exact boundary behavior of the unique positive solution to some singular elliptic problems abstract in this paper we give an exact asymptotic of the unique solution to the following singular boundary value problem u a x g u x u 0 in u 0 here is a c 2 bounded domain in r n n 2 g c 1 0 0 is nonincreasing on 0 with lim t 0 g t 0 t d s g s c g 0 and the function a is in c l o c 0 1 satisfying 0 a 1 lim inf d x 0 a x h d x lim sup d x 0 a x h d x a 2 where h t c t exp t z s s d s 2 c 0 and z continuous on 0 for some 0 such that z 0 0 two applications of this result are also given the first concerns the boundary behavior of the unique solution of u u u 2 a x g u that vanishes on the boundary and the second concerns the behavior of u in the case where the open set is an annular and the behaviors of the function a on the interior boundary and the exterior boundary may be different

asymptotic behavior of positive solutions of a semilinear dirichlet problem abstract in this paper we study the asymptotic behavior of the unique positive classical solution to the following semilinear boundary value problem u a x u 0 x u 0 in u 0 here is a bounded c 1 1 domain 1 and the function a is in c l o c 0 1 such that there exists c 0 satisfying for each x 1 c a x x exp x z t t d t c where 2 d d i a m x d i s t x and z is a continuous function on 0 with z 0 0

on a semilinear fractional dirichlet problem on a bounded domain let d be a bounded c 1 1 domain in r d d 2 0 a 2 and s 1 we are concerned with the existence the uniqueness and the asymptotic behavior of positive continuous solution for the following semilinear fractional differential equation email protected d a 2u a x u s x ind with the boundary dirichlet email protected email protected email protected x 2 au x 0 here email protected d a 2 is the fractional laplacian associated to the subordinate killed brownian motion process in d and d x d x d denotes the euclidean distance between x and d our arguments are based on potential theory associated to the fractional laplacian and some karamata regular variation theory tools

the second expansion of the solution for a singular elliptic boundary value problem abstract in this paper we analyze the second expansion of the unique solution near the boundary to the singular dirichlet problem u b x g u u 0 x u 0 where is a bounded domain with smooth boundary in r n g c 1 0 0 g is decreasing on 0 with lim s 0 g s and g is normalised regularly varying at zero with index 1 b c is positive in may be vanishing on the boundary

the knuth bendix completion procedure and thue systems the knuth bendix completion procedure for term rewriting systems in many cases provides a decision procedure for equational theories and has been found to have many applications in various areas we discuss the application of the knuth bendix procedure to thue systems we use the notion of a reduced thue system and show that for every church rosser thue system there is a unique reduced church rosser thue system equivalent to it furthermore the knuth bendix completion procedure when applied to a thue system t always produces the finite reduced church rosser thue system equivalent to t whenever such a system exists similar results can also be proved for almost confluent thue systems using properties of reduced church rosser systems we develop conditions under which a class of special thue systems have equivalent finite church rosser systems in addition we show that the completion procedure always terminates on finite parenthesized thue systems from which the termination of the completion procedur

the boundary behavior of the unique solution to a singular dirichlet problem abstract in this paper we analyze the boundary behavior of the unique solution to the singular dirichlet problem for a lane emden fowler equation u b x g u u 0 x u 0 where is a bounded domain with smooth boundary in r n g c 1 0 0 lim s 0 g s g is decreasing on 0 which is rapidly varying or normalized regularly varying at zero b c loc is positive in may be vanishing or singular on the boundary and belongs to the kato class k

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

exact behavior of the unique positive solution to some singular elliptic problem in exterior domains in this paper we give an exact behavior on the boundary and at infinity of the unique solution to the following singular boundary value problem u a x g u x u 0 in u 0 and lim x u x 0 here is an exterior domain in rn n 3 with compact c2c2 boundary g c1 0 0 g c1 0 0 is nonincreasing on 0 0 with limt 0g t 0tdsg s cg 0 and the function aa is in cloc 0 10 1 satisfying 0 a1 lim infd x 0a x h d x lim supd x 0a x h d x a2 and 0 b1 lim inf x a x k x lim sup x a x k x b2 where d x d x is the euclidean distance from x x to the boundary h t c1t exp t z s sds 2 2 c1 0c1 0 and zz is continuous on 0 0 for some 0 0 such that z 0 0z 0 0 and k t c2t exp 1ty s sds 2 2 c2 0c2 0 and yy is continuous on 1 1 such that limt y t 0limt y t 0

