Sojourns and extremes of Fourier sums and series with random coefficients

Let X(t) be the trigonometric polynomial Σ^k_j=0a_j(U_tcosjt+V_jsinjt), –∞< t<∞, where the coefficients U_t and V_t are random variables and a_j is real. Suppose that these random variables have a joint distribution which is invariant under all orthogonal transformations of R^2k–2. Then X(t) is stationary but not necessarily Gaussian. Put L_t(u) = Lebesgue measure {s: 0?s?t, X(s) > u}, and M(t) = max{X(s): 0?s?t}. Limit theorems for L_t(u) and $\text{P}\text{(M(t) > u)}$ for u→∞ are obtained under the hypothesis that the distribution of the random norm (Σ^k_j=0(U²_j+V²_j))^{1 2} belongs to the domain of attraction of the extreme value distribution exp{ e^–2}. The results are also extended to the random Fourier series (k=∞).     

Product formulas,nonlinear semigroups,and accretive operators

A special case of our main theorem, when combined with a known result of Brezis and Pazy, shows that in reflexive Banach spaces with a uniformly Gâteaux differentiable norm, resolvent consistency is equivalent to convergence for nonlinear contractive algorithms. (The linear case is due to Chernoff.) The proof uses ideas of Crandall, Liggett, and Baillon. Other applications of our theorem include results concerning the generation of nonlinear semigroups (e.g., a nonlinear Hille-Yosida theorem for “nice” Banach spaces that includes the familiar Hilbert space result), the geometry of Banach spaces, extensions of accretive operators, invariance criteria, and the asymptotic behavior of nonlinear semigroups and resolvents. The equivalence between resolvent consistency and convergence for nonlinear contractive algorithms seems to be new even in Hilbert space. Our nonlinear Hille-Yosida theorem is the first of its kind outside Hilbert space. It establishes a biunique correspondence between m-accretive operators and semigroups on nonexpansive retracts of “nice” Banach spaces and provides affirmative answers to two questions of Kato.     

Crystal Growth and Structure Refinement of K4Ge9

Deep‐red moisture and air sensitive single crystals of K₄Ge₉ were obtained by reacting GeO₂ and elemental Ge with metallic W and K at high temperature in a niobium ampoule. The crystal structure of the compound was determined by single crystal X‐ray diffraction experiments. K₄Ge₉ crystallizes in a polar space group R3c (No. 161), Z = 4 with a = 21.208(1) and c = 25.096(2) Å. The compound contains discrete Ge₉^4? Wade's nido‐clusters which are packed according to a hierarchical atom‐to‐cluster replacement of the Cr₃Si prototype and are separated by K⁺ cations. Two independent [Ge₉]^4? clusters A (at Cr positions) and B (at Si positions) are found with a ratio A:B = 3:1. The B ‐type cluster satisfactorily represents orientational disorder around the three‐fold axis.     

From molecular dynamics to hydrodynamics: a novel Galilean invariant thermostat

This paper proposes a novel thermostat applicable to any particle-based dynamic simulation. Each pair of particles is thermostated either (with probability P) with a pairwise Lowe-Andersen thermostat [C. P. Lowe, Europhys. Lett. 47, 145 (1999)] or (with probability 1-P) with a thermostat that is introduced here, which is based on a pairwise interaction similar to the Nosé-Hoover thermostat. When the pairwise Nosé-Hoover thermostat dominates (low P), the liquid has a high diffusion coefficient and low viscosity, but when the Lowe-Andersen thermostat dominates, the diffusion coefficient is low and viscosity is high. This novel Nosé-Hoover-Lowe-Andersen thermostat is Galilean invariant and preserves both total linear and angular momentum of the system, due to the fact that the thermostatic forces between each pair of the particles are pairwise additive and central. We show by simulation that this thermostat also preserves hydrodynamics. For the (noninteracting) ideal gas at P = 0, the diffusion coefficient diverges and viscosity is zero, while for P > 0 it has a finite value. By adjusting probability P, the Schmidt number can be varied by orders of magnitude. The temperature deviation from the required value is at least an order of magnitude smaller than in dissipative particle dynamics (DPD), while the equilibrium properties of the system are very well reproduced. The thermostat is easy to implement and offers a computational efficiency better than (DPD), with better temperature control and greater flexibility in terms of adjusting the diffusion coefficient and viscosity of the simulated system. Applications of this thermostat include all standard molecular dynamic simulations of dense liquids and solids with any type of force field, as well as hydrodynamic simulation of multiphase systems with largely different bulk viscosities, including surface viscosity, and of dilute gases and plasmas.     

