The oscillating dark energy: future singularity and coincidence problem

We consider the oscillating dark energy with periodic equation of state in two equivalent formulations: ideal fluid or scalar–tensor theory. It is shown that such dark energy suggests the natural way for the unification of early-time inflation with late-time acceleration. We demonstrate how it describes the transition from deceleration to acceleration or from non-phantom to phantom era and how it solves the coincidence problem. The occurrence of finite-time future singularity for the oscillating (phantom) universe is also investigated.     

On partitioning of hypergraphs

We study edge-isoperimetric problems (EIP) for hypergraphs and extend some technique in this area from graphs to hypergraphs. In particular, we establish some new results on a relationship between the EIP and some extremal poset problems, and apply them to obtain an exact solution of the EIP for certain hypergraph families. We also show how to solve the EIP on hypergraphs in some cases when the link to posets does not work. Another outcome of our results is a new series of hypergraphs admitting nested solutions in the EIP.     

Embedding of hyperbolic groups into products of binary trees

We show that every Gromov hyperbolic group Γ admits a quasi-isometric embedding into the product of n+1 binary trees, where n=dim∂_∞Γ is the topological dimension of the boundary at infinity of Γ.     

Synthesis and polymerization of first‐generation dendritic methacrylate macromonomers

First‐generation dendritic macromonomers with a methacryloyl end group on one side, long alkyl chains on the other side, and a biuret system with two urethane groups in the core have been synthesized. The synthesis comprises three steps with hexamethylene diisocyanate uretdione as the starting material. The branching points were introduced via biuret groups and the prepared macromonomers were polymerized by free and controlled radical polymerization. Depending on the reaction conditions linear dendronized polymers as well as branched dendronized polymers and microgels with long alkyl chains were obtained. Scanning force microscopy was used to visualize high molecular weight molecules spincoated on highly oriented pyrolytic graphite. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 614–628, 2007     

On Computing the Size of a Lattice and Related Decision Problems

The problem of determining the size of a finite concept lattice is shown to be #P-complete. Since any finite lattice can be represented as a concept lattice, the problem of determining the size of a lattice given by the ordered sets of its irreducibles is also #P-complete. Some results about NP-completeness or polynomial tractability of decision problems related to concepts with bounded extent, intent, and the sum of both are given. These problems can be reformulated as decision problems about lattice elements generated by a certain amount of irreducibles.     

On optimization of metal forming with adaptable characteristics

In the present paper we consider a problem of choosing the rational way to carry on the metal processing (the problem of stochastic optimization) and the problem of determing the unknown characteristics of parameters described with random variables.     

On the Nature of Ill-Posedness of the Forward-Backward Heat Equation

We study the Cauchy problem with periodic initial data for the forward-backward heat equation defined by a J-self-adjoint linear operator L depending on a small parameter. The problem originates from the lubrication approximation of a viscous fluid film on the inner surface of a rotating cylinder. For a certain range of the parameter we rigorously prove the conjecture, based on numerical evidence, that the complete set of eigenvectors of the operator L does not form a Riesz basis in L²(-p, p)\mathcal{L}^2(-\pi, \pi). Our method can be applied to a wide range of evolution problems given by PT-symmetric operators.