reachability and minimal times for state constrained nonlinear problems without any controllability assumption we consider a target problem for a nonlinear system under state constraints we give a new continuous level set approach for characterizing the optimal times and the backward reachability sets this approach leads to a characterization via a hamilton jacobi equation without assuming any controllability assumption we also treat the case of time dependent state constraints as well as a target problem for a two player game with state constraints our method gives a good framework for numerical approximations and some numerical illustrations are included in the paper

five six and seven term karatsuba like formulae the karatsuba ofman algorithm starts with a way to multiply two 2 term i e linear polynomials using three scalar multiplications there is also a way to multiply two 3 term i e quadratic polynomials using six scalar multiplications these are used within recursive constructions to multiply two higher degree polynomials in subquadratic time we present division free formulae which multiply two 5 term polynomials with 13 scalar multiplications two 6 term polynomials with 17 scalar multiplications and two 7 term polynomials with 22 scalar multiplications these formulae may be mixed with the 2 term and 3 term formulae within recursive constructions leading to improved bounds for many other degrees using only the 6 term formula leads to better asymptotic performance than standard karatsuba the new formulae work in any characteristic but simplify in characteristic 2 we describe their application to elliptic curve arithmetic over binary fields we include some timing data

a viscosity solutions approach to shape from shading the problem of recovering a lambertian surface from a single two dimensional image may be written as a first order nonlinear equation which presents the disadvantage of having several continuous and even smooth solutions a new approach based on hamilton jacobi bellman equations and viscosity solutions theories enables one to study non uniqueness phenomenon and thus to characterize the surface among the various solutions a consistent and monotone scheme approximating the surface is constructed thanks to the dynamic programming principle and numerical results are presented

a generalized fast marching method on unstructured triangular meshes in this paper we extend the generalized fast marching method gfmm presented in e carlini et al siam j numer anal 46 2008 pp 2920 2952 to unstructured meshes the gfmm generalizes the classical fast marching method in the sense that it can be applied to propagate interfaces with time dependent and changing sign velocity the main motivation for this extension is that in many areas such as in fluid dynamics the method for tracking the motion of an interface in time has to be coupled with other solvers based on finite elements or finite volumes typically constructed on triangular grids and on domains with complex geometries we prove an abstract convergence result for the two dimensional 2d case which requires only some properties of the mesh and of the local solvers used to compute the solution on a narrow band in particular this result applies to a class of monotone schemes for 2d problems on triangular grids with acute angles finally some numerical tests in two dimensions illu

well posedness and numerical analysis of a one dimensional non local transport equation modelling dislocations dynamics we consider a situation where dislocations are parallel lines moving in a single plane for this simple geometry dislocations dynamics is modeled by a one dimensional non local transport equation we prove a result of existence and uniqueness for all time of the continuous viscosity solution for this equation a finite difference scheme is proposed to approximate the continuous viscosity solution we also prove an error estimate result between the continuous solution and the discrete solution and we provide some simulations

the semigroup property of value functions in lagrange problems the lagrange problem in the calculus of variations exhibits the principle of optimality in a particularly simple form the binary operation of inf composition applied to the value functions of a lagrange problem equates the principle of optimality with a semigroup property this paper finds the infinitesimal generator of the semigroup by differentiating at t 0 the type of limit is epigraphical convergence in a uniform sense moreover the extent to which a semigroup is uniquely determined by its infinitesimal generator is addressed

comments on five six and seven term karatsuba like formulae for original paper see p l montgomery ibid vol 54 no 3 p 362 369 2005 we show that multiplication complexities of n term karatsuba like formulae of gf 2 x 7 n 19 presented in the above paper can be further improved using the chinese remainder theorem and the construction multiplication modulo x infin w

improved n term karatsuba like formulas in gf 2 it is well known that chinese remainder theorem crt can be used to construct efficient algorithms for multiplication of polynomials over gf 2 in this note we show how to select an appropriate set of modulus polynomials to obtain minimal number of multiplications

a posteriori error estimates for the effective hamiltonian of dislocation dynamics we study an implicit and discontinuous scheme for a non local hamilton jacobi equation modelling dislocation dynamics for the evolution problem we prove an a posteriori estimate of crandall lions type for the error between continuous and discrete solutions we deduce an a posteriori error estimate for the effective hamiltonian associated to a stationary cell problem in dimension one and under suitable assumptions we also give improved a posteriori estimates numerical simulations are provided

