Comparison of Boolean algebras

In this work, some results related to superatomic Boolean interval algebras are presented, and proved in a topological way. Let x be an uncountable cardinal. To each I x, we can associate a superatomic interval Boolean algebra B _I of cardinality x in such a way that the following properties are equivalent: (i) I I x, (ii) B _I is a quotient algebra of B _J, and (iii) there is an homomorphism f from B _J into B _I such that for every atom b of B _I, there is an atom a of B _J satisfying f(a)=b. As a corollary, there are 2 ^x isomorphism types of superatomic interval Boolean algebras of cardinality x. This case is quite different from the countable one.     

Heterocyclic synthesis via enaminones: Novel synthesis of (1H)‐pyridin‐2‐one,pyrazolo[1,5‐a]pyrimidine and isoxazole derivatives incorporating a N‐methylphthalimide and their biological evaluation

Novel synthesis of (1H)‐pyridin‐2‐one, pyrazolo[1,5‐a]pyrimidine and isoxazole derivatives incorporating N‐methylphthalimide moiety are reported. Reaction of enaminone 2 with malononitrile affords 4. Condensation of 2 with cyanothioacetamide or benzoylacetonitrile affords compounds 6 and 7 respectively. Reaction of 2 with hydrazine hydrate afford 2,3‐dihydrophthalazine‐1,4‐dione ( 10 ). Condensation of 2 with hydroxylamine and 3‐aminopyrazole derivatives affords compounds 12 and 15a,b respectively. Antimicrobial and antifungal activity were determined for representative compounds and most of them showed moderate activity as antimicrobial agents, while compounds 2 and 7 show strong activity against Aspergillus niger. The structure of the newly synthesized compounds was elucidated by elemental analyses and ¹H nmr spectra and some cases by ¹³C nmr investigation.     

Substituted γ-lactones: On the structural elucidation of the reaction products from 4-aroyl-3-hydroxy-2(5H)-furanones and 1,2-diamines

The reaction pathway of 4-aroyl-3-hydroxy-2(5H)-furanones 1 with diamines depends on the nature of the amine as well as on the applied reaction conditions. Thus, the reaction of 1a-d with 5,6-diamino-1,3-dimethyluracil 5 led to the formation of two isomeric Schiff bases 7a-d and 8a-d . Conversely type 1 compounds reacted with 4,5-diaminopyrimidine 9 or 2,3-diaminopyridine 10 to form the mono acid-base adducts 11a and 11b respectively. When type 1 compounds were reacted with aliphatic diamines 13a-d or p-phenylenediamine and p-xylenediamine, respectively also an immediate formation of acid-base adducts 15a-f was observed. The reaction of a number of O-methylated type 1 compounds with 1,2-ethylenediamine afforded the novel seven-membered ring compounds 18a-d in good yields. The analogous reaction of O-alkylated 1a with o-phenylenediamine 2 or 2,3-diaminonaphthalene gave the expected tricyclic ring systems 19 or 20 .     

A facile synthesis of novel triazoloquinoxahnones and triazinoquinoxalinones

A series of 1-aryl-s-triazolo[4,3-a]quinoxalin-4-ones, 3 , were synthesized via the pyrolysis of the corresponding hydrazones, 6 . Thus, the cyclodehydrogenation occurred by refluxing them in an inert solvent (e.g. ethylene glycol) to give the triazoloquinoxalin-4-ones in a satisfactory yield. Using DMSO as a solvent for the above transformation afforded as a minor by-product an S-ylid. In contrast to earlier findings, annelation of a six-membered ring was successful and achieved through the pyrolysis of the pyruvate hydrazones derived of the quinoxalin-4-ones at ?230° to give the as-triazino[4,3-a]quinoxalin-5-ones, 4 . The reaction of 5 with acetylacetone afforded 3-(3′,5′-dimethylpyrazol-1-yl)-2(1H)-quinoxalinone, 10 . The structural assignments for the new compounds were based on their elemental analysis and spectroscopic data as well as an independent synthesis.     

Reactions with Carbo(heterylhydrazonoyl) Halides. I. Chemistry of Carbo(3-phenylpyrazol-5-yl-hydrazonoyl) Chlorides

The Carbo(3-phenylpyrazol-5-yl-hydrazonoyl) halides 1a , b react with active methylene compounds to yield the 1-(3-phenylpyrazol-5-yl)-pyrazole derivatives 2a – k (Scheme 1). The acyclic intermediates 3a , b could be isolated from reaction of 1a , b with acetylacetone, thus establishing the substitution mechanism for these reactions. Compounds 1a , b reacted with carbon disulfide, phenyl isothiocyanate, methyl cyanide, and with p-chlorobenzaldehyde to yield the corresponding heterocyclic derivatives 5 – 8 , respectively (Scheme 2). The behaviour of compounds 2 with hydrazine hydrate is reported.     

