comparisons of nonnegative splittings of matrices abstract the comparison of the asymptotic rates of convergence of two iteration matrices induced by two splitting of the same matrix has arisen in the work of many authors in this article some new comparison theorems for two nonnegative splittings a splitting a m n is nonnegative if m 1 exist and m 1 n is nonnegative are derived they extend the known results in the literature in addition we also point out three incorrect conditions in a paper by beauwens furthermore we give some reasonable conditions ensuring the strict inequality between the asymptotic convergence rates this also answers the open question which additional and appropriate conditions should be imposed on miller neumann splittings to obtain strict inequality finally some applications to a class of generalized aor sor and jor iterative methods whose special cases imply block also point aor sor and jor iterative methods for solving linear systems are discussed

nonnegative splittings for rectangular matrices the extension of the nonnegative splitting for rectangular matrices called proper nonnegative splitting is proposed first different convergence and comparison theorems for the proper nonnegative splittings are established the notion of double nonnegative splitting is then generalized to rectangular matrices finally different convergence and comparison results are presented for this decomposition the case for singular square matrices is also studied

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

exact and asymptotic features of the edge density profile for the one component plasma in two dimensions there is a well known analogy between the laughlin trial wave function for the fractional quantum hall effect and the boltzmann factor for the two dimensional one component plasma the latter requires continuation beyond the finite geometry used in its derivation we consider both disk and cylinder geometry and focus attention on the exact and asymptotic features of the edge density at the special coupling gamma q 2 k bt 2 the system is exactly solvable in particular the k point correlation can be written as a k times k determinant allowing the edge density to be computed to first order in gamma 2 a double layer structure is found which in turn implies an overshoot of the density as the edge of the leading support is approached from the interior asymptotic analysis shows that the deviation from the leading order step function value is different for the interior and exterior directions for general gamma a gaussian fluctuation formula is used to study the large deviation form of the density for n large but finite this asymptotic form involves thermodynamic quantities which we independently study and moreover an appropriate scaling gives the asymptotic decay of the limiting edge density outside of the plasma

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

density functional theory study of electric potential saturation planar geometry we investigate the possibility of electrostatic potential saturation which may lead to the phenomenon of effective charge saturation the system under study is a uniformly charged infinite plane immersed in an arbitrary electrolyte made up of several microspecies to describe the electric double layer we use a generic local density functional theory in which the local microionic density profiles are arbitrary functions of the local electrostatic potential a general necessary and sufficient condition is obtained for saturation whereby the electrostatic potential created by the plane becomes independent of its bare charge provided the latter is large enough as a consequence for most situations the following simple and practical sufficient condition follows if as the electric potential psi infinity the local theory predicts that the highest valency counterions density diverges as psi nu with some nu 1 or faster then the electrostatic potential will saturate otherwise if the counterion density diverges as psi or slower or does not diverge as psi infinity the electric potential will not saturate using this condition we investigate the possibility of the saturation phenomenon within the framework of recent theories proposed in the literature to describe electrical double layer beyond the poisson boltzmann description

random walks and random fixed point free involutions a bijection is given between fixed point free involutions of 1 2 2n with maximum decreasing subsequence size 2p and two classes of vicious non intersecting random walker configurations confined to the half line lattice points l 1 in one class of walker configurations the maximum displacement of the rightmost walker is p because the scaled distribution of the maximum decreasing subsequence size is known to be in the soft edge goe random real symmetric matrices universality class the same holds true for the scaled distribution of the maximum displacement of the rightmost walker

pressures for a one component plasma on a pseudosphere the classical i e non quantum equilibrium statistical mechanics of a two dimensional one component plasma a system of charged point particles embedded in a neutralizing background living on a pseudosphere an infinite surface of constant negative curvature is considered in the case of a flat space it is known that for a one component plasma there are several reasonable definitions of the pressure and that some of them are not equivalent to each other in the present paper this problem is revisited in the case of a pseudosphere general relations between the different pressures are given at one special temperature the model is exactly solvable in the grand canonical ensemble the grand potential and the one body density are calculated in a disk and the thermodynamic limit is investigated the general relations between the different pressures are checked on the solvable model

