a short proof of interlacing inequalities on normalized laplacians a short proof of interlacing inequalities on normalized laplacians is given 2000 mathematics subject classification 15a42

the λ coalescent speed of coming down from infinity consider a coalescent that comes down from infinity meani ng that it starts from a configuration containing infinitely many blocks at time 0 yet it has a finite number nt of blocks at any positive time t 0 we exhibit a deterministic function v 0 0 such that nt v t 1 almost surely and in l p for any p 1 as t 0 our approach relies on a novel martingale technique

existence of positive solutions for a system of generalized lidstone problems in this paper we study the existence of positive solutions for the system of generalized lidstone problems for ordinary differential equations 1 mu 2 m f 1 t u u 1 m 1u 2 m 2 v v 1 n 1v 2 n 2 1 nv 2 n f 2 t u u 1 m 1u 2 m 2 v v 1 n 1v 2 n 2 a 0u 2 i 0 b 0u 2 i 1 0 a 1u 2 i 1 b 1u 2 i 1 1 0 i 0 1 m 1 a 0v 2 j 0 b 0v 2 j 1 0 a 1v 2 j 1 b 1v 2 j 1 1 0 j 0 1 n 1 here f 1 f 2 c 0 1 xr m n r r 0 our hypotheses imposed on the nonlinearities f 1 and f 2 are formulated in terms of nonnegative matrices we use fixed point index theory to establish our main results based on a priori estimates of positive solutions

a fourth order compact finite difference method for nonlinear higher order multi point boundary value problems a fourth order compact finite difference method is proposed for a class of nonlinear 2nth order multi point boundary value problems the multi point boundary condition under consideration includes various commonly discussed boundary conditions such as the three or four point boundary condition n 2 point boundary condition and 2 n m point boundary condition the existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear term the convergence and the fourth order accuracy of the method are proved an efficient monotone iterative algorithm is developed for solving the resulting nonlinear finite difference systems various sufficient conditions for the construction of upper and lower solutions are obtained some applications and numerical results are given to demonstrate the high efficiency and advantages of this new approach

approximating spectral invariants of harper operators on graphs abstract we study harper operators and the closely related discrete magnetic laplacians dml on a graph with a free action of a discrete group as defined by sunada sun a main result in this paper is that the spectral density function of dmls associated to rational weight functions on graphs with a free action of an amenable discrete group can be approximated by the average spectral density function of the dmls on a regular exhaustion with either dirichlet or neumann boundary conditions this then gives a criterion for the existence of gaps in the spectrum of the dml as well as other interesting spectral properties of such dmls the technique used incorporates some results of algebraic number theory

triple positive solutions and dependence on higher order derivatives abstract in this paper we consider the lidstone boundary value problem y 2 m t f y t y 2 j t y 2 m 1 t 0 t 1 y 2 i 0 0 y 2 i 1 0 i m 1 where 1 m f 0 growth conditions are imposed on f and inequalities involving an associated green s function are employed which enable us to apply the leggett williams fixed point theorem to cones in ordered banach spaces this in turn yields the existence of at least three positive symmetric concave solutions the emphasis here is that f depends on higher order derivatives

the method of lower and upper solutions for fourth order two point boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u iv f x u x u x u 0 u 1 u 0 u 1 0 where f 0 1 r r r is continuous

self normalized processes exponential inequalities moment bounds and iterated logarithm laws self normalized processes arise naturally in statistical applications being unit free they are not affected by scale changes moreover self normalization often eliminates or weakens moment assumptions in this paper we present several exponential and moment inequalities particularly those related to laws of the iterated logarithm for self normalized random variables including martingales tail probability bounds are also derived for random variables bt 0 and at let yt exp at 2bt2 2 we develop inequalities for the moments of at bt or supt 0at bt log logbt 1 2 and variants thereof when eyt 1 or when yt is a supermartingale for all belonging to some interval our results are valid for a wide class of random processes including continuous martingales with at mt and b t sqrt langle m rangle t and sums of conditionally symmetric variables di with at i 1tdi and b t sqrt sum i 1 t d i 2 a sharp maximal inequality for conditionally symmetric random variables and for continuous local martingales with values in rm m 1 is also established another development in this paper is a bounded law of the iterated logarithm for general adapted sequences that are centered at certain truncated conditional expectations and self normalized by the square root of the sum of squares the key ingredient in this development is a new exponential supermartingale involving i 1tdi and i 1tdi2 a compact law of the iterated logarithm for self normalized martingales is also derived in this connection

