deformation theory of objects in homotopy and derived categories iii abelian categories abstract this is the third paper in a series in part i we developed a deformation theory of objects in homotopy and derived categories of dg categories here we show how this theory can be used to study deformations of objects in homotopy and derived categories of abelian categories then we consider examples from noncommutative algebraic geometry in particular we study noncommutative grassmanians that are true noncommutative moduli spaces of structure sheaves of projective subspaces in projective spaces

caratheodory approximations and stability of solutions to non lipschitz stochastic fractional differential equations of ito doob type abstract the existence and uniqueness theorem of solutions provides an effective tool for the model validation of both deterministic and stochastic equations the objective of this paper is to establish the existence and uniqueness of solutions for a class of ito doob stochastic fractional differential equations under non lipschitz condition which is weaker than lipschitz one and contains it as a special case the solution is constructed with the aid of caratheodory approximation moreover the continuous dependence of solutions on the initial value is investigated in view of the stability of solutions in the sense of mean square finally an example is given to illustrate the theory

the automorphism group of a self dual 72 36 16 code does not contain z_ 4 it has been proven in a series of works that the order of the automorphism group of a self dual binary 72 36 16 code does not exceed five up to equivalence we obtain a parametrization of all self dual binary codes of length 72 with automorphism of order four which can be extremal we use extensive computations in magma and on a supercomputer to show that the automorphism group of an extremal self dual binary code of length 72 does not have an element of order four

successive approximations to solutions of stochastic differential equations abstract in the present paper we shall investigate under what conditions the sequence of stochastic processes constructed by the successive approximations converges uniformly to solutions of a stochastic differential equation of ito type and shall present the local or global existence and uniqueness theorem for solutions of the above mentioned equation under more general conditions we note that lemma 3 in this paper is a generalization of gard s lemma and guarantees the existence of functions which satisfy the conditions of theorems 2 and 3 in this paper respectively theorem 3 includes as a special case a generalization of yamada s theorem which is proved by the method of the successive approximations

deformation theory of objects in homotopy and derived categories ii pro representability of the deformation functor abstract this is the second paper in a series in part i we developed deformation theory of objects in homotopy and derived categories of dg categories here we extend these derived deformation functors to an appropriate bicategory of artinian dg algebras and prove that these extended functors are pro representable in a strong sense

on the classification of extremal 36 18 8 binary self dual codes in this correspondence we give a new recursive method to classify extremal self dual codes as an application we classify all the 41 extremal binary 36 18 8 self dual codes

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

automorphisms of extremal self dual codes let c be a binary extremal self dual code of length n 48 we prove that for each aut c of prime order p 5 the number of fixed points in the permutation action on the coordinate positions is bounded by the number of p cycles it turns out that large primes p i e n p small seem to occur in aut c very rarely examples are the extended quadratic residue codes we further prove that doubly even extended quadratic residue codes of length n p 1 are extremal only in the cases n 8 24 32 48 80 and 104

new extremal self dual codes of lengths 36 and 38 there are two distinct weight enumerators for extremal singly even self dual 36 18 8 codes for each weight enumerator only one extremal self dual code is previously known in this correspondence we construct several new extremal singly even self dual 36 18 8 codes for each weight enumerator we also construct a number of new extremal singly even self dual 38 19 8 codes

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

an algorithm for classification of binary self dual codes an efficient algorithm for classification of binary self dual codes is presented as an application a complete classification of the self dual codes of length 38 is given

deformation theory of objects in homotopy and derived categories i general theory abstract this is the first paper in a series we develop a general deformation theory of objects in homotopy and derived categories of dg categories namely for a dg module e over a dg category we define four deformation functors def h e codef h e def e codef e the first two functors describe the deformations and co deformations of e in the homotopy category and the last two in the derived category we study their properties and relations these functors are defined on the category of artinian not necessarily commutative dg algebras

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

sobolev type fractional stochastic differential equations with non lipschitz coefficients this paper investigates the existence and uniqueness of mild solutions for a class of nonlinear fractional sobolev type stochastic differential equations in hilbert spaces in this work we used the fractional calculus semigroup theory and stochastic analysis techniques for obtaining the required result a new set of sufficient condition is established with the coefficients in the equations satisfying some non lipschitz conditions which include classical lipschitz conditions as special cases more precisely the results are obtained by means of standard picards iteration finally an example is given to illustrate the obtained theory

existence of solutions for nonlinear fractional stochastic differential equations abstract the fractional stochastic differential equations have wide applications in various fields of science and engineering this paper addresses the issue of existence of mild solutions for a class of fractional stochastic differential equations with impulses in hilbert spaces using fractional calculations fixed point technique stochastic analysis theory and methods adopted directly from deterministic fractional equations new set of sufficient conditions are formulated and proved for the existence of mild solutions for the fractional impulsive stochastic differential equation with infinite delay further we study the existence of solutions for fractional stochastic semilinear differential equations with nonlocal conditions examples are provided to illustrate the obtained theory

there is no self dual bbz _ 4 linear code whose gray image has the parameters 72 2 36 16 it is shown that there is no self dual bbz4 linear code whose gray image has the parameters 72 236 16

