On the singular graph and the Weyr characteristic of anM-matrix

LetA be anM-matrix in standard lower block triangular form, with diagonal blocksA _ii irreducible. LetS be the set of indices such that the diagonal blockA is singular. We define the singular graph ofA to be the setS with partial order defined by > if there exists a chain of non-zero blocksA _i, A_ij, , A_l.Let ₁ be the set of maximal elements ofS, and define thep-th level _p ,p = 2, 3, , inductively as the set of maximal elements ofS \( _{1 p-1}). Denote by _p the number of elements in _p . The Weyr characteristic (associated with 0) ofA is defined to be (A) = ( ₁, ₂,, _h ), where ₁ + + _p = dim KerA ^p ,p = 1, 2, , and _h > 0, _h+1 = 0.Using a special type of basis, called anS-basis, for the generalized eigenspaceE(A) of 0 ofA, we associate a matrixD withA. We show that(A) = ( ₁, , _h) if and only if certain submatricesD _p,p+1 ,p = 1, , h – 1, ofD have full column rank. This condition is also necessary and sufficient forE(A) to have a basis consisting of non-negative vectors, which is a Jordan basis for –A. We also consider a given finite partially ordered setS, and we find a necessary and sufficient condition that allM-matricesA with singular graphS have(A) = ( ₁, , _h). This condition is satisfied ifS is a rooted forest.The work of the second-named author was partly supported by the National Science Foundation, under grant MPS-08618 A02.     

Crystal structures and hydrogen bonding schemes in four benzamide derivatives (2-hydroxy-benzamide, 2-hydroxy-thiobenzamide, 2-hydroxy-N,N-dimethyl-benzamide,and 2-Hydroxy-N,N-dimethyl-thiobenzamide)

Summary 2-Hydroxy-benzamide, C₇H₇NO₂; monoclinic; I2/a-C _2h ⁶ ;a=12.901 (2),b=4.982 (1),c=20.987 (3) Å; =91.50 (2)°;Z=8. 2-Hydroxy-thiobenzamide, C₇H₇NOS; monoclinic; P2₁/n-C _2h ⁵ ;a=13.508 (5),b=6.780 (2),c=15.878 (6) Å; =93.74 (5)°;Z=8. 2-Hydroxy-N,N-dimethyl-benzamide, C₉H₁₁NO₂; orthorhombic; Pbca-D _2h ¹⁵ ;a=11.752 (2),b=16.680 (3),c=9.079 (2) Å;Z=8. 2-Hydroxy-N,N-dimethyl-thiobenzamide, C₉H₁₁NOS; monoclinic; P2₁/c-C _2h ⁵ ;a=6.983 (1),b=11.652 (3),c=11.704 (3) Å; =100.02 (2)°;Z=4. The crystal structures of these four compounds were determined (respectively refined: 2-hydroxy-benzamide) by single crystal X-ray data. The refinements of the structure parameters by least squares methods yielded in all casesR<0.056. The hydrogen atoms were located by means of difference Fourier summations. The O-H ... O distances are 2.513 (1) Å in 2-hydroxy-benzamide (intramolecular) and 2.625 (1) Å in 2-hydroxy-N,N-dimethyl-benzamide (intermolecular). The two O-H ... S distances in 2-hydroxy-thiobenzamide are 2.904 (2) Å and 2.918 (2) Å (intramolecular, two molecules in the asymmetric unit) and 3.228 (2) Å in 2-hydroxy-N,N-dimethyl-thiobenzamide (intermolecular). Clear N-H ... O hydrogen bonds with 2.926 (1) Å and 3.006 (1) Å occur only in the structure of 2-hydroxy-benzamide (intermolecular).

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Die Kristallstrukturen und Wasserstoffbrücken-Bindungsschemata in vier Benzamid-Derivaten

Zusammenfassung 2-Hydroxy-benzamid, C₇H₇NO₂; monoklin; I2/a-C _2h ⁶ ;a=12.901 (2),b=4.982 (1),c=20.987 (3) Å; =91.50 (2)°;Z=8. 2-Hydroxy-thiobenzamid, C₇H₇NOS; monoklin; P2₁/n-C _2h ⁵ ;a=13.508 (5),b=6.780 (2),c=15.878 (6) Å; =93.74 (5)°;Z=8. 2-Hydroxy-N,N-dimethyl-benzamid, C₉H₁₁NO₂; orthorhombisch; Pbca-D _2h ¹⁵ ;a=11.752 (2),b=16.680 (3),c=9.079 (2) Å;Z=8. 2-Hydroxy-N,N-dimethyl-thiobenzamid, C₉H₁₁NOS; monoklin; P2₁/c-C _2h ⁵ ;a=6.983 (1),b=11.652 (3),c=11.704 (3) Å; =100.02 (2)°;Z=4. Die Kristallstrukturen dieser vier Verbindungen wurden mittels Röntgen-Einkristalldaten bestimmt (bzw. verfeinert: 2-Hydroxy-benzamid). Die Verfeinerungen der Strukturparameter nach der Methode der kleinsten Quadrate ergab in allen FällenR<0.056. Die Wasserstoffatome konnten anhand von Differenz-Fourier-Summationen belegt werden. Die O-H ... O-Abstände haben folgende Werte: 2.513(1)Å in 2-Hydroxy-benzamid (intramolekular) und 2.625(1) Å in 2-Hydroxy-N,N-dimethyl-benzamid (intermolekular). Die zwei O-H ... S-Abstände sind in 2-Hydroxy-thiobenzamid 2.904(2)Å und 2.918(2)Å (intramolekular, zwei moleküle in der asymmetrischen Einheit) und 3.228(2)Å in 2-Hydroxy-N,N-dimethyl-thiobenzamid(intermollekular). Klar zuzuordnende N-H ... O-Wasserstoffbrücken mit 2.926(1)Å und 3.006(1)Å treten lediglich in der Struktur des 2-Hydroxy-benzamid auf (intermolekular).

