Anosov diffeomorphisms on infra-nilmanifolds associated to graphs

Anosov diffeomorphisms on closed Riemannian manifolds are a type of dynamical systems exhibiting uniform hyperbolic behavior. Therefore, their properties are intensively studied, including which spaces allow such a diffeomorphism. It is conjectured that any closed manifold admitting an Anosov diffeomorphism is homeomorphic to an infra-nilmanifold, that is, a compact quotient of a 1-connected nilpotent Lie group by a discrete group of isometries. This conjecture motivates the problem of describing which infra-nilmanifolds admit an Anosov diffeomorphism. So far, most research was focused on the restricted class of nilmanifolds, which are quotients of 1-connected nilpotent Lie groups by uniform lattices. For example, Dani and Mainkar studied this question for the nilmanifolds associated to graphs, which form the natural generalization of nilmanifolds modeled on free nilpotent Lie groups. This paper further generalizes their work to the full class of infra-nilmanifolds associated to graphs, leading to a necessary and sufficient condition depending only on the induced action of the holonomy group on the defining graph. As an application, we construct families of infra-nilmanifolds with cyclic holonomy groups admitting an Anosov diffeomorphism, starting from faithful actions of the holonomy group on simple graphs.     

On a class of selfadjoint operators in Krein space and their compact perturbations

For a class of selfadjoint operators in a Krein space containing the definitizable selfadjoint operators a funetional calculus and the spectral function are studied. Stability properties of the spectral function with respect to small compact perturbations of the resolvent are proved.     

Uniform estimates for the finite-time ruin probability in the dependent renewal risk model

This paper investigates the finite-time ruin probability in the dependent renewal risk model, where the claim sizes are independent and identically distributed random variables with strongly subexponential tails, and the interarrival times are negatively dependent. We establish an asymptotic estimate, which holds uniformly for the time horizon varying in the positive half line.     

Finite-horizon ruin probability asymptotics in the compound discrete-time risk model

In this paper, we consider the compound discrete-time risk model which is a modification of the classical discrete-time (compound binomial) risk model. In this model, the claims in each fixed subsequent time interval arrive independently, and their number is random. We find the asymptotics of finite-horizon ruin probability in such a model for a subclass of heavy-tailed claim sizes and claim numbers.     

Parallel Nesterov’s method for large-scale minimization of partially separable functions

A parallel Nesterov algorithm, for solving unconstrained minimization of large scale partially separable convex functions, is presented. The problem is first transformed into a linearly constrained minimization of a separable function. A fast projected gradient (Nesterov) method is then applied to obtain a decomposition method with \(O(1/k^2)\) rate of convergence (where k is the iteration number). Preliminary numerical experiments show the efficiency of the proposed approach.     

A projection method to solve linear systems in tensor format

In this paper, we propose a method for the numerical solution of linear systems of equations in low rank tensor format. Such systems may arise from the discretisation of PDEs in high dimensions, but our method is not limited to this type of application. We present an iterative scheme, which is based on the projection of the residual to a low dimensional subspace. The subspace is spanned by vectors in low rank tensor format which—similarly to Krylov subspace methods—stem from the subsequent (approximate) application of the given matrix to the residual. All calculations are performed in hierarchical Tucker format, which allows for applications in high dimensions. The mode size dependency is treated by a multilevel method. We present numerical examples that include high‐dimensional convection–diffusion equations.Copyright © 2012 John Wiley & Sons, Ltd.     

Estimates on the zeros of E_2

We improve the results from El Basraoui (Proc Amer Math Soc 138(7):2289–2299, 2010) about the Eisenstein series $E_2(z)=1-24\sum _{n=1}^\infty \frac{nq^n}{1-q^n}$ . In particular we show that there exists exactly one (simple) zero in each Ford circle and give an approximation to its location.     

Concepts of efficiency for uncertain multi-objective optimization problems based on set order relations

In this paper we present new concepts of efficiency for uncertain multi-objective optimization problems. We analyze the connection between the concept of minmax robust efficiency presented by Ehrgott et al. (Eur J Oper Res, 2014, doi:10.1016/j.ejor.2014.03.013) and the upper set less order relation \(\preceq _s^u\) introduced by Kuroiwa (1998, 1999). From this connection we derive new concepts of efficiency for uncertain multi-objective optimization problems by replacing the set ordering with other set orderings. Those are namely the lower set less ordering (see Kuroiwa 1998, 1999), the set less ordering (see Nishnianidze in Soobshch Akad Nauk Gruzin SSR 114(3):489–491, 1984; Young in Math Ann 104(1):260–290, 1931, doi:10.1007/BF01457934; Eichfelder and Jahn in Vector Optimization. Springer, Berlin, 2012), the certainly less ordering (see Eichfelder and Jahn in Vector Optimization. Springer, Berlin, 2012), and the alternative set less ordering (see Ide et al. in Fixed Point Theory Appl, 2014, doi:10.1186/1687-1812-2014-83; Köbis 2014). We analyze the resulting concepts of efficiency and present numerical results on the occurrence of the various concepts. We conclude the paper with a short comparison between the concepts, and an outlook to further work.     

Ultrapowers as sheaves on a category of ultrafilters

In the paper we investigate the topos of sheaves on a category of ultrafilters. The category is described with the help of the Rudin-Keisler ordering of ultrafilters. It is shown that the topos is Boolean and two-valued and that the axiom of choice does not hold in it. We prove that the internal logic in the topos does not coincide with that in any of the ultrapowers. We also show that internal set theory, an axiomatic nonstandard set theory, can be modeled in the topos.Mathematics Subject Classification (2000): Primary 03G30, 03C20, Secondary 03E05, 03E70, 03H05The author would like to thank the Mittag-Leffler Institute for partial suport.     

On the block counting process and the fixation line of the Bolthausen–Sznitman coalescent

The block counting process and the fixation line of the Bolthausen–Sznitman coalescent are analyzed. It is shown that these processes, properly scaled, converge in the Skorohod topology to the Mittag-Leffler process and to Neveu’s continuous-state branching process respectively as the initial state tends to infinity. Strong relations to Siegmund duality, Mehler semigroups and self-decomposability are pointed out. Furthermore, spectral decompositions for the generators and transition probabilities of the block counting process and the fixation line of the Bolthausen–Sznitman coalescent are provided leading to explicit expressions for functionals such as hitting probabilities and absorption times.