existence uniqueness and l q estimates for the stokes problem in an exterior domain as is well known the study of the properties of solutions to the stokes system plays an essential role in the mathematical theory of viscous fluid flows governed by the navier stokes equations it is not art overstatement to say that existence uniqueness and a priori estimates in sobolev spaces for the stokes system are fundamental in every basic question related to the navier stokes equations such as existence regularity asymptotic behavior in time etc see e g ladyzhenskaya 1969 solonni ov 1977 and te am 1977 so far however the stokes system has been fully analyzed only for bounded domains see sobolevskn 1960 ladvznenskaya 1959 solonnu ov 1960 vorovlch yot oowcr 1961 cattabrma 1961 in particular in the paper of cattabriga a complete treatment is given in the sobotev spaces w l 1 1 q c for space dimension n 3 cattabriga s results essentially state that if 2 is a bounded domain in r a of class c m m max l 2 2 l 1 then given f e wt q y2 3 gc wt l q o q e wl 2 1 q q 8o a 1 q there exists one and only one distributional solution v c w1 z q q 3 p e wt l q o to the stokes system

on the radius of convergence of rayleigh schrodinger perturbative solutions for quantum oscillators in circular and spherical boxes the energy eigenvalues of harmonic oscillators in circular and spherical boxes are obtained through the rayleigh schrodinger perturbative expansion taking the free particle in a box as the non perturbed system the perturbative series is shown to be convergent for small boxes and an upper bound for the radius of convergence is established pade approximant solutions are also constructed for boxes of any size numerical comparison with the exact eigenvalues which are obtained by constructing and diagonalising the hamiltonian in the basis of the eigenfunctions of the free particle in a box corroborates the accuracy and range of validity of the approximate solutions particularly the convergence and the radius of convergence of the perturbative series

on k t subnormal covering codes the concept of a k t subnormal covering code is defined it is discussed how an amalgamated direct sumlike construction can be used to combine such codes the existence of optimal q n m 1 codes c is discussed such that by puncturing the first coordinate of c one obtains a code with q 1 subnorm 2

convergence radii of the rayleigh schrodinger perturbation series for the bounded oscillators the convergence radii of the rayleigh schrodinger perturbation series for some bounded oscillators are calculated the eigenvalues with quantum numbers n 0 and n 2 as well as those with n 1 and n 3 are found to cross at a pair of conjugate branch points of order two in the complex plane higher excited states are briefly discussed

very weak solutions of the stationary navier stokes equations for an incompressible fluid past obstacles abstract we consider the stationary motion of an incompressible navier stokes fluid past obstacles in r 3 subject to the given boundary velocity v b external force f div f and nonzero constant vector k e 1 at infinity our main result is the existence of at least one very weak solution v in k e 1 l 3 for arbitrary large f l 3 2 l 12 7 provided that the flux of v b k e 1 on the boundary of each body is sufficiently small with respect to the viscosity the uniqueness of very weak solutions is proved by assuming that f and v b k e 1 are suitably small moreover we establish weak and strong regularity results for very weak solutions in particular our existence and regularity results enable us to prove the existence of a weak solution v satisfying v l 3 2

modified bounds for covering codes the covering radius of binary codes is studied bounds on k n r the minimum cardinality of any binary code of length n and covering radius r are found modifications of the van wee lower bounds are proved for k n r the minimal number of codewords in any binary code of length n and covering radius r the first of the two van wee bounds is based on studying the hamming spheres of radius 1 centered at the points which have distance r to the code c the points covered by more than one codeword are divided into several classes and better estimates for some of these classes are obtained using a suitable averaging process the lower bound for k n r when r or 2 is improved the second van wee bound studies spheres of radius 2 centered at the points which have distance r 1 or r to the code c these points are divided essentially into two classes the points that are covered by only one codeword of c and the points that are covered by more than one codeword

further results on k t subnormal covering codes the concept of k t subnormal covering codes is discussed generalizing some of the earlier results in a similar way k t normal covering codes are defined using the results including some new constructions upper bounds for covering codes are improved it is shown how simulated annealing can be used to find acceptable partitions for codes

the covering radius of the 2 15 16 reed muller code is at least 16276 two interesting cosets of the first order reed muller code of block length 2 15 are described they provide counterexamples to a conjecture on the covering radius

on the stationary navier stokes equations in exterior domains this paper is concerned with the existence and uniqueness questions on weak solutions of the stationary navier stokes equations in an exterior domain in r3 where the external force is given by divf with f f x fji x i j 1 2 3 first we prove the existence and uniqueness of a weak solution for f l3 2 lp q with 3 2 p 3 and 1 q provided f l3 2 is sufficiently small here lp q denotes the well known lorentz space we next show that weak solutions satisfying the energy inequality are unique for f l3 2 l2 under the same smallness condition on f l3 2 this result provides a complete answer to the uniqueness question of weak solutions satisfying the energy inequality the existence of which was proved by leray in 1933 finally we establish the existence of weak solutions for data f in a very large class for instance in l3 2 l2 which generalizes leray s existence result

