Orthmodular lattices whose MacNeille completions are not orthomodular

The only known example of an orthomodular lattice (abbreviated: OML) whose MacNeille completion is not an OML has been noted independently by several authors, see Adams [1], and is based on a theorem of Ameniya and Araki [2]. This theorem states that for an inner product space V, if we consider the ortholattice ?(V,⊥) = {A \( \subseteq \) V: A = A ^⊥⊥} where A ^⊥ is the set of elements orthogonal to A, then ?(V,⊥) is an OML if and only if V is complete. Taking the orthomodular lattice L of finite or confinite dimensional subspaces of an incomplete inner product space V, the ortholattice ?(V,⊥) is a MacNeille completion of L which is not orthomodular. This does not answer the longstanding question Can every OML be embedded into a complete OML? as L can be embedded into the complete OML ?(V,⊥), where V is the completion of the inner product space V. Although the power of the Ameniya-Araki theorem makes the preceding example elegant to present, the ability to picture the situation is lost. In this paper, I present a simpler method to construct OMLs whose Macneille completions are not orthomodular. No use is made of the Ameniya-Araki theorem. Instead, this method is based on a construction introduced by Kalmbach [7] in which the Boolean algebras generated by the chains of a lattice are glued together to form an OML. A simple method to complete these OMLs is also given. The final section of this paper briefly covers some elementary properties of the Kalmbach construction. I have included this section because I feel that this construction may be quite useful for many purposes and virtually no literature has been written on it.     

Perturbation bounds for theLDL H andLU decompositions

LetA andA+A be Hermitian positive definite matrices. Suppose thatA=LDL ^H and (A+A)=(L+L)(D+D)(L+L)^H are theLDL ^H decompositons ofA andA+A, respectively. In this paper upper bounds on |D| _F and |L| _F are presented. Moreover, perturbation bounds are given for theLU decomposition of a complexn ×n matrix.     

Conformation of BEDT-TTF molecules in the crystal

Based on comparison of low-temperature x-ray diffraction data for BEDT-TTF salts and quantum chemical MNDO calculations, we have shown that the most energetically favorable conformation of the BEDT-TTF molecule is not realized in the crystal. A local energy minimum 0.035 eV above the ground state corresponds to the conformation detected in the crystal. The x-ray diffraction data and the MNDO results suggest weak participation of the heteroatoms of the six-membered rings in the conjugation with the -system of the TTF-fragment in the BEDT-TTF and BEDO-TTF molecules.Institute of Chemical Physics at Chernogolovka, Russian Academy of Sciences, Chernogolovka 142432. Translated from Izvestiya Akademii Nauk, Seriya Khimicheskaya, No. 4, pp. 890–894, April, 1992.     

PEG-400 mediated an efficient eco-friendly synthesis of new isoxazolyl pyrido[2,3-d]pyrimidines and their anti-inflammatory and analgesic activity

Abstract

Polyethylene glycol-400 (PEG-400) has been developed as an efficient and eco-friendly reaction medium for the synthesis of new isoxazolyl pyrido[2,3-d]pyrimidine derivatives 11 from isoxazolyl cyanoacetamide synthon 7. Compound 7 was employed with various aromatic aldehydes 8 and malononitrile 9 in the presence of triethylamine (Et₃N) to afford the corresponding (E)-6-amino-1-(3-methyl-5-styrylisoxazol-4-yl)-2-oxo-4-phenyl-1,2-dihydro- pyridine-3,5-dicarbonitrile 10 at room temperature by using PEG-400 as a solvent medium as well as catalyst. The intermediate 10 on treatment with thiourea in the presence of PEG-400 at 90?°C to give the target compounds isoxazolyl pyrido[2,3-d]pyrimidines 11 in good to excellent yields. The newly synthesized compounds 10 and 11 were characterized by IR, ¹H NMR, ¹³C NMR, and HRMS analysis. The target compounds 11a-x was screened for their anti-inflammatory and analgesic activities. Among the tested compounds, the compounds 11s, 11t, 11u, 11v, 11w, and 11x showed significant anti-inflammatory and potent analgesic activities as that of reference drugs. The advantages of this protocol are operational simplicity, catalyst free, environmental safety, wide substrate scope, good yields, and PEG-400 can be recovered and reused. Most significant of all, this protocol is green.     

