nearest neighbor algorithm for spherical codes from the leech lattice the leech lattice is a regular arrangement of points in 24 dimensional euclidean space that yields an extremely dense packing when equal spheres are centered at these points a subset of the leech lattice can be used as a signal set for the gaussian channel or as representative vectors for a vector quantizer of particular interest are the spherical codes or code books that consist of the points of the leech lattice which lie on a sphere centered at the origin the code points do not have to be stored because they can be obtained from a very small set of basic vectors using permutations of the components in a manner dictated by the words of the extended golay code a nearest neighbor algorithm that works on this is developed to determine the point in the code closet to some arbitrary vector in r sup 24 the performance of this approach when quantizing independent identically distributed gaussian samples is reported

buckling and postbuckling of the lying sheet abstract the buckling and postbuckling of a finite heavy elastic sheet is studied the sheet originally lies on a horizontal plane one end clamped and one end subjected to a compressive load depending on the magnitude of the normalized force f the lying sheet may be regarded long or short stability analyses in both cases show the buckling behavior is quite different from the euler column theory a modified critical buckling load is defined the analytic results compare well with exact numerical integration

a fast encoding method for lattice codes and quantizers in an earlier paper the authors described a very fast method which for the root lattices a n d n e n their duals and certain other lattices finds the closest lattice point to an arbitrary point of the underlying space if the lattices are used as codes for a gaussian channel the algorithm provides a fast decoding procedure or if they are used as vector quantizers the algorithm performs the analog to digital conversion efficiently the present paper offers a solution to the inverse problem for the same lattices the encoding problem for channel codes or the digital to analog part of quantizing namely given an integer k to find the kth code vector and to the closely related problem of finding the index k of a given code vector

a gaussian sinc collocation approach for a whipping cantilever with a follower shear force at its tip a spatial discretization scheme based on a set of gaussian sinc functions is proposed for the temporal projection of a set of partial differential equations pdes describing the non linear dynamics of an elastic plastic hardening softening ephs cantilever subjected to a follower shear force at its tip the dynamics so described correspond to planar whipping of a pipe conveying fluid ruptured near a right angled bend the constitutive ephs moment curvature relationship used here follows the earlier work of reid et al proceedings of the royal society of london series a 1998 454 997 1029 compared to the more classical lagrangian polynomial based collocation functions the gaussian sine functions have better localization properties moreover for a relatively large number of collocation points use of such functions does not lead to numerical overflow or underflow problems often associated with the use of higher order polynomial collocation functions the spatial discretization leads to a set of non linear ordinary differential equations odes in time which are in turn integrated via a fourth order adaptive runge kutta scheme some numerical results for a cantilever whipping in a plane are presented to further illustrate the present approach the method is a step forward towards the development of a mesh free non linear beam element suitable for dynamic analyses of pipe networks and pipe on pipe impact problems

the curved beam deep arch finite ring element revisited an attempt is made to understand the errors arising in curved finite elements which undergo both flexural and membrane deformation it is shown that with elements of finite size i e a practical level of discretization at which reasonably accurate results can be expected there can be errors of a special nature that arise because the membrane strain fields are not consistently interpolated with terms from the two independent field functions that characterize such a problem these lead to errors described here as of the second kind and a physical phenomenon called membrane locking it seems possible to determine optimal integration rules that will allow the extensional deformation of a curved beam deep arch finite ring element to be modelled by independently chosen low order polynomial functions and which will recover the inextensional case in the penalty limit of extreme thinness without spurious locking constraints what is emphasized is that the choice of shape functions or subsequent operations to determine the discretized functionals must consistently model the physical requirements the problem imposes on the field variables

a dynamically adaptive multilevel wavelet collocation method for solving partial differential equations in a finite domain a dynamically adaptive multilevel wavelet collocation method is developed for the solution of partial differential equations the multilevel structure of the algorithm provides a simple way to adapt computational refinements to local demands of the solution high resolution computations are performed only in regions where sharp transitions occur the scheme handles general boundary conditions the method is applied to the solution of the one dimensional burgers equation with small viscosity a moving shock problem and a nonlinear thermoacoustic wave problem the results indicate that the method is very accurate and efficient