on the covering radius of codes the covering radius r of a code is the maximal distance of any vector from the code this work gives a number of new results concerning t n k the minimal covering radius of any binary code of length n and dimension k for example t n 4 and t n 5 are determined exactly and reasonably tight bounds on t n k are obtained for any k when n is large these results are found by using several new constructions for codes with small covering radius one construction the amalgamated direct sum involves a quantity called the norm of a code codes with norm leq 2 r 1 are called normal and may be combined efficiently the paper concludes with a table giving bounds on t n k for n leq 64

new lower bounds for binary covering codes g d chen et al ibid vol it 32 p 680 94 1986 presented two new lower bounds for k n r where k n r denotes the minimum cardinality of a binary code of length n and covering radius r the author shows that a slight modification gives further improvements and some examples are given to confirm the argument codes that have a certain partitioning property are considered

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

constructions of nonlinear covering codes constructions of nonlinear covering codes are given using any nonlinear starting code of covering radius r spl ges 2 these constructions form an infinite family of codes with the same covering radius a nonlinear code is treated as a union of cosets of a linear code new infinite families of nonlinear covering codes are obtained concepts of r l objects r l partitions and r l length are described for nonlinear codes

subcodes and covering radius corresp a sharp lower bound on the covering radius of a linear binary code is given in terms of the covering radius of any subcode of index two an upper bound on the norm of a code is then derived these results hold for all linear codes over gf 2

more binary covering codes are normal it is shown that every optimal binary code with covering radius r 1 is normal this parity proves a conjecture of cohen lobstein and sloane 1986 it is also proved that codes with minimal distance 2r or 2r 1 are normal a generalization of frankl s construction 1987 of abnormal codes is given

further results on the covering radius of codes a number of upper and lower bounds are obtained for k n r the minimal number of codewords in any binary code of length n and covering radius r several new constructions are used to derive the upper bounds including an amalgamated direct sum construction for nonlinear codes this construction works best when applied to normal codes and we give some new and stronger conditions which imply that a linear code is normal an upper bound is given for the density of a covering code over any alphabet and it is shown that k n 2 r 1 leq k n r holds for sufficiently large n

upper bounds for q ary covering codes new methods for constructing q ary covering codes are presented the author introduces the concepts of p seminormal and strongly p seminormal codes and shows how seminormal codes and punctured hamming codes can be combined to construct new covering codes using these methods upper bounds for ternary covering codes are improved the new bounds are k sub 3

very weak solutions and large uniqueness classes of stationary navier stokes equations in bounded domains of r2 extending the notion of very weak solutions developed recently in the three dimensional case to bounded domains r2 we obtain a new class of unique solutions u in lq q 2 to the stationary navier stokes system u u u p f divu k u g with data f k g of low regularity as a main consequence we obtain a new uniqueness class also for classical weak or strong solutions indeed such a solution is unique if its lq norm is sufficiently small or the data satisfy the uniqueness condition of a very weak solution

a 55 16 19 binary goppa code and related codes having large minimum distance a 55 16 19 binary goppa code is used to construct 57 17 17 58 17 18 59 17 19 and 60 17 20 codes the first two codes have smaller redundancy than previously known codes linear or nonlinear of the same length and minimum distance the last two codes have parameters previously attained only by nonlinear codes

a class of solutions to stationary stokes and navier stokes equations with boundary data in w 1 q q we develop a theory for a general class of very weak solutions to stationary stokes and navier stokes equations in a bounded domain with boundary of class c2 1 corresponding to boundary data in the distribution space w 1 q q 1 q these solutions exist and are unique for small data in the nonlinear case in their class of existence and satisfy a correponding estimate in terms of the data moreover they become regular if the data are regular to our knowledge the only existence result for solutions attaining such boundary data is due to giga 16 proposition 2 2 for the stokes case however the methods and the approach used in the present paper are different than giga s and cover more general issues including the nonlinear navier stokes equations and the precise way in which the boundary data are attained by the solutions we also introduce in the last section a further generalization of the solution class

