On the growth of a polymer layer around a catalytic particle: a free boundary problem

The problem studied is a substantial part of a larger mathematical model describing the heterogeneous Ziegler-Natta polymerization process of a gaseous monomer. Here we concentrate on the growth of a polymer layer around a spherical catalytic particle (the whole process taking place in an agglomerate of such particles). The velocity field of the growing layer is determined the absorption rate of the monomer at the catalyst surface. The monomer diffuses through the expanding layer. Its concentration at the outer surface (the free boundary) is supposed to be a known function of time. Global in time existence and uniqueness of a regular solution are proved. Received May 12, 1996     

On the reversibility of queueing networks

This paper relates the reversibility of certain discrete state Markovian queueing networks — the class of quasi-reversible networks — to the reversibility of the underlying switching process. Quasi-reversible networks are characterized by a product form equilibrium state distribution.When the state can be represented by customer totals at each node, the reversibility of the state process is equivalent to the reversibility of the switching process. More complicated quasi-reversible networks require additional conditions, to ensure the reversibility of the network state process.     

Modeling a class of state-dependent routing in flexible manufacturing systems

We develop a closed queueing network model for flexible manufacturing systems (FMSs), where parts routing follows a probabilistic shortest-queue (PSQ) scheme, i.e. parts are routed to the shortest queue (or the most empty station) with the highest probability. We allow limited local buffer at each work station. We prove that with the PSQ routing, the Markovian queue-length process satisfies time reversibility and has a product-form equilibrium distribution. An algorithm is developed to compute the solutions to the model. The model can be used as a performance evaluation tool to study FMSs.     

三状态隐马氏模型观测过程的可逆性(英文)

对于连续时间和离散时间三状态隐马氏模型,给出了观测过程直到三维的似然函数流的显式表达.作为一个应用,证明了观测过程可逆性的充分必要条件.     

Systems of first-order chemical reactions

We consider the classical model in chemical kinetics of a system of n species in which each species is converted to every other species by a first-order reaction. Solutions to the initial-value problem are given in matrix form and the properties of the n × n matrix K representing the system are analysed. For arbitrary (i.e. non-negative) values of the first-order rate constants, zero is an eigenvalue, and the other eigenvalues are complex with negative real parts. Thus, in this case the system generally oscillates to equilibrium. However, if the principle of microscopic reversibility is applied, and if each species is converted directly to every other species, then the system cannot oscillate but must converge “exponentially” to equilibrium. We discuss when K is diagonalizable, and we calculate a bound for the eigenvalues of K. Special forms of K, corresponding to special systems of reactions, are also examined; these include reactions in the configuration of a “chain”, a “cycle”, a “node” and reactions comprising combinations of these. We find again that if the principle of microscopic reversibility is rigorously applied then oscillations cannot take place, but that if this principle is not applied then oscillations may take place. The system of rate equations considered can be used to model various chemical, physical and biological phenomena.     

First passage of time-reversible spectrally negative Markov additive processes

We study the first passage process of a spectrally negative Markov additive process (MAP). The focus is on the background Markov chain at the times of the first passage. This process is a Markov chain itself with a transition rate matrix Λ. Assuming time reversibility, we show that all the eigenvalues of Λ are real, with algebraic and geometric multiplicities being the same, which allows us to identify the Jordan normal form of Λ. Furthermore, this fact simplifies the analysis of fluctuations of a MAP. We provide an illustrative example and show that our findings greatly reduce the computational efforts required to obtain Λ in the time-reversible case.     

Dynamics of a rate equation describing cluster-size evolution

In this paper, we show dynamics of Smoluchowski's rate equation which has been widely applied to studies of aggregation processes (i.e., the evolution of cluster-size distribution) in physics. We introduce dissociation in the rate equation while dissociation is neglected in previous works. We prove the positiveness of solutions of the equation, which is a basic guarantee for the effectiveness of the model since the possibility that some solution may be negative is excluded. For the case of cluster coalesce without dissociation, we show both the equilibrium uniqueness and the equilibrium stability under the condition that the monomer deposition stops. For the case that clusters evolve with dissociation and there is no monomer deposition, we show the equilibrium uniqueness and prove the equilibrium stability if the maximum cluster size is not larger than three while we show the equilibrium stability by numerical simulations if the maximum size is larger than three.     

Symmetric implicational method of fuzzy reasoning

Fuzzy reasoning should take into account the factors of both the logic system and the reasoning model, thus a new fuzzy reasoning method called the symmetric implicational method is proposed, which contains the full implication inference method as its particular case. The previous full implication inference principles are improved, and unified forms of the new method are respectively established for FMP (fuzzy modus ponens) and FMT (fuzzy modus tollens) to let different fuzzy implications be used under the same way. Furthermore, reversibility properties of the new method are analyzed from some conditions that many fuzzy implications satisfy, and it is found that its reversibility properties seem fine. Lastly, the more general α-symmetric implicational method is put forward, and its unified forms are achieved.     

