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:

.     

Descents and one-dimensional characters for classical Weyl groups

This paper examine all sums of the form

where W is a classical Weyl group, X is a one-dimensional character of W, and d(π) is the descent statistic. This completes a picture which is known when W is the symmetric group S_n (the Weyl group A_n−1). Surprisingly, the answers turn out to be simpler and generalize further for the other classical Weyl groups B_n(C_n) and D_n. The B_n, case uses sign-reversing involutions, while the D_n case follows from a result of independent interest relating statistics for all three groups.     

Weighted domination in triangle-free graphs

A weighted graph (G,w) is a graph G together with a positive weight-function on its vertex set w : V(G)→R^>0. The weighted domination number γ_w(G) of (G,w) is the minimum weight w(D)=∑_vDw(v) of a set DV(G) such that every vertex xV(D)−D has a neighbor in D. If ∑_vV(G)w(v)=|V(G)|, then we speak of a normed weighted graph. Recently, we proved that

for normed weighted bipartite graphs (G,w) of order n such that neither G nor the complement has isolated vertices. In this paper we will extend these Nordhaus–Gaddum-type results to triangle-free graphs.     

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.     

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.     

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.     

First passage time for some stationary processes

For a 1-dependent stationary sequence {X_n} we first show that if u satisfies p₁=p₁(u)=P(X₁>u)0.025 and n>3 is such that 88np₁³1, then

P{max(X₁,…,X_n)u}=ν·μⁿ+O{p₁³(88n(1+124np₁³)+561)}, n>3,

where

ν=1−p₂+2p₃−3p₄+p₁²+6p₂²−6p₁p₂,μ=(1+p₁−p₂+p₃−p₄+2p₁²+3p₂²−5p₁p₂)⁻¹

with

p_k=p_k(u)=P{min(X₁,…,X_k)>u}, k1

and

|O(x)||x|.

From this result we deduce, for a stationary T-dependent process with a.s. continuous path {Y_s}, a similar, in terms of P{max_0skTY_s<u}, k=1,2 formula for P{max_0stY_su}, t>3T and apply this formula to the process Y_s=W(s+1)−W(s), s0, where {W(s)} is the Wiener process. We then obtain numerical estimations of the above probabilities.     

Full quadrature sums for pth powers of polynomials with Freud weights

In this paper, we provide a solution of the quadrature sum problem of R. Askey for a class of Freud weights. Let r> 0, b (− ∞, 2]. We establish a full quadrature sum estimate

1 p < ∞, for every polynomial P of degree at most n + rn^1/3, where W² is a Freud weight such as exp(−¦x¦), > 1, λ_jn are the Christoffel numbers, x_jn are the zeros of the orthonormal polynomials for the weight W², and C is independent of n and P. We also prove a generalisation, and that such an estimate is not possible for polynomials P of degree M = m(n) if m(n) = n + ξ_nn^1/3, where ξ_n → ∞ as n → ∞. Previous estimates could sum only over those x_jn with ¦x_jn¦ σx_1n, some fixed 0 < σ < 1.     

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.     

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.     

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.     

An extended result of Kleitman and Saks concerning binary trees

In a recent paper, D.J. Kleitman and M.E. Saks gave a proof of Huang's conjecture on alphabetic binary trees.

Given a set E = {e_i}, I = 0, 1, 2, …, m and assigned positive weights to its elements and supposing the elements are indexed such that w(e₀) ≤ w(e₁) ≤ … ≤w (e_m), where w(e_i) is the weight of e_i, we call the following sequence E^* a ‘saw-tooth’ sequence

E^*=(e₀,e_m,e₁,…,e_j,e_m−j,…).

Huang's conjecture is: E^* is the most expensive sequence for alphabetic binary trees. This paper shows that this property is true for the L-restricted alphabetic binary trees, where L is the maximum length of the leaves and log₂(m + 1) ≤L≤m.     

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.     

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.     

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.     

Exponential boundedness of solutions for impulsive delay differential equations

For an impulsive delay differential equation

exponent estimates of solutions have been obtained.     

New upper and lower bounds on the channel capacity of read/write isolated memory

In this paper, we refine upper and lower bounds for the channel capacity of a serial, binary rewritable medium in which no consecutive locations may store 1's and no consecutive locations may be altered during a single rewriting pass. This problem was originally examined by Cohn (Discrete. Appl. Math. 56 (1995) 1) who proved that C, the channel capacity of the memory, in bits per symbol per rewrite, satisfies

0.50913C0.56029 .

In this paper, we show how to model the problem as a constrained two-dimensional binary matrix problem and then modify recent techniques for dealing with such matrices to derive improved bounds of

0.53500C0.55209 .

     

The second kind Chebyshev–Newton–Cotes quadrature rule (open type) and its numerical improvement

The weighted Newton–Cotes quadrature rules of open type are denoted by

where w(x) is a positive function and is the step size. Various cases can be selected for the weight function of the above formula. In this paper, we consider as the main weight function and study the general formula:

The precision degree of the above formula is n + 1 for even n’s and is n for odd n’s but if one considers its upper and lower bounds as two additional variables, a nonlinear system will be derived whose solution improves the precision degree of above formula up to degree n + 2 numerically. In this way, some examples are given to show the numerical superiority of our idea.