bivariate c1 cubic spline spaces with homogeneous boundary conditions over fvs triangulation in this paper we mainly generalize the results in h w liu d hong d q cao bivariate c 1 cubic spline space over a nonuniform type 2 triangulation and its subspaces with boundary conditions comput math appl 49 2005 1853 1865 from the type 2 triangulation to the so called fvs triangulation a triangulated quadrangulation we study the bivariate c 1 cubic spline spaces s 3 1 0 email protected and s 3 1 1 email protected with homogeneous boundary conditions over an fvs triangulation email protected the dimensions are obtained and the locally supported bases are constructed for these spline spaces furthermore we also study the explicit bezier ordinates of the interpolation basis splines on a representative triangulated quadrilateral the results of this paper can be applied in many fields such as the finite element method for partial differential equation computer aided geometric design numerical approximation and so on

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

construction of wavelets and prewavelets over triangulations constructions of wavelets and prewavelets over triangulations with an emphasis of the continuous piecewise polynomial setting are discussed some recent results on piecewise linear prewavelets and orthogonal wavelets are presented

a note on self dual group codes we classify group algebras over galois rings containing self dual ideals i e ideals c which satisfy c c sup spl perp with respect to the natural nondegenerate bilinear form given on group algebras

linear independence and stability of piecewise linear prewavelets on arbitrary triangulations in this paper we establish linear independence and stability of certain piecewise linear prewavelets over arbitrary bounded triangulations these prewavelets are natural generalizations of the locally supported element constructed by kotyczka and oswald for an infinite three directional mesh

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

c 1 spline wavelets on triangulations in this paper we investigate spline wavelets on general triangulations in particular we are interested in c 1 wavelets generated from piecewise quadratic polynomials by using the powell sabin elements we set up a nested family of spaces of c 1 quadratic splines which are suitable for multiresolution analysis of besov spaces consequently we construct c 1 wavelet bases on general triangulations and give explicit expressions for the wavelets on the three direction mesh a general theory is developed so as to verify the global stability of these wavelets in besov spaces the wavelet bases constructed in this paper will be useful for numerical solutions of partial differential equations

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

the gleason prange theorem the gleason prange theorem describes a nontrivial automorphism of an extended quadratic residue code by using the fourier transform in an appropriate extension field methods of digital signal processing are used to make the proof transparent and the automorphism intuitive the theorem is proved both in a field of characteristic tow and in an arbitrary field and is stated simply as a general property of the discrete fourier transform

bivariate c1 cubic spline space over a nonuniform type 2 triangulation and its subspaces with boundary conditions in this paper we discuss the algebraic structure of bivariate c 1 cubic spline spaces over nonuniform type 2 triangulation and its subspaces with boundary conditions the dimensions of these spaces are determined and their local support bases are constructed

on the automorphism group of a binary self dual doubly even 72 36 16 code we prove that the automorphism group of a binary self dual doubly even 72 36 16 code has order 5 7 10 14 or d where d divides 18 or 24 or it is a4 c3

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

multiresolution analysis over triangles based on quadratic hermite interpolation given a triangulation t of r 2 a recipe to build a spline space s t over this triangulation and a recipe to rene the triangulation t into a triangulation t 0 the question arises whether s t s t 0 i e whether any spline surface over the original triangulation t can also be represented as a spline surface over the rened triangulation t 0 in this paper we will discuss how to construct such a nested sequence of spaces based on powell sabin 6 splits for a regular triangulation the resulting spline space consists of piecewise c 1 quadratics and renement is obtained by subdividing every triangle into four subtriangles at the edge midpoints we develop explicit formulas for wavelet transformations based on quadratic hermite interpolation and give a stability result with respect to a natural norm c 2000 elsevier science b v all rights reserved msc 41a63 65d07 65d17 65t60

on the automorphism group of a putative code a long standing open question in coding theory is whether a binary self dual 72 36 16 code exists it was shown recently that the automorphism group order of such code is equal to 2imiddot3 jmiddot5middot7 2imiddot3jmiddot7 2imiddot3jmiddot5 or 2imiddot3 j for some nonnegative integers i and j we show that the group order is not a multiple of 35 and find that only 22 orders are possible

binary self dual codes with automorphisms of composite order in this paper we present some results concerning a decomposition of binary self dual codes having an automorphism of an order which is the product of two odd prime numbers these results are applied to construct self dual 72 36 12 codes with an automorphism of order 15 furthermore it is proved that the automorphism group of a putative binary extremal self dual 72 36 16 code contains at most 28 nontrivial types of automorphisms of odd order a complete list of these possible types of automorphisms is presented

piecewise linear prewavelets on arbitrary triangulations this paper studies locally supported piecewise linear prewavelets on bounded triangulations of arbitrary topology it is shown that a concrete choice of prewavelets form a basis of the wavelet space when the degree of the vertices in the triangulation is not too high

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

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 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

a counterexample to a 1961 theorem in homological algebra in 1961 jan erik roos published a theorem which says that in an ab4 abelian category lim1 vanishes on mittag leffler sequences see propositions 1 and 5 in 4 this is a theorem that many people since have known and used in this article we outline a counterexample we construct some strange abelian categories which are perhaps of some independent interest these abelian categories come up naturally in the study of triangulated categories a much fuller discussion may be found in 3 here we provide a brief self contained non technical account the idea is to make the counterexample easy to read for all the people who have used the result in their work in the appendix deligne gives another way to look at the counterexample

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

multivariate spline space over cross cut partition in this paper we mainly study the generator bases of the prime module determined by the global conformality condition of multivariate spline over star cross cut partition the results can help us to get the dimension and construct the spline bases for multivariate spline space over cross cut partition quasi cross cut partition and super cross cut partition

on the approximation power of splines on triangulated quadrangulations we study the approximation properties of the bivariate spline spaces srt cal s r 3r hbox diamondsuit kern 0 78em raise0 10786ex hbox of smoothness r and degree 3r defined on triangulations lai hbox diamondsuit kern 0 78em raise0 10786ex hbox which are obtained from arbitrary nondegenerate convex quadrangulations by adding the diagonals of each quadrilateral

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 putative doubly even 72 36 16 code does not have an automorphism of order 9 in this paper we prove that there does not exist a binary self dual doubly even code with an automorphism of order 9 to do so we apply a method for constructing binary self dual codes possessing an automorphism of order for an odd prime