Equations différentielles invariantes sur les groupes et algèbres de Lie réductifs II

Dans un article précédent, nous avons démontré que si D est un opérateur différentiel bi-invariant sur un groupe réductif G vérifiant la condition de Benabdallah-Rouvière, alors on peut résoudre l’équation différentielle Du=v dans l'espace des distributions G-invariantes (par automorphismes intérieurs) d'ordre fini; nous allons montrer ici que, sous la même hypothèse, on peut résoudre cette équation dans l'espace de toutes les distributions G-invariantes. D'autre part, nous donnons un exemple dans qui montre que les équations différentielles invariantes dans les algèbres de Lie réductives ne sont pas toujours résolubles dans l'espace des fonctions indéfiniment différentiables invariantes.     

Thermal buckling of a constrained elastic ring†

The buckling problem of a heated circular elastic ring confined to a uniformly contracting circular boundary is investigated. The buckled configuration is assumed to consist of a shallow detached region with the remainder of the ring in contact with the contracting boundary. The ring is treated as a variational problem with variable end points, by virtue of which, a transversality condition is obtained for the determination of the extent of the detached region. The influence of temperature variation and level of prestressing on the buckling phenomena are studied and a correspondence relationship between the pure contraction and thermal problems is obtained.     

A Generalization of Ritz-Variational Method for Solving a Class of Fractional Optimization Problems

This paper presents an approximate method for solving a class of fractional optimization problems with multiple dependent variables with multi-order fractional derivatives and a group of boundary conditions. The fractional derivatives are in the Caputo sense. In the presented method, first, the given optimization problem is transformed into an equivalent variational equality; then, by applying a special form of polynomial basis functions and approximations, the variational equality is reduced to a simple linear system of algebraic equations. It is demonstrated that the derived linear system has a unique solution. We get an approximate solution for the initial optimization problem by solving the final linear system of equations. The choice of polynomial basis functions provides a method with such flexibility that all initial and boundary conditions of the problem can be easily imposed. We extensively discuss the convergence of the method and, finally, present illustrative test examples to demonstrate the validity and applicability of the new technique.     

Alkylation of aromatic amines by tert-enamides: Direct access to protected aminals

Reaction of aromatic amines with tertiary enamides was performed in n-hexane in the presence of acetic acid as an inexpensive and green catalyst at room temperature. This protocol provides the protected aminals through Markovnikov addition reaction with high to excellent yields and regiospecificity. In addition, this procedure was expanded for the synthesis of aminals from commercially available sulfa drugs such as sulfathiazole and sulfabenzamide.     

Electrochemical Oxidation of Desipramine Drug in the Presence of 4,6‐Dimethylpyrimidine‐2‐thiol Nucleophile in Aqueous Acidic Medium

Electrochemical oxidation of desipramine (DES) has been studied in the presence of 4,6‐dimethylpyrimidine‐2‐thiol (DMPT) as nucleophile in acidic medium by means of cyclic voltammetry, controlled‐potential electrolysis and spectroscopic data, as diagnostic techniques. Voltammetric studies of electro‐oxidation of desipramine were realized in a range of pH 1.0 to 8.0 in the absence and presence of DMPT. The results indicate the participation of the product of electrochemical oxidation of desipramine in the reaction with DMPT with ECEC electrochemical mechanism. However, the voltammetry and coulometry results imply existence of a catalytic (EC′) electrochemical mechanism in parallel with ECEC electrochemical mechanism. The product has been characterized by IR, ¹H NMR, ¹³C NMR and MS methods.     

Dimension group of a non-negative integer irreducible matrix

In this paper we are concerned with a non-negative integer and irreducible matrix $A:{\mathbb{Z}}^d\to {\mathbb{Z}}^d$. The main contribution is to prove that if the matrix satisfies certain spectral and algebraic constraints, the cone:$C=\{v\in {\mathbb{Z}}^d/\exists n⩾0\mathrm{and}{A}^{n}v⩾0\}\subset {\mathbb{Z}}^d$is defined by linear maps ${\varphi }_0\text{,}\dots \text{,}{\varphi }_{k-1}:{\mathbb{Z}}^d\to \mathbb{R}$, in the sense that v ∈ C is equivalent to, ϕ_l(v) ⩾ 0 for all l = 0, … , k − 1 (where k is the index of cyclicity of the irreducible matrix). This result allows us to characterize the dimension group generated by the matrix, it is a subgroup of ${\mathbb{R}}^k$ endowed with an order induced by the positive cone of ${\mathbb{R}}^k$.     

High-resolution surface-plasmon imaging in air and in water: V(z) curve and operating conditions

We present what are believed to be the first images obtained with a far-field high-resolution scanning surface-plasmon microscope in an aqueous medium. Measurements of V(z), the output response of the microscope, versus defocus z give a signature of the surface-plasmon propagation. V(z) is strongly conditioned by the laser beam diameter and the objective's numerical aperture, and we show how the operating mode (in air and in water) must be chosen to maximize the surface-plasmon field and to minimize diffraction (edge) effects.     

Feedback control of congestion in packet switching networks: the case of multiple congested nodes

In this paper, an analytical method for the design of a congestion control scheme in packet switching networks is presented. This scheme is particularly suitable for implementation in ATM switches, for the support of the available bit rate (ABR) service in ATM networks. The control architecture is rate-based with a local feedback controller associated with each switching node. The controller itself is a generalization of the standard proportional-plus-derivative controller, with the difference that extra higher-order derivative terms are involved to accommodate the delay present in high-speed networks. It is shown that, under the specific service discipline introduced here, there exists a set of control gains that result in asymptotic stability of the linearized network model. A method for calculating these gains is given. In addition, it is shown that the resulting steady state rate allocation possesses the so-called max-min fairness property. The theoretical results are illustrated by a simulation example, where it is shown that the controller designed, using the methods developed here, works well for both the service discipline introduced in this paper and for the standard FCFS scheme. © 1997 John Wiley & Sons, Ltd.