a homographic best approximation problem with application to optimized schwarz waveform relaxation we present and study a homographic best approximation problem which arises in the analysis of waveform relaxation algorithms with optimized transmission conditions its solution characterizes in each class of transmission conditions the one with the best performance of the associated waveform relaxation algorithm we present the particular class of first order transmission conditions in detail and show that the new waveform relaxation algorithms are well posed and converge much faster than the classical one the number of iterations to reach a certain accuracy can be orders of magnitudes smaller we illustrate our analysis with numerical experiments

discrete painleve equations coalescences limits and degeneracies abstract starting from the standard form of the five discrete painleve equations we show how one can obtain through appropriate limits a host of new equations which are also the discrete analogues of the continuous painleve equations a particularly interesting technique is the one based on the assumption that some simplification takes place in the autonomous form of the mapping following which the deautonomization leads to a new n dependence and introduces more new discrete painleve equations

how to detect integrability in cellular automata ultra discrete equations are generalized cellular automata in the sense that the dependent and independent variables take only integer values we present a new method for identifying integrable ultra discrete equations which is the equivalent of the singularity confinement property for difference equations and the painleve property for differential equations using this criterion we find integrable ultra discrete equations which include the ultra discrete painleve equations

from stochastic partial difference equations to stochastic cellular automata through the ultra discretization we derive stochastic cellular automata with finite range interactions from stochastic partial difference equations through the ultra discrete limit a typical example for stochastic cellular automata is the domany kinzel model which is equivalent to mixed site bond oriented percolation as a special case

space time continuous analysis of waveform relaxation for the heat equation waveform relaxation algorithms for partial differential equations pdes are traditionally obtained by discretizing the pde in space and then splitting the discrete operator using matrix splittings for the semidiscrete heat equation one can show linear convergence on unbounded time intervals and superlinear convergence on bounded time intervals by this approach however the bounds depend in general on the mesh parameter and convergence rates deteriorate as one refines the mesh motivated by the original development of waveform relaxation in circuit simulation where the circuits are split in the physical domain into subcircuits we split the pde by using overlapping domain decomposition we prove linear convergence of the algorithm in the continuous case on an infinite time interval at a rate depending on the size of the overlap this result remains valid after discretization in space and the convergence rates are robust with respect to mesh refinement the algorithm is in the class of waveform relaxation algorithms based on overlapping multisplittings our analysis quantifies the empirical observation by jeltsch and pohl siam j sci comput 16 1995 pp 40 49 that the convergence rate of a multisplitting algorithm depends on the overlap numerical results are presented which support the convergence theory

dziobek configurations of the restricted n 1 body problem with equal masses in this article we count the number of n 1 dimensional central configurations of the restricted n 1 body problem with equal masses for n 4 5 there are 25 56 central configurations for n 3 or n 6 the number of central configurations is 2n n 1

when is negativity not a problem for the ultradiscrete limit the ultradiscrete limit has provided a link between integrable difference equations and cellular automata displaying soliton like solutions in particular this procedure generally turns strictly positive solutions of algebraic difference equations with positive coefficients into corresponding solutions to equations involving the max operator although it certainly is the case that dropping these positivity conditions creates potential difficulties it is still possible for solutions to persist under the ultradiscrete limit even in their absence to recognize when this will occur one must consider whether a certain expression involving a measure of the rates of convergence of different terms in the difference equation and their coefficients is equal to zero applications discussed include the solution of elementary ordinary difference equations a discretization of the hirota bilinear difference equation and the identification of integrals of motion for ultradiscrete equations

ultradiscrete painleve ii equation and a special function solution ultradiscretization with parity variables which keeps the information of original variables sign is applied to the q painleve ii equation of type a6 q pii a special solution of the resulting ultradiscrete system which corresponds to the special function solution of q pii is constructed ultradiscrete analogues of the q airy equation and its special solutions are also discussed in the process

combinatorial representation of invariants of a soliton cellular automaton abstract the structure of the soliton cellular automaton is studied by means of combinatorial techniques it is shown that the shape of the young tableaux gives an infinite number of time invariants of the automaton the employed combinatorial materials are the dyck language stack representable sequences and the robinson schensted algorithm

