forced lattice vibrations part ii this is the first part of a two part series on forced lattice vibrations in which a semi infinite lattice of one dimensional particles xn n 1 is driven from one end by a particle x0 this particle undergoes a given periodically perturbed uniform motion x0 t at h yt where a and are constants and h has period 2 for a wide variety of restoring forces f i e f 0 numerical calculations indicate the existence of a sequence of thresholds 1 1 a h f 2 2 a h f k k a h f k 0 as k if k k 1 a k phase wave that is well described by the wave form emerges and travels through the lattice the goal of this series is to describe the emergence and calculate some properties of these wave forms in part i the authors first consider the case where f x ex i e toda forces but h is arbitrary and show how to compute a basic diagnostic see j formula 1 26 for the system in terms of the solution of an associated scalar riemann hilbert problem once a certain finite set of numbers is known in another direction the authors consider the case where f x is restoring but arbitrary and h is small here the authors prove a general result asserting that if there exists a sufficiently ample family of traveling wave solutions of the doubly infinite lattice then it is possible to construct time periodic k phase wave solutions with asymptotics in n of type iii for the driven system i in part ii the authors prove that sufficiently ample families of traveling wave solutions of the system iv exist in the cases 1 and 1 2 for general restoring forces f in the case with toda forces f x ex the authors prove that sufficiently ample families of traveling wave solutions

geometry of the modulational instability iii homoclinic orbits for the periodic sine gordon equation in this paper the homoclinic geometric structure of the integrable sine gordon equation under periodic boundary conditions is developed specifically focus is given to orbits homoclinic to n tori simple examples of such homoclinic orbits are constructed and a physical interpretation of these states is given a labeling is provided which identifies and catalogues all such orbits these orbits are related in a one to one manner to linearized instabilities explicit formulas for all homoclinic orbits are given in terms of backlund transformations

spectral and structural analysis of high precision finite difference matrices for elliptic operators abstract in this paper we study the structural properties of matrices coming from high precision finite difference fd formulae when discretizing elliptic or semielliptic differential operators l a u of the form k d k d x k a x d k d x k u x strong relationships with toeplitz structures and linear positive operators lpo are highlighted these results allow one to give a detailed analysis of the eigenvalues localisation distribution of the arising matrices the obtained spectral analysis is then used to define optimal toeplitz preconditioners in a very compact and natural way and in addition to prove szego like and widom like ergodic theorems for the spectra of the related preconditioned matrices a wide numerical experimentation confirming the theoretical results is also reported

spectral behavior of matrix sequences and discretized boundary value problems abstract in this paper we provide theoretical tools for dealing with the spectral properties of general sequences of matrices of increasing dimension more specifically we give a unified treatment of notions such as distribution equal distribution localization equal localization clustering and sub clustering as a case study we consider the matrix sequences arising from the finite difference fd discretization of elliptic and semielliptic boundary value problems bvps the spectral analysis is then extended to toeplitz based preconditioned matrix sequences with special attention to the case where the coefficients of the differential operator are not regular belong to l 1 and to the case of multidimensional problems the related clustering properties allow the establishment of some ergodic formulas for the eigenvalues of the preconditioned matrices

a note on approximate nash equilibria in view of the intractability of finding a nash equilibrium it is important to understand the limits of approximation in this context a subexponential approximation scheme is known richard j lipton evangelos markakis aranyak mehta playing large games using simple strategies in ec 2003 and no approximation better than 14 is possible by any algorithm that examines equilibria involving fewer than logn strategies ingo althofer on sparse approximations to randomized strategies and convex combinations linear algebra and its applications 1994 199 we give a simple linear time algorithm examining just two strategies per player and resulting in a 12 approximate nash equilibrium in any 2 player game for the more demanding notion of approximately well supported nash equilibrium due to constantinos daskalakis paul w goldberg christos h papadimitriou the complexity of computing a nash equilibrium siam journal on computing in press preliminary version appeared in stoc 2006 no nontrivial bound is known we show that the problem can be reduced to the case of win lose games games with all utilities 0 or 1 and that an approximation of 56 is possible contingent upon a graph theoretic conjecture subsequent work extends the 14 impossibility result of ingo althofer s paper as mentioned above to 12 tomas feder hamid nazerzadeh amin saberi approximating nash equilibria using small support strategies in ec 2007 making our 12 approximate nash equilibrium algorithm optimal among the algorithms that only consider mixed strategies of sublogarithmic size support moreover techniques similar to our techniques for approximately well supported nash equilibria are used in spyros kontogiannis paul g spirakis efficient algorithms for constant well supported approximate equilibria in bimatrix games in icalp 2007 for obtaining an efficient algorithm for 0 658 approximately well supported nash equilibria unconditionally

multigrid methods for multilevel circulant matrices we introduce a multigrid technique for the solution of multilevel circulant linear systems whose coefficient matrix has eigenvalues of the form f x j n where f is continuous and independent of n n 1 ldots n d and x j n equiv 2 pi j n 2 pi j 1 n 1 ldots 2 pi j d n d 0 le j r le n r 1 the interest of the proposed technique pertains to the multilevel banded case where the total cost is optimal i e o n arithmetic operations ops n prod r 1 d n r instead of o n log n ops arising from the use of ffts in fact multilevel banded circulants are used as preconditioners for elliptic and parabolic pdes with dirichlet or periodic boundary conditions and for some two dimensional image restoration problems where the point spread function psf is numerically banded so that the overall cost is reduced from o k varepsilon n n log n to o k varepsilon n n where k varepsilon n is the number of pcg iterations to reach the solution within an accuracy of varepsilon several numerical experiments concerning one rank regularized circulant discretization of elliptic 2q differential operators over one dimensional and two dimensional square domains with mixed boundary conditions are performed and discussed

