Asymmetric Binary Covering Codes

An asymmetric binary covering code of length n and radius R is a subset of the n-cube Q_n such that every vector xQ_n can be obtained from some vector c by changing at most R 1's of c to 0's, where R is as small as possible. K⁺(n,R) is defined as the smallest size of such a code. We show K⁺(n,R)Θ(2ⁿ/n^R) for constant R, using an asymmetric sphere-covering bound and probabilistic methods. We show K⁺(n,n− )= +1 for constant coradius iff n ( +1)/2. These two results are extended to near-constant R and , respectively. Various bounds on K⁺ are given in terms of the total number of 0's or 1's in a minimal code. The dimension of a minimal asymmetric linear binary code ([n,R]⁺-code) is determined to be min{0,n−R}. We conclude by discussing open problems and techniques to compute explicit values for K⁺, giving a table of best-known bounds.     

Singularities of secant maps of immersed surfaces

The secant map of an immersion sends a pair of points to the direction of the line joining the images of the points under the immersion. The germ of the secant map of a generic codimension-c immersion $X\!\!:{\mathbb R}^n \to {\mathbb R}^{n+c}The secant map of an immersion sends a pair of points to the direction of the line joining the images of the points under the immersion. The germ of the secant map of a generic codimension-c immersion X:\mathbb Rⁿ ? \mathbb R^n+cX\!\!:{\mathbb R}^n \to {\mathbb R}^{n+c} at the diagonal in the source is a \mathbb Z₂{\mathbb Z}_2 stable map-germ \mathbb R²ⁿ ? \mathbb R^n+c-1{\mathbb R}^{2n} \to {\mathbb R}^{n+c-1} in the following cases: (i) c≥ 2 and (2n,n + c − 1) is a pair of dimensions for which the \mathbb Z₂{\mathbb Z}_2 stable germs of rank at least n are dense, and (ii) for generically immersed surfaces (i.e., n = 2 and any c≥ 1). In the latter surface case the A^{\mathbb Z₂}{\mathcal A}^{{\mathbb Z}_2}-classification of germs of secant maps at the diagonal is described and it is related to the A{\mathcal A}-classification of certain singular projections of the surfaces.     

Convective instability of a rotating layer of ferromagnetic fluid

The instability of a hot horizontal layer of ferromagnetic fluid rotating about a vertical axis has been investigated when the Prandtl numberP < 1. Earlier it was shown that forP > 1 the overstability cannot occur. In this paper the convective and overstable marginal states have been investigated separately forP < 1 and it is found that though convective marginal state is possible for alla, the non-dimensional wave number, and N the Taylor number, the overstability is possible only ifN > (1 +P)π ⁴/(1 −P) and in case the condition is satisfied, overstability is possible for all those values ofa which satisfya ² < [N(1 −P)π ²/(1 +P)] ^1/3 − π². IfR _c ^(con) andR _c ^(o.s) are the critical values of the convective and the overstable marginal states respectively, then it is also found thatR _c ^(con) <R _c ^(o.s) providedN is not sufficiently large.     

Spherical submanifolds in a Euclidean space

In this paper, we are interested in extending the study of spherical curves in R ³ to the submanifolds in the Euclidean space R ^n+p . More precisely, we are interested in obtaining conditions under which an n-dimensional compact submanifold M of a Euclidean space R ^n+p lies on the hypersphere S ^n+p−1(c) (standardly imbedded sphere in R ^n+p of constant curvature c). As a by-product we also get an estimate on the first nonzero eigenvalue of the Laplacian operator Δ of the submanifold (cf. Theorem 3.5) as well as a characterization for an n-dimensional sphere S ⁿ (c) (cf. Theorem 4.1).     

A construction of binary Golay sequence pairs from odd‐length Barker sequences

Binary Golay sequence pairs exist for lengths 2, 10 and 26 and, by Turyn's product construction, for all lengths of the form 2^a10^b26^c where a, b, c are non‐negative integers. Computer search has shown that all inequivalent binary Golay sequence pairs of length less than 100 can be constructed from five “seed” pairs, of length 2, 10, 10, 20 and 26. We give the first complete explanation of the origin of the length 26 binary Golay seed pair, involving a Barker sequence of length 13 and a related Barker sequence of length 11. This is the special case m=1 of a general construction for a length 16m+10 binary Golay pair from a related pair of Barker sequences of length 8m+5 and 8m+3, for integer m≥0. In the case m=0, we obtain an alternative explanation of the origin of one of the length 10 binary Golay seed pairs. The construction cannot produce binary Golay sequence pairs for m>1, having length greater than 26, because there are no Barker sequences of odd length greater than 13. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 478–491, 2009     

