a remark on regularity criterion for the dissipative quasi geostrophic equations abstract this paper concerns with a regularity criterion of solutions to the 2d dissipative quasi geostrophic equations based on a logarithmic sobolev inequality in besov spaces the absence of singularities of in 0 t is derived for a solution on the interval 0 t satisfying the condition l r 0 t b p 0 for 2 p r 4 p this is an extension of earlier regularity results in the serrin s type space l r 0 t l p

computing normal form perfect equilibria for extensive two person games this paper presents an algorithm for computing an equilibrium of an extensive two person game with perfect recall the method is computationally efficient by virtue of using the sequence form whose size is proportional to the size of the game tree the equilibrium is traced on a piecewise linear path in the sequence form strategy space from an arbitrary starting vector if the starting vector represents a pair of completely mixed strategies then the equilibrium is normal form perfect computational experiments compare the sequence form and the reduced normal form and show that only the sequence form is tractable for larger games

efficient numerical differentiation of implicitly defined curves for sparse systems a numerical technique is developed for the efficient numerical differentiation of regular implicitly defined curves existing in high dimensional real space such as those representing homotopies where the system of equations which defines the curve implicitly is assumed to be sparse the calculation is verified numerically through its application to the curve defined implicitly by a homotopy constructed based on a discretization of the equations governing compressible aerodynamic fluid flow consideration is given to computational cost data storage and accuracy this method is applicable to any implicitly defined curves or trajectories which can occur for example in dynamical systems analysis or control applications also exist in the area of homotopy continuation where implicitly defined curves are approximately traced numerically such applications include the analysis of curve traceability and the construction of higher order predictors the latter is investigated numerically and it is found that increasing the order of accuracy of the predictor can significantly improve the curve tracing accuracy within a limited radius

a polynomial time algorithm for computing an arrow debreu market equilibrium for linear utilities we provide the first polynomial time exact algorithm for computing an arrow debreu market equilibrium for the case of linear utilities our algorithm is based on solving a convex program using the ellipsoid algorithm and simultaneous diophantine approximation as a side result we prove that the set of assignments at equilibrium is convex and the equilibrium prices themselves are log convex our convex program is explicit and intuitive which allows maximizing a concave function over the set of equilibria on the practical side ye developed an interior point algorithm lecture notes in comput sci 3521 springer new york 2005 pp 3 5 to find an equilibrium based on our convex program we also derive separate combinatorial characterizations of equilibrium for arrow debreu and fisher cases our convex program can be extended for many nonlinear utilities and production models our paper also makes a powerful theorem theorem 6 4 1 in m grotschel l lovasz and a schrijver geometric algorithms and combinatorial optimization 2nd ed springer verlag berlin heidelberg 1993 even more powerful in theorems 12 and 13 in the area of geometric algorithms and combinatorial optimization the main idea in this generalization is to allow ellipsoids to contain not the whole convex region but a part of it this theorem is of independent interest

global solutions of the super critical 2d quasi geostrophic equation in besov spaces abstract in this paper we study the super critical 2d dissipative quasi geostrophic equation we obtain some regularization effects allowing us to prove a global well posedness result for small initial data lying in critical besov spaces constructed over lebesgue spaces l p with p 1 local results for arbitrary initial data are also given

global well posedness for the critical 2d dissipative quasi geostrophic equation we give an elementary proof of the global well posedness for the critical 2d dissipative quasi geostrophic equation the argument is based on a non local maximum principle involving appropriate moduli of continuity

a strongly polynomial algorithm for a class of minimum cost flow problems with separable convex objectives a well studied nonlinear extension of the minimum cost flow problem is to minimize the objective sum ij in e c ij f ij over feasible flows f where on every arc ij of the network c ij is a convex function we give a strongly polynomial algorithm for the case when all c ij s are convex quadratic functions settling an open problem raised e g by hochbaum math oper res 19 1994 pp 390 409 we also give strongly polynomial algorithms for computing market equilibria in fisher markets with linear utilities and with spending constraint utilities that can be formulated in this framework see shmyrev j appl ind math 3 2009 pp 505 518 birnbaum devanur and xiao proceedings of the 12th acm conference on electronic commerce 2011 pp 127 136 for the latter class this resolves an open question raised by vazirani math oper res 35 2010 pp 458 478 the running time is o m 4 log m for quadratic costs o n 4 n 2 m n log n log n for fisher s markets wi

