characterized the domains of self adjoint differential operators in terms of lagrangian subspaces of complex symplectic spaces in this paper we define dissipative and strictly dissipative subspaces for complex symplectic spaces and characterize the domains of dissipative and strictly dissipative differential operators in terms of these subspaces

critical probabilities for diversity and number of clusters in randomly occupied square lattices we report the occurrence of critical probabilities associated with the maximum of diversity and the maximum number of fragments clusters on a two dimensional square lattice some scaling relations of these two variables are observed in accordance with work on fragmentation processes

fourth order boundary value problems with finite spectrum abstract for every positive integer m we construct a class of regular self adjoint and non self adjoint fourth order boundary value problems with at most 3 m 1 eigenvalues counting multiplicity

the c1 property of carrying simplices for a class of competitive systems of odes abstract in this paper we consider a class of dissipative competitive systems of odes on the nonneptive orthant c we assume among others that the system restricted to any face of c is weakly persistent it is proved that the carrying simplex is a c1 manifold with corners in the proof we analyze the asymptotic behavior of the linearization of the original system

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

self adjoint domains symplectic geometry and limit circle solutions abstract everitt and markus characterized the domains of self adjoint operator realizations of very general even and odd order symmetric ordinary differential equations in terms of lagrangian subspaces of symplectic spaces recently for the even order case with real coefficients wang sun and zettl constructed limit circle lc solutions and hao wang sun and zettl characterized the self adjoint domains in terms of lc solutions these lc solutions are higher order analogues of the celebrated titchmarsh weyl limit circle solutions in the second order case this lc characterization has been used to obtain information about the discrete continuous and essential spectra of these operators in this paper we investigate the connection between these two very different kinds of characterizations and thus add the methods of symplectic geometry to the techniques available for the investigation of the spectrum of self adjoint operators

the spectrum of differential operators and square integrable solutions abstract we give a comprehensive account of the relationship between the square integrable solutions for real values of the spectral parameter and the spectrum of self adjoint even order ordinary differential operators with real coefficients and arbitrary deficiency index d and we solve an open problem stated by weidmann in his well known 1987 monograph according to a well known result if one endpoint is regular and for some real value of the spectral parameter the number of linearly independent square integrable solutions is less than d then is in the essential spectrum of every self adjoint realization of the equation weidmann extends this result to the two singular endpoint case provided an additional condition is satisfied here we prove this result without the additional condition

characterization of domains of self adjoint ordinary differential operators article history the gkn glazman krein naimark theorem characterizes all self adjoint realizations of linear symmetric formally self adjoint ordinary differential equations in terms of maximal domain functions these functions depend on the coefficients and this dependence is implicit and complicated in the regular case an explicit characterization in terms of two point boundary conditions can be given in the singular case when the deficiency index d is maximal the gkn characterization can be made more explicit by replacing the maximal domain functions by a solution basis for any real or complex value of the spectral parameter i n the much more difficult intermediate cases not all solutions contribute to the singular self adjoint conditions in 1986 sun found a representation of the self adjoint singular conditions in terms of certain solutions for nonreal values of in this paper we give a representation in terms of certain solutions for real this leads to a classification of solutions as limit point lp or limit circle lc in analogy with the celebrated weyl classification in the second order case the lc solutions contribute to the singular boundary conditions the lp solutions do not the advantage of using real is not only because it is in general easier to find explicit solutions but more importantly it yields information about the spectrum

limit cycles for competitor competitor mutualist lotka volterra systems abstract it is known that a limit cycle or periodic coexistence can occur in a competitor competitor mutualist lotka volterra system x 1 x 1 r 1 a 11 x 1 a 12 x 2 a 13 x 3 x 2 x 2 r 2 a 21 x 1 a 22 x 2 a 23 x 3 x 3 x 3 r 3 a 31 x 1 a 32 x 2 a 33 x 3 where r i a i j are positive real constants x liang j jiang the dynamical behavior of type k competitive kolmogorov systems and its applications to 3 dimensional type k competitive lotka volterra systems nonlinearity 16 2003 785 801 in this paper we shall construct an example with at least two limit cycles and furthermore we will show that the number of periodic orbits and hence a fortiori of limit cycles is finite it is also shown that contrary to three dimensional competitive lotka volterra systems nontrivial periodic coexistence does occur even if none of the three species can resist invasion from either of the others in this case new amenable conditions are given on the coefficients under which the system has no nontrivial periodic orbits these conditions imply that the positive equilibrium if it exists is globally asymptotically stable

