a numerical verification method for nonlinear functional equations based on infinite dimensional newton like iteration this paper describes a numerical verification of solutions for infinite dimensional functional equations based on residual forms and newton like iteration the method is based upon a verification method previously developed by the authors several computer assisted proofs for differential equations including nonlinear partial differential equations are presented

extreme values and the law of the iterated logarithm ifx takes values in a banach spaceb and is in the domain of attraction of a gaussian law onb thenx satisfies the compact law of the iterated logarithm lil with respect to a regular normalizing sequence n iffx satisfies a certain integrability condition the integrability condition is equivalent to the fact that the maximal term of the sample x 1 x 2 x n does not dominate the partial sums s n and here we examine the precise influence of these maximal terms and its relation to the compactlil in particular it is shown that if one deletes enough of the maximal terms there is always a compactlil with non trivial limit set

an enclosure method of eigenvalues for the elliptic operator linearized at an exact solution of nonlinear problems abstract we consider eigenvalue enclosing of the elliptic operator which is linearized at an exact solution of certain nonlinear elliptic equation this problem is important in the mathematically rigorous analysis of the stability or bifurcation of some solutions for nonlinear problems we formulate such a kind of eigenvalue problem as the nonlinear system which contains both linearized eigenvalue problem and the original nonlinear equation we also consider the indices of eigenvalues especially the first eigenvalue of such a problem in these enclosing procedures the finite dimensional verified computations for linear and nonlinear system of equations play an essential role a numerical example is presented

communication complexity of simultaneous messages in the multiparty communication game cfl game of chandra furst and lipton proceedings of the 15th annual acm symposium on theory of computing boston ma 1983 pp 94 99 k players collaboratively evaluate a function f x0 xk 1 in which player i knows all inputs except xi the players have unlimited computational power the objective is to minimize communication in this paper we study the simultaneous messages sm model of multiparty communication complexity the sm model is a restricted version of the cfl game in which the players are not allowed to communicate with each other instead each of the k players simultaneously sends a message to a referee who sees none of the inputs the referee then announces the function value we prove lower and upper bounds on the sm complexity of several classes of explicit functions our lower bounds extend to randomized sm complexity via an entropy argument a lemma establishing a tradeoff between average hamming distance and range size for transformations of the boolean cube might be of independent interest our lower bounds on sm complexity imply an exponential gap between the sm model and the cfl model for up to log n 1 epsilon players for any epsilon 0 this separation is obtained by comparing the respective complexities of the generalized addressing function gafg k where g is a group of order n we also combine our lower bounds on sm complexity with the ideas of hastad and goldmann comput complexity 1 1991 pp 113 129 to derive superpolynomial lower bounds for certain depth 2 circuits computing a function related to the gaf function we prove some counterintuitive upper bounds on sm complexity we show that sf gaf mathbb z 2 t 3 has sm complexity o n 0 92 when the number of players is at least c log n for some constant c 0 our sm protocol for sf gaf mathbb z 2 t k has polylog n complexity we also examine a class of functions defined by certain depth 2 circuits this class includes the generalized inner product function and majority of majorities when the number of players is at least 2 log n we obtain polylog n upper bounds for this class of functions

existence and instability of steady states for a triangular cross diffusion system a computer assisted proof abstract in this paper we present and apply a computer assisted method to study steady states of a triangular cross diffusion system our approach consist in an a posteriori validation procedure that is based on using a fixed point argument around a numerically computed solution in the spirit of the newton kantorovich theorem it allows to prove the existence of various non homogeneous steady states for different parameter values in some situations we obtain as many as 13 coexisting steady states we also apply the a posteriori validation procedure to study the linear stability of the obtained steady states proving that many of them are in fact unstable

improving the accuracy of computed eigenvalues and eigenvectors this paper describes a computational method for improving the accuracy of a given eigenvalue and its associated eigenvector the method is analogous to iterative improvement for the solution of linear systems an iterative algorithm using working precision arithmetic is applied to increase the accuracy of the eigenpair the only extended precision computation is the residual calculation the method is related to inverse iteration and to newton s method applied to the eigenvalue problem

