Partitioning Boolean lattices into chains of subsets

We prove that the Boolean lattice of all subsets of an n-set can be partitioned into chains of size four if and only if n9.Research supported in part by N.S.F. grant DMS-8401281.Research supported in part by N.S.F. grant DMS-8406451.     

Element neighbourhoods in twofold triple systems

The method of differences is used to establish that every 2-regular multigraph onv– 10,2 (mod 3) points occurs as the neighbourhood graph of an element in a twofold triple system of orderv, with two exceptions: C₂C₃and C₃C₃.Dedicated to Professor Hanfried Lenz on the occasion of his seventieth birthday     

On the nature and origin of complexity in discrete,homogeneous, locally-interacting systems

The observed complexity of nature is often attributed to an intrinsic propensity of matter to self-organize under certain (e.g., dissipative) conditions. In order better to understand and test this vague thesis, we define complexity as logical depth, a notion based on algorithmic information and computational time complexity. Informally, logical depth is the number of steps in the deductive or causal path connecting a thing with its plausible origin. We then assess the effects of dissipation, noise, and spatial and other symmetries of the initial conditions and equations of motion on the asymptotic complexity-generating abilities of statistical-mechanical model systems. We concentrate on discrete, spatially-homogeneous, locally-interacting systems such as kinetic Ising models and cellular automata.     

SINGLET OXYGEN REACTIONS WITH 2,7-DIMETHYL-2,6-OCTADIENE

Abstract— Results concerning Rose Bengal sensitized photoxidation of 2,7-dimethyl-2,6-octadiene (A) are discussed. This diolefin may be considered as a model molecule of polyisoprene of polymerization number two with 1,4-4,1 addition mode.
All the mono- and dihydroperoxides apt to occur are actually obtained. One of the occurring hydroperoxides, 2,7-dimethyl-dihydroperoxy-3,5-octadiene, which has the structure of a conjugated diene, does not lead to an endoperoxide. This is contrary to Kaplan and Kelleher's hypothesis. On the other hand, no trihydroperoxide forms as a result of further addition of singlet oxygen (¹O₂).
A complete kinetic study has allowed us to determine the rate constants of all the reactions which take place. The reactivity of (A) is twice that of 2-methyl-2-pentene and the addition of ¹O₂ to one of the double bonds of (A) induces deactivation of the remaining double bond.     

Isomerisation in the chemistry of platinacyclobutane complexes

The platinacyclobutane complexes PtCl₂L₂(C₃H₅Me)], L  pyridine, CD₃CN, or tetrahydrofuran, exist as mixtures of isomers containing P$\text{tCH}{}_2\text{CHMeC}$H₂ or P$\text{tCHMeCH}{}_2\text{CH}{}_2$ groups in rapid equilibrium. Decomposition occurs in some cases to give [PtCl₂L(CH₃CH₂CHCH₂)]. Stereospecific skeletal isomerisation also occurs in metallocyclobutanes containing the groups P$\text{tCHRCHRC}$H₂  P$\text{tCHRCH}{}_2\text{C}$HR, when R  aryl further decomposition gives ν-allylplatinum complexes.     

Photometrische Antimonbestimmung in Rein- und Elektrolytkupfer

Zusammenfassung Wir zeigen einen Weg zur Bestimmung von Antimon in Rein- bzw. Reinstkupfersorten. Die Erfassungsgrenze liegt bei 0,5 g Metalleinwaage in der Größenordnung zwischen 1 und 30 g Sb/Tonne (0,0001 und 0,0030%).Das Verfahren hat den Vorzug der Schnelligkeit gegenüber allen bisher üblichen Methoden. Die Reproduzierbarkeit der gefundenen Werte kann als befriedigend bezeichnet werden, obgleich die Extinktionskurve nicht ganz geradlinig verläuft.Wenn die oben erwähnte Kupferstammlösung vorliegt, läßt sich die Bestimmung innerhalb von 30 min durchführen.     