a new bernstein s inequality and the 2d dissipative quasi geostrophic equation we show a new bernstein s inequality which generalizes the results of cannone planchon danchin and lemarie rieusset as an application of this inequality we prove the global well posedness of the 2d quasi geostrophic equation with the critical and super critical dissipation for the small initial data in the critical besov space and local well posedness for the large initial data

some computational aspects of the stability analysis of nonlinear structures abstract the perturbation method and the continuation method are the two most popular techniques for the solution of finite element equations that describe instability phenomena this paper presents a summary of the principles involved and describes some trends in the development of these techniques

mixed formulation and locking in path following nonlinear analysis abstract the arc length riks strategy has rapidly become a standard tool for path following analysis of nonlinear structures due to its theoretical ability to surpass limit points the aim of this paper is to show that the failures in convergence that are occasionally experienced are not related to proper defects of the algorithm but come from a subtle locking effect intrinsic to the nonlinear nature of the problem as a consequence its sanitization has to be pursued within a reformulation of the structural model the use of a mixed stress displacement variant of the algorithm in particular appears very promising in this respect the topic is discussed with reference to the analysis of nonlinear frames using a mixed version of the nonlinear beam model discussed in 39 it is shown that with no extra computational cost and only a minor modification in coding with respect to a purely compatible formulation it is possible to achieve a noticeable improvement in convergence and a real gain in both computational time and overall robustness of the algorithm

concave generalized flows with applications to market equilibria we consider a nonlinear extension of the generalized network flow model with the flow leaving an arc being an increasing concave function of the flow entering it as proposed by truemper truemper k 1978 optimal flows in nonlinear gain networks networks 8 1 17 36 and by shigeno shigeno m 2006 maximum network flows with concave gains math programming 107 3 439 459 we give a polynomial time combinatorial algorithm for solving corresponding flow maximization problems finding an e approximate solution in o m m log n log mum e arithmetic operations where m and u are upper bounds on simple parameters and is the complexity of a value oracle query for the gain functions this also gives a new algorithm for linear generalized flows an efficient purely scaling variant of the fat path algorithm by goldberg et al goldberg av plotkin sa tardos e 1991 combinatorial algorithms for the generalized circulation problem math oper res 16 2 351 381 not using any cycle cancellations we sho

nonlinear airfoil limit cycle analysis using continuation method and filtered impulse function in this study an adaptive step size control algorithm incorporated in the continuation method cm is developed and combined with the filtered impulse function method based on aerostructural solve

construction of solution curves for large two dimensional problems of steady state flows of incompressible fluids this work represents a step toward advancing classical methods of bifurcation analysis in conjunction with very large scale scientific computing needed to model realistically processes which involve laminar and steady flows of incompressible fluids a robust method of analysis based on pseudoarclength continuation newton s method and direct solution of linear systems is proposed and applied to the analysis of a representative system of fluid mechanics namely the tilted lid driven cavity accurate solution curves possessing simple singular points were computed for reynolds numbers varying from 0 to 10 000 and for different angles of tilt the results demonstrate that two dimensional models with up to 1 000 000 algebraic equations can be studied feasibly using the methods described here with state of the art vector supercomputers

numerical linear algebra aspects of globally convergent homotopy methods probability one homotopy algorithms are a class of methods for solving nonlinear systems of equations that are globally convergent with probability one these methods are theoretically powerful and if constructed and implemented properly are robust numerically stable accurate and practical the concomitant numerical linear algebra problems deal with rectangular matrices and good algorithms require a delicate balance not always achieved of accuracy robustness and efficiency in both space and time the author s experience with globally convergent homotopy algorithms is surveyed here and some of the linear algebra difficulties for dense and sparse problems are discussed

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

a combinatorial polynomial algorithm for the linear arrow debreu market we present the first combinatorial polynomial time algorithm for computing the equilibrium of the arrow debreu market model with linear utilities our algorithm views the allocation of money as flows and iteratively improves the balanced flow as in 11 for fisher s model we develop new methods to carefully deal with the flows and surpluses during price adjustments our algorithm performs o n 6 log n u maximum flow computations where n is the number of agents and u is the maximum integer utility the flows have to be presented as numbers of bitlength o n log n u to guarantee an exact solution previously 22 29 have given polynomial time algorithms for this problem which are based on solving convex programs using the ellipsoid algorithm and the interior point method respectively

