Neighborhood unions and cyclability of graphs

A graph G is said to be cyclable if for each orientation of G, there exists a set S of vertices such that reversing all the arcs of with one end in S results in a hamiltonian digraph. Let G be a 3-connected graph of order n36. In this paper, we show that if for any three independent vertices x₁, x₂ and x₃, |N(x₁)N(x₂)|+|N(x₂)N(x₃)|+|N(x₃)N(x₁)|2n+1, then G is cyclable.     

Cases of equality for the spectral norm submultiplicativity of the hadamard product

In this note, we characterize those pairs of nonzero r-by-d complex matrices that satisfy N₂(A ∘ B) = N₂(A)N₂(B), in which N₂(·) is the spectral norm and · is the Hadamard product.     

On the Equation n_1n_2= n_3n_4 Restricted to Factor Closed Sets

We study the number of solutions N(B,F) of the diophantine equation n_1n_2 = n_3 n_4,where 1 ≤ n_1 ≤ B,1 ≤ n_3 ≤ B,n_2,n_4 ∈ F and F[1,B] is a factor closed set.We study more particularly the case when F={m = p_1~(ε1)···p_k~(εk),ε_j∈{0,1},1 ≤ j ≤ k},p_1,...,p_k being distinct prime numbers.     

On neighborhood condition for graphs to have [a,b]-factors

Let G be a graph of order n, and let a and b be integers such that 1a<b. Let δ(G) be the minimum degree of G. Then we prove that if δ(G)(k−1)a, n(a+b)(k(a+b)−2)/b, and |N_G(x₁)N_G(x₂)N_G(x_k)|an/(a+b) for any independent subset {x₁,x₂,…,x_k} of V(G), where k2, then G has an [a,b]-factor. This result is best possible in some sense.     

On generalized Boolean functions II

This cycle of papers is based on the concept of generalized Bolean functions introduced by the author in the first article of the series. Every generalized Boolean function f:Bⁿ→B can be written in a manner similar to the canonical disjunctive form using some function defined on A×B, where A is a finite subset of B containing 0 and 1. The set of those functions f is denoted by GBF_n[A]. In this paper the following questions are presented: (1) What is the relationship between GBF_n[A₁] and GBF_n[A₂] when A₁A₂. (2) What can be said about GBF_n[A₁∩A₂] and GBF_n[A₁A₂] in comparison with GBF_n[A₁]∩GBF_n[A₂] and GBF_n[A₁]GBF_n[A₂], respectively.     

On meromorphic solutions of a certain type of nonlinear differential equations

We consider transcendental meromorphic solutions with N(r,f) = S(r,f) of the following type of nonlinear differential equations:f~n + Pn-2(f) = p1(z)e~(α1(z)) +p2(z)e~(α2(z)),where n≥ 2 is an integer, Pn-2(f) is a differential polynomial in f of degree not greater than n-2 with small functions of f as its coefficients, p1(z), p2(z) are nonzero small functions of f, and α1(z), α2(z)are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of meromorphic solutions and their possible forms of the above equation. Our results extend and improve some known results obtained most recently.     

Invariant record processes and applications to best choice modelling

Let X₁, X₂,…be identically distributed random variables from an unknown continuous distribution. Further let I_r(1), I_r(2),…be a sequence of indicator functions defined on X₁, X₂,…by I_r(k) = 0 if k < r, I_r(k) = 1 if X_k is a r-record AND = 0 otherwise. Suppose that we observe X₁, X₂,… at times T₁ < T₂ <… where the T_k's are realisations of some regular counting process (N(τ)) defined on the positive half-line. Having observed [0, τ], say, the problem is to predict the future behaviour of the counting processes (R_r(τ, s))_sτ = # r-records in [τ, s]. More specifically the objective of this paper is to show that these processes can be (inhomogeneous) Poisson processes even if (N(τ))_τ0 has dependent increments.

The strong link between optimal selection and optimal stopping of record sequences or record processes, perhaps not fully recognized so far, is pointed out in this paper. It is shown to lead to a unification of the treatment of problems which, at first sight, are rather different. Moreover the stopping of record processes in continuous time can lead to rigorous and elegant solutions in cases where dynamic programming is bound to fail. Several examples will be given to facilitate a comparison with other methods.     

