C,N‐chelated triorganotin(IV) diesters of 4‐ketopimelic acid and their fungicidal activity

The set of four triorganotin(IV) diesters of 4‐ketopimelic acid containing {2‐[(CH₃)₂NCH₂]C₆H₄}‐ as a C,N‐chelating ligand was prepared. Their structures were studied by the help of IR, NMR and X‐ray crystallographic techniques in the case of {{2‐[(CH₃)₂NCH₂]C₆H₄}SnPh₂}₂[(OOCCH₂CH₂)₂C?]. All these compounds are monomeric both in solid state and solution with five‐coordinated tin atoms and medium strong intramolecular Sn? N connection. The antimycotical activity of these compound was studied and compared with the triorganotin(IV) derivatives of 4‐ketopimelic acid and antimycotical drugs in clinical use. Copyright © 2008 John Wiley & Sons, Ltd.     

Latin bitrades,dissections of equilateral triangles,and abelian groups

Let T=(T^*, T^?) be a spherical latin bitrade. With each a=(a₁, a₂, a₃)∈T^* associate a set of linear equations Eq(T, a) of the form b₁+b₂=b₃, where b=(b₁, b₂, b₃) runs through T^*\{a}. Assume a₁=0=a₂ and a₃=1. Then Eq(T,a) has in rational numbers a unique solution $b_{i}=\bar{b}_{i}Let T=(T^*, T^?) be a spherical latin bitrade. With each a=(a₁, a₂, a₃)∈T^* associate a set of linear equations Eq(T, a) of the form b₁+b₂=b₃, where b=(b₁, b₂, b₃) runs through T^*\{a}. Assume a₁=0=a₂ and a₃=1. Then Eq(T,a) has in rational numbers a unique solution $b_{i}=\bar{b}_{i}$. Suppose that $\bar{b}_{i}\not= \bar{c}_{i}$ for all b, c∈T^* such that $\bar{b}_{i}\not= \bar{c}_{i}$ and i∈{1, 2, 3}. We prove that then T^? can be interpreted as a dissection of an equilateral triangle. We also consider group modifications of latin bitrades and show that the methods for generating the dissections can be used for a proof that T^* can be embedded into the operational table of a finite abelian group, for every spherical latin bitrade T. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 1–24, 2010     

Colored graphs without colorful cycles

A colored graph is a complete graph in which a color has been assigned to each edge, and a colorful cycle is a cycle in which each edge has a different color. We first show that a colored graph lacks colorful cycles iff it is Gallai, i.e., lacks colorful triangles. We then show that, under the operation mon ≡ m + n − 2, the omitted lengths of colorful cycles in a colored graph form a monoid isomorphic to a submonoid of the natural numbers which contains all integers past some point. We prove that several but not all such monoids are realized. We then characterize exact Gallai graphs, i.e., graphs in which every triangle has edges of exactly two colors. We show that these are precisely the graphs which can be iteratively built up from three simple colored graphs, having 2, 4, and 5 vertices, respectively. We then characterize in two different ways the monochromes, i.e., the connected components of maximal monochromatic subgraphs, of exact Gallai graphs. The first characterization is in terms of their reduced form, a notion which hinges on the important idea of a full homomorphism. The second characterization is by means of a homomorphism duality. The first author would like to express his thanks for support by project LN 00A056 of the Ministry of Education of the Czech Republic. The second author would like to express his thanks for support by project LN 00A056 of the Ministry of Education of the Czech Republic, by the NSERC of Canada and by the Gudder Trust of the University of Denver.     

On Existence of Solution, Oscillatory and Non-Oscillatory Properties of Delay Impulsive Differential Equations Containing ‘Maximum’

In this paper, we determined the criteria for the existence of solutions for delay impulsive differential equations containing ‘maximum’ (DIDEM) using the Morales’ fixed point theorem. An impulsive analogue of the Gronwall-Bellman’s inequality with deviating argument is also considered. Some results on the oscillation and non-oscillation of solutions of the (DIDEM) are also obtained. Applications of the oscillatory results obtained are made to four non-linear examples one of which is a military model and upper bounds found for the solutions of the impulsive Fitzhugh-Naguno model and an impulsive nonlinear control system.     

Large deviations forU-statistics

Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol.32, No. 1, pp. 7–19, January–March,1992.     

Estimating the fractal dimension,K 2 -entropy,and the predictability of the atmosphere

The series of mean daily temperature of air recorded over a period of 215 years is used for analysing the dimensionality and the predictability of the atmospheric system. The total number of data points of the series is 78527. Other 37 versions of the original series are generated, including “seasonally adjusted” data, a smoothed series, series without annual course, etc. Modified methods of Grassberger &; Procaccia are applied. A procedure for selection of the “meaningful” scaling region is proposed. Several scaling regions are revealed in the lnC(r) versus Inr diagram. The first one in the range of larger lnr has a gradual slope and the second one in the range of intermediate lnr has a fast slope. Other two regions are settled in the range of small lnr. The results lead us to claim that the series arises from the activity of at least two subsystems. The first subsystem is low-dimensional (d _f =1.6) and it possesses the potential predictability of several weeks. We suggest that this subsystem is connected with seasonal variability of weather. The second subsystem is high-dimensional (d _f >17) and its error-doubling time is about 4–7 days. It is found that the predictability differs in dependence on season. The predictability time for summer, winter and the entire year (T ₂ ≈4.7 days) is longer than for transitionseasons (T ₂ ≈4.0 days for spring,T ₂ ≈3.6 days for autumn). The role of random noise and the number of data points are discussed. It is shown that a 15-year-long daily temperature series is not sufficient for reliable estimations based on Grassberger &; Procaccia algorithms.     

HPTLC Determination of Xanthohumol in Hops (Humulus lupulus L.) and Hop Products

JPC – Journal of Planar Chromatography – Modern TLC - A sensitive thin-layer chromatographic method has been established for quantification of xanthohumol in hops (Humulus lupulus L.)...