existence of some extremal self dual codes two new singly even extremal self dual codes are constructed a 52 26 10 code and a 54 27 10 code

on the rate of convergence of solutions in singular perturbation problems where the obstacle cp is a given smooth function satisfying that rp 0 on dq using a technique in the theory of variational inequalities j l lions 11 proved that 11 min o p i l cnj ce for some constant c we claim that we can never deduce the pointwise convergence of a from the above estimate on the other hand the notion of viscosity solotions was introduced by m g crandall and p l lions 2 for nonlinear first order pdes and then it was extended for nonlinear second order pdes by p l lions 12 141 recently it appeared that a technique in the theory of viscosity solutions could be applied to derive a pointwise convergence for some singular perturbation problems in fact by the same argument as in 7 we can show that u converges to min 0 cp uniformly on 0 243 0022 247x 91 3 00

fractal first order partial differential equations the present paper is concerned with semi linear partial differential equations involving a particular pseudo differential operator it investigates both fractal conservation laws and non local hamilton jacobi equations the idea is to combine an integral representation of the operator and duhamel s formula to prove on the one hand the key a priori estimates for the scalar conservation law and the hamilton jacobi equation and on the other hand the smoothing effect of the operator as far as hamilton jacobi equations are concerned a non local vanishing viscosity method is used to construct a viscosity solution when existence of regular solutions fails and a rate of convergence is provided turning to conservation laws global in time existence and uniqueness are established we also show that our formula allows us to obtain entropy inequalities for the non local conservation law and thus to prove the convergence of the solution as the non local term vanishes toward the entropy solution of the pure conservation law

continuous dependence estimates for viscosity solutions of integro pdes we present a general framework for deriving continuous depen dence estimates for possibly polynomially growing viscosity solutions of fully nonlinear degenerate parabolic integro pdes we use this framework to pro vide explicit estimates for the continuous dependence on the coecients and the levy measure in the bellman isaacs integro pdes arising in stochas tic control dierenti al games moreover these explicit estimates are used to prove regularity results and rates of convergence for some singular perturbation problems finally we illustrate our results on some integro pdes arising when attempting to price european american options in an incomplete stock mar ket driven by a geometric levy process many of the results obtained herein are new even in the convex case where stochastic control theory provides an alternative to our pure pde methods

continuous dependence on the nonlinearity of viscosity solutions of parabolic equations abstract this paper establishes an upper bound for u v where u is a subsolution of u t f u d x u d 2 x u 0and v is a supersolution of v t g v d x v d 2 x v 0in 0 with neumann boundary conditions and where is an open convex set

mf actions and k theoretic dynamics we study the interplay of c dynamics and k theory notions of chain recurrence for transformations groups x and mf actions for non commutative c dynamical systems a are translated into k theoretical language where purely algebraic conditions are shown to be necessary and sufficient for a reduced crossed product to admit norm microstates we are particularly interested in actions of free groups on af algebras in which case we prove that a k theoretic coboundary condition determines whether or not the reduced crossed product is a matricial field mf algebra one upshot is the equivalence of stable finiteness and being mf for these reduced crossed product algebras

erratum a corrected proof of the stochastic verification theorem within the framework of viscosity solutions we correct the proof of theorem 4 1 in gozzi swiech and zhou siam j control optim 43 2005 pp 2009 2019 by imposing additional conditions on the viscosity subsolution u

extremal self dual codes with an automorphism of order 2 a method to design binary self dual codes with an automorphism of order two without fixed points is presented extremal self dual codes with lengths 40 42 44 54 58 68 are constructed many of them have weight enumerators for which extremal codes were previously not known to exist

type ii codes over f sub 2 uf sub 2 the alphabet f sub 2 uf sub 2 is viewed here as a quotient of the gaussian integers by the ideal 2 self dual f sub 2 uf sub 2 codes with lee weights a multiple of 4 are called type ii they give even unimodular gaussian lattices by construction a while type i codes yield unimodular gaussian lattices construction b makes it possible to realize the leech lattice as a gaussian lattice there is a gray map which maps type ii codes into type ii binary codes with a fixed point free involution in their automorphism group combinatorial constructions use weighing matrices and strongly regular graphs gleason type theorems for the symmetrized weight enumerators of type ii codes are derived all self dual codes are classified for length up to 8 the shadow of the type i codes yields bounds on the highest minimum hamming and lee weights

on uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde s we prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations one of them extends some recent uniqueness results by jensen some establish the uniqueness of solutions for second order isaacs equations and hence include the uniqueness results for bellman equations by p l lions our comparison results apply even for discontinuous solutions and so perron s method readily yields the existence of continuous solutions

error bounds for monotone approximation schemes for parabolic hamilton jacobi bellman equations we obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation numerical schemes for parabolic hamilton jacobi bellman equations by introducing a new notion of consistency our results are robust and general they improve and extend earlier results by krylov barles and jakobsen we apply our general results to various schemes including crank nicholson type finite difference schemes splitting methods and the classical approximation by piecewise constant controls in the first two cases our results are new and in the last two cases the results are obtained by a new method which we develop here

