Let G be a group acting freely, properly discontinuously and cellularly on some finite dimensional CW-complex Σ(2n) which has the homotopy type of the 2n-sphere 𝕊2n. Then, that action induces a homomorphism G → Aut(H2n(Σ(2n))). We classify all pairs (G, φ), where G is a virtually cyclic group and φ: G → Aut(ℤ) is a homomorphism, which are realizable in the way above and the homotopy types of all possible orbit spaces as well. Next, we consider the family of all groups which have virtual cohomological dimension one and which act on some Σ(2n). Those groups consist of free groups and semi-direct products F ⋊ ℤ2 with F a free group. For a group G from the family above and a homomorphism φ: G → Aut(ℤ), we present an algebraic criterion equivalent to the realizability of the pair (G, φ). It turns out that any realizable pair can be realized on some Σ(2n) with dim Σ(2n) ≤ 2n + 1.

Triangular algebras, and maximal triangular algebras in particular, have been objects of interest for over 50 years. Rich families of examples have been studied in the context of many w*- and C*-algebras, but there remains a dearth of concrete examples in $B({\cal H})$. In previous work, we described a family of maximal triangular algebras of finite multiplicity. Here, we investigate a related family of maximal triangular algebras with infinite multiplicity, and unearth a new asymptotic structure exhibited by these algebras.

For a von Neumann subalgebra $A \, \subseteq \, {\cal B}({\cal H})$ and any two elements a, b ∈ A with a normal, such that the corresponding derivations da and db satisfy the condition ‖db(x)‖ ≤ ‖da(x)‖ for all x ∈ A, there exist completely bounded (a)ʹ-bimodule map $\varphi : {\cal B}({\cal H}) \rightarrow {\cal B}({\cal H})$ such that db|A = φ da|A=daφ|A. (In particular db(A) ⊆ da(A).) Moreover, if A is a factor, then φ can be taken to be normal and these equalities hold on ${\cal B}({\cal H})$ instead of just on A. This result is not true for general (even primitive) C*-algebras ${\cal A}$.

A connected space is called a C0-space if its rational cup product is trivial. A characterizing property of C0-spaces is obtained. This property is used to calculate the algebraic K0-group K0(C𝔽(X)) of the ring of continuous functions for infinite-dimensional complexes X.

Let G be a finite group, and write cd (G) for the set of degrees of irreducible characters of G. The common-divisor graph Γ(G) associated with G is the graph whose vertex set is cd (G)∖{1} and there is an edge between distinct vertices a and b, if (a, b) > 1. In this paper we prove that if Γ(G) is a k-regular graph for some k ⩾ 0, then for the solvable groups, either Γ(G) is a complete graph of order k + 1 or Γ(G) has two connected components which are complete of the same order and for the non-solvable groups, either k = 0 and cd(G) = cd(PSL2(2f)), where f ⩾ 2 or Γ(G) is a 4-regular graph with six vertices and cd(G) = cd(Alt7) or cd(Sym7).

The algebra of all Dirichlet series that are uniformly convergent in the half-plane of complex numbers with positive real part is investigated. When it is endowed with its natural locally convex topology, it is a non-nuclear Fréchet Schwartz space with basis. Moreover, it is a locally multiplicative algebra but not a Q-algebra. Composition operators on this space are also studied.

In this paper we extend to Kawahara type equations a uniqueness result obtained by C. E. Kenig, G. Ponce, and L. Vega for KdV type equations. We prove that, under certain decay's conditions, the null solution is the unique solution.

This paper concerns a generalization of the Rees algebra of ideals due to Eisenbud, Huneke and Ulrich that works for any finitely generated module over a noetherian ring. Their definition is in terms of maps to free modules. We give an intrinsic definition using divided powers.

We consider a nonlinear Robin problem driven by a non-homogeneous differential operator plus an indefinite potential term. The reaction function is Carathéodory with arbitrary growth near±∞. We assume that it is odd and exhibits a concave term near zero. Using a variant of the symmetric mountain pass theorem, we establish the existence of a sequence of distinct nodal solutions which converge to zero.

In this paper we generalize the notion of the C-numerical range of a matrix to operators in arbitrary tracial von Neumann algebras. For each self-adjoint operator C, the C-numerical range of such an operator is defined; it is a compact, convex subset of ℂ. We explicitly describe the C-numerical ranges of several operators and classes of operators.

