Reconstructing Coefficients of Series from Certain Orthogonal Systems of Functions

We consider a series with respect to a multiplicative Price system or a generalized Haar system and assume that the martingale subsequence of its partial sums converges almost everywhere. In this paper we prove that, under certain conditions imposed on the majorant of this sequence, the series is a Fourier series in the sense of the A-integral (or its generalizations) of the limit function if the series is considered as a series with respect to a system with supp _n < . In similar terms, we also present sufficient conditions for a series to be a Fourier series in the sense of the usual Lebesgue integral. We give an example showing that the corresponding assertions do not hold if supp _n = .     

Order-coherent Archimedean <Emphasis Type="Italic">f</Emphasis>-algebras

Let A be an Archimedean uniformly complete f-algebra with unit element. It is shown that A is a po-coherent ring (in the sense of F. Wehrung) if and only if A is Dedekind σ-complete. Received June 7, 2001; accepted in final form September 1, 2002.     

Ehresmann connection for the canonical foliation on a manifold over a local algebra

For a canonical foliation on a manifoldM ^A over a local algebra, theA-affine horizontal distribution complementary to the leaves, similar to the horizontal distribution of a higher order connection on the fiber bundle ofA-jets, is defined. In the case of a complete manifoldM ^A, theA-affine horizontal distribution is proved to be an Ehresmann connection in the sense of Blumental-Hebda. It is shown that theA-affine horizontal distribution onM ^A exists if and only if the Atiyah class of a certain foliated principal bundle vanishes.Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 303–310, February, 1996.     

Some remarks on normal property of graded subalgebras of polynomial rings

LetR be an integral domain andA a finely gradedR-subalgebra of a polynomial ring overR. There are two topics in this article concerning the normality ofA. The main result in the first topic is thatA is completely normal (or normal) if and only if the corresponding monoid is completely normal (or normal resp.) whenR is a Krull (or Dedekind resp.) domain. The main result in the second topic is that ifR is (completely) normal andA is equipped with a structure of a square-free pre-Hodge algebra overR, thenA is also (completely) normal. We also prove the statement in a sense converse to the one above.     

Positivity and Convexity in Rings of Fractions

Given a commutative ring A equipped with a preordering A⁺ (in the most general sense, see below), we look for a fractional ring extension (= “ring of quotients” in the sense of Lambek et al. [L]) as big as possible such that A⁺ extends to a preordering R⁺ of R (i.e. with A ∩ R⁺ = A⁺) in a natural way. We then ask for subextensions A ⊂ B of A ⊂ R such that A is convex in B with respect to B⁺ : = B ∩ R⁺. Supported by DFG. A short form of this article has been delivered at the conference Carthapos 2006 at Carthago (Tunisia).     

Continuous dependence on data for bilocal difference equations

The continuous dependence on data is studied for a class of second order difference equations governed by a maximal monotone operator A in a Hilbert space. A nonhomogeneous term f appears in the equation and some bilocal boundary conditions a, b are added. One shows that the function which associates to {a, b, A, f} the solution of this boundary value problem is continuous in a specific sense. One uses the convergence of a sequence of operators in the sense of the resolvent. The problem studied here is the discrete variant of a problem from the continuous case.     

Nearly generalized Jordan derivations

Let A be an algebra and let X be an A-bimodule. A ∂-linear mapping d: A → X is called a generalized Jordan derivation if there exists a Jordan derivation (in the usual sense) δ: A → X such that d(a ²) = ad(a)+δ(a)a for all a ∈ A. The main purpose of this paper is to prove the Hyers-Ulam-Rassias stability and superstability of the generalized Jordan derivations.     

Majorant and paley function for series in general Franklin systems

For a general Franklin system generated by a quasidyadic and strongly regular sequence, it is proved, in particular, that any subseries of a Fourier series in the general Franklin system is a Fourier series of an A-integrable function in the sense of A-integration.     

Perturbation of an abstract differential equation containing fractional Riemann-Liouville derivatives

We consider a Cauchy type problem in a Banach space. Under the assumption that the corresponding Cauchy type problem with the operator A is uniformly well-posed and the operator B(t) is subordinate to A in some sense, we prove the unique solvability of the considered problem and its continuous dependence on initial data.     

Bispectral quantum Knizhnik�CZamolodchikov equations for arbitrary root systems

The bispectral quantum Knizhnik–Zamolodchikov (BqKZ) equation corresponding to the affine Hecke algebra H of type A _N-1 is a consistent system of q-difference equations which in some sense contains two families of Cherednik’s quantum affine Knizhnik–Zamolodchikov equations for meromorphic functions with values in principal series representations of H. In this paper, we extend this construction of BqKZ to the case where H is the affine Hecke algebra associated with an arbitrary irreducible reduced root system. We construct explicit solutions of BqKZ and describe its correspondence to a bispectral problem involving Macdonald’s q-difference operators.     