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

some new extremal self dual 58 29 10 codes the authors discuss extremal self dual 60 30 12 codes and then go on to present new extremal self dual 58 29 10 codes

new extremal self dual codes of lengths 36 38 and 58 we construct new extremal self dual binary codes of lengths 36 38 and 58 we show that there are at least 14 inequivalent extremal self dual 38 18 8 codes and that there are at least 368 inequivalent extremal self dual 38 19 8 codes for length 58 we construct 11 extremal self dual 58 29 10 codes whose weight enumerators were previously unknown

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

error estimates for approximate solutions to bellman equations associated with controlled jump diffusions we derive error estimates for approximate viscosity solutions of bellman equations associated to controlled jump diffusion processes which are fully nonlinear integro partial differential equations two main results are obtained i error bounds for a class of monotone approximation schemes which under some assumptions includes finite difference schemes and ii bounds on the error induced when the original levy measure is replaced by a finite measure with compact support an approximation process that is commonly used when designing numerical schemes for integro partial differential equations our proofs use and extend techniques introduced by krylov and barles jakobsen

extremal binary self dual codes in this correspondence we investigate binary extremal self dual codes numerous extremal self dual codes and interesting self dual codes with minimum weight d 14 and 16 are constructed in particular the first extremal type i 86 43 16 code and new extremal self dual codes with weight enumerators which were not previously known to exist for lengths 40 50 52 and 54 are constructed we also determine the possible weight enumerators for extremal type i codes of lengths 66 100

on the structure of binary self dual codes having an automorphism of order a square of an odd prime we describe a method for constructing binary self dual codes having an automorphism of order p sup 2 for an odd prime p using this method we classify the optimal self dual codes of lengths 36 spl les n spl les 44 and n 54 having an automorphism of order 9 we obtain all self dual 56 28 12 58 29 10 and 60 30 12 codes having an automorphism of order 9 without cycles of length 3 some of the constructed codes of lengths 54 58 and 60 have weight enumerators for which the existence of codes was not known before

existence of certain extremal self dual codes two new singly even extremal self dual codes are given these are a 42 21 8 code and a 58 29 10 code

a method for constructing self dual codes with an automorphism of order 2 in this paper we investigate binary self dual codes with an automorphism of order 2 with c cycles and f fixed points a method for constructing such codes using self orthogonal codes of length c and self dual codes of length f is presented we apply this method to construct extremal self dual codes of lengths 40 42 44 52 54 and 58 some of them have weight enumerators for which self dual codes were previously not known to exist we prove that there do not exist self dual 50 25 10 and 96 48 20 codes with an automorphism of order 2 with f fixed points for f 0 in their automorphism groups

construction of self dual codes construction methods for self dual codes are given by using these methods some new extremal self dual 66 33 12 and 68 34 12 codes are obtained

on zero sum stochastic differential games with jump diffusion driven state a viscosity solution framework a zero sum differential game with controlled jump diffusion driven state is considered and studied by a combination of dynamic programming and viscosity solution techniques we prove under certain conditions that the value of the game exists moreover the value function is shown to be the unique viscosity solution of a fully nonlinear integro partial differential equation in addition we formulate and prove a verification theorem for such games within the viscosity solution framework for nonlocal equations

on the rate of convergence of finite difference approximations for bellmans equations with variable coefficients the estimates presented here for parabolic bellman s equations with variable coefficients extend the ones earlier obtained for constant coefficients

fractional semi linear parabolic equations with unbounded data this paper is devoted to the study of semi linear parabolic equations whose principal term is fractional i e is integral and eventually singular a typical example is the fractional laplace operator this work sheds light on the fact that if the initial datum is not bounded assumptions on the non linearity are closely related to its behavior at infinity the sub linear and super linear cases are first treated by classical techniques we next present a third original case if the associated first order hamilton jacobi equation is such that perturbations propagate at finite speed then the semi linear parabolic equation somehow keeps memory of this property by using such a result locally bounded initial data that are merely integrable at infinity can be handled next regularity of the solution is proved eventually strong convergence of gradients as the fractional term disappears is proved for strictly convex non linearity

