Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales

By employing the generalized Riccati transformation technique, we will establish some new oscillation criteria and study the asymptotic behavior of the nonoscillatory solutions of the second-order nonlinear neutral delay dynamic equation

Binormal operator and *-Aluthge transformation

Let T = U｜T｜ be the polar decomposition of a bounded linear operator T on a Hilbert space. The transformation T = ｜T｜^1/2 U｜T｜^1/2 is called the Aluthge transformation and Tn means the n-th Aluthge transformation. Similarly, the transformation T（＊）=｜T＊｜^1/2 U｜T＊｜＆1/2 is called the ＊-Aluthge transformation and Tn^（＊） means the n-th ＊-Aluthge transformation. In this paper, firstly, we show that T（＊） = UV｜T^（＊）｜ is the polar decomposition of T（＊）, where ｜T｜^1/2 ｜T^＊｜^1/2 = V｜｜T｜^1/2 ｜T^＊｜^1/2｜ is the polar decomposition. Secondly, we show that T（＊） = U｜T^（＊）｜ if and only if T is binormal, i.e., [｜T｜, ｜T^＊｜]=0, where [A, B] = AB - BA for any operator A and B. Lastly, we show that Tn^（＊） is binormal for all non-negative integer n if and only if T is centered, and so on.     

Derivations of the even part of the odd hamiltonian superalgebra in modular case

In this paper we mainly study the derivations for even part of the finite-dimensional odd Hamiltonian superalgebra HO over a field of prime characteristic. We first give the generating set of the even part g of HO. Then we compute the derivations from g into the even part m of the generalized Witt superalgebra. Finally, we determine the derivation algebra and outer derivation algebra of and the dimension formulas. In particular, the first cohomology groups H^1（g;m） and H^1（g;g） are determined.     

Rational invariants of certain classical groups over finite fields

Let be a finite field with q elements, where q is a prime power. Let G be a subgroup of the general linear group over and be the rational function field over . We seek to understand the structure of the rational invariant subfield . In this paper, we prove that is rational (or, purely transcendental) by giving an explicit set of generators when G is the symplectic group. In particular, the set of generators we gave satisfies the Dickson property.      

The hardy and bellman operators in spaces connected with H( \mathbb{T} ) and BMO( \mathbb{T} )

Assume that 1 ≤ p < ∞ and a function f ∈ L ^p [0, π] has the Fourier series $ \sum\limits_{n = 1}^\infty {a_n } Assume that 1 ≤ p < ∞ and a function f ∈ L ^p [0, π] has the Fourier series cos nx. According to one result of G.H. Hardy, the series cos nx is the Fourier series for a certain function (f) ∈ L ^p [0, π]. But if 1 < p ≤ ∞ and f ∈ L ^p [0, π], then the series cos nx is the Fourier series for a certain function (f) ∈ L ^p [0, π]. Similar assertions are true for sine series. This allows one to define the Hardy operator on L ^p (), 1 ≤ p < ∞, and to define the Bellman operator on L ^p (), 1 < p ≤ ∞. In this paper we prove that the Bellman operator boundedly acts in VMO(), and the Hardy operator also maps a certain subspace C() onto VMO(). We also prove the invariance of certain classes of functions with given majorants of modules of continuity or best approximations in the spaces H(), L(), VMO() with respect to the Hardy and Bellman operators. Original Russian Text ? S.S. Volosivets and B.I. Golubov, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 5, pp. 4–13.     

Push-outs of derivations

Let be a Banach algebra and let X be a Banach -bimodule. In studying (,X) it is often useful to extend a given derivation D: → X to a Banach algebra containing as an ideal, thereby exploiting (or establishing) hereditary properties. This is usually done using (bounded/unbounded) approximate identities to obtain the extension as a limit of operators b ↦ D(ba) − b.D(a), a ε in an appropriate operator topology, the main point in the proof being to show that the limit map is in fact a derivation. In this paper we make clear which part of this approach is analytic and which algebraic by presenting an algebraic scheme that gives derivations in all situations at the cost of enlarging the module. We use our construction to give improvements and shorter proofs of some results from the literature and to give a necessary and sufficient condition that biprojectivity and biflatness is inherited to ideals.     

Strong Approximation by Cesàro Means with Critical Index in the Hardy Spaces H p (\mathbb{S}^{{d - 1}} ) (0 < p ≤ 1)

Let be a unit sphere of the d–dimensional Euclidean space ℝ ^d and let (0 < p ≤ 1) denote the real Hardy space on For 0 < p ≤ 1 and let E _j (f,H ^p ) (j = 0, 1, ...) be the best approximation of f by spherical polynomials of degree less than or equal to j, in the space Given a distribution f on its Cesàro mean of order δ > –1 is denoted by For 0 < p ≤ 1, it is known that is the critical index for the uniform summability of in the metric H ^p . In this paper, the following result is proved: Theorem Let 0<p<1 and Then for

Deterministic extractors for affine sources over large fields

An (n,k)-affine source over a finite field is a random variable X = (X ₁,..., X _n ) ∈ , which is uniformly distributed over an (unknown) k-dimensional affine subspace of . We show how to (deterministically) extract practically all the randomness from affine sources, for any field of size larger than n ^c (where c is a large enough constant). Our main results are as follows:

Linear connections of a framed distribution of hyperplane elements in conformal space

We study the geometry of affine and normal connections induced by a complete normalization of mutually orthogonal distributions $ \mathcal{M} We study the geometry of affine and normal connections induced by a complete normalization of mutually orthogonal distributions and in conformal space C _n , where is a distribution of hyperplane elements, and is a distribution of line elements. We consider invariant fields of pencils that are parallel with respect to the normal connection along any curve belonging to the distribution . Original Russian Text ? A.M. Matveeva, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 7, pp. 79–84.     