explicit h2 estimates and pointwise bounds for solutions of second order elliptic boundary value problems abstract it is well known that the h 2 norm and the c 0 norm of a function u h 2 where r n is a bounded domain n 3 can be estimated in terms of a given uniformly elliptic second order differential operator l and some boundary operator b applied to u if certain regularity assumptions are satisfied if these bounds shall be used for numerical purposes the constants occurring in the estimates must be known explicitly the main goal of the present article is the computation of such explicit constants for simplicity of presentation we restrict ourselves to the case where l u u c x u as an application we prove an existence and inclusion result for nonlinear boundary value problems

a numerical verification method for solutions of boundary value problems with local uniqueness by banach s fixed point theorem in this paper we propose a method to prove the existence and the local uniqueness of solutions to infinite dimensional fixed point equations using computers choosing a set which possibly includes a solution we transform it by an approximate linearization of the operator appearing in the equation then we calculate the radii of the transformed set in order to check sufficient conditions for banach s fixed point theorem this method is applied to elliptic problems and numerical examples are given

an improved verification algorithm for nonlinear systems of equations based on krawczyk operator abstract in this paper an improved version of a verification algorithm for solving nonlinear systems of equations based on krawczyk operator is presented compared with the original algorithm the improved verification algorithm not only saves computing time but also computes a narrower or at least the same inclusion of the solution to nonlinear systems of equations for certain classes of problems numerical results demonstrate the performance of the proposed algorithms

numerical verification of solutions for obstacle problems this article is an extension of the previous paper numer math 81 1998 305 by the same authors we propose a method to prove the existence of solutions for more general obstacle problems by numerical computations the main task in this paper consists of the numerical determination of some constants which appear in a priori error estimations for the finite element approximation of a simple variational inequality we present a numerical example for verification

a computer assisted proof for the kolmogorov flows of incompressible viscous fluid a computer assisted proof of non trivial steady state solutions for the kolmogorov flows is described the method is based on the infinite dimensional fixed point theorem using newton like operator this paper also proposes a numerical verification algorithm which generates automatically on a computer a set including the exact non trivial solution with local uniqueness all discussed numerical results take into account the effects of rounding errors in the floating point computations

rounds in communication complexity revisited the k round two party communication complexity was studied in the deterministic model by p h papadimitriou and m sipser proc of the 14th stoc 1982 pp 330 337 and p duris z galil and g schnitger proc of the 16th stoc 1984 pp 81 91 and in the probabilistic model by a c yao proc of the 24th focs 1983 pp 420 428 and b halstenberg and r reischuk proc of the 20th stoc 1988 pp 162 172 this paper presents new lower bounds that give 1 randomization is more powerful than determinism in k round protocols and 2 an explicit function which exhibits an exponential gap between its k and k 1 round randomized complexity this paper also studies the three party communication model and exhibits an exponential gap in 3 round protocols that differ in the starting player finally this paper shows new connections of these questions to circuit complexity that motivate further work in this direction

computational fixed point theory for differential delay equations with multiple time lags abstract we introduce a general computational fixed point method to prove existence of periodic solutions of differential delay equations with multiple time lags the idea of such a method is to compute numerical approximations of periodic solutions using newton s method applied on a finite dimensional projection to derive a set of analytic estimates to bound the truncation error term and finally to use this explicit information to verify computationally the hypotheses of a contraction mapping theorem in a given banach space the fixed point so obtained gives us the desired periodic solution we provide two applications the first one is a proof of coexistence of three periodic solutions for a given delay equation with two time lags and the second one provides rigorous computations of several nontrivial periodic solutions for a delay equation with three time lags

rigorous computation of smooth branches of equilibria for the three dimensional cahn hilliard equation in this paper we propose a new general method to compute rigorously global smooth branches of equilibria of higher dimensional partial differential equations the theoretical framework is based on a combination of the theory introduced in global smooth solution curves using rigorous branch following van den berg et al math comput 79 271 1565 1584 2010 and in analytic estimates and rigorous continuation for equilibria of higher dimensional pdes gameiro and lessard j diff equ 249 9 2237 2268 2010 using this method one can obtain proofs of existence of global smooth solution curves of equilibria for large continuous parameter ranges and about local uniqueness of the solutions on the curve as an application we compute several smooth branches of equilibria for the three dimensional cahn hilliard equation

solving obstacle problems with guaranteed accuracy abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for some simple obstacle problems using the finite element approximation and constructive error estimates we construct on a computer a set of solutions which satisfies the hypothesis of the schauder fixed point theorem for a compact map on a certain sobolev space we describe the numerical verification algorithm for solving a two dimensional obstacle problems and report some numerical results

