dependence of eigenvalues of sturm liouville problems with interface conditions this paper investigates sturm liouville problems with interface conditions we obtain that the eigen values depend not only continuously but also smoothly on the coefficient functions boundary conditions and interface conditions moreover the derivative of the eigenvalues with respect to each of these parameters is given

left definite sturm liouville problems abstract left definite regular self adjoint sturm liouville problems with either separated or coupled boundary conditions are studied we give an elementary proof of the existence of eigenvalues for these problems for any fixed equation we establish a sequence of inequalities among the eigenvalues for different boundary conditions and estimate the range of each eigenvalue as a function on the space of boundary conditions some of our results here yield an algorithm for numerically computing the eigenvalues of a left definite problem with an arbitrary coupled boundary condition our inequalities imply that the well known asymptotic formula for the eigenvalues in the separated case also holds in the coupled case moreover we study the continuous and differentiable dependence of the eigenvalues of the general left definite problem on all the parameters in its differential equation and boundary condition

a supersymmetric extension of the kadomtsev petviashvili hierarchy an extension of the kadomtsev petviashvili hierarchy by odd variables is given conservation laws and formal integrability are proved

the study of jacobi and cyclic jacobi matrix eigenvalue problems using sturm liouville theory abstract we study the eigenvalues of matrix problems involving jacobi and cyclic jacobi matrices as functions of certain entries of particular interest are the limits of the eigenvalues as these entries approach infinity our approach is to use the recently discovered equivalence between these problems and a class of sturm liouville problems and then to apply the sturm liouville theory

hypothetical analyses approximate counting in the style of knuth path length in the style of flajolet abstract the first analysis of approximate counting is due to flajolet 1985 whereas the first satisfactory analysis of the average path length in digital search trees has been performed by knuth 1973 both authors have used the mellin integral transform but in rather different ways it was shown by kirschenhofer and prodinger 1991 that both problems are very similar this note contains also an explanation of this phenomenon it is amusing to figure out what flajolet and knuth would have done by considering the exchanged problems the aim of this note is to perform these analyses

dependence of the nth sturm liouville eigenvalue on the problem abstract we consider the nth eigenvalue as a function on the space of self adjoint regular sturm liouville problems with positive leading coefficient and weight functions the discontinuity of the nth eigenvalue is completely characterized

numerical solution of sturm liouville problems via fer streamers we address the numerical challenge of solving regular sturm liouville problems in liouville s normal form with a continuous and piecewise analytic potential and self adjoint separated boundary conditions the novelty of our approach which is based on a non standard truncation of fer expansions which we call fer streamers lies in the construction of a new numerical method which 1 does not impose any restriction on the step size for eigenvalues which are greater than or equal to the minimum of the potential 2 requires only a mild restriction on the step size for the remaining finite number of eigenvalues 3 can attain any convergence rate which grows exponentially with the number of terms and is uniform for every eigenvalue and 4 lends itself to a clear understanding of the manner in which the potential affects the local and global errors we provide our numerical method with its analytical underpinning but emphasize that it is at an early stage of development and that much remains to be done in particular we comment on our investigation of efficient discretization schemes for the integrals which arise in fer streamers

digital search trees again revisited the internal path length perspective this paper studies the asymptotics of the variance for the internal path length in a symmetric digital search tree under the bernoulli model this problem has been open until now it is proved that the variance is asymptotically equal to n cdot0 26600 n cdot delta log 2 n where n is the number of stored records and delta x is a periodic function of mean zero and a very small amplitude this result completes a series of studies devoted to the asymptotic analysis of the variances of digital tree parameters in the symmetric case in order to prove the previous result a number of nontrivial problems concerning analytic continuations and some others of a numerical nature had to be solved in fact some of these techniques are motivated by the methodology introduced in an influential paper by flajolet and sedgewick

eigenvalues of regular sturm liouville problems abstract the eigenvalues of sturm liouville sl problems depend not only continuously but smoothly on the problem an expression for the derivative of an eigenvalue with respect to a given parameter an endpoint a boundary condition a coefficient or the weight function is found

floquet theory for left definite sturm liouville problems abstract we determine the essential spectrum of left definite sturm liouville problems with periodic coefficients and a weight function which changes sign

bi hamiltonian structure of super kp hierarchy the bi hamiltonian structure of the super kp hierarchy is obtained based on the even super kp operator 2 i 2 ui i 1 as a supersymmetric extension of the ordinary kp bi hamiltonian structure it is expected to give rise to a universal super w algebra incorporating all known extended superconformal wn algebras by reduction also constructed is the super bkp hierarchy by imposing a set of anti self dual constraints on the super kp hierarchy

a novel symmetry constraint of the super ckdv system a new 1 1 dimensional integrable system i e the super coupled korteweg de vries ckdv system has been constructed by a super extension of the well known 1 1 dimensional ckdv system for this new system a novel symmetry constraint between the potential and the eigenfunction can be obtained by means of the binary nonlinearization of its lax pairs the constraints for even variables are explicit and the constraints for odd variables are implicit under the symmetry constraint the spacial part and the temporal parts of the equations associated with the lax pairs for the super ckdv system can be decomposed into the super finite dimensional integrable hamiltonian systems on the supersymmetry manifold r4n 2n 2 whose integrals of motion are explicitly given

