Some geometrical aspects of semidefinite linear complementarity problems

This article studies some geometrical aspects of the semidefinite linear complementarity problem (SDLCP), which can be viewed as a generalization of the well-known linear complementarity problem (LCP). SDLCP is a special case of a complementarity problem over a closed convex cone, where the cone considered is the closed convex cone of positive semidefinite matrices. It arises naturally in the unified formulation of a pair of primal-dual semidefinite programming problems. In this article, we introduce the notion of complementary cones in the semidefinite setting using the faces of the cone of positive semidefinite matrices and show that unlike complementary cones induced by an LCP, semidefinite complementary cones need not be closed. However, under R₀-property of the linear transformation, closedness of all the semidefinite complementary cones induced by L is ensured. We also introduce the notion of a principal subtransformation with respect to a face of the cone of positive semidefinite matrices and show that for a self-adjoint linear transformation, strict copositivity is equivalent to strict semimonotonicity of each principal subtransformation. Besides the above, various other solution properties of SDLCP will be interpreted and studied geometrically.     

Preparation,and infrared and thermal properties of six adducts of tetrakis-(8-hydroxyquinolinato)thorium(IV)

Six adducts of the tctrakis(8-hydroxyquinolinato) complex of thorium(IV) with 8-hydroxyquinoline, pyridine, urea, dimethylsulphoxide, dimethylformamide and 1,10-phenanthroline were prepared and characterized, the infrared bands for the room-temperature forms are given and their particular thermal properties are reported. The thermal analyses were performed mostly in the presence of air, but in some cases an argon atmosphere was used for comparison. The adducts were lost from the principal complex at temperatures varying between 110 and 290°. Intermediate oxygenated complexes were formed in each case at about 400°. The thermal data suggest the standardization of a procedure for the formation of Th(C₉H₆NO)₄.

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Zusammenfassung Sechs Addukte des Tetrakis(8-hydroxyquinolinato)-Komplexes von Thorium(IV) mit 8-Hydroxyquinolin, Pyridin, Harnstoff, Dimethylsulphoxid, Dimethylformamid und 1,10-Phenanthrolin wurden dargestellt und charakterisiert. Infrarotbanden und spezielle thermische Eigenschaften der Raumtemperatur-Formen sind angegeben. Die thermischen Analysen wurden meist in Luft ausgeführt, in einigen Fällen zu Vergleichszwecken aber auch in Argonatmosphäre. Die Addukte wurden vom Komplex bei Temperaturen zwischen 110 und 290 °C abgegeben. Intermediäre oxydierte Komplexe wurden in jedem Fall bei etwa 400 °C gebildet. Die thermischen Daten ermöglichen die Standardisierung eines Verfahrens zur Darstellung von Th(C₉H₆NO)₄.

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-(8- ) (IV) 8- , , , - , 1,10-. - , . , , . 110–290°. 400° . Th(C₉H₆NO)₄.

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We are grateful to the Bangladesh University Grants Commission for a Fellowship Grant to AR.     

Antimicrobial triterpenes from Debregeasia salicifolia

New triterpene, 3beta-(trans-cinnamoyloxy)-19alpha-hydroxy-urs-12-ene (1) has been isolated from the methanolic fraction of Debregeasia salicifolia, along with uvaol (2), 3beta,19alpha-dihydroxy-urs-12-ene (3), ursolic acid (4), pomolic acid (5), pomolic acid methyl ester (6) and tormentic acid (7) reported for the first time from this species. The compounds (1), (3) and (6) showed significant antimicrobial activity. The structure elucidation was made with the help of extensive 2D NMR spectroscopic techniques.     

Spectrophotometric determination of ziram, ferbam and zineb with diphenylcarbazone

A procedure has been developed for the determination of ziram (zinc dimethyldithiocarbamate), ferbam (ferric dimethyldithiocarbamate) and zineb (zinc ethylenebisdithiocarbamate) after their decomposition and extraction of the diphenylcarbazone complexes of the zinc or iron into isobutyl methyl ketone. These complexes absorb strongly at 520 nm. The method is rapid, sensitive and selective and can be used for the determination of these dithiocarbamates in commercial and synthetic mixtures.     

Variational bounds on the radius of convergence of the born series

The radius of convergence of the Born series for potential scattering problems is discussed in terms of the eigen-values of the scattering integral equation. Variational procedures yielding upper and lower bounds for these eigen-values are introduced. These bounds axe calculated using various trial functions and compared to the exact values for the square well, exponential repulsive, and Yukawa potentials.     

2,5-Dihydroxyacetophenone oxime as an analytical reagent for the amperometric determination of copper,palladium, and nickel

Summary Copper, palladium and nickel can be titrated amperometrically with 2,5-dihydroxyacetophenone oxime at –0.15, –0.4 and, –1.2 V vs. S. C. E. respectively, the optimum p_H values being 4.6, 4.0, and 9.0. Alkali and alkaline earth metals, Al, Bi, Cd, Co, Cr. Fe(II), Hg, Mn, Ni, Pb, Th, Zn, UO, W, and Zr do not interfere with the estimation of copper and palladium. The interferences due to the presence of silver and iron(III) can be eliminated by adding excess of KCl and KF respectively. Most heavy metals interfere with the estimation of nickel and should be eliminated before titration.

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Zusammenfassung Kupfer, Palladium und Nickel können mit 2,5-Dihydroxyaeetophenonoxim bei –0,15, –0,4 bzw. –1,2 V gegen eine gesättigte Kalomelelektrode amperometrisch titriert werden, wobei man das p_H am besten auf 4,6, 4,0 bzw. 9,0 hält. Alkalimetalle, Erdalkalien, Al, Bi, Cd, Co, Cr, Fe(II), Hg, Mn, Ni, Pb, Th, Zn, UO₂, W und Zr stören die Bestimmung des Kupfers und des Palladiums. Silber und Eisen können mit Kaliumchlorid gefällt bzw. mit Kaliumfluorid maskiert werden. Die meisten Schwermetalle stören die Nickelbestimmung und müssen daher vor der Titration entfernt werden.

     

Convex hull of powers of a complex number,trinomial equations and the Farey sequence

The color of a complex number is defined as the number of vertices of the convex hull of powers of that number. This induces a coloring of the unit disk. The structure of the set of points where the color changes is investigated here. It is observed that there is a connection between this fractal set and some family of trinomial equations. Three algorithms for coloring the unit disk are described, the last one (related to the Farey sequence) arising out of a conjecture. This conjecture is formulated and proved in this presentation.     

Electric field gradient at the Hf site and the phase transition in HfV2

The electric quadrupole interaction at the Hf site in the cubic HfV₂ compound, which shows lattice instability when cooled below 120 K, has been studied using the 482 keV (5/2⁺) state in¹⁸¹Ta. The time differential perturbed angular correlation (TDPAC) studies involving (133+136)-482 keV - cascade at room temperature show that only a fraction of¹⁸¹Ta nuclei see a cubic symmetry while the remaining experience a weak, randomly oriented, electric quadrupole interaction described by a Gaussian frequency distribution with _Q=6.1±1.0 Mrad/sec and the relative width =0.5. There is a dramatic change in TDPAC pattern indicating a phase transition when the compound is cooled down to 77 K. At this temperature the quadrupole interaction frequency is found to be _Q=40.6±3.0 Mrad/sec with =0.35. The experimentally observed EFG both at the Hf and the V sites are compared to the theoretical estimates based on point charge model.