j self adjoint extensions of a class of hamiltonian differential systems abstract this paper is concerned with j self adjoint extensions for a class of hamiltonian differential systems the domains of the corresponding minimal and maximal operators are described a classification of limit types of the systems at each of endpoints is given a property of the regularity field of the corresponding minimal operator is derived and an expression of the j defect index of the minimal operator by those of the left and right minimal operators is presented based on them a characterization of all j self adjoint extensions for the hamiltonian system is obtained in terms of square integrable solutions of a certain equation which is generated by this hamiltonian system and its formal adjoint as a consequence characterizations of all j self adjoint extensions are given for the systems in some special cases in particular all j self adjoint extensions are characterized in terms of square integrable solutions of the system itself in certain special cases

on the convexity of the carrying simplex of planar lotka volterra competitive systems in this paper we prove that a result by zeeman and zeeman on the convexity of carrying simplices in competitive lotka volterra systems differential equations dynamical systems and control science a festschift in honor to laurence markus 1994 pp 353 364 is partially wrong and we find necessary and sufficient conditions for which the carrying simplex of a planar competitive lotka volterra system is twice continuously differentiable in particular the main result by mierczynski j differential equations 111 1994 385 409 is sharp

cluster size entropy in the axelrod model of social influence small world networks and mass media we study the axelrod s cultural adaptation model using the concept of cluster size entropy s c which gives information on the variability of the cultural cluster size present in the system using networks of different topologies from regular to random we find that the critical point of the well known nonequilibrium monocultural multicultural order disorder transition of the axelrod model is given by the maximum of the s c q distributions the width of the cluster entropy distributions can be used to qualitatively determine whether the transition is first or second order by scaling the cluster entropy distributions we were able to obtain a relationship between the critical cultural trait q c and the number f of cultural features in two dimensional regular networks we also analyze the effect of the mass media external field on social systems within the axelrod model in a square network we find a partially ordered phase whose largest cultural cluster is not aligned with the external field in contrast with a recent suggestion that this type of phase cannot be formed in regular networks we draw a q b phase diagram for the axelrod model in regular networks

self adjoint domains of products of differential expressions under the assumption that the product l2 of the formally symmetric differential expression l of order n on a is partially separated in l2 a we present a new characterization of self adjoint boundary conditions for l2 for two differential operators t1 l and t2 l associated with l we show that the product t2 l t1 l is self adjoint if and only if t2 l t 1 l it extends the previous result in 1 where both t1 l and t2 l are self adjoint singular limit circle sturm liouville operators furthermore we also characterize the boundary conditions of the friedrichs extension of the minimal operator generated by l2

continuous spectrum and square integrable solutions of differential operators with intermediate deficiency index we explore the connection between square integrable solutions for real values of the spectral parameter and the continuous spectrum of self adjoint ordinary differential operators with arbitrary deficiency index d we show that if for all in an open interval i there are d of linearly independent square integrable solutions then for every extension of dmin the point spectrum is nowhere dense in i and there is a self adjoint extension of smin which has no continuous spectrum in i this analysis is based on our construction of limit point lp and limit circle lc solutions obtained recently in an earlier paper

the finite element galerkin method for singular self adjoint differential equations we investigate the finite element galerkin method for singular self adjoint second order differential expressions the weak formulation of the problem involves integration by parts which allows the use of the usual piecewise linear functions our analysis shows that the method produces the solution corresponding to a particular self adjoint realization of the differential expression we also propose two algorithms to approximate the solution of any self adjoint realization numerical examples are given to illustrate the analysis as well as the proposed algorithms

