On Osgood theorem in Banach spaces

Let X be a real Banach space, ω : [0, +∞) → ? be an increasing continuous function such that ω(0) = 0 and ω(t + s) ≤ ω(t) + ω(s) for all t, s ∈ [0, +∞). According to the infinite dimensional analog of the Osgood theorem if ∫¹₀ (ω(t))^?1 dt = ∞, then for any (t₀, x₀) ∈ ?×X and any continuous map f : ?×X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all t ∈ ?, x, y ∈ X, the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has a unique solution in a neighborhood of t₀. We prove that if X has a complemented subspace with an unconditional Schauder basis and ∫¹₀ (ω(t))^?1 dt < ∞ then there exists a continuous map f : ? × X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all (t, x, y) ∈ ? × X × X and the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has no solutions in any interval of the real line.     

3-(v, 4, 1) covering designs with chromatic numbers 2 and 3

A t-(v, k, λ) covering design is a pair (X, B) where X is a v-set and B is a collection of k-sets in X, called blocks, such that every t element subset of X is contained in at least λ blocks of B. The covering number, C_λ(t, k, v), is the minimum number of blocks a t-(v, k, λ) covering design may have. The chromatic number of (X, B) is the smallest m for which there exists a map φ: X → Z_m such that ∣φ((β)∣ ≥2 for all β ∈ B, where φ(β) = {φ(x): x ∈ β}. The system (X, B) is equitably m-chromatic if there is a proper coloring φ with minimal m for which the numbers ∣φ^?1(c)∣ c ∈ Z_m differ from each other by at most 1. In this article we show that minimum, (i.e., ∣B∣ = C λ (t, k, v)) equitably 3-chromatic 3-(v, 4, 1) covering designs exist for v ≡ 0 (mod 6), v ≥ 18 for v ≥ 1, 13 (mod 36), v ≡ 13 and for all numbers v = n, n + 1, where n ≡ 4, 8, 10 (mod 12), n ≥ 16; and n = 6.5^a 13^b 17^c ?4, a + b + c > 0, and n = 14, 62. We also show that minimum, equitably 2-chromatic 3-(v, 4, 1) covering designs exist for v ≡ 0, 5, 9 (mod 12), v ≥ 0, v = 2.5^a 13^b 17^c + 1, a + b + c > 0, and v = 23. © 1993 John Wiley & Sons, Inc.     

Uniqueness properties in finite-continuous additive measurement

Let ? be a binary relation on A×X, and suppose that there are real valued functions f on A and g on X such that, for all ax, by ∈ A×X, ax ? by if and only if f (a)+g(x) ? f(b)+g(y). This paper establishes uniqueness properties for f and g when A is a finite set, X is a real interval with g increasing on X, and for any a, b and x there is a y for which f(a)+g(x)=f(b)+g(y). The resultant uniqueness properties occupy an intermediate position among uniqueness properties for other structural cases of two-factor additive measurement.It is shown that f is unique up to a positive affine transformation (αf+β₁ with α > 0), but that g is unique up to a similar positive affine transformation (αg+β₂) if and only if the ratio [f(a)?f(b)]/[f(a)?f(c)] is irrational for some a, b, c ∈ A. When the f ratios are rational for all cases where they are defined, there will be a half-open interval (x₀, x₁) in X such that the restriction of g on (x₀, x₁) can be any increasing function for which sup {g(x)?g(x₀): x₀ ? x < x₁} does not exceed a specified bound, and, when g is thus defines on (x₀, x₁), it will be uniquely determined on the rest of X. In general, g must be continuous only in the ‘irrational’ case.     

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.     

Centralizers of generalized derivations on multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2, with Utumi quotient ring U and extended centroid C, δ a nonzero derivation of R, G a nonzero generalized derivation of R, and f(x ₁, …, x _n ) a noncentral multilinear polynomial over C. If δ(G(f(r ₁, …, r _n ))f(r ₁, …, r _n )) = 0 for all r ₁, …, r _n ∈ R, then f(x ₁, …, x _n )² is central-valued on R. Moreover there exists a ∈ U such that G(x) = ax for all x ∈ R and δ is an inner derivation of R such that δ(a) = 0.     

Hitting time of a moving boundary for a diffusion

We obtain asymptotic estimates for the quantity r = log P[T_f[rang]t] as t → ∞ where T_f = inf\s{s : |X(s)|[rang]f(s)\s} and X is a real diffusion in natural scale with generator a(x) d²(·)/dx² and the ‘boundary’ f(s) is an increasing function. We impose regular variation on a and f and the result is expressed as r = ∫^t₀ λ₁ (f(s) ds(1 + o(1)) where λ₁(f) is the smallest eigenvalue for the process killed at ±f.     