verified numerical computations for multiple and nearly multiple eigenvalues of elliptic operators in this paper we propose a numerical method to verify bounds for multiple eigenvalues for elliptic eigenvalue problems we calculate error bounds for approximations of multiple eigenvalues and base functions of the corresponding invariant subspaces for matrix eigenvalue problems rump linear algebra appl 324 2001 209 recently proposed a validated numerical method to compute multiple eigenvalues in this paper we extend his formulation to elliptic eigenvalue problems combining it with a method developed by one of the authors jpn j indust appl math 16 1998 307

automatic differentiation for fourier series and the radii polynomial approach abstract in this work we develop a computer assisted technique for proving existence of periodic solutions of nonlinear differential equations with non polynomial nonlinearities we exploit ideas from the theory of automatic differentiation in order to formulate an augmented polynomial system we compute a numerical fourier expansion of the periodic orbit for the augmented system and prove the existence of a true solution nearby using an a posteriori validation scheme the radii polynomial approach the problems considered here are given in terms of locally analytic vector fields i e the field is analytic in a neighborhood of the periodic orbit hence the computer assisted proofs are formulated in a banach space of sequences satisfying a geometric decay condition in order to illustrate the use and utility of these ideas we implement a number of computer assisted existence proofs for periodic orbits of the planar circular restricted three body problem pcrtbp

analytic estimates and rigorous continuation for equilibria of higher dimensional pdes abstract in this paper we extend the ideas of the so called validated continuation technique to the context of rigorously proving the existence of equilibria for partial differential equations defined on higher dimensional spatial domains for that effect we present a new set of general analytic estimates these estimates are valid for any dimension and are used together with rigorous computations to construct a finite number of radii polynomials these polynomials provide a computationally efficient method to prove via a contraction argument the existence and local uniqueness of solutions for a rather large class of nonlinear problems we apply this technique to prove existence and local uniqueness of equilibrium solutions for the cahn hilliard and the swift hohenberg equations defined on two and three dimensional spatial domains

numerical inclusion methods of solutions for variational inequalities we consider a numerical method that enables us to verify the existence of solutions for variational inequalities this method is based on the infinite dimensional fixed point theorems and explicit error estimates for finite element approximations using the finite element approximations and explicit a priori error estimates we present an effective verification procedure that through numerical computation generates a set which includes the exact solution copyright 2002 john wiley sons ltd

an efficient numerical verification method for the kolmogorov problem of incompressible viscous fluid some computer assisted proofs of nontrivial steady state solutions for the kolmogorov flows are presented the method is based on the infinite dimensional fixed point theorem using a newton like operator with a numerical verification algorithm that automatically generates a set that includes the exact nontrivial solution when discussing the numerical results we consider the effects of rounding errors in the floating point computations this is a continuation of our study that was presented in watanabe 2009

constructive a priori error estimates for a full discrete approximation of the heat equation in this paper we consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation our method is based on the finite element galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space the present estimates play an essential role in the numerical verification method of exact solutions for the nonlinear parabolic equations this implies that by utilizing the present results we could get the guaranteed a posteriori error estimates for various kinds of nonlinear evolutional problems our results can also be considered as an explicit optimal estimate with the limited regularity of solutions which should be unknown up to the present

the computational complexity of universal hashing any implementation of carter wegman universal hashing from n bit strings to m bit strings requires a time space tradeoff of ts nm the bound holds in the general boolean branching program model and thus in essentially any model of computation as a corollary computing a b c in any field f requires a quadratic time space tradeoff and the bound holds for any representation of the elements of the field other lower bounds on the complexity of any implementation of universal hashing are given as well quadratic at2 bound for vlsi implementation logn parallel time bound on a crew pram and exponential size for constant depth circuits

global smooth solution curves using rigorous branch following in this paper we present a new method for rigorously computing smooth branches of zeros of nonlinear operators f l 1 b 1 l 2 b 2 where b 1 and b 2 are banach spaces the method is first introduced for parameter continuation and then generalized to pseudo arclength continuation examples in the context of ordinary partial and delay differential equations are given

some results on the lil in banach space with applications to weighted empirical processes we examine the cluster set of s n a n for banach space valued random variables and investigate the relationship between the central limit theorem and the law of the iterated logarithm in this setting in the case of hilbert space valued random variables necessary and sufficient conditions are given for the law of the iterated logarithm some interesting examples are also included we then apply our results to weighted empiricals both in the supremum norm and the l 2 lbrack 0 1 rbrack norm

