Almost sure central limit theorem for partial sums and maxima

Let X, X₁, X₂, … be i.i.d. random variables with nondegenerate common distribution function F, satisfying EX = 0, EX² = 1. Let X_i and M_n = max{X_i, 1 ≤ i ≤ n }. Suppose there exists constants a_n > 0, b_n ∈ R and a nondegenrate distribution G (y) such that Then, we have almost surely, where f (x, y) denotes the bounded Lipschitz 1 function and Φ(x) is the standard normal distribution function (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundedness in nonlinear oscillations

In this paper, the boundedness of all solutions of the nonlinear differential equation (φ_p(x′))′ + αφ_p(x⁺) – βφ_p(x^–) + f(x) = e(t) is studied, where φ_p(u) = |u|^p–2 u, p ≥ 2, α, β are positive constants such that = 2w^–1 with w ∈ ?⁺\?, f is a bounded C⁵ function, e(t) ∈ C⁶ is 2π_p‐periodic, x⁺ = max{x, 0}, x^– = max{–x, 0}. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Anti‐periodic solutions for evolution equations with mappings in the class (S+)

In this paper, we study the existence of anti‐periodic solutions for the first order evolution equation in a Hilbert space H, where G : H → ? is an even function such that ?G is a mapping of class (S₊) and f : ? → ? satisfies f(t + T) = –f(t) for any t ∈ ? with f(·) ∈ L²(0, T; H). (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a nonlinear stochastic integral equation with application to control systems

The aim of the present paper is to study a nonlinear stochastic integral equation of the form

Regressions and monotone chains: A ramsey-type extremal problem for partial orders

Aregression is a functiong from a partially ordered set to itself such thatg(x)≦x for allz. Amonotone k-chain is a chain ofk elementsx ₁<x ₂ <...<x _k such thatg(x ₁)≦g(x ₂)≦...≦g(x _k ). If a partial order has sufficiently many elements compared to the size of its largest antichain, every regression on it will have a monotone (k + 1)-chain. Fixingw, letf(w, k) be the smallest number such that every regression on every partial order with size leastf(w, k) but no antichain larger thanw has a monotone (k + 1)-chain. We show thatf(w, k)=(w+1) ^k . Dedicated to Paul Erdős on his seventieth birthday Research supported in part by the National Science Foundation under ISP-80-11451.     

Spherical Maximal Operators on Radial Functions

Let A_tf(x) denote the mean of f over a sphere of radius t and center x. We prove sharp estimates for the maximal function M_E f(X) = sup_t∈_E |Atf(x)| where E is a fixed set in IR⁺ and f is a radial function ∈ L^p(IR^d). Let P_d = d/(d?1) (the critical exponent for Stein's maximal function). For the cases (i) p < p_d, d ? 2, and (ii) p = p_d, d ? 3, and for p ? q ? ∞ we prove necessary and sufficient conditions on E for M_E to map radial functions in L^p to the Lorentz space L^P,q.     

On nonautonomous linear systems of differential and difference equations with -symmetric coefficient matrices

Let denote the set of continuous n×n matrices on an interval . We say that is a nontrivial k-involution if where ζ=e^-2πi/k, d₀+d₁++d_k-1=n, and with . We say that is R-symmetric if R(t)A(t)R^-1(t)=A(t), , and we show that if A is R-symmetric then solving x^′=A(t)x or x^′=A(t)x+f(t) reduces to solving k independent d_ℓ×d_ℓ systems, 0ℓk-1. We consider the asymptotic behavior of the solutions in the case where . Finally, we sketch analogous results for linear systems of difference equations.     

Semigroup approach for identification of the unknown diffusion coefficient in a quasi‐linear parabolic equation

This article presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(u(x,t)) in the quasi‐linear parabolic equation u_t(x,t)=(k(u(x,t))u_x(x,t))_x, with Dirichlet boundary conditions u(0,t)=ψ₀, u(1,t)=ψ₁. The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[?]:?? →C¹[0,T], Ψ[?]:??→C¹[0,T] via semigroup theory. In this paper, it is shown that if the null space of the semigroup T(t) consists of only zero function, then the input–output mappings Φ[?] and Ψ[?] have the distinguishability property. It is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of these mappings. Moreover, under the light of measured output data (boundary observations) f(t):=k(u(0,t))u_x(0,t) or/and h(t):=k(u(1,t))u_x(1,t), the values k(ψ₀) and k(ψ₁) of the unknown diffusion coefficient k(u(x,t)) at (x,t)=(0,0) and (x,t)=(1,0), respectively, can be determined explicitly. In addition to these, the values k_u(ψ₀) and k_u(ψ₁) of the unknown coefficient k(u(x,t)) at (x,t)=(0,0) and (x,t)=(1,0), respectively, are also determined via the input data. Furthermore, it is shown that measured output data f(t) and h(t) can be determined analytically by an integral representation. Hence the input–output mappings Φ[?]:??→ C¹[0,T], Ψ[?]:??→C¹[0,T] are given explicitly in terms of the semigroup. Copyright © 2007 John Wiley & Sons, Ltd.     

