Trading crossings for handles and crosscaps

Let c_k = cr_k (G) denote the minimum number of edge crossings when a graph G is drawn on an orientable surface of genus k. The (orientable) crossing sequence c_o, c₁,c₂…encodes the trade‐off between adding handles and decreasing crossings. We focus on sequences of the type c_o > c₁ > c₂ = 0; equivalently, we study the planar and toroidal crossing number of doubly‐toroidal graphs. For every ? > 0 we construct graphs whose orientable crossing sequence satisfies c₁/c_o > 5/6??. In other words, we construct graphs where the addition of one handle can save roughly 1/6th of the crossings, but the addition of a second handle can save five times more crossings. We similarly define the non‐orientable crossing sequence ?₀,?₁,?₂, ··· for drawings on non‐orientable surfaces. We show that for every ?₀ > ?₁ > 0 there exists a graph with non‐orientable crossing sequence ?₀, ?₁, 0. We conjecture that every strictly‐decreasing sequence of non‐negative integers can be both an orientable crossing sequence and a non‐orientable crossing sequence (with different graphs). © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 230–243, 2001     

Packing of Incongruent Circles on the Sphere

 In this paper we provide an upper bound to the density of a packing of circles on the sphere, with radii selected from a given finite set. This bound is precise, e.g. for the system of incircles of Archimedean tilings (4, 4, n) with n ? 6. A generalisation to the weighted density of packing is applied to problems of solidity of a packing of circles. The simple concept of solidity was introduced by L. Fejes Toóth [6]. In particular, it is proved that the incircles of the faces of the Archimedean tilings (4,6,6), (4,6,8) and (4, 6, 10) form solid packings.     

Mathematical and physical aspect of the solution of the Cauchy problem for a system of equations describing a nonlocal model of propagation of heat with finite speed

In our paper we present a new system of equations describing a nonlocal model of propagation of heat with finite speed in three-dimensional space. Such a system of equations is described by a system of integral – differential equations. At first using the modiffied Cagniard de Hoop method, we construct the fundamental solution of this system of equations. On the basis of the constructed fundamental solution we obtain the explicite formulate of the solution of the Cauchy problem for this system of equations and applying the method of Sobolev and Biesov spaces, we get L^p – L^q time decay estimate for the solution of the Cauchy problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Determination of organic acids in tobacco by capillary isotachophoresis

Nine organic acids and phosphate have been separated and quantified in tobacco by capillary isotachophoresis. Two operating systems for the separation were found: hydrochloric acid (10 mmol l(-1)) adjusted with beta-alanine to pH 2.9 plus 0.1% poly(vinylpyrrolidone) was used as the leading electrolyte and 10 mmol l(-1) nicotinic acid as the terminating electrolyte and hydrochloric acid (10 mmol l(-1)) including 5.5 mmol l(-1) 1,3-bis[tris(hydroxymethyl)methylamino]propane plus 0.1% poly(vinylpyrrolidone) was used as the leading electrolyte and 5 mmol l(-1) 2-morpholinoethanesulfonic acid as the terminating electrolyte. Linearity was observed from 0.008 to 0.100 mmol l(-1) with a coefficient of determination (r2) of 0.999. The separation of anions was achieved in less than 16 min. The minimal sample pretreatment and relatively low running cost make isotachophoresis a good alternative to existing methods.     

Influence of a Magnetic Fluxon on the Vacuum Energy of Quantum Fields Confined by a Bag

We study the simultaneous influence of boundary conditions and external fields on quantum fluctuations by considering vacuum zero-point energies for quantum fields in the presence of a magnetic fluxon confined by a bag, circular and spherical for bosons and circular for fermions. The Casimir effect is calculated in a generalized cut-off regularization after applying zeta-function techniques to eigenmode sums and using recent techniques about Bessel zeta functions at negative arguments. Received: 25 March 1997 / Accepted: 3 September 1997     

Roothaan INDO CI spin density calculation

Spin densities have been calculated by the Roothaan INDO CI method in the parametrization of Pople and Kaufman. The dependence of the spin densities on both the type and the number of configurations in CI has been followed. Results have been compared with the UHF and spin-projected UHF methods.     

Electrochemical,chemical and spectrophotometric investigation of the copper(II)-bicinchoninic acid reagent used for protein measurements

The use of a reagent containing copper (II), bicinchoninic acid (BCA) and tartrate buffered at pH 11.25 was studied voltammetrically, coulometrically, spectrophotometrically and chemically. The reagent exhibits three cathodic waves at rotating platinum disk and rotating glassy carbon electrodes. The two more-positive cathodic waves correspond to electrochemical reduction to copper (I)-bisbicinchoninate, Cu(BCA)₂^3?. The third cathodic wave is caused by reduction to metallic copper. A reaction mechanism is proposed that shows the major chemical species in the solution and the electrochemical reaction products. Voltammetric and chemical studies indicate that the reagent should be used with care for protein assays because it is subject to multiple chemical interferences.