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

optimal control with state space constraint i optimal control of piecewise deterministic processes with state space constraint is studied under appropriate assumptions it is shown that the optimal value function is the only viscosity solution on the open domain which is also a supersolution on the closed domain finally the uniform continuity of the value function is obtained under a condition on the deterministic drift

continuous dependence results for non linear neumann type boundary value problems abstract we obtain estimates on the continuous dependence on the coefficient for second order non linear degenerate neumann type boundary value problems our results extend previous work of cockburn et al jakobsen and karlsen and gripenberg to problems with more general boundary conditions and domains a new feature here is that we account for the dependence on the boundary conditions as one application of our continuous dependence results we derive for the first time the rate of convergence for the vanishing viscosity method for such problems we also derive new explicit continuous dependence on the coefficients results for problems involving bellman isaacs equations and certain quasilinear equation

fractal hamilton jacobi kpz equations nonlinear and nonlocal evolution equations of the form u t lu u q where c is a pseudodifferential operator representing the infinitesimal generator of a levy stochastic process have been derived as models for growing interfaces in the case when the continuous brownian diffusion surface transport is augmented by a random hopping mechanism the goal of this paper is to study properties of solutions to this equation resulting from the interplay between the strengths of the diffusive linear and hyperbolic nonlinear terms posed in the whole space r n and supplemented with nonnegative bounded and sufficiently regular initial conditions

the existence of certain extremal 54 27 10 self dual codes some new extremal binary 54 27 10 self dual codes are constructed using automorphisms of order 7

new extremal self dual codes of length 68 we give two computational results on binary self dual codes a number of extremal singly even self dual 68 34 12 codes with weight enumerators not known to exist are constructed for k spl les 10 the relationship between extremal doubly even self dual codes of length 24 k and certain singly even self dual codes of length 24 k 2 is presented

an analytic approach to purely nonlocal bellman equations arising in models of stochastic control abstract given a bounded domain r d and two integro differential operators l 1 l 2 of the form l j u x p v u x u y k j x y x y d y we study the fully nonlinear bellman equation 0 1 max j 1 2 l j u x a j x u x f j x 0 in with dirichlet boundary conditions here a j f j r are non negative functions we prove the existence of a non negative function u r which satisfies 0 1 almost everywhere the main difficulty arises through the nonlocality of l j and the absence of regularity near the boundary

some new extremal self dual codes with lengths 44 50 54 and 58 we construct extremal self dual codes with lengths 44 50 54 and 58 they have weight enumerators for which extremal codes were previously not known to exist two methods are used for constructing the codes using self dual codes of same or smaller length to obtain the codes we use a combinatorial optimization search

error bounds for monotone approximation schemes for hamilton jacobi bellman equations we obtain error bounds for monotone approximation schemes of hamilton jacobi bellman equations these bounds improve previous results of krylov and the authors the key step in the proof of these new estimates is the introduction of a switching system which allows the construction of approximate almost smooth supersolutions for the hamilton jacobi bellman equation

continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations using the maximum principle for semicontinuous functions 3 4 we prove a general continuous dependence on the nonlinearities estimate for bounded holder continuous viscosity solutions of fully nonlinear degenerate elliptic equations furthermore we provide existence uniqueness and holder continuity results for bounded viscosity solutions of such equations our results are general enough to encompass hamilton jacobi bellman isaacs s equations of zero sum two player stochastic dierential games an immediate conse quence of the results obtained herein is a rate of convergence for the vanishing viscosity method for fully nonlinear degenerate elliptic equations

new extremal self dual codes of length 62 and related extremal self dual codes new extremal self dual codes of length 62 are constructed with weight enumerators of three different types two of these types were not represented by any known code up till now all these codes possess an automorphism of order 15 some of them are used to construct extremal self dual codes of length 60 by the method of subtracting by additional subtracting an extremal self dual 58 29 10 code was obtained having a weight enumerator which does not correspond to any code known so far

some restrictions on weight enumerators of singly even self dual codes in this correspondence we give some restrictions on weight enumerators of singly even self dual n n 2 d codes whose shadows have minimum weight d 2 as a consequence we determine the weight enumerators for which there is an extremal singly even self dual 40 20 8 code and an optimal singly even self dual 50 25 10 code

a new upper bound on the minimal distance of self dual codes it is shown that the minimal distance d of a binary self dual code of length n or 74 is at most 2 n 6 10 this bound is a consequence of some new conditions on the weight enumerator of a self dual code obtained by considering a particular translate of the code called its shadow these conditions also enable one to find the highest possible minimal distance of a self dual code for all n or 60 to show that self dual codes with d or 22 with d or 8 exist precisely for n 24 32 and n or 26 and with d or 10 exist precisely for n or 46 and to show that there are exactly eight self dual codes of length 32 with d 8 several of the self dual codes of length 34 have trivial group this appears to be the smallest length where this can happen