Estimation error for occupation time functionals of stationary Markov processes

The approximation of integral functionals with respect to a stationary Markov process by a Riemann sum estimator is studied. Stationarity and the functional calculus of the infinitesimal generator of the process are used to explicitly calculate the estimation error and to prove a general finite sample error bound. The presented approach admits general integrands and gives a unifying explanation for different rates obtained in the literature. Several examples demonstrate how the general bound can be related to well-known function spaces.     

Bounding the distance among longest paths in a connected graph

It is easy to see that in a connected graph any $2$ longest paths have a vertex in common. For $k\ge 7$, Skupień in 1966 obtained a connected graph in which some $k$ longest paths have no common vertex, but every $k?1$ longest paths have a common vertex. It is not known whether every $3$ longest paths in a connected graph have a common vertex and similarly for 4, 5, and $6$ longest path. Fujita et al. in 2015 give an upper bound on distance among $3$ longest paths in a connected graph. In this paper we give a similar upper bound on distance between $4$ longest paths and also for $k$ longest paths, in general.     

Boundary regularity of flows under perfect slip boundary conditions

We investigate boundary regularity of solutions of generalized Stokes equations. The problem is complemented with perfect slip boundary conditions and we assume that the nonlinear elliptic operator satisfies non-standard ?-growth conditions. We show the existence of second derivatives of velocity and their optimal regularity.     

Stability of Hereditary Graph Classes Under Closure Operations

If is a subclass of the class of claw‐free graphs, then is said to be stable if, for any , the local completion of G at any vertex is also in . If is a closure operation that turns a claw‐free graph into a line graph by a series of local completions and is stable, then for any . In this article, we study stability of hereditary classes of claw‐free graphs defined in terms of a family of connected closed forbidden subgraphs. We characterize line graph preimages of graphs in families that yield stable classes, we identify minimal families that yield stable classes in the finite case, and we also give a general background for techniques for handling unstable classes by proving that their closure may be included into another (possibly stable) class.     

Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

Several classical constructions illustrate the fact that the chromatic number of a graph may be arbitrarily large compared to its clique number. However, until very recently no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set $X$ X in $\mathbb{R }^2$ R 2 that is not an axis-aligned rectangle and for any positive integer $k$ k produces a family $\mathcal{F }$ F of sets, each obtained by an independent horizontal and vertical scaling and translation of $X$ X , such that no three sets in $\mathcal{F }$ F pairwise intersect and $\chi (\mathcal{F })>k$ χ ( F ) > k . This provides a negative answer to a question of Gyárfás and Lehel for L-shapes. With extra conditions we also show how to construct a triangle-free family of homothetic (uniformly scaled) copies of a set with arbitrarily large chromatic number. This applies to many common shapes, like circles, square boundaries or equilateral L-shapes. Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-line coloring of intervals on the line.     

Generalized Specker lattice-ordered groups and two types of distributivity

Let A be a lattice-ordered group, B a generalized Boolean algebra. The Boolean extension A _B of A has been investigated in the literature; we will refer to A _B as a generalized Specker lattice-ordered group (namely, if A is the linearly ordered group of all integers, then A _B is a Specker lattice-ordered group). The paper establishes that some distributivity laws extend from A _B to both A and B, and (under certain circumstances) also conversely.     

Winding up of the wave-function phase by an insulator-to-superfluid transition in a ring of coupled Bose-Einstein condensates

We study phase transition from the Mott insulator to superfluid in a periodic optical lattice. Kibble-Zurek mechanism predicts buildup of winding number through random walk of BEC phases, with the step size scaling as a third root of transition rate. We confirm this and demonstrate that this scaling accounts for the net winding number after the transition.     

Raman spectroscopy of hydrogen arsenate group (AsO3OH)2− in solid‐state compounds: cobalt‐containing zinc arsenate mineral,koritnigite (Zn,Co)(AsO3OH)·H2O

Raman spectra of two well‐defined types of koritnigite crystals from the Jáchymov ore district, Czech Republic, were recorded and interpreted. No substantial differences were observed between both crystal types. The observed Raman bands were attributed to the (AsO₃OH)²⁻ stretching and bending vibrations as well as stretching and bending vibrations of water molecules and hydroxyl ions. The non‐interpreted Raman spectra of koritnigite from the RRUFF database and the published infrared spectra of cobaltkoritnigite were used for comparison. The O H···O hydrogen bond lengths in the crystal structure of koritnigite were inferred from the Raman spectra and compared with those derived from the X‐ray single‐crystal refinement. The presence of (AsO₃OH)²⁻ units in the crystal structure of koritnigite was proved from the Raman spectra, which supports the conclusions of the X‐ray structure analysis. Copyright © 2010 John Wiley & Sons, Ltd.     

A Raman spectroscopic study of the different vanadate groups in solid‐state compounds—model case: mineral phases vésigniéite [BaCu3(VO4)2(OH)2] and volborthite [Cu3V2O7(OH)2·2H2O]

Raman spectroscopy has been used to study vanadates in the solid state. The molecular structure of the vanadate minerals vésigniéite [BaCu₃(VO₄)₂(OH)₂] and volborthite [Cu₃V₂O₇(OH)₂·2H₂O] have been studied by Raman spectroscopy and infrared spectroscopy. The spectra are related to the structure of the two minerals. The Raman spectrum of vésigniéite is characterized by two intense bands at 821 and 856 cm⁻¹ assigned to ν₁ (VO₄)³⁻ symmetric stretching modes. A series of infrared bands at 755, 787 and 899 cm⁻¹ are assigned to the ν₃ (VO₄)³⁻ antisymmetric stretching vibrational mode. Raman bands at 307 and 332 cm⁻¹ and at 466 and 511 cm⁻¹ are assigned to the ν₂ and ν₄ (VO₄)³⁻ bending modes. The Raman spectrum of volborthite is characterized by the strong band at 888 cm⁻¹, assigned to the ν₁ (VO₃) symmetric stretching vibrations. Raman bands at 858 and 749 cm⁻¹ are assigned to the ν₃ (VO₃) antisymmetric stretching vibrations; those at 814 cm⁻¹ to the ν₃ (VOV) antisymmetric vibrations; that at 508 cm⁻¹ to the ν₁ (VOV) symmetric stretching vibration and those at 442 and 476 cm⁻¹ and 347 and 308 cm⁻¹ to the ν₄ (VO₃) and ν₂ (VO₃) bending vibrations, respectively. The spectra of vésigniéite and volborthite are similar, especially in the region of skeletal vibrations, even though their crystal structures differ. Copyright © 2011 John Wiley & Sons, Ltd.