Multiphase saturation curves of the oxoglutarate carrier: A mathematical model

Experimental data on binding and initial rate of the oxoglutarate-malate exchanger of rat-heart mitochondria exhibit a multiphase shape characterized by 3 intermediary plateaus in the saturation curve of external oxoglutarate [1]. In this paper, we present a quantitative analysis of these data on the basis of a phenomenological model which consists of a sum of Hill equations. This model leads to the assumption that there exist five iso-carriers of different sizes: one monomer exhibiting a Michaelian behaviour and four non-Michaelian oligomers. The monomer possesses a high half-saturation constant K₁ = 14.05 μM and the highest rate constant k₁ = 200s⁻¹, whereas the dimer exhibits the lowest half-saturation constant K₂ = 71 nM, but the lowest rate constant k₂ = 0.91s⁻¹. The half-saturation constants of the other oligomers monotonously increase with their complexity. The relations observer among the amounts of the various iso-carriers suggests that the oligomers are in equilibrium with the monomer.     

Modelling the semi-batch vinyl acetate emulsion polymerization in a real-life industrial reactor

A detailed dynamic model of the polymerization reaction of vinyl acetate in a real-life industrial reactor is presented. With the recipe and the operating procedures observed in the factory as inputs, the model predicts with reasonable accuracy the final conversion, the average particle diameter, the solid content and the viscosity. The manual and semi-manual operations, and the decision process followed by the operator, are also modelled in order to replicate closely the process carried out in the factory. The model provides further insight into the reaction kinetics and allows us to make knowledge-based decisions. The model is intended to be used for the optimization of the policy of adding monomer in order to reduce the batch time.     

On the order of tandem queues

We consider the optimal order of servers in a tandem queueing system withm stages, an unlimited supply of customers in front of the first stage, and a service buffer of size 1 but no intermediate storage buffers between the first and second stages. Service times depend on the servers but not the customers, and the blocking mechanism at the first two stages is manufacturing blocking. Using a new characterization of reversed hazard rate order, we show that if the service times for two servers are comparable in the reversed hazard rate sense, then the departure process is stochastically earlier if the slower server is first and the faster server is second than if the reverse is true. This strengthens earlier results that considered individual departure times marginally. We show similar results for the last two stages and for other blocking mechanisms. We also show that although individual departure times for a system with servers in a given order are stochastically identical to those when the order of servers is reversed, this reversibility property does not hold for the entire departure process.     

模糊推理三Ⅰ算法的还原性

模糊推理算法的还原性是判断蕴涵算子与推理方法配合效果的一个重要标准,只有蕴涵算子与推理方法搭配适当,才能使模糊推理有一个好的效果。本文对模糊推理三I算法具备还原性的条件进行了研究。首先,当与蕴涵算子相伴随的三角模为连续三角模时,给出了FM P问题三I算法具有还原性的充要条件;其次,当蕴涵算子为连续的正则蕴涵算子时,给出了FM T问题的三I算法具有还原性的充要条件;最后,当正则蕴涵算子关于补运算满足对合律时,给出了FM T问题三I算法满足还原性的一个充分条件。     

Understanding Qualitative Calculus: A Structural Synthesis of Learning Research

More than a decade of research and innovation in using computer-based graphing and simulation environments has encouraged many of us in the research community to believe important dimensions of calculus-related reasoning can be successfully understood by young learners. This paper attempts to address what kinds of calculus-related insights seem to typify this early form of calculus reasoning. The phrase “qualitative calculus” is introduced to frame the analysis of this “other” calculus. The learning of qualitative calculus is the focus of the synthesis. The central claim is that qualitative calculus is a cognitive structure in its own right and that qualitative calculus develops or evolves in ways that seem to fit with important general features of Piaget's analyses of the development of operational thought. In particular, the intensification of rate and two kinds of reversibility between what are called “how much” (amount) and “how fast” (rate) quantities are what interactively, and collectively,characterize and help to define understanding qualitative calculus. Although sharing a family resemblance with traditional expectations of what it might mean to learn calculus, qualitative calculus does not build from ratio- or proportion-based ideas of slope as they are typically associated with defining rate. The paper does close, however, with a discussion of how understanding qualitative calculus can support and link to the rate-related literature of slope, ratio and proportion. Additionally, curricular connections and implications are discussed throughout to help illustrate and explore the significance of learning qualitative calculus. This revised version was published online in July 2006 with corrections to the Cover Date.     

