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.     

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).     

Survey of mathematical programming models in air pollution management

This paper surveys the current state of the literature in management science/operations research approaches to air pollution management. After introducing suitable background we provide some of the institutional and legal framework needed to understand the continuing regulatory efforts in United States. Attention is then turned to mathematical programming models ranging from fairly simple deterministic linear programs to quite sophisticated stochastic models which have appeared in the literature dealing with these topics. This is followed by extensions reflecting some of the work we have undertaken in association with the Texas Natural Resource Conservation Commission, a regulatory agency in Texas. Application and potential use of models is the central theme of this survey. Issues for future research are presented at the end and an extensive list of publications is provided in the references at the end of the article.Principal air quality issues of local, national, and international concern are listed below in increasing order of difficulty based on the number of different types of pollutants and problems in quantification of the risks the pollutants pose:

• 1.1. Stratospheric ozone depletion: one relatively easily controllable class of trace gases - ozone depleting chemicals, or ODCs, principally chloroflurocarbons (CFCs) — with relatively well quantified risks;
• 2.2. Criteria pollutants: six common pollutants — ozone (O₃), carbon monoxide (CO), sulfur dioxide (SO₂), nitrogen dioxide (NO₂), lead (Pb), and particulate matter less than 10 microns in size (PM10) — regulated since 1970 in the U.S. and presenting relatively well quantified risks;
• 3.3. Acid precipitation: two relatively easily controllable classes of trace gases — oxides of nitrogen (NO_x) and oxides of sulfur (SO_x) with relatively well quantified risks;
• 4.4. Global warming/climate change: a few difficult to control trace gases — principally carbon dioxide (CO₂), methane (CH₄), nitrous oxide (N₂O), and CFCs — with highly uncertain risks;
• 5.5. Toxics or HAPS (hazardous air pollutants): hundreds of types of gaseous chemicals and particles with uncertain risks;
• 6.6. Somewhat dated, but nevertheless useful, is the following reference: Glossary on Air Pollution (Copenhagen, World Health Organization, 1980).

     

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.     

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.

     

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¯.

     

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.     

A decomposition of the Brownian path

The Brownian path {ω(s); 0 ⩽ s ⩽ t} is dissected and then reassembled in such a way that

• (i) the last visit γ_t at the origin, as well as the fragment {ω(s); γ_t ⩽ s ⩽ t}, are left invariant;
• (ii) on [0, γ_t], local time becomes maximum-to-date and occupation time of|R₊ becomes location of maximum; and
• (iii) the resulting process is again Brownian.

Characterizations of conditional processes are employed to establish the result. Several consequences of the latter are discussed.     

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.

     

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.     

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.

     

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.     

Transient solutions of a software model with imperfect debugging and generation of errors by two servers

In this paper, we obtain the transient solution of probabilities of error in the software, mean number of faults and the expected number of failures remaining at time t, under the assumption that the number of faults is finite, the failure rate is proportional to the number of faults present in the software at any time, debugging is imperfect and error generation will never lead the software to have infinite errors. Moreover, the software is tested by two servers with the first M errors being debugged by first server and the remaining errors (M +1 ≤n ≤N) by the second server. Also, when a failure occurs, instantaneously repair starts with the following probabilities.

• 1.(a) The fault content is reduced by one by the first (second) server with probability μ₁(μ₂),μ₂ ≥μ₁
• 2.(b) The fault content remains unchanged with probability Ψ.
• 3.(c) The fault content is increased by one by the first (second) server with probability λ₁(λ₂), λ₁ ≥λ₂ where μ₁ + Ψ + λ₁ = 1, μ₂ + Ψ + λ₂ = 1, μ₁ ⪢ Ψ ⪢ λ₁, μ₂ ⪢ Ψ ⪢ λ₂. Finally, a numerical example is presented for the transient probabilities for the number of errors in the software, mean number of faults and the expected number of failures remaining in the software.

     

Single machine scheduling with common due date assignment in a group technology environment

We consider a scheduling problem in which n jobs are grouped into F groups and are to be processed on a single machine. A machine setup time is required when the machine switches from one group of jobs to the other. All jobs have a common due date that needs to be determined. The objective is to find an optimal common due date and an optimal sequence of jobs to minimize the sum of the cost of tardy jobs and the cost related to the common due date. We consider two cases:

• 1.(i) the jobs have to be processed in groups; and
• 2.(ii) the jobs do not have to be processed in groups.

Analytical results are presented and computational algorithms are developed.     

Some results on entire functions of finite lower order

Letf(z) be an entire function of order λ and of finite lower order μ. If the zeros off(z) accumulate in the vicinity of a finite number of rays, then

1. λ is finite;
2. for every arbitrary numberk ₁>1, there existsk ₂>1 such thatT(k ₁ r,f)≤k ₂ T(r,f) for allr≥r ₀. Applying the above results, we prove that iff(z) is extremal for Yang's inequalityp=g/2, then
3. every deficient values off(z) is also its asymptotic value;
4. every asymptotic value off(z) is also its deficient value;
5. λ=μ;
6. $\sum\limits_{a \ne \infty } {\delta (a,f) \leqslant 1 - k(\mu ).} $

     

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.

     

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.