boundary triples and weyl m functions for powers of the jacobi differential operator abstract the abstract theory of boundary triples is applied to the classical jacobi differential operator and its powers in order to obtain the weyl m function for several self adjoint extensions with interesting boundary conditions separated periodic and those that yield the friedrichs extension these matrix valued nevanlinna herglotz m functions are to the best knowledge of the author the first explicit examples to stem from singular higher order differential equations the creation of the boundary triples involves taking pieces determined in 26 of the principal and non principal solutions of the differential equation and putting them into the sesquilinear form to yield maps from the maximal domain to the boundary space these maps act like quasi derivatives which are usually not well defined for all functions in the maximal domain of singular expressions however well defined regularizations of quasi derivatives are produced by putting the pieces of the non principal solutions through a modified gram schmidt process

carrying simplices in nonautonomous and random competitive kolmogorov systems abstract the purpose of this paper is to investigate the asymptotic behavior of positive solutions of nonautonomous and random competitive kolmogorov systems via the skew product flows approach it is shown that there exists an unordered carrying simplex which attracts all nontrivial positive orbits of the skew product flow associated with a nonautonomous random competitive kolmogorov system

the classification of self adjoint boundary conditions separated coupled and mixed there are three basic types of self adjoint regular and singular boundary conditions separated coupled and mixed for even order problems with real coefficients one regular endpoint and arbitrary deficiency index d we give a construction for each type and determine the number of possible conditions of each type under the assumption that there are d linearly independent square integrable solutions for some real value of the spectral parameter in the separated case our construction yields non real conditions for all orders greater than two it is well known that no such conditions exist in the second order case our construction gives a direct alternative to the recent construction of everitt and markus which uses the theory of symplectic spaces we believe our construction will prove useful in the spectral analysis of these operators and in obtaining canonical forms of self adjoint boundary conditions such forms are known only in the second order i e sturm liouville case even for regular problems of order four no such forms are available

on the self adjoint extensions of symmetric ordinary differential operators in direct sum spaces in this paper we give the characterizations of self adjoint domains for singular symmetric operators by describing the boundary conditions of domain of conjugate differential operator with interior singular points or in direct sum spaces last we show that there exist other self adjoint extension of t0 m which cannot be expressed as the direct sum of the self adjoint extensions over the various subintervals

dynamics alternatives and generic convergence for c1 smooth strongly monotone discrete dynamical systems abstract for c 1 smooth strongly monotone discrete time dynamical systems we prove dynamics alternatives which concludes that any compact orbit is either asymptotic to a linearly stable cycle or manifestly unstable for this purpose we improve several properties of the exponential separation for continuous maps the generic convergence to cycles is obtained as a by product of the dynamics alternatives

positive self adjoint operators generated by products of differential expressions abstract under the assumption of the product l l of the regular differential expression l and its adjoint l being well formed a complete characterization of all the positive self adjoint extensions of the minimal operator generated by in terms of boundary conditions is given

boundary conditions for general ordinary differential operators and their adjoints a characterisation is obtained of all the regularly solvable operators and their adjoints generated by a general differential expression in the domains of these operators are described in terms of boundary conditions involving the solutions of mu wu and the adjoint equation the results include those of sun jiong 15 concerning self adjoint realisations of a symmetric m when the minimal operator has equal deficiency indices if the deficiency indices are unequal the maximal symmetric operators are determined by the results herein another special case concerns the j self adjoint operators where j denotes complex conjugation and for this we recover the results of zai jiu shang in 16

a symmetric gkn type theorem and symmetric differential operators abstract we characterize the two point boundary conditions which determine symmetric operators in hilbert space and prove a symmetric gkn type theorem this for general symmetric differential expressions of any order even or odd and for arbitrary deficiency index the self adjoint operators are a special case

an expanded phillips theory and its application to differential operators abstract a new characterization of maximal accretive extensions of accretive operators is presented it is based on expanded spaces of the abstract boundary spaces developed by r s phillips and is therefore called the expanded phillips theory in particular when the operator is symmetric a concrete realization of the expanded spaces is given in applications of the expanded phillips theory to ordinary differential equations we characterize all maximal accretive extensions of singular symmetric ordinary differential operators of order 2 n