some recent developments of numerov s method abstract this paper is a survey of some recent developments of numerov s method for solving nonlinear two point boundary value problems the survey consists of three different parts the existence uniqueness of a solution computational algorithm for computing a solution and some extensions of numerov s method the sufficient conditions for the existence and uniqueness of a solution are presented some of them are best possible various iterative methods are reviewed including picard s iterative method modified newton s iterative method monotone iterative method and accelerated monotone iterative method in particular two more direct monotone iterative methods are presented to save computational work each of these iterative methods not only gives a computational algorithm for computing a solution but also leads to an existence and uniqueness theorem the estimate on the rate of convergence of the iterative sequence is given the extensions of numerov s method to a coupled problem and a general problem are addressed the numerical results are presented to validate the theoretical analysis

debye huckel theory for charge symmetric coulomb systems it is proven that the pressure density and correlation functions of a classical charge symmetric coulomb system are asymptotic as the plasma parameter e tends to zero to the approximations predicted by the debye huckel theory these approximations consist of the ideal gas term plus a term of one lower order in e the sine gordon transformation and some new correlation inequalities for the associated functional integrals are used

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

mean field theory for coulomb systems we study a classical charge symmetric system with an external charge distributionq in three dimensions in the limit that the plasma parameter zero we prove that ifq is scaled appropriately then the correlation functions converge pointwise to those of an ideal gas in the external mean field x where is given by 2z sinh q this is the mean field equation of debye and huckel the proof uses the sine gordon transformation the mayer expansion and a correlation inequality

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

comparison theorems for splittings of matrices this paper investigates the comparisons of asymptotic rates of convergence of two iteration matrices on the basis of nonnegative matrix theory comparisons between two nonnegative splittings and between two parallel multisplitting methods are derived when the coefficient matrix a is hermitian positive semi definite comparison theorems about two p regular splittings and two parallel multisplitting methods are proved

comparisons of regular splittings of matrices in this article new comparison theorems for regular splittings of matrices are derived in so doing the initial results of varga in 1960 on regular splittings of matrices and the subsequent unpublished results of wo nicki in 1973 on regular splittings of matrices will be seen to be special cases of these new comparison theorems

a note on comparison theorems for splittings and multisplittings of hermitian positive definite matrices abstract we discuss iterative methods for the solution of the linear system ax b which are based on a single splitting or a multisplitting of a in order to compare different methods it is common to compare the spectral radius of the iterative matrix for m matrices a and weak regular splittings there exist well known comparison theorems here we give a comparison theorem for splittings of hermitian positive definite matrices furthermore we establish a comparison theorem for multisplittings of a hermitian positive definite matrix

finite size corrections to the free energy of coulomb systems with a periodic boundary condition classical coulomb systems ind dimensions d 2 with a periodic boundary condition periodw in the directionx d are considered with the other directions of the confining volume of lengthl it is shown that if the system is in a conducting phase then the strip free energyktf w f w liml l d 1 log z has the large w expansion f w sim wf infty frac d 2 1 gamma d 2 1 pi d 2 w d 1 zeta d o left frac 1 w d 1 right wherektf is the bulk free energy per unit volume x denotes the riemann zeta function and x denotes the gamma function with 1 w identified askt this result is precisely the low temperature behavior of the free energy of a d 1 dimensional debye solid this fact is explained in terms of an equivalence between the coulomb gas and quantum fields also the expansion is verified for some exactly solved models of coulomb systems in two dimensions

an invariant structure of the multi particle correlations of the two dimensional one component plasma the model under consideration is that of a two dimensional one component plasma confined to the surface of a sphere in the canonical format the system is mapped onto a discrete one dimensional anticommuting field theory a unitary transformation of anticommuting variables is revealed to generate a correlation hierarchy truncated rigorously at each hierarchical level in the thermodynamic limit of the fluid regime this truncation provides a specific invariant structure of multi particle densities of the plasma formulated on an infinite plane