l q estimates for the stationary oseen equations on exterior domains we study the dirichlet problem for the stationary oseen equations on exterior smooth domains in rn n 2 our main results are the existence and uniqueness of weak and very weak solutions of the oseen equations satisfying appropriate lq estimates the uniqueness of very weak solutions is shown by utilizing recent results for very weak solutions of the stokes equations on bounded domains then for n 1 n 1 q the existence and lq estimates of very weak solutions in lq n is deduced by a duality argument from our existence result and d2 r estimates for strong solutions these results combined with some regularity results for very weak solutions enable us to prove the existence uniqueness and d1 r estimates of weak solutions in d1 r n l n 1 r n 1 r n where n 1 n r n 1

lower bounds on t n k from linear inequalities the linear inequality method for covering codes is used to improve the lower bounds of t n k the smallest covering radius of any n k binary linear code to make better use of the strength of this method the relation between the covering radius of a code and the minimum distance of its dual code is studied the authors obtained 65 improved lower bounds for t n k within the range of n

on binary linear codes which satisfy the two way chain condition general constructions of binary linear codes satisfying the two way chain condition are found sufficient conditions for their existence are determined some of these codes are proven to be z sub 4 linear an expression for their state complexity profile is also found cases when such codes have covering radius r t n k are shown

do most binary linear codes achieve the goblick bound on the covering radius corresp the following two problems are dealt with p1 finding the smallest rate r of a binary code of length n admitting a prescribed covering radius rho n p2 discovering whether a majority of codes with any rate larger than r admits the given covering radius for the class of unrestricted nonlinear codes a solution to both problems is obtained by an elementary averaging argument the solution to p1 is r 1 h rho o n 1 log n and the answer to p2 is positive as for the more interesting class of linear codes goblick s extension method shows that the solution to p1 is the same as in the unrestricted case in contrast p2 seems to remain an open question a simple derivation of goblick s result is presented and a discussion is made of the positive conjecture concerning p2 for linear codes

inequalities for covering codes any code c with covering radius r must satisfy a set of linear inequalities that involve the lloyd polynomial l sub r x these generalize the sphere bound syndrome graphs associated with a linear code c are introduced to help keep track of low weight vectors in the same coset of c if there are too many such vectors c cannot exist illustrations show that t 17 10 3 and t 23 15 3 where t n k is the smallest covering radius of any n k code

the covering radius of cyclic codes of length up to 31 corresp the covering radius is given for all binary cyclic codes of length less than or equal to 31 many of these codes are optimal in the sense of having the smallest possible covering radius of any linear code of that length and dimension

new covering codes from an ads like construction a covering code construction is presented using this construction it is shown that t 52 39 3 t 36 21 4 t 58 32 7 k 32 2 spl les 62 spl middot 2 sup 18 and k 62 5 spl les 31 spl middot 2 sup 37 where t n k is the minimum covering radius among all binary n k codes and k n r is the minimum cardinality of a binary code of length n and covering radius r four new linear codes found by computer search are also given these include a 23 9 5 code a 32 8 10 code a 51 41 2 code and a 45 20 8 code

constructions for perfect mixed codes and other covering codes a construction for an infinite family of perfect mixed codes with covering radius 2 is presented these are the first known nontrivial perfect mixed codes with covering radius greater than 1 based on mixed codes constructions for binary covering codes that lead to a considerable improvement of upper bounds on the sizes of covering codes are presented these codes and some other codes can be obtained by the blockwise direct sum construction two infinite families of codes are of special interest they are quasi perfect nonlinear union of their disjoint translates covers the space and their density as covering codes is remarkably low

on the string consensus problem and the manhattan sequence consensus problem abstract we study the manhattan sequence consensus problem msc problem in which we are given k integer sequences each of length l and we are to find an integer sequence x of length l called a consensus sequence such that the maximum manhattan distance of x from each of the input sequences is minimized a related problem with hamming distance instead of manhattan distance is called hamming string consensus hsc also known under the names of string center problem or closest string problem for binary sequences manhattan distance coincides with hamming distance hence in this case hsc is a special case of msc we design a practically efficient o l time algorithm solving msc for k 5 sequences it improves upon the quadratic algorithm by amir et al 2012 1 for hsc for k 5 binary strings similarly as in the algorithm of amir et al we use a column based framework we replace the implied general integer linear programming by its easy special cases due to combinatorial properties of msc for k 5 practicality of our algorithms has been verified experimentally we also show that for a general parameter k any instance can be reduced in linear time to a kernel of size k so the problem is fixed parameter tractable nevertheless for k 4 this is still too large for any naive solution to be feasible in practice this is a full version of an article published at spire 2014 15

the existence and uniqueness of very weak solutions of the stationary boussinesq system abstract we consider the dirichlet problem for the stationary boussinesq system in a bounded smooth domain in r n where n 2 or 3 we first prove the existence of very weak solutions for arbitrarily large boundary data in w 1 q q w 1 r r provided that 3 q r if n 3 and 2 q r if n 2 the uniqueness of very weak solutions is also shown for sufficiently small data

on the optimal linear receiving filter for digital transmission over nonlinear channels a recent paper by e biglieri et al see ibid vol 35 p 620 5 may 1989 shows that the matched filter tapped delay line structure is optimum not only for linear pulse modulated signals and linear channel distortion but also for nonlinear finite alphabet pulse modulation and some nonlinear channel distortion this has important practical applications therefore its connection with other work reported in the literature is presented