Schur multiplier characterization of a class of infinite matrices

Let B _w (ℓ ^p ) denote the space of infinite matrices A for which A(x) ∈ ℓ ^p for all x = {x _k } _k=1^∞ ∈ ℓ ^p with |x _k | ↘ 0. We characterize the upper triangular positive matrices from B _w (ℓ ^p ), 1 < p < ∞, by using a special kind of Schur multipliers and the G. Bennett factorization technique. Also some related results are stated and discussed.     

Pre Stieltjes Functions

In this paper we characterize the class C_k{{\mathcal{C}_k}} of functions f on (0,∞) for which f(x), . . . ,(x ^k f(x))^(k) are completely monotonic for given k. In the limit we obtain the well-known characterization of the class of Stieltjes functions as those functions f defined on the positive half line for which (x ^k f(x))^(k) is completely monotonic on (0,∞) for all k ≥ 0.     

C-Cosine Functions and the Abstract Cauchy Problem, II

For a bounded linear injectionCon a Banach spaceXand a closed linear operatorA : D(A) X → Xwhich commutes withCwe prove that (1) the abstract Cauchy problem,u″(t) = Au(t),t R,u(0) = Cx,u′(0) = Cy, has a unique strong solution for everyx,y D(A) if and only if (2)A₁ = AD(A²) generates aC₁-cosine function onX₁(D(A) with the graph norm), if (and only if, in caseAhas nonempty resolvent set) (3)Agenerates aC-cosine function onX. HereC₁ = CX₁. Under the assumption thatAis densely defined andC⁻¹AC = A, statement (3) is also equivalent to each of the following statements: (4) the problemv″(t) = Av(t) + C(x + ty) + ∫^t₀ Cg(r) dr,t R,v(0) = v′(0) = 0, has a unique strong solution for everyg L¹_locandx, y X; (5) the problemw″(t) = Aw(t) + Cg(t),t R,w(0) = Cx,w′(0) = Cy, has a unique weak solution for everyg L¹_locandx, y X. Finally, as an application, it is shown that for any bounded operatorBwhich commutes withCand has range contained in the range ofC,A + Bis also a generator.     

Legendre polynomial kernel estimation of a density function with censored observations and an application to clinical trials

Let f(x), x ∈ ?^M, M ≥ 1, be a density function on ?^M, and X₁, …., X_n a sample of independent random vectors with this common density. For a rectangle B in ?^M, suppose that the X's are censored outside B, that is, the value X_k is observed only if X_k ∈ B. The restriction of f(x) to x ∈ B is clearly estimable by established methods on the basis of the censored observations. The purpose of this paper is to show how to extrapolate a particular estimator, based on the censored sample, from the rectangle B to a specified rectangle C containing B. The results are stated explicitly for M = 1, 2, and are directly extendible to M ≥ 3. For M = 2, the extrapolation from the rectangle B to the rectangle C is extended to the case where B and C are triangles. This is done by means of an elementary mapping of the positive quarter‐plane onto the strip {(u, v): 0 ≤ u ≤ 1, v > 0}. This particular extrapolation is applied to the estimation of the survival distribution based on censored observations in clinical trials. It represents a generalization of a method proposed in 2001 by the author [2]. The extrapolator has the following form: For m ≥ 1 and n ≥ 1, let K_{m, n}(x) be the classical kernel estimator of f(x), x ∈ B, based on the orthonormal Legendre polynomial kernel of degree m and a sample of n observed vectors censored outside B. The main result, stated in the cases M = 1, 2, is an explicit bound for E|K_{m, n}(x) ? f(x)| for x ∈ C, which represents the expected absolute error of extrapolation to C. It is shown that the extrapolator is a consistent estimator of f(x), x ∈ C, if f is sufficiently smooth and if m and n both tend to ∞ in a way that n increases sufficiently rapidly relative to m. © 2006 Wiley Periodicals, Inc.     

