High-density percolation: Exact solution on a Bethe lattice

A new percolation problem is posed where the sites on a lattice are randomly occupied but where only those occupied sites with at least a given numberm of occupied neighbors are included in the clusters. This problem, which has applications in magnetic and other systems, is solved exactly on a Bethe lattice. The classical percolation critical exponents=gg=1 are found. The percolation thresholds vary between the ordinary percolation thresholdp _c (m=1)=l/(z – 1) andp _c(m=z) =[l/(z – 1)]¹/(z–1). The cluster size distribution asymptotically decays exponentially withn, for largen, p p _c .Supported in part by National Science Foundation grant DMR78-10813.     

THE EFFECT OF MOLECULAR POLARIZATION ON THE ELECTROCHROMISM OF CAROTENOIDS—I. THE INFLUENCE OF A CARBOXYLIC GROUP

Abstract— Electrochromism of oriented all- trans -β-apo-8'-carotenoic acid is studied in thin capacitors. The linear electrochromism is very strong, in contrast to that of symmetrical carotenoids. It is proportional to the first derivative of the absorption spectrum. The quadratic electrochromism can be described as a superposition of fractions proportional to the first and second derivatives of the absorption spectrum. The permanent dipole moment difference between the ground state and the excited state of the carotenoic acid molecule is Δμ= 3.6 × 10^-29 C·m (±20%) (10.7 Debyes). The polarizability difference parallel to the long axis of the molecule is Δα|| = 1.17 × 10^-37 C·m²·V^-1 (±20%) (1050 ų). Furthermore, the relative permittivity of the solid carotenoic ethyl ester is _r= 3.5 ± 0.2.
Δμ is due to the polarizing force of the carboxylic group. This force is equivalent to a mean local electric field of F _t≅3 × 10⁶V/cm. Such a "local field" may also be exerted on a symmetrical carotenoid in the membrane of photosynthesis, e.g. by asymmetrical complex formation with a polarizing molecule. To obtain an effective permanent field of F _p≅ 2 × 10⁶V/cm across the membrane, as postulated in photosynthesis, a local field of F _l≅ 5.5 × 10⁵ V/cm would be sufficient. F _p is shown to be directed from inside to outside of the thylakoid. To realize this, e.g. a positive polar (i.e. electron-attracting) complex partner of the carotenoid, located more to the inside of the thylakoid, can be postulated.     

Neue Oxonium-Bromochalkogenate(IV) — Darstellung,Struktur und Eigenschaften von [H3O][TeBr5] · 3C4H8O2 und [H3O]2[SeBr6]

New Oxonium Bromochalcogenates(IV) — Synthesis, Structure, and Properties of [H₃O][TeBr₅] · 3 C₄H₈O₂ and [H₃O]₂[SeBr₆] Dark red crystals of the composition [H₃O][TeBr₅] · 3 C₄H₈O₂ ( 1 ) were isolated from a saturated solution of TeBr₄ in 1,4-dioxane containing a small amount of water. In this compound (space group P2₁/m, a = 8.922(4) Å, b = 13.204(7) Å, c = 9.853(5) Å, β = 91.82(4)° at 150 K) a square pyramidal [TeBr₅]^? anion has been isolated for the first time. The coordination sphere of the anion is completed to a distorted octahedron by weak interaction with a dioxane molecule of the cationic system. The [H₃O]⁺ cations are connected to chains by dioxane molecules. At room temperature the compound is stable only in its mother liquor. Crystalline [H₃O]₂[SeBr₆] ( 2 ) (space group Fm3m, a = 10.421(1) Å at 170 K) is a bromoselenous acid of high symmetry. The [H₃O]⁺ ion is only weakly coordinated by Br atoms of the anion. The anions are isolated octahedral [SeBr₆]^2? units. The structure is isotypic to the K₂[PtCl₆] structure. Despite being a halogenochalcogen(IV) acid, 2 exhibits a remarkable thermal stability. Both oxonium compounds were characterized by single-crystal X-ray structure analyses. Vibrational spectra of 2 are reported.     