A combinatorial condition for the existence of polyhedral 2-manifolds

LetP denote a polyhedral 2-manifold, i.e. a 2-dimensional cell-complex inR ^d (d≧3) having convex facets, such that set (P) is homeomorphic to a closed 2-dimensional manifold. LetE be any subset of odd valent vertices ofP, andc _E its cardinality. Then for the numberc _P(E) of facets containing a vertex ofE the inequality 2c _P(E)≧c_E+1 is proved. This local combinatorial condition shows that several combinatorially possible types of polyhedral 2-manifolds cannot exist.     

The Łojasiewicz exponent of an analytic function at an isolated zero

Let f be a real analytic function defined in a neighborhood of 0 ? \Bbb Rⁿ 0 \in {\Bbb R}^n such that f^-1(0)={0} f^{-1}(0)=\{0\} . We describe the smallest possible exponents !, #, / for which we have the following estimates: |f(x)| 3 c|x|^a |f(x)|\geq c|x|^{\alpha} , |grad f(x)| 3 c|x|^b |{\rm grad}\,f(x)|\geq c|x|^{\beta} , |grad f(x)| 3 c|f(x)|^q |{\rm grad}\,f(x)|\geq c|f(x)|^{\theta} for x near zero with c > 0 c > 0 . We prove that a = b+1 \alpha=\beta+1, q = b/a\theta=\beta/\alpha . Moreover b = N+a/b \beta=N+a/b where $ 0 h a < b h N^{n-1} $ 0 h a < b h N^{n-1} . If f is a polynomial then |f(x)| 3 c|x|^(degf-1)n+1 |f(x)|\geq c|x|^{(\deg f-1)^n+1} in a small neighborhood of zero.     

Geometrical solutions of some quadratic equations with non-real roots

This note gives geometrical/graphical methods of finding solutions of the quadratic equation ax ² + bx + c = 0, a p 0, with non-real roots. Three different cases which give rise to non-real roots of the quadratic equation have been discussed. In case I a geometrical construction and its proof for finding the solutions of the quadratic equation ax ² + bx + c = 0, a p 0, when a,b,c ] R, the set of real numbers, are presented. Case II deals with the geometrical solutions of the quadratic equation ax ² + bx + c = 0, a p 0, when b ] R, the set of real numbers; and a,c ] C, the set of complex numbers. Finally, the solutions of the quadratic equation ax ² + bx + c = 0, a p 0, when a,c ] R, the set of real numbers, and b ] C, the set of complex numbers, are presented in case III.     

On the correctness of the Cauchy problem for a linear differential system on an infinite interval

The conditions ensuring the correctness of the Cauchy problem $$\frac{{dx}}{{dt}} = P(t)x + q(t),x(t_0 ) = c_0$$ on the nonnegative half-axisR ₊ are found, whereP:R ₊→R ^n×n andq:R ₊→R ⁿ are locally summable matrix and vector functions, respectively,t ₀∈R ₊ andc ₀∈R ⁿ .     

On (p a,p, p a,p a−1 )-relative difference sets

Abelian relative difference sets of parameters (m, n, k, )=(p ^a , p, p ^a , p ^a–1 )are studied in this paper. In particular, we show that for an abelian groupG of orderp ^2c+1 and a subgroupN ofG of orderp, a (p ^2c , p, p ^2c , p ^2c–1 )-relative difference set exists inG relative toN if and only if exp (G)p ^c+1 .Furthermore, we have some structural results on (p ^2c p, p ^2c , p ^2c–1 )-relative difference sets in abelian groups of exponentp ^c+1 . We also show that for an abelian groupG of order 2^2c+2 and a subgroupN ofG of order 2, a (2^2c+1, 2, 2^2c+1, 2^2c )-relative difference set exists inG relative toN if and only if exp(G)2 ^c+2 andN is contained in a cyclic subgroup ofG of order 4. New constructions of (p ^2c+1 , p, p ^2c+1 , p ^2c )-relative difference sets, wherep is an odd prime, are given. However, we cannot find the necessary and sufficient condition for this case.     

On a Diophantine Inequality Over Primes (II)

In this short note we prove that if 1 < c < 81/40, c ≠ 2, N is a large real number, then the Diophantine inequality |p₁^c+p₂^c+p₃^c+p₄^c+p₅^c-N| < log^-1 N \vert p_1^c+p_2^c+p_3^c+p_4^c+p_5^c-N\vert < \log^{-1} N is solvable, where p ₁,···,p ₅ are primes.     