A wavelike functional equation of Pexider type

Summary. Let (G, +) and (H, +) be abelian groups such that the equation 2u = v 2u = v is solvable in both G and H. It is shown that if f₁, f₂, f₃, f₄, : G ×G ? H f_1, f_2, f_3, f_4, : G \times G \longrightarrow H satisfy the functional equation f₁(x + t, y + s) + f₂(x - t, y - s) = f₃(x + s, y - t) + f₄(x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , then f₁, f₂, f₃, and f₄ are given by f₁ = w + h, f₂ = w - h, f₃ = w + k, f₄ = w - k where w : G ×G ? H w : G \times G \longrightarrow H is an arbitrary solution of f (x + t, y + s) + f (x - t, y - s) = f (x + s, y - t) + f (x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , and h, k : G ×G ? H h, k : G \times G \longrightarrow H are arbitrary solutions of D_y,t³g(x,y) = 0 \Delta_{y,t}^{3}g(x,y) = 0 and D_x,t³g(x,y) = 0 \Delta_{x,t}^{3}g(x,y) = 0 for all x, y, s, t ? G x, y, s, t \in G .     

An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, u∈H ¹ ₀(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ ^s ₀ f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.     

Geometric computation of the numerical radius of a matrix

We use methods of geometric computing combined with hermitean matrix eigenvalue/eigenvector evaluations to find the numerical radius w(A) of a real or complex square matrix A simply, quickly, and accurately. The numerical radius w(A) is defined as the maximal distance of points in the field of values F(A) = { x^* A x | ||x||₂ = 1 }F(A) = \{ x^* A x \mid \|x\|_2 = 1 \} from zero in ℂ. Its value is an indicator of the transient behavior of the discrete dynamical system f _k + 1 = Af _k . We describe and test a MATLAB code for solving this optimization problem that can have multiple and even infinitely many solutions with maximal distance.     

A Modification of the Lyons-Sullivan Discretization of Positive Harmonic Functions

A modification of the Lyons-Sullivan discretization of positive harmonic functions on a Riemannian manifold M is proposed. This modification, depending on a choice of constants C = {C _n :n = 1,2,..}, allows for constructing measures n_x^C, x ? M\nu_x^\mathbf{C},\ x\in M, supported on a discrete subset Γ of M such that for every positive harmonic function f on M

Global bifurcation theorem for a class of boundary conditions for ordinary differential equations of second order

In this paper we deal with boundary value problems where l : C¹([a, b], ?^k) → ?^k × ?^k is continuous, μ ≤ 0 and φ is a Caratheodory map. We define the class S of maps l, for which a global bifurcation theorem holds for the problem (+), with φ(t, x, y, λ) = λ(|x₁|, …, |x^k|) + o(|x| + |y|). We show that the class S contains Sturm‐Liouville boundary conditions. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Linkage between Adiabatic Invariance and Symmetry of Solutions

In this article, for the symmetric pendulum equation and the symmetric bisuperlinear equation respectively, we show that there are two one-parameter families of solutions, y_s and y_a, so that one is adiabatically symmetric, y_s(?t)=y_s(t)+o(ε^k) for all k≥0, and the other adiabatically antisymmetric, y_a(?t)=?y_a(t)+o(ε^k) for all k≥0. By using the techniques of exponential asymptotics to calculate y′_s(0) and y_a(0), we demonstrate that, in general, they are not genuinely symmetric or antisymmetric, because these quantities are in fact exponentially small. Finally, after establishing a relationship between the total change in the leading-order adiabatic invariant and the quantity y′_s(0) for the family of solutions y_s of the bisuperlinear equation, we are able to reveal explicitly how the behavior of the adiabatic invariant depends on the complex singularities of the equation.      

Generalized Derivations with Engel Conditions on Polynomials

Let R be a noncommutative prime ring with extended centroid C, and let D: R → R be a nonzero generalized derivation, f(X ₁,…, X _t ) a nonzero polynomial in noncommutative indeterminates X ₁,…, X _t over C with zero constant term, and k ≥ 1 a fixed integer. In this article, D and f(X ₁,…, X _t ) are characterized if the Engel identity is satisfied: [D(f(x ₁,…, x _t )), f(x ₁,…, x _t )] _k  = 0 for all x ₁,…, x _t  ∈ R.