numerical solution method for the dbar equation in the plane a fast method for solving equations of the form v tv is presented where v and t are complex valued functions of two real variables the multigrid method of vainikko int soc anal appl comput 5 2000 is adapted to the problem with a fft implementation convergence with rate o h is proved for the method applied to equations of the form above one grid and two grid versions of the method are implemented and their effectiveness is demonstrated on an application arising in electrical impedance tomography eit

the semiclassical limit of the defocusing nls hierarchy we establish the semiclassical limit of the one dimensional defocusing cubic nonlinear schrodinger nls equation complete integrability is exploited to obtain a global characterization of the weak limits of the entire nls hierarchy of conserved densities as the field evolves from reflectionless initial data under all the associated commuting flows consequently this also establishes the zero dispersion limit of the modified korteweg de vries equation that resides in that hierarchy we have adapted and clarified the strategy introduced by lax and levermore to study the zero dispersion limit of the korteweg de vries equation expanding it to treat entire integrable hierarchies and strengthening the limits obtained a crucial role is played by the convexity of the underlying log determinant with respect to the times associated with the commuting flows c 1999 john wiley sons inc

discrete fixed points models complexities and applications we study three discrete fixed point concept sperner dpzp brouwer under two different models the polynomial time function model and the oracle function model we fully characterize the computational complexities of these three problems the computational complexity unification of the above problems gives us more choices in the study of different applications as an example by a reduction from dpzp we derive asymptotically equal lower and upper bound for tucker in the oracle model the same reduction also allows us to derive a single proof for the ppad completeness of tucker in any constant dimension which is significantly simpler than the recent proofs

spectral approach to d bar problems we present the first numerical approach to d bar problems having spectral convergence for real analytic rapidly decreasing potentials the proposed method starts from a formulation of the problem in terms of an integral equation that is numerically solved with fourier techniques the singular integrand is regularized analytically the resulting integral equation is approximated via a discrete system that is solved with krylov methods as an example the d bar problem for the davey stewartson ii equations is considered the result is used to test direct numerical solutions of the pde 2017 wiley periodicals inc

distribution results on the algebra generated by toeplitz sequences a finite dimensional approach abstract let a be an l 1 symbol defined on q d q with d 1 and let us consider the multi indexed sequence of toeplitz matrices t n a with n n d it is well known that t n a is spectrally distributed as a in the sense of the singular values in this paper we prove that a sequence as t n a is spectrally distributed in the sense of the singular values as the measurable function a for a l 1 and and ranging in any finite set of values

stability of the calderon problem for less regular conductivities abstract in these notes we prove log type stability for the calderon problem with conductivities in c 1 e we follow the lines of a recent work by haberman and tataru in which they prove uniqueness for c 1

intrinsic robustness of the price of anarchy the price of anarchy defined as the ratio of the worst case objective function value of a nash equilibrium of a game and that of an optimal outcome quantifies the inefficiency of selfish behavior remarkably good bounds on this measure are known for a wide range of application domains however such bounds are meaningful only if a game s participants successfully reach a nash equilibrium this drawback motivates inefficiency bounds that apply more generally to weaker notions of equilibria such as mixed nash equilibria and correlated equilibria and to sequences of outcomes generated by natural experimentation strategies such as successive best responses and simultaneous regret minimization we establish a general and fundamental connection between the price of anarchy and its seemingly more general relatives first we identify a canonical sufficient condition for an upper bound on the price of anarchy of pure nash equilibria which we call a smoothness argument second we prove an extension theorem every bound on the price of anarchy that is derived via a smoothness argument extends automatically with no quantitative degradation in the bound to mixed nash equilibria correlated equilibria and the average objective function value of every outcome sequence generated by no regret learners smoothness arguments also have automatic implications for the inefficiency of approximate equilibria for bicriteria bounds and under additional assumptions for polynomial length best response sequences third we prove that in congestion games smoothness arguments are complete in a proof theoretic sense despite their automatic generality they are guaranteed to produce optimal worst case upper bounds on the price of anarchy

a complementary pivot algorithm for market equilibrium under separable piecewise linear concave utilities using lemke s scheme we give a complementary pivot algorithm for computing an equilibrium for arrow debreu markets under separable piecewise linear concave splc utilities despite the polynomial parity argument on directed graphs ppad completeness of this case experiments indicate that our algorithm is practical on randomly generated instances the number of iterations it needs is linear in the total number of segments i e pieces in all the utility functions specified in the input our paper settles a number of open problems 1 eaves 1976 gave an lcp formulation and a lemke type algorithm for the linear arrow debreu model we generalize both to the splc case hence settling the relevant part of his open problem 2 our path following algorithm for splc markets together with a result of todd 1976 gives a direct proof of membership of such markets in ppad and settles a question of vazirani and yannakakis 2011 3 we settle a question of devanur and kannan 2008 of obtaining a sy