lidstone polynomials and boundary value problems abstract we define lidstone polynomials provide their explicit representations give their relations with bernoulli polynomials obtain best possible error estimates in lidstone interpolation and establish several best possible sharp inequalities which compare favourably with the known results next we use these results to study even order differential equations together with lidstone boundary conditions

solutions of 2nth lidstone boundary value problems and dependence on higher order derivatives abstract in this paper we consider the lidstone boundary value problem 1 n u 2n x f x u x u x u 2 n 1 x 0 u 2i 0 u 2i 1 0 0 i n 1 some monotone conditions are imposed on f which enable us to apply a new maximum principle to deduce the existence of solution the emphasis here is that f depends on higher order derivatives two examples are also included to dwell upon the importance of the results obtained

spectral gaps and discrete magnetic laplacians abstract the aim of this article is to give a simple geometric condition that guarantees the existence of spectral gaps of the discrete laplacian on periodic graphs for proving this we analyse the discrete magnetic laplacian dml on the finite quotient and interpret the vector potential as a floquet parameter we develop a procedure of virtualising edges and vertices that produces matrices whose eigenvalues written in ascending order and counting multiplicities specify the bracketing intervals where the spectrum of the laplacian is localised we prove higuchi shirai s conjecture for z periodic trees and apply our technique in several examples like the polypropylene or the polyacetylene to show the existence of spectral gaps

difference equations isoperimetric inequality and transience of certain random walks the difference laplacian on a square lattice in rn has been stud ied by many authors in this paper an analogous difference operator is studied for an arbitrary graph it is shown that many properties of the laplacian in the continuous setting e g the maximum principle the harnack inequality and cheeger s bound for the lowest eigenvalue hold for this difference oper ator the difference laplacian governs the random walk on a graph just as the laplace operator governs the brownian motion as an application of the theory of the difference laplacian it is shown that the random walk on a class of graphs is transient the random walks we consider are defined as follows let k be a connected graph i e a one dimensional simplicial complex for a vertex x e k let m x denote the number of edges emanating from x the probability that a particle moves from x to another vertex y e k is l m x if x and y are connected by an edge and it is zero otherwise as observed by courant friedrichs and lewy cfl for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the laplacian

fourth order two point boundary value problems estimates by two sided bounds on presente une methode simple pour transformer des problemes aux deux points limites d ordre 4 en problemes d ordre 2

on the spectrum of the normalized laplacian for signed graphs interlacing contraction and replication abstract we consider the normalized laplacian matrix for signed graphs and derive interlacing results for its spectrum in particular we investigate the effects of several basic graph operations such as edge removal and addition and vertex contraction on the laplacian eigenvalues we also study vertex replication whereby a vertex in the graph is duplicated together with its neighboring relations this operation causes the generation of a laplacian eigenvalue equal to one we further generalize to the replication of motifs i e certain small subgraphs and show that the resulting signed graph has an eigenvalue 1 whenever the motif itself has eigenvalue 1

spectral preorder and perturbations of discrete weighted graphs in this article we introduce a geometric and a spectral preorder relation on the class of weighted graphs with a magnetic potential the first preorder is expressed through the existence of a graph homomorphism respecting the magnetic potential and fulfilling certain inequalities for the weights the second preorder refers to the spectrum of the associated laplacian of the magnetic weighted graph these relations give a quantitative control of the effect of elementary and composite perturbations of the graph deleting edges contracting vertices etc on the spectrum of the corresponding laplacians generalising interlacing of eigenvalues we give several applications of the preorders we show how to classify graphs according to these preorders and we prove the stability of certain eigenvalues in graphs with a maximal d clique moreover we show the monotonicity of the eigenvalues when passing to spanning subgraphs and the monotonicity of magnetic cheeger constants with respect to the geometric preorder finally we prove a refined procedure to detect spectral gaps in the spectrum of an infinite covering graph

on 2nth order lidstone boundary value problems this paper is concerned with the solutions of a class of 2nth order lidstone boundary value problems sufficient conditions for the existence and uniqueness of a solution are given a monotone iteration is developed so that the iteration sequence converges monotonically to a maximal or a minimal solution the approach to the problem is by the method of upper and lower solutions with a new maximum principle