elliptic solutions to difference non linear equations and related many body problems we study algebro geometric finite gap and elliptic solutions of fully discretized kp or 2d toda equations in bilinear form they are hirota s difference equation for functions starting from a given algebraic curve we express the function and the baker akhiezer function in terms of the riemann theta function we show that the elliptic solutions when the function is an elliptic polynomial form a subclass of the general algebro geometric solutions we construct the algebraic curves of the elliptic solutions the evolution of zeros of the elliptic solutions is governed by the discrete time generalization of the ruijsenaars schneider many body system the zeros obey equations which have the form of nested bethe ansatz equations known from integrable quantum field theories we discuss the lax representation and the action angle type variables for the many body system we also discuss elliptic solutions to discrete analogues of kdv sine gordon and 1d toda equations and describe the loci of the zeros

casorati determinant solutions for the discrete painleve ii equation we present a class of solutions to the discrete painleve ii equation for particular values of its parameters it is shown that these solutions can be expressed in terms of casorati determinants whose entries are discrete airy functions the analogy between the tau function for the discrete pii and that of the discrete toda molecule equation is pointed out

constructing solutions to the ultradiscrete painleve equations we investigate the nature of particular solutions to the ultradiscrete painleve equations we start by analysing the autonomous limit and show that the equations possess an explicit invariant which leads naturally to the ultradiscrete analogue of elliptic functions for the ultradiscrete painleve equations ii and iii we present special solutions reminiscent of the casorati determinant ones which exist in the continuous and discrete cases finally we analyse the discrete painleve equation i and show how it contains both the continuous and the ultradiscrete ones as particular limits

a continuum of periodic solutions to the planar four body problem with two pairs of equal masses abstract in this paper we apply the variational method with structural prescribed boundary conditions spbc to prove the existence of periodic and quasi periodic solutions for the planar four body problem with two pairs of equal masses m 1 m 3 and m 2 m 4 a path q t on 0 t satisfies the spbc if the boundaries q 0 a and q t b where a and b are two structural configuration spaces in r 2 4 and they depend on a rotation angle 0 2 and the mass ratio m 2 m 1 r we show that there is a region 0 2 r such that there exists at least one local minimizer of the lagrangian action functional on the path space satisfying the spbc q t h 1 0 t r 2 4 q 0 a q t b for any the corresponding minimizing path of the minimizer can be extended to a non homographic periodic solution if is commensurable with or a quasi periodic solution if is not commensurable with in the variational method with the spbc we only impose constraints on the boundary and we do not impose any symmetry constraint on solutions instead we prove that our solutions that are extended from the initial minimizing paths possess certain symmetries the periodic solutions can be further classified as simple choreographic solutions double choreographic solutions and non choreographic solutions among the many stable simple choreographic orbits the most extraordinary one is the stable star pentagon choreographic solution when 4 5 1 remarkably the unequal mass variants of the stable star pentagon are just as stable as the equal mass choreographies

on exact solutions with periodic structure of the ultradiscrete toda equation with parity variables exact solutions of the ultradiscrete toda equation with parity variables are constructed from the soliton solutions of the discrete toda lattice equation the solution has a periodic phase constant and describes a traveling pulse with a periodic variation its behavior is different from that of the usual soliton solution

central configurations of the 4 body problem with masses m 1 m 2 m 3 m 4 m 0 and m small in this paper we give a complete description of the families of central configurations of the planar 4 body problem with two pairs of equals masses and two equal masses sufficiently small in particular we give an analytical proof that this particular 4 body problem has exactly 34 different classes of central configurations moreover for this problem we prove the following two conjectures there is a unique convex planar central configuration of the 4 body problem for each ordering of the masses in the boundary of its convex hull which appears in albouy and fu 2007 3 we also prove the conjecture there is a unique convex planar central configuration having two pairs of equal masses located at the adjacent vertices of the configuration and it is an isosceles trapezoid finally the families of central configurations of this 4 body problem are numerically continued to the 4 body problem with four equal masses

tropical geometric interpretation of ultradiscrete singularity confinement using the interpretation of the ultradiscretization procedure as a non archimedean valuation we use results of tropical geometry to show how roots and poles manifest themselves in piece wise linear systems as points of non differentiability this will allow us to demonstrate a correspondence between singularity confinement for discrete integrable systems and ultradiscrete singularity confinement for ultradiscrete integrable systems