boolean circuits tensor ranks and communication complexity we investigate two methods for proving lower bounds on the size of small depth circuits namely the approaches based on multiparty communication games and algebraic characterizations extending the concepts of the tensor rank and rigidity of matrices our methods are combinatorial but we think that our main contribution concerns the algebraic concepts used in this area tensor ranks and rigidity our main results are following i an o n bit protocol for a communication game for computing shifts which also gives an upper bound of o n 2 on the contact rank of the tensor of multiplication of polynomials this disproves some earlier conjectures a related probabilistic construction gives an o n upper bound for computing all permutations and an o n log log n upper bound on the communication complexity of pointer jumping with permutations ii a lower bound on certain restricted circuits of depth 2 which are related to the problem of proving a superlinear lower bound on the size of logarithmic depth circuits this bound has interpretations both as a lower bound on the rigidity of the tensor of multiplication of polynomials and as a lower bound on the communication needed to compute the shift function in a restricted model iii an upper bound on boolean circuits of depth 2 for computing shifts and more generally all permutations this shows that such circuits are more efficient than the model based on sending bits along vertex disjoint paths

a method for separating nearly multiple eigenvalues for hermitian matrix in this paper we propose a numerical method to verify for nearly multiple eigenvalues of a hermitian matrix not being strictly multiple eigenvalues from approximate eigenvalues computed it seems to be difficult to distinguish whether they are strictly multiple eigenvalues or simple ones and if they are very close each other the verification method for simple eigenvalues may fail to enclose them separately because of singularity of the system in the verification there are several methods for enclosing multiple and nearly multiple eigenvalues e g rump computational error bounds for multiple or nearly multiple eigenvalues linear algebra appl 324 2001 209 226 for such cases there is no result to decide the enclosed eigenvalues are nearly multiple or strictly multiple up to now so for enclosed eigenvalues we propose a numerical method to separate nearly multiple eigenvalues

numerical method with guaranteed accuracy of a double turning point for a radially symmetric solution of the perturbed gelfand equation we describe a numerical method with guaranteed accuracy to enclose a double turning point for a radially symmetric solution of the perturbed gelfand equation we use nakao s method and a system of equations defined on an extended space to enclose a double turning point we describe verification procedures in detail and give a numerical example

numerical verification of solutions for elasto plastic torsion problems abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for the elasto plastic torsion problems governed by the variational inequality based upon the finite element approximations and the explicit a priori error estimates for a simple problem we present an effective verification procedure that through numerical computation generates a set which includes the exact solution this paper is an extension of the previous paper 1 in which we mainly dealt with the obstacle problems but some special techniques are utilized to verify the solutions for nondifferentiable nonlinear equations concerned with the present problem a numerical example is illustrated

numerical existence and uniqueness proof for solutions of nonlinear hyperbolic equations abstract we consider a numerical method to verify the existence and uniqueness of the solutions of nonlinear hyperbolic problems with guaranteed error bounds using a c 1 finite element solution and an inequality constituting a bound on the norm of the inverse operator of the linearized operator we numerically construct a set of functions which satisfy the hypothesis of banach s fixed point theorem for a continuous map on l p space in a computer we present detailed verification procedures and give some numerical examples

a law of the iterated logarithm for distributions in the generalized domain of attraction of a nondegenerate gaussian law when operators t n exist such that for sums s n of n i i d copies of a finite dimensional random vector x we have t n s n is shift convergent in distribution to a standard gaussian law a necessary and sufficient condition on the distribution of x is given for the appropriate law of the iterated logarithm using the operators t n to hold our result extends certain well known real line l i l s it utilizes a necessary and sufficient condition due to hahn and klass for t n to exist giving a gaussian limit law and employs a second moment technique due to kuelbs and zinn

computational error bounds for multiple or nearly multiple eigenvalues abstract in this paper bounds for clusters of eigenvalues of non selfadjoint matrices are investigated we describe a method for the computation of rigorous error bounds for multiple or nearly multiple eigenvalues and for a basis of the corresponding invariant subspaces the input matrix may be real or complex dense or sparse the method is based on a quadratically convergent newton like method it includes the case of defective eigenvalues uncertain input matrices and the generalized eigenvalue problem computational results show that verified bounds are still computed even if other eigenvalues or clusters are nearby the eigenvalues under consideration

