On deviations between empirical and quantile processes for mixing random variables

Let {X_n} be a strictly stationary φ-mixing process with Σ_j=1^∞ φ^1/2(j) < ∞. It is shown in the paper that if X₁ is uniformly distributed on the unit interval, then, for any t [0, 1], |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log log n)^3/4) a.s. and sup_0≤t≤1 |F_n⁻¹(t) − t + F_n(t) − t| = (O(n^−3/4(log n)^1/2(log log n)^1/4) a.s., where F_n and F_n⁻¹(t) denote the sample distribution function and tth sample quantile, respectively. In case {X_n} is strong mixing with exponentially decaying mixing coefficients, it is shown that, for any t [0, 1], |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log n)^1/2(log log n)^3/4) a.s. and sup_0≤t≤1 |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log n)(log log n)^1/4) a.s. The results are further extended to general distributions, including some nonregular cases, when the underlying distribution function is not differentiable. The results for φ-mixing processes give the sharpest possible orders in view of the corresponding results of Kiefer for independent random variables.     

Some Properties of Nonstar Steps in Addition Chains and New Cases Where the Scholz Conjecture Is True

Let ℓ(n) be the smallest possible length of addition chains for a positive integer n. Then Scholz conjectured that ℓ(2ⁿ − 1) ≤ n + ℓ(n) − 1, which still remains open. It is known that the Scholz conjecture is true when ν(n) ≤ 4, where ν(n) is the number of 1's in the binary representation of n. In this paper, we give some properties of nonstar steps in addition chains and prove that the Scholz conjecture is true for infinitely many new integers including the case where ν(n) = 5.     

An Error Expansion for some Gauss–Turán Quadratures and <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Estimates of the Remainder Term

Our aim in this paper is to obtain error expansions in the Gauss–Turán quadrature formula ∫₋₁¹f(t)w(t) dt=∑_ν=1ⁿ∑_i=0^2sA_i,νf⁽ⁱ⁾(τ_ν)+R_n,s(f), in the case when f is an analytic function in some region of the complex plane containing the interval [−1,1] in its interior. Using a representation of the remainder term R_n,s(f) in the form of contour integral over confocal ellipses, we obtain R_n,1(f) for the four Chebyshev weights and R_n,2(f) for the Chebyshev weight of the first kind. Also, we get a few new L¹-estimates of the remainder term, which are stronger than the previous ones. Some numerical results, illustrations and comparisons are also given. AMS subject classification (2000) 41A55, 65D30, 65D32.Received January 2004. Accepted October 2004. Communicated by Lothar Reichel.M. M. Spalević: This work was supported in part by the Serbian Ministry of Science and Environmental Protection (Project: Applied Orthogonal Systems, Constructive Approximation and Numerical Methods, grant number 2002).     

Oscillations of first-order neutral delay differential equations

Consider the neutral delay differential equation (*) (d/dt)[y(t) + py(t − τ)] + qy(t − σ) = 0, t t₀, where τ, q, and σ are positive constants, while p ε (−∞, −1) (0, + ∞). (For the case p ε [−1, 0] see Ladas and Sficas, Oscillations of neutral delay differential equations (to appear)). The following results are then proved. Theorem 1. Assume p < − 1. Then every nonoscillatory solution y(t) of Eq. (*) tends to ± ∞ as t → ∞. Theorem 2. Assume p < − 1, τ > σ, and q(σ − τ)/(1 + p) > (1/e). Then every solution of Eq. (*) oscillates. Theorems 3. Assume p > 0. Then every nonoscillatory solution y(t) of Eq. (*) tends to zero as t → ∞. Theorem 4. Assume p > 0. Then a necessary condition for all solutions of Eq. (*) to oscillate is that σ > τ. Theorem 5. Assume p > 0, σ > τ, andq(σ − τ)/(1 + p) > (1/e). Then every solution of Eq. (*) oscillates. Extensions of these results to equations with variable coefficients are also obtained.     