an interval step control for continuation methods the authors present a step control for continuation methods that is deterministic in the sense that i it computationally but rigorously verifies that the corrector iteration will converge to a point on the same curve as the previous point i e the predictor corrector iteration will never jump across paths and ii each predictor step is as large as possible subject to verification that the curve is unique with the given interval extension the authors present performance data and comparisons with an approximate step control method pitcon version 6 1 a comparison of plots obtained from both step controls reveals that an approximate step control will behave erratically in situations where the interval step control leads to orderly progression along the curve this is true even if the maximum allowable stepsize for the approximate method is set to be smaller than many of the steps actually taken by the interval algorithm limitations of interval step controls are also discussed

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

an automatic load stepping algorithm with error control summary this paper presents an algorithm for controlling the error in non linear finite element analysis which is caused by the use of finite load steps in contrast to most recent schemes the proposed technique is non iterative and treats the governing load deflection relations as a system of ordinary differential equations this permits the governing equations to be integrated adaptively where the step size is controlled by monitoring the local truncation error the latter is measured by computing the difference between two estimates of the displacement increments for each load step with the initial estimate being found from the first order euler scheme and the improved estimate being found from the second order modified euler scheme if the local truncation error exceeds a specified tolerance then the load step is abandoned and the integration is repeated with a smaller load step whose size is found by local extrapolation local extrapolation is also used to predict the size of the next load step following a successful update in order to control not only the local load path error but also the global load path error the proposed scheme incorporates a correction for the unbalanced forces overall the cost of the automatic error control is modest since it requires only one additional equation solution for each successful load step because the solution scheme is non iterative and founded on successful techniques for integrating systems of ordinary differential equations it is particularly robust to illustrate the ability of the scheme to constrain the load path error to lie near a desired tolerance detailed results are presented for a variety of elastoplastic boundary value problems

market equilibrium via a primal dual algorithm for a convex program we give the first polynomial time algorithm for exactly computing an equilibrium for the linear utilities case of the market model defined by fisher our algorithm uses the primal dual paradigm in the enhanced setting of kkt conditions and convex programs we pinpoint the added difficulty raised by this setting and the manner in which our algorithm circumvents it

geometrically non linear and elastoplastic three dimensional shear flexible beam element of von mises type hardening material a three dimensional elastoplastic beam element being capable of incorporating large displacement and large rotation is developed and examined elastoplastic constitutive equations are applied to the beam element based upon the assumption of small deformational strain leading to a material formulation which is completely objective for the application of stress update procedures the continuum type equations of plastic model of j2 mixed hardening are transformed into the beam equations by satisfying beam hypotheses an effective stress update algorithm is proposed to integrate elastoplastic rate equations by means of the so called multistep method which is a method of successive control of residuals on yield surfaces it avoids severe divergence when the displacement increments become large which is usual for the continuation methods material tangent stiffness matrix is derived by using consistent elastoplastic modulus resulting from the integration algorithm and is combined with geometric tangent stiffness matrix different from other elements the present element is shear flexible and can satisfy the plasticity condition in a pointwise fashion a great number of numerical examples are analysed and compared with the literature the proposed beam element is verified to be not only quite accurate but also very effective for the analyses of pre buckling and large deflection collapse of spatial framed structures

a perfect price discrimination market model with production and a rational convex program for it recent results showing ppad completeness of the problem of computing an equilibrium for fisher s market model under additively separable piecewise linear concave plc utilities have dealt a serious blow to the program of obtaining efficient algorithms for computing equilibria in traditional market models and has prompted a search for alternative models that are realistic as well as amenable to efficient computation in this paper we show that introducing perfect price discrimination into the fisher model with plc utilities renders its equilibrium polynomial time computable moreover its set of equilibria are captured by a convex program that generalizes the classical eisenberg gale program and always admits a rational solution we also give a combinatorial polynomial time algorithm for computing an equilibrium next we introduce production into our model and again give a rational convex program that captures its equilibria we use this program to obtain surprisingly simple proofs of both welfare theorems for this model finally we also give an application of our price discrimination market model to online display advertising marketplaces

predictor and steplength selection in continuation methods for the navier stokes equations abstract continuation methods are in common use for solving the nonlinear equations resulting from discretizations of the navier stokes equations an imoportant ingrediet in these algorithms is the selection of the step size in the continuation parameter for certain values of the reynolds number it is shown that the step size in the reynolds number may be chosen independently of the particular predictor step used within the continuation algorithm estimates for the path derivatives are derived and used to show that the allowable step sizes are substantially greater than would be predicted by theories that apply to general nonlinear mappings