market equilibrium under separable piecewise linear concave utilities we consider fisher and arrow debreu markets under additively separable piecewise linear concave utility functions and obtain the following results for both market models if an equilibrium exists there is one that is rational and can be written using polynomially many bits there is no simple necessary and sufficient condition for the existence of an equilibrium the problem of checking for existence of an equilibrium is np complete for both market models the same holds for existence of an e approximate equilibrium for e e o n 5 under standard mild sufficient conditions the problem of finding an exact equilibrium is in ppad for both market models finally building on the techniques of chen et al 2009a we prove that under these sufficient conditions finding an equilibrium for fisher markets is ppad hard

universality for the focusing nonlinear schrodinger equation at the gradient catastrophe point rational breathers and poles of the tritronquee solution to painleve i the semiclassical zero dispersion limit of solutions to the one dimensional focusing nonlinear schrodinger equation nls is studied in a scaling neighborhood d of a point of gradient catastrophe x 0 t 0 we consider a class of solutions specified in the text that decay as the neighborhood d contains the region of modulated plane wave with rapid phase oscillations as well as the region of fast amplitude oscillations spikes in this paper we establish the following universal behaviors of the nls solutions q near the point of gradient catastrophe i each spike has height and uniform shape of the rational breather solution to the nls scaled to the size ii the location of the spikes is determined by the poles of the tritronquee solution of the painleve i p1 equation through an explicit map between d and a region of the painleve independent variable iii if but lies away from the spikes the asymptotics of the nls solution is given by the plane wave approximation with the correction term being expressed in terms of the tritronquee solution of p1 the relation with the conjecture of dubrovin grava and klein about the behavior of solutions to the focusing nls near a point of gradient catastrophe is discussed we conjecture that the p1 hierarchy occurs at higher degenerate catastrophe points and that the amplitudes of the spikes are odd multiples of the amplitude at the corresponding catastrophe point our technique is based on the nonlinear steepest descent method for matrix riemann hilbert problems and discrete schlesinger isomonodromic transformations 2013 wiley periodicals inc

high precision numerical approach for davey stewartson ii type equations for schwartz class initial data we present an efficient high precision numerical approach for daveystewartson ds ii type equa tions treating initial data from the schwartz class of smooth rapidly decreasing functions as wit

query complexity of approximate nash equilibria we study the query complexity of approximate notions of nash equilibrium in games with a large number of players n our main result states that for n player binary action games and for constant e the query complexity of an e well supported nash equilibrium is exponential in n as a consequence of this result we get an exponential lower bound on the rate of convergence of adaptive dynamics to approximate nash equilibria

spectral approach to the scattering map for the semi classical defocusing davey stewartson ii equation abstract the inverse scattering approach for the defocusing davey stewartson ii equation is given by a system of d bar equations we present a numerical approach to semi classical d bar problems for real analytic rapidly decreasing potentials we treat the d bar problem as a complex linear second order integral equation which is solved with discrete fourier transforms complemented by a regularization of the singular parts by explicit analytic computation the resulting algebraic equation is solved either by fixed point iterations or gmres several examples for small values of the semi classical parameter in the system are discussed

on the complexity of an expanded tarski s fixed point problem under the componentwise ordering abstract let be a finite lattice of integer points in a box of r n and f an increasing mapping in terms of the componentwise ordering from to itself the well known tarski s fixed point theorem asserts that f has a fixed point in a simple expansion of f from to a larger lattice c of integer points in a box of r n yields that the smallest point in c is always a fixed point of f an expanded tarski s fixed point problem by introducing an integer labeling rule and applying a cubic triangulation of the euclidean space we prove in this paper that the expanded tarski s fixed point problem is in the class ppa when f is given as an oracle it is shown in this paper that nash equilibria of a bimatrix game can be reformulated as fixed points different from the smallest point in c of an increasing mapping from c to itself this implies that the expanded tarski s fixed point problem has at least the same complexity as that of the nash equilibrium problem as a byproduct we also present a homotopy like simplicial method to compute a tarski fixed point of f the method starts from an arbitrary lattice point and follows a finite simplicial path to a fixed point of f

a uniqueness theorem for an inverse boundary value problem in electrical prospection we show that a near constant conductivity of a two dimensional body can be uniquely determined by steady state direct current measurements at the boundary mathematically we show that the coefficient in the operator is uniquely determined by its dirichlet integrals

action of a scattering map on weighted sobolev spaces in the plane abstract we consider a scattering map that arises in the approach to the scattering theory for the davey stewartson ii equation and show that the map is an invertible map between certain weighted l 2 sobolev spaces

d bar method for electrical impedance tomography with discontinuous conductivities the effects of truncating the approximate scattering transform in the d bar reconstruction method for two dimensional electrical impedance tomography are studied the method is based on the uniqueness proof of nachman ann of math 2 143 1996 pp 71 96 that applies to twice differentiable conductivities however the reconstruction algorithm has been successfully applied to experimental data which can be characterized as piecewise smooth conductivities the truncation is shown to stabilize the method against measurement noise and to have a smoothing effect on the reconstructed conductivity thus the truncation can be interpreted as regularization of the d bar method numerical reconstructions are presented demonstrating that features of discontinuous high contrast conductivities can be recovered using the d bar method further a new connection between calderon s linearization method and the d bar method is established and the two methods are compared numerically and analytically

large oscillations arising in a dispersive numerical scheme abstract we study the oscillatory behavior that arises in solutions of a dispersive numerical scheme for the hopf equation whenever the classical solution of the equation develops a singularity modulation equations are derived that describe period two oscillations so long as the solution of those equations takes values for which the equations are hyperbolic however those equations have an elliptic region that may be entered by its solutions in a finite time after which the corresponding period two oscillations are seen to break down this kind of phenomenon has not been observed for integrable schemes the generation and propagation of period two oscillations are asymptotically analyzed and a matching formula is found for the transition between oscillatory and nonoscillatory regions modulation equations are also presented for period three oscillations numerical experiments are carried out that illustrate our analysis