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

finite size corrections for coulomb systems in the debye huckel regime it has been argued that for a finite two dimensional classical coulomb system of characteristic size r in its conducting phase as r the total free energy times the inverse temperature admits an expansion of the form where is the euler characteristic of the manifold where the system exists the first two terms represent the bulk free energy and the surface free energy respectively the coefficients a and b are non universal but the coefficient of ln r is universal it does not depend on the detail of the microscopic constitution of the system particle densities temperature etc by doing the usual legendre transform this universal finite size correction is also present in the grand potential the explicit form of the expansion has been checked for some exactly solvable models for a special value of the coulombic coupling in this paper we present a method for obtaining these finite size corrections in the debye huckel regime it is based on the sine gordon field theory to find an expression for the grand canonical partition function in terms of the spectrum of the laplace operator as an example we find explicitly the grand potential expansion for a coulomb system confined in a disc and in an annulus with ideal conductor walls

on exact solutions to the cylindrical poisson boltzmann equation with applications to polyelectrolytes using exact results from the theory of completely integrable systems of the painleve toda type we examine the consequences for the theory of polyelectrolytes in the nonlinear poisson boltzmann approximation

periodic boundary conditions on the pseudosphere we provide a framework for building periodic boundary conditions on the pseudosphere or hyperbolic plane the infinite two dimensional riemannian space of constant negative curvature starting from the common case of periodic boundary conditions in the euclidean plane we introduce all the required mathematical notions and sketch a classification of periodic boundary conditions on the hyperbolic plane we stress the possible applications in statistical mechanics for studying the bulk behavior of physical systems and illustrate how to implement such periodic boundary conditions in two examples the dynamics of particles on the pseudosphere and the study of classical spins on hyperbolic lattices

sine gordon theory for the equation of state of classical hard core coulomb systems ii high temperature expansion we perform a high temperature expansion of the grand potential of the restrictive primitive model of electrolytes in the frame of the extended sine gordon theory exposed in the companion paper we recover a result already obtained by stell an lebowitz j chem phys 49 3706 1968 by means of diagrammatic expansions

numerov s method for strongly nonlinear two point boundary value problems abstract by a proper transformation a class of strongly nonlinear two point boundary value problems are transformed into semilinear problems so that the well known numerov s method can be applied some applications and numerical results are presented to demonstrate the efficiency of the approach

comparisons of weak regular splittings and multisplitting methods comparison results for weak regular splittings of monotone matrices are derived as an application we get upper and lower bounds for the convergence rate of iterative procedures based on multisplittings this yields a very simple proof of results of neumann plemmons on upper bounds and establishes lower bounds which has in special cases been conjectured by these authors

convergence and comparison theorems for double splittings of matrices in this paper some convergence theorems for the double splitting of a monotone matrix or a hermitian positive definite matrix are presented two comparison theorems for two double splittings of a monotone matrix are obtained meanwhile we establish a new sufficient condition for convergence of the gauss seidel double sor method for an h matrix

spectral functions special functions and the selberg zeta function the functional determinant of an eigenvalue sequence as defined by zeta regularization can be simply evaluated by quadratures we apply this procedure to the selberg trace formula for a compact riemann surface to find a factorization of the selberg zeta function into two functional determinants respectively related to the laplacian on the compact surface itself and on the sphere we also apply our formalism to various explicit eigenvalue sequences reproducing in a simpler way classical results about the gamma function and the barnesg function concerning the latter our method explains its connection to the selberg zeta function and evaluates the related glaisher kinkelin constanta

general considerations on the finite size corrections for coulomb systems in the debye huckel regime we study the statistical mechanics of classical coulomb systems in a low coupling regime debye huckel regime in a confined geometry with dirichlet boundary conditions for the electric potential we use a method recently developed by the authors which relates the grand partition function of a coulomb system in a confined geometry with a certain regularization of the determinant of the laplacian on that geometry with dirichlet boundary conditions we study several examples of fully confining geometry in two and three dimensions and semi confined geometries where the system is confined only in one or two directions of the space we also generalize the method to study systems confined in arbitrary geometries with smooth boundary we find a relation between the expansion for small argument of the heat kernel of the laplacian and the large size expansion of the grand potential of the coulomb system this allow us to find the finite size expansion of the grand potential of the system in general we recover known results for the bulk grand potential in two and three dimensions and the surface tension for two dimensional systems we find the surface tension for three dimensional systems for two dimensional systems our general calculation of the finite size expansion gives a proof of the existence a universal logarithmic finite size correction predicted some time ago at least in the low coupling regime for three dimensional systems we obtain a prediction for the curvature correction to the grand potential of a confined system

