The Schur or projective Schur group of a field consists of the classes of central simple algebras which occur in the decomposition of a group algebra or a twisted group algebra. For number fields, the projective Schur group has been determined in [8], whereas the Schur group is extensively studied in [25]. Recently, some authors have generalized these concepts to commutative rings. One then studies the classes of Azumaya algebras which are epimorphic images of a group ring or a twisted group ring. Though several properties of the Schur or projective Schur group defined in this way have been obtained, they remain rather obscure objects.

We construct a one-parameter automorphism group of the injective type II1 factor with Connes spectrum ﹛0﹜ which is not stably conjugate to an infinite tensor product action. We construct a countable family of one-parameter automorphism groups of the injective type II1 factor such that all are stably conjugate but no two are cocycle conjugate.

One of the important invariants of a р-block B of a group algebra is ℓ (B), the number of non-isomorphic simple B-modules. A number of authors calculated ℓ (B) for various types of defect groups of B. In particular, by Olsson [6], it has been proved that if p = 2 and the defect groups of the block B are dihedral or semi-dihedral or generalized quaternion, then ℓ (B) is at most 3. In this paper, we restrict our attention to the principal p-block B0 of a finite р-solvable group with ℓ (B0) ≤ 3. Let Γ be a finite р-solvable group and k a splitting field for Γ with characteristic р.

The purpose of this paper is to introduce the notion of a CW complex over a topological category. The main theorem of this paper gives an equivalence between the homotopy theory of diagrams of spaces based on a topological category and the homotopy theory of CW complexes over the same base category.

A brief description of the paper goes as follows: in Section 1 we introduce the homotopy category of diagrams of spaces based on a fixed topological category. In Section 2 homotopy groups for diagrams are defined. These are used to define the concept of weak equivalence and J-n equivalence that generalize the classical definition. In Section 3 we adapt the classical theory of CW complexes to develop a cellular theory for diagrams. In Section 4 we use sheaf theory to define a reasonable cohomology theory of diagrams and compare it to previously defined theories. In Section 5 we define a closed model category structure for the homotopy theory of diagrams. We show this Quillen type homotopy theory is equivalent to the homotopy theory of J-CW complexes. In Section 6 we apply our constructions and results to prove a useful result in equivariant homotopy theory originally proved by Elmendorf by a different method.

Berg's interchange technique is generalized to the context of certain new objects called pseudo-actions. This is used to find a more geometric proof of the Pimsner-Voiculescu theorem on the AF embedding of the irrational rotation algebras. Connections with Berg's original results are briefly examined.

Embedding diagrams are introduced to provide a uniform way of describing embeddings of transformation group C*-algebras C(X) ⋊ ℤ into AF algebras. Pimsner has classified the transformation group C* -algebras which can be AF embedded. We present a new proof of this result using embedding diagrams and pseudo-actions. The need to calculate the join of an open cover with its iterates under the transformation has been eliminated.

In this paper we show that every locally finite strongly Abelian variety satisfies the Hamiltonian property. An algebra is Hamiltonian if every one of its subuniverses is a block of some congruence of the algebra. A counterexample is provided to show that not all strongly Abelian varieties are Hamiltonian.

Assume ƒ is continuous on the closed disk D1 : |z| ≤ 1, analytic in |z| ≤ 1, but not analytic on D1. Our concern is with the behavior of the zeros of the polynomials of best uniform approximation to ƒ on D1. It is known that, for such ƒ, every point of the circle |z| = 1 is a cluster point of the set of all zeros of Here we show that this property need not hold for every subsequence of the Specifically, there exists such an f for which the zeros of a suitable subsequence all tend to infinity. Further, for near-best polynomial approximants, we show that this behavior can occur for the whole sequence. Our examples can be modified to apply to approximation in the Lq-norm on |z|= 1 and to uniform approximation on general planar sets (including real intervals).

Explicit solutions to the recurrence relation for associated continuous Hahn polynomials are derived using 3F2 contiguous relations. These solutions are used to obtain a new continued fraction and the associated absolutely continuous measure. An exceptional case is shown to yield entry 33 in Chapter 12 of Ramanujan's second notebook.

For all algebras G, such that is an algebra mentioned in the title, the modules of dimension ≤ p2 are determined. The module homomorphisms from the tensor product of these modules into a third module of the same type are described. We also give the central extensions of the algebras .

Let dbe a positive square-free integer and let Od denote the ring of integers in . The groups PSL2(Od) are collectively known as the Bianchi groups and have been widely studied from the viewpoints of group theory, number theory and low-dimensional topology. The interest of the present article is in geometric Fuchsian subgroups of the groups PSL2(Od). Clearly PSL2 is such a subgroup; however results of [18], [19] show that the Bianchi groups are rich in Fuchsian subgroups.