A two-sided estimate in the Hsu-Robbins-Erdös law of large numbers

Let X₁, X₂, … be independent identically distributed random variables. Then, Hsu and Robbins (1947) together with Erdös (1949, 1950) have proved that

,

if and only if E[X²₁] < ∞ and E[X₁] = 0. We prove that there are absolute constants C₁, C₂ (0, ∞) such that if X₁, X₂, … are independent identically distributed mean zero random variables, then

c₁λ⁻² E[X₁²·1_{|X1|λ}]S(λ)C₂λ⁻² E[X₁²·1_{|X1|λ}]

,

for every λ > 0.     

Asymptotic behavior of a nonlinear delay difference equation

This paper considers a class of nonlinear difference equations

Δ₃y_n + ƒ(n, y_n, y_n−r) = 0, n N (n₀)

. A necessary and sufficient condition for the existence of a bounded nonoscillatory solution is given.     

Bilevel control of degraded machining system with warm standbys, setup and vacation

In this paper, we study (N, L) switch-over policy for machine repair model with warm standbys and two repairmen. The repairman (R₁) turns on for repair only when N-failed units are accumulated and starts repair after a set up time which is assumed to be exponentially distributed. As soon as the system becomes empty, the repairman (R₁) leaves for a vacation and returns back when he finds the number of failed units in the system greater than or equal to a threshold value N. Second repairman (R₂) turns on when there are L(>N) failed units in the system and goes for a vacation if there are less than L failed units. The life time and repair time of failed units are assumed to be exponentially distributed. The steady state queue size distribution is obtained by using recursive method. Expressions for the average number of failed units in the queue and the average waiting time are established.     

考虑非周期检测的视情维修与备件订购联合策略优化

针对单部件系统/关键部件提出视情维修与备件订购联合策略,其中系统退化服从两阶段延迟时间过程且采用非周期检测策略,退化初期以检测间隔T₁检查系统状态,而在第一次识别缺陷状态时,缩短检测周期为T₂、订购备件且进行不完美维修;若系统在随后的退化中被识别处于缺陷状态,执行不完美维修直至超过阈值次数N_max并采取预防性更换,但若在检测周期内发生故障则进行更换。根据系统状态和备件状态分析各种可能更新事件及相应的联合决策,利用更新报酬理论构建最小化单位时间内期望成本的目标函数,优化T₁,T₂, N_max。与对比模型策略相比,算例结果表明所提出的联合策略能有效降低单位时间内的期望成本。     

On the sum of two N0-matrices

In this paper, we study the Fan product of two N₀-matrices and inequalities for two N₀-matrices A,B with A ≤ B. Necessary and sufficient conditions are given for the sum of two symmetric N₀-matrices to be an N₀-matrix.     

Additive Maps Preserving Nilpotent Perturbation of Scalars

Let X be a Banach space over F(= R or C) with dimension greater than 2. Let N(X) be the set of all nilpotent operators and B_0(X) the set spanned by N(X). We give a structure result to the additive maps on FI + B_0(X) that preserve rank-1 perturbation of scalars in both directions. Based on it, a characterization of surjective additive maps on FI + B_0(X) that preserve nilpotent perturbation of scalars in both directions are obtained. Such a map Φ has the form either Φ(T) = cAT A~(-1)+ φ(T)I for all T ∈ FI + B_0(X) or Φ(T) = cAT*A~(-1)+ φ(T)I for all T ∈ FI + B_0(X), where c is a nonzero scalar,A is a τ-linear bijective transformation for some automorphism τ of F and φ is an additive functional.In addition, if dim X = ∞, then A is in fact a linear or conjugate linear invertible bounded operator.     

π-Armendariz rings relative to a monoid

Let Mbe a monoid. A ring Ris called M-π-Armendariz if whenever α = a₁g₁+ a₂g₂+ · · · + a_ng_n, β = b₁h₁+ b₂h₂+ · · · + b_mh_m ∈ R[M] satisfy αβ ∈ nil(R[M]), then aibj ∈ nil(R) for all i, j. A ring R is called weakly 2-primal if the set of nilpotent elements in R coincides with its Levitzki radical. In this paper, we consider some extensions of M-π-Armendariz rings and further investigate their properties under the condition that R is weakly 2-primal. We prove that if R is an M-π-Armendariz ring then nil(R[M]) = nil(R)[M]. Moreover, we study the relationship between the weak zip-property (resp., weak APP-property, nilpotent p.p.-property, weak associated prime property) of a ring R and that of the monoid ring R[M] in case R is M-π-Armendariz.     

A note on the stationarity of a threshold first-order bilinear process

In the present note we study the threshold first-order bilinear model

X(t)=aX(t−1)+(b₁1_{X(t−1)<c}+b₂1_{X(t−1)c})X(t−1)e(t−1)+e(t), tεN

where {e(t), tεN} is a sequence of i.i.d. absolutely continuous random variables, X(0) is a given random variable and a, b₁, b₂ and c are real numbers. Under suitable conditions on the coefficients and lower semicontinuity of the densities of the noise sequence, we provide sufficient conditions for the existence of a stationary solution process to the present model and of its finite moments of order p.     