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)     

A Numerical solution to the linear and nonlinear Fredholm integral equations using Legendre wavelet functions

Two methods for solving the Fredholm integral equation of the second kind in linear case, i.e. f (x) – λ ∫_a^b K (x,y)f (y)dy = g (x), and nonlinear case, i.e., f (x) = g (x) + λ ∫_a^b K (x,y)F (f (y))dy, are proposed. In order to solve the linear equation, the kernel K (x,y) as well as the functions f and g are initially approximated through Legendre wavelet functions. This leads to a system of linear equations its solution culminates in a solution to the Fredholm integral equation. In nonlinear case only K (x,y) is approximated by Legendre wavelet base functions. This leads to a separable kernel and makes it possible to employ a number of earlier methods in solving nonlinear Fredholm integral equation with separable kernels. Another feature of the proposed method is that it finds the solution as a function instead of specific solution points, what is done by the majority of the existing methods. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Algorithm for the computation of Bessel function integrals

A fortran subroutine is given for the computation of integrals of the form ∫^c₀f(x)J_v(αx)dx, where v = 0, 1,…,10.     

Self-converse Mendelsohn designs with block size 4t + 2

A Mendelsohn design MD(v, k, λ) is a pair (X, B) where X is a v-set together with a collection B of cyclic k-tuples from X such that each ordered pair from X, as adjacent entries, is contained in exactly λk-tuples of B. An MD(v, k, λ) is said to be self-converse, denoted by SCMD(v, k, λ) = (X, B, f), if there is an isomorphic mapping from (X, B) to (X, B⁻¹), where B⁻¹ = {B⁻¹ = 〈x_k, x_k−1, … x₂, x₁〉; B = 〈x₁, … ,x_k〉 ∈ B.}. The existence of SCMD(v, 3, λ) and SCMD(v, 4, 1) has been settled by us. In this article, we will investigate the existence of SCMD(v, 4t + 2, 1). In particular, when 2t + 1 is a prime power, the existence of SCMD(v, 4t + 2, 1) has been completely solved, which extends the existence results for MD(v, k, 1) as well. © 1999 John Wiley & Sons, Inc. J. Combin Designs 7: 283–310, 1999     

Dynamics of Rational Surface Automorphisms: Linear Fractional Recurrences

We consider the family f _a,b (x,y)=(y,(y+a)/(x+b)) of birational maps of the plane and the parameter values (a,b) for which f _a,b gives an automorphism of a rational surface. In particular, we find values for which f _a,b is an automorphism of positive entropy but no invariant curve. The Main Theorem: If f _a,b is an automorphism with an invariant curve and positive entropy, then either (1) (a,b) is real, and the restriction of f to the real points has maximal entropy, or (2) f _a,b has a rotation (Siegel) domain. Research supported in part by the NSF.     

On a fractional partial differential equation with dominating linear part

It is proved that there is a (weak) solution of the equation u_t=a*u_xx+b*g(u_x)_x+f, on ℝ⁺ (where * denotes convolution over (−∞, t)) such that u_x is locally bounded. Emphasis is put on having the assumptions on the initial conditions as weak as possible. The kernels a and b are completely monotone and if a(t)=t^−α, b(t)=t^−β, and g(ξ)∼sign(ξ)∣ξ∣^γ for large ξ, then the main assumption is that α>(2γ+2)/(3γ+1)β+(2γ−2)/(3γ+1). © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Vanishing Derivations and Co-Centralizing Generalized Derivations on Multilinear Polynomials in Prime Rings

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, and f(x₁,…, x_n) be a multilinear polynomial over C, which is not central valued on R. Suppose that F and G are two generalized derivations of R and d is a nonzero derivation of R such that d(F(f(r))f(r) ? f(r)G(f(r))) = 0 for all r = (r₁,…, r_n) ∈ Rⁿ, then one of the following holds:

1. There exist a, p, q, c ∈ U and λ ∈C such that F(x) = ax + xp + λx, G(x) = px + xq and d(x) = [c, x] for all x ∈ R, with [c, a ? q] = 0 and f(x₁,…, x_n)² is central valued on R;

2. There exists a ∈ U such that F(x) = xa and G(x) = ax for all x ∈ R;

3. There exist a, b, c ∈ U and λ ∈C such that F(x) = λx + xa ? bx, G(x) = ax + xb and d(x) = [c, x] for all x ∈ R, with b + αc ∈ C for some α ∈C;

4. R satisfies s₄ and there exist a, b ∈ U and λ ∈C such that F(x) = λx + xa ? bx and G(x) = ax + xb for all x ∈ R;

5. There exist a′, b, c ∈ U and δ a derivation of R such that F(x) = a′x + xb ? δ(x), G(x) = bx + δ(x) and d(x) = [c, x] for all x ∈ R, with [c, a′] = 0 and f(x₁,…, x_n)² is central valued on R.