when is a monoid a group the church rosser case is tractable abstract the following question is shown to be tractable given a finite monadic church rosser thue system t and a finite set a is the submonoid generated by a of the monoid presented by t a group

code enumerators and tutte polynomials it is proved that the set of higher weight enumerators of a linear code over a finite field is equivalent to the tutte polynomial associated to the code an explicit expression for the tutte polynomial is given in terms of the subcode weights generalizations of these results are proved and are applied to codeword m tuples these general results are used to prove a very general macwilliams type identity for linear codes that generalizes most previous extensions of the macwilliams identity in addition a general and very useful matrix framework for manipulating weight and support enumerators of linear codes is presented

boundary behaviour of the unique solution to a singular dirichlet problem with a convection term by karamata regular variation theory and constructing comparison functions we derive that the boundary behaviour of the unique solution to a singular dirichlet problem u b x g u u q u 0 x u 0 which is independent of u q w here is a bounded domain with smooth boundary in r n r q 0 2 lims 0 g s a ndb is nonnegative on which may be vanishing on the boundary

multi parameter bifurcation and asymptotics for the singular lane emden fowler equation with a convection term to the second variable and g is unbounded around the origin the asymptotic behaviour of the solution around the bifurcation point is also established provided g u behaves like u around the origin for some 0 1 our approach relies on finding explicit sub and supersolutions combined with various techniques related to the maximum principle for elliptic equations the analysis we develop in this paper shows the key role played by the convection term u p

second order expansion for the solution to a singular dirichlet problem in this paper we analyze the second order expansion for the unique solution near the boundary to the singular dirichlet problem u b x g u u 0 x u 0 where is a bounded domain with smooth boundary in r n g c1 0 0 g is decreasing on 0 with lim s 0 g s and g is normalized regularly varying at zero with index 1 b c l o c 0 1 is positive in may be vanishing or singular on the boundary and belongs to the kato class k our analysis is based on the sub supersolution method and karamata regular variation theory

the exact boundary behavior of the unique solution to a singular dirichlet problem with a nonlinear convection term abstract in this paper we analyze the exact boundary behavior of the unique solution to the singular nonlinear dirichlet problem u b x g u u q u 0 x u 0 where is a bounded domain with smooth boundary in r n q 0 2 0 g c 1 0 0 lim s 0 g s g is decreasing on 0 and b c loc is positive in may be vanishing or singular on the boundary we reveal that the nonlinear convection term u q does not affect the first expansion of the unique solution near the boundary to the problem

boundary behavior of solutions of monge ampere equations with singular righthand sides abstract in this paper by karamata regular variation theory and constructing super and subsolutions we obtain the asymptotic boundary behavior of convex solutions of monge ampere equations with singular righthand sides

the automorphism group of a binary self dual doubly even 72 36 16 code is solvable in this correspondence we prove that the automorphism group of a putative binary self dual doubly even 72 36 16 code is solvable moreover its order is 5 7 10 14 56 or a divisor of 72

gilman s conjecture abstract we prove a conjecture made by gilman in 1984 that the groups presented by finite monadic confluent rewriting systems are precisely the free products of free and finite groups

monadic thue systems abstract certain infinite thue systems over a finite alphabet are studied in particular systems s e such that for each a e the set u u a s is a context freelanguage the syntactic structure of sets of ancestors and sets of descendants is considered as well as that of unions of congruence classes taken over infinite context free languages or regular sets the common descendant problem is shown to be tractable while the common ancestor problem is shown to be undecidable even for finite systems the word problem for confluent systems of this type is shown to be tractable the question of whether an infinite system of this type is confluent is shown to be undecidable as is the question of whether the congruence generated by such a system has a confluent presentation

the exact asymptotic behaviour of the unique solution to a singular nonlinear dirichlet problem abstract by karamata regular varying theory a perturbed argument and constructing comparison functions we show the exact asymptotic behaviour of the unique solution u c 2 c near the boundary to a singular dirichlet problem u b x g u f u u 0 x u 0 which is independent on f u and we also show the existence and uniqueness of solutions to the problem where is a bounded domain with smooth boundary in r n 0 g c 1 0 0 and there exists 1 such that lim t 0 g t g t 1 0 f c loc 0 0 the function f s s s 0 is decreasing on 0 for some s 0 0 and b is nonnegative nontrivial on which may be vanishing on the boundary