dynamics of the periodic type k competitive kolmogorov systems abstract for time periodic dissipative and irreducible type k competitive kolmogorov systems it is proved that there is a canonically defined countable family f of unordered disjoint invariant sets with the property that for every persistent trajectory whose limit set is not a cycle there exists a unique trajectory in some element of f such that these two trajectories are asymptotic and the corresponding points in these two trajectories are k related

cluster diversity and entropy on the percolation model the lattice animal identification algorithm we present an algorithm to identify and count different lattice animals la s in the site percolation model this algorithm allows a definition of clusters based on the distinction of cluster shapes in contrast with the well known hoshen kopelman algorithm in which the clusters are differentiated by their sizes it consists in coding each unit cell of a cluster according to the nearest neighbors nn and ordering the codes in a proper sequence in this manner a la is represented by a specific code sequence in addition with some modification the algorithm is capable of differentiating between fixed and free la s the enhanced hoshen kopelman algorithm j hoshen m w berry and k s minser phys rev e 56 1455 1997 is used to compose the set of nn code sequences of each cluster using monte carlo simulations on planar square lattices up to 2000x2000 we apply this algorithm to the percolation model we calculate the cluster diversity and cluster entropy of the system which leads to the determination of probabilities associated with the maximum of these functions we show that these critical probabilities are associated with the percolation transition and with the complexity of the system

three dimensional competitive lotka volterra systems with no periodic orbits the following conjecture of m l zeeman is proved if three interacting species modeled by a competitive lotka volterra system can each resist invasion at carrying capacity then there can be no coexistence of the species indeed two of the species are driven to extinction it is also proved that in the other extreme if none of the species can resist invasion from either of the others then there is stable coexistence of at least two of the species in this case if the system has a fixed point in the interior of the positive cone in r3 then that fixed point is globally asymptotically stable representing stable coexistence of all three species otherwise there is a globally asymptotically stable fixed point in one of the coordinate planes of r3 representing stable coexistence of two of the species

from local to global behavior in competitive lotka volterra systems in this paper we exploit the linear quadratic monotone and geometric structures of competitive lotka volterra systems of arbitrary dimension to give geometric algebraic and computational hypotheses for ruling out nontrivial recurrence we thus deduce the global dynamics of a system from its local dynamics the geometric hypotheses rely on the introduction of a split liapunov function we show that if a system has a fixed point p int r n and the carrying simplex of the system lies to one side of its tangent hyperplane at p then there is no nontrivial recurrence and the global dynamics are known we translate the geometric hypotheses into algebraic hypotheses in terms of the definiteness of a certain quadratic function on the tangent hyperplane finally we derive a computational algorithm for checking the algebraic hypotheses and we compare this algorithm with the classical volterra liapunov stability theorem for lotka volterra systems

on invariant hypersurfaces of strongly monotone maps forck k 1 strongly monotone discrete time dynamical systems we present simple criteria which ensure that nonmonotone invariant manifolds areck

scaling and critical probabilities for cluster size and la diversity on randomly occupied square lattices using monte carlo simulations we report the behaviour of the total number of clusters the cluster size diversity and the lattice animals la diversity on randomly occupied square lattices the critical probability associated with the maximum of these variables is determined in comparison with the percolation probability pc our simulations indicate that pc and the critical probability of the maximum cluster size diversity pc d s max occur at the same point as indicated in a previous paper tsang i r and tsang i j 1997 j phys a math gen 30 l239 the probability for the maximum number of clusters is obtained at a lower value pc n max 0 27 0 01 we describe the cluster identification algorithm used to count the different la and determined the critical probability for the maximum la diversity pc d f max 0 45 0 02 we derive the exponents characterizing the relation between d s max d f max and n max and several scaling relations between the variables measured the lattice size and the probability of occupation p in addition we show the scaling behaviour of la diversity versus cluster size diversity for each value of p

eigenvalues below the lower bound of minimal operators of singular hamiltonian expressions for singular hamiltonian systems in the limit point case this paper characterizes the number of eigenvalues of self adjoint extensions below the lower bound of the minimal operator in terms of corresponding self adjoint boundary conditions and the m l matrix for the friedrichs extension

