Electrochemical synthesis of a thiophene-containing cyclo[9]pyrrole

A new member of the cyclo[n]pyrrole class of expanded porphyrins could be prepared from the corresponding thiophene-containing terpyrrole precursor through use of a mild electrochemical oxidative procedure. The isolated macrocycle, featuring nine heterocyclic subunits directly connected through their α,α'-positions, is the largest cyclo[n]pyrrole derivative reported to date (see figure).     

On the Nonlinear Schrödinger Equation on the Half Line with Homogeneous Robin Boundary Conditions

Boundary value problems for the nonlinear Schrödinger equations on the half line with homogeneous Robin boundary conditions are revisited using Bäcklund transformations. In particular: relations are obtained among the norming constants associated with symmetric eigenvalues; a linearizing transformation is derived for the Bäcklund transformation; the reflection‐induced soliton position shift is evaluated and the solution behavior is discussed. The results are illustrated by discussing several exact soliton solutions, which describe the soliton reflection at the boundary with or without the presence of self‐symmetric eigenvalues.     

Conservative methods for the Toda lattice equations

We are concerned with the numerical integration of the Todalattice equations by using different conservative methods. Numericalexperiments suggest that the global error for isospectral schemesdecreases exponentially with time but it is almost constantfor either symplectic or more general integrators. We providea theoretical explanation for these experimental findings.     

Littlewood-Paley Inequalities on Manifolds with Ends

Potential Analysis - Let M be a manifold with ends $\mathbb {R}^{m}\sharp \mathcal {R}^{n}$ with m > n >?2 which is a non-doubling manifold. In this paper we prove a...     

New analytical solutions for modified polarization saturation models in piezoelectric materials

Meccanica - In this paper, we present new analytical solutions for modified polarization saturation (PS) models for arbitrary polarized and semipermeable 2-D piezoelectric media. The PS model is...     

Characterizations of weighted Besov and Triebel-Lizorkin spaces via temperatures

Hardy-Littlewood type characterizations (via temperatures on a half-space) of weighted Besov and Triebel-Lizorkin spaces studied recently by the author are given. In the proof of the sufficient parts of the results, a “sub-mean-value property” of temperatures is used in an elegant way to get control of certain terms by the Hardy maximal function or its vector-valued version. As a byproduct of the results, another characterization of weighted Hardy spaces as well as characterizations of the Gauss-Weierstrass integrals on a strip domain of distributions in the above weighted spaces is obtained also.     

Measurements of anomalous dispersion in rayleigh scattering of characteristic X-ray fluorescence

Summary When using characteristic fluorescence X-rays or γ-rays of energy under 100 keV to study the Rayleigh scattering of photons from atoms the effect of anomalous dispersion is not in general negligible. On the contrary, it may cause the atomic cross-section values to vary with respect to the tabulated values—usually calculated in nonresonant approximation—of an order of magnitude in the most critical cases. In this work we present experimental results of Rayleigh scattering where this effect is well evident. We used as sources of excitation a radioisotope (²⁴¹Am,E _γ 59.6 keV) and a X-ray tube with a series of different secondary anodes and solid-state detectors to detect the scattered radiation. We reviewed the atomic form factor formalism which describes the phenomenon in a nonrelativistic treatment and we show its strict analogy with QEDS-matrix theory. We obtained numerical values for the cross-sections by using the tabulated values of the photoelectric cross-sections and we made a comparison with the experimental values. The error of the measurements (∼3%) derives from the superposition of the Rayleigh scattering with the characteristic X-ray fluorescence lines. Within the error limits the agreement with the values from the form factor approximation is very satisfactory.

---

Riassunto Quando si utilizza la radiazione X di fluorescenza caratteristica o la radiazione γ di energia minore di 100 keV per studiare la diffusione elastica da parte degli atomi (diffusione di Rayleigh), l'effetto della diffusione anomala non è in generale trascurabile. Al contrario, essa può causare variazioni anche di un ordine di grandezza dei valori delle sezioni d'urto atomiche rispetto ai valori tabulati che sono usualmente calcolati nell'approssimazione non risonante. In questo lavoro si presentano i risultati sperimentali di alcune misure di diffusione di Rayleigh nei quali questo effetto è ben evidente. Come sorgenti di radiazione sono stati utilizzati un isotopo radioattivo (²⁴¹Am,E _γ=59.6 keV) e una serie di anodi secondari associati a un tubo X. Per la rivelazione della radiazione X sono stati utilizzati rivelatori a stato solido. Si è eseguita una revisione del formalismo dei fattori di forma atomici per la descrizione del fenomeno in una trattazione non relativistica e si mostra la stretta analogia con la teoria della matriceS dell'elettrodinamica quantistica. Si sono ottenuti i risultati numerici per le sezioni d'urto di diffusione elastica utilizzando i valori tabulati per le sezioni d'urto fotoelettriche e si sono confrontati con i valori ottenuti sperimentalmente. L'errore delle misure (∼3%) deriva dalla sovrapposizione della diffusione di Rayleigh con le righe di fluorescenza caratteristica. Entro questi limiti di errore l'accordo con i valori ricavati nell'approssimazione dei fattori di forma è molto soddisfacente.

---

Резюме При использовании характеристической рентгеновской или γ-лучевой флуоресценции с энергией до 100 кэВ для исследования рэлеевского рассеяния фотонов на атомах, влияние аномальной дисперсии не является в общем случае пренебрежимо малым. Наоборот, эффект аномальной дисперсии может изменить значения атомных поперечных сечений по сравнению с табулированными значениями, обычно вычисляемыми в нерезонансном приближении, в наиболее критических случаях на порядок величины. В этой работе приводятся резулятаты для рэлеевского рассеяния, где эффект аномальной дисперсии является существенным. Мы используем в качестве источников возбуждения радиоизотоп (²⁴¹Am,E _γ=59.6 кэВ) и рентгеновскую трубку с различными вторичными анодами и твердотельные детекторы для регистрации рассеянного излучения. Мы используем формализм атомного форм-фактора, который описывает явление в нерелятивистском приближении. Мы отмечаем точную аналогию с теориейS-матрицы в квантовой электродинамике. Получаются численные значения для поперечных сечений, используя табулированные величины для фотоэлектрических поперечных сечений. Проводится сравнение полученных значений с экспериментальными данными. Погрешность измерений (∼3%) определяется из суперпозиции рэлеевского рассеяния с линиями характеристической рентгеновской Флуоресценции. В пределах погрешностей получается удовлетворительное согласие с результатами, полученными из приближения атомного форм-фактора.

     

The subgroups of the special linear group over a skew field that contain the group of diagonal matrices

For any (noncommutative) skew field T, the lattice of subgroups of the special linear group Λ=SL(n,T) that contain the subgroup Δ=SD(n,T) of diagonal matrices (with Dieudonné determinants equal to 1) is studied. It is established that for any subgroup H, Δ≤H≤Λ, there exists a uniquely determined unital net σ such that Λ(σ)≤H≤N(σ), where Λ(σ) is the net subgroup associated with the net σ and N(σ) is its normalizer in Λ. Bibliography: 11 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 91–103. Translated by Bui Xuan Hai.