Lead tetraacetate mediated one-pot multistage transformations: theoretical studies on the diverging behavior in the Hajos-Parrish and Wieland-Miescher series

A computational approach to the proposed mechanistic rationale of the lead tetraacetate mediated oxidative cleavage of selected unsaturated diols in the hydrindenediol (derived from the Hajos-Parrish ketone, abbreviated as HP) and octalindiol (derived from the Wieland-Miescher ketone, abbreviated as WM) series is presented. These two unexpected cascade-type transformations have been found to yield structurally different products despite the similarities in the starting compounds 1-HP and 1-WM, which differ only by the number of carbon atoms in the cycloalkane rings. The final product in the HP series is the complex ring-expanded molecule iii-6, while the final product in the WM series is the ring-expanded bisacetoxy diacetal 7. Calculations carried out at the B3LYP/6-31G level have revealed the fact that the transformations of i-6 to ii-6 and of i-7 to ii-7 are endothermic by 19.93 and 4.20 kcal/mol, respectively. The isomerization of compounds i to ii is more endothermic in the case of the HP series. Furthermore, calculations have shown the transformation of i to ii to be kinetically less favored in the HP series than in its analogue WM series. The activation barrier for the transformation of i-6 to ii-6 is calculated to be 22.09 kcal/mol, whereas the corresponding barrier in the WM series is 13.77 kcal/mol. The thermodynamic features of the reactions support the experimental findings as well: 7 is calculated to be more stable than iii-7, and iii-6 is more stable than 6, as expected. The diverging behavior in the mechanism of the interconversion of i to ii has been attributed to the stabilization of the positive charge by a carbonyl oxygen in ii-7 due to the flexibility of the seven-membered ring in contrast to the rigidity of the six-membered ring.     

Two results in metric fixed point theory

We establish two fixed point theorems for certain mappings of contractive type. The first result is concerned with the case where such mappings take a nonempty, closed subset of a complete metric space X into X, and the second with an application of the continuation method to the case where they satisfy the Leray–Schauder boundary condition in Banach spaces.     

On Ovoids of Parabolic Quadrics

It is known that every ovoid of the parabolic quadric Q(4, q), q=p ^h , p prime, intersects every three-dimensional elliptic quadric in 1 mod p points. We present a new approach which gives us a second proof of this result and, in the case when p=2, allows us to prove that every ovoid of Q(4, q) either intersects all the three-dimensional elliptic quadrics in 1 mod 4 points or intersects all the three-dimensional elliptic quadrics in 3 mod 4 points. We also prove that every ovoid of Q(4, q), q prime, is an elliptic quadric. This theorem has several applications, one of which is the non-existence of ovoids of Q(6, q), q prime, q>3. We conclude with a 1 mod p result for ovoids of Q(6, q), q=p ^h , p prime.     

Electronic and ESR Spectra of Copper(II) Complexes with N,N-Dialkyl Amino Acids

Five N,N-dialkylamino acids were synthesised [4]: N,N-dimethylisoleucine (DMIL), N,N-dimethyl-alloisoleucine (DMAIL), N,N-dimethylvaline (DMV), N,N-diethyl-alanine (DEA), and N,N-diethylvaline (DEV), as well as their complexes with copper(II): Cu(DMIL)₂, Cu(DMAIL)₂, Cu(DMV)₂, Cu(DMIL)(DMV), Cu(DEA)₂, and Cu(DEV)₂. UV/VIS spectra of the complexes were recorded in CH₂Cl₂ solutions and the ESR spectra in CH₂Cl₂, CD₃OD, and D₂O solutions. By applying the analysis of variance, ESR spectra were shown to be influenced by the solvent and substituent differences, the former effect being related to the degree of covalency of the copper-ligand bonds. The effect of the ligand's spatial configuration (diastereoisomerism) upon the ESR spectrum of the complex was demonstrated and explained.