On critical graphs with chromatic index 4

It is shown that the number of vertices of valency 2 in a critical graph with chromatic index 4 is at most 1/3 of the total number of vertices, and that there exist critical graphs with just one vertex of valency 2, but none with exactly two vertices of valency 2. From this bounds for the number of edges are deduced. The paper ends with a presentation of a catalogue of all critical graphs with chromatic index 4 and at most 8 vertices, and all simple critical graphs with chromatic index 4 and at most 10 vertices.     

Catalytic Asymmetric [4+2]‐Cycloadditions Using Tropolones: Developments,Scope, Transformations,and Bioactivity

An organocatalyzed asymmetric [4+2]‐cycloaddition between tropolones and electron‐deficient dienophiles is presented. Complex and biologically interesting dihydrohomobarrelenone scaffolds are formed through a Diels–Alder reaction utilizing bifunctional Brønsted‐base catalysis, affording the corresponding bridged bicyclic cycloadducts in up to quantitative yields with good enantio‐ (up to 92 % ee) and diastereoselectivity (up to >20:1 d.r.). The synthetic value of the obtained products is explored by downstream transformations, including photoisomerizations, and their biological relevancy by in vivo testing in MCF‐7 cancer cells.     

In Situ Total X‐Ray Scattering Study of WO3 Nanoparticle Formation under Hydrothermal Conditions

Pair distribution function analysis of in situ total scattering data recorded during formation of WO₃ nanocrystals under hydrothermal conditions reveal that a complex precursor structure exists in solution. The WO₆ polyhedra of the precursor cluster undergo reorientation before forming the nanocrystal. This reorientation is the critical element in the formation of different hexagonal polymporphs of WO₃.     

Stochastic Relations and the Problem of Prior in the Principle of Maximum Entropy

In this paper we discuss the problem of prior for the maximum entropy principle. We show that stochastic relations can be used to constrain priors and in some case uniquely determine them. The principle of maximum entropy turns stochastic relations into (over)determined systems of partial difference equations for the partition function. All statistical consequences of the stochastic relations are determined by the space of solutions of the system.     

Experimental investigation of cutoff phenomena in nonlinear photonic crystal fibers

Modal cutoff is investigated experimentally in a series of high-quality nonlinear photonic crystal fibers. We demonstrate a suitable measurement technique with which to determine the cutoff wavelength and verify the technique by inspecting the near field of the modes that may be excited below and above the cutoff. We observe a double-peak structure in the cutoff spectra, which is attributed to splitting of the higher-order modes. The cutoff is measured for seven different fiber geometries with different pitches and relative hole sizes, and very good agreement with recent theoretical work is found.     

A Remark on the Equivalence between Poisson and Gaussian Stochastic Partial Differential Equations

We discuss the connection between Gaussian and Poisson noise Wick-type stochastic partial differential equations.