Optimal interleaving schemes for correcting two-dimensional cluster errors

We present an elementary theory of optimal interleaving schemes for correcting cluster errors in two-dimensional digital data. It is assumed that each data page contains a fixed number of, say n, codewords with each codeword consisting of m code symbols and capable of correcting a single random error (or erasure). The goal is to interleave the codewords in the m×n array such that different symbols from each codeword are separated as much as possible, and consequently, an arbitrary error burst with size up to t can be corrected for the largest possible value of t. We show that, for any given m, n, the maximum possible interleaving distance, or equivalently, the largest size of correctable error bursts in an m×n array, is given by if n?⌈m²/2⌉, and t=m+⌊(n-⌈m²/2⌉)/m⌋ if n?⌈m²/2⌉. Furthermore, we develop a simple cyclic shifting algorithm that can provide a systematic construction of an m×n optimal interleaving array for arbitrary m and n. This extends important earlier work on the complementary problem of constructing interleaving arrays that, given the burst size t, minimize the interleaving degree, that is, the number of different codewords in a 2-D (or 3-D) array such that any error burst with given size t can be corrected. Our interleaving scheme thus provides the maximum burst error correcting power without requiring prior knowledge of the size or shape of an error burst.     

Some necessary and sufficient conditions for existence of positive solutions for third order singular super-linear multi-point boundary value problems

We mainly study the existence of positive solutions for the following third order singular super-linear multi-point boundary value problem $$ \left \{ \begin{array}{l} x^{(3)}(t)+ f(t, x(t), x'(t))=0,\quad0 where \(0\leq\alpha_{i}\leq\sum_{i=1}^{m_{1}}\alpha_{i}<1\) , i=1,2,…,m ₁, \(0<\xi_{1}< \xi_{2}< \cdots<\xi_{m_{1}}<1\) , \(0\leq\beta_{j}\leq\sum_{i=1}^{m_{2}}\beta_{i}<1\) , j=1,2,…,m ₂, \(0<\eta_{1}< \eta_{2}< \cdots<\eta_{m_{2}}<1\) . And we obtain some necessary and sufficient conditions for the existence of C ¹[0,1] and C ²[0,1] positive solutions by means of the fixed point theorems on a special cone. Our nonlinearity f(t,x,y) may be singular at t=0 and t=1.     

Global center manifolds in singular systems

The problem of existence of aglobal center manifold for a system of O.D.E. like (*) $$\left\{ {\begin{array}{*{20}c} {\dot x = A(y)x + F(x,y)} \\ {\dot y = G(x,y), (x,y) \in \mathbb{R}^n \times \mathbb{R}^m ,} \\ \end{array} } \right.$$ is considered. We give conditions onA(y), F(x, y), G(x, y) in order that a functionH: ? ^m →? ⁿ , with the same smoothness asA(y), F(x, y), G(x, y), exists and is such that the manifoldC={(x,y)∈? ⁿ ×? ^m ∣x=H(y),y∈? ^m } is an invariant manifold for (*), and there exists ρ>0 such that any solution of (*) satisfying sup _t∈?∣x(t)∣ <ρ must belong toC. This is why we callC global center manifold. Applications are given to the problem of existence of heteroclinic orbits in singular systems.     

Existence and multiplicity of solutions to 2mth-order ordinary differential equations

In this paper, the existence and multiplicity of solutions are obtained for the 2mth-order ordinary differential equation two-point boundary value problems u(2(m−i))(t)=f(t,u(t)) for all t∈[0,1] subject to Dirichlet, Neumann, mixed and periodic boundary value conditions, respectively, where f is continuous, ai∈R for all i=1,2,…,m. Since these four boundary value problems have some common properties and they can be transformed into the integral equation of form , we firstly deal with this nonlinear integral equation. By using the strongly monotone operator principle and the critical point theory, we establish some conditions on f which are able to guarantee that the integral equation has a unique solution, at least one nonzero solution, and infinitely many solutions. Furthermore, we apply the abstract results on the integral equation to the above four 2mth-order two-point boundary problems and successfully resolve the existence and multiplicity of their solutions.     

The number of distinct symbols in sections of rectangular arrays

We investigate transversals of rectangular arrays. For positive integers m and n, where 2?m?n an m by n array consists of mn cells arranged in m rows and n columns. Each cell contains one symbol. When m=n we speak of an array of order n. A section in the array consists of m cells, one from each row and no two from the same column. A transversal is a section whose m symbols are distinct. A partial transversal is a subset of a transversal. We investigate the existence in an array of a section with many different symbols, in particular the existence of a transversal.     