Metal chelates of heterocyclic nitrogen-containing ketones,Part XXII. Spectroscopic studies on rhodium(III) complexes of phenyl-2-picolylketone-2-pyridyl hydrazone

Summary Mono, bis and tris complexes of rhodium(III) with phenyl-2-picolylketone-2-pyridyl hydrazone (PPKPyH) have been characterized. In every case, the imino-proton of PPKPyH shows marked acidity associated with the coordination to rhodium(III). Electronic spectra show that all complexes are octahedral. The B-values suggest a strong covalency in the metal-ligand -bond and the Dq-values indicate a medium-strong ligand field. The magnetic susceptibility indicates that PPKPyH forms low-spin complexes with rhodium(III).¹H n.m.r. spectra show that the tris(ligand) complexes arecis isomers. I.r. spectra show that the ligand is neutral or monobasic tridentate or bidentate. Far i.r. studies show that [Rh(PPKPyH)X₃] · 2 H₂O (X = Cl, Br or I) aremer isomers. The effect of pH variation on the rection products is also discussed.     

On the Cauchy problem for some abstract nonlinear differential equations

In the present paper, we study the Cauchy problem in a Banach spaceE for an abstract nonlinear differential equation of form $$\frac{{d^2 u}}{{dt^2 }} = - A\frac{{du}}{{dt}} + B(t)u + f(t,W)$$ whereW = (A ₁(t)u,A ₂(t)u,?,A _?(t)u), (A _i (t),i = 1, 2, ?,?), (B(t),t ∈I = [0,b]) are families of closed operators defined on dense sets inE intoE, f is a given abstract nonlinear function onI ×E ^? intoE and ?A is a closed linear operator defined on dense set inE intoE, which generates a semi-group. Further, the existence and uniqueness of the solution of the considered Cauchy problem is studied for a wide class of the families (A _i(t),i = 1, 2, ?,?), (B(t),t ∈I). An application and some properties are also given for the theory of partial diferential equations.     

Curves of marginal stability, and weak- and strong-coupling BPS spectra in N = 2 supersymmetric QCD

We explicitly determine the global structure of the SL(2, ) bundle over the Coulomb branch of the moduli space of asymptotically free N = 2 supersymmetric Yang-Mills theories with gauge group SU(2) when massless hypermultiplets are present. For each relevant number of flavours, we show that there is a curve of marginal stability of the Coulomb branch, diffeomorphic to a circle, across which the BPS spectrum is discontinuous. We determine rigorously and completely the BPS spectra inside and outside the curve. In all cases, the spectrum inside the curve consists of only those BPS states that are responsible for the singularities of the low energy effective action (in addition to the massless abelian gauge multiplet which is always present). The predicted decay patterns across the curve of marginal stability are perfectly consistent with all quantum numbers carried by the BPS states. As a byproduct, we also show that the electric and magnetic quantum numbers of the massless states at the singularities proposed by Seiberg and Witten are the only possible ones.     

Polyselenide mit langkettigen Tetraalkylammoniumionen Die Kristallstruktur von Trimethyl-tetradecyl-ammonium-hexaselenid

Polyselenides with Long-chain Tetraalkylammonium Ions. Crystal Structure of Trimethyltetradecyl-ammonium Hexaselenide Na₂Se₂ and Na₂Se react with various tetraalkylammonium halides in ethanol and in presence of grey selenium and catalytic quantities of iodine forming different polyselenides Se_n^2? (n = 3, 5—9). In the solutions equilibria of polyselenides seem to occur; cooling of saturated solutions causes crystallization of polyselenides with a composition depending on the cation. Tri- and pentaselenide are dark green. The higher members form black crystals, all compounds are sensitive to oxygen. The i.r. spectra are reported. [(CH₃)₃N(CH₂)₁₃CH₃]₂Se₆ is characterized by a crystallographic structure determination with X-ray data: space group P2₁2₁2, Z = 4, a = 5043, b = 734.2, c = 600.3 pm (986 observed independent reflexions. R = 0.072). The compound consists of trimethyl tetradecylammonium ions and angular Se₆^2? chains of symmetry C₂ with Se? Se bond lengths of 227 and 235 pm.