on a type of stochastic differential equations driven by countably many brownian motions consider the one dimensional sde xt x i 1 0t i xs dwsi 0tb xs ds where wi is an infinite sequence of independent standard brownian motions i 1 2 3 we prove two theorems on the existence and uniqueness of solutions with non lipschitz coefficients and give a non contact property and a strong comparison theorem for solutions

upper bounds for the lengths of s extremal codes over bbf _ 2 bbf _ 4 and bbf _ 2 u bbf _ 2 our purpose is to find an upper bound for the length of an s extremal code over f2 resp f4 when d equiv 2 mod 4 resp d odd this question is left open in a bound for certain s extremal lattices and codes archiv der mathematik vol 89 no 2 pp 143 151 2007 resp s extremal additive f4 codes advances in mathematics of communications vol 1 no 1 pp 111 130 2007 more precisely we show that if there is an n n 2 d s extremal type i binary self dual code with d 6 and d equiv 2 mod 4 then n 1 and d equiv 1 mod 2 then n 13d 26 we also define s extremal self dual codes over f2 uf2 and derive an upper bound for the length of an s extremal self dual code over f2 uf2 using the information on binary s extremal codes

existence uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay in this paper we obtain the existence and uniqueness of solutions to neutral stochastic functional differential equations with infinite delay at phase space bc 0 r d which denotes the family of bounded continuous r d value functions f defined on 0 with norm email protected sup q 0 f q under non lipschitz condition with lipschitz condition being considered as a special case and a weakened linear growth condition the solution is constructed by the successive approximation furthermore we give the continuous dependence of solutions on the initial value by means of the corollary of bihari inequality

classification of extremal and s extremal binary self dual codes of length 38 in this paper we classify all extremal and s extremal binary self dual codes of length 38 there are exactly 2744 extremal self dual codes two s extremal codes and 1730 s extremal codes we obtain our results from the use of a recursive algorithm used in the recent classification of all extremal self dual codes of length 36 and from a generalization of this recursive algorithm for the shadow the classification of extremal codes permits to achieve the classification of all extremal codes with

singly even self dual codes with minimal shadow in this paper extremal singly even self dual codes with minimal shadow are investigated nonexistence of such codes for particular parameters is proved by a result of rains the length of extremal singly even self dual codes is bounded explicit bounds are given in case the shadow is minimal

homological mirror symmetry for curves of higher genus abstract this paper is devoted to homological mirror symmetry conjecture for curves of higher genus it was proposed by katzarkov as a generalization of original kontsevich s conjecture a version of this conjecture in the case of the genus two curve was proved by seidel 25 based on the paper of seidel we prove the conjecture in the same version for curves of genus g 3 namely we relate the fukaya category of a genus g curve to the category of singularities of zero fiber in the mirror dual landau ginzburg model we also prove a kind of reconstruction theorem for hypersurface singularities namely formal type of hypersurface singularity i e a formal power series up to a formal change of variables can be reconstructed with some technical assumptions from its d z 2 g category of landau ginzburg branes the precise statement is theorem 1 2

existence and stability of solutions to non lipschitz stochastic differential equations driven by levy noise in this paper successive approximation method is applied to investigate the existence and uniqueness of solutions to stochastic differential equations sdes driven by levy noise under non lipschitz condition which is a much weaker condition than lipschitz one the stability of solutions to non lipschitz sdes driven by levy noise is also considered and the stochastic stability is obtained in the sense of mean square

a note on the existence and uniqueness of the solution to neutral stochastic functional differential equations with infinite delay in this note we prove the existence and uniqueness of the solution to neutral stochastic functional differential equations with infinite delay insfdes in short in which the initial value belongs to the phase space bc 0 r d which denotes the family of bounded continuous r d value functions f defined on 0 with norm f sup q 0 f q under some caratheodory type conditions on the coefficients by means of the successive approximation especially we extend the results appeared in ren et al y ren s lu n xia remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay j comput appl math 220 2008 364 372 ren and xia y ren n xia existence uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay appl math comput 210 2009 72 79 and zhou and xue s zhou m xue the existence and uniqueness of the solutions for neutral stochastic functional differential equations with infinite delay math appl 21 2008 75 83

existence result for fractional neutral stochastic integro differential equations with infinite delay this paper is concerned with the existence of mild solutions for a class of fractional neutral stochastic integro differential equations with infinite delay in hilbert spaces a sufficient condition for the existence is obtained under non lipschitz conditions by means of sadovskii s fixed point theorem an example is given to illustrate the theory

shadow bounds for self dual codes conway and sloane 1990 have previously given an upper bound on the minimum distance of a singly even self dual binary code using the concept of the shadow of a self dual code we improve their bound finding that the minimum distance of a self dual binary code of length n is at most 4 n 24 4 except when n mod 24 22 when the bound is 4 n 24 6 we also show that a code of length a multiple of 24 meeting the bound cannot be singly even the same technique gives similar results for additive codes over gf 4 relevant to quantum coding theory