on the time fractional navier stokes equations abstract this paper is concerned with the navier stokes equations with time fractional derivative of order 0 1 this type of equations can be used to simulate anomalous diffusion in fractal media we establish the existence and uniqueness of local and global mild solutions in h q meanwhile we also give local mild solutions in j q moreover we prove the existence and regularity of classical solutions for such equations in j q

soft decoding techniques for codes and lattices including the golay code and the leech lattice two kinds of algorithms are considered 1 if is a binary code of length n a soft decision decoding algorithm for changes an arbitrary point of r n into a nearest codeword nearest in euclidean distance 2 similarly a decoding algorithm for a lattice lambda in r n changes an arbitrary point of r n into a closest lattice point some general methods are given for constructing such algorithms ami are used to obtain new and faster decoding algorithms for the gosset lattice e 8 the golay code the leech lattice

classification of positive definite lattices 1 classification of positive norm vectors 525 2 vectors in the lattice ii1 25 528 3 lattices with no roots 534 the primitive norm 0 vectors of ii1 25 537 the norm 2 vectors of ii1 25 539 the norm 4 vectors of ii1 25 544

the monster lie algebra abstract we calculate the multiplicities of all the roots of the monster lie algebra this gives an example of a lie algebra all of whose simple roots and root multiplicities are known and which is not finite dimensional or an affine kac moody algebra there seem to be several similar infinite dimensional lie algebras which to a sympathetic eye appear to correspond to some of the sporadic simple groups

uniform global well posedness of the navier stokes coriolis system in a new critical space we prove global well posedness for the navier stokes coriolis system nsc in a critical space whose definition is based on fourier transform namely the fourier besov morrey space mathcal fn 1 mu infty mu 1 with 0 mu 3 the smallness condition on the initial data is uniform with respect to the angular velocity omega our result provides a new class for the uniform global solvability of nsc and covers some previous ones for mu 0 nsc is ill posedness in mathcal fn 1 mu infty mu 1 which shows the optimality of the results with respect to the space parameter mu 0 the lack of hausdorff young inequality in morrey spaces suggests that there are no inclusions between mathcal fn 1 mu infty mu 1 and the largest previously known existence classes of kozono yamazaki besov morrey space and koch tataru textit bmo 1 for navier stokes equations 3dns so taking in particular omega 0 we obtain a critical initial data class that seems to be new for global existence of solutions of 3dns

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

a new upper bound for the minimum of an integral lattice of determinant 1 let be an n dimensional integral lattice of determinant 1 we show that for all sufficiently large n the minimal nonzero squared length in does not exceed n 6 10 this bound is a consequence of some new conditions on the theta series of these lattices these conditions also enable us to find the greatest possible minimal squared length in all dimensions n 33 in particular we settle the no roots problem there is a determinant 1 lattice containing no vectors of squared length 1 or 2 precisely when n 23 n 25 there are also analogues of all these results for codes

a three dimensional element free galerkin elastic and elastoplastic formulation a small strain three dimensional elastic and elastoplastic element free galerkin efg formulation is developed singular weight functions are utilized in the moving least squares mls determination of shape functions and shape function derivatives allowing accurate direct nodal imposition of essential boundary conditions a variable domain of influence efg method is introduced leading to increased efficiency in computing the mls shape functions and their derivatives the elastoplastic formulations are based on the consistent tangent operator approach and closely follow the incremental formulations for non linear analysis using finite elements several linear elastic and small strain elastoplastic numerical examples are presented to verify the accuracy of the numerical formulations copyright 1999 john wiley sons ltd

