Thermostaten

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Literatur

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Focal points and bargaining

This is a contribution to the growing experimental literature on how trial-and-error adjustment processes can establish a convention for coordination on an equilibrium in a game. A simple bargaining game introduced by Nash is used for this purpose. Subjects are conditioned in different treatments to use four different bargaining solutions. The stability of the conditioning is then studied as the bargaining game is played over an extended period. The data obtained is unusually sharp. In the long run, the median subject behaves as though optimizing often down to a fraction of a penny. The results are therefore not supportive of the view that strategic considerations in such situations can be neglected in favor of a study of fairness norms. Indeed, the equilibrium actually achieved in a session turns out to be a very good predictor of what the median subject says is fair in the game after play is over.We are grateful to the National Science Foundation for funding the experiments reported in this paper under Grant NSF-SES-8821521. We also gratefully acknowledge funding from the University of Michigan to set up the Michigan Economics Laboratory, where the experiments were conducted. We would also like to thank Richard Stallman and the Free Software Foundation for developing EMACS, Luke Tierney for developing LISPSTAT, and Hal Varian for showing us how to use both.     

Imaginary numbers are not real—The geometric algebra of spacetime

This paper contains a tutorial introduction to the ideas of geometric algebra, concentrating on its physical applications. We show how the definition of a geometric product of vectors in 2-and 3-dimensional space provides precise geometrical interpretations of the imaginary numbers often used in conventional methods. Reflections and rotations are analyzed in terms of bilinear spinor transformations, and are then related to the theory of analytic functions and their natural extension in more than two dimensions (monogenics), Physics is greatly facilitated by the use of Hestenes' spacetime algebra, which automatically incorporates the geometric structure of spacetime. This is demonstrated by examples from electromagnetism. In the course of this purely classical exposition many surprising results are obtained—results which are usually thought to belong to the preserve of quantum theory. We conclude that geometric algebra is the most powerful and general language available for the development of mathematical physics.The title of this paper is inspired by David Hestenes, who is known to have a fondness for deliberate ambiguity.⁽¹⁾ Supported by a SERC studentship.     

Zirconium(IV) complexes of oxydiacetic acid in aqueous solution and in the solid state as studied by multinuclear NMR and X-ray crystallography

The structures of complexes of Zr(IV) and oxydiacetate (ODA2-) in aqueous solutions of pH 0-7 were investigated with the use of 1H, 13C, and 17O NMR spectroscopy. Equilibria of mononuclear [Zr(oda)]2+, [Zr(oda)2], and [Zr(oda)3]2- complexes have been observed. In all complexes ODA2- is bound in a tridentate fashion through the two carboxylate groups and the ether oxygen. No di- or oligonuclear species containing ODA2- were observed. An excess of free Zr(IV) remains in solution, probably as a result of weak electrostatic interactions between negatively charged Zr-ODA complexes or free ODA2- and a positively charged cyclic tetranuclear hydroxy zirconium complex. CP-MAS 13C NMR spectra of solid compounds isolated from the samples indicated that the structures of the [Zr(oda)2] and [Zr(oda)3]2- complexes in solution are similar to those in the solid state. This is corroborated by the single-crystal X-ray structure of Na2[Zr(oda)3] x 5.5 H2O, which was obtained from a solution containing exclusively the [Zr(oda)3]2- complex. In this structure Zr(IV) is nine-coordinate with the three ODA2- ligands bound in a tricapped trigonal prismatic geometry. The negative charge of this [Zr(oda)3]2complex is balanced by two Na+ ions, one of which is on a center of symmetry between delta and lambda enantiomers of [Zr(oda)3]2-. This Na+ is octahedrally coordinated to six (non Zr(IV)-bound) carboxylate oxygen atoms of six different [Zr(oda)3]2- units.