Closed polylines inscribed in a closed space curve

Let n be an odd positive integer. It is proved that if n + 2 is a power of a prime number and C is a regular closed non-self-intersecting curve in \mathbbRⁿ {\mathbb{R}^n} ,then C contains vertices of an equilateral (n + 2)-link polyline with n + 1 vertices lying in a hyperplane. It is also proved that if C is a rectifiable closed curve in \mathbbRⁿ {\mathbb{R}^n} ,then C contains n + 1 points that lie in a hyperplane and divide C into parts one of which is twice as long as each of the others. Bibliography: 6 titles.     

Lie subalgebras in a certain operator Lie algebra with involution

We show in a certain Lie*-algebra, the connections between the Lie subalgebra G ⁺:= G + G* + [G, G*], generated by a Lie subalgebra G, and the properties of G. This allows us to investigate some useful information about the structure of such two Lie subalgebras. Some results on the relations between the two Lie subalgebras are obtained. As an application, we get the following conclusion: Let A ⊂ B(X) be a space of self-adjoint operators and ℒ:= A ⊕ iA the corresponding complex Lie*-algebra. G ⁺ = G + G* + [G, G*] and G are two LM-decomposable Lie subalgebras of ℒ with the decomposition G ⁺ = R(G ⁺) + S, G = R _G + S _G , and R _G ⊆ R(G ⁺). Then G ⁺ is ideally finite iff R _G ⁺:= R _G + R* _G ^* + [R _G , R _G ^*] is a quasisolvable Lie subalgebra, S _G ⁺:= S _G + S _G ^* + [S _G , S _G ^*] is an ideally finite semisimple Lie subalgebra, and [R _G , S _G ] = [R _G ^*, S _G ] = {0}.     

Spherical uniformity and some characterizations of the Cauchy distribution

Given a fixed line L (in Rⁿ) and a uniform distribution of points (c) on the unit sphere, L(t_c), the point of intersection of L and the hyperplane P · c = 0, leads to a mapping X_n : Rⁿ → R, which is shown to have a Cauchy distribution.     

A representation for the remainder of the Maclaurin quadrature formula

Summary We show that the remainder of the Maclaurin quadrature formula belonging to oddn (n+1 is the number of nodes) can be represented asR _n (f)=c _n f ⁽ⁿ⁺¹⁾ (), wheneverf ⁽ⁿ⁺¹⁾ exists and is continuous The corresponding problem for evenn has already been settled by A. Walther in 1925.     

On the Solutions of a Second Order Differential Equation Arising in the Theory of Resonant Oscillations in a Tank

The properties of the solutions of the differential equation y″ = y² ? x² ? c subject to the condition that y is bounded for all finite x discussed. The arguments of Holmes and Spence have been used by Ockendon, Ockendon, and Johnson to show that there are no solutions if c is large and negative. Numerically we find that solutions exist provided c is greater than a critical value c* and estimate this value to be c* = ?…. As x tends to + ∞ the solutions are asymptotic to . The relation between A₊ and ?₊ are found analytically as A₊ → ∞. This problem arises as a connection problem in the theory of resonant oscillations of water waves.     

Oscillations of Higher Order Neutral Differential Equations with Variable Coefficients

We obtain sufficient conditions for the oscillation of all solutions of the higher order neutral differential equation dⁿ/d^m[y(t) + P(t) y(t - μ)] + Q(t) y(t ?σ) = 0, t ≧ t₀ where n ≧ 1, P ? C[t₀, ∞), R ], Q ? C[t₀, ∞), R ] and τ, μ ? R ⁺. Our results extend and improve several known results in the literature.     

Minimax approximation by rational fractions of the inverse polynomial type

Rational fractions of the formR(x)/(c ₁ +c ₂ x +c ₃ x ² + ...) ^r are useful for approximating decay type functions over infinite and semi-infinite domains. A procedure is given which produces the optimal coefficients with no more effort than for linear approximations. No initial guess is needed for the values of the coefficients nor for the maximum error of approximation.     

Transforming binary sequences using priority queues

A priority queue transforms an input sequence into an output sequence which is a re-ordering of the sequence . The setR of all such related pairs is studied in the case that is a binary sequence. It is proved thatR is a partial order and that ¦R¦=c _n+1, the (n+1)th Catalan number. An efficient (O(n ²)) algorithm is given for computing the number of outputs achievable from a given input.