on the automorphism group of a doubly even 72 36 16 code we prove that a putative doubly even 72 36 16 code has no automorphism of order 3 with fixed points moreover such a code has no automorphism of an odd order greater than 7 eventually we prove that 25 and 49 do not divide the order of its automorphism group

geometric approach to higher weights the notion of higher or generalized weights of codes is just as natural as that of the classical hamming weight the authors adopt the geometric point of view and always treat the q ary case some results and proofs being new the main goal is to present a clear picture of what is known on the subject

positive solution of singular dirichlet boundary value problems for second order differential equation system this paper investigates the existence of positive solutions of singular dirichlet boundary value problems for second order differential system a necessary and sufficient condition for the existence of c 0 1 c 0 1 positive solutions as well as c1 0 1 c1 0 1 positive solutions is given by means of the method of lower and upper solutions and the fixed point theorems our nonlinearity fi t x1 x2 may be singular at x1 0 x2 0 t 0 and or t 1 i 1 2

an existence and uniqueness result for the singular lane emden fowler equation abstract using a fixed point theorem of general concave operators we present in this paper criteria which guarantee the existence and uniqueness of positive solutions for the singular lane emden fowler equation

positive solutions of lane emden systems with negative exponents existence boundary behavior and uniqueness abstract we study the existence boundary behavior and uniqueness of solutions for the singular elliptic system u u p v q v u r v s u 0 v 0 x u v 0 where is a bounded domain with smooth boundary in r n p s 0 and q r 0 our results are obtained in a range of p q r s different from those in m ghergu lane emden systems with negative exponents j funct anal 258 2010 3295 3318

existence and estimates of solutions to a singular dirichlet problem for the monge ampere equation given a strictly convex smooth and bounded domain in rn we establish the existence of a negative convex solution in c c with zero boundary value to the singular monge ampere equation det d2u p x g u an associated dirichlet problem will be employed to provide a necessary and sufficient condition for the solvability of the singular boundary value problem estimates of solutions will also be given and regularity of solutions will be deduced from the estimates

the second expansion of the unique vanishing at infinity solution to a singular elliptic equation abstract this paper is considered with the second expansion of the unique vanishing at infinity solution to u b x g u x r n n 3 where b c r n is nonnegative and may be supercritical attenuation or critical attenuation at infinity g c 1 0 0 is non increasing on 0 with lim s 0 g s and g is normalized regularly varying at zero with index 1

computing automorphism groups of error correcting codes an algorithm is described for computing the automorphism group of an error correcting code the algorithm determines the order of the automorphism group and produces a set of monomial permutations which generate the group it has been implemented on a computer and has been used successfully on a great number of codes of moderate length

general uniqueness results and variation speed for blow up solutions of elliptic equations let omega be a smooth bounded domain in r n we prove general uniqueness results for equations of the form delta u au b x f u in omega subject to u infty on partial omega our uniqueness theorem is established in a setting involving karamata s theory on regularly varying functions which is used to relate the blow up behavior of u x with f u and b x where b equiv 0 on partial omega and a certain ratio involving b is bounded near partial omega a key step in our proof of uniqueness uses a modification of an iteration technique due to safonov

asymptotic behavior of positive solutions of a nonlinear dirichlet problem abstract we take up the existence and the asymptotic behavior of a classical solution to the following semilinear dirichlet problem u a x g u x u 0 in u 0 where is a c 1 1 bounded domain in r n n 2 and the function a belongs to c l o c 0 1 such that there exist c 1 c 2 0 satisfying for each x c 1 x 1 exp x z 1 s s d s a x c 2 x 2 exp x z 2 s s d s where d i a m x d i s t x 1 2 2 and for i 1 2 z i is a continuous function on 0 with z i 0 0 our arguments are based on the sub supersolution method with karamata regular variation theory

boundary behavior of k convex solutions for singular k hessian equations abstract we discuss the existence and boundary behavior of k convex solution to the singular k hessian problem s k d 2 u x b x f u x x u x 0 x where s k d 2 u k 1 2 n is the k hessian operator r n n 2 is a smooth bounded strictly convex domain here the weight function b x is not necessarily bounded on another interest is that f u as u 0 our approach mainly relies on karamata s regular variation theory and the construction of suitable sub and super solutions

