duality for modules over finite rings and applications to coding theory this paper sets a foundation for the study of linear codes over finite rings the finite frobenius rings are singled out as the most appropriate for coding theoretic purposes because two classical theorems of mac williams the extension theorem and the mac williams identities generalize from finite fields to finite frobenius rings it is over frobenius rings that certain key identifications can be made between the ring and its complex characters

generic mean curvature flow i generic singularities it has long been conjectured that starting at a generic smooth closed embedded surface in r 3 the mean curvature ow remains smooth until it arrives at a singularity in a neighborhood of which the ow looks like concentric spheres or cylinders that is the only singularities of a generic ow are spherical or cylindrical we will address this conjecture here and in a sequel the higher dimensional case will be addressed elsewhere the key to showing this conjecture is to show that shrinking spheres cylinders and planes are the only stable self shrinkers under the mean curvature ow we prove this here in all dimensions an easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away

triangular norms position paper ii general constructions and parameterized families this second part out of three of a series of position papers on triangular norms for part i see triangular norms position paper i basic analytical and algebraic properties fuzzy sets and systems in press deals with general construction methods based on additive and multiplicative generators and on ordinal sums also included are some constructions leading to non continuous t norms and a presentation of some distinguished families of t norms

on the algebraic structure of quasi cyclic codes i finite fields a new algebraic approach to quasi cyclic codes is introduced the key idea is to regard a quasi cyclic code over a field as a linear code over an auxiliary ring by the use of the chinese remainder theorem crt or of the discrete fourier transform dft that ring can be decomposed into a direct product of fields that ring decomposition in turn yields a code construction from codes of lower lengths which turns out to be in some cases the celebrated squaring and cubing constructions and in other cases the u spl upsi u spl upsi and vandermonde constructions all binary extended quadratic residue codes of length a multiple of three are shown to be attainable by the cubing construction quinting and septing constructions are introduced other results made possible by the ring decomposition are a characterization of self dual quasi cyclic codes and a trace representation that generalizes that of cyclic codes

how many weights can a quasi cyclic code have we investigate the largest number of nonzero weights of quasi cyclic codes in particular we focus on the function gamma q n ell k q that is defined to be the largest number of nonzero weights a quasi cyclic code of index gcd ell n length n and dimension k over mathbb f q can have and connect it to similar functions related to linear and cyclic codes we provide several upper and lower bounds on this function using different techniques and studying its asymptotic behavior moreover we determine the smallest index for which a q ary reed muller code is quasi cyclic a result of independent interest

curvature evolution of nonconvex lens shaped domains we study the curvature flow of planar nonconvex lens shaped domains con sidered as special symmetric networks with two triple junctions we show that the evolv ing domain becomes convex in finite time then it shrinks homothetically to a point as proved in 22 our theorem is the analog of the result of grayson 13 for curvature flow of closed planar embedded curves

triangular norms position paper iii continuous t norms this third and last part of a series of position papers on triangular norms for parts i and ii see e p klement r mesiar e pap triangular norms position paper i basic analytical and algebraic properties fuzzy sets and systems in press e p klement r mesiar e pap triangular norms position paper ii general constructions and parameterized families submitted for publication presents the representation of continuous archimedean t norms by means of additive generators and the representation of continuous t norms by means of ordinal sums with archimedean summands both with full proofs finally some consequences of these representation theorems in the context of comparison and convergence of continuous t norms and of the determination of continuous t norms by their diagonal sections are mentioned

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

asymptotic behavior and stability of mean curvature flow with a conical end abstract if the initial hypersurface of an immortal mean curvature flow is asymptotic to a regular cone whose entropy is small the flow will become asymptotically self expanding moreover the expander that gives rise to the limiting flow is asymptotically stable as an equilibrium solution of the normalized mean curvature flow

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

cyclic codes and self dual codes over f sub 2 uf sub 2 we introduce linear cyclic codes over the ring f sub 2 uf sub 2 0 1 u u u 1 where u sup 2 0 and study them by analogy with the z sub 4 case we give the structure of these codes on this new alphabet self dual codes of odd length exist as in the case of z sub 4 codes unlike the z sub 4 case here free codes are not interesting some nonfree codes give rise to optimal binary linear codes and extremal self dual codes through a linear gray map