Sine-wave polarographic determination of zirconium or niobium in aqueous media

The sine-wave polarographic determination of zirconium in aqueous media was investigated using solutions which were 0.55 – 5.5·10^-3M in zirconyl chloride and 1 M in potassium chloride and had been adjusted to pH 2.0 with hydrochloric acid. It was possible to determine zirconium in the concentration range of 0.05 to 0.4 mg per ml. The sine-wave polarographic behavior of zirconium in aqueous solutions in the pH range 2–3 is discussed. The sine-wave polarographic determination of niobium in aqueous media was investigated using concentrated sulfuric acid containing 5 to 0.1 mg of niobium per ml in a supporting electrolyte of citric acid; the determination of niobium was possible down to 0.1 mg of niobium per ml of concentrated sulfuric acid although the D.C. polarographic method was impractical for the determination of less than 0.5 mg of niobium per ml.     

A Study of the 13C NMR Shifts in Two Tetradehydrocyclodecabiphenylenes and Related Aryl and Biphenylenylacetylenes

The ¹³C spectra of 5,6,9,10-tetradehydrocyclodeca[1,2,3,4-def]-benzo [7,8]biphenylene, 1, and 5,6,9,10-tetradehydrocyclodeca [1,2,3,4-def]-naphtho [2,3-7,8]biphenylene, 2, are reported as are those of a number of simpler acetylenic hydrocarbons used as spectral references. Most of the shifts can be assigned unambiguously. The acetylenic shift assignments were verified by ortho-proton, sp-carbon (¹H(1)-¹³C_sp(3)) decoupling experiments. A simple additive shift correlation is found for the hydrocarbons containing unstrained acetylenic groups. However, significant discrepencies are found for the ¹³C shifts for the strained hydrocarbons 1, 2, 1,2-bis(phenylethynyl)-benzene, 12, and 2,3-bis(phenylethynyl)-naphthalene, 13. The discrepencies are particularily large for carbons near the triple bonds and are attributed to a combination of strain, rehybridization, and other proximity effects related to the interaction between the ortho-substituted acetylenic carbons.     

Preparation and reactions of the (dicarbonyl)(η5-cyclopentadienyl)(tetrahydrofuran)iron cation: A convenient route to (dicarbonyl)(η5-cyclopentadiemyl)(η2-olefin)iron cations and related complexes

The versatile reagent [η⁵-C₅H₅)Fe(CO)₂(THF)]BF₄ has been isolated from the reaction of (η₅-C₅H₅)Fe(CO)₂I and AgBF₄ in THF and shown to react in CH₂Cl₂ with olefins to yield [(η⁵-C₅H₅)Fe(CO)₂(η²-olefin)]BF₄ complexes. For most olefins the yields are high. The yield in these reactions can be increased by treating the CH₂Cl₂ solution of [(η⁵-C₅H₅)Fe(Co)₂(THF)]BF₄ and olefin with gaseous BF₃ in order to complex the THF as the BF₃-THF adduct. Most striking is the increase in yield for the cyclohexene complex from 17% to 92%.     

4-carboxy-1,2-cyclohexanedionedioxime complexes of nickel(ii) in alkaline media

The reactions of 4-carboxy-1,2-cyclohexanedionedioxime and nickel(II) were studied in alkaline media. Spectrophotometric studies indicate the presence of a 1:1 complex ion, NiD^-. Magnetic susceptibility measurements on a series of solutions of varying ratios of vic-dioxime and nickel(II) showed that the 1:1 complex ion was diamagnetic and that two paramagnetic complexes, probably NiD₂^4- and NiD₃^7-, are present in solution. The stability constants for the three complexes were calculated from spectrophotometric and magnetic susceptibility data. The log K values were found to be log K₁ = 28.74 ± 0.60, log K₂ = 0.76 ± 0.15, and log K₃ = 3.67 ± 0.73, respectively.