17 does not divide the order of the group of a 72 36 16 doubly even code corresp it is an interesting open question whether an extremal 72 36 16 doubly even code c exists in 3 the odd prime numbers which can divide the order of the group of c were determined the largest of these 23 was eliminated by finding weight 12 vectors in 384 codes 8 the next largest prime remaining is 17 it is shown that 17 is also not possible by reducing the problem to the consideration of 16 17 3 codes and then finding a weight 12 vector by computer in each of these codes

the decoding error probability of linear codes over the erasure channel in this paper we study the decoding error probability of linear codes over the erasure channel under the list decoding the notion of the q ell incorrigible sets of linear codes is introduced to characterize its decoding error probability under the list decoding or the maximum likelihood decoding by calculating the q ell incorrigible set distributions the decoding error probability of a linear code over the erasure channel under the list decoding or the maximum likelihood decoding is expressed by its support weight distributions for the ensemble of all n k linear codes the average decoding error probability under the maximum likelihood decoding and the average unsuccessful decoding probability under unambiguous decoding are determined furthermore the error exponent of the average unsuccessful decoding probability under the unambiguous decoding is determined for the ensemble of all n nr linear codes

the second order expansion of solutions to a singular dirichlet boundary value problem abstract in this paper we mainly study the second order expansion of classical solutions in a neighborhood of to the singular dirichlet problem u b x g u a x f u u 0 x u 0 where is a bounded domain with smooth boundary in r n 0 the weight functions b a c loc are positive in and both may be vanishing or be singular on the boundary the function g c 1 0 0 satisfies lim t 0 g t and f c 0 0 we show that the nonlinear term a x f u does not affect the second order expansion of solutions in a neighborhood of to the problem for some kinds of functions b and a

on a singular nonlinear semilinear elliptic problem where k x c2 b v9 a p 0 1 and l is a real parameter such singular elliptic problems arise in the contexts of chemical heterogeneous catalysts nonnewtonian fluids and also the theory of heat conduction in electrically conducting materials see 3 5 8 9 for a detailed discussion obviously 1 1 cannot have a solution u c2 v9 if k x is not vanishing near v however under various appropriate assumptions on k x we will obtain classical solutions of 1 1 for l belonging to a certain range and we will also obtain some uniqueness criteria here a classical solution is a solution u of 1 1 which belongs to c2 v mc v9 with u 0 in v we also study the boundary behaviour of solutions of 1 1 and we will show that the solution u of 1 1 lies in a certain holder class the special case when k x is negative and l 0 has been studied by several authors the existence and uniqueness of the solution were established by crandall rabinowitz and tartar 6 del pino 7 gomes 10 and lazer and mckenna

existence and optimal estimates of solutions for singular nonlinear dirichlet problems abstract by applying the properties of the unique classical solution to the singular boundary value problem on half line p s g p s p s 0 s 0 p 0 0 lims p s b 0 and constructing the new comparison functions they show the existence and the optimal global estimates of solutions to singular nonlinear dirichlet problems u k x g u u 0 x u 0 where is a bounded domain with smooth boundary in rn g s is nonincreasing and positive in 0 1 g t dt and lim s 0 g s k c is positive in and may be singular or zero on the boundary

existence and estimates of solutions for singular nonlinear elliptic problems for both bounded and unbounded domain in r d 3 with smooth boundary the function 0 0 is measurable and satisfies some appropriate hypotheses detailed below the problem 1 has been studied extensively for the special nonlinearity x t p x t 0 indeed in 7 edelson proved the existence of a positive solution u c2 loc r decaying as x 2 n as x provided that p c loc r 0 0 for x 0 and satisfying the growth condition

new fixed point theorems for mixed monotone operators and local existence uniqueness of positive solutions for nonlinear boundary value problems abstract in this article we present a new fixed point theorem for a class of general mixed monotone operators which extends the existing corresponding results moreover we establish some pleasant properties of nonlinear eigenvalue problems for mixed monotone operators based on them the local existence uniqueness of positive solutions for nonlinear boundary value problems which include neumann boundary value problems three point boundary value problems and elliptic boundary value problems for lane emden fowler equations is proved the theorems for nonlinear boundary value problems obtained here are very general

