Common knowledge revisited

We consider the common-knowledge paradox raised by Halpern and Moses: common knowledge is necessary for agreement and coordination, but common knowledge is unattainable in the real world because of temporal imprecision. We discuss two solutions to this paradox:

• 1.(1) modeling the world with a coarser granularity, and
• 2.(2) relaxing the requirements for coordination.

     

An optimal time to sleep for an auto-sleep system considering multi-usage states

An auto-sleep system is defined by the following two properties:

• 1.(i) a call for the system occurs randomly and intermittently
• 2.(ii) the system automatically goes to sleep if there occurs no call during a prespecified time T.

It considers four states:

• 1.(a) sleep
• 2.(b) warm-up
• 3.(c) nonusage
• 4.(d) usage.

For such a system, the time to sleep has been discussed based on suitable criteria. This study extends the model for an auto-sleep system so that the model can deal with multi-usage states. With a view to determining an optimal time to sleep under the extended model, the expected energy consumed per unit time is formulated as a criterion to be minimized. The existence of an optimal time to sleep is examined under a general call distribution. Numerical examples are also provided for a Weibull as well as a log-normal call distribution.     

Warranty cost analysis for second-hand products

For second-hand products sold with warranty, the expected warranty cost for an item to the manufacturer, depends on

• 1.(i) the age and/or usage as well as the maintenance history for the item
• 2.(ii) the terms of the warranty policy.

The paper develops probabilistic models to compute the expected warranty cost to the manufacturer when the items are sold with free replacement or pro rata warranties.     

On the computational complexity of the theory of Abelian groups

We develop a series of Ehrenfeucht games and prove the following results:

• 1.(i) The first order theory of the divisible and indecomposable p-group, the first order theory of the group of rational numbers with denominators prime to p and the first order theory of a cyclic group of prime power order can be decided in 2^{2cn log n} Turing time.
• 2.(ii) The first order theory of the direct sum of countably many infinite cyclic groups, the first order theory of finite Abelian groups and the first order theory of all Abelian groups can be decided in 2^2dn Turing space.

     

The dynamic programming method in the generalized traveling salesman problem

A procedure of the dynamic programming (DP) for the discrete-continuous problem of a route optimization is considered. It is possible to consider this procedure as a dynamic method of optimization of the towns choice in the well-known traveling salesman problem. In the considered version of DP, elements of a dynamic optimization are used. Two variants of the function of the aggregations of losses are investigated:

• 1.(1) the additive functions;
• 2.(2) the function characterizing the aggregation of losses in the bottle-neck problem.

     

System dynamics and feedback control problem formulations for real time dynamic traffic routing

This paper formulates the Dynamic Traffic Routing (DTR) problem as a real-time feedback control problem. Three different forms of the formulation are presented:

• 1.(1) distributed parameter system form derived from the conservation law;
• 2.(2) space discretized continuous lumped parameter form;
• 3.(3) space and time discretized lumped parameter form.

These formulations can be considered as the starting points for development of feedback control laws for the different control problems stated in this paper. This paper presents the feedback control problems, and does not discuss in detail the methodology of solution techniques which could be used to solve these problems. However, for the sake of completeness a brief treatment of the three forms are included in this paper to show possible ways to design the controllers.     

Mathematical modelling of the flotation deinking process

The overall flotation deinking process can be divided into four basic microprocesses:

• 1.(1) collision or capture of an (ink) particle by an air bubble
• 2.(2) adhesion of an (ink) particle to the air bubble by sliding
• 3.(3) development of a three-phase contact at the air bubble/water/particle interface, and
• 4.(4) bubble/particle stability or instability after an aggregate is formed each of these microprocesses have an associated probability that they will occur successfully in a flotation cell.

In this paper, the associated probabilities of each microprocess are employed in the development of a kinetic- or population balance-type model of the overall flotation process. The overall model contains two kinetic constants: the first, k₁ governs the overall probability of a free ink particle successfully intercepting and adhering to an air bubble; the second, k₂ is a measure of the probability that a bubble/particle aggregate pair will become unstable and split to yield a “new” free ink particle.The solution to the kinetic model is presented in terms of k₁ and k₂, which are themselves functions of system parameters such as bubble and particle physical properties (e.g., diameter, density), fluid properties (e.g., viscosity, surface tension), etc. From this solution, a definition of a theoretical flotation efficiency, as well as other system performance parameters are presented.     

Cellularity of first countable spaces

In this paper, we find subspaces of the Pixley-Roy space on the irrationals which are

• 1.(1) a first countable ccc space which does not have a σ-linked base,
• 2.(2) for each n>1, a first countable space which has a σ-n-linked base but which does not have a (σ−n+1)-linked base and
• 3.(3) a first countable space which has, for each n>1, a σ-n-linked base but which does not have a σ-centered base.

It is consistent with ¬CH that (1) and (2) have cardinality ℵ₁. (3) is constructed from a graph G on the continuum c which is not the union of countably many complete subgraphs but has no uncountable pairwise incompatible family of finite complete subgraphs (complete subgraphs A and B are compatible if there is a complete subgraph C which contains A and B).     

A Note on Old Babylonian Computational Techniques

Analysis of the errors in two Old Babylonian “algebraic” problems shows

• •that the computations were performed on a device where additive contributions were no longer identifiable once they had entered the computation;
• •that this device must have been some kind of counting board or abacus where numbers were represented as collections of calculi;
• •that units and tens were represented in distinct ways, perhaps by means of different calculi.

