Clusters of extremes: modeling and examples

We study clusters of threshold exceedances caused by dependence in time series. The clusters are defined as conglomerates containing consecutive threshold exceedances of the series separated by return intervals with consecutive non-exceedances. We derive asymptotic distributions of the cluster and inter-cluster sizes for processes with the extremal index equal to zero, the asymptotic expectation of the inter-cluster size and an exponential rate of convergence of the distribution tail of the return interval between clusters to the stable distribution tail. Distributions of the cluster and inter-cluster sizes of ARMAX, MM and AR(1) processes are obtained.     

Controllability of linear systems with positive controls: Geometric considerations

The problem of 0-controllability of linear systems with control restraint set Ω such that 0 ? int Ω is investigated.     

LCNO Sturm-Liouville problems: computational difficulties and examples

Summary. In this paper we give a new proof of a theorem of Bailey, Everitt and Zettl on the convergence of truncated approximations to limit circle (LC) Sturm-Liouville problems, both non-oscillatory (LCNO) and oscillatory (LCO). The proof gives an error bound not previously available. We prove a theorem on the conditioning of LCNO problems with respect to non-Friedrichs boundary conditions. We present numerical experiments which illustrate how the theorem successfully predicts the conditioning of LCNO problems. Our work may also explain the performance of the code SLEIGN2 recently reported by Bailey et al. [1] on a number of problems. Received January 22, 1993 / Revised version received April 22, 1994     

Solving algebra problems with interactive diagrams: Demonstration and construction of examples

We investigated how students use the representation of data in a given example appearing in an interactive diagram (ID) and how they create additional examples with the ID. Students who worked with the ID that offered limited representations and tools (illustrating ID) looked for ways to bypass the designed constraints: they changed the representation of the data or built new representations, but did not create new examples in any form. Working with another type of diagram (narrating ID), students treated the specific given example as a generic example and were able to reach a generalization in a process of systematic change and comparison. The variety of tools and representations offered in the design of the third type of ID (elaborating ID) yielded diverse strategies: students constructed, without guidance, various examples and initiated further inquiry that sometimes resulted in the systematic construction of examples, although the ID provided no tools for systematic change.     

Word maps and word maps with constants of simple algebraic groups

In the present paper, we consider word maps w: G ^m → G and word maps with constants w _Σ: G ^m → G of a simple algebraic group G, where w is a nontrivial word in the free group F _m of rank m, w _Σ = w ₁ σ ₁ w ₂ ··· w _r σ _r w _{r + 1}, w ₁, …, w _{r + 1} ∈ F _m , w ₂, …, w _r ≠ 1, Σ = {σ ₁, …, σ _r | σ _i ∈ G Z(G)}. We present results on the images of such maps, in particular, we prove a theorem on the dominance of “general” word maps with constants, which can be viewed as an analogue of a well-known theorem of Borel on the dominance of genuine word maps. Besides, we establish a relationship between the existence of unipotents in the image of a word map and the structure of the representation variety R(Γ_w, G) of the group Γ_w = F _m /<w>.     

Left-divisibility and word problems in single relation monoids

LetM be a monoid presented by <Σ;u=v> whereu andv are words on the finite alphabet Σ./ Deciding the world problem forM is still an open question, though it seems decidable and partial results exist. The remaining cases to solve are presentations of the form <a, b; bva=aua>,u, v∈{a, b}^⋆ The word problem is then closely related to the left-divisibility problem, as shown by S.I. Adjan who introduced a procedure that “almost” allows to decide the problem. The main contribution, due to Adjan and Oganesjan, states that ifbva is an unbordered factor ofaua then the word problem is deciable. We restrict Adjan's procedure to study the case whenbva is unbordered, which allows us to extend Adjan and Oganesjan's theorem. More specifically, we show (Proposition 4.24) that the word problem is decidable for presentations <a, b; bva=aua> with the only following condition: Inbva, the leftmost train ofb is strictly longer than the others. The following corollary naturally holds: The word problem is decidable for presentations of the form <a, b; b ^m a ⁿ =aua>,u∈{a, b}^⋆,m, n>0     

Children’s use of realistic considerations in problem solving: some English evidence

This is the second of two papers exploring children’s responses to an extended version of a division-with-remainder problem intended to elicit general rather than particular realistic considerations during mathematical problem solving. Responses to two problems are analyzed. The first is a ‘realistically’ contextualised item drawn from national tests in England whose ambiguities have been previously discussed (Cooper, 1992); the second is a version of this problem revised to encourage a wider range of realistic responses. In Cooper and Harries (2002), the responses of children at the end of their first year of secondary schooling were analyzed. Here the responses of children at the end of their primary schooling are analyzed and compared with the previous results. It is shown that many children, given suitable encouragement, are willing and able to enter into an extended form of realistic thinking during problem solving, although the original test item renders this invisible.     

Optimum production and inspection modeling with minimal repair and rework considerations

This paper extends an integrated model of economic production quantity (EPQ) and preventive maintenance (PM) to incorporate possibilities of minimal repair and rework. Our model determines simultaneously the optimal number of inspections, the inspection interval, the EPQ, and the PM level. Numerical examples of Weibull shock models are given to show that allowing minimal repair and rework will raise the expected profit. Our analyses demonstrate that both minimal repair and rework significantly influence the optimal policy.     