a decomposition algorithm for general equilibrium computation with application to international trade models in this paper we outline the computation of general equilibrium in a pure exchange economy via a fixed point decomposition procedure for general equilibrium models of the required structure a full equilibrium may be computed through the solution of a sequence of smaller scale sub equilibrium problems the text contains a presentation of the methods involved along with a discussion of initial computational experience for some numerical examples in this paper we describe the computation of general equilibrium in a pure exchange economy via a fixed point decomposition procedure similar in spirit to the dantzig wolfe decomposition algorithm for the solution of linear programming problems dantzig and wolfe 2 the method involves the generation of labels for vertices on a master simplex through the separate solution of subequilibrium problems whose parameters are determined by the coordinates of the vertex on the master simplex for general equilibrium models of the required structure it is possible to compute a full equilibrium through the solution of a sequence of smaller scale sub equilibrium problems the analogues to the common constraints in the dantzig wolfe procedure are common commodities with common prices and the block diagonal structure on non common constraints in dantzig wolfe is replaced by an analogous block diagonal pattern of demands and endowments of agents over non common goods a natural application of the method is to international trade models with traded and non traded goods traded goods are common to all countries non traded goods are traded only within the country involved we report execution times for numerical examples using merrill s 5 algorithm for solution of both full dimensional problems and the same problems by the decomposition procedure we do not discuss the application of these procedures to economies with production although it seems likely to us that a similar procedure can be applied if a comparable block diagonal structure characterizes the production set the economic interpretation we offer for our procedure draws on the partition of the full list of commodities in a general equilibrium model into common goods traded among all agents and non common goods traded among a subset of agents the assignment of non common goods to agents is represented in a block diagonal pattern of demands and asset ownership by agent agents have

conjugate gradient methods for continuation problems ii we study linear conjugate gradient cg methods for large sparse continuation problems first we show how the cg methods can be incorporated as linear solvers in the context of a continuation method next we indicate how the preconditioned generalized minimal residual algorithm may be used to test for simple and multiple bifurcations without solving the bordered linear systems sample numerical results are reported some concluding remarks concerning the performance of three term cg methods are given

on steplength algorithms for a class of continuation methods the continuation methods considered here are algorithms for the computational analysis of the regular parts of the solution field of equations of the form fx b f d subset r n 1 to r n for given b in r n while these methods are similar in structure to those used for ode solvers their errors are independent of the history of the process and are solely determined by the termination criterion of the corrector at the current step this suggests the use of a posteriors estimates of the convergence radii of the corrector it is proved here that such estimates cannot be obtained from the sequence of corrector iterates alone but that they require some global information about f however it is shown that a finite sequence of corrector iterates does allow for the computation of effective estimates of the convergence quality of certain types of correctors this is used for the design of various step algorithms for continuation processes two of them are based on a newton corrector while the third o

computational error estimates and adaptive processes for some nonlinear structural problems abstract in earlier papers the authors introduced and analyzed the calculation of reliable a posteriori error estimates for finite element solutions and discussed the design and effectivity of adaptive procedures based upon them while most of this work concerned linear problems this paper is intended to show that the same approaches remain also highly effective in the nonlinear case as a model problem a planar elastic rod is considered involving both material as well as geometrical nonlinearities but as far as possible the discussion is kept general for nonlinear problems of this type interest centers on an analysis of the shape and features of the equilibrium surface after some general remarks about such surfaces and the related computational problems a general continuation process is summarized and some of its extensions for determining limit points and tracing stability paths are discussed then estimators are introduced for the error along the solution paths and of the computed critical values experimental results for the model problem show the effectivity of these estimators and of the adaptive procedures based upon them then some aspects relating to the occurrence and identification of spurious solutions are discussed in order to illustrate desirable generalizations of these results the paper ends with an outline of some results for linear problems of the type that should be achievable also in the nonlinear case

commutator estimate in terms of partial derivatives of solutions for the dissipative quasi geostrophic equation through littlewood paley decomposition argument a commutator estimate in terms of partial derivatives of solutions for the critical and supercritical dissipative 2d quasi geostrophic equation is established as an application of this estimate we obtain some new a priori estimates and prove the existence and uniqueness of solution for the small initial data in critical besov spaces

