Random subgraphs of properly edge-coloured complete graphs and long rainbow cycles

A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph Kn has a rainbow Hamiltonian path. Although this conjecture turned out to be false, it was widely believed that such a colouring always contains a rainbow cycle of length almost n. In this paper, improving on several earlier results, we confirm this by proving that every properly edge-coloured Kn has a rainbow cycle of length n − O(n 3/4 ). One of the main ingredients of our proof, which is of independent interest, shows that the subgraph of a properly edge-coloured Kn formed by the edges a random set of colours has a similar edge distribution as a truly random graph with the same edge density. In particular it has very good expansion properties

Inequivalent representations of commutator or anticommutator rings of field operators and their applications

Hamiltonian of a system in quantum field theory can give rise to infinitely many partition functions which correspond to infinitely many inequivalent representations of the canonical commutator or anticommutator rings of field operators. This implies that the system can theoretically exist in infinitely many Gibbs states. The system resides in the Gibbs state which corresponds to its minimal Helmholtz free energy at a given range of the thermodynamic variables. Individual inequivalent representations are associated with different thermodynamic phases of the system. The BCS Hamiltonian of superconductivity is chosen to be an explicit example for the demonstration of the important role of inequivalent representations in practical applications. Its analysis from the inequivalent representations' point of view has led to a recognition of a novel type of the superconducting phase transition.Comment: 25 pages, 6 figure

Partitioning multi-dimensional sets in a small number of ``uniform'' parts

In this paper we prove that every finite subset of ZxZ can be partitioned into a small number of subsets so that, in each part all vertical sections have aproximately the same size and all horyzontal sections have aproximately the same size. The generalization of this statement is used to give a combinatorial interpretation to every information inequality

Neue Methoden zur Bekämpfung des Echten Mehltaus im Ökologischen Landbau

Blattoberflächen-modifizierende Salzapplikationen sollten als Bekämpfungsmittel gegen Echten Mehltau an Tomaten und Gurken getestet werden hinsichtlich folgender Eigenschaften: - Wirkung verschiedener Salze - Blattverträglichkeit - Untersuchung der unterschiedlichen Wirkungsmechanismen mittels REM - Übertragbarkeit in praxisnahe Versuche. Die verschiedenen Salzlösungen und Kombinationen aus diesen Salzen unterschieden sich deutlich in ihrer Wirkung auf Gurkenmehltau. Die beste Wirkung ergab sich mit allenfalls geringer Schädigung des Blattes mit 0,1% MnCl2, 1 % Patentkali (K2SO4/MgSO4) und 1% Knöterichextrakt. MnCl2 liegt auf Grund seines niedrigen Deliqueszenzpunktes ständig gelöst auf der Blattoberfläche vor, wogegen sich Knöterichextrakt und Patentkali auf der Blattoberfläche in einer Art „Salzkriechen“ verteilen durch häufigen Wechsel zwischen Trocknen und Wiederverflüssigung. Salzschäden ließen sich verringern durch Zusatz von Netzmitteln zur Sprühlösung. Rasterelektronische Visualisierung zeigte für alle untersuchten Lösungen einen direkt toxischen Effekt auf die Mehltausporen und ihre Hyphen in jeder Entwicklungsphase des Pilzes. Eine Verbesserung des Nährstoffstatus der Pflanzen scheidet als Wirkungsmechanismus aus, da alle Pflanzen optimal ernährt waren und die Behandlungen zu keinen signifikanten Veränderungen der Nährstoffgehalte in den Blättern führten. Besonders chloridische Salze führten z.T. zu erheblicher Nekrotisierung der Blätter. Da diese Salzlösungen jedoch auch in geringer Konzentration in der Lage waren, die Ausbreitung des Echten Mehltau effektiv zu verhindern, sollten Möglichkeiten zur Mikrodosierung dieser Salze geprüft werden. Extreme Witterungsbedingungen im Sommer 2003 führten dazu, dass Praxistests nicht ausgeführt oder normal beendet werden konnten. Es gilt jedoch als sicher, dass einige der untersuchten Salzmischungen in geringen Konzentrationen (zw. 0,1 und 1%) den Befall mit Echten Mehltaupilzen effektiv reduzieren können, ohne Blattoberflächen zu schädigen

