Über die Mittel von orthogonalen Funktionen. IV

Analysis Mathematica - Пусть {λ n } — неубываю щая...     

Analytical possibilities for determining the demand for the optimal quantity of deoxidants influencing favourably the morphology of nonmetallic inclusions in steels

Summary The analytical methods, which give information directly or at least at a speed corresponding to the technology about the momentary composition of the steel bath, are indispensable for the optimization of the metallurgical processes. Especially the determination of oxygen concentration plays a very important role among the socalled quick analytical methods.The sources of reoxidation have the most serious pernicious effect, because the oxygen level of steel increases again for the effect of reoxidation after the deoxidation.The determination of oxygen content of the samples taken from the steel bath by means of classical method is not suitable for controlling the metallurgical processes, because the time necessary for the traditional analytical methods is unfavourably long and they measure the total oxygen content of the steel bath. It is, however, more important to know the so-called disposable oxygen concentration, for controlling the metallurgical reactions.The principle of determining the active oxygen content is the measurement of the electromotive force of the solid ZrO₂ electrolyte cell. The possibility of the direct measurement of active oxygen content seems to be a milestone in adjusting the oxygen level of the metallurgical processes, and by this influencing favourable the morphology of the nonmetallic inclusions of steel to improve the quality of steel.The quantity of Al necessary theoretically for the final deoxidation, if the oxygen level is known, is not enough, but it is important to know the quantity of the dissolved Al, too. Therefore, it became necessary to work out such methods which determine the Al content after a very quick dissolution by means of atomic absorption spectrometry, or photometry and the period of determination is some minutes only.The development of the inclusion–separation methods is considered also very important, because it completes the analytical examinations controlling and following the steelmaking technology with valuable data.The paper has been presented at 12th Colloquium on Materials Analysis, Institute for Analytical Chemistry, Technical University in Vienna, May 13–15, 1985.     

Bemerkungen zum Gesetz der grossen Zahlen

Ohne ZusammenfassungDem Andenken von Prof.A. Rényi gewidmet     

Geometry optimization of crystals by the quasi-independent curvilinear coordinate approximation

The quasi-independent curvilinear coordinate approximation (QUICCA) method [K. Nemeth and M. Challacombe, J. Chem. Phys. 121, 2877 (2004)] is extended to the optimization of crystal structures. We demonstrate that QUICCA is valid under periodic boundary conditions, enabling simultaneous relaxation of the lattice and atomic coordinates, as illustrated by tight optimization of polyethylene, hexagonal boron nitride, a (10,0) carbon nanotube, hexagonal ice, quartz, and sulfur at the Gamma-point RPBE/STO-3G level of theory.     

Über die Lebesgueschen Funktionen. III

Ohne ZusammenfassungMitglied der Akademie     

Typical faces of extremal polytopes with respect to a thin three-dimensional shell

Summary Given <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>r>1$, we search for the convex body of minimal volume in $\mathbb{E}^3$ that contains a unit ball, and whose extreme points are of distance at least $r$ from the centre of the unit ball. It is known that the extremal body is the regular octahedron and icosahedron for suitable values of $r$. In this paper we prove that if $r$ is close to one then the typical faces of the extremal body are asymptotically regular triangles. In addition we prove the analogous statement for the extremal bodies with respect to the surface area and the mean width.     

Rigid Ball-Polyhedra in Euclidean 3-Space

A ball-polyhedron is the intersection with non-empty interior of finitely many (closed) unit balls in Euclidean $3$ -space. One can represent the boundary of a ball-polyhedron as the union of vertices, edges, and faces defined in a rather natural way. A ball-polyhedron is called a simple ball-polyhedron if at every vertex exactly three edges meet. Moreover, a ball-polyhedron is called a standard ball-polyhedron if its vertex–edge–face structure is a lattice (with respect to containment). To each edge of a ball-polyhedron, one can assign an inner dihedral angle and say that the given ball-polyhedron is locally rigid with respect to its inner dihedral angles if the vertex–edge–face structure of the ball-polyhedron and its inner dihedral angles determine the ball-polyhedron up to congruence locally. The main result of this paper is a Cauchy-type rigidity theorem for ball-polyhedra stating that any simple and standard ball-polyhedron is locally rigid with respect to its inner dihedral angles.     

Expectation of intrinsic volumes of random polytopes

Let K be a convex body in ℝ ^d , let j ∈ {1, …, d−1}, and let K(n) be the convex hull of n points chosen randomly, independently and uniformly from K. If ∂K is C ₊², then an asymptotic formula is known due to M. Reitzner (and due to I. Bárány if ∂K is C ₊³) for the difference of the jth intrinsic volume of K and the expectation of the jth intrinsic volume of K(n). We extend this formula to the case when the only condition on K is that a ball rolls freely inside K. Funded by the Marie-Curie Research Training Network “Phenomena in High-Dimensions” (MRTN-CT-2004-511953).     

Influence of intermolecular interactions on the Mössbauer quadrupole splitting of organotin(IV) compounds as studied by DFT calculations

The influence of intermolecular interactions on the M?ssbauer quadrupole splitting (Delta) of 119Sn was investigated in detail by density functional theory (DFT) calculations. Six organotin(IV) complexes [Me2Sn(acac)2 (1), Ph3SnCl (2), Me3Sn-succinimide (3), Me3Sn-phthalimide (4), Me3SnCl (5), and cHex3SnCl (6)] of known solid-state structures and quadrupole splittings were selected. Theoretical Delta values were calculated for both fully optimized geometries and experimental solid-state structures of different size, and the results were compared to the experimental Delta values. Compared to a synthetic procedure described in the literature for compound 4, a more convenient synthesis is reported here. The experimental Delta of this compound has also been redetermined at 80 K. For compounds with negligible intermolecular interactions in the solid state, calculated Delta values obtained did not vary significantly. In contrast, the calculated Delta values turned out to be very sensitive to the size of the supramolecular moiety considered in the crystal lattice. The crystal structure of compound 2 shows no significant intermolecular interactions; however, the calculated and the experimental Delta values remained very different, even when the supramolecular moiety considered was extended. Distortion of the coordination sphere of tin in the molecule of 2 toward a trigonal bipyramidal geometry was considered, and a possible weak intermolecular Sn...Cl interaction was included in the model. Steps of the distortion followed the new structure correlation function, which was found for the R3SnCl (R=alkyl, aryl) compounds. The experimental Delta value could be approached by this method. These results suggest that compound 2 is involved in some unexpected intermolecular interaction at 80 K.     

Violation of distribution symmetry in statistical evaluation of absolute enantioselective synthesis

Enantiomeric excesses obtained in absolute enantioselective synthesis by chiral autocatalysis (Soai-reaction) were statistically analyzed. Two sets of parallel experiments, which were performed under chemically different conditions, are available. One group contains 37, while the other contains 84 preparative results. The former group shows some interesting tendencies but does not give conclusive statistical results. The sample of 84 parallel experiments, providing 39 R- and 45 S-excesses have shown that these data represent two distinct, non-symmetric sets with different non-Gaussian distributions. Clear S preference was found.