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

parameter selection rules for path following with singular embedding singular embedding methods require appropriately adjusted parameters to guarantee the contraction of locally quadratically convergent iterative methods firstly some general rule for the parameter selection is proposed and its rate of convergence is analyzed secondly some modifications in the case of polynomial error and contraction bounds are studied finally these results are applied to the embedding of an elliptic boundary value problem with discontinuous nonlinearities into a family of smooth problems here the regularization is done in such a way that the solutions of the resulting auxiliary problems require only one step of newton s method

a maximum principle applied to quasi geostrophic equations we study the initial value problem for dissipative 2d quasi geostrophic equations proving local existence global results for small initial data in the super critical case decay of lp norms and asymptotic behavior of viscosity solution in the critical case our proofs are based on a maximum principle valid for more general flows

higher order predictors and adaptive steplength control in path following algorithms for tracing the solution path of an underdetermined nonlinear system f y 0 with f r n 1 to r n predictor corrector methods such as the euler newton continuation method have proved to be reliable tools in order to reduce the computational work a more sophisticated pc method is proposed and investigated the method is based on secant length as the local parameter and takes either the second order taylor polynomial or the third order 2 1 pade approximation as the predictor whereas in both cases the directional derivatives containing f and f are efficiently approximated by a few function values it uses a newton type corrector iteration that takes into account the special parametrization and applies an adaptive steplength control strongly related to the higher order predictors numerical tests show that the new algorithm performs very efficiently and essentially reduces the number of evaluations and factorizations of jacobian matrices compared with other pc methods known from lite

tracing the buckling of a rectangular plate with the block gmres method we investigate bifurcation scenario of the von karman equations with partially clamped boundary conditions defined in rectangular domains first we study how the preconditioned block gmres method can be used in the context of continuation methods for tracing solution curves of large systems of nonlinear equations next we discuss linear stabilities of the von karman equations with partially clamped boundary conditions in particular we calculate the first seven eigenvalues and its associated eigenfunctions of the linearized von karman equations via computer algebra the block gmres method is used to solve linear systems and to detect singularity along solution paths of the discrete problem sample numerical results are reported

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

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

efficient numerical path following beyond critical points the paper presents a numerical pathfollowing technique based on an implicit change of parametrization using qr decomposition an automatic steplength control for tangent continuation is derived on a theoretical basis in the suggested approach turning points do not play any exceptional role for the actual computation of simple bifurcation points a special algorithm based on the augmented system of moore is developed this treatment includes imperfect bifurcations and permits an easy way of branch switching an efficient linear equation solver is worked out in connection with the qr decomposition finally numerical comparisons are included that clearly document the efficiency of the newly developed algorithms

market equilibrium under piecewise leontief concave utilities abstract leontief is one of the most widely used and extensively studied function in economic modeling for both production and preferences however it lacks the desirable property of diminishing returns in this paper we consider piecewise leontief concave p leontief utility function which consists of a set of leontief type segments with decreasing returns and an upper limit on the utility on each segment and study complexity of computing equilibria in both fisher and exchange market models leontief is a special case of p leontief with exactly one segment with no upper limit while it is well known that equilibrium computation in fisher markets with leontief utilities is polynomial time in case of p leontief even with two segments we show that the problem is ppad hard on the other hand for the case when coefficients across segments are scaled versions of each other we give a polynomial time algorithm as a corollary we obtain a non trivial new class of exchange markets with leontief utilities where an equilibrium can be computed in a polynomial time whereas the general problem is known to be ppad hard for p leontief exchange markets with pairing economy we show that equilibria are rational and we obtain a simplex like algorithm to compute one using the classic lemke howson scheme thereby extending the results of ye tcs 2007 and codenotti saberi varadarajan and ye soda 2006 for leontief to p leontief utilities

the random product homotopy and deficient polynomial systems most systems ofn polynomial equations inn unknowns arising in applications aredeficient in the sense that they have fewer solutions than that predicted by the total degree of the system we introduce the random product homotopy an efficient homotopy continuation method for numerically determining all isolated solutions of deficient systems in many cases the amount of computation required to find all solutions can be made roughly proportional to the number of solutions

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

the method of resultants for computing real solutions of polynomial systems a new method for determining the real solutions to a set of polynomial equations is presented it is based on the theory of multiresultants the inherently unstable calculation of the determinant is replaced by a stable minimization procedure which is able to take advantage of the sparseness of the resultant matrix two numerical examples illustrate the method the paper contains preliminary work which demonstrates the feasibility of the given approach