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

semidefinite programming for min max problems and games we consider two min max problems 1 minimizing the supremum of finitely many rational functions over a compact basic semi algebraic set and 2 solving a 2 player zero sum polynomial game in randomized strategies with compact basic semi algebraic sets of pure strategies in both problems the optimal value can be approximated by solving a hierarchy of semidefinite relaxations in the spirit of the moment approach developed in lasserre siam j optim 11 796 817 2001 math program b 112 65 92 2008 this provides a unified approach and a class of algorithms to compute nash equilibria and min max strategies of several static and dynamic games each semidefinite relaxation can be solved in time which is polynomial in its input size and practice on a sample of experiments reveals that few relaxations are needed for a good approximation and sometimes even for finite convergence a behavior similar to what was observed in polynomial optimization

macroscopic lattice dynamics abstract fully nonlinear modulation equations are presented that govern the slow evolution of single phase harmonic wavetrains under a large family of spatially discrete nonintegrable flows among which is the discrete nonlinear schrodinger equation dnls these modulation equations are a pair of partial differential equations in conservation form that are hyperbolic for some data and elliptic for other data in some cases these equations are capable of dynamically changing type from hyperbolic to elliptic a phenomenon that has been associated with the modulational instability of the underlying wavetrain by putting the modulation equations in riemann invariant form one can select initial data that avoid this dynamic change of type numerical experiments demonstrating the theoretical results are presented

the conditioning of fd matrix sequences coming from semi elliptic differential equations abstract in this paper we are concerned with the study of spectral properties of the sequence of matrices a n a coming from the discretization using centered finite differences of minimal order of elliptic or semielliptic differential operators l a u of the form 1 d d x a x d d x u x f x on 0 1 dirichlet b c on where the nonnegative bounded coefficient function a x of the differential operator may have some isolated zeros in more precisely we state and prove the explicit form of the inverse of a n a and some formulas concerning the relations between the orders of zeros of a x and the asymptotic behavior of the minimal eigenvalue condition number of the related matrices as a conclusion and in connection with our theoretical findings first we extend the analysis to higher order semi elliptic differential operators and then we present various numerical experiments showing that similar results must hold true in 2d as well

on total functions existence theorems and computational complexity abstract nondeterministic multivalued functions with values that are polynomially verifiable and guaranteed to exist form an interesting complexity class between p and np we show that this class which we call tfnp contains a host of important problems whose membership in p is currently not known these include besides factoring local optimization brouwer s fixed points a computational version of sperner s lemma bimatrix equilibria in games and linear complementarity for p matrices

the behavior of the weyl function in the zero dispersion kdv limit the moment formulas that globally characterize the zero dispersion limit of the korteweg devries kdv equation are known to be expressed in terms of the solution of a maximization problem here we establish a direct relation between this maximizer and the zero dispersion limit of the logarithm of the jost functions associated with the inverse spectral transform all the kdv conserved densities are encoded in the spatial derivative of these functions known as weyl functions we show the weyl functions are densities of measures that converge in the weak sense to a limiting measure this limiting measure encodes all of the weak limits of the kdv conserved densities moreover we establish the weak limit of spectral measures associated with the dirichlet problem

on the inverse conductivity problem it is shown here that for the boundary value problem div email protected u email protected x inr n u x 0as x in order to identify the coefficient a one needs on additional data u x x g x x where email protected email protected 1 x email protected 2 c 1 c 2 are two open nonempty subsurfaces of email protected w is a domain with analytic boundary which contains a bounded set v and email protected h 2 pp n 2 and a satisfies conditions of lemma 2 here we also prove that the uniqueness of a entering to the problem the additional data

analysis of preconditioning strategies for collocation linear systems in the past few years several authors have studied the preconditioning of collocation matrices by finite differences fds matrices arising from the associated collocation points here we discuss how to solve in an efficient way nonuniform grid fd linear systems including those related to a generic fd collocation preconditioner the main idea is based on a further step of preconditioning defined in terms of diagonal and toeplitz matrices first we identify the limit spectral distributions of the involved fd collocation matrix sequences and then we prove that the proposed toeplitz based preconditioners assure a clustering at the unity with respect to the eigenvalues in the 1d case in the 2d case the situation is different so that more appropriate strategies are discussed a wide numerical experimentation emphasizing the correctness of the theoretical results is also reported

a general equidistribution theorem for the roots of orthogonal polynomials abstract it is well known that the roots of any two orthogonal polynomials are distributed equally if the weights satisfy the szeg condition in this paper we propose a general equidistribution theorem that does not use the szeg condition and admits an elementary proof

