V-positivity,poverses and the economic global unicity theorems of Gale and Nikaido

Summary Strong theorems concerning globally unique solutions to nonlinear inequalities have been obtained byGale andNikaido via P-matrix characterizations of theJacobians of the mappings involved. We introduce two new concepts here: v-positivity and the poverse. These permit us to state and prove significant generalizations of the theorems just mentioned and, equally important, provide access to preliminary results concerning linear inequalities by the powerful direct techniques of linear programming theory.

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Zusammenfassung Gale undNikaido haben starke Sätze über globale, eindeutige Lösungen nichtlinearer Ungleichungen erhalten für den Fall, daß dieJacobi- Matrix der auftretenden Abbildung eineP-Matrix ist. Darauf aufbauend werden zwei neue Konzepte vorgestellt: v-positivity und die poverse. Diese erlauben es, Verallgemeinerungen der erwähnten Sätze aufzustellen und zu beweisen und, was von gleicher Bedeutung ist, die Verbindung zu früheren Ergebnissen über lineare Ungleichungen durch die wirkungsvollen direkten Verfahren der Theorie der Linearen Optimierung herzustellen.

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This Research was partly supported by Project No. NR 047-021, ONR Contracts NO 0014-67-A-0126-0008 and NO 0014-67-A-0126-0009 with the Center for Cybernetic Studies, The University of Texas.     

An Experimental Study of a Flux Modulated Solid State Metathesis Reaction

A solid state metathesis (SSM) reaction was investigated with respect to the formation of rare‐earth carbodiimides, the role of the co‐produced salt (LiCl), and the eutectic flux medium (LiCl/KCl). A SSM reaction is characterized by an exothermic reaction in which a salt (often LiCl) is coproduced. When the salt melts, it can serve as a useful medium for the crystallization of a desired product. An improved crystal growth can be observed by using an eutectic flux. However, the composition of an eutectic LiCl/KCl flux is altered when LiCl is produced during the reaction. The thermal effects concerning the endothermic melting of the flux and the exothermic ingnition of the SSM reaction may compensate each other, which is not necessarily a drawback for the reaction to proceed.     

Magnetic Anisotropy of Dichlorobis(η5‐cyclopentadienyl) Complexes of Vanadium,Niobium, and Tantalum

Abstract. The magnetic behavior of the mononuclear nd¹ systems MCp₂Cl₂ (M = V⁴⁺[3d¹], Nb⁴⁺[4d¹], Ta⁴⁺[5d¹], space group P2₁/c, pseudosymmetry of the molecules C_2v) deviates from pure single ion spin magnetism on account of ligand field effect (H_lf), spin‐orbit coupling (H_so), and intermolecular spin‐spin exchange interactions (H_ex). For both VCp₂Cl₂ and NbCp₂Cl₂ excellent adaptations to the measured susceptibility data were obtained (2 K ≤ T ≤ 300 K) on the basis of spectroscopic data (lf, so) and cooperative metal–metal interactions (ex) of antiferromagnetic nature [molecular field model (mf)]. For TaCp₂Cl₂ experimental term structure data are not available. Therefore, Jørgensen's spectroscopical series (g‐factor of the central ion) was applied to extrapolate the data set for TaCp₂Cl₂. H_lf, H_so, and H_ex (antiferromagnetic) increase in the order 3d¹ → 4d¹ → 5d¹ leading, with rising atomic number of the metals, to a distinct enhancement of the magnetic anisotropy. At 4 K the μ_eff components μ_eff,y (oriented perpendicular to the cg–M–cg plane; “cg” = center of gravity of the Cp ring), μ_eff,z (oriented along the twofold pseudoaxis), and μ_eff,x are 1.73, 1.69, 1.68 (V), 1.73, 1.62, 1.59 (Nb), and 1.71, 1.59, 1.49 (Ta). While μ_eff,y is independent of T, both μ_eff,z and μ_eff,x decrease with decreasing T.