Stability in the Erdős–Gallai Theorem on cycles and paths,II

The Erd?s–Gallai Theorem states that for $k\ge 3$, any $n$-vertex graph with no cycle of length at least $k$ has at most $\frac{1}{2}(k?1)(n?1)$ edges. A stronger version of the Erd?s–Gallai Theorem was given by Kopylov: If $G$ is a 2-connected $n$-vertex graph with no cycle of length at least $k$, then $e\left(G\right)\le max\{h(n,k,2),h(n,k,?\frac{k?1}{2}?)\}$, where $h(n,k,a)?\left(\genfrac{}{}{0ex}{}{k?a}{2}\right)+a(n?k+a)$. Furthermore, Kopylov presented the two possible extremal graphs, one with $h(n,k,2)$ edges and one with $h(n,k,?\frac{k?1}{2}?)$ edges.In this paper, we complete a stability theorem which strengthens Kopylov’s result. In particular, we show that for $k\ge 3$ odd and all $n\ge k$, every $n$-vertex 2-connected graph $G$ with no cycle of length at least $k$ is a subgraph of one of the two extremal graphs or $e\left(G\right)\le max\{h(n,k,3),h(n,k,\frac{k?3}{2})\}$. The upper bound for $e\left(G\right)$ here is tight.     

A Radon–Nikodym type theorem for forms

The Lebesgue decomposition theorem and the Radon–Nikodym theorem are the cornerstones of the classical measure theory. These theorems were generalized in several settings and several ways. Hassi, Sebestyén, and de Snoo recently proved a Lebesgue type decomposition theorem for nonnegative sesquilinear forms defined on complex linear spaces. The main purpose of this paper is to formulate and prove also a Radon–Nikodym type result in this setting. As an application, we present a Lebesgue type decomposition theorem and solve a special case of the infimum problem for densely defined (not necessarily bounded) positive operators.     

Complement of forms

The notions of the parallel sum, the parallel difference, and the complement of two nonnegative sesquilinear forms were introduced and studied by Hassi, Sebestyé and de Snoo in Hassi et al. (Oper Theory Adv Appl 198:211–227, 2010) and Hassi et al. (J Funct Anal 257(12):3858–3894, 2009). In this paper we continue these investigations. The Galois correspondence induced by the map ${\mathfrak{m} \mapsto \mathfrak{m}_\mathfrak{t}}$ (where ${\mathfrak{m}_\mathfrak{t}}$ denotes the ${\mathfrak{t}}$ -complement of ${\mathfrak{m}}$ ) is also studied. Inspired by the work of Eriksson and Leutwiler Eriksson and Leutwiler (Math Ann 274:301–317, 1986), we introduce the notion of quasi-unit for nonnegative sesquilinear forms. The quasi-units are characterized by means of the complement and the disjoint part. It is also shown that the ${{\mathfrak{t}}}$ -quasi-units coincide with the extreme points of the convex set ${\mathfrak{z}: 0 \leq \mathfrak{z} \leq \mathfrak{t}\}}$ .     

Sprays metrizable by Finsler functions of constant flag curvature

In this paper we characterize sprays that are metrizable by Finsler functions of constant flag curvature. By solving a particular case of the Finsler metrizability problem, we provide the necessary and sufficient conditions that can be used to decide whether or not a given homogeneous system of second order ordinary differential equations represents the geodesic equations of a Finsler function of constant flag curvature. The conditions we provide are tensorial equations on the Jacobi endomorphism. We identify the class of homogeneous SODE where the Finsler metrizability is equivalent with the metrizability by a Finsler function of constant flag curvature.     

Cycle‐Saturated Graphs with Minimum Number of Edges

A graph G is called H‐saturated if it does not contain any copy of H, but for any edge e in the complement of G, the graph contains some H. The minimum size of an n‐vertex H‐saturated graph is denoted by . We prove holds for all , where is a cycle with length k. A graph G is H‐semisaturated if contains more copies of H than G does for . Let be the minimum size of an n‐vertex H‐semisaturated graph. We have We conjecture that our constructions are optimal for . © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 203–215, 2013     

Isometric embeddings into Heisenberg groups

We study isometric embeddings of a Euclidean space or a Heisenberg group into a higher dimensional Heisenberg group, where both the source and target space are equipped with an arbitrary left-invariant homogeneous distance that is not necessarily sub-Riemannian. We show that if all infinite geodesics in the target are straight lines, then such an embedding must be a homogeneous homomorphism. We discuss a necessary and certain sufficient conditions for the target space to have this ‘geodesic linearity property’, and we provide various examples.     

Efficient copper-free Sonogashira coupling of aryl chlorides with palladium on charcoal

Palladium on charcoal serves as an efficient and reusable solid supported catalyst for the Sonogashira coupling of aryl chlorides with terminal acetylenes in the presence of a bulky, electron-rich biphenyl type ligand (XPhos), without copper co-catalyst.     

Ethylcyclopentane ring opening reaction over PtGe/Al2O3 catalysts prepared by controlled surface reaction

Platinum-germanium catalysts supported on a non-acidic Al₂O₃ have been prepared by adding Ge in amounts corresponding nominally to 1/8 (PtGe1/8/Al₂O₃); 1/2 (PtGe1/2/Al₂O₃); 1 (PtGe1/Al₂O₃) and 2 (PtGe2/Al₂O₃) monolayers by controlled surface reaction of Ge(n-C₄H₉)₄ to Pt/Al₂O₃. These catalysts were characterized by electron microscopy (TEM), FTIR of CO adsorption and H₂ chemisorption. The ring opening of ethylcyclopentane (ECP) was studied as a test reaction between 543 and 633 K. PtGe1/8/Al₂O₃ catalyst produced the most ring opening products (ROP) in the whole temperature range. A good agreement with statistical values of ROP was observed at low temperature, but at higher temperature, the opening became selective, producing mostly heptane. Bimetallic catalysts PtGe1/Al₂O₃ and PtGe2/Al₂O₃ led to a nonselective hydrogenolysis, similar to the monometallic platinum catalyst Pt/Al₂O₃. The catalysts PtGe1/Al₂O₃ and PtGe2/Al₂O₃ produced ROP with the lowest selectivity; instead, much aromatics and fragments were formed, in increasing amounts above 600 K.