on dispersive effect of the coriolis force for the stationary navier stokes equations abstract the dispersive effect of the coriolis force for the stationary and non stationary navier stokes equations is investigated existence of a unique solution is shown for arbitrary large external force provided the coriolis force is large enough in addition to the stationary case counterparts of several classical results for the non stationary navier stokes problem have been proven the analysis is carried out in a new framework of the fourier besov spaces

well posedness for the navier stokes equations where u is the velocity and p is the pressure it is well known that the navierstokes equations are locally well posed for smooth enough initial data as long as one imposes appropriate boundary conditions on the pressure at for instance it is easy to see see 9 for much more general results that if s n 2 then for any h initial data there exists a unique c 0 t h r local solution with a pressure p c 0 t h r in the sequel we consider solutions for less regular initial data this has to be understood in the sense that the map from the initial data to the solution extends continuously to rougher function spaces the question we are interested in is the global well posedness for small data and local well posedness for large data with respect to a certain space of

orbit and coset analysis of the golay and related codes let b be a code of length n over a field f with automorphism group g b sub w denotes the subset of codewords of weight w the goal is to classify the vectors of f sup n into orbits under g and to determine their distances from the various subcodes b sub w this is done for the first order reed muller nordstrom robinson and hamming codes of length 16 the golay and shortened golay codes of lengths 22 23 24 and the ternary golay code of length 12

on the covering radius of z sub 4 codes and their lattices in this correspondence we investigate the covering radius of codes over z sub 4 for the lee and euclidean distances in relation with those of binary nonlinear codes and lattices obtained by the gray map and construction a sub 4 respectively we give several upper and lower bounds on covering radii including z sub 4 analogs of the sphere covering bound the packing radius bound the delsarte bound and the redundancy bound we show that any euclidean optimal type ii code of length 24 has covering radius 8 with respect to the euclidean distance we determine the covering radius of the klemm codes with respect to the lee distance we derive lower bounds on the covering radii of the niemeier lattices

the covering radius of the leech lattice we investigate the points in 24 dimensional space that are at maximal distance from the leech lattice i e the deep holes in that lattice the distance of such a point from the leech lattice is 1 sqrt 2 times the minimal distance between the lattice points furthermore there are 23 inequivalent types of deep hole one for each of the 23 even unimodular 24 dimensional lattices found by niemeier

the automorphism group of the 26 dimensional even unimodular lorentzian lattice abstract the group mentioned in the title is shown to be a certain infinitely generated coxeter group extended by the negative of the identity operation and the group of all automorphisms of the notorious leech lattice

voronoi regions of lattices second moments of polytopes and quantization if a point is picked at random inside a regular simplex octahedron 600 cell or other polytope what is its average squared distance from the centroid in n dimensional space what is the average squared distance of a random point from the closest point of the lattice a n or d n e n a n ast or d n ast the answers are given here together with a description of the voronoi or nearest neighbor regions of these lattices the results have applications to quantization and to the design of signals for the gaussian channel for example a quantizer based on the eight dimensional lattice e8 has a mean squared error per symbol of 0 0717 cdots when applied to uniformly distributed data compared with 0 08333 cdots for the best one dimensional quantizer

convergence of the sinc method for a fourth order ordinary differential equation with an application we analytically show the exponential convergence of the sinc method applied to fourth order ordinary differential equations odes linear numerical examples are included to confirm our analytical results the solution obtained with the sinc method is compared with the one obtained with a second and a sixth order finite difference scheme and with the ode boundary value solver colnew in the case of the linear equations we then compute the steady state solution of the one dimensional cahn hilliard equation to illustrate our results

global solutions of two dimensional navier stokes and euler equations long time solutions to the navier stokes ns and euler e equations of incompressible flow in the whole plane are constructed under the assumption that the initial vorticity is in l1 2 for ns and in l1 2 lr 2 for some r 2 for e it is shown that the solution to ns is unique smooth and depends continuously on the initial data and that the velocity solution to e is holder continuous in the space and time coordinates it is shown that as the viscosity vanishes there is a subsequence of solutions to ns converging to a solution of e