Finite extensions of complex commutative Banach algebras are naturally related to corresponding finite covering maps between the carrier spaces for the algebras. In the case of function rings, the finite extensions are induced by the corresponding finite covering maps, and the topological properties of the coverings are strongly reflected in the algebraic properties of the extensions and conversely. Of particular interest to us is the class of finite covering maps for which the induced extensions of function rings admit primitive generators. This is exactly the class of polynomial covering maps and the extensions are algebraic extensions defined by the underlying Weierstrass polynomials.

The purpose of this paper is to develop a suitable Galois theory for finite extensions of function rings induced by finite covering maps and to apply it in the case of Weierstrass polynomials and polynomial covering maps.

In the paper [KIS2], C. Kiselman studied the boundary smoothness of the vector sum of two smoothly bounded convex sets A and B in . He discovered the startling fact that even when A and B have real analytic boundary the set A + B need not have boundary smoothness exceeding C20/3 (this result is sharp). When A and B have C∞ boundaries, then the smoothness of the sum set breaks down at the level C5 (see [KIS2] for the various pathologies that arise).

A lattice polytope is a polytope in whose vertices are all in . The volume of a lattice polytope P containing exactly k ≥ 1 points in d in its interior is bounded above by . Any lattice polytope in of volume V can after an integral unimodular transformation be contained in a lattice cube having side length at most n˙n ! V. Thus the number of equivalence classes under integer unimodular transformations of lattice poly topes of bounded volume is finite. If S is any simplex of maximum volume inside a closed bounded convex body K in having nonempty interior, then K⊆ ( n + 2)S — (n+ l)s where mS denotes a nomothetic copy of S with scale factor m, and s is the centroid of S.

We give mild sufficient conditions on a nonlinear functional to have eigenvalues. These results are intended for the study of boundary value problems for semilinear elliptic equations.

Martin's axiom and the Continuum Hypothesis are studied here using the notion of a ccc partition i.e., a partition of the form where K0 has the following properties:

(a) K0 contains subsets of its elements as well as all singletons of X.

(b) Every uncountable subset of K0 contains two elements whose union is in K0.

The following special function of two real variables x2 and x1 is considered: and its connections with the incomplete Gaussian sums where ω are intervals of length |ω| ≤1. In particular, it is proved that for each fixed x2 and uniformly in X2 the function H(x2, x1) is of weakly bounded 2-variation in the variable x1 over the period [0, 1]. In terms of the sums W this means that for collections Ω = {ωk}, consisting of nonoverlapping intervals ωk ∪ [0,1) the following estimate is valid: where card denotes the number of elements, and c is an absolute positive constant. The exact value of the best absolute constant к in the estimate (which is due to G. H. Hardy and J. E. Littlewood) is discussed.

Let X be a complex Banach space and A a linear operator from X into X with dense domain. We construct the minimal and maximal operators of the operator A and prove that they are equal under reasonable hypotheses on the space X and operator A. As an application, we obtain the existence and regularity of weak solutions of linear equations on the space X. As another application we obtain a criterion for a symmetric operator on a complex Hilbert space to be essentially self-adjoint. An application to pseudo-differential operators of the Weyl type is given.

We give complex weight functions with respect to which the Jacobi, Laguerre, little q-Jacobi and Askey-Wilson polynomials are orthogonal. The complex functions obtained are weight functions in a wider range of parameters than the real weight functions. They also provide an alternative to the recent distributional weight functions of Morton and Krall, and the more recent hyperfunction weight functions of Kim.

In the early 70's Dennis Sullivan applied his theory of minimal models and surgery to the classification of 1-connected closed smooth manifolds of dimension ≥ 5 up to finite ambiguity [Su]. To a diffeomorphism class of such a manifold M he assigns the isomorphism class given by the real minimal model ℳ (M), the integral structure in form of various lattices and the real Pontryagin classes. If one controls the torsion of the manifolds by some bound, his result is that the map given by the triple above is finite-to-one ([Su], Theorem 13.1). He also proves a realization result for the rational minimal model and the Pontryagin classes but not for the lattices ([Su], Theorem 13.2).

Nous allons considérer ici un opérateur de Cauchy à deux variables Fuchsiennes de la forme: Puis nous allons chercher, sous certaines conditions d'hyperbolicité sur les racines caractéristiques Fuchsiennes, des solutions de l'équation 𝓟u = f, dans le cas o ù ƒ est régulière par rapport aux variables Fuchsiennes et analytique par rapport aux autres.

Baouendi et Goulaouic ont traité dans [1] ce même type d'opérateurs dans le cas d'une seule variable Fuchsienne. Dans [3] a été étudié le cas de plusieurs variables Fuchsiennes pour certains opérateurs de Goursat.