On a resonance energy model based on expansion in terms of acyclic moments: Exact results

Summary The new concept of the resonance energy in conjugated hydrocarbons introduced by Jiang Y, Zhang H (1989) Theor Chim Acta 75:279 is further developed. This model is based on expansion of the -electron energy in terms of moments which are also equal to numbers of closed walks in a molecular graph. The reference system is established by counting only acyclic walks, i.e. those tracing only on acyclic subgraphs. Because acyclic walks could be counted only up to some finite length, the energy of the reference system has been evaluated by truncating higher terms in the expansion. In this paper a finite expression for the energy of the same reference system is derived, thus allowing its exact evaluation. The exact values differ significantly from the truncated ones. This difference, as well as the discrepancy between exact results and chemical experience, are discussed.     

The generalized order complementarity problem

Given an ordered Banach Space (E,K) andm functionsf ₁,f ₂,...,f _m:EE, the generalized order complementarity problem associated with {f _i} andK is to findx ₀K such thatf _i(x ₀)K,i=1,...,m, and (x ₀,f ₁(x ₀),...,f _m(x ₀))=0. The problem is shown to be equivalent to several fixed-point problems and equivalent to the order complementarity problem studied by Borwein and Dempster and by Isac. Existence and uniqueness of solutions and least-element theory are shown in the spacesC(, ) andL _p(, ). For general locally convex spaces, least-element theory is derived, existence is proved, and an algorithm for computing a solution is presented. Applications to the mixed lubrication theory of fluid mechanics are described.     

Improving the rate of convergence of interior point methods for linear programming

This paper proposes a procedure for improving the rate of convergence of interior point methods for linear programming. If (x ^k ) is the sequence generated by an interior point method, the procedure derives an auxiliary sequence ( ). Under the suitable assumptions it is shown that the sequence ( ) converges superlinearly faster to the solution than (x ^k ). Application of the procedure to the projective and afflne scaling algorithms is discussed and some computational illustration is provided.     

A review ofL=λW and extensions

A fundamental principle of queueing theory isL=W (Little's law), which states that the time-average or expected time-stationary number of customers in a system is equal to the product of the arrival rate and the customer-average or expected customer-stationary time each customer spends in the system. This principle is now well known and frequently applied. However, in recent years there have been extensions, such as H=G and the continuous, distributional, ordinal and central-limit-theorem versions, which show that theL=W relation, when viewed properly, has much more power than was previously realized. Moreover, connections have been established between H=G and other fundamental relations, such as the rate conservation law and PASTA (Poisson arrivals see time averages), which show that there is a much greater unity in the overall theory than was previously realized. This paper provides a review.This paper is dedicated to the memory of our colleague Professor Peter Franken (1937–1989), who contributed greatly to the subject of this paper and to queueing theory more generally.     

Wave chaos in quantum systems with point interaction

We study perturbations of the quantized version ₀ of integrable Hamiltonian systems by point interactions. We relate the eigenvalues of to the zeros of a certain meromorphic function . Assuming the eigenvalues of ₀ are Poisson distributed, we get detailed information on the joint distribution of the zeros of and give bounds on the probability density for the spacings of eigenvalues of . Our results confirm the wave chaos phenomenon, as different from the quantum chaos phenomenon predicted by random matrix theory.SFB 237 Essen-Bochum-Düsseldorf     

High-density properties of hard spheres within a modified Percus-Yevick theory: The role of thermodynamic consistency

Using an integral-equation approach based upon an approximation for the tail function, the equilibrium properties of a system of hard spheres are studied with special concern for the behavior in the region of close packing. The closure adopted is such that full, internal consistency is ensured in the thermodynamics of the model with respect to both the two zero-separation theorems as well as to the more standard virial and fluctuation routes to the equation of state. The scheme also makes use of the continuity properties of the tail function and of the cavity distribution function at contact. These properties are explictly tested in the low-density limit up to the fourth derivative. The theory generates an equilibrium branch bounded on the high-density side by a point corresponding to a packing fraction0.78, a value which closely matches Rogers' least upper bound for the densest packing of spheres. The pair structure of the fluid in the state of random close packing is also compared to the type of local order predicted by the theory at similar densities.     

Exact solution of the Boltzmann equation in the homogeneous color conductivity problem

An exact solution of the Boltzmann equation for a binary mixture of colored Maxwell molecules is found. The solution corresponds to a nonequilibrium homogeneous steady state created by a nonconservative external force. Explicit expressions for the moments of the distribution function are obtained. By using information theory, an approximate velocity distribution function is constructed, which is exact in the limits of small and large field strengths. Comparison is made between the exact energy flux and the one obtained from the information theory distribution.     

NonstableK-Theory for operator algebras

We show that the homotopy groups of the group of quasi-unitaries inC ^*-algebras form a homology theory on the category of allC ^*-algebras which becomes topologicalK-theory when stabilized. We then show how this functorial setting, in particular the half-exactness of the involved functors, helps to calculate the homotopy groups of the group of unitaries in a series ofC ^*-algebras. The calculations include the case of all AbelianC ^*-algebras and allC ^*-algebras of the formAB, whereA is one of the Cuntz algebras O_n n=2, 3, ..., an infinite dimensional simpleAF-algebra, the stable multiplier or corona algebra of a-unitalC ^*-algebra, a properly infinite von Neumann algebra, or one of the projectionless simpleC ^*-algebras constructed by Blackadar.     

Transport phenomena in a plane shock wave

With the use of the nonpolynomial closure 1/ _z in the Mott-Smith approximation of the solution of the Boltzmann equation, we obtain a value of the density gradient in the limit of a very weak shock wave that is close to the correct value. For the determination of the transverse temperature gradient we calculated the _x ² / _z moment of the Mott-Smith collision integral. The effective values of viscosity and thermal conductivity in the limit of a very weak shock wave were calculated for inverse-power potentials and found to agree almost exactly with the Chapman-Enskog values. Such a comparison can serve as a criterion for the evaluation of different bimodal theories. Various bimodal theories give different values of viscosity and thermal conductivity, but all of them give 33 % too high a value of the Eucken ratio.