Positive solutions of higher-order singular boundary value problems

Some results of existence of positive solutions for singular boundary value problem $$\left\{\begin{array}{l}\displaystyle (-1)^{m}u^{(2m)}(t)=p(t)f(u(t)),\quad t\in(0,1),\\[2mm]\displaystyle u^{(i)}(0)=u^{(i)}(1)=0,\quad i=0,\ldots,m-1,\end{array}\right.$$ are given, where the function p(t) may be singular at t=0,1. Our analysis relies on the variational method.     

On proper oscillatory and vanishing-at-infinity solutions of differential equations with a deviating argument

Sufficient conditions are found for the existence of multiparametric families of proper oscillatory and vanishing-at-infinity solutions of the differential equation $$u^{(n)} (t) = g\left( {t, u(\tau _0 (t)), \ldots ,u^{(m - 1)} (\tau _{m - 1} (t))} \right)$$ , wheren≥4,m is the integer part of π/2,g:R ₊×R ^m →R is a function satisfying the local Carathéodory conditions, and τ _i :R ₊→R(i=0,...,m?1) are measurable functions such that τ _i (t) →+∞ fort→+∞(i=0,...,m?1).     

Construction of universal one-way hash functions: Tree hashing revisited

We present a binary tree based parallel algorithm for extending the domain of a universal one-way hash function (UOWHF). For t?2, our algorithm extends the domain from the set of all n-bit strings to the set of all ((2^t-1)(n-m)+m)-bit strings, where m is the length of the message digest. The associated increase in key length is 2m bits for t=2; m(t+1) bits for 3?t?6 and m×(t+⌊log₂(t-1)⌋) bits for t?7.     

Existence and comparison theorems for nonlinear diffusion systems

In this paper we prove existence, uniqueness, and regularity results for systems of nonlinear second order parabolic equations with boundary conditions of the Dirichlet, Neumann, and regular oblique derivative types. Let K(t) consist of all functions (v¹(x), v²(x),…, v^m(x)) from $\text{Ω ? R}{}^{\mathrm{n}}$ into R^m which satisfy ψⁱ(x, t) ? vⁱ(x) ? θⁱ(x, t) for all $\text{x ? Ω}\text{and}\text{1 ? i ? m}$, where ψⁱand θⁱ are extended real-valued functions on $\text{}\text{?}\text{gW × [0, T)}$. We find conditions which will ensure that a solution U(x, t) ≡ (u¹(x, t), u²(x, t),…, u^m(x, t)) which satisfies U(x, 0) ?K(0) will also satisfy U(x, t) ?K(t) for all 0 ? t < T. This result, which has some similarity to the Gronwall Inequality, is then used to prove a global existence theorem.     

On incomplete orthogonal arrays

First we prove that, if an incomplete orthogonal array (1, r, s, k, t) does exist, then s ?(r ? t + 1)k. Next, we establish a relation between the existence of incomplete orthogonal arrays and the existence of orthogonal arrays. From this relation, we may bring out the upper bounds of the maximum number of contraints.     

Existence of positive solutions of Neumann boundary value problem via a cone compression-expansion fixed point theorem of functional type

This paper is devoted to study the existence of positive solutions of second-order boundary value problem $$-u''+m^2u=h(t)f(t,u),\quad t\in (0,1)$$ with Neumann boundary conditions $$u'(0)=u'(1)=0,$$ where m>0, f∈C([0,1]×?⁺,?⁺), and h(t) is allowed to be singular at t=0 and t=1. The arguments are based only upon the positivity of the Green function, a fixed point theorem of cone expansion and compression of functional type, and growth conditions on the nonlinearity f.     

The method of lower and upper solutions for fourth order singular m-point boundary value problems

This paper investigates the existence of positive solutions for fourth order singular m-point boundary value problems. Firstly, we establish a comparison theorem, then we define a partial ordering in C²[0,1]∩C⁴(0,1) and construct lower and upper solutions to give a necessary and sufficient condition for the existence of C²[0,1] as well as C³[0,1] positive solutions. Our nonlinearity f(t,x,y) may be singular at x, y, t=0 and/or t=1.     

Existence results for some polyharmonic problems in the half-space

We study the existence of positive solutions of the m-polyharmonic nonlinear elliptic equation m(−Δ)u+f(⋅,u)=0 in the half-space , n?2 and m?1. Our purpose is to give two existence results for the above equation subject to some boundary conditions, where the nonlinear term f(x,t) satisfies some appropriate conditions related to a certain Kato class of functions .     