the iterative solutions of 2nth order nonlinear multi point boundary value problems abstract the aim of this paper is to investigate the existence of iterative solutions for a class of 2 n th order nonlinear multi point boundary value problems the multi point boundary condition under consideration includes various commonly discussed boundary conditions such as three or four point boundary condition n 2 point boundary condition and 2 n m point boundary condition the existence problem is based on the method of upper and lower solutions and its associated monotone iterative technique a monotone iteration is developed so that the iterative sequence converges monotonically to a maximal solution or a minimal solution depending on whether the initial iteration is an upper solution or a lower solution two examples are presented to illustrate the results

a fourth order compact finite difference method for higher order lidstone boundary value problems a compact finite difference method is proposed for a general class of 2nth order lidstone boundary value problems the existence and uniqueness of the finite difference solution is investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear term a monotone iteration process is provided for solving the resulting discrete system efficiently and a simple and easily verified condition is obtained to guarantee a geometric convergence of the iterations the convergence of the finite difference solution and the fourth order accuracy of the proposed method are proved numerical results demonstrate the high efficiency and advantages of this new approach

boundary value problems occurring in plate deflection theory abstract the paper deals with several iterative methods for solving some boundary value problems occurring in plate deflection theory making it possible to avoid the lengthy problem preparation which is required to solve these problems by finite difference methods

general lidstone problems multiplicity and symmetry of solutions abstract for the 2mth order lidstone boundary value problem y 2 m t f y t y t y 2 i t y 2 m 1 t t 0 1 y 2 i 0 y 2 i 1 0 0 i m 1 where 1 mf r m 0 thinsp is continuous growth conditions are imposed on f which yield the existence of at least three symmetric positive solutions this generalizes earlier papers which have applied avery s generalization of the leggett williams theorem to lidstone problems we then prove the analogous result for difference equations

existence and uniqueness of positive solutions for a higher order boundary value problem the goal of this paper is to study the existence and uniqueness of positive solutions for the 2nth order boundary value problem 1 nu 2 n f t u u 1 n 1u 2 n 2 a 0u 2 i 0 email protected 0u 2 i 1 0 0 i 0 1 n 1 a 1u 2 i 1 email protected 1u 2 i 1 1 0 i 0 1 n 1 the krein rutman theorem and the krasnoselskii zabreiko fixed point theorem are the main tools that have been used to develop our work

the method of lower and upper solutions for a bending of an elastic beam equation abstract in this paper by combining a new maximum principle and the theory of the two parameter linear eigenvalue problem we develop the monotone method in the presence of lower and upper solutions for the problem u 4 x f x u x u x 0 x 1 u 0 u 1 u 0 u 1 0 where f 0 1 r r r is continuous

positive solutions of a 2nth order boundary value problem involving all derivatives via the order reduction method this paper is mainly concerned with the existence multiplicity and uniqueness of positive solutions for the 2nth order boundary value problem 1 nu 2 n f t u u 1 i 2 u i 1 n 1u 2 n 1 u 2 i 0 u 2 i 1 1 0 i 0 1 n 1 where n 2 and f c 0 1 xr 2 n r r 0 we first use the method of order reduction to transform the above problem into an equivalent initial value problem for a first order integro differential equation and then use the fixed point index theory to prove the existence multiplicity and uniqueness of positive solutions for the resulting problem based on a priori estimates achieved by developing spectral properties of associated parameterized linear integral operators finally as a byproduct our main results are applied for establishing the existence multiplicity and uniqueness of symmetric positive solutions for the lidstone problem involving all derivatives

bifurcation techniques for lidstone boundary value problems abstract by employing bifurcation techniques this paper investigates the existence of nontrivial solutions single and multiple for lidstone boundary value problems depending on higher order derivatives our results improve on those in the literature

the method of lower and upper solutions for second third fourth and higher order boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u n t t u t u i a u i b i r i 0 n 1 where is a caratheodory function we obtain necessary and sufficient conditions in to guarantee the existence of solutions between a lower solution and an upper solution for n 2 if n 3 either or and n 4 if furthermore we obtain sufficient conditions in for n 2 k 6 when