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1.
Restricted strong partially balanced t-designs were first formulated by Pei, Li, Wang and Safavi-Naini in investigation of authentication codes with arbitration. In this article, we will prove that splitting authentication codes that are multi-fold perfect against spoofing can be characterized in terms of restricted strong partially balanced t-designs. We will also investigate the existence of restricted strong partially balanced 3-designs RSPBD 3-(v, b, 3 × 2; λ1, λ2, 1, 0)s, and show that there exists an RSPBD 3-(v, b, 3 × 2; λ1, λ2, 1, 0) for any v o 9 (mod 16){v\equiv 9\ (\mbox{{\rm mod}}\ 16)} . As its application, we obtain a new infinite class of 3-fold perfect splitting authentication codes.  相似文献   

2.
In this note we investigate the sharpness of Bruen’s bound on the size of a t-fold blocking set in \(AG(n,q)\) with respect to the hyperplanes. We give a construction for t-fold blocking sets meeting Bruen’s bound with \(t=q-n+2\) . This construction is used further to find the minimal size of a t-fold affine blocking set with \(t=q-n+1\) . We prove that for blocking sets in the geometries \(AG(n,2)\) the difference between the size of an optimal t-fold blocking set and tn exceeds any given number. In particular, we deviate infinitely from Bruen’s bound as n goes to infinity. We conclude with a construction that gives t-fold blocking sets with \(t=q-n+3\) whose size is close to the lower bounds known so far.  相似文献   

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4.
An ${(N;n,m,\{w_1,\ldots, w_t\})}$ -separating hash family is a set ${\mathcal{H}}$ of N functions ${h: \; X \longrightarrow Y}$ with ${|X|=n, |Y|=m, t \geq 2}$ having the following property. For any pairwise disjoint subsets ${C_1, \ldots, C_t \subseteq X}$ with ${|C_i|=w_i, i=1, \ldots, t}$ , there exists at least one function ${h \in \mathcal{H}}$ such that ${h(C_1), h(C_2), \ldots, h(C_t)}$ are pairwise disjoint. Separating hash families generalize many known combinatorial structures such as perfect hash families, frameproof codes, secure frameproof codes, identifiable parent property codes. In this paper we present new upper bounds on n which improve many previously known bounds. Further we include constructions showing that some of these bounds are tight.  相似文献   

5.
In this paper we are concerned with the study of the existence and multiplicity of connecting orbits for a singular planar Newtonian system ${\ddot{q} + V_q(t, q) = 0}$ with a periodic strong force V q (t, q), an infinitely deep well of Gordon's type at one point and two stationary points at which a potential V (t, q) achieves a strict global maximum. To this end we minimize the corresponding actiön functional over the classes of functions in the Sobolev space ${W^{1, 2}_{\rm loc}(\mathbb{R}, \mathbb{R}^2)}$ that turn a given number of times around the singularity.  相似文献   

6.
We prove that there are 0/1 polytopes ${P \subseteq \mathbb{R}^{n}}$ that do not admit a compact LP formulation. More precisely we show that for every n there is a set ${X \subseteq \{ 0,1\}^n}$ such that conv(X) must have extension complexity at least ${2^{n/2\cdot(1-o(1))}}$ . In other words, every polyhedron Q that can be linearly projected on conv(X) must have exponentially many facets. In fact, the same result also applies if conv(X) is restricted to be a matroid polytope. Conditioning on ${\mathbf{NP}\not\subseteq \mathbf{P_{/poly}}}$ , our result rules out the existence of a compact formulation for any ${\mathbf{NP}}$ -hard optimization problem even if the formulation may contain arbitrary real numbers.  相似文献   

7.
A proper t-coloring of a graph G is a mapping ${\varphi: V(G) \rightarrow [1, t]}$ such that ${\varphi(u) \neq \varphi(v)}$ if u and v are adjacent vertices, where t is a positive integer. The chromatic number of a graph G, denoted by ${\chi(G)}$ , is the minimum number of colors required in any proper coloring of G. A linear t-coloring of a graph is a proper t-coloring such that the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number of a graph G, denoted by ${lc(G)}$ , is the minimum t such that G has a linear t-coloring. In this paper, the linear t-colorings of Sierpiński-like graphs S(n, k), ${S^+(n, k)}$ and ${S^{++}(n, k)}$ are studied. It is obtained that ${lc(S(n, k))= \chi (S(n, k)) = k}$ for any positive integers n and k, ${lc(S^+(n, k)) = \chi(S^+(n, k)) = k}$ and ${lc(S^{++}(n, k)) = \chi(S^{++}(n, k)) = k}$ for any positive integers ${n \geq 2}$ and ${k \geq 3}$ . Furthermore, we have determined the number of paths and the length of each path in the subgraph induced by the union of any two color classes completely.  相似文献   

8.
We consider the perturbed Thomas–Fermi equation $$\begin{array}{ll} x^{\prime \prime}\, =\, p(t)|x|^{\gamma-1}x\, +\, q(t)|x|^{\delta-1}x, \qquad \qquad \qquad (A) \end{array}$$ where δ and γ are positive constants with \({\delta < 1 < \gamma}\) and p(t) and q(t) are positive continuous functions on \({[a,\infty), a > 0}\) . Our aim here is to establish criteria for the existence of positive solutions of (A) decreasing to zero as \({t \to \infty}\) in the case where p(t) and q(t) are regularly varying functions (in the sense of Karamata). Generalization of the obtained results to equations of the form $$\begin{array}{ll} \left(r(t)x^{\prime}\right)^{\prime}\, =\, p(t)|x|^{\gamma-1}x \,+ \,q(t)|x|^{\delta-1}x, \qquad \qquad \qquad (B) \end{array}$$ is also given.  相似文献   

9.
Let m, m′, r, r′, t, t′ be positive integers with r, r′ ? 2. Let \(\mathbb{L}_r \) denote the ring that is universal with an invertible 1×r matrix. Let \(M_m (\mathbb{L}_r^{ \otimes t} )\) denote the ring of m × m matrices over the tensor product of t copies of \(\mathbb{L}_r \) . In a natural way, \(M_m (\mathbb{L}_r^{ \otimes t} )\) is a partially ordered ring with involution. Let \(PU_m (\mathbb{L}_r^{ \otimes t} )\) denote the group of positive unitary elements. We show that \(PU_m (\mathbb{L}_r^{ \otimes t} )\) is isomorphic to the Brin-Higman-Thompson group tV r,m ; the case t=1 was found by Pardo, that is, \(PU_m (\mathbb{L}_r )\) is isomorphic to the Higman-Thompson group V r,m . We survey arguments of Abrams, Ánh, Bleak, Brin, Higman, Lanoue, Pardo and Thompson that prove that t′V r′,m′ tV r,m if and only if r′ =r, t′ =t and gcd(m′, r′?1) = gcd(m, r?1) (if and only if \(M_{m'} (\mathbb{L}_{r'}^{ \otimes t'} )\) and \(M_m (\mathbb{L}_r^{ \otimes t} )\) are isomorphic as partially ordered rings with involution).  相似文献   

10.
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