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## The facets of the polyhedral set determined by the Gale—Hoffman inequalities

### Mathematical Programming (1993-02-01) 62: 215-222 , February 01, 1993

The Gale—Hoffman inequalities characterize feasible external flow in a (capacitated) network. Among these inequalities, those that are redundant can be identified through a simple arc-connectedness criterion.

## A problem related to Foulkes’s conjecture

### Graphs and Combinatorics (1993-06-01) 9: 117-134 , June 01, 1993

In this paper we study a class of symmetric matrices*T* indexed by positive integers m≥ n≥2 and defined as follows: for any positive integers*p* and*q* let ℬ_{p,q} be the set of partitions of*U* = {1,2,3, ...,pq} into p blocks each of size*q*. Let*m* ≥*n* ≥ 2 be positive integers. By a*transversal* of α = A_{1}/A_{2}/.../A_{n} ∈ ℬ_{n,m} we mean a partition*ß* = B_{1}/B_{2}/.../B_{m}*∈**ℬ*_{m,n} such that ‖*A*_{i}∩*B*_{j} = 1 for every i= 1,2, ...,n and every*j* = 1,2, ...,m. Let*M* be the zero-one matrix with rows indexed by the elements of ℬ_{n,m} and columns indexed by the elements of ℬ_{m,n} such that M_{αß} = 1 iff*ß* is a transversal of α. We are interested in finding the eigenvalues and eigenspaces of the symmetric matrix*T* = MM^{t}. The nonsingularity of*T* implies Foulkes’s Conjecture (for these values of m and*n*). In the case*n* = 2 we completely determine the eigenvalues and eigenspaces of T and in so doing demonstrate the non-singularity of*T.* For*n* = 3 we develop a fast algorithm for computing the eigenvalues of*T*, and give numerical results in the cases m = 3,4, 5, 6.

## On the control of a circular membrane. I

### Acta Mathematica Hungarica (1993-09-01) 61: 303-325 , September 01, 1993

## Practical Use of Bootstrap in Regression

### Computer Intensive Methods in Statistics (1993-01-01): 150-166 , January 01, 1993

The usefulness of bootstrap in statistical analysis of regression models is demonstrated. Surveying earlier results, four specific problems are considered:

the computation of confidence intervals for parameters in a nonlinear regression model,

the computation of calibration sets in calibration analysis, when the standard curve is described by a nonlinear function,

the estimation of the covariance matrix of the parameter estimates for an incomplete analysis of variance model, in the presence of an interaction term,

the computation of confidence intervals for the value of the regression function, when a nonparametric heteroscedastic model is considered.

Theoretical properties of the proposed bootstrap procedures, as well as indications about their actual efficiency based on simulation results, are given.

## Polyhedral study of the capacitated vehicle routing problem

### Mathematical Programming (1993-06-01) 60: 21-52 , June 01, 1993

The capacitated vehicle routing problem (CVRP) considered in this paper occurs when goods must be delivered from a central depot to clients with known demands, using*k* vehicles of fixed capacity. Each client must be assigned to exactly one of the vehicles. The set of clients assigned to each vehicle must satisfy the capacity constraint. The goal is to minimize the total distance traveled. When the capacity of the vehicles is large enough, this problem reduces to the famous traveling salesman problem (TSP). A variant of the problem in which each client is visited by at least one vehicle, called the graphical vehicle routing problem (GVRP), is also considered in this paper and used as a relaxation of CVRP. Our approach for CVRP and GVRP is to extend the polyhedral results known for TSP. For example, the subtour elimination constraints can be generalized to facets of both CVRP and GVRP. Interesting classes of facets arise as a generalization of the comb inequalities, depending on whether the depot is in a handle, a tooth, both or neither. We report on the optimal solution of two problem instances by a cutting plane algorithm that only uses inequalities from the above classes.

## Computational Logic and Proof Theory

### Computational Logic and Proof Theory (1993-01-01): 713 , January 01, 1993

## General interpolation schemes for the generation of irregular surfaces

### Constructive Approximation (1993-12-01) 9: 525-542 , December 01, 1993

We introduce an interpolation scheme to generate a class of irregular surfaces. The analysis is first carried out for a triangle*T*. We define the function ϕ on a subset*X*, dense in*T*. In terms of the construction parameters of ϕ, we establish sufficient conditions for its uniform continuity so that it would be possible to extend it to a continuous function on the whole of*T*. We do the same analysis in the case of a rectangle*R*.

## Stability in a model of a delayed neural network

### Journal of Dynamics and Differential Equations (1993-10-01) 5: 607-623 , October 01, 1993

The stability of the null solution in a system of coupled cells is investigated. Each cell evolves according to Hopfield's equation for an analog circuit, with a delay incorporated to account for finite switching speed of amplifiers. A necessary and sufficient condition on the connection matrix is obtained for delay-induced oscillations to be possible in a general (not necessarily symmetric) network.

## Back Matter - Hamiltonian Mechanical Systems and Geometric Quantization

### Hamiltonian Mechanical Systems and Geometric Quantization (1993-01-01): 260 , January 01, 1993

## Equivalence Relations and Quotient Spaces

### Quantum Field Theory and Topology (1993-01-01) 307: 233-234 , January 01, 1993

In many situations in physics and mathematics it is reasonable to consider two different objects as equivalent in some sense. For example, in quantum mechanics the state of a particle or system of particles can be described by a nonzero vector in a complex Hilbert space (the state vector). But two vectors *ψ* and *ψ*′ proportional to each other are physically equivalent, that is, they describe the same state. Likewise, an electromagnetic field can be described by a vector potential, but two potentials *A*_{μ}^{′}
(*x*) and *A*_{µ}
(*x*) that differ by a gauge transformation (that is, that satisfy *A*_{μ}^{′}
(*x*) = *A*_{μ}(*x*) + *∂*_{μ}*λ*(*x*)) are physically equivalent.