On the zeros of polynomials over arbitrary finite dimensional algebras

In [3] it was shown that a polynomial of degree n with coefficients in an associative division algebra, which is d-dimensional over its center, has either infinitely many or at most n^d zeros. In this paper we raise the same question for arbitrary m-ary F-algebras A which are d-dimensional over the algebraically closed field F. Our main result states that in the affine space of m-ary algebras of dimension d there is a non-empty Zariski-open set whose elements A have the following property: in the space of polynomial of precise degree n with coefficients in A there is a non-empty Zariski-open set whose elements have precisely n^d zeros. It is shown that all simple algebras, all semi-simple associative algebras, all semisimple Jordan algebras (char F2), all semi-simple Lie algebras (char F=0), and the generic algebra possess this property.     

Symmetrische Permutationsmengen

A permutation set (M, I) consisting of a setM and a set of permutations ofM, is calledsymmetric, if for any two permutations, the existence of anx M with (x) (x) and ^–1 (x) = ^–1 (x) implies ^–1 = ^–1 , andsharply 3-transitive, if for any two triples (x ₁,x ₂,x ₃), (y ₁,y ₂,y ₃) M ³ with|{x ₁,x ₂,x ₃ }| = |{y ₁,y ₂,y ₃ }| = 3 there is exactly one permutation with(x ₁) =y ₁,(x ₂) =y ₂,(x ₃) =y ₃. The following theorem will be proved.THEOREM.Let (M, ) be a sharply 3-transitive symmetric permutation set with |M|3, such that contains the identity. Then is a group and there is a commutative field K such that and the projective linear group PGL(2, K) are isomorphic.     

Gasförmige Acetate

Gaseous Acetates Thermoanalytical and mass-spectrometrical observations are undertaken with some acetates and oxiacetates. The volatilization of copper(I) acetate takes place like that of the silver acetate as M₂Ac₂⁺ (besides the deposition of Ag). On the volatilization of the anhydrous compounds Cu₂Ac₄, Cr₂Ac₄, Rh₂Ac₄, and Mo₂Ac₄ in the vacuum of a mass spectrometer is observed that Cu₂Ac₄ vaporizes dissociative as Cu₂Ac₂⁺ (+ 2 “Ac”), while the other compounds vaporize as M₂Ac₄⁺ and simultaneously is formed an oxidic (e.g. Cr₂O₄) or metallic residue. PdAc₂ vaporizes in the mass spectrometer as a trimeric molecule Pd₃Ac₆. M₄OAc₆, which is formed from the dihydrates, vaporizes in a mass spectrometer with M ? Co, Mn as M₄OAc₆⁺. Other complexes of the same type appear as Be₄OAc₅⁺, Mg₄OAc₅⁺, and Zn₄OAc₅⁺.     

Stereoselective synthesis of pyrrolo[2,3-d]pyrimidine α- and β-D-ribonucleosides from anomerically pure D-ribofuranosyl chlorides: Solid-Liquid Phase-Transfer Glycosylation and 15N-NMR Spectra

Solid-liquid phase-transfer glycosylation (KOH, tris[2-(2-methoxyethoxy)ethye]amine ( = TDA-1), MeCN) of pyrrolo[2,3-d]pyrimidines such as 3a and 3b with an equimolar amount of 5-O-[(1,1 -dimethylethyl)dimethylsilyl]-2,3-O-(1-methylethylidene)-α-D -ribofuranosyl chloride (1) [6] gave the protected β-D -nucleosides 4a and 4b , respectively, stereoselectively (Scheme). The β-D -anomer 2 [6] yielded the corresponding α-D -nucleosides 5a and 5b with traces of the β-D -compounds. The 6-substituted 7-deazapurine nucleosides 6a , 7a , and 8 were converted into tubercidin (10) or its α-D -anomer (11) . Spin-lattice relaxation measurements of anomeric ribonucleosides revealed that T₁ values of H? C(8) in the α-D -series are significantly increased compared to H? C(8) in the β-D -series while the opposite is true for T₁ of H? C(1′). ¹⁵N-NMR data of 6-substituted 7-deazapurine D -ribofuranosides were assigned and compared with those of 2′-deoxy compounds. Furthermore, it was shown that 7-deaza-2′deoxyadenosine ( = 2′-deoxytubercidin; 12 ) is protonated at N(1), whereas the protonation site of 7-deaza-2′-deoxyguanosine ( 20 ) is N(3).     

Metallcarbonyl-Synthesen,XIV. Neuartige Vanadium-, Niob- und Tantal-Komplexe mit Wasserstoff-Brücken

Syntheses of Metal Carbonyls, XIV. Novel Vanadium, Niobium, and Tantalum Complexes Having Hydrido Bridges The novel homo- and heterodinuclear organometallic μ-hydrido compounds 3a – c of vanadium, niobium, and tantalum have been synthesized by light-induced reactions of the hydridoniobium complex (η⁵-C₅H₅)₂NbH₃ ( 1 ) with the half-sandwich complexes (η⁵-C₅H₅)M(CO)₄ ( 2a : M = V; 2b : M = Nb; 2c : M = Ta). These compounds have the general composition L_xM – H – Nb(CO)(η⁵-C₅H₅)₂. The formation of the M – H – Nb moieties is a dark-reaction preceded by two photoreactions that are independent from each other elimination of CO from 2a – c and elimination of H₂ from 1 , with the extruded carbon monoxide being transfered to the (η⁵-C₅H₅)₂NbH fragment; subsequent fixation of the species (η⁵-C₅H₅)₂Nb(CO)H thus generated to the photogragments (η⁵-C₅H₅)M(CO)₃ results in donor stabilization of these latter groups. The structural architecture of the derivatives 3a und b was established by X-Ray diffraction. The hydrogen bridges are to be considered as three-center two-electron functions that are responsible for a serious lengthening of the otherwise by at least 40 pm shorter metal-to-metal distances amounting to 371.3 pm in 3a and 373.3 pm in 3b (mean values).     