Classically, Nevanlinna showed that functions from the complex upper half plane into itself which satisfy nice asymptotic conditions are parametrized by finite measures on the real line. Furthermore, the higher order asymptotic behaviour at infinity of a map from the complex upper half plane into itself is governed by the existence of moments of its representing measure, which was the key to his solution of the Hamburger moment problem. Agler and McCarthy showed that an analogue of the above correspondence holds between a Pick function f of two variables, an analytic function which maps the product of two upper half planes into the upper half plane, and moment-like quantities arising from an operator theoretic representation for f. We apply their ‘moment’ theory to show that there is a fine hierarchy of levels of regularity at infinity for Pick functions in two variables, given by the Löwner classes and intermediate Löwner classes of order N, which can be exhibited in terms of certain formulae akin to the Julia quotient.

Using that integrable derivations of symmetric algebras can be interpreted in terms of Bockstein homomorphisms in Hochschild cohomology, we show that integrable derivations are invariant under the transfer maps in Hochschild cohomology of symmetric algebras induced by stable equivalences of Morita type. With applications in block theory in mind, we allow complete discrete valuation rings of unequal characteristic.

We introduce full diffeomorphism-invariant Colombeau algebras with added ε-dependence in the basic space. This unites the full and special settings of the theory into one single framework. Using locality conditions we find the appropriate definition of point values in full Colombeau algebras and show that special generalized points suffice to characterize elements of full Colombeau algebras. Moreover, we specify sufficient conditions for the sheaf property to hold and give a definition of the sharp topology in this framework.

We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties enjoyed by standard smooth functions on compact sets into the framework of generalized functions. Based on this concept, we introduce spaces of compactly supported generalized smooth functions that are close analogues to the test function spaces of distribution theory. We then develop the topological and functional–analytic foundations of these spaces.

We classify pointed Hopf algebras with finite Gelfand–Kirillov dimension whose infinitesimal braiding has dimension 2 but is not of diagonal type, or equivalently is a block. These Hopf algebras are new and turn out to be liftings of either a Jordan or a super Jordan plane over a nilpotent-by-finite group.

In 1973, Williams [D. Williams, On Rényi's ‘record’ problem and Engel's series, Bull. London Math. Soc.5 (1973), 235–237] introduced two interesting discrete Markov processes, namely C-processes and A-processes, which are related to record times in statistics and Engel's series in number theory respectively. Moreover, he showed that these two processes share the same classical limit theorems, such as the law of large numbers, central limit theorem and law of the iterated logarithm. In this paper, we consider the large deviations for these two Markov processes, which indicate that there is a difference between C-processes and A-processes in the context of large deviations.

We study an abstract second order inclusion involving two nonlinear single-valued operators and a nonlinear multi-valued term. Our goal is to establish the existence of solutions to the problem by applying numerical scheme based on time discretization. We show that the sequence of approximate solution converges weakly to a solution of the exact problem. We apply our abstract result to a dynamic, second-order-in-time differential inclusion involving a Clarke subdifferential of a locally Lipschitz, possibly non-convex and non-smooth potential. In the two presented examples the Clarke subdifferential appears either in a source term or in a boundary term.

We consider equivariant continuous families of discrete one-dimensional operators over arbitrary dynamical systems. We introduce the concept of a pseudo-ergodic element of a dynamical system. We then show that all operators associated to pseudo-ergodic elements have the same spectrum and that this spectrum agrees with their essential spectrum. As a consequence we obtain that the spectrum is constant and agrees with the essential spectrum for all elements in the dynamical system if minimality holds.

Let p ≥ 3 be a prime. A generalized multi-edge spinal group

$$G = \langle \{ a\} \cup \{ b_i^{(j)} {\rm \mid }1 \le j \le p,\, 1 \le i \le r_j\} \rangle \le {\rm Aut}(T)$$

$b^{(j)}_{1}, \ldots, b^{(j)}_{r_{j}}$

The Weierstrass function σ(u) associated with an elliptic curve can be generalized in a natural way to an entire function associated with a higher genus algebraic curve. This generalized multivariate sigma function has been investigated since the pioneering work of Felix Klein. The present paper shows Hurwitz integrality of the coefficients of the power series expansion around the origin of the higher genus sigma function associated with a certain plane curve, which is called an (n, s)-curve or a plane telescopic curve. For the prime (2), the expansion of the sigma function is not Hurwitz integral, but its square is. This paper clarifies the precise structure of this phenomenon. In Appendix A, computational examples for the trigonal genus 3 curve ((3, 4)-curve) y3 + (μ1x + μ4)y2 + (μ2x2 + μ5x + μ8)y = x4 + μ3x3 + μ6x2 + μ9x + μ12 (where μj are constants) are given.