on the complexity of nash equilibria and other fixed points we reexamine what it means to compute nash equilibria and more generally what it means to compute a fixed point of a given brouwer function and we investigate the complexity of the associated problems specifically we study the complexity of the following problem given a finite game gamma with 3 or more players and given epsilon 0 compute an approximation within epsilon of some actual nash equilibrium we show that approximation of an actual nash equilibrium even to within any nontrivial constant additive factor epsilon 1 2 in just one desired coordinate is at least as hard as the long standing square root sum problem as well as a more general arithmetic circuit decision problem that characterizes p time in a unit cost model of computation with arbitrary precision rational arithmetic thus placing the approximation problem in p or even np would resolve major open problems in the complexity of numerical computation we show similar results for market equilibria it is hard to estimate with any nontrivial accuracy the equilibrium prices in an exchange economy with a unique equilibrium where the economy is given by explicit algebraic formulas for the excess demand functions we define a class fixp which captures search problems that can be cast as fixed point computation problems for functions represented by algebraic circuits straight line programs over basis max min with rational constants we show that the exact or approximate computation of nash equilibria for 3 or more players is complete for fixp the price equilibrium problem for exchange economies with algebraic demand functions is another fixp complete problem we show that the piecewise linear fragment of fixp equals ppad many other problems in game theory economics and probability theory can be cast as fixed point problems for such algebraic functions we discuss several important such problems computing the value of shapley s stochastic games and the simpler games of condon extinction probabilities of branching processes probabilities of stochastic context free grammars and termination probabilities of recursive markov chains we show that for some of them the approximation or even exact computation problem can be placed in ppad while for others they are at least as hard as the square root sum and arithmetic circuit decision problems

spectral analysis of nonsymmetric quasi toeplitz matrices with applications to preconditioned multistep formulas the eigenvalue spectrum of a class of nonsymmetric preconditioned matrices arising in time dependent partial differential equations is analyzed and discussed the matrices generated by the underlying numerical integrators are small rank perturbations of block toeplitz matrices circulant like preconditioners based on the former are considered the eigenvalue distribution of the preconditioned matrix influences often crucially the convergence of krylov iterative accelerators due to several reasons lack of symmetry band structure and coefficients depending on the size the classical approach based on smooth generating functions gives very little insight here therefore to characterize the eigenvalues a difference equation approach exploiting the band toeplitz and circulant patterns generalizing the well known results of trench is proposed

market equilibria with hybrid linear leontief utilities we introduce a new family of utility functions for exchange markets this family provides a natural and continuous hybridization of the traditional linear and leontief utilities and might be useful in understanding the complexity of computing approximating market equilibria although computing an equilibrium in a market with this family of utility functions this is ppad hard in general in this paper we present an algorithm for finding an approximate arrow debreu equilibrium when the leontief components of the market are grouped finite and well conditioned

dispersive approximations in fluid dynamics we have compared the results of a von neumann richtmyer calculation with the weak limit of calculations performed by von neumann s original method although there are discernible differences they are surprisingly small we analyze von newmann s algorithm for solving the initial value problem for the lagrangian equations of compressible flow

fourth order time stepping for kadomtsev petviashvili and davey stewartson equations purely dispersive partial differential equations such as the korteweg de vries equation the nonlinear schrodinger equation and higher dimensional generalizations thereof can have solutions which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blow up to numerically study such phenomena fourth order time stepping in combination with spectral methods is beneficial in resolving the steep gradients in the oscillatory region we compare the performance of several fourth order methods for the kadomtsev petviashvili and the davey stewartson equations two integrable equations in 2 1 dimensions these methods are exponential time differencing integrating factors time splitting implicit runge kutta and driscoll s composite runge kutta method the accuracy in the numerical conservation of integrals of motion is discussed

on the asymptotic spectrum of finite element matrix sequences we derive a new formula for the asymptotic eigenvalue distribution of stiffness matrices obtained by applying p 1 finite elements with standard mesh refinement to the semielliptic pde of second order in divergence form nabla k nabla t u f on o u g on partial omega here omega subset r 2 and k is supposed to be piecewise continuous and pointwise symmetric semipositive definite the symbol describing this asymptotic eigenvalue distribution depends on the pde but also both on the numerical scheme for approaching the underlying bilinear form and on the geometry of triangulation of the domain our work is motivated by recent results on the superlinear convergence behavior of the conjugate gradient method which requires the knowledge of such asymptotic eigenvalue distributions for sequences of matrices depending on a discretization parameter h when h to 0 we compare our findings with similar results for the finite difference method which were published in recent years in particular we observe that our sequence of stiffness matrices is part of the class of generalized locally toeplitz sequences for which many theoretical tools are available this enables us to derive some results on the conditioning and preconditioning of such stiffness matrices

spectral analysis of coupled pdes and of their schur complements via generalized locally toeplitz sequences in 2d spectral analysis of coupled pdes and of their schur complements via generalized locally toeplitz sequences in 2d

fast algorithms for rank 1 bimatrix games the rank of a bimatrix game is the matrix rank of the sum of the two payoff matrices this paper comprehensively analyzes games of rank one and shows the following 1 for a game of rank r the set of its nash equilibria is the intersection of a generically one dimensional set of equilibria of parameterized games of rank r 1 with a hyperplane 2 one equilibrium of a rank 1 game can be found in polynomial time 3 all equilibria of a rank 1 game can be found by following a piecewise linear path in contrast such a path following method finds only one equilibrium of a bimatrix game 4 the number of equilibria of a rank 1 game may be exponential 5 there is a homeomorphism between the space of bimatrix games and their equilibrium correspondence that preserves rank it is a variation of the homeomorphism used for the concept of strategic stability of an equilibrium component

