A linear algorithm for nonhomogeneous spectra of numbers

Given a sequence of integers [a_i]_i=1ⁿ, an O(n) iterative algorithm is presented which decides whether there exist real numbers α and β such that a_i = [iα + β] (1 ? i ? n). In fact, the linear algorithm computes the partial quotients of the continued fraction expansions of $\text{d}$ and $\text{d}$ such that $\text{d < α <}\text{d}$ if and only if a_i = [iα + β] (1 ? i ? n) for suitable β = β(α).     

Aggregation and C-N rotation of the lithium salt of N,N-dimethyldiphenylacetamide

[formula: see text] Two methyl 1H NMR signals for the Li salt of N,N-dimethyldiphenylacetamide are observed at low temperature and assigned to the monomer and dimer. From line shape analysis, the dimerization constant (K1,2) is 40 +/- 10 M-1 at 200 K (delta G degree = 1.5 kcal mol-1, delta H degree = 0.8 kcal mol-1, delta S degree = 12 eu) and the activation parameters are delta H++ = 5.5 kcal mol-1 and delta S++ = -18 eu. The C-N bond rotation is too fast to observe on the NMR time scale, indicating a rotation barrier of less than 10 kcal mol-1.     

Die Bestimmung der Hydrolyse

Ohne Zusammenfassung     

Theoretical interpretation of the limiting electric conductivity in ionic solution

The physical essence of the limiting equivalent ionic conductivity in solution, λ⁰_i, has been a continuing challenge over almost a century. Here I briefly present an ab initio theoretical treatment providing (1) a new insight into the nature of λ⁰_i, and (2) a mathematical formula for computing λ⁰_i. In the new treatment, one assumes that any chosen ion i is surrounded by a spherical body of oriented solvent dipoles carrying the charge of the counterion, and the bulk solvent is a continuum with no molecular detail. λ⁰_i is thus the result of the tandem operation, at hydrodynamic equilibrium, of the dipole body's electrophoretic and relaxation forces exerted on the drifting ion. λ⁰_i is found to be proportional to the radius of ion i, and independent of the ionic charge. From experimental λ⁰_i's, the ion radius can be computed as ‘electric radius.’ An electric ion-radius scale so derived compares well with other ion-size scales. The current theory expresses λ⁰_i using only universal constants and unitary factors of the ionic solution, and it sheds new light on the fundamental nature of ion and charge transport in a polar liquid medium.     

Semihard scattering unraveled from collective dynamics by two-pion azimuthal correlations in 158A GeV/c Pb+Au collisions

Elliptic flow and two-particle azimuthal correlations of charged hadrons and high-p(T) pions (p(T)>1 GeV/c) have been measured close to midrapidity in 158A GeV/c Pb+Au collisions by the CERES experiment. Elliptic flow (v(2)) rises linearly with p(T) to a value of about 10% at 2 GeV/c. Beyond p(T) approximately 1.5 GeV/c, the slope decreases considerably, possibly indicating a saturation of v(2) at high p(T). Two-pion azimuthal anisotropies for p(T)>1.2 GeV/c exceed the elliptic flow values by about 60% in midcentral collisions. These nonflow contributions are attributed to nearside and back-to-back jetlike correlations, the latter exhibiting centrality dependent broadening.     

Axial chirality in 1,4-disubstituted (ZZ)-1,3-dienes. Surprisingly low energies of activation for the enantiomerization in synthetically useful fluxional molecules

Trialkylsilyltrialkylstannes (R(3)Si-SnR'(3)) add to 1,6-diynes in the presence of Pd(0) and tris-pentaflurophenylphosphine to give 1,2-dialkylidenecyclopentanes with terminal silicon and tin substituents. The (ZZ)-geometry of these s-cis-1,3-dienes, resulting from the organometallic reaction mechanisms involved, forces the silicon and tin groups to be nonplanar, thus making the molecules axially chiral. There is rapid equilibration between the two helical forms at room-temperature irrespective of the size of the Si and Sn substituents. However, the two forms can be observed by 1H, 13C, and 119Sn NMR spectroscopy at low temperature. The rates of enantiomerization, which depend on the Si and Sn substituents, and the substitution pattern of the cylopentane ring can be studied by dynamic NMR spectroscopy using line shape analysis. The surprisingly low energies of activation (DeltaG++ = 52-57 kJ mol(-1)) for even the bulky Si and Sn derivatives may be attributed to a widening of the exo-cyclic bond-angles of the diene carbons.