A new multifractal network traffic model

In this paper, a new multifractal traffic model to capture the multifractal nature of modern Internet traffic was developed. Employing the algorithm of network traffic analysis (binomial inverse cascade process) to analyze the multifractal feature of traffic data and adopting the algorithm of network traffic synthesis (binomial cascade process) to model the network traffic, this approach gave an easy and efficient way to infer the model parameters from the measured traffic traces. Moreover, the traffic was simulated and analyzed using obtained parameters. It was found that the simulated traffic data were in a close fit to the real trace statistics. The analysis results showed that this model could capture the real network traffic very well.     

The Equilibrium Behavior of Reversible Coagulation-Fragmentation Processes

The coagulation-fragmentation process models the stochastic evolution of a population of N particles distributed into groups of different sizes that coagulate and fragment at given rates. The process arises in a variety of contexts and has been intensively studied for a long time. As a result, different approximations to the model were suggested. Our paper deals with the exact model which is viewed as a time-homogeneous interacting particle system on the state space _N, the set of all partitions of N. We obtain the stationary distribution (invariant measure) on _N for the whole class of reversible coagulation-fragmentation processes, and derive explicit expressions for important functionals of this measure, in particular, the expected numbers of groups of all sizes at the steady state. We also establish a characterization of the transition rates that guarantee the reversibility of the process. Finally, we make a comparative study of our exact solution and the approximation given by the steady-state solution of the coagulation-fragmentation integral equation, which is known in the literature. We show that in some cases the latter approximation can considerably deviate from the exact solution.     

Time dependence of the strength of polyvinylene carbonate and its low-molecular-weight analog

The temperature-time dependences of the strength of polyvinylene carbonate (PVCa) and its solid hydrogenated monomeric analog, ethylene carbonate (EC), have been investigated. The parameters of these relations have been determined graphically and by calculation. The fracture energy barriers of the polymer and the monomer are compared and the sublimation energy of the monomer is determined. It is concluded that the polymer fracture process is directly related to the rupture of both chemical and intermolecular bonds.Mekhanika Polimerov, Vol. 3, No. 6, pp. 1054–1059, 1967     

Towards an integrated theory of mathematics conceptual learning and instructional design: The Learning Through Activity theoretical framework

We discuss the theoretical framework of the Learning Through Activity research program. The framework includes an elaboration of the construct of mathematical concept, an elaboration of Piaget’s reflective abstraction for the purpose of mathematics pedagogy, further development of a distinction between two stages of conceptual learning, and a typology of different reverse concepts. The framework also involves instructional design principles built on those constructs, including steps for the design of task sequences, development of guided reinvention, and ways of promoting reversibility of concepts. This article represents both a synthesis of prior work and additions to it.     

State matrix recursion method and monomer–dimer problem

The exact enumeration of pure dimer coverings on the square lattice was obtained by Kasteleyn, Temperley and Fisher in 1961. In this paper, we consider the monomer–dimer covering problem (allowing multiple monomers) which is an outstanding unsolved problem in lattice statistics. We have developed the state matrix recursion method that allows us to compute the number of monomer–dimer coverings and to know the partition function with monomer and dimer activities. This method proceeds with a recurrence relation of so-called state matrices of large size. The enumeration problem of pure dimer coverings and dimer coverings with single boundary monomer is revisited in partition function forms. We also provide the number of dimer coverings with multiple vacant sites. The related Hosoya index and the asymptotic behavior of its growth rate are considered. Lastly, we apply this method to the enumeration study of domino tilings of Aztec diamonds and more generalized regions, so-called Aztec octagons and multi-deficient Aztec octagons.     

An empirically-based trajectory for fostering abstraction of equivalent-fraction concepts: A study of the Learning Through Activity research program

Promoting deep understanding of equivalent-fractions has proved problematic. Using a one-on-one teaching experiment, we investigated the development of an increasingly sophisticated, sequentially organized set of abstractions for equivalent fractions. The article describes the initial hypothetical learning trajectory (HLT) which built on the concept of recursive partitioning (anticipation of the results of taking a unit fraction of a unit fraction), analysis of the empirical study, conclusions, and the resulting revised HLT (based on the conclusions). Whereas recursive partitioning proved to provide a strong conceptual foundation, the analysis revealed a need for more effective ways of promoting reversibility of concepts. The revised HLT reflects an approach to promoting reversibility derived from the empirical and theoretical work of the researchers.     

Polymerization of vinyl monomers

A systematic study of the kinetics of photosensitized polymerization of vinyl monomers with the system Dye (erythrosin or acriflavine)-reducing agent (ascorbic acid), in the presence of oxygen and under buffered conditions (pH 6) in aqueous solution has been made. The determinations of the monomer disappearance, dye disappearance and of chainlengths of the polymers obtained against the different variables like light intensity, concentrations of monomer, dye, ascorbic acid, etc., have been made. A kinetic mechanism proposed for the overall reaction is discussed in the light of the experimental results and certain rate constants have also been evaluated.