Howard J. Weiner

A Review of the packing problem

The random pazking problem has been of interest to inoestigsccrs in seveal disciplines , Physical chemists have investigaced such models in two and three dimensions, Because of aralytical difficulties, one-dimensional analogacs have been explored and theseare referred to as the parking problem, A number of results areexplored and attempts are made to tie them together, Applicationsare also highlighted.

Age Dependent Branching Processes with Two Types of Immigration

Abstract A critical age-dependent (Bellman-Harris) branching process with state-dep endent immigration at 0 is denoted Z(t). At IID times of an independent renewal process, IID {Zi(t)} processes are introduced. Asymptotic moments, and a limit th eorem for the overall process are given.

Distributed Wiener-Poisson control †

A one-dimensional Wiener plus independent Poisson control problem, with the state governed by a partial differential equation, has an integrated discounted quadratic cost function and asymmetric bounds on the control, which is a function of the current state. A Bellman equation and the maximum principle for partial differential equations are used to obtain the optimal closed-loop control in bang-bang form. The finite and infinite integral quadratic cost functions are treated separately.

On asymmetric stochastic bang-bang control†

Three stochastic bang-bang control problems, the predicted miss, the linear regulator, and a complete observation model, are shown to have formally similar solutions under an asymmetric boundedness condition on the control u(t), 0

Critical age-dependent branching processes with sibling correlation

Abstract Asymptotic moments of a one dimensional critical age-dependent branching process with symmetric lifetime and offspring dependence among siblings are found. An application is to show that the number alive at time t divided by bt, given that the population has survived, is exponentially distributed for t → ∞, where b is explicitly obtained. Various extensions are indicated.

A Multi-type Critical Emigration Branching Process

Abstract A multi-type age-dependent Bellman-Harris process is defined such that each cell is subject to emigration in accord with a simple renewal process, independent of other cells, with the renewal process depending only on its type. This age-dependent emigration process may be made to act similar to a critical multi-type Bellman-Harris branching process. Limit theorems analogous to those for critical multi-type Bellman-Harris processes are given. A multi-type critical emigration-immigration model is also considered.

A Critical Emigration Branching Process

Abstract A Bellman–Harris process is defined with each cell subject to emigration in accord with a simple renewal distribution. The emigration process may be made to act similar to a critical Bellman–Harris branching process under simple conditions. Under these conditions, the emigration process is called a critical emigration process. Various limit theorems, analogous to those for critical Bellman–Harris processes, are given for this process. Proofs are largely by analogy with an appropriate critical Bellman–Harris process. A critical emigration–immigration Bellman–Harris process is also considered.

Joint Moments of the Maximum in a Critical Branching Process

Abstract The asymptotic order of magnitude of the joint moments of the maxima in a critical Galton Watson process are given.

A Random Motion Problem

Abstract The limiting rate of Markov chain motion of the center of gravity of N tokens at equilibrium, as N → ∞, is computed for a random mechanism suggested by the average behavior of certain local improvement algorithms of linear programming. The equilibrium generating function of the cells of tokens is given, as N → ∞.

Random Covering and Packing on the Line

Abstract The asymptotic expected mean of the minimum number of randomly, uniformly placed unit intervals needed to cover [0, x) is obtained for the models of Renyi and Solomon. Asymptotic variances are indicated, and other models are given. The mean number of unit length intervals which may pack or cover one-dimensional abacus grid is given.