some results on type iv codes over z4 dougherty gaborit harada munemasa and sole see ibid vol 45 p 2345 60 1999 have previously given an upper bound on the minimum lee weight of a type iv self dual z sub 4 code using a similar bound for the minimum distance of binary doubly even self dual codes we improve their bound finding that the minimum lee weight of a type iv self dual z sub 4 code of length n is at most 4 n 12 except when n 4 and n 8 when the bound is 4 and n 16 when the bound is 8 we prove that the extremal binary doubly even self dual codes of length n spl ges 24 n spl ne 32 are not z sub 4 linear we classify type iv i codes of length 16 we prove that all type iv codes of length 24 have minimum lee weight 4 and minimum hamming weight 2 and the euclidean optimal type iv i codes of this length have minimum euclidean weight 8

symmetry of minimizers of a gaussian isoperimetric problem we study an isoperimetric problem described by a functional that consists of the standard gaussian perimeter and the norm of the barycenter the second term is in competition with the perimeter balancing the mass with respect to the origin and because of that the solution is not always the half space we characterize all the minimizers of this functional when the volume is close to one by proving that the minimizer is either the half space or the symmetric strip depending on the strength of the barycenter term as a corollary we obtain that the symmetric strip is the solution of the gaussian isoperimetric problem among symmetric sets when the volume is close to one as another corollary we obtain the optimal constant in the quantitative gaussian isoperimetric inequality

construction of an extremal self dual code of length 62 we construct an extremal singly even self dual 62 31 12 code the existence of such a self dual code was not previously known combined with known results the above code also determines all lengths for which there are self dual codes with minimum weight 12

optimal self dual codes over f sub 2 spl times f sub 2 with respect to the hamming weight in this paper we study optimal self dual codes and type iv self dual codes over the ring f sub 2 spl times f sub 2 of order 4 we give improved upper bounds on minimum hamming and lee weights for such codes using the bounds we determine the highest minimum hamming and lee weights for such codes of lengths up to 30 we also construct optimal self dual codes and type iv self dual codes

compactness of self shrinkers in r3 with fixed genus abstract we prove the compactness of self shrinkers in r 3 with bounded entropy and fixed genus as a corollary we show that numbers of ends of such surfaces are uniformly bounded by the entropy and genus

currents and flat chains associated to varifolds with an application to mean curvature flow we prove under suitable hypotheses that convergence of integral varifolds implies convergence of associated mod 2 flat chains and subsequential convergence of associated integer multiplicity rectifiable currents the convergence results imply restrictions on the kinds of singularities that can occur in mean curvature flow

there is no 24 12 10 self dual quaternary code an attempt at directly constructing a 24 12 10 extremal quaternary code is reported the attempt at directly constructing the generator matrix of the code is detailed it was found that no such code exists

decoding of cyclic codes over f sub 2 uf sub 2 we give a simple decoding algorithm to decode linear cyclic codes of odd length over the ring r f sub 2 uf sub 2 0 1 u u u 1 where u sup 2 0 a spectral representation of the cyclic codes over r is given and a bch like bound is given for the lee distance of the codes the ring r shares many properties of z sub 4 and f sub 4 and admits a linear gray map

type iv self dual codes over rings we study type iv self dual codes over the commutative rings of order 4 gleason type theorems of type iv codes and their shadow codes are investigated a mass formula of type iv codes over these rings are given we give a classification of type tv codes over z sub 4 and f2 uf sub 2 for reasonable lengths we also construct a number of optimal type tv codes

mass formulas for self dual codes over z sub 4 and f sub q uf sub q rings we give a mass formula for quaternary self dual and type ii codes and for self dual codes over f sub q uf sub q rings we define type ii codes and compute the number of distinct type ii codes we also give formulas for the number of doubly even binary codes necessary to compute n sub d n the number of distinct codes of length n for any dimension n and conclude that the classification of quaternary self dual codes could be reasonably extended to length 14

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

uniqueness of self similar shrinkers with asymptotically conical ends in this paper we show the uniqueness of smooth embedded selfshrinkers asymptotic to generalized cylinders of infinite order also we construct non rotationally symmetric self shrinking ends asymptotic to generalized cylinders with rate as fast as any given polynomial

parabolic equations for curves on surfaces 2 intersections blow up and generalized solutions we describe a theory for parabolic equations for immersed curves on surfaces which generalizes the curve shortening or flow by mean curvature problem as well as several models in the theory of phase transitions in two dimensions a class of equations is described for which the initial value problem is well posed for rough initial data for which one can give a description of the way a smooth solution becomes singular and for which one can define generalized solutions i e solutions which are smooth except at a discrete set of times the methods which are used in this paper are more geometrical than those of part i by comparing arbitrary solutions with certain special solutions and by considering the way they intersect we derive estimates for the curvature and the tangent which allow one to study the initial value problem and the way solutions become singular