generalized locally toeplitz sequences spectral analysis and applications to discretized partial differential equations abstract starting from the finite difference discretization of an elliptic second order pde as i j 1 d x i a i j x x j u x b x over a bounded domain we introduce the notion of generalized locally toeplitz sequence of matrices the singular value distribution the eigenvalue distribution in the hermitian case is studied and characterized for generalized locally toeplitz sequences in terms of weighted multidimensional szeg formulas this extends preceding results attributed to tilli which concern the unilevel case the application of this theoretic analysis to the numerical solution of pdes is finally discussed

the complexity of non monotone markets we introduce the notion of non monotone utilities which covers a wide variety of utility functions in economic theory we then prove that it is ppad hard to compute an approximate arrow debreu market equilibrium in markets with linear and non monotone utilities building on this result we settle the long standing open problem regarding the computation of an approximate arrow debreu market equilibrium in markets with ces utility functions by proving that it is ppad complete when the constant elasticity of substitution parameter is any constant less than 1

constant rank two player games are ppad hard finding a nash equilibrium in a two player normal form game 2 nash is one of the most extensively studied problems within mathematical economics as well as theoretical computer science such a ga

solution of a tridiagonal operator equation let h be a separable hilbert space with an orthonormal basis en n n t be a bounded tridiagonal operator on h and tn be its truncation on span e1 e2 en we study the operator equation tx y through its finite dimensional truncations tnxn yn it is shown that if tn 1en and tn 1en are bounded then t is invertible and the solution of tx y can be obtained as a limit in the norm topology of the solutions of its finite dimensional truncations this leads to uniform boundedness of the sequence tn 1 we also give sufficient conditions for the boundedness of tn 1en and tn 1en in terms of the entries of the matrix of t

inverse boundary value problems at the boundary continuous dependence we use the methods of microlocal analysis to give a new proof of a theorem of kohn and vogelius showing that the boundary values of a continuous isotropic conductivity can be recovered from voltage and current measurements at the boundary moreover we prove sharp estimates to establish the continuous dependence of the boundary values of the conductivity on the voltage to current maps

stability estimates for the radon transform with restricted data and applications in this article we prove a stability estimate going from the radon transform of a function with limited angle distance data to the l p norm of the function itself under some conditions on the support of the function we apply this theorem to obtain stability estimates for an inverse boundary value problem with partial data

the glt class as a generalized fourier analysis and applications recently the class of generalized locally toeplitz glt sequences has been introduced as a generalization both of classical toeplitz sequences and of variable coefficient differential operators and for every sequence of the class it has been demonstrated that it is possible to give a rigorous description of the asymptotic spectrum in terms of a function the symbol that can be easily identified this generalizes the notion of a symbol for differential operators discrete and continuous or for toeplitz sequences for which it is identified through the fourier coefficients and is related to the classical fourier analysis the glt class has nice algebraic properties and indeed it has been proven that it is stable under linear combinations and products in this paper we prove that the considered class is closed under inversion as well when the sequence which is inverted shows a sparsely vanishing symbol sparsely vanishing symbol a symbol which vanishes at most in a set of zero lebesgue measure furthermore we show that the glt class virtually includes any finite difference or finite element discretization of pdes and based on this we demonstrate that our results on glt sequences can be used in a pde setting in various directions 1 as a generalized fourier analysis for the study of iterative and semi iterative methods when dealing with variable coefficients non rectangular domains non uniform gridding or triangulations 2 in order to provide a tool for the stability analysis of pde numerical schemes e g a necessary von neumann criterium for variable coefficient systems of pdes is obtained uniformly with respect to the boundary conditions 3 for a multigrid analysis of convergence and for providing spectral information on large preconditioned systems in the variable coefficient case etc the final part of the paper deals indeed with problems 1 3 and other possible directions in which the glt analysis can be conveniently employed

the linear complementarity problems with a few variables per constraint in this paper we consider the sparse linear complementarity problem denoted by k lcp the coefficient matrices are restricted to have at most k nonzero entries per row it is known that the 1 lcp is solvable in linear time and the 3 lcp is strongly np hard we show that the 2 lcp is strongly np hard and it can be solved in polynomial time if it is sign balanced i e each row of the matrix has at most one positive and one negative entry our second result matches the currently best known complexity bound for the corresponding sparse linear feasibility problem in addition we show that an integer variant of the sign balanced 2 lcp is weakly np hard and pseudo polynomially solvable and the generalized 1 lcp is strongly np hard