     

Syntheses and crystal structures of [Mg(HF)2](SbF6)2 and [Ca(HF)2](SbF6)2: new examples of HF acting as a ligand to metal centres

[Mg(HF)₂](SbF₆)₂ and [Ca(HF)₂](SbF₆)₂ monocrystals were grown from the corresponding hexafluoroantimonates(V) dissolved in anhydrous hydrogen fluoride. [Mg(HF)₂](SbF₆)₂ crystallizes in the space group Pnma (no. 62) with a=1249.1(4) pm, b=1230.2(4) pm, c=699.1(2) pm, V=1.0742(6) nm³, Z=4. Magnesium is octahedrally coordinated by six fluorine atoms from which two belong to two HF molecules. The structure can be represented by alternating rows of magnesium and antimony atoms running parallel to the c-axis. Magnesium atoms are connected by cis bridging Sb(2)F₆ units along the a-axis and by trans bridging Sb(1)F₆ units along the b-axis. In this way a three-dimensional network is formed.[Ca(HF)₂](SbF₆)₂ crystallizes in the space group P2₁/n (no. 14) with a=935.2(3) pm, b=1088.7(3) pm, c=1104.8(3) pm, β=106.697(5)°, V=1.0774(5) nm³, Z=4. The coordination sphere around the calcium atom consists of eight fluorine atoms which define the vertices of an Archimedean antiprism. The two HF molecules directly coordinate the calcium atom and their fluorine atoms are placed in the corners of different square faces of the Archimedean antiprism. The Ca-F(HF) distances are shorter than the Ca-F(Sb) distances. The Sb(1)F₆ and Sb(2)F₆ groups have four equatorial bridging fluorine atoms, while the Sb(3)F₆ groups have only two bridging trans F ligands. The Ca atoms in the [−1,0,1] plane are connected by equatorial F ligands of Sb(1)F₆ and Sb(2)F₆ units, forming a [Ca(SbF₆)⁺]_n layer. These layers are connected by trans bridging Sb(3)F₆ groups. HF molecules occupy the space between these layers and additionally contribute to the connection between the layers by hydrogen bonding.     

Synthesis and properties of new soluble aromatic polyamides and polyimides on the basis of N,N′‐bis(3‐aminobenzoyl)‐N,N′‐diphenyl‐1,4‐phenylenediamine

A new N‐phenylated amide (N‐phenylamide) unit containing aromatic diamine, N,N′‐bis(3‐aminobenzoyl)‐N,N′‐diphenyl‐1,4‐phenylenediamine, was prepared by the condensation of N,N′‐diphenyl‐1,4‐phenylenediamine with 3‐nitrobenzoyl chloride, followed by catalytic reduction. Two series of organosoluble aromatic poly(N‐phenylamide‐imide)s and poly(N‐phenylamide‐amide)s with inherent viscosities of 0.58–0.82 and 0.56–1.21 dL/g were prepared by a conventional two‐stage method and the direct phosphorylation polycondensation, respectively, from the diamine with various aromatic dianhydrides and aromatic dicarboxylic acids. All polyimides and polyamides are amorphous and readily soluble in many organic solvents such as N,N‐dimethylacetamide and N‐methyl‐2‐pyrrolidone. These polymers could be solution cast into transparent, tough, and flexible films with high tensile strengths. These polyimides and polyamides had glass‐transition temperatures in the ranges of 230–258 and 196–229 °C, respectively. Decomposition temperatures of the polyimides for 10% weight loss all occurred above 500 °C in both nitrogen and air atmospheres. © 2002 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 40: 2564–2574, 2002     

Crystal structure and spectroscopic study of di-μ(1,1)-azido-di-μ(0,0-nitrato)tetrakis(3,5-lutidine)dicopper(II), [Cu(N3) (NO3) (3,5-lutidine)2]2

A mixed ligand 12 complex of copper(II) azide with 3,5-lutidine, namely di-(1,1)-azido-di(0,0)-nitrato)tetrakis(3,5-lutidine)dicopper(II) has been prepared and characterized by X-ray crystallographic and spectroscopic methods. The dimeric molecule, which possesses a crystallographic inversion center, contains two(1,1) bridging azido ligands. Each copper(II) atom in the cyclic Cu₂N₂ unit is further coordinated by two oxygen atoms from two(0,0-nitrato) bridges at Cu-O distances of 2.465(3) and 2.568(3) Å and two nitrogen atoms from the lutidine molecules [Cu-N=2.003(3) and 2.012(3) Å] to give a distorted tetragonal bipyramid. Both azido ligands are linear [N-N-N angle=179.0(4) and asymmetric N-N=1.207(4) and N-N=1.135(5) Å]. The azido bridges produce a rather short Cu Cu distance of 3.013(2) Å. Infrared and electronic data are presented and discussed.     

