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.      

Approximation by the Bézier variant of the MKZ-Kantorovich operators in the case α < 1

The pointwise approximation properties of the Bézier variant of the MKZ-Kantorovich operators for α ≥ 1 have been studied in [Comput. Math. Appl., 39 (2000), 1-13]. The aim of this paper is to deal with the pointwise approximation of the operators for the other case 0 < α < 1. By means of some new techniques and new inequalities we establish an estimate formula on the rate of convergence of the operators for the case 0 < α < 1. In the end we propose the q-analogue of MKZK operators.      

Compact corigid objects in triangulated categories and co-<Emphasis Type="Italic">t</Emphasis>-structures

In the work of Hoshino, Kato and Miyachi, [11], the authors look at t-structures induced by a compact object, , of a triangulated category, , which is rigid in the sense of Iyama and Yoshino, [12]. Hoshino, Kato and Miyachi show that such an object yields a non-degenerate t-structure on whose heart is equivalent to Mod(End()^op). Rigid objects in a triangulated category can the thought of as behaving like chain differential graded algebras (DGAs). Analogously, looking at objects which behave like cochain DGAs naturally gives the dual notion of a corigid object. Here, we see that a compact corigid object, , of a triangulated category, , induces a structure similar to a t-structure which we shall call a co-t-structure. We also show that the coheart of this non-degenerate co-t-structure is equivalent to Mod(End()^op), and hence an abelian subcategory of .      

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

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.     

Metrics in the sphere of a C*-module

Given a unital C*-algebra and a right C*-module over , we consider the problem of finding short smooth curves in the sphere = {x ∈ : 〈x, x〉 = 1}. Curves in are measured considering the Finsler metric which consists of the norm of at each tangent space of . The initial value problem is solved, for the case when is a von Neumann algebra and is selfdual: for any element x ₀ ∈ and any tangent vector ν at x ₀, there exists a curve γ(t) = e ^tZ (x ₀), Z ∈ , Z* = −Z and ∥Z∥ ≤ π, such that γ(0) = x ₀ and (0) = ν, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x ₀, x ₁ ∈ , find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denote by f ₀ the selfadjoint projection I − x ₀ ⊗ x ₀, if the algebra f ₀ f ₀ is finite dimensional, then there exists a curve γ joining x ₀ and x ₁, which is minimizing along its path.      

<Emphasis Type="Italic">K</Emphasis><Subscript>0</Subscript>-group and approximate similarity invariants of strongly irreducible operators

Let H olenote a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator T ∈ L(H) is said to be strongly irreducible if T does not commute with any nontrivial idempotent. Herrero and Jiang showed that the norm-closure of the class of all strongly irreducible operators is the class of all operators with connected spectrum. This result can be considered as an approximate inverse of the Riesz decomposition theorem. In the paper, we give a more precise charact...     

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.     

Holomorphic automorphisms and collective compactness in J*-algebras of operator

Let G be the Banach-Lie group of all holomorphic automorphisms of the open unit ball in a J^*-algebra of operators. Let be the family of all collectively compact subsets W contained in . We show that the subgroup F ⊂ G of all those g ∈ G that preserve the family is a closed Lie subgroup of G and characterize its Banach-Lie algebra. We make a detailed study of F when is a Cartan factor.      

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.     

Perfect difference sets constructed from Sidon sets

A set of positive integers is a perfect difference set if every nonzero integer has a unique representation as the difference of two elements of . We construct dense perfect difference sets from dense Sidon sets. As a consequence of this new approach we prove that there exists a perfect difference set such that

The closure diagram for nilpotent orbits of the split real form of E8

Let and be adjoint nilpotent orbits in a real semisimple Lie algebra. Write ≥ if is contained in the closure of . This defines a partial order on the set of such orbits, known as the closure ordering. We determine this order for the split real form of the simple complex Lie algebra, E ₈. The proof is based on the fact that the Kostant-Sekiguchi correspondence preserves the closure ordering. We also present a comprehensive list of simple representatives of these orbits, and list the irreeducible components of the boundaries and of the intersections .     

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.      

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:

Maximal subclasses of local fitting classes

A Fitting class $ \mathfrak{F} A Fitting class is said to be π-maximal if is an inclusion maximal subclass of the Fitting class of all finite soluble π-groups. We prove that is a π-maximal Fitting class exactly when there is a prime p ∊ π such that the index of the -radical in G is equal to 1 or p for every π-subgroup of G. Hence, there exist maximal subclasses in a local Fitting class. This gives a negative answer to Skiba’s conjecture that there are no maximal Fitting subclasses in a local Fitting class (see [1, Question 13.50]). Original Russian Text Copyright ? 2008 Savelyeva N. V. and Vorob’ev N. T. __________ Vitebsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 6, pp. 1411–1419, November–December, 2008.     

Nonlinear maps of convex sets in Hilbert spaces with application to kinetic equations

Let be a separable Hilbert space, an open convex subset, and f: a smooth map. Let Ω be an open convex set in with , where denotes the closure of Ω in . We consider the following questions. First, in case f is Lipschitz, find sufficient conditions such that for ɛ > 0 sufficiently small, depending only on Lip(f), the image of Ω by I + ɛf, (I + ɛf)(Ω), is convex. Second, suppose df(u): is symmetrizable with σ(df(u)) ⊆ (0,∞), for all u ∈ , where σ(df(u)) denotes the spectrum of df(u). Find sufficient conditions so that the image f(Ω) is convex. We establish results addressing both questions illustrating our assumptions and results with simple examples. We also show how our first main result immediately apply to provide an invariance principle for finite difference schemes for nonlinear ordinary differential equations in Hilbert spaces. The main application of the theory developed in this paper concerns our second result and provides an invariance principle for certain convex sets in an L ²-space under the flow of a class of kinetic transport equations so called BGK model.      

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     

Normality concerning shared values

Let be a family of meromorphic functions in a plane domain D, and a and b be finite non-zero complex values such that . If for and , then is normal. We also construct a non-normal family of meromorphic functions in the unit disk Δ={|z|<1} such that for every and in Δ, where m is a given positive integer. This answers Problem 5.1 in the works of Gu, Pang and Fang. This work was supported by National Natural Science Foundation of China (Grant Nos. 10671093, 10871094) and the Natural Science Foundation of Universities of Jiangsu Province of China (Grant No. 08KJB110001), the Qing Lan Project of Jiangsu, China and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry     

On the distribution of the residues of small multiplicative subgroups of $$ \mathbb{F}_p $$

Distributional properties of small multiplicative subgroups of are obtained. In particular, it is shown that if H < is of size larger than polylogarithmic in p, then, letting β < 1 be a fixed exponent, most elements of any coset aH (a ∈ , arbitrary) will not fall into the interval [−p ^β, p ^β] ∈ . The arguments are based on the theory of heights and results from additive combinatoric.