A new extension theorem for linear codes

For an [n,k,d]_q code with k3, gcd(d,q)=1, the diversity of is defined as the pair (Φ₀,Φ₁) with

All the diversities for [n,k,d]_q codes with k3, d−2 (mod q) such that A_i=0 for all i0,−1,−2 (mod q) are found and characterized with their spectra geometrically, which yields that such codes are extendable for all odd q5. Double extendability is also investigated.     

Non-simultaneous blow-up of n components for nonlinear parabolic systems

This paper deals with non-simultaneous and simultaneous blow-up for radially symmetric solution (u₁,u₂,…,un) to heat equations coupled via nonlinear boundary (i=1,2,…,n). It is proved that there exist suitable initial data such that ui(i∈{1,2,…,n}) blows up alone if and only if qi+1<pi. All of the classifications on the existence of only two components blowing up simultaneously are obtained. We find that different positions (different values of k, i, n) of ui−k and ui leads to quite different blow-up rates. It is interesting that different initial data lead to different blow-up phenomena even with the same requirements on exponent parameters. We also propose that ui−k,ui−k+1,…,ui blow up simultaneously while the other ones remain bounded in different exponent regions. Moreover, the blow-up rates and blow-up sets are obtained.     

Likelihood ratio orderings of spacings of heterogeneous exponential random variables

Let X₁,X₂,…,X_n be independent exponential random variables such that X_i has failure rate λ for i=1,…,p and X_j has failure rate λ^* for j=p+1,…,n, where p≥1 and q=n-p≥1. Denote by D_i:n(p,q)=X_i:n-X_i-1:n the ith spacing of the order statistics , where X_0:n≡0. It is shown that D_i:n(p,q)?_lrD_i+1:n(p,q) for i=1,…,n-1, and that if λ?λ^* then , and for i=1,…,n, where ?_lr denotes the likelihood ratio order. The main results are used to establish the dispersive orderings between spacings.     

Labeling graphs with two distance constraints

Given a graph G and integers p,q,d₁ and d₂, with p>q, d₂>d₁?1, an L(d₁,d₂;p,q)-labeling of G is a function f:V(G)→{0,1,2,…,n} such that |f(u)−f(v)|?p if d_G(u,v)?d₁ and |f(u)−f(v)|?q if d_G(u,v)?d₂. A k-L(d₁,d₂;p,q)-labeling is an L(d₁,d₂;p,q)-labeling f such that max_v∈V(G)f(v)?k. The L(d₁,d₂;p,q)-labeling number ofG, denoted by , is the smallest number k such that G has a k-L(d₁,d₂;p,q)-labeling. In this paper, we give upper bounds and lower bounds of the L(d₁,d₂;p,q)-labeling number for general graphs and some special graphs. We also discuss the L(d₁,d₂;p,q)-labeling number of G, when G is a path, a power of a path, or Cartesian product of two paths.     

On the minimum number of completely 3-scrambling permutations

A family P={π₁,…,π_q} of permutations of [n]={1,…,n} is completely k-scrambling [Spencer, Acta Math Hungar 72; Füredi, Random Struct Algor 96] if for any distinct k points x₁,…,x_k∈[n], permutations π_i's in P produce all k! possible orders on π_i(x₁),…,π_i(x_k). Let N^*(n,k) be the minimum size of such a family. This paper focuses on the case k=3. By a simple explicit construction, we show the following upper bound, which we express together with the lower bound due to Füredi for comparison.

     

Counting lattice chains and Delannoy paths in higher dimensions

Lattice chains and Delannoy paths represent two different ways to progress through a lattice. We use elementary combinatorial arguments to derive new expressions for the number of chains and the number of Delannoy paths in a lattice of arbitrary finite dimension. Specifically, fix nonnegative integers n₁,…,n_d, and let L denote the lattice of points (a₁,…,a_d)∈Z^d that satisfy 0≤a_i≤n_i for 1≤i≤d. We prove that the number of chains in L is given by where . We also show that the number of Delannoy paths in L equals Setting n_i=n (for all i) in these expressions yields a new proof of a recent result of Duchi and Sulanke [9] relating the total number of chains to the central Delannoy numbers. We also give a novel derivation of the generating functions for these numbers in arbitrary dimension.     

A volume formula for asymptotic hyperbolic tetrahedra with an application to quantum field theory

In this paper, a volume formula for hyperbolic tetrahedra with one vertex at infinity is derived. This yields a new representation of the four-dimensional three-point function of Euclidean quantum field theory, i.e., of ∝_R⁴Π³_{k = 1} (q₁² + … + q₄² + b_1kq₁ + … + b_4kq₄ + d_k)⁻¹dq₁…dq₄, through 15 dilogarithms containing inverse trigonometric functions which are symmetric in the parameters b_jk and d_k, j = 1,…,4, k = 1,2,3.     