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

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

comparison theorems for weak splittings of bounded operators comparison theorems for weak splittings of bounded operators are presented these theorems extend the classical comparison theorem for regular splittings of matrices by varga the less known result by wo nicki and the recent results for regular and weak regular splittings of matrices by neumann and plemmons elsner and lanzkron rose and szyld the hypotheses of the theorems presented here are weaker and the theorems hold for general banach spaces and rather general cones hypotheses are given which provide strict inequalities for the comparisons it is also shown that the comparison theorem by alefeld and volkmann applies exclusively to monotone sequences of iterates and is not equivalent to the comparison of the spectral radius of the iteration operators

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

coulomb systems seen as critical systems ideal conductor boundaries when a classical coulomb system has macroscopic conducting behavior its grand potential has universal finite size corrections similar to the ones which occur in the free energy of a simple critical system the massless gaussian field here the coulomb system is assumed to be confined by walls made of an ideal conductor material this choice corresponds to simple dirichlet boundary conditions for the gaussian field for ad dimensional d 2 coulomb system confined in a slab of thicknessw the grand potential in units ofkbt per unit area has the universal term d 2 d 2d d 2wd 1 for a two dimensional coulomb system confined in a disk of radiusr the grand potential in units ofkbt has the universal term 1 6 lnr these results of general validity are checked on two dimensional solvable models

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

the two dimensional one component plasma at γ 2 the semiperiodic strip the one component two dimensional plasma is studied in a strip of finite width replicated periodically parallel to the long axis of the strip exact results for the one and two particle distribution functions are found at coupling q2 kt 2 the system is inhomogeneous the one and two particle distribution functions show long range order

block monotone iterations for numerical solutions of fourth order nonlinear elliptic boundary value problems this paper is concerned with monotone iterative methods for numerical solutions of a class of nonlinear fourth order elliptic boundary value problems in a two dimensional domain the boundary value problem is discretized by the finite difference method and two iterative processes called block jacobi and block gauss seidel monotone iterations are presented for the computation of solutions of the finite difference system using either an upper solution or a lower solution as the initial iteration it is shown that the sequence of iterations converges monotonically to a maximal solution or a minimal solution if the nonlinear function is quasi monotone nondecreasing a sufficient condition is given to ensure that the maximal and minimal solutions coincide and their common value is the unique solution of the finite difference system similar results are obtained for quasi monotone nonincreasing functions an analytical comparison relation between the block jacobi and block gauss seidel monotone iterations is obtained it is also shown that the finite difference solution converges to the continuous solution as the mesh size tends to zero numerical results by the block monotone iterative schemes are given for two model problems and are compared with the known analytical solutions for accuracy also compared are the rates of convergence of the two types of block monotone iterations as well as similar types of pointwise monotone iterations

sine gordon theory for the equation of state of classical hard core coulomb systems iii loopwise expansion we present an exact field theoretical representation of an ionic solution made of charged hard spheres the action of the field theory is obtained by performing a hubbard stratonovich transform of the configurational boltzmann factor it is shown that the stillinger lovett sum rules are satisfied if and only if all the field correlation functions are short range functions the mean field gaussian and two loops approximations of the theory are derived and discussed the mean field approximation for the free energy constitutes an exact lower bound for the exact free energy while the mean field pressure is an exact upper bound the one loop order approximation is shown to be identical with the random phase approximation of the theory of liquids finally at the two loop order and in the pecular case of the restricted primitive model one recovers results obtained in the framework of the mode expansion theory

convergence and comparison theorems for a generalized alternating iterative method using as a principal tool the convergence results of standard iterative process for the solution of linear systems alternating iterative methods are studied we extend the convergence theorem for the stationary alternating iterative method of benzi and szyld numererische mathematik 76 1997 309 for weak nonnegative splittings of the first type of a monotone matrix to weak nonnegative splitting of the second type on the other hand we introduce a more general method the nonstationary alternating iterative method establishing convergence results for weak nonnegative splittings of a monotone matrix and for p regular splittings of a symmetric positive definite matrix

another derivation of a sum rule for the two dimensional two component plasma abstract in a two dimensional two component plasma the second moment of the number density correlation function has the simple value 12 1 4 2 1 where is the dimensionless coupling constant this result is derived directly by using diagrammatic methods

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