     

Gâteaux derivatives and their applications to approximation in Lorentz spaces Γp,w

We establish the formulas of the left‐ and right‐hand Gâteaux derivatives in the Lorentz spaces Γ_p,w = {f: ∫₀^α (f **)^p w < ∞}, where 1 ≤ p < ∞, w is a nonnegative locally integrable weight function and f ** is a maximal function of the decreasing rearrangement f * of a measurable function f on (0, α), 0 < α ≤ ∞. We also find a general form of any supporting functional for each function from Γ_p,w , and the necessary and sufficient conditions for which a spherical element of Γ_p,w is a smooth point of the unit ball in Γ_p,w . We show that strict convexity of the Lorentz spaces Γ_p,w is equivalent to 1 < p < ∞ and to the condition ∫₀^∞ w = ∞. Finally we apply the obtained characterizations to studies the best approximation elements for each function f ∈ Γ_p,w from any convex set K ? Γ_p,w (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A New Approach to the Creation and Propagation of Exponential Moments in the Boltzmann Equation

We study the creation and propagation of exponential moments of solutions to the spatially homogeneous d-dimensional Boltzmann equation. In particular, when the collision kernel is of the form |v ? v _*|^β b(cos (θ)) for β ∈ (0, 2] with cos (θ) = |v ? v _*|^?1(v ? v _*)·σ and σ ∈ 𝕊 ^d?1, and assuming the classical cut-off condition b(cos (θ)) integrable in 𝕊 ^d?1, we prove that there exists a > 0 such that moments with weight exp (amin {t, 1}|v|^β) are finite for t > 0, where a only depends on the collision kernel and the initial mass and energy. We propose a novel method of proof based on a single differential inequality for the exponential moment with time-dependent coefficients.     

Multipliers of classes of cauchy transforms

The analytic map g on the unit disk D is said to induce a multiplication operator L from the Banach space X to the Banach space Y if L(f)=f·g∈Y for all f∈X. For z ∈ D and α>0 the families of weighted Cauchy transforms F_α are defined by ?(z) = ∫_T K_x ^α (z)dμ(x) where μ(x) is complex Borel measures, x belongs to the unit circle T and the kernel K_x (z) = (1- xz)^?1. In this article we will explore the relationship between the compactness of the multiplication operator L acting on F ₁ and the complex Borel measures μ(x). We also give an estimate for the essential norm of L     

Analytic Extension of Non Quasi - Analytic Whitney Jets of Beurling Type

Let (M_r)_r∈_?0 be a logarithmically convex sequence of positive numbers which verifies M₀ = 1 as well as M_r ≥ 1 for every r ∈ ? and defines a non quasi - analytic class. Let moreover F be a closed proper subset of ?ⁿ. Then for every function f on ?ⁿ belonging to the non quasi - analytic (M_r)-class of Beurling type, there is an element g of the same class which is analytic on ?,ⁿ F and such that D^αf(x) = D^αg(x) for every α ∈ ?ⁿ₀ and x ∈ F.     

Generalized potentials in variable exponent Lebesgue spaces on homogeneous spaces

We consider generalized potential operators with the kernel on bounded quasimetric measure space (X, μ, d) with doubling measure μ satisfying the upper growth condition μB(x, r) ? Kr^N, N ∈ (0, ∞). Under some natural assumptions on a(r) in terms of almost monotonicity we prove that such potential operators are bounded from the variable exponent Lebesgue space L^p(?)(X, μ) into a certain Musielak‐Orlicz space L^p(X, μ) with the N‐function Φ(x, r) defined by the exponent p(x) and the function a(r). A reformulation of the obtained result in terms of the Matuszewska‐Orlicz indices of the function a(r) is also given. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Numerical analysis and orthogonal polynomials

Two topics of general interest are investigated. First an iterative improvement algorithm is given to reduce the accumulation of truncation errors which may occur when recursion formulae are utilized. Then some properties of orthogonal polynomials are derived that allow a successful construction of Gaussian type integration formulae employing the improvement algorithm. As special examples integrals of the ∫^b_aexp(-x²)f(x)dx and ∫^b_aexp(-|x|)f(x)dx are considered, where a and/o r b may be infinite.     

Vertical versus horizontal Poincaré inequalities on the Heisenberg group

Let ? = 〈a, b|a[a, b] = [a, b]a ∧ b[a, b] = [a, b]b〉 be the discrete Heisenberg group, equipped with the left-invariant word metric d _W (·, ·) associated to the generating set {a, b, a ^?1, b ^?1}. Letting B _n = {x ∈ ?: d _W (x, e _?) ? n} denote the corresponding closed ball of radius n ∈ ?, and writing c = [a, b] = aba ^?1 b ^?1, we prove that if (X, ‖ · ‖X) is a Banach space whose modulus of uniform convexity has power type q ∈ [2,∞), then there exists K ∈ (0, ∞) such that every f: ? → X satisfies $$\sum\limits_{k = 1}^{{n^2}} {\sum\limits_{x \in {B_n}} {\frac{{\left\| {f(x{c^k}) - f(x)} \right\|_X^q}}{{{k^{1 + q/2}}}}} } \leqslant K\sum\limits_{x \in {B_{21n}}} {(\left\| {f(xa) - f(x)} \right\|_X^q + \left\| {f(xb) - f(x)} \right\|_X^q)} $$ . It follows that for every n ∈ ? the bi-Lipschitz distortion of every f: B _n → X is at least a constant multiple of (log n)^1/q , an asymptotically optimal estimate as n → ∞.