Tropical Arithmetic and Matrix Algebra

This article introduces a new structure of commutative semiring, generalizing the tropical semiring, and having an arithmetic that modifies the standard tropical operations, i.e., summation and maximum. Although our framework is combinatorial, notions of regularity and invertibility arise naturally for matrices over this semiring; we show that a tropical matrix is invertible if and only if it is regular.     

Preparation and characterization of novel inorganic-organic hybrid materials containing rare, mixed-halide anions of bismuth(III)

Three new hybrid inorganic-organic salts containing novel mixed haloanions of bismuth were synthesized by the solvothermal reaction of bismuth iodide with a haloacid, HX (X = Cl or Br), and the alkylamine 4,4'-trimethylenedipiperidine (TMDP). All three compounds were structurally characterized by single-crystal X-ray diffraction. Reaction of TMDP and BiI(3) with HCl yielded two crystalline products: [H(2)TMDP](2)[(Bi(2)I(9))(BiCl(2)I(2))] (1, major yield) and [H(2)TMDP](2)[Bi(2)Cl(10-x)I(x)] (2, x = 3.83, minor yield). Compound 1 crystallizes in the monoclinic space group Cc (a = 22.8586(11) A, b = 15.5878(7) A, c = 17.6793(9) A, beta = 118.7010(10) degrees , Z = 4) and contains the mononuclear mixed-halide anion BiCl(2)I(2)(-) in addition to a face-sharing bioctahedral Bi(2)I(9)(3)(-) anion and two independent H(2)TMDP(2+) cations. The BiCl(2)I(2)(-) anion has a sawhorse geometry (equatorially vacant trigonal bipyramidal geometry) that is not commonly observed in bismuth chemistry. Compound 2 crystallizes in the monoclinic space group P2(1)/c (a = 14.9471(7) A, b = 12.7622(6) A, c = 13.3381(7) A, beta = 116.1030(10) degrees , Z = 2) and contains an edge-sharing bioctahedral mixed-halide anion in which iodide occupies one and chloride occupies two of the five crystallographically independent halide sites. The remaining two sites have mixed-chloride and -iodide occupancy. Reaction of TMDP and BiI(3) with HBr yielded the crystalline product [H(2)TMDP][BiBr(5-x)I(x)] (3, x = 0.99), which contains, in addition to the organic cation, a polymeric, mixed-haloanion of bismuth(III). Compound 3 crystallizes in the chiral, orthorhombic space group P2(1)2(1)2(1) (a = 8.5189(5) A, b = 14.8988(9) A, c = 17.9984(11) A, Z = 4) and consists of an H(2)TMDP(2+) cation in addition to the anion, which is built up of corner-sharing BiX(6) octahedra. Of the five crystallographically independent halide sites in this anion, two are occupied solely by Br and the remaining three have mixed-bromide and -iodide occupancy. Other anion stoichiometries have been observed crystallographically for 3, as the specific stoichiometry is dependent on the relative concentration of the haloacid starting material used.     

Asymmetric aldol reactions on a soluble polymeric support

[reaction: see text] The combinatorial synthesis of small, nonpeptidic compounds is of increasing interest in current medicinal chemistry. To meet this demand, efficient entries, preferentially on polymeric supports, to pharmacologically interesting classes of compounds such as polyketides are necessary. Therefore, we have developed a synthetic protocol allowing for the asymmetric synthesis of diketides on the soluble support MeOPEG-5000. The strategy employed mainly allows for repeated aldolizations, thus providing access to functionalized polyketides of varying degree of oligomerization.     

Infrarotspektroskopische Charakterisierung von Carbons?uren durch ihre Trimethylsilylester

Zusammenfassung 91 Carbonsäuren wurden mit Dialkoxytrimethylsiloxy-N-trimethylsilylphosphoriminen (besonders Diäthoxytrimethylsiloxy-N-trimethylsilylphosphorimin, DASPI) oder mit Trimethylsilylmethyl acetamid, TMSMA, silyliert. Die C=O-Frequenzen der entsprechenden Trimethylsilylester können zur Ermittlung des Substitutionstyps am-C-Atom der Carbonsäuren dienen.-Aminosäuren bilden eine Ausnahme. Diese Säuren wurden wegen ungenügender Silylierung mit DASPI nur mit TMSMA silyliert. Folgende Frequenzbereiche der Silylesterbande wurden gefunden: aliphatisch, aliphatisch-verzweigt 1713 bis 1733 (1722±4) cm^–1;-heterosubstituiert, einschließlich-Acylamino- 1721–1754 (1737±11) cm^–1, in den meisten Fällen von einer zweiten, längerwelligen Bande begleitet im Abstand von 10–25 cm^–1:-unge-sättigt 1687–1736 (1701±13) cm^–1; aromatisch 1673–1721 (1705+10) cm^–1;-Amino- 1702–1731 (1719±7) cm^–1. (In Klammern: Mittelwerte mit Standardabweichung.)

