Some results on the nonstationary ideal

The strength of precipitousness, presaturatedness and saturatedness of NS_κ and NS _κ ^λ is studied. In particular, it is shown that:

1. The exact strength of “ $NS_{\mu ^ + }^\lambda $ for a regularμ > max(λ, ?₁)” is a (ω,μ)-repeat point.
2. The exact strength of “NS_κ is presaturated over inaccessible κ” is an up-repeat point.
3. “NS_κ is saturated over inaccessible κ” implies an inner model with ?αo(α) =α ⁺⁺.

     

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.     

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.

     

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.     

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.

     

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.     

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).     

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.

     

Gradients of inner functions in strictly pseudoconvex domains

We investigate the unbalanced ordinary partition relations of the form λ → (λ, α)² for various values of the cardinal λ and the ordinal α. For example, we show that for every infinite cardinal κ, the existence of a κ+-Suslin tree implies κ⁺ ? (κ⁺, log _κ (κ⁺) + 2)². The consistency of the positive partition relation b → (b, α)2 for all α < ω₁ for the bounding number b is also established from large cardinals.     

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.

     

Cichoń’s diagram for uncountable cardinals

We develop a version of Cichoń’s diagram for cardinal invariants on the generalized Cantor space 2 ^κ or the generalized Baire space κ ^κ , where κ is an uncountable regular cardinal. For strongly inaccessible κ, many of the ZFC-results about the order relationship of the cardinal invariants which hold for ω generalize; for example, we obtain a natural generalization of the Bartoszyński–Raisonnier–Stern Theorem. We also prove a number of independence results, both with < κ-support iterations and κ-support iterations and products, showing that we consistently have strict inequality between some of the cardinal invariants.     

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¯.

     

On the existence of free topological groups

Given a Tychonoff space X and classes U and V of topological groups, we say that a topological group G = G(X, U, V) is a free (U,V)-group over X if (a) X is a subspace of G, (b) G ϵ U, and (c) every continuous f: X → H with H ϵ V extends uniquely to a continuous homomorphism f̄: G→H. For certain classes U and V, we consider the question of the existence of free (U,V)- groups. Our principal results are the following. Let PA and CA denote, respectively, the class ofpseudocompact Abelian groups and the class of compact Abelian groups. Then

• 1.(a) there is a free (PA,PA)-group over X iff; X=Ø and
• 2.(b) there is for each X a free (PA,CA)-group over X in which X is closed.

     

Free energy and solutions of the Vlasov-Poisson-Fokker-Planck system: external potential and confinement (Large time behavior and steady states)

This paper is devoted to the characterization of external electrostatic potentials for which the Vlasov-Poisson-Fokker-Planck system satisfies one of the following properties:

• (i) the system admits stationary solutions,
• (ii) any solution to the evolution problem converges to a stationary solution, or, equivalently, no mass vanishes for large times,
• (iii) the free energy is bounded from below, We give conditions under which these different notions of confinement are equivalent.

     

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.