box and ball system as a realization of ultradiscrete nonautonomous kp equation cellular automata which are realized by dynamics of several kinds of balls in an infinite array of boxes are investigated they show soliton patterns even in the case when each box has arbitrary capacity the analytical expression for the soliton patterns are obtained using ultradiscretization of the nonautonomous discrete kp equation

optimization of transmission conditions in waveform relaxation techniques for rc circuits waveform relaxation techniques have become increasingly important with the wide availability of parallel computers with a large number of processors a limiting factor for classical waveform relaxation however is the convergence speed for an important class of problems especially if long time windows are considered in contrast the optimized waveform relaxation algorithm discussed in this paper is well suited to address this problem today several numerical analyses have shown that optimized waveform relaxation algorithms can overcome slow convergence over long time windows however the optimized waveform relaxation techniques require the determination of optimized parameters in this paper we present a theoretical foundation for the determination of the optimized parameters for an important class of rc circuits

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

lax pairs for ultra discrete painleve cellular automata ultra discrete versions of the discrete painleve equations are well known however evidence for their integrability has so far been restricted in this letter we show that their lax pairs can be constructed and furthermore that compatibility conditions of the result yield the ultra discrete painleve equation for conciseness we restrict our attention to a new d piii

super central configurations of the n body problem in this paper we consider the inverse problem of central configurations of the n body problem for a given q q1 q2 qn rd n let s q be the admissible set of masses by s q m m1 mn mi r q is a central configurationfor m for a given m s q let sm q be the permutational admissible set about m m1 m2 mn by sm q m m s q m m and m is apermutation of m here q is called a super central configuration if there exists m such that sm q is nonempty for any q in the planar four body problem q is not a super central configuration as an immediate consequence of a theorem proved by macmillan and bartky permanent configurations in the problem of four bodies trans am math soc 34 838 1932 the main discovery in this paper is the existence of super central configurations in the collinear three body problem we proved that for any q in the collinear three body problem and any m s q sm q has at most one element and the detailed classification of sm q is provided

waveform relaxation as a dynamical system in this paper the properties of waveform relaxation are studied when applied to the dynamical system generated by an autonomous ordinary differential equation in particular the effect of the waveform relaxation on the invariant sets of the flow is analysed windowed waveform relaxation is studied whereby the iterative technique is applied on successive time intervals of length t and a fixed finite number of iterations taken on each window this process does not generate a dynamical system on r since two different applications of the waveform algorithm over different time intervals do not in general commute in order to generate a dynamical system it is necessary to consider the time t map generated by the relaxation process this is done and c 1 closeness of the resulting map to the time t map of the underlying ordinary differential equation is established using this various results from the theory of dynamical systems are applied and the results discussed

bilinear discrete painleve equations based on the results of singularity confinement we derive bilinear expressions for the discrete painleve equations in these cases where a bilinear expression is not sufficient we obtain trilinear or higher multilinear expressions we show that the bilinear approach provides a natural framework for the derivation of backlund and schlesinger transforms for the discrete painleve equations

ultradiscretization of elliptic functions and its applications to integrable systems it is shown that there exist three kinds of ultradiscrete analogues of jacobi s elliptic functions in this process the asymptotic behaviour of the poles and the zeros of the functions plays a crucial role using the ultradiscrete analogues and an addition formula exact solutions to the ultradiscrete kp equation are constructed and their relation to the ultradiscrete qrt system is discussed

multiple addition theorem for discrete and continuous nonlinear problems the addition relation for the riemann theta functions and for its limits which leads to the appearance of exponential functions in soliton type equations is discussed the form of addition property presented resolves itself to the factorization of the n tuple product of the shifted functions and it seems to be useful for analysis of soliton type continuous and discrete processes in the n 1 space time a close relation with the natural generalization of bi and tri linear operators into multiple linear operators concludes the paper