Routing through a generalized switchbox

We present an algorithm for the routing problem for two-terminal nets in generalized switchboxes. A generalized switchbox is any subset R of the planar rectangular grid with no nontrivial holes, i.e., every finite face has exactly four incident vertices. A net is a pair of nodes of nonmaximal degree on the boundary of R. A solution is a set of edge-disjoint paths, one for each net. Our algorithm solves standard generalized switchbox routing problems in time O(n(log n)²) where n is the number of vertices of R, i.e., it either finds a solution or indicates that there is none. A problem is standard if deg(ν) + ter(ν) is even for all vertices ν where deg(ν) is the degree of ν and ter(ν) is the number of nets which have ν as a terminal. For nonstandard problems we can find a solution in time O(n(log n)² + |U|²) where U is the set of vertices ν with deg(ν) + ter(ν) is odd.     

Global bifurcation of a class of periodic boundary-value problems

We prove that λ=0 is a global bifurcation point of the second-order periodic boundary-value problem (p(t)x^′(t))^′−λx(t)−λ²x^′(t)−f(t,x(t),x^′(t),x^″(t));x(0)=x(1),x^′(0)=x^′(1). We study this equation under hypotheses for which it may be solved explicitly for x^″(t). However, it is shown that the explicitly solved equation does not satisfy the usual conditions that are sufficient to conclude global bifurcation. Thus, we need to study the implicit equation with regard to global bifurcation.     

On the approximate behavior of the posterior distribution for an extreme multivariate observation

The behavior of the posterior for a large observation is considered. Two basic situations are discussed; location vectors and natural parameters.Let X = (X₁, X₂, …, X_n) be an observation from a multivariate exponential distribution with that natural parameter Θ = (Θ₁, Θ₂, …, Θ_n). Let θ_x^* be the posterior mode. Sufficient conditions are presented for the distribution of Θ − θ_x^* given X = x to converge to a multivariate normal with mean vector 0 as |x| tends to infinity. These same conditions imply that E(Θ | X = x) − θ_x^* converges to the zero vector as |x| tends to infinity.The posterior for an observation X = (X₁, X₂, …, X_n is considered for a location vector Θ = (Θ₁, Θ₂, …, Θ_n) as x gets large along a path, γ, in Rⁿ. Sufficient conditions are given for the distribution of γ(t) − Θ given X = γ(t) to converge in law as t → ∞. Slightly stronger conditions ensure that γ(t) − E(Θ | X = γ(t)) converges to the mean of the limiting distribution.These basic results about the posterior mean are extended to cover other estimators. Loss functions which are convex functions of absolute error are considered. Let δ be a Bayes estimator for a loss function of this type. Generally, if the distribution of Θ − E(Θ | X = γ(t)) given X = γ(t) converges in law to a symmetric distribution as t → ∞, it is shown that δ(γ(t)) − E(Θ | X = γ(t)) → 0 as t → ∞.     

Positive Solutions of the Equation (t) = −c(t)x(t − τ) in the Critical Case

The equation (t) − c(t)x(t − τ) is considered in the critical case. For it, the asymptotic behavior of dominant and subdominant solutions is studied. A generalization is made and connections with known results are discussed.     

Asymptotic distributions of regression and autoregression coefficients with martingale difference disturbances

In this paper a form of the Lindeberg condition appropriate for martingale differences is used to obtain asymptotic normality of statistics for regression and autoregression. The regression model is y_t = Bz_t + v_t. The unobserved error sequence {v_t} is a sequence of martingale differences with conditional covariance matrices {Σ_t} and satisfying sup_{t=1,…, n} {v′_tv_tI(v′_tv_t>a) |z_t, v_t−1, z_t−1, …} 0 as a → ∞. The sample covariance of the independent variables z₁, …, z_n, is assumed to have a probability limit M, constant and nonsingular; max_t=1,…,nz′_tz_t/n 0. If (1/n)Σ_t=1ⁿΣ_t Σ, constant, then √nvec( _n−B) N(0,M⁻¹Σ) and _n Σ. The autoregression model is x_t = Bx_{t − 1} + v_t with the maximum absolute value of the characteristic roots of B less than one, the above conditions on {v_t}, and (1/n)Σ_t=max(r,s)+1(Σ_tv_t−1−rv′_t−1−s) δ_rs(ΣΣ), where δ_rs is the Kronecker delta. Then √nvec( _n−B) N(0,Γ⁻¹Σ), where Γ = Σ_{s = 0}^∞B^sΣ(B′)^s.     

