Segments in a planar nearring

Suppose (N,+,·) is a finite (left) planar nearring. Define the (line) segment in N with endpoints a and b by

We prove that under many circumstances we have

which is the same phenomenon as in the Euclidean plane.     

Nonexistence of positive solutions for 2 order integro-differential inequalities

In this paper, we shall show that under suitable conditions on f and K, the inequalities

imply that the integro-differential inequalities

have no positive solutions, respectively.     

An extension theorem for t-designs

In 1994, van Trung (Discrete Math. 128 (1994) 337–348) [9] proved that if, for some positive integers d and h, there exists an S_λ(t,k,v) such that

then there exists an S_λ(v−t+1)(t,k,v+1) having v+1 pairwise disjoint subdesigns S_λ(t,k,v). Moreover, if B_i and B_j are any two blocks belonging to two distinct such subdesigns, then d|B_i∩B_j|<k−h. In 1999, Baudelet and Sebille (J. Combin. Des. 7 (1999) 107–112) proved that if, for some positive integers, there exists an S_λ(t,k,v) such that

where m=min{s,v−k} and n=min{i,t}, then there exists an

having pairwise disjoint subdesigns S_λ(t,k,v). The purpose of this paper is to generalize these two constructions in order to produce a new recursive construction of t-designs and a new extension theorem of t-designs.     

On the convergence of generalized moments in almost sure central limit theorem

Let {ζ_k} be the normalized sums corresponding to a sequence of i.i.d. variables with zero mean and unit variance. Define random measures

and let G be the normal distribution. We show that for each continuous function h satisfying ∫ hdG<∞ and a mild regularity assumption, one has

a.s.     

Law of the iterated logarithm for Lévy''s area process composed with Brownian motion

Let {A(t)}−∞<t<∞ be Lévy's stochastic area process and assume {W(t)}t0 is an independent Brownian motion. Then we prove the following local law of the iterated logarithm for the composed process {A(W(t))}t;0:

.     

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.     

The integrity of a cubic graph

Integrity, a measure of network reliability, is defined as

where G is a graph with vertex set V and m(G−S) denotes the order of the largest component of G−S. We prove an upper bound of the following form on the integrity of any cubic graph with n vertices:

Moreover, there exist an infinite family of connected cubic graphs whose integrity satisfies a linear lower bound I(G)>βn for some constant β. We provide a value for β, but it is likely not best possible. To prove the upper bound we first solve the following extremal problem. What is the least number of vertices in a cubic graph whose removal results in an acyclic graph? The solution (with a few minor exceptions) is that n/3 vertices suffice and this is best possible.     

A characterization for the boundedness of geometric mean operator

We give a characterization for the geometric mean inequality

to hold for the case 0 < q < p ≤ ∞, p > 1, where f is positive a.e. on (0, ∞), and C > 0 independent of f.     

Two theorems on Euclidean distance matrices and Gale transform

We present a characterization of those Euclidean distance matrices (EDMs) D which can be expressed as D=λ(E−C) for some nonnegative scalar λ and some correlation matrix C, where E is the matrix of all ones. This shows that the cones

where is the elliptope (set of correlation matrices) and is the (closed convex) cone of EDMs.

The characterization is given using the Gale transform of the points generating D. We also show that given points , for any scalars λ₁,λ₂,…,λ_n such that

∑_j=1ⁿλ_jp^j=0, ∑_j=1ⁿλ_j=0,

we have

∑_j=1ⁿλ_jpⁱ−p^j2= forall i=1,…,n,

for some scalar independent of i.     

A parallel computing scheme for minimizing a class of large scale functions

This paper gives a parallel computing scheme for minimizing a twice continuously differentiable function with the form

where x = (x^T₁,…,x^T_m)^T and x_i R^n_i, ∑^m_{i = 1}n_i = n, and n a very big number. It is proved that we may use m parallel processors and an iterative procedure to find a minimizer of ƒ(x). The convergence and convergence rate are given under some conditions. The conditions for finding a global minimizer of ƒ(x by using this scheme are given, too. A similar scheme can also be used parallelly to solve a large scale system of nonlinear equations in the similar way. A more general case is also investigated.     

A formula for the number of Latin squares

We establish an explicit formula for the number of Latin squares of order n:

, where B_n is the set of n×n(0,1) matrices, σ₀(A is the number of zero elements of the matrix A and per A is the permanent of the matrix A.     

On the Glivenko–Cantelli theorem for generalized empirical processes based on strong mixing sequences

Given \s{X_i, i 1\s} as non-stationary strong mixing (n.s.s.m.) sequence of random variables (r.v.'s) let, for 1 i n and some γ ε [0, 1],

F₁(x)=γP(X_i<x)+(1-γ)P(X_ix)

and

I_i(x)=γI(X_i<x)+(1-γ)I(X_ix)

. For any real sequence \s{C_i\s} satisfying certain conditions, let

.

In this paper an exponential type of bound for P(D_n ), for any >0, and a rate for the almost sure convergence of D_n are obtained under strong mixing. These results generalize those of Singh (1975) for the independent and non-identically distributed sequence of r.v.'s to the case of strong mixing.     

Homogeneous renewal equation

Let F be a non-arithmetic distribution on the line , and W be the class of bounded functions w without discontinuity of the second kind such that

.In this paper, we show that the solution of the homogeneous renewal equation w = w F in the class W is a constant-function.     

Uniqueness of positive solutions for quasilinear boundary value problems

Sufficient conditions for the uniqueness of positive solutions of boundary value problems for quasilinear differential equations of the type

(|u′|^m−2u′)′ + f(t,u,u′)=0, m 2

are established. These problems arise, for example, in the study of the m-Laplace equation in annular regions.     

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.     

Oscillation of higher-order nonlinear difference equations of neutral type

We shall establish some new criteria for the oscillation of all solutions of higher-order difference equations of the form

δ^m(x_n-x_n-r)+q_nf(x_n-g=0, m1

     

Exponential boundedness of solutions for impulsive delay differential equations

For an impulsive delay differential equation

exponent estimates of solutions have been obtained.     

Analytical and numerical investigation of two families of Lorenz-like dynamical systems

We investigate two families of Lorenz-like three-dimensional nonlinear dynamical systems (i) the generalized Lorenz system and (ii) the Burke–Shaw system. Analytical investigation of the former system is possible under the assumption (I) which in fact concerns four different systems corresponding to  = ±1, m = 0, 1.

(I)

The fixed points and stability characteristics of the Lorenz system under the assumption (I) are also classified. Parametric and temporal (t → ∞) asymptotes are also studied in connection to the memory of both the systems. We calculate the Lyapunov exponents and Lyapunov dimension for the chaotic attractors in order to study the influence of the parameters of the Lorenz system on the attractors obtained not only when the assumption (I) is satisfied but also for other values of the parameters σ, r, b, ω and m.     

The oscillation of higher-ordernonlinear difference equations

The authors discuss the relation of the oscillation of the following two difference equations,

where m ≥ 2, τ : N → N, N isthe set of integers, |n − τ(n)| ≤ Mfor n N₀, M is a positive integer, is nondecreasing in x, xf(n, x)> 0, as x ≠ 0. Wewill show some relations of the oscillation of the above two equations. Especially, for m to be even, we establish the equivalenceof the oscillation of the above two difference equations.