Ein neues photometrisches Auswertungsverfahren bei komplexometrischen Titrationen kleiner Metallmengen

Zusammenfassung Zur photometrischen Auswertung komplexometrischer Titrationen von Metallspuren in sehr verdünnten Lösungen mit Hilfe von Metallindicatoren wird aus mathematischen Überlegungen ein Verfahren abgeleitet, das auf der Messung von Extinktionsänderungen während der Titration beruht. Im Vergleich zu herkömmlichen photometrischen Methoden wird die Empfindlichkeit etwa um den Faktor 30 verbessert.Auszug aus der von der Fakultät für Allgemeine Wissenschaften der Technischen Hochschule München genehmigten Dissertation.Herrn Prof. Dr. F. Hegemann möchte der Verfasser für die Anregung und Förderung der vorliegenden Arbeit sehr danken. Der Deutschen Forschungsgemeinschaft, Bad. Godesberg, und der Fraunhofer-Gesellschaft zur Förderung der angewandten Forschung e. V., München, sei für die gewährte Unterstützung auch an dieser Stelle verbindlichst gedankt.     

Untersuchungen an Hydriden im Bereich der Phase Ti2CO1—x Fe x (x=0–0,5)

The hydrogen absorption of the phase Ti₆₄Co_32–x Fe _x (x=0...16) and its influence on the magnetic properties have been investigated. Measurements of the vapour-pressure, wide-line NMR and magnetic susceptibility have been performed. Substitution of Co by Fe does not change the amount of absorbed hydrogen. However the reaction rate of hydriding process, the activation energy of diffusion and the magnetic quantities change upon this substitution.

     

Bildung oktaedrischer Komplexe durch cis-Addition an planar-quadratisches Bis(oxamidoximato)nickel(II): drei Strukturbeispiele

Formation of Octahedral Complexes via cis-Addition to Square Planar Bis (oxamideoximato)nickel(II): Three Structure Examples In the reaction of orange square planar bis(oxamide oximato)nickel(II) with acids, blue to blue-green octahedral complexes are formed with neutral oxamide oximide ligands and two acid anions in cis-positions. Three compounds are described: cis-dichlorobis(oxamide oxime)nickel(II) ( 1 ), NiCl₂(C₂H₆N₄O₂)₂, M_r = 365.81, monoclinic P2₁/n, a = 6.641(2), b = 14.086(4), c = 13.473(3) Å, β = 96.26(2)°, V = 1 252.8 ų, Z = 4, d_c = 1.94 gcm^?3, final R_w = 0.031 for 4090 reflections. In cis-di(sulfanilato)bis(oxamide oxime)nickel(II) dihydrate ( 2 ) one sulfanilic anion coordinates via the sulfonic acid group, the other one via the amino group; Ni(C₆H₆NO₃S)₂(C₂H₆N₄O₂)₂ · 2 H₂O, M_r = 675.30, monoclinic P2₁, a = 6.879(3), b = 14.305(5), c = 13.930(5) Å, β = 103.62(4)°, V = 1332 Å, Z = 2, d_c = 1.68 gcm^?3, R = 0.067 for 2693 reflections. In catena-μ-(phthalato)bis(oxamide oxime)nickel(II) tetrahydrate ( 3 ) bidentate bridging phthalate anions lead to chain formation; Ni(C₈H₄O₄)(C₂H₆N₄O₂)₂ · 4H₂O, M_r = 531.09, monoclinic P2₁/c, a = 10.633(8), b = 11.324(5), c = 17.680(14) Å, β = 98.25(7)°, V = 2107 ų, Z = 4, d_c = 1.67 gcm^?3. Final R = 0.110 for 3290 reflections.     

Dissoziative ionisierung von 2-trimethylsilyl-1-phenoxyethan. Nachweis des nicht-klassischen ethylen-trimethylsilanium-ions in der gasphase

The mass spectrometric investigation of specifically deuterium and ¹³C labelled 2-trimethylsilyl-l-phenoxyethanes proves that the dissociative ionization of β-silyl-substituted ethane derivatives (loss of PhO^?; p-CH₃C₆H₄O^?; and C₄H^?₉ from PhOCH₂CH₂SiMe₃, p-MeC₆H₄OCH₂CH₂SiMe₃ and CH₃CH₂CH(CH₃)CH₂-CH₂SiMe₃, respectively) yields the non-classical bridge ethylene trimethylsilanium ion and not the open-chain isomer. Other stable C₅H₁₃Si^+? ions, characterised by collisional activation mass spectrometry, are the dimethyl n-propyl silicenium ion and the l-trimethylsilyl ethyl cation, both generated from the molecular ions of CH₃CH₂CH₂Si(Cl)Me₂ and CH₃CH(Cl)SiMe₃ via unimolecular loss of Cl^?.