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

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

commutator estimate and its application to regularity criteria of the dissipative quasi geostrophic equation abstract a new commutator estimate with respect to a nonlinear convection upper bounded by a single partial derivative component in hilbert spaces is obtained as an application regularity criteria on the supercritical quasi geostrophic equation are obtained provided that solution growth conditions are assumed to involve a single partial derivative component

unified approach to flutter equations a general formulation of the flutter equation includes gyroscopics viscous damping active controls mach and complex reduced frequency based unsteady aerodynamics quasi linear approximated structural nonlinearities and parameterized matrices solving the flutter equation usually involves a search for neutral stability points the boundary between stability and instability followed by variations in the neutral stability point with various parameters a technique for solving the flutter equation under these conditions must necessarily be capable of solving systems of nonlinear parameterized equations over a range of parameter values continuation methods are specifically designed for solving such problems a continuation method is presented that is capable of solving general formulations of the flutter equation for neutral stability parameter variations optimization model tuning and describing function flutter analyses including an efficient technique for assessing limit cycle stability

lower bounds for an integral involving fractional laplacians and the generalized navier stokes equations in besov spaces when estimating solutions of dissipative partial differential equations in l p related spaces we often need lower bounds for an integral involving the dissipative term if the dissipative term is given by the usual laplacian lower bounds can be derived through integration by parts and embedding inequalities however when the laplacian is replaced by the fractional laplacian the approach of integration by parts no longer applies in this paper we obtain lower bounds for the integral involving by combining pointwise inequalities for with bernstein s inequalities for fractional derivatives as an application of these lower bounds we establish the existence and uniqueness of solutions to the generalized navier stokes equations in besov spaces the generalized navier stokes equations are the equations resulting from replacing in the navier stokes equations by

arclength numerical continuation in free boundary flow abstract using helmholtz s wake model we reduce the study of the free boundary flow past an obstacle consisting of an arc of circle to the investigation of a hammerstein nonlinear integral equation depending on a real parameter the papers dedicated to this problem investigated the case 0 which corresponds to a convex obstacle with respect to the incoming fluid herein we apply for the first time in the literature the arclength continuation method for the case 0 corresponding to a concave arc of a circle for 0 the existence and the uniqueness of the solution was demonstrated but for 0 depending on its value compared to the one of a turning point the integral equation has either no solution or two distinct solutions corresponding to two different obstacles we numerically calculate the free lines the velocity field and the stream lines a diagram of the drag coefficient versus the arc measure for both convex and concave obstacles suggests us to draw some conclusions concerning the optimization of the blades of a vertical axis savonius wind turbine

existence and uniqueness of the solution to the dissipative 2d quasi geostrophic equations in the sobolev space we study the two dimensional dissipative quasi geostrophic equations in the sobolev space open image in new window existence and uniqueness of the solution local in time is proved in h s when s 2 1 existence and uniqueness of the solution global in time is also proved in h s when s 2 1 and the initial data open image in new window is small for the case s 2 1 we also obtain the unique large global solution in h s provided that open image in new window is small enough

matching algorithmic bounds for finding a brouwer fixed point we prove a new discrete fixed point theorem for direction preserving functions defined on integer points based on a novel characterization of boundary conditions for the existence of fixed points the theorem allows us to derive an improved algorithm for finding such a fixed point we also develop a new lower bound proof technique together they allow us to derive an asymptotic matching bound for the problem of finding a fixed point in a hypercube of any constantly bounded finite dimension exploring a linkage with the approximation version of the continuous fixed point problem we obtain asymptotic matching bounds for the complexity of the approximate brouwer fixed point problem in the continuous case for lipschitz functions it settles a fifteen years old open problem of hirsch papadimitriou and vavasis by improving both the upper and lower bounds our characterization for the existence of a fixed point is also applicable to functions defined on nonconvex domains which makes it a potentially useful tool for the design and analysis of algorithms for fixed points in general domains

hard to solve bimatrix games the lemke howson algorithm is the classical method for finding one nash equilibrium of a bimatrix game this paper presents a class of square bimatrix games for which this algorithm takes even in the best case an exponential number of steps in the dimension d of the game using polytope theory the games are constructed using pairs of dual cyclic polytopes with 2d suitably labeled facets in d space the construction is extended to nonsquare games where in addition to exponentially long lemke howson computations finding an equilibrium by support enumeration takes on average exponential time