categorical resolution of singularities abstract building on the concept of a smooth dg algebra we define the notion of a smooth derived category we then propose the definition of a categorical resolution of singularities our main example is the derived category d x of quasi coherent sheaves on a scheme x we prove that d x has a canonical categorical resolution if the base field is perfect and x is a separated scheme of finite type with a dualizing complex

a note on the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations with non lipschitz coefficients in this note we study the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations under some caratheodory type conditions on the coefficients by means of the successive approximation in particular we generalize and improve the results that appeared in govindan t e govindan almost sure exponential stability for stochastic neutral partial functional differential equations stochastics 77 2005 139 154 and bao and hou j bao z hou existence of mild solutions to stochastic neutral partial functional differential equations with non lipschitz coefficients comput math appl 59 2010 207 214

new self dual codes over gf 4 with the highest known minimum weights the purpose of this correspondence is to construct new hermitian self dual codes over gf 4 of lengths 22 24 26 32 and 34 which have the highest known minimum weights in particular for length 22 we construct eight new extremal self dual 22 11 8 codes over gf 4 which do not have a nontrivial automorphism of odd order the existence of such codes has been left open since 1991 by huffman

successive approximation to solutions of stochastic differential equations with jumps in local non lipschitz conditions in this paper by using successive approximation the existence and uniqueness of initial value problem for stochastic differential equations driven by both wiener process and poisson process are studied under a local non lipschitz conditions moreover the numerical solutions are shown to converge uniformly to the analytical solutions of the stochastic differential equation with jumps

the extended quadratic residue code is the only 48 24 12 self dual doubly even code an extremal self dual doubly even binary n k d code has a minimum weight d 4 spl lfloor n 24 spl rfloor 4 of such codes with length divisible by 24 the golay code is the only 24 12 8 code the extended quadratic residue code is the only known 48 24 12 code and there is no known 72 36 16 code one may partition the search for a 48 24 12 self dual doubly even code into three cases a previous search assuming one of the cases found only the extended quadratic residue code we examine the remaining two cases separate searches assuming each of the remaining cases found no codes and thus the extended quadratic residue code is the only doubly even self dual 48 24 12 code

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

the automorphism group of a self dual 72 36 16 binary code does not contain elements of order 6 the existence of an extremal code of length 72 is a long standing open problem let c be a putative extremal code of length 72 and suppose that c has an automorphism g of order 6 we show that c as an bbf2 g module is the direct sum of two modules it is easy to determine one of them while the other one has a very restrictive structure we use this fact to do an exhaustive search and we do not find an extremal code this proves that the automorphism group of an extremal code of length 72 does not contain elements of order 6

noncommutative deformations and flops we prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a 33 fold is representable and hence we associate to every such curve a noncommutative deformation algebra aconacon this new invariant extends and unifies known invariants for flopping curves in 33 folds such as the width of reid and the bidegree of the normal bundle it also applies in the settings of flips and singular schemes we show that the noncommutative deformation algebra aconacon is finite dimensional and give a new way of obtaining the commutative deformations of the curve allowing us to make explicit calculations of these deformations for certain 3 1 3 1 curves we then show how our new invariant aconacon also controls the homological algebra of flops for any flopping curve in a projective 33 fold with only gorenstein terminal singularities we construct an autoequivalence of the derived category of the 33 fold by twisting around a universal family over the noncommutative deformation algebra aconacon and prove that this autoequivalence is an inverse of bridgeland s flop flop functor this demonstrates that it is strictly necessary to consider noncommutative deformations of curves in order to understand the derived autoequivalences of a 33 fold and thus the bridgeland stability manifold

a note on the neutral stochastic functional differential equation with infinite delay and poisson jumps in an abstract space in this note we prove the existence and uniqueness of the solution to neutral stochastic functional differential equation with infinite delay and poisson jumps in the phase space b with non lipschitz coefficients the known results that appeared in the studies by ren and xia appl math comput 210 72 2009 ren and xiaappl math comput 214 457 2009 and xu and hu acta appl math unpublished are generalized and improved an example is provided to illustrate the theory

stochastic fractional differential equations driven by levy noise under caratheodory conditions in this manuscript we initiate a study on a class of stochastic fractional differential equations driven by levy noise the existence and uniqueness theorem of solutions to equations of this class is established under global and local caratheodory conditions our analysis makes use of the caratheodory approximation as well as a stopping time technique the results obtained here generalize the main results from pedjeu and ladde chaos solitons fractals 45 279 293 2012 xu et al appl math comput 263 398 409 2015 and abouagwa et al appl math comput 329 143 153 2018 finally an application to the stochastic fractional burgers differential equations is designed to validate the theory obtained in this manuscript we initiate a study on a class of stochastic fractional differential equations driven by levy noise the existence and uniqueness theorem of solutions to equations of this class is established under global and local caratheodory conditions our analysis makes use of the caratheodory approximation as well as a stopping time technique the results obtained here generalize the main results from pedjeu and ladde chaos solitons fractals 45 279 293 2012 xu et al appl math comput 263 398 409 2015 and abouagwa et al appl math comput 329 143 153 2018 finally an application to the stochastic fractional burgers differential equations is designed to validate the theory obtained

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