ultradiscrete qrt maps and tropical elliptic curves it is shown that the polygonal invariant curve of the ultradiscrete qrt uqrt map which is a two dimensional piecewise linear integrable map is the complement of the tentacles of a tropical elliptic curve on which the curve has a group structure in analogy to classical elliptic curves through the addition formula of a tropical elliptic curve a tropical geometric description of the uqrt map is then presented this is a natural tropicalization of the geometry of the qrt map found by tsuda moreover the uqrt map is linearized on the tropical jacobian of the corresponding tropical elliptic curve in terms of the abel jacobi map finally a formula concerning the period of a point in the uqrt map is given and an exact solution to its initial value problem is constructed by using the ultradiscrete elliptic theta function

convergence analysis of two level space time additive schwarz method for parabolic equations the additive schwarz as method was originally designed for preconditioning elliptic problems and recently the method was extended successfully as a coupled space time preconditioner for parabolic problems however the existing theory for the additive schwarz method doesn t apply directly to the space time discretized problems in this paper we develop an optimal convergence theory for the two level space time additive schwarz preconditioner we obtain lower and upper bounds for the spectrum of the as preconditioned operator and their dependency on the fine coarse mesh sizes in space and time the number of subdomains and the window size i e the number of coupled time steps some numerical experiments are reported to confirm the theory

inversible max plus algebras and integrable systems we present an extended version of max plus algebra which includes the inverse operator of max this algebra enables us to ultradiscretize the system including subtractions and obtain new ultradiscrete equations the known ultradiscrete equations can also be recovered by this construction

symmetry of planar four body convex central configurations we study the relationship between the masses and the geometric properties of central configurations we prove that in the planar four body problem a convex central configuration is symmetric with respect to one diagonal if and only if the masses of the two particles on the other diagonal are equal if these two masses are unequal then the less massive one is closer to the former diagonal finally we extend these results to the case of non planar central configurations of five particles

periodic multiwave solutions to the toda type cellular automaton an ultradiscretization of the riemann theta function is proposed the ultradiscretization satisfies an addition formula which is an ultradiscrete analogue of an addition formula for the riemann theta function using the addition formula periodic multiwave solutions to the toda type cellular automaton are obtained

space time domain decomposition for parabolic problems we analyze a space time domain decomposition iteration for a model advection diffusion equation in one and two dimensions the discretization of this iteration is the block red black variant of the waveform relaxation method and our analysis provides new convergence results for this scheme the asymptotic convergence rate is super linear and it is governed by the diffusion of the error across the overlap between subdomains hence it depends on both the size of this overlap and the diffusion coefficient in the equation however it is independent of the number of subdomains provided the size of the overlap remains fixed the convergence rate for the heat equation in a large time window is initially linear and it deteriorates as the number of subdomains increases the duration of the transient linear regime is proportional to the length of the time window for advection dominated problems the convergence rate is initially linear and it improves as the the ratio of advection to diffusion increases moreover it is independent of the size of the time window and of the number of subdomains numerical calculations illustrate our analysis

some notes on four body co circular central configurations abstract there are two symmetric families of four body co circular central configurations namely the kite and isosceles trapezoid using mutual distances as coordinates we prove that if the four body central configuration is an isosceles trapezoid then the diagonals of the isosceles trapezoid cannot be perpendicular to each other furthermore we show that for any four body co circular central configuration the diagonals of the quadrilateral cannot be perpendicular except that the configuration is a kite

large deviations and one sided scaling limit of randomized multicolor box ball system the basic kappa color box ball bbs system is an integrable cellular automaton on one dimensional lattice whose local states take 0 1 ldots kappa with 0 regarded as an empty box the time evolution is defined by a combinatorial rule of quantum group theoretical origin and the complete set of conserved quantities is given by a kappa tuple of young diagrams in the randomized bbs a probability distribution on 0 1 ldots kappa to independently fill the consecutive n sites in the initial state induces a highly nontrivial probability measure on the kappa tuple of those invariant young diagrams in a recent work kuniba et al nucl phys b 937 240 271 2018 their large n equilibrium shape has been determined in terms of schur polynomials by a markov chain method and also by a very different approach of thermodynamic bethe ansatz tba in this paper we establish a large deviations principle for the row lengths of the invariant young diagrams as a corollary they are shown to converge almost surely to the equilibrium shape at an exponential rate we also refine the tba analysis and obtain the exact scaling form of the vacancy the row length and the column multiplicity which exhibit nontrivial factorization in a one parameter specialization