singular left definite sturm liouville problems abstract we study singular left definite sturm liouville problems with an indefinite weight function the existence of eigenvalues is established based on the existence of eigenvalues of corresponding right definite problems furthermore for each singular left definite problem with limit circle non oscillatory endpoints we construct a regular left definite problem with the same eigenvalues and use it to obtain properties of eigenvalues and eigenfunctions inequalities among eigenvalues recently established for regular left definite problems are extended to the singular case

on the kadomtsev petviashvili hierarchy ŵ algebra and conformal sl 2 r u 1 model i the classical case in this article the interrelationship between the integrable kadomtsev petviashvili kp hierarchy nonlinear w algebra and the conformal noncompact sl 2 r u 1 coset model at the classical level is studied the poisson brackets of the second hamiltonian structure of the kp hierarchy is derived first explicitly then the w1 algebra and its reduction w are defined then it is shown that the latter is realized in the sl 2 r u 1 coset model as a hidden current algebra through a free field realization of w in closed form for all higher spin currents in terms of two bosons an immediate consequence is the existence of an infinite number of kp flows in the coset model which preserve the w current algebra

dependence of eigenvalues of a class of fourth order sturm liouville problems on the boundary in this paper we consider the dependence of eigenvalues of a class of fourth order sturm liouville problems on the boundary we show that the eigenvalues depend not only continuously but smoothly on boundary points and that the derivative of the nth eigenvalue as a function of an endpoint satisfies a first order differential equation in addition we prove that as the length of the interval shrinks to zero all higher fourth order dirichlet eigenvalues march off to plus infinity this is also true for the first i e lowest eigenvalue

regular third order boundary value problems abstract in this paper we consider some boundary value problems generated by third order formally symmetric self adjoint regular differential expression and separated real coupled and complex coupled boundary conditions it is shown that these problems generate self adjoint operators moreover the dependence of eigenvalues of these problems on the data are studied and some derivatives of the eigenvalues with respect to some elements of data are introduced

generalized approximate counting revisited a large class of q distributions is defined on the stochastic model of bernoulli trials in which the probability of success advancing to the next level depends geometrically on the number of trials and the level already reached if the dependency is only on the level already reached this is an algorithm called approximate counting two random variables x n level reached after n trials and y k number of trials to reach level k are of interest we rederive known results and obtain new ones in a consistent way based on generating functions we also discuss asymptotics the classical instance of approximate counting is more interesting from a mathematical point of view on the other hand if the number of trials also decreases the probability of success advancing to the next level then the limits are constants which are straightforward to compute

existence of multiple positive solutions for sturm liouville like four point boundary value problem with p laplacian abstract in this paper we consider the sturm liouville like four point boundary value problem with p laplacian p u t f t u t 0 t 0 1 u 0 u 0 u 1 u 0 where p s s p 2 s p 1 by means of a fixed point theorem for operators on a cone the existence of multiple at least three positive solutions to the above boundary value problem is obtained

an approach to generate superextensions of integrable systems an algorithm to generate integrable systems is extended to the super case some new examples of superextensions of integrable systems are illustrated we also generalize the trace identity due to tu to the super case and use it to establish hamiltonian structures of superextensions of integrable systems under consideration

approximate counting with m counters a detailed analysis the classical algorithm approximate counting has been recently modified by cichon and macyna instead of one counter m counters are used and the assignment of an incoming item to one of the counters is random the parameter of interest is the sum of the values of all the counters we analyse expectation and variance getting explicit and asymptotic formulae

on domains of cal p cal t symmetric operators related to y x 1 nx2ny x in recent years a generalization of hermiticity has been investigated using a complex deformation h p2 x2 ix of the harmonic oscillator hamiltonian where is a real parameter these complex hamiltonians possessing symmetry the product of parity and time reversal can have a real spectrum we will consider the most simple case even in this paper we describe all self adjoint hermitian and at the same time symmetric operators associated with h p2 x2 ix surprisingly it turns out that there is a large class of self adjoint operators associated with h p2 x2 ix which are not symmetric

classical n 2 w superalgebras from superpseudodifferential operators we study the supersymmetric gelfand dickey algebras associated with the superpseudodifferential operators of positive as well as negative leading order we show that upon the usual constraint these algebras contain the n 2 super virasoro algebra as a subalgebra when the leading order is odd the decompositions of the coefficient functions into n 1 primary fields are then obtained by covariantizing the superpseudodifferential operators we discuss the problem of identifying n 2 supermultiplets and work out a couple of supermultiplets by explicit computations

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

mellin transforms and asymptotics finite differences and rice s integrals high order differences of simple number sequences may be analysed asymptotically by means of integral representations residue calculus and contour integration this technique akin to mellin transform asymptotics is put in perspective and illustrated by means of several examples related to combinatorics and the analysis of algorithms like digital tries digital search trees quadtrees and distributed leader election

dependence of eigenvalues of sturm liouville problems on the boundary abstract the eigenvalues of sturm liouville sl problems depend not only continuously but smoothly on boundary points the derivative of the n th eigenvalue as a function of an endpoint satisfies a first order differential equation this for arbitrary separated or coupled self adjoint regular boundary conditions in addition as the length of the interval shrinks to zero all higher eigenvalues march off to plus infinity this is also true for the first i e lowest dirichlet eigenvalue but not for the lowest neumann eigenvalue the latter has a finite limit