Certain C*-algebras on generators and relations are associated to directed graphs. For a finite graph γ, C*-algebra is canonically isomorphic to Cuntz-Krieger algebra corresponding to the adjacency matrix of γ. It is shown that if a countably infinite graph γ is strongly connected, γ is simple and purely infinite.

We show that mono-unary algebras have rank at most two and are thus strongly dualizable. We provide an example of a strong duality for a mono-unary algebra using an alter ego with (partial) operations of arity at most two. This mono-unary algebra has rank two and generates the same quasivariety as an injective, hence rank one, mono-unary algebra.

It is shown that no finite group containing a non-abelian nilpotent subgroup is dualizable. This is in contrast to the known result that every finite abelian group is dualizable (as part of the Pontryagin duality for all abelian groups) and to the result of the authors in a companion article that every finite group with cyclic Sylow subgroups is dualizable.

In this paper, the variety of three-valued closure algebras, that is, closure algebras with the property that the open elements from a three-valued Heyting algebra, is investigated. Particularly, the structure of the finitely generated free objects in this variety is determined.

For a real-valued ergodic process X with strictly stationary increments satisfying some measurability and continuity assumptions it is proved that the long-run ‘average behaviour’ of all its increments over finite intervals replicates the distribution of the corresponding increments of X in a strong sense. Moreover, every Lévy process has a version that possesses this ergodic path property.

A harmonic NA group is a suitable solvable extension of a two-step nilpotent Lie group N of Heisenberg type by R+, which acts on N by anisotropic dilations. A hypergroup is a locally compact space for which the space of Borel measures has a convolution structure preserving the probability measures and satisfying suitable conditions. We describe a class of hypergroups associated to NA groups.

We prove that the lattice points on the circles x2 + y2 = n are well distributed for most circles containing lattice points.

The graph product of a family of groups lies somewhere between their direct and free products, with the graph determining which pairs of groups commute. We show that the graph product of quasi-lattice ordered groups is quasi-lattice ordered, and, when the underlying groups are amenable, that it satisfies Nica's amenability condition for quasi-lattice orders. The associated Toeplitz algebras have a universal property, and their representations are faithful if the generating isometries satisfy a joint properness condition. When applied to right-angled Artin groups this yields a uniqueness theorem for the C*-algebra generated by a collection of isometries such that any two of them either *-commute or else have orthogonal ranges. The analogous result fails to hold for the nonabelian Artin groups of finite type considered by Brieskorn and Saito, and Deligne.

Recently, Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. Afterwards, we dealt with similar problems for submanifolds in complex space forms.

In the present paper, we obtain sharp inequalities between the Ricci curvature and the squared mean curvature for submanifolds in Sasakian space forms. Also, estimates of the scalar curvature and the k-Ricci curvature respectively, in terms of the squared mean curvature, are proved.

We construct irreducible unitary representations of a finitely generated free group which are weakly contained in the left regular representation and in which a given linear combination of the generators has an eigenvalue. When the eigenvalue is specified, we conjecture that there is only one such representation. The representation we have found is described explicitly (modulo inversion of a certain rational map on Euclidean space) in terms of a positive definite function, and also by means of a quasi-invariant probability measure on the combinatorial boundary of the group.

The Bernstein-von Mises theorem, concerning the convergence of suitably normalized and centred posterior density to normal density, is proved for a certain class of linearly parametrized parabolic stochastic partial differential equations (SPDEs) as the number of Fourier coefficients in the expansion of the solution increases to infinity. As a consequence, the Bayes estimators of the drift parameter, for smooth loss functions and priors, are shown to be strongly consistent, asymptotically normal and locally asymptotically minimax (in the Hajek-Le Cam sense), and asymptotically equivalent to the maximum likelihood estimator as the number of Fourier coefficients become large. Unlike in the classical finite dimensional SDEs, here the total observation time and the intensity of noise remain fixed.