Second-order linearity of the general signed-rank statistic

Let X₁,…, X_n be i.i.d. random variables symmetric about zero. Let R_i(t) be the rank of |X_i − tn^−1/2| among |X₁ − tn^−1/2|,…, |X_n − tn^−1/2| and T_n(t) = Σ_{i = 1}ⁿφ((n + 1)⁻¹R_i(t))sign(X_i − tn^−1/2). We show that there exists a sequence of random variables V_n such that sup_{0 ≤ t ≤ 1} |T_n(t) − T_n(0) − tV_n| → 0 in probability, as n → ∞. V_n is asymptotically normal.     

Oscillation of Second-Order Differential Equations with Mixed Argument

Our aim in this paper is to present a sufficient condition for the oscillation of the second order differential equation with mixed argument [formula] We compare (*) with the first order advanced equation of the form z′(t) − q₂(t)z(σ(t)) = 0.     

On the graph of a function over a prime field whose small powers have bounded degree

Let f be a function from a finite field with a prime number p of elements, to . In this article we consider those functions f(X) for which there is a positive integer with the property that f(X)ⁱ, when considered as an element of , has degree at most p−2−n+i, for all i=1,…,n. We prove that every line is incident with at most t−1 points of the graph of f, or at least n+4−t points, where t is a positive integer satisfying n>(p−1)/t+t−3 if n is even and n>(p−3)/t+t−2 if n is odd. With the additional hypothesis that there are t−1 lines that are incident with at least t points of the graph of f, we prove that the graph of f is contained in these t−1 lines. We conjecture that the graph of f is contained in an algebraic curve of degree t−1 and prove the conjecture for t=2 and t=3. These results apply to functions that determine less than directions. In particular, the proof of the conjecture for t=2 and t=3 gives new proofs of the result of Lovász and Schrijver [L. Lovász, A. Schrijver, Remarks on a theorem of Rédei, Studia Sci. Math. Hungar. 16 (1981) 449–454] and the result in [A. Gács, On a generalization of Rédei’s theorem, Combinatorica 23 (2003) 585–598] respectively, which classify all functions which determine at most 2(p−1)/3 directions.     

Error bounds for least squares approximation by polynomials

Let f ε Cⁿ⁺¹[−1, 1] and let H[f](x) be the nth degree weighted least squares polynomial approximation to f with respect to the orthonormal polynomials q_k associated with a distribution dα on [−1, 1]. It is shown that if q_n+1/q_n max(q_n+1(1)/q_n(1), −q_n+1(−1)/q_n(−1)), then f − H[f] f^{n + 1} · q_n+1/q_{n + 1}^{(n + 1)}, where · denotes the supremum norm. Furthermore, it is shown that in the case of Jacobi polynomials with distribution (1 − t)^α (1 + t)^β dt, α, β > −1, the condition on q_n+1/q_n is satisfied when either max(α,β) −1/2 or −1 < α = β < −1/2.     

Asymptotic expansions in the conditional central limit theorem

Let X_n, n , be i.i.d. with mean 0, variance 1, and E(¦X_n¦^r) < ∞ for some r 3. Assume that Cramér's condition is fulfilled. We prove that the conditional probabilities P(1/√n Σ_{i = 1}ⁿ X_i t¦B) can be approximated by a modified Edgeworth expansion up to order o(1/n^{(r − 2)/2)}), if the distances of the set B from the σ-fields σ(X₁, …, X_n) are of order O(1/n^{(r − 2)/2)}(lg n)^β), where β < −(r − 2)/2 for r and β < −r/2 for r . An example shows that if we replace β < −(r − 2)/2 by β = −(r − 2)/2 for r (β < −r/2 by β = −r/2 for r ) we can only obtain the approximation order O(1/n^{(r − 2)/2)}) for r (O(lg lgn/n^{(r − 2)/2)}) for r ).     