improved method for finding optimal formulas for bilinear maps in a finite field in 2012 barbulescu detrey estibals and zimmermann proposed a new framework to exhaustively search for optimal formulae for evaluating bilinear maps such as strassen or karatsuba formulae the main contribution of this work is a new criterion to aggressively prune useless branches in the exhaustive search thus leading to the computation of new optimal formulae in particular for the short product modulo x 5 and the circulant product modulo x 5 1 moreover we are able to prove that there is essentially only one optimal decomposition of the product of 3 x 2 by 2 x 3 matrices up to the action of some group of automorphisms

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

optimal karatsuba like formulae for certain bilinear forms in gf 2 we propose two exhaustive search type methods for the construction of karatsuba like algorithms for fast computation of certain bilinear forms in gf 2 the computation is done via an explicit construction of trilinear decompositions using heuristica search algorithms using that approach several old and new algorithms for the fast computation of bilinear forms were obtained

multiplication of polynomials modulo xn let n be positive integers with email protected 2n 1 let r be an arbitrary nontrivial ring not necessarily commutative and not necessarily having a multiplicative identity and r x be the polynomial ring over r in this paper we give improved upper bounds on the minimum number of multiplications needed to multiply two arbitrary polynomials of degree at most n 1 modulo x n over r moreover we introduce a new complexity notion the minimum number of multiplications needed to multiply two arbitrary polynomials of degree at most n 1 modulo x over r this new complexity notion provides improved bounds on the minimum number of multiplications needed to multiply two arbitrary polynomials of degree at most n 1 modulo x n over r

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

a convergent scheme for a non local hamilton jacobi equation modelling dislocation dynamics we study dislocation dynamics with a level set point of view the model we present here looks at the zero level set of the solution of a non local hamilton jacobi equation as a dislocation in a plane of a crystal the front has a normal speed depending on the solution itself we prove existence and uniqueness for short time in the set of continuous viscosity solutions we also present a first order finite difference scheme for the corresponding level set formulation of the model the scheme is based on monotone numerical hamiltonian proposed by osher and sethian the non local character of the problem makes it not monotone we obtain an explicit convergence rate of the approximate solution to the viscosity solution we finally provide numerical simulations

bounds on steady states for a non systemically autocatalysed reaction diffusion system bounds on parameter values that lead to different types of solutions for a non systemically autocatalysed reaction diffusion system are derived and show the existence of a critical concentration beyond which the stability properties undergo change additionally bifurcating solutions at the zero eigenvalue are obtained

remarks on hamilton jacobi equations with measurable time dependent hamiltonians on considere les equations de hamilton jacobi de la forme suivante u t h x t u u 0 dans r n 0 t ou h x t s p est continu en x s p et mesurable en t on considere une extension de la notion de solution viscosite au cas de h mesurable dependant du temps

a note on the paper by murat cenk and ferruh ozbudak multiplication of polynomials modulo xn theoret comput sci 412 2011 3451 3462 we recall some bibliography on fast polynomial multiplication related to the recent progress in the paper by cenk and ozbudak of 2011

dislocation dynamics short time existence and uniqueness of the solution we study a mathematical model describing dislocation dynamics in crystals we consider a single dislocation line moving in its slip plane the normal velocity is given by the peach koehler force created by the dislocation line itself the mathematical model is an eikonal equation with a velocity which is a non local quantity depending on the whole shape of the dislocation line we study the special case where the dislocation line is assumed to be a graph or a closed loop in the framework of discontinuous viscosity solutions for hamilton jacobi equations we prove the existence and uniqueness of a solution for small time we also give physical explanations and a formal derivation of the mathematical model finally we present numerical results based on a level sets formulation of the problem these results illustrate in particular the fact that there is no general inclusion principle for this model

a short time existence uniqueness result for a nonlocal topology preserving segmentation model motivated by a prior applied work of vese and the second author dedicated to segmentation under topological constraints we derive a slightly modified model phrased as a functional minimization problem and propose to study it from a theoretical viewpoint the mathematical model leads to a second order nonlinear pde with a singularity at nabla u 0 and containing a nonlocal term a suitable setting is thus the one of the viscosity solution theory and in this framework we establish a short time existence uniqueness result as well as a lipschitz regularity result for the solution