twisted angles for central configurations formed by two twisted regular polygons abstract in this paper we study the necessary conditions and sufficient conditions for the twisted angles of the central configurations formed by two twisted regular polygons in particular we prove that for the 2n body problem the twisted angles must be 0 or n

new airy type solutions of the ultradiscrete painleve ii equation with parity variables the q difference painleve ii equation admits special solutions written in terms of determinants whose entries are the general solution of the q airy equation an ultradiscrete limit of the special solutions is studied using the procedure of ultradiscretization with parity variables then we obtain new airy type solutions of the ultradiscrete painleve ii equation with parity variables and the solutions have a richer structure than the known solutions

a tropical analogue of fay s trisecant identity and the ultra discrete periodic toda lattice we introduce a tropical analogue of fay s trisecant identity for a special family of hyperelliptic tropical curves we apply it to obtain the general solution of the ultra discrete toda lattice with periodic boundary conditions in terms of the tropical riemann s theta function

conserved quantities and generalized solutions of the ultradiscrete kdv equation we construct generalized solutions to the ultradiscrete kdv equation including the so called negative solition solutions the method is based on the ultradiscretization of soliton solutions to the discrete kdv equation with gauge transformation the conserved quantities of the ultradiscrete kdv equation are shown to be constructed in a similar way to those for the box ball system

casorati determinant solutions for the discrete painleve iii equation the discrete painleve iii equation is investigated based on the bilinear formalism it is shown that it admits the solutions expressed by the casorati determinant whose entries are given by the discrete bessel functions moreover based on the observation that these discrete bessel functions are transformed to the q bessel functions by a simple variable transformation we present a q difference analog of the painleve iii equation

convergence analysis of schwarz waveform relaxation with convolution transmission conditions it is well known that the optimal transmission conditions tcs give in theory the best convergence for the schwarz waveform relaxation swr algorithm these tcs are of convolution type and are not systematically explored yet partly because numerical computation of convolutions is expensive however if the governing equation is also of convolution type such as the volterra integro partial differential equations with weakly singular kernels studied in this paper this kind of tcs seems a very natural choice for a class of convolution tcs including the optimal one derived from the space time continuous analysis we perform a convergence analysis for the swr algorithm at the fully discretized level in the framework of convolution quadratures which permits us to precisely capture the effects of the discretizations on the convergence rate by optimizing the parameter in the tcs we show that the convergence factor can be reduced to a value around 0 0730 if an a stable convolution quadrature is used th

collinear central configurations and singular surfaces in the mass space for a given m m1 mn r n let p and q r3 n be two central configurations for m then we call p and qequivalent and write p q if they differ by an so 3 rotation followed by a scalar multiplication as well as by a permutation of bodies denote by l n m the set of equivalent classes of n body collinear central configurations in r3 for any given mass vector m m1 mn r n the main discovery in this paper is the existence of a union h3 of three non empty algebraic surfaces in the mass half space m1 m2 m1 m3 m2 r r2 besides the planes generated by equal masses which decreases the number of collinear central configurations the union h3 in r r2 is explicitly constructed by three 6 degree homogeneous polynomials in three variables such that for any mass vector m m1 m2 m3 r 3 l 3 m 3 if m1 m2 and m3 are mutually distinct and m1 m2 m1 m3 m2 h3 l 3 m 2 if m1 m2 and m3 are mutually distinct and m1 m2 m1 m3 m2 h3 l 3 m 2 if two of m1 m2 and m3 are equal but not the third l 3 m 1 if m1 m2 m3 we give also a sharp upper bound on l n m for any positive mass vector m r n

piecewise linear soliton equations and piecewise linear integrable maps we study piecewise linear integrable systems the associated piecewise linear solitons piecewise linear integrable maps and piecewise linear lax representation are discussed