Combined study of thermochemical properties of nitroform and its salts

A combined method for determination of the standard enthalpies of formation of nitroform and its salts was proposed. The enthalpies of dissolution of nitroform and its ammonium, hydrazinium, guanidinium, and potassium salts in water were measured. The enthalpy of combustion of hydrazinium salt of nitroform was determined by a combustion calorimetry, and its standard enthalpy of formation was calculated. The enthalpy of formation of trinitromethyl anion in indefinitely diluted aqueous solution −24.94±0.79 kJ mol⁻¹, was calculated on the basic of the data obtained for enthalpies of formation and dissolution of ammonium and hydrazinium salts of nitroform. Standard enthalpies of formation of nitroform and its salts were calculated. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 12, pp. 2451–2454, December, 1998.     

Structural Studies on Host + Guest Recognition Sensory Systems

Rhodamine B-ethylenediamine--cyclodextrins (RhB--CDen) and rhodamine B--cyclodextrins (RhB--CD) can form inclusive complexes with many guest molecules, a reaction which can be used as a nucleic acid probe. In this paper, the most stable conformations of RhB--CDen and RhB--CD have been determined by fluorescence experiments and analyzed by molecular modeling simulation. The interaction between RhB--CDen and two guest molecules, 1-borneol and cyclohexanol, has also been investigated. The results showed that RhB--CDen has a stronger interaction with 1-borneol than with cyclohexanol. Borneol could push the three aromatic-rings of rhodamine B out of the CD cavity, while the cyclohexanol could not. The interactive sites of host and guest are also presented.     

Reaction ofN-(3,3,3-trifluoro-2-trifluoromethylprop-1-enyl)dimethylamine with MgSO4·7H2O. Synthesis of 4,4-difluoro-5-trifluoromethyl-2-(2,2,2-trifluoro-1-trifluoromethylethyl)-4H-1,3-dioxine andcis/trans-3-dimethylamino-2-trifluoromethacryloyl fluoride

4,4-Difluoro-5-trifluoromethyl-2-(2,2,2-trifluoro-1-trifluoromethylethyl)-4H-1,3-dioxine andcis/trans-3-dimethylamino-2-trifluoromethacryloyl fluoride were obtained by reaction ofN-(3,3,3-trifluoro-2-trifluoromethylprop-1-enyl)dimethylamine with MgSO₄·7H₂O. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 5, pp. 1069–1071, May, 1997.     

New sterically hindered di-o-quinones of the biphenyl series

New di-o-quinones of the biphenyl series, namely, 2,2′-dialkyl-5,5′-di-tert-butylbiphenyl-3,4,3′,4′-diquinones, were synthesized. Their structures were established by IR and NMR spectroscopy. The molecular structure of 2,2′-dimethyl-5,5′-di-tert-butylbiphenyl-3,4,3′,4′-diquinone was established by X-ray structural analysis. The structure is characterized by orthogonal (the torsion angle is 82.9°) mutual arrangement ofo-benzoquinone fragments. ESR studies demonstrated that chemical reduction of diquinone proceeds in four oneelectron stages to form paramagnetic mono- and trianions as intermediates. Quinopyrocatechols, which are intermediates in the synthesis of di-o-quinones, were isolated and characterized. Translated fromIzvestiya Akademii Nauk, Seriya Khimicheskaya, No. 4, pp. 804–809, April, 1997.     

Einflu? der Wertigkeit bei der flammenabsorptionsspektrometrischen Bestimmung des Chroms

Zusammenfassung Die Intensität des flammenabsorptionsspektrometrischen Cr-Signals hängt in einer sehr komplexen Weise von der Wertigkeit des Chroms und der Flammentemperatur ab. Ferner ist ein ausgeprägter Kalium-Einfluß vorhanden. Alle diese Schwierigkeiten können am besten dadurch eliminiert werden, daß in der Lachgasflamme gearbeitet und die Analysenlösung mit einer höheren K-Konzentration gepuffert wird.

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Effect of valency on the determination of chromium by flame absorption spectrometry

The intensity of the chromium signal in AAS depends in a very complex manner strongly on the valency of the Cr and the flame temperature. It is furthermore influenced by potassium, if present. The best way to overcome all these interferences is to use the nitrous oxide flame and to buffer the solution with a higher concentration of potassium.