existence of viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games recently m g crandall and p l lions trans amer math soc 277 1983 1 42 introduced the class of viscosity solutions of these equations and proved uniqueness within this class this paper discusses the existence of these solutions under assumptions closely related to the ones which guarantee the uniqueness

on systems of reaction diffusion equations with a balance law the sequel abstract this paper is a sequel of the review paper on reaction diffusion systems with a balance law by martin and pierre 1990 here we present the results obtained after martin and pierre 1990 not only for classical systems but also for systems with either time fractional derivatives or space fractional derivatives accounting for anomalous diffusions

a convergent scheme for a non local coupled system modelling dislocations densities dynamics in this paper we study a non local coupled system that arises in the theory of dislocations densities dynamics within the framework of viscosity solutions we prove a long time existence and uniqueness result for the solution of this model we also propose a convergent numerical scheme and we prove a crandall lions type error estimate between the continuous solution and the numerical one as far as we know this is the first error estimate of crandall lions type for hamilton jacobi systems we also provide some numerical simulations

diffusive instability near hopf bifurcation for exponentially autocatalyzed reaction diffusion system the analysis of an exponentially autocatalyzed reaction diffusion system near the hopf bifurcation point has been carried out using a reductive perturbation approach to obtain a description in terms of the ginzburg landau equation the conditions for the occurrence of instability in the presence and absence of diffusion leading to hopf bifurcation are also derived the nature of the governing equations leads to multi valued instability conditions and eventually results in more than one region in parameter space where instability of uniform oscillations due to diffusion is possible

kam theory for quasi periodic equilibria in 1d quasi periodic media ii long range interactions we consider frenkel kontorova models corresponding to one dimensional quasi crystal with non nearest neighbor and many body interactions we formulate and prove a kam type theorem which establishes the existence of quasi periodic solutions the interactions we consider do not need to be of finite range or involve finitely many particles but have to decay sufficiently fast with respect to the distance of the position of the atoms the kam theorem we present has an a posteriori format we do not need to assume that the system is close to integrable we just assume that there is an approximate solution for the functional equation which satisfies some non degeneracy conditions

on existence and uniqueness of solutions of hamilton jacobi equations abstract the theory of scalar first order nonlinear partial differential equations has been enjoying a rapid development in the last few years this development occurred because the authors established uniqueness criteria for generalized solutions called viscosity solutions which correctly identify the solutions sought in areas of application including control theory differential games and the calculus of variations the concept of viscosity solutions is relatively easy to work with and many formally heuristic or difficult proofs have been made rigorous or simple using this concept a feedback process has begun and the experience recently gained in working with viscosity solution has suggested new existence and uniqueness results the current paper continues this interaction by establishing new existence ane uniqueness results in a natural generality suggested by earlier proofs it is also felt that the presentation of the comparison results which imply uniqueness continuous dependence and are used to estimate moduli of continuity has something to offer over earlier presentations in special cases author

the optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws we are interested in the rate of convergence in l1 of the approximate solution of a conservation law generated by a monotone finite difference scheme kuznetsov has proved that this rate is 1 2 ussr comput math math phys 16 1976 pp 105 119 and topics numer anal iii in proc roy irish acad conf dublin 1976 pp 183 197 and recently teng and zhang have proved this estimate to be sharp for a linear flux siam j numer anal 34 1997 pp 959 978 we prove by constructing appropriate initial data for the cauchy problem that kuznetsov s estimates are sharp for a nonlinear flux as well

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

convergence of a generalized fast marching method for an eikonal equation with a velocity changing sign we present a new fast marching algorithm for an eikonal equation with a velocity changing sign this first order equation models a front propagation in the normal direction the algorithm is an extension of the fast marching method in two respects the first is that the new scheme can deal with a time dependent velocity and the second is that there is no restriction on its change in sign we analyze the properties of the algorithm and we prove its convergence in the class of discontinuous viscosity solutions finally we present some numerical simulations of fronts propagating in mathbb r 2

convergence of approximation schemes for nonlocal front propagation equations we provide a convergence result for numerical schemes approximating nonlocal front propagation equations our schemes are based on a recently investigated notion of a weak solution for these equations we also give examples of such schemes for a dislocation dynamics equation and for a fitzhugh nagumo type system