the inverse kinematics problem of spatial 4p3r robot manipulator by the homotopy continuation method with an adjustable auxiliary homotopy function abstract in this paper the concept of an adjustable auxiliary homotopy function for the homotopy continuation method is presented by means of adjusting the auxiliary function we can solve non linear equations and guarantee the solutions exactly without divergence rather than the traditional numerical methods such as the newton raphson method and so on

higher order commutator estimates and local existence for the non resistive mhd equations and related models this paper establishes the local in time existence and uniqueness of strong solutions in hs for s n 2 to the viscous non resistive magnetohydrodynamics mhd equations in rn n 2 3 as well as for a related model where the advection terms are removed from the velocity equation the uniform bounds required for proving existence are established by means of a new estimate which is a partial generalisation of the commutator estimate of kato ponce comm pure appl math 41 7 891 907 1988

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

the maximum principle and the global attractor for the dissipative 2d quasi geostrophic equations the long time behavior of the solutions to the two dimensional dissipative quasi geostrophic equations is studied we obtain a new positivity lemma which improves a previous version of a cordoba and d cordoba 10 and 11 as an application of the new positivity lemma we obtain the new maximum principle i e the decay of the solution in l p for any p 2 when f is zero as a second application of the new positivity lemma for the sub critical dissipative case with open image in new window the existence of the global attractor for the solutions in the space h s for any s 2 1 is proved for the case when the time independent f is non zero therefore the global attractor is infinitely smooth if f is this significantly improves the previous result of berselli 2 which proves the existence of an attractor in some weak sense for the case 1 the global attractor exists in h s for any s 0 and the estimate of the hausdorff and fractal dimensions of the global attractor is also available

on the regularity criteria of a surface quasi geostrophic equation abstract we obtain regularity criteria for a quasi geostrophic equation that depends more on one direction than the others in particular we show that in the critical case the global regularity depends only on a partial derivative rather than a gradient of the solution

spending constraint utilities with applications to the adwords market the notion of a market has undergone a paradigm shift with the internet totally new and highly successful markets have been defined and launched by internet companies which already form an important part of today s economy and are projected to grow considerably in the future another major change is the availability of massive computational power for running these markets in a centralized or distributed manner in view of these new realities the study of market equilibria an important though essentially nonalgorithmic theory within mathematical economics needs to be revived and rejuvenated via an inherently algorithmic approach such a theory should not only address traditional market models but also define new models for some of the new markets we present a new natural class of utility functions that allow buyers to explicitly provide information on their relative preferences as a function of the amount of money spent on each good these utility functions offer considerable expressivity especially in google s adwords market in addition they lend themselves to efficient computation while retaining some of the nice properties of traditional models the latter include weak gross substitutability and the uniqueness and continuity of equilibrium prices and utilities

existence and uniqueness results for the 2 d dissipative quasi geostrophic equation abstract this paper concerns itself with besov space solutions of the 2 d quasi geostrophic qg equation with dissipation induced by a fractional laplacian the goal is threefold first to extend a previous result on solutions in the inhomogeneous besov space b 2 q r j wu global solutions of the 2d dissipative quasi geostrophic equation in besov spaces siam j math anal 36 2004 2005 1014 1030 to cover the case when r 2 2 second to establish the global existence of solutions in the homogeneous besov space b p q r with general indices p and q and third to determine the uniqueness of solutions in any one of the four spaces b 2 q s b p q r l q 0 t b 2 q s 2 q and l q 0 t b p q r 2 q where s 2 2 and r 1 2 2 p

a fptas for computing a symmetric leontief competitive economy equilibrium in this paper we consider a linear complementarity problem lcp arisen from the nash and arrow debreu competitive economy equilibria where the lcp coefficient matrix is symmetric we prove that the decision problem to decide whether or not there exists a complementary solution is np complete under certain conditions an lcp solution is guaranteed to exist and we present a fully polynomial time approximation scheme fptas for approximating a complementary solution although the lcp solution set can be non convex or non connected our method is based on approximating a quadratic social utility optimization problem qp and showing that a certain kkt point of the qp problem is an lcp solution then we further show that such a kkt point can be approximated with a new improved running time complexity o frac n 4 epsilon log log frac 1 epsilon arithmetic operation in accuracy epsilon in 0 1 we also report preliminary computational results which show that the method is highly effective applications in competitive market model problems with other utility functions are also presented including global trading and dynamic spectrum management problems