the complexity of computing a nash equilibrium in 1951 john f nash proved that every game has a nash equilibrium ann of math 2 54 1951 pp 286 295 his proof is nonconstructive relying on brouwer s fixed point theorem thus leaving open the questions is there a polynomial time algorithm for computing nash equilibria and is this reliance on brouwer inherent many algorithms have since been proposed for finding nash equilibria but none known to run in polynomial time in 1991 the complexity class ppad polynomial parity arguments on directed graphs for which brouwer s problem is complete was introduced c papadimitriou j comput system sci 48 1994 pp 489 532 motivated largely by the classification problem for nash equilibria but whether the nash problem is complete for this class remained open in this paper we resolve these questions we show that finding a nash equilibrium in three player games is indeed ppad complete and we do so by a reduction from brouwer s problem thus establishing that the two problems are computationally equivalent our reduction simulates a stylized brouwer function by a graphical game m kearns m littman and s singh graphical model for game theory in 17th conference in uncertainty in artificial intelligence uai 2001 relying on gadgets graphical games performing various arithmetic and logical operations we then show how to simulate this graphical game by a three player game where each of the three players is essentially a color class in a coloring of the underlying graph subsequent work x chen and x deng setting the complexity of 2 player nash equilibrium in 47th annual ieee symposium on foundations of computer science focs 2006 established by improving our construction that even two player games are ppad complete here we show that this result follows easily from our proof

direct reconstructions of conductivities from boundary measurements the problem of reconstructing an unknown electric conductivity from boundary measurements has applications in medical imaging geophysics and nondestructive testing a nachman ann of math 2 143 1996 pp 71 96 proved global uniqueness for the two dimensional inverse conductivity problem using a constructive method of proof based on this proof siltanen mueller and isaacson inverse problems 16 2000 pp 681 699 presented a new numerical reconstruction method that solves the nonlinear problem directly without iteration the method was verified with nonnoisy rotationally symmetric examples in this paper the method is extended by introducing a new regularization scheme which is analyzed theoretically and tested on symmetric and nonsymmetric numerical examples containing computer simulated noise

ranking games that have competitiveness based strategies an extensive literature in economics and social science addresses contests in which players compete to outperform each other on some measurable criterion often referred to as a player s score or output players incur costs that are an increasing function of score but receive prizes for obtaining higher score than their competitors in this paper we study finite games that are discretized contests and the problems of computing exact and approximate nash equilibria our motivation is the worst case hardness of nash equilibrium computation and the resulting interest in important classes of games that admit polynomial time algorithms for games that have a tie breaking rule for players scores we present a polynomial time algorithm for computing an exact equilibrium in the 2 player case and for multiple players a characterization of nash equilibria that shows an interesting parallel between these games and unrestricted 2 player games in normal form when ties are allowed via a reduction from these games to a subclass of anonymous games we give approximation schemes for two special cases constant sized set of strategies and constant number of players

global uniqueness for a two dimensional inverse boundary value problem we show that the coefficient y x of the elliptic equation vie qyvu 0 in a two dimensional domain is uniquely determined by the corresponding dirichlet to neumann map on the boundary and give a reconstruction pro

morse and melnikov functions for nls pde s the theory of the focusing nls equation under periodic boundary conditions together with the floquet spectral theory of its associated zakharov shabat linear operator hat l is developed in sufficient detail for later use in studies of perturbations of the nls equation counting lemmas for the non selfadjoint operator hat l are established which control its spectrum and show that all of its eccentricities are finite in number and must reside within a finite discd in the complex eigenvalue plane the radius of the discd is controlled by theh 1 norm of the potential overrightarrow q for this integrable nls hamiltonian system unstable tori are identified and backlund transformations are then used to construct global representations of their stable and unstable manifolds whiskered tori for the nls pde the floquet discriminant delta lambda overrightarrow q is used to introduce a natural sequence of nls constants of motion f j overrightarrow q equiv delta lambda lambda j c overrightarrow q overrightarrow q where denotes thej th critical point of the floquet discriminant a taylor series expansion of the constants f j overrightarrow q with explicit representations of the first and second variations is then used to study neighborhoods of the whiskered tori in particular critical tori with hyperbolic structure are identified through the first and second variations of f j overrightarrow q which themselves are expressed in terms of quadratic products of eigenfunctions of hat l the second variation permits identification within the disc d of important bifurcations in the spectral configurations of the operator hat l the constant f j overrightarrow q as the height of the floquet discriminant over the critical point admits a natural interpretation as a morse function for nls isospectral level sets this morse interpretation is studied in some detail it is valid globally for the infinite tail f j overrightarrow q j n which is associated with critical points outside the discd within this disc the interpretation is only valid locally with the same obstruction to its global validity as to a global ordering of the spectrum nevertheless this local morse theory together with the backlund representations of the whiskered tori produces extremely clear pictures of the stratification of nls invariant sets near these whiskered tori pictures which are useful in the study of perturbations of nls finally a natural connection is noted between the constants f j overrightarrow q of the integrable theory and melnikov functions for the theory of perturbations of the nls equation this connection generates a simple but general representations of the melnikov functions

some theorems on linear positive operators and functionals and their applications abstract in this paper we prove two theorems concerning linear positive operators and functionals in a very general sense we show how these results can be used to derive new results in a wide range of applications from approximation theory to the numerical treatment of differential equations and to the numerical analysis connected with signals and images

g convergence dirichlet to neumann maps and invisibility we establish optimal conditions under which the g convergence of linear elliptic operators implies the convergence of the corresponding dirichlet to neumann maps as an application we show that the approximate cloaking isotropic materials from 19 are independent of the source