repeatedly smooting discrete scale space evolution and dominant point detection in this paper the fourier analysis is used to derive the properties of an evolving curve an arbitrary kernel repeatedly smoothing akrs evolution of a curve is then introduced it is shown that when the repeated number is large the akrs evolution of a curve is an approximately discrete implementation of the scale based evolution of this curve in the euclidean space as a special case an exponential repeatedly smoothing is proposed to implement the scale based evolution it is shown that in addition to its simple implementation and its desired approximation to a gaussian kernel an exponential function is a function that when it is selected as a repeatedly smoothing kernel the motion both magnitude and direction of a point on a curve from the i 1 th to ith instant is equal to the curvature of the ith smoothed curve at this point finally a perimeter controlled evolution method is proposed to extract dominant points it is shown experimentally that the proposed method is robust to noise object rotation and object changes in sizes

fuzzy t neighbourhood spaces part 1 t proximities this is the first one of three articles that are intended to be published consecutively in this journal together they constitute a three part work on the unified subject of fuzzy t neighbourhood spaces and their fuzzy t proximities where t stands for any continuous triangular norm these new notions generalize to arbitrary t corresponding ones due to artico hohle lowen and moresco in this part we define and study notions of fuzzy t proximity one for each t their definition subsumes that of fuzzy proximity due to artico and moresco fuzzy sets and systems 21 1987 85 as our min proximity in particular we study proximity maps the fuzzy topological space associated with a t proximity which in the second part will be proved a fuzzy t neighbourhood space t proximities as fuzzy relations in ordinary power sets t proximal neighbourhood systems and relationships to the hohle lowen fuzzy t uniformities j math anal appl 82 1981 370 manuscripta math 38 1982 289 the second part of this three part work will be concerned with the categories of fuzzy t neighbourhood spaces one for each t whereas we devote the third part to our t separation axioms in those categories

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

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

constructions in the category of fuzzy t proximity spaces abstract this paper is devoted to study the initial fuzzy t proximity structures of the fuzzy t proximity spaces defined by hashem and morsi in 2002 where t stands for any continuous triangular norm in this paper we show that all initial and final lifts in the category t fps of these fuzzy t proximity spaces exist and hence the initial and final fuzzy t proximity structures exist we introduce a characterization for the initial fuzzy t proximity structures so as a special initial fuzzy t proximity spaces the subspaces and product spaces of these fuzzy t proximity spaces can also be characterized we also show that the fuzzy topology associated to the initial fuzzy t proximity structure of a family of fuzzy t proximity spaces coincides with the initial fuzzy topology of the family of fuzzy topologies associated to these fuzzy t proximity spaces

some results on cyclic codes over f _ 2 v f _ 2 in this paper we investigate the structure and properties of cyclic codes over the ring f 2 vf 2 we first study the relationship between cyclic codes over f 2 vf 2 and binary cyclic codes then we prove that cyclic codes over the ring are principally generated and give the generator polynomial of cyclic codes over the ring finally we obtain the unique idempotent generators for cyclic codes of odd length and determine the number of cyclic codes for a given length n over f 2 vf 2

construction of optimal type iv self dual codes over f sub 2 uf sub 2 previously type iv self dual codes over the ring f sub 2 uf sub 2 of order 4 have been introduced in this paper we construct five optimal type iv self dual codes which are the first examples of such codes for that length we also give several new optimal type iv self dual codes and type iv self dual codes which improve the upper bounds for minimum weights

a sharp lower bound for the entropy of closed hypersurfaces up to dimension six the entropy is a natural geometric quantity which measures the complexity of a hypersurface in mathbb r n 1 it is non increasing along the mean curvature flow and so plays a significant role in analyzing the dynamics of this flow in colding et al j differ geom 95 1 53 69 2013 colding ilmanen minicozzi white showed that within the class of closed smooth self shrinking solutions of the mean curvature flow in mathbb r n 1 the entropy is uniquely minimized at the round sphere they conjectured that for 2 le n le 6 the round sphere minimizes the entropy among all closed hypersurfaces using an appropriate weak mean curvature flow we prove their conjecture for these dimensions our approach also gives a new proof of the main result of colding et al j differ geom 95 1 53 69 2013 and extends its conclusions to compact singular self shrinking solutions