Molecular structure and spectral properties of thionaphthalimides

The molecular structure and absorption spectra of monothio- and dithio-naphthalimides were compared to their naphthalimide analogues using AM1, PM3 and ZINDO/S semiempirical quantum chemical methods. The substitution of the 4R-naphthalimide oxygen atoms by sulphur atoms resulted in a red-shift of the absorption spectra by Δλ_max60-65 and 100-140 nm, respectively. The thionated naphthalimide derivatives do not show observable fluorescence due to intersystem crossing to the triplet -states localised at the CS groups. The -absorption bands of monothioimides are located at 525-580 nm (ε=60-80) and those for dithioimides at 535-560 nm (ε=140-390) and 628-686 nm (ε=34-68). None of these transitions are solvent sensitive. The -transitions of N-phenylthioimides have also a small contribution from -states due to a partial conjugation between CS group and π-electronic system of the N-phenyl ring. As a result, the bands of aromatic substituted N-phenylthioimides are red-shifted as compared to those of the aliphatic N-methyl-thioimides.     

C_(70)R_2(R=OH,CH_3)的结构和电子光谱

INDO方法研究了C70R2（R＝OH，CH3）4种异构体的结构和稳定性，表明1，9－C70（OH）2比7，8－C70（OH）2稳定，两者能量差为38．5kJ／mol，而7，8－C70（CH3）2比1，9－C70（CH3）2能量低23．0kJ／mol．以优化构型为基础，对C70R2（R＝OH，CH3）的电子光谱进行了理论预测．     

Synthesen,Einkristall-Röntgenstrukturanalysen und 29Si-Festkörper-NMR-Untersuchungen eines zwitterionischen λ5-Spirosilicats und eines käfigartigen Octa(silasesquioxans)

Syntheses, Single-Crystal X-Ray Analyses and Solid-State ²⁹Si NMR Studies of a Zwitterionic λ⁵-Spirosilicate and a Cage-like Octa(silasesquioxane) The zwitterionic λ⁵-spirosilicate bis[2,3-naphthalenediolato(2 ?)][2-(dimethylammonio)phenyl]silicate ( 1 ; isolated as 1 · 1/2 CH₃CN) was synthesized by reaction of the [2-(dimethylamino)phenyl]dimethoxyorganosilanes 5, 6 and 7 [2-(Me₂N)C₆H₄Si(OMe)₂R: R = Ph ( 5 ), cyclo? C₆H₁₁ ( 6 ), Me ( 7 )] with 2,3-dihydroxynaphthalene in acetonitrile at room temperature. Reaction of 1 · 1/2 CH₃CN or [2-(dimethylamino)phenyl]trimethoxysilane ( 3 ) with water in acetonitrile yielded the cage-like octa{[2-(dimethylamino)phenyl]silasesquioxane} ( 2 ). The crystal structures of 1 · 1/2 CH₃CN and 2 were studied by X-ray diffraction. In addition, 1 · 1/2 CH₃CN and 2 were characterized by solid-state (²⁹Si CP/MAS) and solution NMR studies (¹H, ¹³C, ²⁹Si).     