analytical properties of ultradiscrete burgers equation and rule 184 cellular automaton in this paper we propose an ultradiscrete burgers equation of which all the variables are discrete the equation is derived from a discrete burgers equation under an ultradiscrete limit and reduces to an ultradiscrete diffusion equation through the cole hopf transformation moreover it becomes a cellular automaton ca under appropriate conditions and is identical to rule 184 ca in a specific case we show shock wave solutions and asymptotic behaviours of the ca exactly via the diffusion equation finally we propose a particle model expressed by the ca and discuss a mean flux of particles

ultradiscretization of solvable one dimensional chaotic maps we consider the ultradiscretization of a solvable one dimensional chaotic map which arises from the duplication formula of the elliptic functions it is shown that the ultradiscrete limit of the map and its solution yield the tent map and its solution simultaneously a geometric interpretation of the dynamics of the tent map is given in terms of the tropical jacobian of a certain tropical curve generalization to the maps corresponding to the mth multiplication formula of the elliptic functions is also discussed

four body central configurations with some equal masses we prove firstly that any convex non collinear central configuration of the planar 4 body problem with equal opposite masses 0 such that the diagonal corresponding to the mass is not shorter than that corresponding to the mass must possess a symmetry and therefore must be a kite then by a recent result of bernat llibre and perez chavela this kite is actually a rhombus secondly we prove that a convex non collinear planar 4 body central configuration with three equal masses must be a kite too we also prove that the concave central configuration with three equal masses forming a triangle and the fourth one with any given mass in the interior must be either an equilateral triangle with the fourth mass at its geometric center or an isosceles triangle with the fourth mass on the symmetry axis

the golden ratio and super central configurations of the n body problem abstract in this paper we consider the problem of central configurations of the n body problem with the general homogeneous potential 1 r a configuration q q 1 q 2 q n is called a super central configuration if there exists a positive mass vector m m 1 m n such that q is a central configuration for m with m i attached to q i and q is also a central configuration for m where m m and m is a permutation of m the main discovery in this paper is that super central configurations of the n body problem have surprising connections with the golden ratio let r be the ratio q 3 q 2 q 2 q 1 of the collinear three body problem with the ordered positions q 1 q 2 q 3 on a line q is a super central configuration if and only if 1 r 1 r r 1 and r 1 where r 1 1 is a continuous function such that lim 0 r 1 the golden ratio the existence and classification of super central configurations are established in the collinear three body problem with general homogeneous potential 1 r super central configurations play an important role in counting the number of central configurations for a given mass vector which may decrease the number of central configurations under geometric equivalence

determinant structure of the rational solutions for the painleve ii equation two types of determinant representations of the rational solutions for the painleve ii equation are discussed by using the bilinear formalism one of them is a representation by the devisme polynomials and another one is hankel determinant representation they are derived from the determinant solutions of the kp hierarchy and toda lattice respectively

waveform relaxation with overlapping splittings in this paper an extension of the waveform relaxation algorithm for solving large systems of ordinary differential equations is presented the waveform relaxation algorithm is well suited for parallel computation because it decomposes the solution space into several disjoint subspaces allowing the subspaces to overlap i e dropping the assumption of disjointedness an extension of this algorithm is obtained this new algorithm the so called multisplitting algorithm is also well suited for parallel computation as numerical examples demonstrate this overlapping of the subsystems heavily reduces the computation time

box and ball system with a carrier and ultradiscrete modified kdv equation a new soliton cellular automaton is proposed it is defined by an array of an infinite number of boxes a finite number of balls and a carrier of balls moreover it reduces to a discrete equation obtained from the discrete modified korteweg de vries equation through a limit an algebraic expression of soliton solutions is also proposed

the symmetric central configurations of the 4 body problem with masses m1 m2 m3 m4 we characterize the planar central configurations of the 4 body problem with masses m 1 m 2 m 3 m 4 which have an axis of symmetry it is known that this problem has exactly two classes of convex central configurations one with the shape of a rhombus and the other with the shape of an isosceles trapezoid we show that this 4 body problem also has exactly two classes of concave central configurations with the shape of a kite this proof is assisted by computer

classification of four body central configurations with three equal masses it is known that a central configuration of the planar four body problem consisting of three particles of equal mass possesses a symmetry if the configuration is convex or is concave with the unequal mass in the interior we use analytic methods to show that besides the family of equilateral triangle configurations there are exactly one family of concave and one family of convex central configurations which completely classifies such central configurations