regular approximation of linear hamiltonian operators with two singular endpoints abstract in this paper we investigate regular approximation of linear hamiltonian operators with two singular endpoints for any given singular self adjoint hamiltonian operator its spectrum can be approximated with the eigenvalues of a sequence of regular operators the inherited restriction operators are constructed by the limit circle solutions and the limit point solutions and proved to be self adjoint via the abstract operator theory we obtain the strongly resolvent convergence of the inherited regular operators and spectral inclusion of spectrum moreover for the case with both endpoints being limit circle spectral exactness of spectrum is obtained by hilbert schmidt norm convergence in the case of friedrichs extension of singular hamiltonian systems spectral exactness also holds due to variational principle of eigenvalues

a gkn theorem for symmetric operators abstract we prove a gkn theorem for symmetric operators the well known gkn theorem for self adjoint operators is a special case of this result

formation and maintenance of complex systems a complex system is often identified by the absence of a characteristic length e g as in a fractal a very large system subject to a fragmentation and or aggregation dynamics passes through such complex configurations we study statistically creation and maintenance of such configurations in space dimensions d 1 to 5 and find that they are easily created maintained for small large d an intermediate d such as d 3 seems to be ideal for the creation and maintenance of complex systems this has consequences in a statistical description of the universe

a general left definite theory for certain self adjoint operators with applications to differential equations we show that any self adjoint operator a bounded or unbounded in a hilbert space h v that is bounded below generates a continuum of hilbert spaces hr r 0 and a continuum of self adjoint operators ar r 0 for reasons originating in the theory of differential operators we call each hr the rth left definite space and each ar the rth left definite operator associated with h a each space hr can be seen as the closure of the domain d ar of the self adjoint operator ar in the topology generated from the inner product arx y x y d ar furthermore each ar is a unique self adjoint restriction of a in hr we show that the spectrum of each ar agrees with the spectrum of a and the domain of each ar is characterized in terms of another left definite space the hilbert space spectral theorem plays a fundamental role in these constructions we apply these results to two examples including the classical laguerre differential expression l in which we explicitly find the left definite spaces and left definite operators associated with a the self adjoint operator generated by l in l2 0 t e t having the laguerre polynomials as eigenfunctions

kolmogorov vector fields with robustly permanent subsystems the following results are proven all subsystems of a dissipative kolmogorov vector field i xifi x are robustly permanent if and only if the external lyapunov exponents are positive for every ergodic probability measure with support in the boundary of the nonnegative orthant if the vector field is also totally competitive its carrying simplex is c1 applying these results to dissipative lotka volterra systems robust permanence of all subsystems is equivalent to every equilibrium x satisfying fi x 0 whenever xi 0 if in addition the lotka volterra system is totally competitive then its carrying simplex is c1

properties and decompositions of domains for powers of the jacobi differential operator abstract we set out to build a framework for self adjoint extension theory for powers of the jacobi differential operator that does not make use of classical deficiency elements instead we rely on simpler functions that capture the impact of these elements on extensions but are defined by boundary asymptotics this new perspective makes calculations much more accessible and allows for a more nuanced analysis of the associated domains the maximal domain for n th composition of the jacobi operator is characterized in terms of a smoothness condition for each derivative and the endpoint behavior of functions in the underlying hilbert space can then be classified for j n 0 by 1 x j 1 x j 1 x j and 1 x j most of these behaviors can only occur when functions are in the associated minimal domain and this leads to a formulation of the defect spaces with a convenient basis self adjoint extensions including the important left definite domain are then given in terms of the new basis functions for the defect spaces using gkn theory comments are made for the laguerre operator as well

symmetrical differential operators and their friedrichs extension abstract symmetric operator realizations of ordinary regular differential expressions are characterized explicitly by boundary conditions for any such operator which is bounded below the boundary condition determining its friedrichs extension is identified