Alkylidinphosphane und -arsane. I [P ≡ C?S]−[Li(dme)3]+ – Synthese und Struktur

Alkylidynephosphanes and -arsanes. I [P ≡ C? S]^?[Li(dme)₃]⁺ – Synthesis and Structure O,O′-Diethyl thiocarbonate and bis(tetrahydrofuran)-lithium bis(trimethylsilyl)phosphanide dissolved in 1,2-dimethoxyethane, react below 0°C to give ethoxy trimethylsilane and tris(1,2-dimethoxyethane-O,O′)lithium 2λ³-phosphaethynylsulfanide – [P≡C? S]^? [Li(dme)₃]⁺ – ( 1a ). Apart from bis(trimethylsilyl)sulfane or carbon oxide sulfide, dark red concentrated solutions of λ³-phosphaalkyne 1 are also obtained from reactions of carbon disulfide with bis(tetrahydrofuran)-lithium bis(trimethylsilyl)phosphanide or with the homologous lithoxy-methylidynephosphane ( 2 ) [1]. The ir spectrum shows two absorptions at 1762 and 747 cm^?1 characteristic for the P≡C and C? S stretching vibrations. The nmr parameters {δ(³¹P) ? 121.3; δ(¹³C) 190.8 ppm; ¹J_CP 18.2 Hz} resemble much more values of diorganylamino-2λ³-phosphaalkynes than those of bis(1,2-dimethoxyethane-O,O′)lithoxy-methylidyne-phosphane ( 2a ). As found by an X-ray structure analysis (P2₁/c; a = 1192.6(16); b = 1239.1(19); c = 1414.8(26) pm; β = 105.91(13)° at ?100 ± 3°C; Z = 4 formula units; wR = 0.064) of pale yellow crystals (mp. + 16°C) isolated from the reaction with O,O′-diethyl thiocarbonate, the solid is built up of separate [P≡C? S]^? and [Li(dme)₃]⁺ ions. Typical bond lengths and angles are: P≡C 155.5(11); C? S 162.0(11); Li? O 206.4(17) to 220.3(20) pm; P≡C? S 178.9(7)°.     

Synthesis of new pyridazino[4′,3′:4,5]-thieno[3,2-d]-1,2,3-triazine and pyrimido[4′,5′:4,5]thieno[2,3-c]pyridazine derivatives

Summary 8,9-Diphenylpyridazino[4,3:4,5]thieno[3,2-d]-1,2,3-triazin-4(3H)-one (2), 3-substituted 8,9-diphenylpyridazino[4,3:4,5]thieno[3,2-d]-1,2,3-triazin-4(3H)-ones (3a–c), 3,4-diphenylpyrimido[4,5:4,5]thieno[2,3-c]pyridazin-8(7H)-one (4), 8-chloro-3,4-diphenylpyrimido[4,5:4,5]thieno[2,3-c]pyridazine (5), 8-substituted 3,4-diphenylpyrimido[4,5:4,5]thieno[2,3-c]pyridazines (6a–h) and 7-substituted 3,4-diphenylpyrimido[4,5:4,5]thieno[2,3-c]pyridazin-8(7H)-ones(7a–c) were synthesized from 5-amino-3,4-diphenylthieno[2,3-c]pyridazine-6-carboxamide (1).

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Synthese neuer Pyridazino[4,3:4,5]thieno[3,2-d]-1,2,3-triazin-und Pyrimido[4,5:4,5]thieno[2,3-c]pyridazin-Derivate

Zusammenfassung Folgende Verbindungen wurden ausgehend von 5-Amino-3,4-diphenylthieno[2,3-c]pyridazin-6-carboxamid (1) synthetisiert: 8,9-Diphenylpyridazino[4,3:4,5]thieno[3,2-d]-1,2,3-triazin-4(3H)-on (2), 3-substituierte 8,9-Diphenylpyridazino[4,3:4,5]thieno[3,2-d]-1,2,3-triazin-4(3H)-one (3a–c), 3,4-Diphenylpyrimido[4,5:4,5]thieno[2,3-c]pyridazin-8[7H]-on (4), 8-Chlor-3,4-diphenylpyrimido[4,5:4,5]thieno[2,3-c]pyridazin (5), 8-substituierte 3,4-Diphenylpyrimido[4,5:4,5]thieno[2,3-c]pyridazine (6a–h) und 7-substituierte 3,4-Diphenylpyrimido[4,5:4,5]thieno[2,3-c]pyridazin-8(7H)-one (7a–c).