Global existence and nonexistence of solution for Cauchy problem of multidimensional double dispersion equations

In this paper we consider the Cauchy problem of multidimensional generalized double dispersion equations utt−Δu−Δutt+Δ²u=Δf(u), where f(u)=ap|u|. By potential well method we prove the existence and nonexistence of global weak solution without establishing the local existence theory. And we derive some sharp conditions for global existence and lack of global existence solution.     

On Frankl and Füredi’s conjecture for 3-uniform hypergraphs

Frankl and Füredi in [1] conjectured that the r-graph with m edges formed by taking the first m sets in the colex ordering of N^(r) has the largest Lagrangian of all r-graphs with m edges. Denote this r-graph by C _r,m and the Lagrangian of a hypergraph by λ(G). In this paper, we first show that if \(\leqslant m \leqslant \left( {\begin{array}{*{20}{c}}t \\ 3 \end{array}} \right)\), G is a left-compressed 3-graph with m edges and on vertex set [t], the triple with minimum colex ordering in G ^c is (t ? 2 ? i)(t ? 2)t, then λ(G) ≤ λ(C _3,m ). As an implication, the conjecture of Frankl and Füredi is true for \(\left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right) - 6 \leqslant m \leqslant \left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right)\).     

Existence of optimal control without convexity and a bang-bang theorem for linear volterra equations

We consider a Mayer problem of optimal control monitored by an integral equation of Volterra type: $$x(t) = x(t_1 ) + \int_{t_1 }^t { [h(t,s)x(s) + g(t, s)f(s, u(s))] ds,} $$ where the measurable control functionu satisfies a constraint of the formu(t) ∈U(t) ?E ^m,t ₁≤t≤t ₂, andg is a continuous kernel. Using the resolvent kernel associated with the kernelh, we prove the existence of an optimal usual solution for orientor fields without convexity assumptions. Further, ifU is a fixed compact set, we show the existence of an optimal bang-bang control.     

Positive solutions to singular semipositone m-point n-order boundary value problems

In this paper, we consider the existence of positive solutions to the following Singular Semipositone m-Point n-order Boundary Value Problems (SBVP): $$\left\{\begin{array}{l@{\quad}l}(-1)^{(n-k)}x^{(n)}(t)=\lambda f(t,x(t)),&0<t<1,\\[4pt]x(1)=\sum_{i=1}^{m-2}a_ix(\eta_i),\qquad x^{(i)}(0)=0,&0\leq i\leq k-1,\\[4pt]x^{(j)}(1)=0,&1\leq j\leq n-k-1,\end{array}\right.$$ where m≥3, λ>0, a _i ∈[0,∞),(i=1,2,…,m?2),0<η ₁<η ₂<???<η _m?2<1 are constants, f:(0,1)×[0,+∞)→R is continuous and may have singularity at t=0 and/or 1. Without making any monotone-type assumption, we obtain the positive solution of the problem for λ lying in some interval, based on fixed-point index theorem in a cone.     

Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance

We consider a nonlinear system of two-dimensional Klein-Gordon equations with masses m₁, m₂ satisfying the resonance relation m₂=2m₁>0. We introduce a structural condition on the nonlinearities under which the solution exists globally in time and decays at the rate O(|t|⁻¹) as t→±∞ in L^∞. In particular, our new condition includes the Yukawa type interaction, which has been excluded from the null condition in the sense of J.-M. Delort, D. Fang and R. Xue [J.-M. Delort, D. Fang, R. Xue, Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions, J. Funct. Anal. 211 (2004) 288-323].     

Existence results for KS3(v; 2, 4)s

A Kirkman square with index λ, latinicity μ, block size k and v points, KS_k(v; μ, λ), is a t × t array (t = λ(v − 1)/μ(k − 1)) defined on a v-set V such that (1) each point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a k subset of V, and (3) the collection of blocks obtained from the nonempty cells of the array is a (v, k, λ)-BIBD. The existence question for KS₂(v; μ, λ) has been completely selttled. We are interested in the next case k = 3. The case k = 3 and μ = λ = 1 appears to be quite difficult, although some existence results are available. For λ > 1 and μ ⩾ 1, the problem is more tractable. In this paper, we prove the existence of KS₃(v; 2, 4) for v ≡ 3 (mod 12), v ≡ 6 (mod 60) and v ≡ 9 (mod 96).