Reaping Numbers of Boolean Algebras

A subset A of a Boolean algebra B is said to be (n,m)-reapedif there is a partition of unity p B of size n such that |{b p:b a 0}| m for all a A. The reaping number r_n,m (B) ofa Boolean algebra B is the minimum cardinality of a set A B\{0}which cannot be (n,m)-reaped. It is shown that for each n, thereis a Boolean algebra B such that r_n+1,2(B) r_n,2(B). Also, {r_n,m(B):mn } consists of at most two consecutive cardinals. The existenceof a Boolean algebra B such that r_n,m (B) r_n',m' (B) is equivalentto a statement in finite combinatorics which is also discussed.     

Electroweak tests with τ leptons at LEP

We report on two sensitive tests of lepton universality carried out by the 4 LEP experiments at the Z⁰ pole. From measurements of the τ polarization in e⁺ e^?→τ ⁺ τ ^?, the ratios of the vector and axial vector coupling constants of the electron and the tau lepton to the weak neutral current are obtained to beg _{v e} /g _{a e} =0.066±0.015 andg _{V τ} /g _{A τ} =0.070±0.009 respectively. From measurement of the τ lifetime and the τ leptonic branching ratios, the ratio of the coupling constants describing weak leptonic decays of the τ and the μ is measured to beG _τ /G _μ =0.996±0.008.     

Standard<Emphasis Type="Italic">SU</Emphasis>(5) supergravity grand unification model and predictions

A review is given of the StandardSU(5) supergravity model. This model has passed an important check regarding unification of the electro-weak and the strong couplings using high precision LEP data. It is shown that for a significant domain of the parameter space the model also satisfies constraints on the SUSY spectrum from CDF and LEP, as well as proton stability and cosmological relic density constraints.     

Structures of two ketals derived from a pentacyclic cage diketone

The structure of a compound derived from a symmetrical cage diketone in which one of the ketone C=O groups has been converted into a ketal is reported. An investigation into reasons for the selective functionalization resulted in the isolation of acis, cisoid, cis linear triquinane derivative in which both ketones are converted to ketals that share a transannular oxygen atom. Crystal data:2, monoclinic,P2₁/c,a=7.6298(9),b=7.673(3),c=22.125(3)Å,=97.59(1)°,V=1283.9(5)ų, andR=0.048 (1001 reflections);4, triclinic,P¯1,a=7.3864(9),b=9.6362(9),c=10.3889(9)Å,=75.749(6),=76.300(8), =83.100(8)°,V=694.8(4)ų, andR=0.088 (1284 reflections).     

Local solvability of PDE with constant coefficients

The fundamental result of the paper consists in the following assertion: Let D be an open set in Rⁿ,f £ L₂(D), and let be a distribution in Rⁿ with a compact support, whose Fourier transform satisfies the condition

Substituent effects in the fragmentation reactions of acetanilides and phenyl acetates

The mass spectra of a group of m- and p-substituted acetanilides were measured to determine the effect of the substituents, if any, on the rate of C₂H₂O expulsion (rearrangement) vs. the rate of [CH₃CO]⁺ formation (simple cleavage). Our results agree with observations of others in that substituents which raise the ionization potential of the aromatic ring appear to localize, on the average, less charge density on this locus, and conversely. The atomic composition of the substituent, however, irrespective of its position, seems to determine the relative rates of the competing reaction. Although in several cases Z_m/Z_p values were close to unity, rearrangement of isomeric molecular ions to a common species is shown not to occur.     

Yields of Br* (4P1/2) as a function of wavelength in the photodissociation of Br2 and IBr

A flashlamp-pumped tunable dye laser has been used to study the photochemical production of Br* (4²P_1/2) atoms from Br₂ and IBr exc     

β-lithiations of carboxamides. Synthesis of β-substituted-α,β-unsaturated amides

N-Methyl-2-methyl-3-(benzotriazol-l-yl)propanamide, on treatment with butyllithium forms a dianion which on treatment with alkyl and benzyl halides, aldehydes and ketones affords monosubstituted products; with ethyl p-toluate, a lactam is formed. The alkylated derivatives eliminate benzotriazole in the presence of base to afford trisubstituted α,β-unsaturated amides.     

Homopolar- and Heteropolar Bond Dissociation Energies and Heats of Formation of Radicals and Ions in the Gas Phase. I. Data on organic molecules

The literature data on heteropolar and homopolar 2-center bond dissociation energies in organic molecules in the gas phase and the corresponding heats of formation of radicals and ions have been critically evaluated. Data for more than 500 bonds are represented in tabular form together with the pertinent literature references. Selected electron affinities and π-bond dissociation energies have also been incorporated. The follow-up paper will discuss some empirical general aspects of these data particularly regarding the effect of structure on the bond dissociation energies.