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

semi boundedness of ordinary differential operators abstract the boundedness from above and from below of very general symmetric quasi differential operators s is studied here we show that i if s is regular of even order and has positive leading coefficient then it is bounded below ii if the order of s which may be regular or singular is odd then s is unbounded both above and below and iii if s is of even order regular or singular but with a leading coefficient which changes sign then s is unbounded both above and below

on the game of life population and its diversity abstract one of the most important features of biological life in all levels is its astounding diversity in this work we study the well known game life due to conway analysing the statistics of cluster population n t and cluster diversity d t we have performed simulations on life for dimensions d 1 and 2 starting with an uncorrelated distribution of live and dead sites at t 0 for d 2 we study the effect of different neighbourhood relations in identifying and counting clusters an interesting scaling relation connecting the maxima of n t and d t is found

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

the classification of self adjoint boundary conditions of differential operators with two singular endpoints abstract for general even order linear ordinary differential equations with real coefficients and endpoints which are regular or singular and for arbitrary deficiency index d the self adjoint domains are determined by d linearly independent boundary conditions these conditions are of three types separated coupled and mixed we give a construction for all conditions of each type and determine the number of conditions of each type possible for a given self adjoint domain our construction gives a direct alternative to the recent construction of everitt and markus which uses the theory of symplectic spaces we believe our construction will prove useful in the spectral analysis of these operators and in obtaining canonical forms of self adjoint boundary conditions such forms are known only in the second order i e sturm liouville case even for regular problems of order four no such forms are available in the case when all d conditions are separated this construction yields explicit non real conditions for all orders greater than two it is well known that no such conditions exist in the second order case

friedrichs extension of operators defined by even order sturm liouville equations on time scales in this paper we characterize the friedrichs extension of operators associated with the 2n th order sturm liouville dynamic equations on time scales with using the time reversed symplectic systems and its recessive system of solutions a nontrivial example is also provided

numerical simulations for fragmentation dynamics a geometrical model for the study of fragmentation dynamics is introduced detailed statistical analysis of the fragment distribution for percolation clusters sierpinski carpets and menger sponges on square and cubic lattices is given and static and dynamic critical exponents are calculated

dissipative operators symplectic geometry and limit circle solutions for a general symmetric ordinary differential expression m and a positive weight function w the authors characterized the dissipative and strictly dissipative extensions of the minimal operator generated by m in terms of subspaces of a symplectic geometry space here we characterize these subspaces in terms of lc solutions namely we characterize the boundary conditions of the dissipative and strictly dissipative extensions by lc solutions in a hilbert space

smoothness of the carrying simplex for discrete time competitive dynamical systems a characterization of neat embedding abstract the paper is concerned with the question of smoothness of the carrying simplex s for a discrete time dissipative competitive dynamical system we give a necessary and sufficient criterion for s being a c 1 submanifold with corners neatly embedded in the nonnegative orthant formulated in terms of inequalities between lyapunov exponents for ergodic measures supported on the boundary of the orthant this completes one thread of investigation occasioned by a question posed by m w hirsch in 1988 besides amenable conditions are presented to guarantee the c r r 1 smoothness of s in the time periodic competitive kolmogorov systems of odes examples are also presented one in which s is of class c 1 but not neatly embedded the other in which s is not of class c 1

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

gkn theory for linear hamiltonian systems abstract in this paper we give the definition of maximal and minimal operators for linear hamiltonian systems and investigate the relationship between the conjugate scalar product in a weighted hilbert space and the skew symmetric boundary form of the associated singular hamiltonian operator namely the one to one correspondence between the set of self adjoint extensions of the minimal operator and the set of lagrangian symplectic subspaces these results extend and improve the classical glazman krein naimark gkn theory for quasi differential operators

characterization of self adjoint extensions for discrete symplectic systems all self adjoint extensions of minimal linear relation associated with the discrete symplectic system are characterized especially for the scalar case on a finite discrete interval some equivalent forms and the uniqueness of the given expression are discussed and the krein von neumann extension is described explicitly in addition a limit point criterion for symplectic systems is established the result partially generalizes even the classical limit point criterion for the second order sturm liouville difference equations

matrix representations of fourth order boundary value problems with finite spectrum abstract we show that a class of regular self adjoint fourth order boundary value problems is equivalent to a certain class of matrix problems equivalent here means that they have exactly the same eigenvalues such an equivalence was previously known only in the second order case