convergence analysis of two grid methods for elliptic toeplitz and pdes matrix sequences summary the solution of large toeplitz systems with nonnegative generating functions by multigrid methods was proposed in previous papers 13 14 22 the technique was modified in 6 36 and a rigorous proof of convergence of the tgm two grid method was given in the special case where the generating function has only a zero at x 0 0 of order at most two here by extending the latter approach we perform a complete analysis of convergence of the tgm under the sole assumption that f is nonnegative and with a zero at x 0 0 of finite order an extension of the same analysis in the multilevel case and in the case of finite difference matrix sequences discretizing elliptic pdes with nonconstant coefficients and of any order is then discussed

a korovkin based approximation of multilevel toeplitz matrices with rectangular unstructured blocks via multilevel trigonometric matrix spaces in this paper we are concerned with the approximation of multilevel toeplitz matrices generated by multivariate rectangular matrix valued continuous also l2 functions f i p rightarrow cal c s times t with i pi pi we first define a class cal a t of multilevel trigonometric matrix spaces with unstructured s times t blocks second we introduce a multi sequence of linear approximation operators cal p n x n n 1 n 2 ldots n p acting on complex matrices of dimensions n 1 cdots n p s times n 1 cdots n p t n j in cal n and giving the minimizer of the distance of x from a fixed representative of cal a t with regard to the frobenius norm then by stating and proving new versions of the korovkin theorem for f ranging in the banach space bf c i p cal c s times t cdot infty and by linking these theorems to the operators cal p n we prove the goodness of the proposed approximation in terms of the strong and weak equidistribution of the eigenvalues singular values of a n f and cal p n a n f the discussed results provide a very uniform tool for treating the preconditioning problem with applications to a wide variety of structures as analyzed in the twin paper s serra capizzano calcolo in press

stability of the inverse conductivity problem in the plane for less regular conductivities abstract the dirichlet to neumann map or voltage to current map takes dirichlet data u f to the conormal derivatives u at the boundary where u is a solution to the elliptic equation div u 0 on a plane lipchitz domain r2 we consider the inverse conductivity problem that consists of the recovery of in the interior of from the knowledge of we prove stability for this inverse map assuming an isotropic conductivity c1 e we reduce the equation div u 0 to an equivalent system and use the scattering transform this approach relaxes the number of derivatives of required in former works

on the spectrum of stiffness matrices arising from isogeometric analysis we study the spectral properties of stiffness matrices that arise in the context of isogeometric analysis for the numerical solution of classical second order elliptic problems motivated by the applicative interest in the fast solution of the related linear systems we are looking for a spectral characterization of the involved matrices in particular we investigate non singularity conditioning extremal behavior spectral distribution in the weyl sense as well as clustering of the eigenvalues to a certain compact subset of mathbb c c all the analysis is related to the notion of symbol in the toeplitz setting and is carried out both for the cases of 1d and 2d problems

singular solutions of elliptic equations and the determination of conductivity by boundary measurements abstract we improve some results on uniqueness and stability in the determination of the coefficient a in the equation i div a grad u 0 in when all possible pairs of dirichlet and neumann data on are known we also treat cases of anisotropic equations our method relies on the construction of solutions to i having an isolated singularity with prescribed asymptotic behaviour

kac moody lie algebras and soliton equations ii lax equations associated with a1 1 the soliton equations associated with sl 2 eigenvalue problems polynomial in the eigenvalue parameter are given a unified treatment they are shown to be generated by a single family of commuting hamiltonians on a subalgebra of the loop algebra of sl 2 the conserved densities and fluxes of the usual anks hierarchy are identified with conserved densities and fluxes for the polynomial eigenvalue problems the hamiltonian structures of the soliton equations associated with the polynomial eigenvalue problems are given a unified treatment

an inverse scattering transform for the davey stewartson ii equations ii the rigorous treatment of the direct and inverse problems is carried out for an inverse scattering transform that can be used to solve the davey stewartson ii equations

an inverse scattering transform for the davey stewartson ii equations iii abstract the cauchy problem of the davey stewartson ii equations is solved by the method of an inverse scattering transform results of existence and uniqueness of global solutions regularity spatial and long time decay are obtained

on dispersive difference schemes i etude du comportement oscillatoire et de la convergence faible des solutions des approximations aux differences dispersives de u t 1 2 u 2 x 0

the korteweg de vries equation with small dispersion higher order lax levermore theory we utilize the inverse scattering transformation to study the rapid oscillations which arise in the solution of the initial value problem begin gathered u t 6u u x in 2 u xxx 0 hfill u left x 0 right v left x right hfill end gathered when 0 we refine the lax levermore theory which gives the weak limit of the solution as 0 by introducing a quantum condition this allows us to calculate the waveform of the local oscillations up to phase shifts when is small

settling the complexity of computing two player nash equilibria we prove that bimatrix the problem of finding a nash equilibrium in a two player game is complete for the complexity class ppad polynomial parity argument directed version introduced by papadimitriou in 1991 our result building upon the work of daskalakis et al 2006a on the complexity of four player nash equilibria settles a long standing open problem in algorithmic game theory it also serves as a starting point for a series of results concerning the complexity of two player nash equilibria in particular we prove the following theorems bimatrix does not have a fully polynomial time approximation scheme unless every problem in ppad is solvable in polynomial time the smoothed complexity of the classic lemke howson algorithm and in fact of any algorithm for bimatrix is not polynomial unless every problem in ppad is solvable in randomized polynomial time our results also have a complexity implication in mathematical economics arrow debreu market equilibria are ppad hard to compute