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

explicit codes with low covering radius a series of explicit low rate binary linear codes that have relatively low covering radius and can be rapidly decoded is exhibited these codes can be derived from higher dimensional analogs of the gale berlekamp switching game conjectures of independent interest involving hadamard matrices are given that could yield semiexplicit covering codes of very low density

a removable isolated singularity theorem for the stationary navier stokes equations abstract we show that an isolated singularity at the origin 0 of a smooth solution u p of the stationary navier stokes equations is removable if the velocity u satisfies u l n or u x o x 1 as x 0 here n 3 denotes the dimension as a byproduct of the proof we also obtain a new interior regularity theorem

an improved sphere covering bound for the codes with n 3r 2 let c be a binary code not necessarily linear with covering radius r and length n 3r 2 the sphere covering bound on the cardinality of c is improved considerably provided c has minimal distance d 2 some new results on the function t n k the smallest covering radius of any binary linear code with length n and dimension k t 38 6 or 13 t 47 7 or 16 t 59 8 or 20 are given

singular points from taylor series a simple and accurate method is developed for calculating singular points from taylor series it consists of finding the least squares deviation of the taylor coefficients from a proposed asymptotic form sequences are obtained that converge quickly to the closest singularity to the origin some simple mathematical examples and physically interesting eigenvalue problems are discussed to illustrate the procedure the branch points of the eigenvalues for the solutions of period 2 of the mathieu equation and those of the stark shifts for rigid symmetric top molecules which were not obtained before are shown

bounds for abnormal binary codes with covering radius one the normality of binary codes is studied the minimum cardinality of a binary code of length n with covering radius r is denoted by k n r it is assumed that c is an n m r code that is a binary code of length n with m codewords and covering radius r it is shown that if c is an n m 1 code then it is easy to find a normal n m 1 code by changing c in a suitable way and that all the optimal n m 1 codes i e those for which m k n 1 are normal and their every coordinate is acceptable it is shown that if c is an abnormal n m code then n or 9 and an abnormal 9118 1 code which is the smallest abnormal code known at present is constructed lower bounds on the minimum cardinality of a binary abnormal code of length n with covering radius 1 are derived and it is shown that if an n m 1 code is abnormal then m or 96

existence and regularity of very weak solutions of the stationary navier stokes equations we consider the stationary navier stokes equations in a bounded domain in rn with smooth connected boundary where n 2 3 or 4 in case that n 3 or 4 existence of very weak solutions in ln is proved for the data belonging to some sobolev spaces of negative order moreover we obtain complete lq regularity results on very weak solutions in ln if n 2 then similar results are also proved for very weak solutions in l q 0 omega with any q0 2 we impose neither smallness conditions on the external force nor boundary data for our existence and regularity results

stationary stokes oseen and navier stokes equations with singular data the concept of very weak solution introduced by giga math z 178 287 329 1981 for the stokes equations has hardly been studied in recent years for either the navier stokes equations or the navier stokes type equations we treat the stationary stokes oseen and navier stokes systems in the case of a bounded open set connected of class mathcal c 1 1 of mathbb r 3 taking up once again the duality method introduced by lions and magenes problemes aus limites non homogenes et applications vols 1 2 dunod paris 1968 and giga math z 178 287 329 1981 for open sets of class mathcal c infty see also chapter 4 of necas les methodes directes en theorie des equations elliptiques french masson et cie ed paris academia editeurs prague 1967 which considers the hilbertian case p 2 for general elliptic operators we give a simpler proof of the existence of a very weak solution for stationary oseen and navier stokes equations when data are not regular enough based on density arguments and a functional framework adequate for defining more rigourously the traces of non regular vector fields in the stationary navier stokes case the results will be valid for external forces not necessarily small which lets us extend the uniqueness class of solutions for these equations considering more regular data regularity results in fractional sobolev spaces will also be discussed for the three systems all these results can be extended to other dimensions

on normal and subnormal q ary codes the authors extend to the q ary case the notions of a normal code a subnormal code and the amalgamated direct sum construction in order to investigate problems related to the covering radius of codes for example the authors prove that every nonbinary nontrivial perfect code is absubnormal they also include some linear programming lower bounds on ternary codes with covering radius 2 or 3