invariant foliations near normally hyperbolic invariant manifolds for semiflows let m be a compact c1 manifold which is invariant and normally hyperbolic with respect to a c1 semiflow in a banach space then in an neighborhood of m there exist local c1 center stable and center unstable manifolds w cs and w cu respectively here we show that w cs and w cu may each be decomposed into the disjoint union of c1 submanifolds leaves in such a way that the semiflow takes leaves into leaves of the same collection furthermore each leaf intersects m in a single point which determines the asymptotic behavior of all points of that leaf in either forward or backward time

regular approximation of singular sturm liouville problems with transmission conditions in this paper we study regular approximation of singular sturm liouville problems with transmission conditions we construct all self adjoint extensions of the minimal operators and the induced restriction operators using the abstract operator theory in the hilbert space we obtain that the spectrum of singular sturm liouville problems with transmission conditions can be approximated by eigenvalues of a sequence of regular sturm liouville problems with transmission conditions

friedrichs extensions for singular hamiltonian operators with intermediate deficiency indices abstract for singular hamiltonian operators in the intermediate deficiency indices case we give a complete characterization of friedrichs extensions of minimal hamiltonian operators which unifies and generalizes some known results in the literature the exact boundary conditions for the friedrichs extensions are constructed via the principal solutions the main approach in this paper is the friedrichs construction by way of the refined lc type solutions at singular endpoints

real parameter square integrable solutions and the spectrum of differential operators abstract we continue to investigate the connection between the spectrum of self adjoint ordinary differential operators with arbitrary deficiency index d and the number of linearly independent square integrable solutions for real values of the spectral parameter we show that if for all in an open interval i there are d linearly independent square integrable solutions then there is no continuous spectrum in i this for any self adjoint realization with boundary conditions which may be separated coupled or mixed the proof is based on a new characterization of self adjoint domains and on limit point lp and limit circle lc solutions established in an earlier paper

on j selfadjoint extensions of j symmetric ordinary differential operators on donne une caracterisation des conditions aux limites qui determinent le domaine d une extension j autoadjointe d un operateur differentiel j symetrique t 0 qui a une extremite singuliere

complex j symplectic geometry characterization for j symmetric extensions of j symmetric differential operators we give complex j symplectic geometry characterizations for j symmetric extensions of j symmetric ordinary differential operators

linearization and invariant manifolds on the carrying simplex for competitive maps abstract a result due to m w hirsch states that most competitive maps admit a carrying simplex i e an invariant hypersurface of codimension one which attracts all nontrivial orbits the common approach in the study of these maps is to focus on the dynamical behavior on the carrying simplex however this manifold is normally non smooth therefore not every tool coming from differential geometry can be applied in this paper we prove that the restriction of the map to the carrying simplex in a neighborhood of an interior fixed point is topologically conjugate to the restriction of the map to its pseudo unstable manifold by an invariant foliation this implies that the linearization techniques are applicable for studying the local dynamics of the interior fixed points on the carrying simplex we further construct the stable and unstable manifolds on the carrying simplex our results give partial responses to hirsch s problem regarding the smoothness of the carrying simplex we discuss some applications in classical models of population dynamics

the higher order differential operators in direct sum spaces w n everitt and a zettl l introduced the concept of differential operator into the direct sum space and provided an operator theoretic framework for the study of two generalized sturm liouville differential operators together m defined on an interval i and m defined on i in particular they characterized self adjoint extensions of the minimal operator in the direct sum space in terms of boundary conditions the aim of this paper is to study the same problem for the higher order differential operators in the direct sum spaces since sturm liouville differential operators always have equal deficiency indices the minimal operators in the direct sum spaces also have equal deficiency indices and self adjoint extensions but for higher order differential operators on an interval z as is well known the values of the deficiency index are very complicated e g see 7 chap 61 hence the minimal operator in the direct sum space has equal deficiency indices only when the deficiency indices of two higher order differential operators satisfy certain conditions in this paper we use cao s and sun s methods 2 31 to give conditions which characterize the domains of self adjoint extensions of minimal operators