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.     

Limit cycles of generalized Liénard equations

The generalized Liénard equations of the form:

where F, g, and h are polynomials, are examined. It has been found that the results given by Blows, Lloyd and Lynch [1–5] for Liénard equations hold also for the generalized systems. A new result is also presented within this article.     

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.     

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.     

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.     

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.     

Periodic solutions of the higher-dimensional non-autonomous systems

In this paper, the new existence theorem, unique theorem of periodic solution of the periodic system

and new stationary oscillation theorem of the periodic system

are derived by using functional analysis method, algebraic method and the new estimated formulas of solution of the homogenous linear system. Our results extend and improve some main results related to references. In addition, these criteria are of great interest in many applications such as computation.     

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.     

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.     

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.     

Exponential boundedness of solutions for impulsive delay differential equations

For an impulsive delay differential equation

exponent estimates of solutions have been obtained.     

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.     

Impulsive effects on global existence of solutions for degenerate semilinear parabolic equations

Let q be a nonnegative real number, and λ and σ be positive constants. This article studies the following impulsive problem: for n = 1, 2, 3,…,

. The number λ^* is called the critical value if the problem has a unique global solution u for λ < λ^*, and the solution blows up in a finite time for λ > λ^*. For σ < 1, existence of a unique λ^* is established, and a criterion for the solution to decay to zero is studied. For σ > 1, existence of a unique λ^* and three criteria for the blow-up of the solution in a finite time are given respectively. It is also shown that there exists a unique T^* such that u exists globally for T> T^*, and u blows up in a finite time for T < T^*.     

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.     

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.     

Fourteen limit cycles in a cubic Hamiltonian system with nine-order perturbed term

We study limit cycles of the following system:

with a>c>0, ac>1, 0<1, m,l,λ are real parameters and n is a positive integer. When n=2, J.B. Li and Z.R. Liu [Publ. Math. 35 (1991) 487] showed that the system has 11 limit cycles. When n=6, H.J. Cao, Z.R. Liu and Z.J. Jing [Chaos, Solitons & Fractals 11 (2000) 2293] showed the system has 13 limit cycles. Using the same method of detection function, we first show that the system and others four systems have the same bifurcation diagrams of limit cycle. Then we demonstrate that any one of the five systems has 14 limit cycles for n=8. The positions of the 14 limit cycles are given by numerical exploration.     

Investigation of the volume effect on a simple batch fermentation process

In this paper two models, one structured and the other unstructured, for the simple batch fermentation process, based on the kinetic scheme

, are compared quantitatively and qualitatively. One model assumes that the volume of the cells in the fermentation process is approximately constant while the other model incorporates the changing cell volume. It is shown that the specific growth rates for each model differ by approximately a factor of 2. Qualitatively, the shape of the trajectories for each model in the specific growth rate–substrate phase plane are very similar. One model is the traditional fixed volume model where the intra- and extracellular substrate levels are equal, while the “new” model incorporates the intracellular substrate (without cellular transport) with changing cell volumes.     

On infinite divisibility of normed product of Cauchy densities

It is shown that the following probability density function

is an infinitely divisible density.     

Coupling and harmonic functions in the case of continuous time Markov processes

Consider two transient Markov processes (X^v_t)_tεR·, (X^μ_t)_tεR· with the same transition semigroup and initial distributions v and μ. The probability spaces supporting the processes each are also assumed to support an exponentially distributed random variable independent of the process.

We show that there exist (randomized) stopping times S for (X^v_t), T for (X^μ_t) with common final distribution, L(X^v_S|S < ∞) = L(X^μ_T|T < ∞), and the property that for t < S, resp. t < T, the processes move in disjoint portions of the state space. For such a coupling (S, T) it is shown

where denotes the bounded harmonic functions of the Markov transition semigroup. Extensions, consequences and applications of this result are discussed.