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Infrared-spectroscopic characterisation of carboxylic acids by their trimethylsilyl esters

91 Carboxylic acids were silylated by dialkyltrimethylsilyl-N-trimethylsilyl phosphorimidates (mainly diethyltrimethylsilyl-N-trimethylsilyl phosphorimidate, DESPI) or trimethylsilylmethyl acetamide, TMSMA, and the C=O frequencies of the corresponding trimethylsilyl esters were investigated. Exceptions were -amino acids. Due to uncomplete silylation by DESPI these were silylated by TMSMA, but investigations in the region lower than 1690 cm^–1 could not be made because of the strong absorption by the amide I band of TMSMA.The following frequency regions of the silyl ester band were found: aliphatic, aliphatic-branched 1713–1733 (1722±4) cm^–1;-heterosubstituted, including-acylamino- 1721–1754 (1737±11) cm^–1 and, in most cases a second, lower frequency band at a distance of 10–25 cm^–1 originating from conformational isomers:-unsa-turated 1687–1736 (1701±13) cm^–1; aromatic 1673–1721 (1705±10) cm^–1;-amino-1702–1731 (1719±7) cm^–1. (In brackets: mean value ± standard deviation.)

     

Self-assembled nanotubes that reversibly bind acetic acid guests

A large bis-urea macrocycle was synthesized and assembled into columnar nanotubes containing a sizable cavity. This purely organic nanotube is held together primarily by hydrogen bonding and yet shows remarkable thermal stability up to 180 degrees C in the presence and absence of acetic acid guest. This enables the nanotube to be used as reusable organic zeolite.     

Polynomials with two values

This paper investigates the minimal degree of polynomialsfR[x] that take exactly two values on a given range of integers {0,...n}. We show that thegap, defined asn-deg(f), isO(n ⁵⁴⁸). The maximal gap forn128 is 3. As an application, we obtain a bound on the Fourier degree of symmetric Boolean functions.     

Structural Diversity and Thermochromic Properties of Iodobismuthate Materials Containing d-Metal Coordination Cations: Observation of a High Symmetry [Bi(3)I(11)](2-) Anion and of Isolated I(-) Anions

Six new inorganic-organic salts, all containing iodobismuthate anions and d-metal coordination cations, were synthesized solvothermally from reactions of bismuth iodide, a transition metal (M) nitrate salt (M = Co, Fe or Zn), and a heterocyclic, chelating organic ligand: 1,10-phenanthroline (1,10-phen), 3,4,7,8-tetramethyl-1,10-phenanthroline (TMphen), or 2,2':6',2'-terpyridine (tpy). All six compounds were structurally analyzed by single crystal X-ray diffraction, including variable temperature crystallographic analysis to monitor for structural changes. Furthermore, those containing novel anions and achieved in high yield were additionally characterized by solid-state UV visible spectroscopy at room temperature. [Co(1,10-phen)(3)][Bi(3)I(11)] (1), [Fe(1,10-phen)(3)][Bi(3)I(11)] (2), and [Zn(1,10-phen)(3)][Bi(3)I(11)] (3) are isostructural. They crystallize in the monoclinic space group P2(1)/n and contain the unprecedented iodobismuthate anion, [Bi(3)I(11)](2-), which exhibits near D(3h) symmetry and has an unusual arrangement of three cis face-sharing BiI(6) octahedra. [Co(TMPhen)(3)](2)[Bi(2)I(9)][I] (4), which crystallizes in the trigonal space group P-31c, and [Co(tpy)(2)](2)[Bi(2)I(9)][I] (5) and [Zn(tpy)(2)](2)[Bi(2)I(9)][I] (6), which are isostructural and crystallize in the monoclinic space group C2/c, contain the discrete binuclear [Bi(2)I(9)](3-) anion, common in previously reported iodobismuthate compounds. In addition they contain unusual isolated I(-) anions, which are rarely encountered in iodobismuthate phases. Compounds 1-6 show constitutional similarities while utilizing different organic ligands and illustrate the sensitive dependence of reaction conditions on the identity of the halometalate anion formed. Additionally, all six compounds and the starting material BiI(3) are thermochromic; the origin of this behavior is spectroscopically and crystallographically investigated.     

Supertropical algebra

We develop the algebraic polynomial theory for “supertropical algebra,” as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of “ghost elements,” which also play the key role in our structure theory. Here, we work somewhat more generally over an ordered monoid, and develop a theory which contains the analogs of several basic theorems of classical commutative algebra. This structure enables one to develop a Zariski-type algebraic geometric approach to tropical geometry, viewing tropical varieties as sets of roots of (supertropical) polynomials, leading to an analog of the Hilbert Nullstellensatz.Particular attention is paid to factorization of polynomials. In one indeterminate, any polynomial can be factored into linear and quadratic factors, and although unique factorization may fail, there is a “preferred” factorization that is explained both geometrically and algebraically. The failure of unique factorization in several indeterminates is explained by geometric phenomena described in the paper.     

Crystal growth of novel osmium-containing triple perovskites

Single crystals of two new osmium-containing triple perovskites, Ba(3)LiOs(2)O(9) and Ba(3)NaOs(2)O(9), were grown from reactive molten hydroxide fluxes in sealed silver tubes. They crystallize in the space group P6(3)/mmc with lattice parameters of a = 5.8025(1) A, c = 14.1468(4) A for Ba(3)LiOs(2)O(9) and a = 5.8858(1) A, c = 14.3451(5) A for Ba(3)NaOs(2)O(9). The magnetic susceptibility of these osmates indicates significant Os-Os coupling within the octahedra pair.