Synthesen von Heterocyclen, 135. Mitt.: Chinolizine und Indolizine,VI: eine Synthese von 2-Hydroxy-benzo[a] chinolizin-4-onen

Zusammenfassung 2-Hydroxy-6,7-dihydro-4H-benzo[a]chinolizin-4-one (3a—p) werden durch Kondensation von 1-Alkyl-3,4-dihydro-isochinolinen (1a—g) mit Malonsäure-bis-2,4,6-trichlorphenylestern (2a—e) erhalten. Die Ausbeuten sowie die erforderlichen Reaktionszeiten und Temperaturen sind stark von der Art der Substituenten abhängig.

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Syntheses of heterocycles, CXXXV: Quinolizines and indolizines, VI. A synthesis of 2-hydroxy-4H-benzo[a]quinolizin-4-ones

The condensation of 1-alkyl-3.4-dihydro-isoquinolines (1a tog) with 2.4.6-trichlorophenyl malonates (2a—e) yields 2-hydroxy-6,7-dihydro-4H-benzo[a]quinolizin-4-ones (3a—p). Yields, required reaction-periods and temperatures are depending on the nature of the substituents present in the malonyl and isoquinoline residue.

     

Sign changes in the prime number theorem

The Ramanujan Journal - Let V(T) denote the number of sign changes in $$\psi (x) - x$$ for $$x\in [1, T]$$ . We show that $$\liminf _{T\rightarrow \infty } V(T)/\log T\ge \gamma _{1}/\pi +...     

Rare Events in Random Geometric Graphs

This work introduces and compares approaches for estimating rare-event probabilities related to the number of edges in the random geometric graph on a Poisson point process. In the one-dimensional setting, we derive closed-form expressions for a variety of conditional probabilities related to the number of edges in the random geometric graph and develop conditional Monte Carlo algorithms for estimating rare-event probabilities on this basis. We prove rigorously a reduction in variance when compared to the crude Monte Carlo estimators and illustrate the magnitude of the improvements in a simulation study. In higher dimensions, we use conditional Monte Carlo to remove the fluctuations in the estimator coming from the randomness in the Poisson number of nodes. Finally, building on conceptual insights from large-deviations theory, we illustrate that importance sampling using a Gibbsian point process can further substantially reduce the estimation variance.

     

Forced lattice vibrations: Part I

This is the First part of a two-part series on forced lattice vibrations in which a semi-infinite lattice of one-dimensional particles {x_n}_n≧1 is driven from one end by a particle x₀. This particle undergoes a given, periodically perturbed, uniform motion, _x0(_t) = _at + _h(_yt), where a and γ are constants and h(·) has period 2π. For a wide variety of restoring forces F (i.e., F′ > 0), numerical calculations indicate the existence of a sequence of thresholds γ₁ = γ₁(a, h, F) > γ₂ = γ₂(a,h,F) > … > γ_k = γ_k(a,h,F) > …, γ_k → 0, as k → ∞. If γ_k > γ > γ_k+1, a k-phase wave that is well described by the wave form, emerges and travels through the lattice. The goal of this series is to describe the emergence and calculate some properties of these wave forms. In Part I the authors first consider the case where F(x) = e^x (i.e., Toda forces) but h is arbitrary, and show how to compute a basic diagnostic (see J(λ), formula (1.26)) for the system in terms of the solution of an associated scalar Riemann-Hilbert problem, once a certain finite set of numbers is known. In another direction, the authors consider the case where F(x) is restoring but arbitrary, and h is small. Here the authors prove a general result, asserting that if there exists a sufficiently ample family of traveling-wave solutions of the doubly infinite lattice, then it is possible to construct time-periodic k-phase wave solutions with asymptotics in n of type (iii) for the driven system (i). In Part II, the authors prove that sufficiently ample families of traveling-wave solutions of the system (iv) exist in the cases γ > γ₁ and γ₁ > γ > γ₂ for general restoring forces F. In the case with Toda forces, F(x) = e^x, the authors prove that sufficiently ample families of traveling-wave solutions.     

Positive Lyapunov exponents for a class of deterministic potentials

LetV() be a smooth, non-constant function on the torus and letT be a hyperbolic toral automorphism. Consider a discrete one dimensional Schrödinger operatorH, whose potential at sitej is given bygV _j =gV(T ^j ). We prove that wheng0 is small andg ^1/2 |E|2–g ^1/2 , the Lyapunov exponent for the cocycle generated byH-E is proportional tog ². The proof relies on a formula of Pastur and Figotin and on symbolic dynamics.