a sinc collocation method for initial value problems a collocation procedure is developed for the initial value problem u t f t u t u 0 0 using the globally defined sinc basis functions it is shown that this sinc procedure converges to the solution at an exponential rate i e o m 2 exp m where 0 and 2m basis functions are used in the expansion problems on the domains r and r 0 are used to illustrate the implementation and accuracy of the procedure

complex and integral laminated lattices in an earlier paper we studied real laminated lattices or z modules a where a is the lattice of even integers and a is obtained by stacking layers of a suitable n l dimensional lattice a as densely as possible keeping the same minimal norm the present paper considers two generalizations a complex and quaternionic lattices obtained by replacing z module by module where j may be the eisenstein gaussian or hurwitzian integers etc and b integral laminated lattices in which a is required to be an integral lattice with the prescribed minimal norm this enables us to give a partial answer to a question of j g thompson on integral lattices and to explain some of the computer generated results of plesken and pohst on this problem also a number of familiar lattices now arise in a canonical way for example the coxeter todd lattice is the 6 dimensional integral laminated lattice over z to of minimal norm 2 the paper includes tables of the best real integral lattices in up to 24 dimensions

the leech lattice new proofs of several known results about the leech lattice are given in particular i prove its existence and uniqueness and prove that its covering radius is the square root of 2 i also give a uniform proof that the 23 holy constructions of the leech lattice all work

covering radius survey and recent results all known results on covering radius are presented as well as some new results there are a number of upper and lower bounds including asymptotic results a few exact determinations of covering radius some extensive relations with other aspects of coding theory through the reed muller codes and new results on the least covering radius of any linear n k code there is also a recent result on the complexity of computing the covering radius

the fujita kato approach to the navier stokes equations in the rotational framework consider the navier stokes equations in the rotational framework it is proved that these equations possess a unique global mild solution for arbitrary speed of rotation provided the initial data u 0 is small enough in the h frac12 sigma mathbb r 3 norm

numerical methods based on whittaker cardinal or sinc functions this paper summarizes the results known to date for using sine functions composed with other functions as bases for approximations in numerical analysis described in this paper are methods of interpolation and approximation of functions and their derivatives quadrature the approximate evaluation of transforms hilbert fourier laplace hankel and mellin and the approximate solution of differential and integral equations the methods have many advantages over classical methods which use polynomials as bases in addition all of the methods converge at an optimal rate if singularities on the boundary of approximation are ignored

global solutions for the navier stokes equations in the rotational framework the existence of global unique solutions to the navier stokes equations with the coriolis force is established in the homogeneous sobolev spaces dot h s mathbb r 3 3 for 1 2 s 3 4 if the speed of rotation is sufficiently large this phenomenon is so called the global regularity the relationship between the size of initial datum and the speed of rotation is also derived the proof is based on the space time estimates of the strichartz type for the semigroup associated with the linearized equations in the scaling critical space dot h frac 1 2 mathbb r 3 3 the global regularity is also shown

the 290 fixed point sublattices of the leech lattice abstract we determine the orbits of fixed point sublattices of the leech lattice with respect to the action of the conway group co 0 there are 290 such orbits detailed information about these lattices the corresponding coinvariant lattices and the stabilizing subgroups is tabulated in several tables

a new table of constant weight codes a table of binary constant weight codes of length nl28 is presented explicit constructions are given for most of the 600 codes in the table the majority of these codes are new the known techniques for constructing constant weight codes are surveyed and a table of unrestricted binary codes of length nl28 is given

twenty three constructions for the leech lattice in the previous chapter we classified the points at maximal distance from the leech lattice the deep holes in that lattice and showed that there are 23 classes of such holes the classes being in one to one correspondence with the 23 niemeier lattices in 24 dimensions we now present 23 constructions for the leech lattice one for each class of hole or niemeier lattice two of these are the usual constructions of the leech lattice from the golay codes over f 2 and f 3