Multiple blocking sets in finite projective spaces and improvements to the Griesmer bound for linear codes

Belov, Logachev and Sandimirov construct linear codes of minimum distance d for roughly 1/q ^k/2 of the values of d < q ^k-1. In this article we shall prove that, for q = p prime and roughly \frac38{\frac{3}{8}}-th’s of the values of d < q ^k-1, there is no linear code meeting the Griesmer bound. This result uses Blokhuis’ theorem on the size of a t-fold blocking set in PG(2, p), p prime, which we generalise to higher dimensions. We also give more general lower bounds on the size of a t-fold blocking set in PG(δ, q), for arbitrary q and δ ≥ 3. It is known that from a linear code of dimension k with minimum distance d < q ^k-1 that meets the Griesmer bound one can construct a t-fold blocking set of PG(k−1, q). Here, we calculate explicit formulas relating t and d. Finally we show, using the generalised version of Blokhuis’ theorem, that nearly all linear codes over \mathbb F_p{{\mathbb F}_p} of dimension k with minimum distance d < q ^k-1, which meet the Griesmer bound, have codewords of weight at least d + p in subcodes, which contain codewords satisfying certain hypotheses on their supports.     

The topological type of the α-sections of convex sets

An α=(α₁,…,αk)(0?αi?1) section of a family {K₁,…,Kk} of convex bodies in Rd is a transversal halfspace H⁺ for which Vold(Ki∩H⁺)=αi⋅Vold(Ki) for every 1?i?k. Our main result is that for any well-separated family of strictly convex sets, the space of α-sections is diffeomorphic to Sd−k.     

Rotations of Periodic Strings and Short Superstrings

This paper presents two simple approximation algorithms for the shortest superstring problem with approximation ratios 2 ( ≈ 2.67) and( ≈ 2.596). The framework of our improved algorithms is similar to that of previous algorithms in the sense that they construct a superstring by computing some optimal cycle covers on the distance graph of the given strings and then break and merge the cycles to finally obtain a Hamiltonian path, but we make use of new bounds on the overlap between two strings. We prove that for each periodic semiinfinite string α = a₁a₂··· of periodq, there exists an integerk, such that forany(finite) stringsof periodpwhich isinequivalentto α, the overlap betweensand therotationα[k] = a_ka_{k + 1}··· is at mostp + q. Moreover, ifp ≤ q, then the overlap betweensand α[k] is not larger than (p + q). The bounds are tight. In the previous shortest superstring algorithmsp + qwas used as the standard (tight) bound on overlap between two strings with periodspandq.     

The asymptotic number of spanning trees in circulant graphs

Let T(G) be the number of spanning trees in graph G. In this note, we explore the asymptotics of T(G) when G is a circulant graph with given jumps.The circulant graph is the 2k-regular graph with n vertices labeled 0,1,2,…,n−1, where node i has the 2k neighbors i±s₁,i±s₂,…,i±s_k where all the operations are . We give a closed formula for the asymptotic limit as a function of s₁,s₂,…,s_k. We then extend this by permitting some of the jumps to be linear functions of n, i.e., letting s_i, d_i and e_i be arbitrary integers, and examining

     

Gaps in semigroups

In this paper we investigate the behaviour of the gaps in numerical semigroups. We will give an explicit formula for the ith gap of a semigroup generated by k+1 consecutive integers (generalizing a result due to Brauer) as well as for a special numerical semigroup of three generators. It is also proved that the number of gaps of the numerical semigroup generated by integers p and q with g.c.d.(p,q)=1, in the interval [pq-(k+1)(p+q),…,pq-k(p+q)] is equals to

     

Kolmogorov's problem on the class of multiply monotone functions

In this paper we give necessary and sufficient conditions for the system of positive numbers _Mk1,_Mk2,…,_Mkd$M_{k_1},M_{k_2},\dots ,M_{k_d}$, 0≤_k1<…<_kd≤r$0\le k_1<\dots <k_d\le r$, to guarantee the existence of an r  -monotone function defined on the negative half-line _R−${\mathbb{R}}_{-}$ and such that _{‖x(ki)‖∞}=_Mki${\Vert x^{(k_i)}\Vert }_{\infty }=M_{k_i}$, i=1,2,…,d$i=1,2,\dots ,d$. We also discuss some applications of the obtained results and connections with other problems.     