Flawless 0-sequences and Hilbert functions of Cohen-Macaulay integral domains

Let A = A₀A₁ be a commutative graded ring such that (i) A₀ = k a field, (ii) A = k[A₁] and (iii) dim_k A₁ < ∞. It is well known that the formal power series ∑^∞_{n = 0} (dim_kA_n)λ ⁿ is of the form (h₀ + h₁λ + + h_sλ^s)/(1 − λ)^dimA with each h_iε . We are interested in the sequence (h₀, h₁,…,h_s), called the h-vector of A, when A is a Cohen–Macaulay integral domain. In this paper, after summarizing fundamental results (Section 1), we study h-vectors of certain Gorenstein domains (Section 2) and find some examples of h-vectors arising from integrally closed level domains (Sections 3 and 4).     

Jacobi-Sobolev-type orthogonal polynomials: Second-order differential equation and zeros

We obtain an explicit expression for the Sobolev-type orthogonal polynomials {Q_n} associated with the inner product

, where p(x) = (1 − x)(1 + x)^β is the Jacobi weight function, ,β> − 1, A₁,B₁,A₂,B₂0 and p, q P, the linear space of polynomials with real coefficients. The hypergeometric representation (₆F₅) and the second-order linear differential equation that such polynomials satisfy are also obtained. The asymptotic behaviour of such polynomials in [−1, 1] is studied. Furthermore, we obtain some estimates for the largest zero of Q_n(x). Such a zero is located outside the interval [−1, 1]. We deduce his dependence of the masses. Finally, the WKB analysis for the distribution of zeros is presented.     

Asymptotic bounds for some bipartite graph: complete graph Ramsey numbers

The Ramsey number r(H,K_n) is the smallest integer N so that each graph on N vertices that fails to contain H as a subgraph has independence number at least n. It is shown that r(K_2,m,K_n)(m−1+o(1))(n/log n)² and r(C_2m,K_n)c(n/log n)^m/(m−1) for m fixed and n→∞. Also r(K_2,n,K_n)=Θ(n³/log² n) and .     

Gowers norms and pseudorandom measures of subsets

Let $A \subset {{\Bbb Z}_N}$, and ${f_A}(s) = \left\{ {\begin{array}{*{20}{l}}{1 - \frac{{|A|}}{N},}&{{\rm{for}}\;s \in A,}\\{ - \frac{{|A|}}{N},}&{{\rm{for}}\;s \notin A.}\end{array}} \right.$ We define the pseudorandom measure of order k of the subset A as follows, P_k(A, N) = $\begin{array}{*{20}{c}}{\max }\\D\end{array}$|$\mathop \Sigma \limits_{n \in {\mathbb{Z}_N}}$f_A(n + c₁)f_A(n + c₂) … f_A(n + c_k)|, where the maximum is taken over all D = (c₁, c₂, . . . , c_k) ∈ ${\mathbb{Z}^k}$ with 0 ≤ c₁ < c₂ < … < c_k ≤ N - 1. The subset A ⊂ ${{\mathbb{Z}_N}}$ is considered as a pseudorandom subset of degree k if P_k(A, N) is “small” in terms of N. We establish a link between the Gowers norm and our pseudorandom measure, and show that “good” pseudorandom subsets must have “small” Gowers norm. We give an example to suggest that subsets with “small” Gowers norm may have large pseudorandom measure. Finally, we prove that the pseudorandom subset of degree L(k) contains an arithmetic progression of length k, where L(k) = 2·lcm(2, 4, . . . , 2|$\frac{k}{2}$|), for k ≥ 4, and lcm(a₁, a₂, . . . , a_l) denotes the least common multiple of a₁, a₂, . . . , a_l.     

Oscillation theorems for second-order half-linear differential equations

Oscillation criteria for the second-order half-linear differential equation

[r(t)|ξ′(t)|⁻¹ ξ′(t)]′ + p(t)|ξ(t)|⁻¹ξ(t)=0, t t₀

are established, where > 0 is a constant and exists for t [t₀, ∞). We apply these results to the following equation:

where , D = (D₁,…, D_N), Ω_a = x ^N : |x| ≥ a} is an exterior domain, and c C([a, ∞), ), n > 1 and N ≥ 2 are integers. Here, a > 0 is a given constant.