Uniform Estimates for a Class of Evolution Equations

L^∞ estimates are derived for the oscillatory integral ∫⁺₀^∞e^{−i(xλ + (1/m) tλm)}a(λ) dλ, where 2 ≤ m and (x, t) × ₊. The amplitude a(λ) can be oscillatory, e.g., a(λ) = e^{it (λ)} with (λ) a polynomial of degree ≤ m − 1, or it can be of polynomial type, e.g., a(λ) = (1 + λ)^k with 0 ≤ k ≤ (m − 2). The estimates are applied to the study of solutions of certain linear pseudodifferential equations, of the generalized Schrödinger or Airy type, and of associated semilinear equations.     

On Hilbert''s Integral Inequality

In this paper, we generalize Hilbert's integral inequality and its equivalent form by introducing three parameterst,a, andb.Iff, g L²[0, ∞), then[formula]where π is the best value. The inequality (1) is well known as Hilbert's integral inequality, and its equivalent form is[formula]where π²is also the best value (cf. [[1], Chap. 9]). Recently, Hu Ke made the following improvement of (1) by introducing a real functionc(x),[formula]wherek(x) = 2/π∫^∞₀(c(t²x)/(1 + t²)) dt − c(x), 1 − c(x) + c(y) ≥ 0, andf, g ≥ 0 (cf. [[2]]). In this paper, some generalizations of (1) and (2) are given in the following theorems, which are other than those in [ [2]].     

On existence of singular solutions

In the paper sufficient conditions are given under which the differential equation y⁽ⁿ⁾=f(t,y,…,y⁽ⁿ⁻²⁾)g(y⁽ⁿ⁻¹⁾) has a singular solution y :[T,τ)→R, τ<∞ fulfilling

     

Randomized online graph coloring

We study the problem of coloring graphs in an online manner. The only known deterministic online graph coloring algorithm with a sublinear performance function was found by [9.], 319–325). Their algorithm colors graphs of chromatic number χ with no more than (2χn)/log^* n colors, where n is the number of vertices. They point out that the performance can be improved slightly for graphs with bounded chromatic number. For three-chromatic graphs the number of colors used, for example, is O(n log log log n/log log n). We show that randomization helps in coloring graphs online. We present a simple randomized online algorithm to color graphs with expected number of colors O(2^χχ²n^{(χ−2)/(χ−1)}(log n)^1/(χ−1)). For three-colorable graphs the expected number of colors our algorithm uses is . All our algorithms run in polynomial time. It is interesting to note that our algorithm compares well with the best known polynomial time offline algorithms. For instance, the best polynomial time algorithm known for three-colorable graphs, due to [4.] pp. 554–562). We also prove a lower bound of Ω((1/(χ − 1))((log n/(12(χ + 1))) − 1)^χ−1) for the randomized model. No lower bound for the randomized model was previously known. For bounded χ, our result improves even the best known lower bound for the deterministic case: Ω((log n/log log n)^χ−1), due to Noga Alon (personal communication, September 1989).     

On positive solutions of some nonlinear fourth-order beam equations

The existence, uniqueness and multiplicity of positive solutions of the following boundary value problem is considered:

u⁽⁴⁾(t)−λf(t,u(t))=0, for 0<t<1,u(0)=u(1)=u″(0)=u″(1)=0,

where λ>0 is a constant, f :[0,1]×[0,+∞)→[0,+∞) is continuous.     

Trace cocharacters and the Kronecker products of Schur functions

It follows from the theory of trace identities developed by Procesi and Razmyslov that the trace cocharacters arising from the trace identities of the algebra M_r(F) of r×r matrices over a field F of characteristic zero are given by TC_r,n=∑_λΛr(n)χ^λχ^λ where χ^λχ^λ denotes the Kronecker product of the irreducible characters of the symmetric group associated with the partition λ with itself and Λ_r(n) denotes the set of partitions of n with r or fewer parts, i.e. the set of partitions λ=(λ₁λ_k) with kr. We study the behavior of the sequence of trace cocharacters TC_r,n. In particular, we study the behavior of the coefficient of χ^(ν,n−m) in TC_r,n as a function of n where ν=(ν₁ν_k) is some fixed partition of m and n−mν_k. Our main result shows that such coefficients always grow as a polynomial in n of degree r−1.