multiple positive solutions to a singular boundary value problem for a superlinear emden fowler equation a multiplicity result for the singular ordinary differential equation y x 2y 0 posed in the interval 0 1 with the boundary conditions y 0 0 and y 1 where 1 0 and 0 are real parameters is presented using a logarithmic transformation and an integral equation method we show that there exists 0 2 such that a solution to the above problem is possible if and only if 1 for 0 1 there are multiple positive solutions while if 1 1 1 the problem has a unique positive solution which is monotonic increasing the asymptotic behavior of y x as x 0 is also given which allows us to establish the absence of positive solution to the singular dirichlet elliptic problem u d x 2u in where rn n 2 is a smooth bounded domain and d x dist x

classical and weak solutions of a singular semilinear elliptic problem the singular semilinear elliptic equation u p x f u 0 is shown to have a unique positive classical solution inrnthat decays to zero at ifp x is simply a nontrivial nonnegative continuous function satisfying 0 t max x t p x dt providedfis a non increasing continuously differentiable function on 0 it is also shown that the equation has a unique weakh10 solution on a bounded domain provided e0 f s ds andp x l2

sublinear singular elliptic problems with two parameters abstract we establish several existence and nonexistence results for the boundary value problem u k x g u f x u h x in u 0 on where is a smooth bounded domain in r n and are positive parameters h is a positive function while f has a sublinear growth the main feature of this paper is that the nonlinearity g is assumed to be unbounded around the origin our analysis shows the importance of the role played by the decay rate of g combined with the signs of the extremal values of the potential k x on the proofs are based on various techniques related to the maximum principle for elliptic equations

chains demi matroids and profiles it is shown that each chain of linear codes has an associated demi matroid a combinatorial structure that extends the notion of a vector matroid of a linear code these demi matroids are proven to determine important properties of the chain and it is shown that linear code chain duality is represented by demi matroid duality in a natural and yet surprising way profiles are defined for arbitrary demi matroids and thus for linear code chains which generalizes previous results in the literature and provides simple and transparent proofs thereof finally applications are given to encryption of messages through wire tap channels of type ii

duality and support weight distributions we show how to compute the support weight distribution a sub i sup r for r spl ges k d sub 2 sup spl perp 3 where d sub 2 sup spl perp is the second minimum support weight of a code provided the weight enumerator of the dual code is known

exact asymptotic behavior near the boundary to the solution for singular nonlinear dirichlet problems abstract we take up the existence and the exact asymptotic behavior near the boundary of the unique classical solution to a singular dirichlet problem u a x g u x u 0 in u 0 where is a c 1 1 bounded domain in r n n 2 and the function a is singular near the boundary and belongs to the kato class k

generalized hamming weights for linear codes motivated by cryptographical applications the algebraic structure of linear codes from a new perspective is studied by viewing the minimum hamming weight as a certain minimum property of one dimensional subcodes a generalized notion of higher dimensional hamming weights is obtained these weights characterize the code performance on the wire tap channel of type ii basic properties of generalized weights are derived the values of these weights for well known classes of codes are determined and lower bounds on code parameters are obtained several open problems are also listed

the higher weight enumerators of the doubly even self dual 48 24 12 code an efficient algorithm is presented for calculating higher weight enumerators of linear codes given generator matrices by this algorithm the higher weight enumerators of the unique doubly even self dual code are calculated the algorithm is based on a previously shown relationship between tutte polynomials and higher weight enumerators

asymptotic behavior of positive solutions of a singular nonlinear dirichlet problem abstract let be a c 1 1 bounded domain in r n for n 2 in this paper we are concerned with the asymptotic behavior of the unique positive classical solution to the singular boundary value problem u a x u 0 in u 0 where 0 a is a nonnegative function in c loc 0 1 and there exists c 0 such that 1 c a x x k 1 m log k x k c here 2 k r is a positive constant and x dist x

complexity combinatorial group theory and the language of palutators abstract after a brief survey of the known results about group languages we prove that many of the familiar classes of groups have word problems whose complexity is linear time we also consider the complexity of extensions and hnn extensions finally in an attempt to find languages that have complexity which is greater than linear time we discuss the language of palutators where a palutator is defined as a generalization of a palindrome and a commutator

groups presented by finite two monadic church rosser thue systems it is shown that a group g can be defined by a monoid presentation of the form 2 7 where t is a finite two monadic church rosser thue system over 2 if and only if g is isomorphic to the free product of a finitely generated free group with a finite number of finite groups