The effect of light intensity on the production of oat (Avena sativa L.) doubled haploids through oat × maize crosses

Oat haploid embryos were obtained by wide crossing with maize. The effect of light intensity during the growing period of donor plants (450 and 800 µmol m−2 s−1) and in vitro cultures (20, 40, 70 and 110 µmol m−2 s−1) was examined for the induction and development of oat DH lines. Oat florets (26008) from 32 genotypes were pollinated with maize and treated with 2,4-dichlorophenoxyacetic acid. All the tested genotypes formed more haploid embryos when donor plants were grown in a greenhouse (9.4%) compared to a growth chamber (6.1%). The light intensity of 110 µmol m−2 s−1 during in vitro culture resulted in the highest percentage of embryo germination (38.9%), conversion into plants (36.4%) and DH line production (9.2%) when compared with lower light intensities (20, 40 and 70 µmol m−2 s−1). The results show that the growth conditions of the donor plant and light intensity during in vitro culture can affect the development of haploid embryos. This fact may have an impact on oat breeding programs using oat × maize crosses

The Power of Pivoting for Exact Clique Counting

Clique counting is a fundamental task in network analysis, and even the simplest setting of $3$-cliques (triangles) has been the center of much recent research. Getting the count of $k$-cliques for larger $k$ is algorithmically challenging, due to the exponential blowup in the search space of large cliques. But a number of recent applications (especially for community detection or clustering) use larger clique counts. Moreover, one often desires \textit{local} counts, the number of $k$-cliques per vertex/edge. Our main result is Pivoter, an algorithm that exactly counts the number of $k$-cliques, \textit{for all values of $k$}. It is surprisingly effective in practice, and is able to get clique counts of graphs that were beyond the reach of previous work. For example, Pivoter gets all clique counts in a social network with a 100M edges within two hours on a commodity machine. Previous parallel algorithms do not terminate in days. Pivoter can also feasibly get local per-vertex and per-edge $k$-clique counts (for all $k$) for many public data sets with tens of millions of edges. To the best of our knowledge, this is the first algorithm that achieves such results. The main insight is the construction of a Succinct Clique Tree (SCT) that stores a compressed unique representation of all cliques in an input graph. It is built using a technique called \textit{pivoting}, a classic approach by Bron-Kerbosch to reduce the recursion tree of backtracking algorithms for maximal cliques. Remarkably, the SCT can be built without actually enumerating all cliques, and provides a succinct data structure from which exact clique statistics ($k$-clique counts, local counts) can be read off efficiently.Comment: 10 pages, WSDM 202

The influence of continuous casting and extrusion processes on the properties and structure of CuNi2Si alloy and the morphology of the Ni-Si phase precipitates

CuNiSi alloys are widely used in various mechanical and electrical applications. These group of materials, due to the phenomena of precipitation hardening, are able to obtain high mechanical properties with also relatively high electrical properties. In the article authors compare two different types of products, made from the CW111C alloy i.e. rods which were continuously cast on the horizontal laboratory casting set-up (low degree of structure refinement) and rods commercially extruded with high degree of structure refinement. The presented results of experimental work characterize the tested materials in terms of their chemical composition, mechanical and electrical properties depending on the manufacturing process, as well as reveal their structures and the effect of heat treatment on the morphology of the Ni-Si precipitates

The influence of continuous casting and extrusion processes on the properties and structure of CuNi2Si alloy and the morphology of the Ni-Si phase precipitates

CuNiSi alloys are widely used in various mechanical and electrical applications. These group of materials, due to the phenomena of precipitation hardening, are able to obtain high mechanical properties with also relatively high electrical properties. In the article authors compare two different types of products, made from the CW111C alloy i.e. rods which were continuously cast on the horizontal laboratory casting set-up (low degree of structure refinement) and rods commercially extruded with high degree of structure refinement. The presented results of experimental work characterize the tested materials in terms of their chemical composition, mechanical and electrical properties depending on the manufacturing process, as well as reveal their structures and the effect of heat treatment on the morphology of the Ni-Si precipitates