Parameter Estimates in Random Intercept Mixed Effects Model for Repeated Measures

In this article the following random intercept mixed effects model will be considered： yij = vi =v^τijβ＋ εij,i=1,…,m;j=1,2,…,ni, where {vi} are i.i.d, random effects with mean α 2. 2 and finite variance σ^2 v, {εij} are i.i.d, random errors with finite variance ε^2 ε. Here we will estimate α,σ^2 v,σ^2 ε,β and study their large sample properties, such as strong consistency, strong convergence rates and asymptotic normality.     

On long time existence for small solutions of semi-linear Klein-Gordon equations on the torus

We prove that small smooth solutions of weakly semi-linear Klein-Gordon equations on the torus (d ≥ 2) exist over a larger time interval than the one given by local existence theory, for almost every value of the mass. We use a normal form method for the Sobolev energy of the solution. The difficulty, in comparison with previous results obtained on the sphere, comes from the fact that the set of differences of eigenvalues of on (d ≥ 2) is dense in ℝ. This work was partially supported by the ANR project Equa-disp.     

Parabolic twists for the linear algebras <Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis>−1</Subscript>

New solutions of twist equations for the universal enveloping algebras U (A_n−1) are found. These solutions can be represented as products of full chains of extended Jordanian twists Abelian factors (“rotations”) , and sets of quasi-Jordanian twists . The latter are generalizations of Jordanian twists (with carrier b²) for special deformed extensions of the Hopf algebra U (b²). The carrier subalgebra for the composition is a nonminimal parabolic subalgebra in A _n−1 such that . The parabolic twisting elements are obtained in an explicit form. Details of the construction are illustrated by considering the examples n = 4 and n = 11. Bibliography: 21 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 187–213.     

Nonparametric inference in a simple change-point model

In this paper, we consider a change point model allowing at most one change, X() = f() + e(), where f(t) = α + θ (t), 0 ≤ t ≤ 1, {e(), ..., e()} is a sequence of i.i.d. random variables distributed as e with 0 being the median of e. For this change point model, hypothesis test problem about the change-point t0 is studied and a test statistic is constructed. Furthermore, a nonparametric estimator of t0 is proposed and shown to be strongly consistent. Finally, we give an estimator of jump θ and obtain it’s asymptotic property. Performance of the proposed approach is investigated by extensive simulation studies. Research partially supported by National Natural Science Foundation of China (Grant No. 10471136), Ph.D. Program Foundation of the Ministry of Education of China, and Special Foundations of the Chinese Academy of Sciences and USTC     

Three-dimensional sets with small sumset

We describe the structure of three dimensional sets of lattice points, having a small doubling property. Let be a finite subset of ℤ³ such that dim = 3. If and , then lies on three parallel lines. Moreover, for every three dimensional finite set that lies on three parallel lines, if , then is contained in three arithmetic progressions with the same common difference, having together no more than terms. These best possible results confirm a recent conjecture of Freiman and cannot be sharpened by reducing the quantity υ or by increasing the upper bounds for .     

Boundedness of Some Marcinkiewicz Integral Operators Related to Higher Order Commutators on Hardy Spaces

In this paper, the authors study the boundedness properties of μΩ↑m,b generated by the function b ∈Lipβ（R^n）（0 〈β≤ 1/m） and the Marcinkiewicz integrals operator μΩ. The boundednesses are established on the Hardy type spaces Hb^m^p,n（R^n） and the Herz Hardy type spaces Hbm Kq^α,p（R^b）.     

A new class of weights adapted to rough singular integrals and Marcinkiewicz integrals

In this paper, we introduce a new class of weights Ap （Rn） which retains many fine properties of the classical Muchenhoupt weights Ap （Rn）. While Ap （Rn） is too big a class to obtain the weighted norm inequalities for rough singular integrals and Marcinkiewicz integrals, our new class Ap （Rn） adapts well to these rough operators. As applications, we improve some known weighted estimates.     

On polynomial automorphisms of spheres

Let K be an infinite field with characteristic different from two and (K) the n-sphere over K. We show that ambient polynomial automorphisms of (K) preserve the quadratic form x ₀² + ⋯ + x _n ² and the group Aut ((K ⁿ⁺¹, (K)) of such automorphisms of (K) is isomorphic to the (n + 1)-orthogonal group O(n + 1, K) provided K is real. Next, the restriction map Aut (K ³, (K)) → Aut ( (K)) yields a surjection provided K is an algebraically closed field as well. Furthermore, for any such a field K, there is an imbedding

On some new criteria for infinite differentiability of periodic functions

We study the set of infinitely differentiable periodic functions in terms of generalized -derivatives defined by a pair of sequences ψ ₁ and ψ ₂. It is shown that every function ƒ from the set has at least one derivative whose parameters ψ ₁ and ψ ₂ decrease faster than any power function and, at the same time, for any function ƒ ∈ different from a trigonometric polynomial, there exists a pair ψ whose parameters ψ ₁ and ψ ₂ have the same rate of decrease and for which the -derivative no longer exists. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 10, pp. 1399–1409, October, 2007.     

Periodic subgroups of projective linear groups in positive characteristic

We classify the maximal irreducible periodic subgroups of PGL(q, ), where is a field of positive characteristic p transcendental over its prime subfield, q = p is prime, and ^× has an element of order q. That is, we construct a list of irreducible subgroups G of GL(q, ) containing the centre ^×1 _q of GL(q, ), such that G/ ^×1 _q is a maximal periodic subgroup of PGL(q, ), and if H is another group of this kind then H is GL(q, )-conjugate to a group in the list. We give criteria for determining when two listed groups are conjugate, and show that a maximal irreducible periodic subgroup of PGL(q, ) is self-normalising.