© 2002 Elsevier Science (USA).Eine Analyse der Rechenfehler in zwei altbabylonischen “algebraischen” Aufgaben läßt mehrere Rückschlüsse auf ein Hilfsmittel zu, das zur Durchführung von Rechnungen benutzt worden sein kann:

• •Additive Beiträge waren nach ihrer Eintragung in die Rechnung nicht länger identifizierbar.
• •Das Gerät war eine Art Rechenbrett, auf welchem Zahlen als Haufen von Rechensteinen erschienen.
• •Einer und Zehner wurden in verschiedener Weise, evtl. mittels verschiedener Rechensteine repräsentiert.

© 2002 Elsevier Science (USA).MSC subject classification: 01A17.     

Ultrafilters,independent collections,and applications to topology

The following are consequences of the main results in this paper:

• 1.(1) The number of countably compact, completely regular spaces of density κ is 2^22κ.
• 2.(2) There are 2^2κ points in U(κ) (= space of uniform ultrafilters on κ), each of which has tightness 2^κ in U(κ) and is a limit point of a countable subset of U(κ).
• 3.(3) There are 2^2κ points in U(κ), each of which has tightness 2^κ and is a weak P-point of κ¹.
• 4.(4) For each λ ⩽ κ there are at least 2^2λ · κ points in βκ, each of which has tightness 2^λ in β κ and is a weak P-point of κ¹. Moreover, under GCH there are at least 2^2λ · κ^λ such points.

     

Scheduling unit processing time jobs on an <Emphasis Type="Italic">m</Emphasis>-machine flow-shop

The sequential production of identical jobs and the flow-shop machine setting are extremely common in real-life applications. We study a scheduling problem that combines these two elements: jobs of identical processing time, with job-dependent weights, and a given common due date processed on an m-machine flow-shop. The (just-in-time) objective is to minimize the maximum earliness/tardiness cost. We introduce a polynomial time solution in both cases of (i) a non-restrictive (ie, sufficiently large) due date, and (ii) a restrictive due date (which restricts the number of early jobs).     

Analysis of performance of an iron-bath reactor using computational fluid dynamics

The performance of an iron-bath reactor has been studied using a comprehensive numerical model that combines a computational fluid dynamics approach for the gas phase and a heat and mass balance model for the bath. The model calculates:

• •coal, ore, flux and oxygen consumption;
• •post-combustion ratio (PCR);
• •heat-transfer efficiency (HTE);
• •off-gas temperature and composition;
• •heat transfer and chemical reactions between gas and iron and slag droplets; and
• •heat transfer between gas and bath, refractories and lance.

The model was validated with data reported by the Nippon Steel Corporation for a 100 t pilot plant, and the calculated and measured data are in good agreement. Modelling results showed that the dominant mechanisms of heat transfer from the gas to the bath are radiation to the slag surface and convection heat transfer to droplets.     

A model for an age-structured population with two time scales

In the modelisation of the dynamics of a sole population, an interesting issue is the influence of daily vertical migrations of the larvae on the whole dynamical process. As a first step towards getting some insight on that issue, we propose a model that describes the dynamics of an age-structured population living in an environment divided into N different spatial patches. We distinguish two time scales: at the fast time scale, we have migration dynamics and at the slow time scale, the demographic dynamics. The demographic process is described using the classical McKendrick model for each patch, and a simple matrix model including the transfer rates between patches depicts the migration process. Assuming that the migration process is conservative with respect to the total population and some additional technical assumptions, we proved in a previous work that the semigroup associated to our problem has the property of positive asynchronous exponential growth and that the characteristic elements of that asymptotic behaviour can be approximated by those of a scalar classical McKendrick model. In the present work, we develop the study of the nature of the convergence of the solutions of our problem to the solutions of the associated scalar one when the ratio between the time scales is ε (0 < ε ⪡ 1). The main result decomposes the action of the semigroup associated to our problem into three parts:

• 1.(1) the semigroup associated to a demographic scalar problem times the vector of the equilibrium distribution of the migration process;
• 2.(2) the semigroup associated to the transitory process which leads to the first part; and
• 3.(3) an operator, bounded in norm, of order ε.

     

Dynamic graph models

Research in graph theory has focused on studying the structure of graphs with the assumption that they are static. However, in many applications, the graphs that arise change with time, i.e., they are dynamic in nature. This is especially true of applications involving graph models in computer science. We present an expository study of dynamic graphs with the main driving force being practical applications. We first develop a formal classification of dynamic graphs. This taxonomy in the form of generalizations and extensions will in turn suggest new areas of application. Next, we discuss areas where dynamic graphs arise in computer science such as compilers, databases, fault-tolerance, artificial intelligence, and computer networks. Finally, we propose approaches that can be used for studying dynamic graphs. The main objective in any study of dynamic graphs should be to

• 1.(i) extend results developed for static graph theory to dynamic graphs,
• 2.(ii) study the properties that describe how a dynamic graph changes,
• 3.(iii) investigate problems and issues in dynamic graph theory that are raised by practical applications of dynamic graphs in computer science.

     

Stability and boundedness of a kind of third-order delay differential system

The paper studies equation (1.1) in two cases:

• •(i)p ≡ 0,
• •(ii)p ≠ 0.

In Case (i), the asymptotic stability of the solution x = 0 is studied; in Case (ii), the uniform boundedness and uniform ultimate boundedness of all solutions of (1.1) are proved.     

All Lobsters with Perfect Matchings are Graceful

Given a tree, T, consider one of its longest paths, P_T, not necessarily unique. We define T to be m–distant if no vertices of T are a distance greater than m away from P_T. We will show that all 3–distant graphs are graceful, providing they satisfy the following properties.

• 1.They have perfect matchings.
• 2.They can be constructed by attaching paths of length 2 to the vertices of a 1–distant tree (caterpillar), by the identification of their end vertices.

A consequence of this is that all 2–distant trees (lobsters) having perfect matchings are graceful.