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.     

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.     

Self-intersecting geodesics and entropy of the geodesic flow

Given a symmetric Finsler metric on T^2 whose geodesic flow has zero topological entropy, we show that the lift in the universal covering R^2 →T^2 of any closed geodesic on T^2 must be an embedded curve in R^2.     

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

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.      

A superalgebraic interpretation of the quantization maps of Weil algebras

Let G be a Lie group whose Lie algebra g is quadratic. In the paper ＂the non-commutative Weil algebra＂, Alekseev and Meinrenken constructed an explicit G-differential space homomorphism ￡, called the quantization map, between the Well algebra Wg = S（g^＊） χ∧A（g^＊） and Wg= U（g） χ Cl（g） （which they call the noncommutative Weil algebra） for g. They showed that ￡ induces an algebra isomorphism between the basic cohomology rings Hbas^＊（Wg） and Hbas^＊（Wg）. In this paper, we will interpret the quantization map .~ as the super Duflo map between the symmetric algebra S（Tg[1]） and the universal enveloping algebra U（Tg[1]） of a super Lie algebra T9[1] which is canonically associated with the quadratic Lie algebra g. The basic cohomology rings Hbas^＊（Wg） and Hbas^＊（Wg） correspond exactly to S（Tg[1]）^inv and U（Tg[1]）^inv, respectively. So what they proved is equivalent to the fact that the super Duflo map commutes with the adjoint action of the super Lie algebra, and that the super Duflo map is an algebra homomorphism when restricted to the space of invariants.     

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.     

<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...     

On some types of radical classes

Let be an infinite cardinal. We denote by the collection of all -representable Boolean algebras. Further, let be the collection of all generalized Boolean algebras B such that for each b ∈ B, the interval [0, b] of B belongs to . In this paper we prove that is a radical class of generalized Boolean algebras. Further, we investigate some related questions concerning lattice ordered groups and generalized MV-algebras. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-51-032002. This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence-Physics of Information.     

Vanishing of the top local cohomology modules over Noetherian rings

Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let be an ideal of R and denote the intersection of all prime ideals . It is shown that

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     

The number of rooted essential maps on surfaces

In this paper, general functional equations of rooted essential maps on surfaces （orientable and nonorientable） are deduced and their formal solutions are presented. Further, three explicit for- mulae for counting essential maps on S2,N3 andN4 are given. In the same time, some known results can be derived.     

On the existence of Fagnano trajectories in convex polygonal billiards

We prove that an n-gon is pedal strong Fagnano trajectory in some convex polygonal billiard table if and only if it is a convex Poncelet n-gon.      

Hyperbolic unit groups and quaternion algebras

We classify the quadratic extensions and the finite groups G for which the group ring [G] of G over the ring of integers of K has the property that the group of units of augmentation 1 is hyperbolic. We also construct units in the ℤ-order of the quaternion algebra , when it is a division algebra.     

On ideals and quotients of A \mathcal{T} -algebras

Some results on A -algebras are given. We study the problem when ideals, quotients and hereditary subalgebras of A -algebras are A -algebras or A -algebras, and give a necessary and sufficient condition of a hereditary subalgebra of an A -algebra being an A -algebra.     

Quantization of orbit bundles in \mathfrakg\mathfrakln* (\mathbbC) \mathfrak{g}\mathfrak{l}_n^* (\mathbb{C})

Let G be the complex general linear group and its Lie algebra equipped with a factorizable Lie bialgebra structure; let U_ħ() be the corresponding quantum group. We construct explicit U_ħ()-equivariant quantization of Poisson orbit bundles O _λ → O _μ in *.     

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 .     

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:

Unconditional ideals of finite rank operators

Let X be a Banach space. We give characterizations of when is a u-ideal in for every Banach space Y in terms of nets of finite rank operators approximating weakly compact operators. Similar characterizations are given for the cases when is a u-ideal in for every Banach space Y, when is a u-ideal in for every Banach space Y, and when is a u-ideal in for every Banach space Y.     

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.