quaternary quadratic residue codes and unimodular lattices we construct new self dual and isodual codes over the integers module 4 the binary images of these codes under the gray map are nonlinear but formally self dual the construction involves hensel lifting of binary cyclic codes quaternary quadratic residue codes are obtained by hensel lifting of the classical binary quadratic residue codes repeated hensel lifting produces a universal code defined over the 2 adic integers we investigate the connections between this universal code and the codes defined over z sub 4 the composition of the automorphism group and the structure of idempotents over z sub 4 we also derive a square root bound on the minimum lee weight and explore the connections with the finite fourier transform certain self dual codes over z sub 4 are shown to determine even unimodular lattices including the extended quadratic residue code of length q 1 where q spl equiv 1 mod8 is a prime power when q 23 the quaternary golay code determines the leech lattice in this way this is perhaps the simplest construction for this remarkable lattice that is known

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

min max theory and the willmore conjecture in 1965 t j willmore conjectured that the integral of the square of the mean curvature of a torus immersed in r 3 is at least 2 2 we prove this conjecture using the min max theory of minimal surfaces

a local regularity theorem for mean curvature flow this paper proves curvature bounds for mean curvature flows and other related flows in regions of spacetime where the gaussian densities are close to 1

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

a mass formula and rank of spl zopf sub 4 cyclic codes of length 2 sup e in this correspondence we study cyclic codes of length n 2 sup e over the ring r sub 4 spl zopf sub 4 x x sup n 1 in particular we derive a closed form expression for the number of these codes for a given length n we also study the rank of these codes and derive an expression for that furthermore we give an example in which we study all cyclic codes of length 8 we also study all self dual codes of length 8 and 16 and classify them according to their type

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

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

a linear construction for certain kerdock and preparata codes the nordstrom robinson kerdock and slightly modified preparata codes are shown to be linear over z4 the integers mod 4 the kerdock and preparata codes are duals over z4 and the nordstrom robinson code is self dual all these codes are just extended cyclic codes over z4 this provides a simple definition for these codes and explains why their hamming weight distributions are dual to each other firstand second order reed muller codes are also linear codes over z4 but hamming codes in general are not nor is the golay code

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

embedded self similar shrinkers of genus 0 we conrm a well known conjecture that the round sphere is the only compact embedded self similar shrinking solution of mean curvature ow in r 3 with genus 0 more generally we show that the only properly embedded self similar shrinkers in r 3 with vanishing intersection form are the sphere the cylinder and the plane this answers two questions posed by t ilmanen

maximum distance codes over rings of order 4 in this correspondence we study bounds on the euclidean hamming lee and bachoc weights of codes over rings of order 4 similar to the singleton bound and investigate the relationship between these bounds moreover we give some characterizations of the codes meeting these bounds

symmetries and invariant solutions for the geometric heat flows we study lie symmetries and invariant solutions of the geometric heat flows the basic similarity reductions for the ghe are performed reduced equations and exact solutions associated with the symmetries are obtained group invariant solutions and reductions for the affine case are also discussed in a special case

uniqueness of compact tangent flows in mean curvature flow we show for mean curvature flows in euclidean space that if one of the tangent flows at a given space time point consists of a closed multiplicity one smoothly embedded self similar shrinker then it is the unique tangent flow at that point that is the limit of the parabolic rescalings does not depend on the chosen sequence of rescalings furthermore given such a closed multiplicity one smoothly embedded self similar shrinker we show that any solution of the rescaled flow which is sufficiently close to with gaussian density ratios greater or equal to that of stays for all time close to and converges to a possibly different self similarly shrinking solution the central point in the argument is a direct application of the ojasiewicz simon inequality to huisken s monotone gaussian integral for mean curvature flow de gruyter 2014

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

motion of level sets by mean curvature ii we give a new proof of short time existence for the classical motion by mean curvature of a smooth hypersurface our method consists in studying a fully nonlinear uniformly parabolic equation satisfied by the signed distance function to the surface

on the stability of m fold circles and the dynamics of generalized curve shortening flows abstract in this paper we study the asymptotic and exponential stability of the m fold circle as a solution of the p curve shortening flow p 1 an integer

cyclic codes over r f p uf p u k 1 f p with length p s n in this paper all cyclic codes with length p sn n prime to p over the ring r f p uf p u k 1f p are classified it is first proved that tor j c is an ideal of s f p m w so that the structure of ideals over extension ring s u k m w gr u k m w is determined then an isomorphism between r x and a direct sum h is u k m h w can be obtained using discrete fourier transform the generator polynomial representation of the corresponding ideals over f p uf p u k 1f p is calculated via the inverse isomorphism moreover torsion codes ms polynomial and inversion formula are described

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