Perfect matchings in large uniform hypergraphs with large minimum collective degree

We define a perfect matching in a k-uniform hypergraph H on n vertices as a set of ⌊n/k⌋ disjoint edges. Let δk−1(H) be the largest integer d such that every (k−1)-element set of vertices of H belongs to at least d edges of H.In this paper we study the relation between δk−1(H) and the presence of a perfect matching in H for k?3. Let t(k,n) be the smallest integer t such that every k-uniform hypergraph on n vertices and with δk−1(H)?t contains a perfect matching.For large n divisible by k, we completely determine the values of t(k,n), which turn out to be very close to n/2−k. For example, if k is odd and n is large and even, then t(k,n)=n/2−k+2. In contrast, for n not divisible by k, we show that t(k,n)∼n/k.In the proofs we employ a newly developed “absorbing” technique, which has a potential to be applicable in a more general context of establishing existence of spanning subgraphs of graphs and hypergraphs.     

Adaptive finite element methods in electrochemistry

In this article, we review some of our previous work that considers the general problem of numerical simulation of the currents at microelectrodes using an adaptive finite element approach. Microelectrodes typically consist of an electrode embedded (or recessed) in an insulating material. For all such electrodes, numerical simulation is made difficult by the presence of a boundary singularity at the electrode edge (where the electrode meets the insulator), manifested by the large increase in the current density at this point, often referred to as the edge effect. Our approach to overcoming this problem has involved the derivation of an a posteriori bound on the error in the numerical approximation for the current that can be used to drive an adaptive mesh-generation algorithm, allowing calculation of the quantity of interest (the current) to within a prescribed tolerance. We illustrate the generic applicability of the approach by considering a broad range of steady-state applications of the technique.     

Monochromatic Matchings in the Shadow Graph of Almost Complete Hypergraphs

Edge colorings of r-uniform hypergraphs naturally define a multicoloring on the 2-shadow, i.e., on the pairs that are covered by hyperedges. We show that in any (r – 1)-coloring of the edges of an r-uniform hypergraph with n vertices and at least (1-e)( *20c nr)(1-\varepsilon)\left( {\begin{array}{*{20}c} n\\ r\\ \end{array}}\right) edges, the 2-shadow has a monochromatic matching covering all but at most o(n) vertices. This result confirms an earlier conjecture and implies that for any fixed r and sufficiently large n, there is a monochromatic Berge-cycle of length (1 – o(1))n in every (r – 1)-coloring of the edges of K^(r)_n{K^{(r)}_{n}}, the complete r-uniform hypergraph on n vertices.     

Friendship Two-Graphs

A friendship graph is a graph in which every two distinct vertices have exactly one common neighbor. All finite friendship graphs are known, each of them consists of triangles having a common vertex. We extend friendship graphs to two-graphs, a two-graph being an ordered pair G = (G ₀, G ₁) of edge-disjoint graphs G ₀ and G ₁ on the same vertex-set V(G ₀) = V(G ₁). One may think that the edges of G are colored with colors 0 and 1. In a friendship two-graph, every unordered pair of distinct vertices u, v is connected by a unique bicolored 2-path. The pairs of adjacency matrices of friendship two-graphs are solutions to the matrix equation AB + BA = J − I, where A and B are n × n symmetric 0 − 1 matrices, J is an n × n matrix with every entry being 1, and I is the identity n × n matrix. We show that there is no finite friendship two-graph with minimum vertex degree at most two. However, we construct an infinite such graph, and this construction can be extended to an infinite (uncountable) family. Also, we find a finite friendship two-graph, conjecture that it is unique, and prove this conjecture for the two-graphs that have a dominating vertex.     

On triazoles. XXVII . The acylation,carbamoylation and thiocarbamoylation of tetrahydropyrido[4,3-d][1,2,4]triazolo[1,5-a]pyrimidinones

Isomeric 6,7,8,9-tetrahydropyrido[4,3-d][1,2,4]triazolo[1,5-a]pyrimidine-5(10H)-ones 3 , and -6(10H)-ones 4 were synthesised. Isomers 3 were converted to their 7-acyl 7 , 7-carbamoyl 8 and 7-thiocarbamoyl 9 derivatives.     

