Optimal error estimates for the hp-version interior penalty discontinuous Galerkin finite element method

We consider the hp-version interior penalty discontinuous Galerkinfinite-element method (hp-DGFEM) for second-order linear reaction–diffusionequations. To the best of our knowledge, the sharpest knownerror bounds for the hp-DGFEM are due to Rivière et al.(1999,Comput. Geosci., 3, 337–360) and Houston et al.(2002,SIAM J. Numer. Anal., 99, 2133–2163). These are optimalwith respect to the meshsize h but suboptimal with respect tothe polynomial degree p by half an order of p. We present improvederror bounds in the energy norm, by introducing a new functionspace framework. More specifically, assuming that the solutionsbelong element-wise to an augmented Sobolev space, we deducefully hp-optimal error bounds.     

On the number of flats spanned by a set of points in PG(d, q)

It is shown that for fixed 1 r s < d and > 0, if X PG(d, q) contains (1 + )q^s points, then the number of r-flats spanned by X is at least c()q^{(r+1)(s+1−r)}, i.e. a positive fraction of the number of r-flats in PG(s + 1,q).     

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.

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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.

     

On the Pósa-Seymour conjecture

Paul Seymour conjectured that any graph G of order n and minimum degree at least contains the k^th power of a Hamilton cycle. We prove the following approximate version. For any ϵ ≥ 0 and positive integer k, there is an n₀ such that, if G has order n ≥ n₀ and minimum degree at least $(\frac{k}{k+1} + \epsilon )n$, then G contains the k^th power of a Hamilton cycle. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 167–176, 1998     

Ichthyosan in fish eye

Ichthyosan A and V are two highly elastoviscous glycan complexes present in the aqueus and vitreus [here aqueus and vitreus are used as nouns as was suggested by Balazs and Denlinger in The eye, vol 1A. Vegetative physiology and biochemistry, 3rd edn. Academic Press, New York, pp 533–589, 1984] of the fish eye. Ichthyosan A, with its high elastic properties, surrounds and stabilizes the lens of the eye. Ichthyosan V, within the collagen fibers, serves as a structure stabilizer of the gel vitreus. These two molecular complexes are non-covalent aggregates composed of hyaluronan, a chondroitin-proteoglycan (sulfate free), and a keratan-like molecule. The ratio of hyaluronan to chondroitin–proteoglycan varies in the two ichthyosans. Electrophoretic separation methods (both free and gel electrophoresis) demonstrate that the hyaluronan–proteoglycan aggregates move as one molecular entity. The average molecular weight of the ichthyosan varies from 5.2 to 13.0 million in various species. Aquatic mammals do not have ichthyosan in their eyes.     

Uniqueness of embedding into a Gaussian semigroup on a nilpotent Lie group

This research was partially supported by the Hungarian Foundation for Scientific Researchers under grant number OTKA-T4047/1992.