Error Correcting Sequence and Projective De Bruijn Graph

Let X be a finite set of q elements, and n, K, d be integers. A subset C ⊂ X ⁿ is an (n, K, d) error-correcting code, if #(C) = K and its minimum distance is d. We define an (n, K, d) error-correcting sequence over X as a periodic sequence {a _i } _i=0,1,... (a _i ∈ X) with period K, such that the set of all consecutive n-tuples of this sequence form an (n, K, d) error-correcting code over X. Under a moderate conjecture on the existence of some type of primitive polynomials, we prove that there is a error correcting sequence, such that its code-set is the q-ary Hamming code with 0 removed, for q > 2 being a prime power. For the case q = 2, under a similar conjecture, we prove that there is a error-correcting sequence, such that its code-set supplemented with 0 is the subset of the binary Hamming code [2 ^m  − 1, 2 ^m  − 1 − m, 3] obtained by requiring one specified coordinate being 0. Received: October 27, 2005. Final Version received: December 31, 2007     

An Erdo&#x030B;s–Ko–Rado Theorem for Direct Products

Letn_i, k_ibe positive integers,i=1, ..., d,satisfyingn_i≥2k_i.LetX₁, ..., X_dbe pairwise disjoint sets with |X_i| =n_i.Letbe the family of those (k₁+···+k_d)-element sets which have exactlyk_ielements inX_i, i=1,..., d.It is shown that if⊂is an intersecting family then ||/||≤max_ik_i/n_i,and this is best possible. The proof is algebraic, although in thed=2 case a combinatorial argument is presented as well.     

A recurrence formula for leaping convergents of non-regular continued fractions

Given a continued fraction [a₀;a₁,a₂,…], p_n/q_n=[a₀;a₁,…,a_n] is called the n-th convergent for n=0,1,2,…. Leaping convergents are those of every r-th convergent p_rn+i/q_rn+i (n=0,1,2,…) for fixed integers r and i with r?2 and i=0,1,…,r-1. This leaping step r can be chosen as the length of period in the continued fraction. Elsner studied the leaping convergents p_3n+1/q_3n+1 for the continued fraction of and obtained some arithmetic properties. Komatsu studied those p_3n/q_3n for (s?2). He has also extended such results for some more general continued fractions. Such concepts have been generalized in the case of regular continued fractions. In this paper leaping convergents in the non-regular continued fractions are considered so that a more general three term relation is satisfied. Moreover, the leaping step r need not necessarily to equal the length of period. As one of applications a new recurrence formula for leaping convergents of Apery’s continued fraction of ζ(3) is shown.     

Classifying Subspaces of Hamming Spaces

A linear code in F ⁿ _q with dimension k and minimum distance at least d is called an [n, k, d] _q code. We here consider the problem of classifying all [n, k, d] _q codes given n, k, d, and q. In other words, given the Hamming space F ⁿ _q and a dimension k, we classify all k-dimensional subspaces of the Hamming space with minimum distance at least d. Our classification is an iterative procedure where equivalent codes are identified by mapping the code equivalence problem into the graph isomorphism problem, which is solved using the program nauty. For d = 3, the classification is explicitly carried out for binary codes of length n 14, ternary codes of length n 11, and quaternary codes of length n 10.     

Global flow of the anisotropic Kepler problem on negative energy levels whenμ>9/8

In a previous study of the orbits of the anisotropic Kepler problem for>9/8 it had been shown that there are orbits which visit both theq ₂>0 and theq ₂<0 half-plane infinite times and that, denoting byc _n the number of times an orbit crosses theq ₂ axis before the half-plane where it is to be found changes forn-th time, every sequence of positive integersc _n is realized by at least one orbit. In this paper it is shown that these crossings do not occur in arbitrary regions of theq ₂ axis and the spatial order to which they obey is found: both half-axes,q ₂>0 andq ₂<0, may be divided into a sequence of contiguous segmentsp _n such that, for every family ofc _k succesive crossings, thei-th crossing occurs atp _i, fori going from 1 toc _k.

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Resumé Ce travail compléte la caractérization des orbites du probléme anisotropique de Kepler pour>9/8 et prend comme point de départ l'existence déjá connue d'orbites qui visitent les deux semiplansq ₂>0 etq ₂<0 une infinité de fois et qui réalisent toute successionc _n de nombres entiers positifs, désignant parc _n le nombre de fois que l'orbite coupe l'axe desq ₂ avant de changer de semiplan pour lan-éme fois. Nous démontrons que ces intersections de l'axeq ₂ sont ordonnées de la façon suivante: on peut définir une division de chaque semiaxeq ₂>0 etq ₂<0 en une succession de ségments contigusp _n telle que, pour toute famille dec _k croisements succéssifs et pouri entre 1 etc _k, lei-éme croisement coupe le ségment p_i.

     

Maps preserving the spectrum of generalized Jordan product of operators

Let A₁,A₂ be standard operator algebras on complex Banach spaces X₁,X₂, respectively. For k?2, let (i₁,…,i_m) be a sequence with terms chosen from {1,…,k}, and define the generalized Jordan product

     

On the directions problem in <Emphasis Type="Italic">AG</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)

We prove that if q = p ^h , p a prime, do not exist sets U í AG(n,q){U {\subseteq} AG(n,q)}, with |U| = q ^k and 1 < k < n, determining N directions where