Dirichletsche Reihen auf arithmetischen Progressionen

A characterisation of all real Δ≠0 is given, such that $$\overline {f(\sigma + i\Delta \mathbb{N})} = \overline {f(\sigma + i\mathbb{R})} $$ holds for all Dirichlet seriesf(s)=Σa _n n ^?s and all σ greater than the abscissa of absolute convergence off. For only countably many Δ relation (*) is not valid. By using transcendental results it is shown, that (*) holds in the special case, Δ=1.     

The strength of parallel-displaced arene-arene interactions in chloroform

[reaction: see text] Triptycene-derived compounds have been prepared to serve as conformational equilibrium reporters for direct measurements of arene-arene interactions in the parallel-displaced orientation. A series of such compounds bearing arenes with different substituents were synthesized, and the ratios of the syn and anti conformers were determined by variable-temperature NMR spectroscopy. The syn conformer allows attached arenes to interact with each other while the anti conformer does not. The free energies derived from the syn/anti ratios in chloroform range from slightly positive (0.2 kcal/mol) to considerably negative (-0.98 kcal mol) values. The interactions between the arenes bearing electron-donating groups (EDG) are either negligible or slightly repulsive, while the interactions between arenes bearing electron-withdrawing groups (EWG) are attractive. Intermediate free energy values are obtained for those compounds bearing arenes with one EDG and one EWG.     

4H-Chromenone Glycosides from Eranthis hyemalis (L.) SALISBURY

Investigation of the tubers of Eranthis hyemalis (Ranunculaceae) afforded six chromenone glycosides. Their structures have been elucidated mainly by spectroscopic (FAB-MS, 2D-NMR techniques) and chemical methods (acidic and enzymatic hydrolysis) as 9-{[(β-D -glucopyranosyl)oxy]methyl}-8,11-dihydro-5-hydroxy-2-methyl-4H-pyrano[2,3-g][1]benzoxepin-4-one ( 1 ), 9-{[(β-D -gentiobiosyl)oxy]methyl}-8,11-dihydro-5-hydroxy-2-methyl-4H-pyrano[2,3-g][1]benzoxepin-4-one( 2 ), 9-{[(β-D -glucopyranosvl)oxy]melhyl}-8,11-dihydro-5-hydroxy-2-(hydroxy-methyl)-4H-pyrano[2,3-g][1]benzoxepin-4-one( 3 ), 8-{(2E)-4-[(β-D -glucopyranosyl)oxy]-3-methylbut-2-enyl}-5,7-dihydroxy-2-methyl-4H-1-benzopyran-4-one ( 4 ), 8-{(2E)-4-[(β-D -glucopyranosyi)oxy]-3-methylbut-2-enyl}-5,7-dihydroxy-2-(hydroxymethyl)-4H-1-benzopyran-4-one ( 5 ), and 7-{[(β-D -glucopyranosy1)oxy]methyl}-2,3-dihydro-2-(l-hydroxy-1-methylethyl)-4-methoxy-5H-furo[3,2-g][1]benzopyran-5-one ( 6 ). Compound 2 exhibited negative inotropic activity.     

The EPR Spectra of Manganese(II) on Sulfonated Polystyrene Resins Cross-linked with Pure Divinylbenzene Isomers

The EPR spectra of manganese(II) ions on sulfonated polystyrene cross-linked with either pure meta or pure para divinylbenzene show distinctions indicative of ionic environments of different symmetry.     

Convergence of non-periodic infinite products of orthogonal projections and nonexpansive operators in Hilbert space

We provide sufficient conditions for strong and uniform (on bounded subsets of initial points) convergence of infinite products of orthogonal projections and other (possibly nonlinear) nonexpansive operators in a Hilbert space. Our main tools are new estimates of the inclination of a finite tuple of closed linear subspaces.     

Finite element approximations of a nonlinear diffusion model with memory

The convergence of a finite element scheme approximating a nonlinear system of integro-differential equations is proven. This system arises in mathematical modeling of the process of a magnetic field penetrating into a substance. Properties of existence, uniqueness and asymptotic behavior of the solutions are briefly described. The decay of the numerical solution is compared with both the theoretical and finite difference results.