large sparse continuation problems we study and develop efficient and versatile predictor corrector continuation methods for large sparse problems the first object is to show how special solving methods for a large sparse linear systems can be incorporated into the basic steps of a continuation method next we describe how to use a special nonlinear conjugate gradient method to perform the corrector phase it is shown how such methods can be used to detect bifurcation points and how to trace bifurcating solution branches by using local perturbations finally a numerical example involving bifurcating branches of a nonlinear eigenvalue problem is given

spectral collocation methods using sine functions for a rotating bose einstein condensation in optical lattices we study spectral galerkin methods sgm and spectral collocation methods scm for parameter dependent problems where the fourier sine functions are used as the basis functions when the sgm and the scm are incorporated in the context of a taylor predictor inexact newton corrector continuation algorithm for tracing solution curves of the gross pitaevskii equation gpe they can efficiently provide accurate numerical solutions for the gpe we show how the inexact newton method outperforms the classical newton method in the continuation algorithm in our numerical experiments the centered difference method cdm the sgm and scm are exploited to compute energy levels and wave functions of a rotating bose einstein condensation bec and a rotating bec in optical lattices in 2d sample numerical results are reported

an e relaxation method for separable convex cost generalized network flow problems cover title the extended abstract of this article appeared in the proceedings of the 5th international ipco conference vancouver june 1996 p 1

the notion of a rational convex program and an algorithm for the arrow debreu nash bargaining game we introduce the notion of a rational convex program rcp and we classify the known rcps into two classes quadratic and logarithmic the importance of rationality is that it opens up the possibility of computing an optimal solution to the program via an algorithm that is either combinatorial or uses an lp oracle next we define a new nash bargaining game called adnb which is derived from the linear case of the arrow debreu market model we show that the convex program for adnb is a logarithmic rcp but unlike other known members of this class it is nontotal our main result is a combinatorial polynomial time algorithm for adnb it turns out that the reason for infeasibility of logarithmic rcps is quite different from that for lps and quadratic rcps we believe that our ideas for surmounting the new difficulties will be useful for dealing with other nontotal rcps as well we give an application of our combinatorial algorithm for adnb to an important fair throughput allocation problem on a wireless channel finally we present a number of interesting questions that the new notion of rcp raises

existence and regularity of rotating global solutions for the generalized surface quasi geostrophic equations motivated by the recent work of hassainia and hmidi z hassainia t hmidi on the vstates for the generalized quasi geostrophic equations arxiv preprint arxiv 1405 0858 we close the question of the existence of convex global rotating solutions for the generalized surface quasigeostrophic equation for 2 1 2 we also show c 1 regularity of their boundary for all 2 0 2

parametrization via secant length and application to path following a possible way for parametrizing the solution path of the nonlinear systemh u 0 h n 1 n consists of using the secant length as parameter this idea leads to a quadratic constraint by which the parameter is introduced a newton like method for computing the solution for a given parameter is proposed where the nonlinear system is linearized at each iterate but the quadratic parametrizing equation is exactly satisfied the localq quadratic convergence of the method is proved and some hints for implementing the algorithm are given

proportional response dynamics in the fisher market we show that the proportional response dynamics a utility based distributed dynamics converges to the market equilibrium in the fisher market with constant elasticity of substitution ces utility functions by the proportional response dynamics each buyer allocates his budget proportional to the utility he receives from each good in the previous time period unlike the tatonnement process and its variants the proportional response dynamics is a large step discrete dynamics and the buyers do not solve any optimization problem at each step in addition the goods are always cleared and assigned to the buyers proportional to their bids at each step despite its simplicity the dynamics converges fast for strictly concave ces utility functions matching the best upper bound of computing the market equilibrium via the solution of a global convex optimization problem

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

a parallel and adaptive continuation method for semilinear bifurcation problems in this paper we propose an efficient continuation method for the following of solution branches of parametrized semilinear elliptic problems in order to limit the time required for the computation of a whole branch of accurate solutions we combine a mesh adaptation strategy with parallel programming on a computer network the first computer is devoted to the continuation on a coarse mesh the others share out the correction of every roughly generated solution on a sequence of a posteriori refined meshes a numerical example illustrating the performance of the method is given