Noetherian Connected Graded Algebras of Global Dimension 3

Let A be a noetherian connected graded algebra of global dimension 3. We show that A is regular in the sense of Artin and Schelter if one of the following conditions holds: (1) A is generated by two elements; (2) the graded simple module has a standard resolution; (3) the degrees of minimal relations are the same (this includes the quadratic case). Some general properties of A are studied without assuming regularity.     

On group stable commutative separable semisimple subalgebras

In a G-algebra A of a p-group G over a perfect ground field there is a commutative separable semisimple G-subalgebra such that any traces on relative projectivity in A can be controlled in the subalgebra; and in a residually central sense, the maximal such subalgebras form exactly one -conjugacy class. Applying to modules, the induced module of an indecomposable module of a subnormal subgroup of p- primary index is characterized. Received: 13 April 2002 / Published online: 2 December 2002     

On Consistent Operators and Reflexivity

We study Hilbert space operators A=?_{i ? \mathbbN} A_i{A=\oplus_{i \in \mathbb{N} } A_i} which are consistent in the sense that each A _i+1 contains a copy of A _i . The formal definition is reminiscent of the classical ordering on projections in a von Neumann algebra. It is shown that if the powers of A are simultaneously consistent, then A must be reflexive. This is applied to study reflexivity of power partial isometries.     

Chaos induced by weak A-coupled-expansion of non-autonomous discrete dynamical systems

This paper focuses on chaos induced by weak A-coupled-expansion of non-autonomous discrete systems in compact subsets of metric spaces and in bounded and closed subsets of complete metric spaces, separately. A new concept of weak A-coupled-expansion for non-autonomous discrete systems, whose condition is weaker than that of A-coupled-expansion, is introduced, and several new criteria of chaos induced by weak A-coupled-expansion of non-autonomous discrete systems are established. By applying some close relationships between chaotic dynamical behaviours of the original system and its induced systems, two criteria of chaos are established. One example is provided for illustration.     

C*-Nuclearity implies CPAP

In this paper we show: A C^*-algebra A with identity e is C^*-nuclear (i.e. “nuclear” in the sense of LANCE [3, p. 159] or “property T” in the sense of TAKESAKI [6] if and only if the identity operator id_A on A can be approximated in the strong operator topology by completely positive linear operators V from A into A of finite rank with norm one and V(e)=e. From which follows that every C^*-nuclear C^*-algebra (possible without identity) possesses the CPAP of LANCE [3, def. 3.5.]. This answers questions of LANCE [3, p. 173] and SAKAI [5, p. 64].     

The ordinary quivers of Hochschild extension algebras for self-injective Nakayama algebras

Let T be a Hochschild extension algebra of a finite dimensional algebra A over a field K by the standard duality A-bimodule Hom_K(A, K). In this paper, we determine the ordinary quiver of T if A is a self-injective Nakayama algebra by means of the ?-graded second Hochschild homology group HH₂(A) in the sense of Sköldberg.     

Each four-celled animal tiles the plane

An animal A is a set of unit squares in the plane, parallel to the axes, and with corners at integer lattice points. We show that any animal A with four cells tiles the plane, in the sense that infinitely many copies of A, translated by integer vectors and possibly rotated through 90°, 180°, or 270°, can be placed so as to fill plane exactly without overlap.     

Analyticity of nonlinear semigroups

The Cauchy problemdu/dt =Au(t),u(0) =u ₀∈D(A) has analytic solutions whenA has first and second Gateaux derivatives along the solution curve in a certain weak sense. HereA is a maximal monotone operator in a complex Hilbert space.     

The Kurosh problem,height theorem,nilpotency of the radical,and algebraicity identity

This paper is devoted to relations between the Kurosh problem and the Shirshov height theorem. The central point and main technical tool is the identity of algebraicity. The main result of this paper is the following. Let A be a finitely generated PI-algebra and Y be a finite subset of A. For any Noetherian associative and commutative ring {ie125-01}, let any factor of R ⊗ A such that all projections of elements from Y are algebraic over π(R) be a Noetherian R-module. Then A has bounded essential height over Y. If, furthermore, Y generates A as an algebra, then A has bounded height over Y in the Shirshov sense. This paper also contains a new proof of the Razmyslov-Kemer-Braun theorem on radical nilpotence of affine PI-algebras. This proof allows one to obtain some constructive estimates. The main goal of the paper is to develop a “virtual operator calculus.” Virtual operators (pasting, deleting, and transfer) depend not only on an element of the algebra but also on its representation. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 2, pp. 3–29, 2007.