Algebra word problem solving approaches in a chemistry context: Equation worked examples versus text editing

Text editing directs students’ attention to the problem structure as they classify whether the texts of word problems contain sufficient, missing or irrelevant information for working out a solution. Equation worked examples emphasize the formation of a coherent problem structure to generate a solution. Its focus is on the construction of three equation steps each of which comprises essential units of relevant information. In an experiment, students were randomly assigned to either text editing or equation worked examples condition in a regular classroom setting to learn how to solve algebra word problems in a chemistry context. The equation worked examples group outperformed the text editing group for molarity problems, which were more difficult than dilution problems. Empirical evidence supports the theoretical rationale in using equation worked examples to facilitate students’ construction of a coherent problem structure so as to develop problem skills and expertise to solve molarity problems.     

Conservative parabolic problems: Nondegenerated theory and degenerated examples from population dynamics

We consider partial differential equations of drift‐diffusion type in the unit interval, supplemented by either 2 conservation laws or by a conservation law and a further boundary condition. We treat 2 different cases: (1) uniform parabolic problems and (ii) degenerated problems at the boundaries. The former can be treated in a very general and complete way, much as the traditional boundary value problems. The latter, however, brings new issues, and we restrict our study to a class of forward Kolmogorov equations that arise naturally when the corresponding stochastic process has either 1 or 2 absorbing boundaries. These equations are treated by means of a uniform parabolic regularisation, which then yields a measure solution in the vanishing regularisation limit. Two prototypical problems from population dynamics are treated in detail. For these problems, we show that the structure of measure‐valued solutions is such that they are absolutely continuous in the interior. However, they will also include Dirac masses at the degenerated boundaries, which appear, irrespective of the regularity of the initial data, at time t=0⁺. The time evolution of these singular masses is also explicitly described and, as a by‐product, uniqueness of this measure solution is obtained.     

Two-stage stochastic equilibrium problems with equilibrium constraints: modeling and numerical schemes

This article presents a two-stage stochastic equilibrium problem with equilibrium constraints (SEPEC) model. Some source problems which motivate the model are discussed. Monte Carlo sampling method is applied to solve the SEPEC. Convergence analysis on the statistical estimators of Nash equilibria and Nash stationary points are presented.     

Nearest feasible paths in optimal control problems: Theory,examples, and counterexamples

Many infinite-horizon optimal control problems in management science and economics have optimal paths that approach some stationary level. Often, this path has the property of being the nearest feasible path to the stationary equilibrium. This paper obtains a simple multiplicative characterization for a single-state single-control problem to have this property. By using Green's theorem it is shown that the property is observed as long as the stationary level is sustainable by a feasible control. If not, the property is, in general, shown to be false. The paper concludes with an important theorem which states that even in the case of multiple equilibria, the optimal path is a nearest feasible path to one of them.The author thanks J. Bona for his helpful comments.     

Two evasion problems in a plane with separate controls and non-convex terminal sets

Two different pursuit-evasion games are considered from the evader's point of view. The phase space is a plane, each of the two players controlling the motion of a point only along its own coordinate. The terminal sets are not convex; in the first problem, the set is an arc of a circle, in the second, the union of tow segments. In both games evasion cannot the achieved by means of programmed controls, but it can be achieved using feedback control. However, the strategies, which are continuous functions of the phase vector, have different properties in each problem. In the first, they cannot guarantee evasion (which is typical for the linear-convex case as well), but in the second they can (which is impossible in linear-convex games with a fixed final time). Verification that evasion is unachievable using such strategies reduces here to proving the solvability of a certain initial-value problem for an advanced differential equation, to which the Schauder principle is applicable.     

The maximum principle in optimal control problems with concentrated and distributed delays in controls

In the present work there has been posed and studied a general nonlinear optimal problem and a quasi-linear optimal problem with fixed time and free right end. It contains absolutely continuous monotone delays in phase coordinates and absolutely continuous monotone and distributed delays in controls. For these problems the necessary and, respectively, sufficient conditions of optimality in the form of the maximum principle have been proved.     

Financial systems: a few theoretical and algebraic considerations for their modeling

In this paper I suggest that observable entities, usually named ‘financial systems’, may be related to the general conceptual framework of systems theory. Starting from the requisite properties of a (general) system, I derive a strong and operational concept for specific financial systems (Section 1). Then a general modeling procedure is proposed, mainly based upon graph theory (with an additional and complementary use of linear systems analysis), through which it is possible to establish the general static and almost dynamic properties of these specific systems and their implications for financial analysis itself (Section 2). A numerical example (Section 3) illustrates most of the concepts and ideas introduced throughout the paper.     

Optimality conditions for extremal controls in bilinear and quadratic problems

We consider the optimization problem for a bilinear functional with respect to a linear phase system with a modularly constrained control. On the base of exact formulas for the functional increment we establish sufficient optimality conditions for extremal controls. These conditions are stated as inequalities for one-dimensional functions on a time interval. They supplement the maximum principle, keeping the implementation complexity at the same level. The optimization problem for a quadratic functional is reduced to the bilinear case with the help of the matrix conjugate function.