properly embedded minimal planar domains in 1997 collin proved that any properly embedded minimal surface in r 3 with nite topology and more than one end has

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

embedded minimal surfaces with an infinite number of ends we prove the existence of an infinite family of periodic properly embedded minimal surfaces with an infinite number of annular ends 25 refs 29 figs

on singly periodic minimal surfaces with planar ends the spaces of nondegenerate properly embedded minimal surfaces in quotients of r3 by nontrivial translations or by screw motions with nontrivial rotational part fixed finite topology and planar type ends are endowed with natural structures of finite dimensional real analytic manifolds this nondegeneracy is defined in terms of jacobi functions riemann s minimal examples are characterized as the only nondegenerate surfaces with genus one in their corresponding spaces we also give natural immersions of these spaces into certain complex euclidean spaces which turn out to be lagrangian immersions with respect to the standard symplectic structures

degree theory on oriented infinite dimensional varieties and the morse number of minimal surfaces spanning a curve in ℝn the algebraic number of disc minimal surfaces spanning a wire in 3 is defined and shown to be equal to one

null controllability of a class of functional differential systems we consider a class of functional differential equations with both discrete and distributed delays distributed delays are represented by integrals whose bounds are commensurate with the discrete delays and by weight functions which are impulse responses of finite dimensional time invariant linear systems the properties of such systems can be studied by using simple algebraic techniques polynomials in one variable under the natural assumption of spectral controllability we give an explicit algebraic algorithm to compute a control function which steers any given initial datum to the equilibrium position in finite time this algorithm also constitutes a direct constructive and simple proof that spectral controllability implies null controllability for the class of systems considered the algorithm requires after the computation of the roots of a polynomial in one variable a finite number of steps

on the existence of a certain 64 32 12 extremal code j h conway and n j a sloane see ibid vol 36 no 6 p 1319 33 nov 1990 give weight enumerators of several self dual codes with the highest possible minimal distance whose existence was not known a generator matrix for one of these a type i 64 32 12 code is given proving its existence the method of construction is described

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

the structure of singly periodic minimal surfaces the only properly embedded minimal surface with more than one topological end and a cont inuous g roup of symmetries is the catenoid considering periodic minimal surfaces with more than one end a surface in n 3 is periodic if it is invariant under an infinite discrete g roup of isometrics that act freely on 1r3 the only classical examples are those discovered by riemann the riemann examples possess a quite special set of properties 1 they have an infinite number of flat annular ends 1 2 they are invariant under a nontrivial t ranslat ion t 3 the surfaces t h a v e genus 1 and 2 ends 4 the surfaces t have total curvature equal to 8 zr mot iva ted by the riemann examples we established the existence of an infinite family of proper ly embedded periodic minimal surfaces d k each with an infinite number of ends 1 these surfaces which are discussed below see sect 1 2 on rigidity also have an infinite number of flat annular ends and are invariant under a t ranslat ion t the quot ient of these surfaces by t is a surface of genus 2 k 1 with two ends and total curvature 8 r e k 1 k l we have been able to twis t each of these new surfaces that is we construct surfaces with the same properties that are invariant under screw motions t r where r is a rota t ion a round an axis in the direction of t see sect 1 3 below together with the catenoid and the examples of riemann these new surfaces comprise all the known properly embedded minimal surfaces with more than one topological end whose symmetry g roup contains an infinite discrete subgroup these examples share m a n y of the properties of the riemann examples a major goal of this paper is to show that this is not accidental

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

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

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

cubic self dual binary codes we study binary self dual codes with a fixed point free automorphism of order three all binary codes of that type can be obtained by a cubic construction that generalizes turyn s we regard such cubic codes of length 3 spl lscr as codes of length spl lscr over the ring f sub 2 spl times f sub 4 classical notions of type ii codes shadow codes and weight enumerators are adapted to that ring two infinite families of cubic codes are introduced new extremal binary codes in lengths spl les 66 are constructed by a randomized algorithm necessary conditions for the existence of a cubic 72 36 16 type ii code are derived

the strong halfspace theorem for minimal surfaces d hoffman and w h meeks iii department of mathematics university of massachusetts amherst ma 01003 usa in 6 jorge and xavier constructed examples of complete nonplanar minimal surfaces contained between two parallel planes in 3 recently rosenberg and toubiana have found complete minimally immersed annuli that are proper in an open slab 1 12 on the basis of these and other results the question has arisen as to whether or not there could exist a properly immersed minimal surface in 3 that was contained between two parallel planes the answer is negative an immediate consequence of the following theorem

the classical theory of minimal surfaces we present here a survey of recent spectacular successes in classical minimal surface theory we highlight this article with the theorem that the plane the helicoid the catenoid and the one parameter family frtgt2 0 1 of riemann minimal examples are the only complete properly embedded minimal planar domains in r 3 the proof of this result depends primarily on work of colding

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

riemann bilinear relations on minimal surfaces from a physical point of view minimal surfaces in r are objects submitted to a balanced force system consisting in the forces associated to non zero onedimensional homology classes in the surface several geometric properties can be studied in terms of those forces as embeddedness symmetries and deformations leading up to uniqueness and non existence results see 11 13 more precisely each closed curve in a minimal surface s r carries a force that expresses the stress produced by an unit conormal vector field along on the whole surface the action of this conormal field provides a tendency of translation or linear momentum f and another one of rotation around an axis or angular momentum m the first one is given by the force vector

some uniqueness and nonexistence theorems for embedded minimal surfaces thanks to the works of callahan costa hoffman karcher meeks rosenberg wei etc see for example 1 2 3 5 8 11 12 16 20 we dispose now of a large number of properly embedded minimal surfaces in the euclidean space ir other than the classical examples all those surfaces can be viewed as minimal surfaces with finite total curvature properly embedded in complete flat three manifolds the most basic invariants associated to a surface of this type are its topology and its periodicity it is an interesting problem to decide if the simplest examples like the catenoid the helicoid the scherk s surfaces or the riemann example can be characterized in terms of some of these invariants in the non periodic case there are two important uniqueness theorems in this direction the first one obtained by schoen 17 characterizes the catenoid as the only complete minimal surface embedded

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

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 covering radius of extremal self dual code d11 and its application d11 constructed by conway and sloane is an extremal self dual 56 28 12 code with weight enumerator 1 8190y sup 12 622314y sup 16 11699688y sup 20 we find that the covering radius of d11 is 10 this shows that the inequality r c sub 56 spl les 10 cannot be improved any more where r c sub 56 is the covering radius of any extremal doubly even 56 28 12 code c sub 56 and we use it to get ten new 58 29 10 codes

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