Symplectic geometry,minors and graph Laplacians

In my talk, I will present some works done in the nineties on Laplacians on graphs: from eigenvalue problems to inverse problem for resistor networks. I will focus on the motivations and the main results as well as on the main ideas:

• •A differential topology point of view on the minor relation: a nice stratification associated to a finite graph Γ whose strata are associated to the minors of Γ
• •“Discrete” (graphs) versus “continuous” (Riemannian manifolds)
• •Stability of spectra with respect to singular limits: a finite dimensional theory of operators with domains (Von Neumann theory).

The link with topology will appear in some results about my graph parameter μ, in particular the planarity and the linkless embedding properties.     

On degree sequences of undirected,directed, and bidirected graphs

Bidirected graphs generalize directed and undirected graphs in that edges are oriented locally at every node. The natural notion of the degree of a node that takes into account (local) orientations is that of net-degree. In this paper, we extend the following four topics from (un)directed graphs to bidirected graphs:

• –Erdős–Gallai-type results: characterization of net-degree sequences,
• –Havel–Hakimi-type results: complete sets of degree-preserving operations,
• –Extremal degree sequences: characterization of uniquely realizable sequences, and
• –Enumerative aspects: counting formulas for net-degree sequences.

To underline the similarities and differences to their (un)directed counterparts, we briefly survey the undirected setting and we give a thorough account for digraphs with an emphasis on the discrete geometry of degree sequences. In particular, we determine the tight and uniquely realizable degree sequences for directed graphs.     

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.     

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.

     

Eccentric graphs

For any graph G we define the eccentric graph G_e on the same set of vertices, by joining two vertices in G_e if and only if one of the vertices has maximum possible distance from the other. The following results are given in this paper:

• (1)A few general properties of eccentric graphs.
• (2)A characterization of graphs G with G_e = K_p and with G_e = pK₂.
• (3)A solution of the equation G_e = G¯.

     

Social theory and system dynamics practice

This paper explores the social theories implicit in system dynamics (SD) practice. Groupings of SD practice are observed in different parts of a framework for studying social theories. Most are seen to be located within `functionalist sociology'. To account for the remainder, two new forms of practice are discussed, each related to a different paradigm. Three competing conclusions are then offered:

• 1.The implicit assumption that SD is grounded in functionalist sociology is correct and should be made explicit.
• 2.Forrester's ideas operate at the level of method not social theory so SD, though not wedded to a particular social theoretic paradigm, can be re-crafted for use within different paradigms.
• 3.SD is consistent with social theories which dissolve the individual/society divide by taking a dialectical, or feedback, stance. It can therefore bring a formal modelling approach to the `agency/structure' debate within social theory and so bring SD into the heart of social science.

The last conclusion is strongly recommended.     

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.

     

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.     

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.

     

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.     

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).     

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.     

Improved OR education through the use of spreadsheet models

The first and major part of this presentation shows how OR education may be improved through the use of spreadsheets, with several examples:

• •in decision theory, where the drawing capabilities of Excel are used to represent decision trees and where window switching and macros allow the educator to animate his presentation;
• •in linear programming, where the solver is used (remarks are made relating to the adequacy of the solver's solutions), and where the spreadsheet's graphing capabilities are used both to represent the feasible set and the objective function, and to interactively animate the objective function or constraint movements;
• •in dynamic programming, deterministic or stochastic, where it is shown that several standard spreadsheet functions facilitate the development of specialized models;
• •in simulation, where it will be seen how stochastic simulation may be facilitated through the use of an add-in, and how a decision support system may thus be built from scratch in less than 10 minutes.

A creative use of the spreadsheet in business OR models is presented in the second part of this presentation, with an example in process simulation: airport luggage traffic simulation. A short third part presents the possibilities offered by spreadsheet add-ins.     

Methods for routing with time windows

Consider a set of trips where each trip is specified a priori by a place of origin, a destination, a duration, a cost and a time interval within which the trip must begin. The trips may include visits to one or more specific points. Our problem is to determine the number of vehicles required together with their routes and schedules, so that each trip begins within his given time interval, while the fixed costs related to the number of vehicles, and the travel costs between trips are minimized. The problem is a generalization of the m-travelling salesman problem.We compare numerical results for 3 algorithms developed by our research team:

• 1.(1) Column generation on a set partitioning problem solved by simplex and branch-and-bound; columns are generated by a shortest path algorithm with time constraints on the nodes.
• 2.(2) Adaptation of the Carpaneto-Toth algorithm for the asymmetric travelling salesman problem: solution of network problems by relaxing scheduling constraints, and branch-and-bound on flow variables.
• 3.(3) Solution of network problems by relaxing scheduling constraints and branch-and-bound based on dividing the time windows.

     

Determining the optimal place and capacity of spare parts warehouses in the construction industry

In the construction industry, places, capacities and levels of demand in basic spare parts are changing in relatively short periods of time. This creates an optimization problem of the following form.We are given the following:o

1. (i)The location and the level of demand for each basic spare part in each work site for a specific time period.
2. (ii)The places and the levels of demand can be altered.
3. (iii)There are more than one supplier of each part geografically distributed.
4. (iv)The number of basic equipment spare parts.
5. (v)The transportation cost per load of spare parts.
6. (vi)The purchasing and functioning cost of the various air houses used as warehouses of spare parts.

In this paper we present an algorithm which determines the place, capacity and number of supply points to minimize the total cost of the supply system under the constraints (i) to (vi) above.     

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.