Pulsed laser crystallization of hydrogen-free a-Si thin films for high-mobility poly-Si TFT fabrication

The possibility to fabricate high-mobility polysilicon TFTs by nanosecond pulsed laser crystallization of unhydrogenated amorphous Si thin films has been investigated. Two types of lasers have been used: a large area ( 1 cm²) single ArF excimer laser pulse and a small diameter ( 100 m) frequency-doubled Nd:YAG laser beam, working in the scanning regime. Processed films have been characterized in detail by different optical and microscopic techniques. Device performances indicate that the best results are achieved with the excimer laser leading to high mobility values (up to 140 cm²/Vs) which are much larger than in polysilicon TFTs fabricated onto the same quartz substrates by low-temperature thermal (630° C) crystallization of amorphous Si films (_fe55 cm²/Vs).     

High-Performance Liquid Chromatographic Analysis of Triterpenoids in Commercial Frankincense

A reversed-phase high-performance liquid chromatographic procedure has been developed: it is a simple and specific method for the determination of fifteen pentacyclic triterpenic compounds (α-boswellic acid, 3-O-acetyl-α-boswellic acid, β-boswellic acid, 3-O-acetyl-β-boswellic acid, α-amyrin, β-amyrin, lupeol, 3-epi-α-amyrin, 3-epi-β-amyrin, 3-epi-lupeol, α-amyrenone, β-amyrenone, lupenone, lupeolic acid and 3-O-acetyl-lupeolic acid) found in the most commonly traded frankincense, usually called “Eritrean-type” olibanum. In addition, the chromatographic comparison between fresh commercial resins and botanically certified ones was described in order to determine the geographical and/or the botanical origins of commercial frankincense. According to previous botanical studies, it appears difficult to make an unequivocal distinction between Boswellia carteri and B. sacra. On the other hand, Boswellia frereana (considered as a source of high-grade frankincense) shows a characteristic chromatogram and could be unambiguously distinguished from the other producing species of commercial frankincense. In a chemical point of view, Boswellia carteri and B. sacra were more especially characterized by the presence of lupeolic acid, boswellic acids and their respective O-acetyl derivatives, whereas 3-epi-lupeol was the major compound in B. frereana methanolic extracts.     

Studies in the field of pyridazine compounds, 20. 6H-1,2,4-Triazino[4,3—b]1,2,4-triazolo[3,4—f]pyridazine,a novel angular ring system

The novel ring system, 6H-1,2,4-triazino[4,3—b]1,2,4-triazolo[3,4—f]-pyridazine was prepared either by ring closure of1-(1-ethoxycarbonylethylene)-2-(1,2,4-triazolo[4,3—b]pyridazinyl-6)-hydrazine derivatives (4, 5, 7) in polyphosphoric acid or of a hydrazine15 derived from pyridazino-1,2,4-triazine under the action of triethyl orthoformate. Compound8 showed a positive inotropic effect.

---

Über Pyridazinring enthaltende Verbindungen, 20. 6H-1,2,4-Triazino[4,3—b]1,2,4-triazolo[3,4—f]pyridazin, ein neues angulares Ringsystem

Zusammenfassung Das neue Ringsystem 6H-1,2,4-Triazino[4,3—b]1,2,4-triazolo[3,4—f]-pyridazin wurde entweder durch Ringschluß der1-(1-Ethoxycarbonylethylen)-2-(1,2,4-triazolo[4,3—b]pyridazinyl-6)-hydrazin-Derivate(4, 5, 7) 4, 5, 7 in Polyphosphorsäure, oder aus einem von Pyridazino-1,2,4-triazin abgeleiteten Hydrazin15 durch Ringschluß mit Orthoameisensäuretriethylester hergestellt. Verbindung8 zeigte einen positiven inotropen Effekt.

     

Some aspects of the chemistry of pyrimido[1,2-B]pyridazinones

On reacting the 3-aminopyridazines 1a,d,e with dimethyl acetylenedicarboxylate (DMAD), the pyrimido[1,2-b]pyridazin-2-(2H)-ones 2e-g , whereas starting from 1f , the 4(4H)-ones 5a and 3b,d were prepared. In the 2(2H)-one series, the reactions of 2b with various amino compounds resulted in various types of products. The reaction of N-methylaminopyridazines 1g,h with DMAD led to the endo-N-substituted derivatives 8a,b , whereas 1h with diethyl ethoxymethylenemalonate (DEM) gave the exo-N-substituted compound 1k. The constitution of the compounds was proved by spectroscopic and chemical evidences.