Invariant Shen connections and geodesic orbit spaces

Summary The geodesic graph of Riemannian spaces all geodesics of which are orbits of 1-parameter isometry groups was constructed by J. Szenthe in 1976 and it became a basic tool for studying such spaces, called g.o.\ spaces. This infinitesimal structure corresponds to the reductive complement <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\mathfrak m$ in the case of naturally reductive spaces. The systematic study of Riemannian g.o. spaces was started by O. Kowalski and L.~Vanhecke in 1991, when they introduced the most important definitions, classified the low-dimensional examples and described the basic constructions of this theory. The aim of this paper is to investigate a connection theoretical analogue of the concept of the geodesic graph.     

Sprays metrizable by Finsler functions of constant flag curvature

In this paper we characterize sprays that are metrizable by Finsler functions of constant flag curvature. By solving a particular case of the Finsler metrizability problem, we provide the necessary and sufficient conditions that can be used to decide whether or not a given homogeneous system of second order ordinary differential equations represents the geodesic equations of a Finsler function of constant flag curvature. The conditions we provide are tensorial equations on the Jacobi endomorphism. We identify the class of homogeneous SODE where the Finsler metrizability is equivalent with the metrizability by a Finsler function of constant flag curvature.     

Inverse problem of the calculus of variations on Lie groups

This article studies the inverse problem of the calculus of variations for the special case of the geodesic flow associated to the canonical symmetric bi-invariant connection of a Lie group. Necessary background on the differential geometric structure of the tangent bundle of a manifold as well as the Fröhlicher-Nijenhuis theory of derivations is introduced briefly. The first obstructions to the inverse problem are considered in general and then as they appear in the special case of the Lie group connection. Thereafter, higher order obstructions are studied in a way that is impossible in general. As a result a new algebraic condition on the variational multiplier is derived, that involves the Nijenhuis torsion of the Jacobi endomorphism. The Euclidean group of the plane is considered as a working example of the theory and it is shown that the geodesic system is variational by applying the Cartan-Kähler theorem. The same system is then reconsidered locally and a closed form solution for the variational multiplier is obtained. Finally some more examples are considered that point up the strengths and weaknesses of the theory.     

Conductometric titration of thiosulfate using new-type conductivity cells

The conductometric titration of thiosulfate with silver ions using non-conventional conductivity cells is described. For this purpose conductivity cells with different cell constants and electrode constructions, equipped with silver, amalgamated silver, stainless-steel and polished platinum electrodes were used. Two well-resolved break-points were observed at 1:1 and 2:1 silver/thiosulfate stoichiometries. Accurate conductometric determinations can be made, using the second break-points of the titration curves as equivalence points, in the range of thiosulfate concentrations 10(-4)-10(-2) M. Reverse titrations are less accurate.     

On the projective Finsler metrizability and the integrability of Rapcsák equation

A. Rapcsák obtained necessary and sufficient conditions for the projective Finsler metrizability in terms of a second order partial differential system. In this paper we investigate the integrability of the Rapcsák system and the extended Rapcsák system, by using the Spencer version of the Cartan-Kähler theorem. We also consider the extended Rapcsák system completed with the curvature condition. We prove that in the non-isotropic case there is a nontrivial Spencer cohomology group in the sequences determining the 2-acyclicity of the symbol of the corresponding differential operator. Therefore the system is not integrable and higher order obstruction exists.     

On the problem of linearizability of a 3-web

In this paper we study the linearizability problem for 3-webs on a two-dimensional manifold. With an explicit computation we examine a 3-web whose linearizability was claimed in [J. Grifone, Z. Muzsnay, J. Saab, On the linearizability of 3-webs, Nonlinear Anal. 47 (2001) 2643–2654] and was contested later in [V.V. Goldberg, V.V. Lychagin, On the Blaschke conjecture for 3-webs, J. Geom. Anal. 16 (1) (2006) 69–115] and [V.V. Goldberg, V.V. Lychagin, On linearization of planar three-webs and Blaschke’s conjecture, C. R. Acad. Sci. Paris, Ser. I. 341 (3) (2005)]. On the basis of the theories of [J. Grifone, Z. Muzsnay, J. Saab, On the linearizability of 3-webs, Nonlinear Anal. 47 (2001) 2643–2654], we give an effective method for computing the linearizability criterion, and we prove that this particular web is linearizable by finding explicitly the affine deformation tensor and the corresponding flat linear connection.     

Invariant Metrizability and Projective Metrizability on Lie Groups and Homogeneous Spaces

In this paper, we study the invariant metrizability and projective metrizability problems for the special case of the geodesic spray associated to the canonical connection of a Lie group. We prove that such canonical spray is projectively Finsler metrizable if and only if it is Riemann metrizable. This result means that this structure is rigid in the sense that considering left invariant metrics, the potentially much larger class of projective Finsler metrizable canonical sprays, corresponding to Lie groups, coincides with the class of Riemann metrizable canonical sprays. Generalisation of these results for geodesic orbit spaces are given.

     

Direct current conductometry with non-polarizable external electrodes

Summary The d.c. conductometric titration cell formerly proposed by the authors was improved by using an ion exchange separation membrane between the cell and the electrode-system. Some properties of the exchanger and the design of the new titration cell are described. The electrical behaviour of the new cell in the circuit was investigated and it was stated that the cell slightly changes its resistance with time, but this is not significant for the relatively short period of an actual analytical determination. Some titration curves are given and the results are discussed.

---

Zusammenfassung Die von den Verfassern früher beschriebene Titrationszelle für konduktometrische Titrationen mit Gleichstrom wurde dadurch verbessert, daß zwischen der Zelle und den Elektroden Ionenaustauschermembranen angebracht wurden. Die Eigenschaften des Austauschers und der Aufbau der Zelle werden beschrieben. Das elektrische Verhalten der Zelle wurde untersucht und festgestellt, daß sich ihr Widerstand mit der Zeit geringfügig verändert, was jedoch für die kurze Zeit einer Bestimmung ohne Bedeutung ist. Einige Titrationskurven werden wiedergegeben und die Ergebnisse diskutiert.

---

---

Part I: Muzsnay, Cs., and L. Kékedy ⁵.