Quantum Monodromy and Bohr–Sommerfeld Rules

This article is based on a talk given in Dijon for the 2000 summer school Topological and Geometrical Methods: Applications to Dynamical Systems. The standard definitions of the monodromy invariant for completely integrable classical systems are reviewed, and the link to the quantum monodromy observed in the joint spectrum of commuting operators is explained. The mathematical treatment relies on a modern and attractive version of the Bohr–Sommerfeld quantisation rules.     

Generalized Quantum Mechanics for Separated Systems

We give seven necessary physical conditions on a property lattice for to describe two quantum systems when they are separated.     

On the Influence of the Lowest Terms on Dirichlet—Type Problems for Elliptic Systems Degenerate at a Point

We consider Dirichlet–type problems for weakly connected systems of elliptic equations of second order with a strong degeneracy at an inner point of the domain, when, in a neighborhood of this point, the asymptotics of a solution is additionally given. The form of the given asymptotics essentially depends on the properties of the coefficients at the first–order derivatives. We prove the existence and uniqueness of solutions of the problems considered in Hölder function classes.     

On the Geometry of Inner Approximations to the Set of Null Controllability for Constrained Linear Systems

The problem of obtaining inner approximations to the set of null controllability for linear systems is considered, and the geometry of certain sets arising in the approximation of using linear feedback is explored. A two-dimensional example is worked in detail in order to delineate the limitations to the use of linear feedback.     

Design of Civil Engineering Frame Structures Using a Monomial/Newton Hybrid Method

This paper presents an application of a monomial approximation method for solving systems of nonlinear equations to the design of civil engineering frame structures. This is accomplished by solving a set of equations representing the state known as fully-stressed design, where each member of the structure is stressed to the maximum safe allowable level under at least one of the loading conditions acting on it. The monomial approximation method is based on the process of condensation, which has its origin in geometric programming theory. A monomial/Newton hybrid method is presented which permits some of the design variables to be free in sign, while others are strictly positive. This hybrid method is well suited to the structural design application since some variables are naturally positive and others are naturally free. The proposed method is compared to the most commonly used fully-stressed design method in practice. The hybrid method is shown to find solutions that the conventional method cannot find, while doing so with less computational effort. The impact of this approach on the activity of structural design is discussed.     

On the use of Laplace's equation for global predictions of internuclear separation and dissociation energy

Properties of target species can be estimated by various means including interpolations in periodic charts. Interpolation is equivalent to numerical solution of the Laplace equation. A test of this equivalence, within some confidence level, for any N-atomic molecule surrounded by 4N nearest neighbors: the sum of the second differences of the data in all directions must be zero. Since very few molecules have 4N neighbors with known data, the test becomes: the sum of the averages of the second differences must be zero. The validity of these tests is explored. For radii of main-group atoms, and for internuclear separations of their diatomic combinations, the averages are different from zero and the sums of the averages are zero to within one if second-nearest neighbors are used. Dissociation potentials pass the tests but with larger scatter. Predictions for dissociation potentials, using iterative interpolation within boundaries on which there are known data, are reviewed.     

INITIAL-BOUNDARY VALUE PROBLEM FOR THE UNSATURATED LANDAU-LIFSHITZ SYSTEM

51. IntroductionLet fi C Re (n = 1,2) be a bounded smooth domain. Consider the following nonhomogeneous initial-boundary value problem for the unsaturated Landau-Lifshitz systems offerromagnetic spin chain with Gilbert damping constant afl > 0,where adZ is the exchange constallt, u = (u', u', u'), "o(x) is smooth and satisfies the unsaturated condition, i.e., Ilo(~)I gi constant, and in IVuol' < co, "o(x)IOn = op(x). DenoteW(x) = luo(x)I. We assume 0 < m = mane < M ~ mpxW. Throughout t…     

Production systems with interruptions, arbitrary topology and finite buffers

We consider a production system with finite buffers and arbitrary topology where service time is subject to interruptions in one of three ways, viz. machine breakdown, machine vacations or Npolicy. We develop a unified approximation (analytical) methodology to calculate the throughput of the system using queueing networks together with decomposition, isolation and expansion techniques. The methodology is rigorously tested covering a large experimental region. Orthogonal arrays are used to design the experiments in order to keep the number of experiments manageable. The results obtained using the approximation methodology are compared to the simulation results. The ttests carried out to investigate the differences between the two results show that they are statistically insignificant. Finally, we test the methodology by applying it to several arbitrary topology networks. The results show that the performance of the approximation methodology is consistent, robust and produces excellent results in a variety of experimental conditions.     

Tactical Decompositions of Steiner Systems and Orbits of Projective Groups

Block's lemma states that the numbers m of point-classes and n of block-classes in a tactical decomposition of a 2-(v, k, ) design with b blocks satisfy m n m + b – v. We present a strengthening of the upper bound for the case of Steiner systems (2-designs with = 1), together with results concerning the structure of the block-classes in both extreme cases. Applying the results to the Steiner systems of points and lines of projective space PG(N, q), we obtain a complete classification of the groups inducing decompositions satisfying the upper bound; answering the analog of a question raised by Cameron and Liebler (P.J. Cameron and R.A. Liebler, Lin. Alg. Appl. 46 (1982), 91–102) (and still open).