validated continuation for equilibria of pdes one of the most efficient methods for determining the equilibria of a continuous parameterized family of differential equations is to use predictor corrector continuation techniques in the case of partial differential equations this procedure must be applied to some finite dimensional approximation which of course raises the question of the validity of the output we introduce a new technique that combines the information obtained from the predictor corrector steps with ideas from rigorous computations and verifies that the numerically produced equilibrium for the finite dimensional system can be used to explicitly define a set which contains a unique equilibrium for the infinite dimensional partial differential equation using the cahn hilliard and swift hohenberg equations as models we demonstrate that the cost of this new validated continuation is less than twice the cost of the standard continuation method alone

a numerical verification method for the existence of weak solutions for nonlinear boundary value problems abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for nonlinear two point boundary value problems bvp we formulate the problem as a fixed point of a newton like operator and present a verification algorithm by computer based on sadovskii s fixed point theorem in order to compute the newton type iterative sequences in infinite dimensional space two concepts the rounding and the rounding error are introduced also some numerical examples are illustrated we emphasize that since the present method uses the weak formulation of bvp and the finite element approximations instead of the existing validation techniques it is aslo applicable in principle to the bvp for partial differential equations as it stands

existence and enclosure results for continua of solutions of parameter dependent nonlinear boundary value problems we propose a numerical method for computing enclosures for continua of solutions of nonlinear elliptic boundary value problems depending on a real parameter at the same time proving the existence of a smooth solution continuum the method is extended to turning point problems by change of parameters

on the parameterized complexity of approximating dominating set we study the parameterized complexity of approximating the k dominating set domset problem where an integer k and a graph g on n vertices are given as input and the goal is to find a dominating set of size at most f k k whenever the graph g has a dominating set of size k when such an algorithm runs in time t k poly n i e fpt time for some computable function t it is said to be an f k fpt approximation algorithm for k domset whether such an algorithm exists is listed in the seminal book of downey and fellows 2013 as one of the most infamous open problems in parameterized complexity this work gives an almost complete answer to this question by showing the non existence of such an algorithm under w 1 fpt and further providing tighter running time lower bounds under stronger hypotheses specifically we prove the following for every computable functions t f and every constant e 0 assuming w 1 fpt there is no f k fpt approximation algorithm for k domset assuming the exponential time hypothesis eth there is no f k approximation algorithm for k domset that runs in t k no k time assuming the strong exponential time hypothesis seth for every integer k 2 there is no f k approximation algorithm for k domset that runs in t k nk e time assuming the k sum hypothesis for every integer k 3 there is no f k approximation algorithm for k domset that runs in t k n k 2 e time previously only constant ratio fpt approximation algorithms were ruled out under sf w 1 fpt and log1 4 m k fpt approximation algorithms were ruled out under eth chen and lin focs 2016 recently the non existence of an f k fpt approximation algorithm for any function f was shown under gap eth chalermsook et al focs 2017 note that to the best of our knowledge no running time lower bound of the form nd this allows us to easily apply known techniques to solve them

numerical verification of solutions for variational inequalities of the second kind abstract the purpose of this paper is to present an approach to the numerical proof of existence of solutions for the problem of the flow of a viscous plastic fluid in a pipe using the finite element approximations and the explicit a priori error estimates for the problem of the flow of a viscous plastic fluid in a pipe we present an effective verification procedure that through numerical computation generates a set which includes the exact solution further a numerical example is presented

a hermite spectral method for the computation of homoclinic orbits and associated functionals we present a spectral method for the computation of homoclinic orbits in ordinary differential equations the method is based on hermite fourier expansions of the complete homoclinic solution and exhibits exponential convergence in addition our method can be used to approximate nonlinear functionals which depend on the complete homoclinic solution this is demonstrated using examples from phase separation dynamics and metastability

numerical method for verifying the existence and local uniqueness of a double turning point for a radially symmetric solution of the perturbed gelfand equation a numerical verification method to confirm the existence and local uniqueness of a double turning point for a radially symmetric solution of the perturbed gelfand equation is presented using certain systems of equations corresponding to a double turning point we derive a sufficient condition for its existence whose satisfaction can be verified computationally we describe verification procedures and give a numerical example as a demonstration