23 does not divide the order of the group of a 72 36 16 doubly even code corresp an interesting open question is whether a 72 36 16 doubly even code c exists in 3 the odd prime numbers which can divide the order of the group of c were determined and 23 is the largest of these twenty three is eliminated by reducing the problem to the consideration of 348 codes each of which is shown to have minimum weight 12 or less one of these codes denoted by c arises from the a x b x a b x construction where a and b are in one quadratic residue code and x is in the other the weight distribution of c is given

a sum operator equation and applications to nonlinear elastic beam equations and lane emden fowler equations article history this paper is concerned with an operator equation ax bx cx x on ordered banach spaces where a is an increasing concave operator b is an increasing sub homogeneous operator and c is a homogeneous operator the existence and uniqueness of its positive solutions is obtained by using the properties of cones and a fixed point theorem for increasing general concave operators as applications we utilize the fixed point theorems obtained in this paper to study the existence and uniqueness of positive solutions for two classes nonlinear problems which include fourth order two point boundary value problems for elastic beam equations and elliptic value problems for lane emden fowler equations

asymptotic behavior of positive solutions of a semilinear polyharmonic problem in the unit ball abstract let m be a positive integer we investigate the existence and the asymptotic behavior of positive continuous solutions to the following semilinear polyharmonic boundary value problem in the unit ball b of r n n 2 m u a x u lim x 1 u x 1 x m 1 0 where 1 1 and a is a positive measurable function in b such that there exists c 0 satisfying for each x b 1 c a x 1 x exp 1 x z s s d s c 1 m 1 1 and z is a continuous function on 0 with z 0 0

the asymptotic behaviour of the unique solution for the singular lane emden fowler equation abstract by karamata regular variation theory and constructing comparison functions we show the exact asymptotic behaviour of the unique classical solution u c 2 c near the boundary to a singular dirichlet problem u k x g u u 0 x u 0 where is a bounded domain with smooth boundary in r n g c 1 0 0 lim t 0 g t g t for each 0 for some 0 and k c loc for some 0 1 is nonnegative on which is also singular near the boundary

the asymptotic behaviour of solutions with blow up at the boundary for semilinear elliptic problems abstract by constructing the comparison functions and the perturbed method it is showed that any solution u c 2 to the semilinear elliptic problems u k x g u x u satisfies lim d x 0 u x z d x 2 2 2 c 0 2 1 where is a bounded domain with smooth boundary in r n lim d x 0 k x d x c 0 2 c 0 0 2 2 g c 1 0 g 0 and g s s is increasing on 0 there exists 0 such that lim s g s g s 0 z s d t 2 g t s g t 0 t g s d s

a 72 36 16 doubly even code does not have an automorphism of order 11 corresp we prove that there does not exist a 72 36 16 doubly even code with an automorphism of order 11 if such a code exists it can be decomposed as a direct sum of two codes one can be viewed as a self dual 12 6 4 binary code and the other a self dual 6 3 4 code over f 1024 this is shown to be impossible

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

automorphisms of codes with applications to extremal doubly even codes of length 48 general results on automorphisms of self dual binary codes are given these results are applied to the study of extremal self dual doubly even binary codes of length 48 the main theorem proved is that an extremal self dual doubly even code of length 48 with a nontrivial automorphism of odd order is equivalent to the extended quadratic residue code interesting constructions of the binary extended golay code as well as a conjecture about a possible connection between an extremal self dual doubly even code of length 72 and an extremal quaternary code of length 24 arc yielded by techniques used in the proof

the third support weight enumerators of the doubly even self dual 32 16 8 codes we present combinatorial methods for computing the third support weight enumerators of the five doubly even self dual 32 16 8 codes the methods exploit relationships that exist between support weight enumerators and complete coset weight enumerators of a self dual code

existence theory for single and multiple solutions to singular positone boundary value problems abstract in this paper we present some old and new existence results for singular boundary value problems our nonlinearity may be singular in its dependent variable

on extremal self dual codes of length 96 let c be a binary extremal self dual code of length 96 we prove that an automorphism of c of order 3 has 6 or no fixed points and an automorphism of order 5 has 6 fixed points moreover if all automorphisms of order 3 are fixed point free then aut c is solvable and its order divides 253 or 255 or aut c is the alternating group a5 which is the only possible group of order 60 furthermore aut c 20 or 40 cannot occur

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