Metric Diophantine approximation and ‘absolutely friendly’ measures

Let W(ψ) denote the set of ψ-well approximable points in and let K be a compact subset of which supports a measure μ. In this short article, we show that if μ is an ‘absolutely friendly’ measure and a certain μ-volume sum converges then The result obtained is in some sense analogous to the convergence part of Khintchine’s classical theorem in the theory of metric Diophantine approximation. The class of absolutely friendly measures is a subclass of the friendly measures introduced in [2] and includes measures supported on self-similar sets satisfying the open set condition. We also obtain an upper bound result for the Hausdorff dimension of      

Sets of exact ‘logarithmic’ order in the theory of Diophantine approximation

For each real number , let denote the set of real numbers with exact order . A theorem of Güting states that for the Hausdorff dimension of is equal to . In this note we introduce the notion of exact t–logarithmic order which refines the usual definition of exact order. Our main result for the associated refined sets generalizes Güting's result to linear forms and moreover determines the Hausdorff measure at the critical exponent. In fact, the sets are shown to satisfy delicate zero-infinity laws with respect to Lebesgue and Hausdorff measures. These laws are reminiscent of those satisfied by the classical set of well approximable real numbers, for example as demonstrated by Khintchine's theorem. Received: 12 December 2000 / Published online: 25 June 2001     

Conjugacy and dynamics in Thompson’s groups

We give a unified solution to the conjugacy problem for Thompson’s groups \(F, \,T\), and \(V\). The solution uses “strand diagrams”, which are similar in spirit to braids and generalize tree-pair diagrams for elements of Thompson’s groups. Strand diagrams are closely related to piecewise-linear functions for elements of Thompson’s groups, and we use this correspondence to investigate the dynamics of elements of \(F\). Though many of the results in this paper are known, our approach is new, and it yields elegant proofs of several old results.     

Holistic oil field value management: using system dynamics for ‘intermediate level’ and ‘value-based’ modelling in the oil industry

System dynamics has been seen primarily as a strategic tool, most effectively used at the highest level of strategy to identify robust policy interventions under a wide range of scenarios. However, an alternative, complementary and powerful role is emerging. This is at an ‘intermediate level’ in organisations to coordinate and integrate policies across the value chain. It is at this level where business value, as defined by the discounted value of future free cash flow, is both created and destroyed. This paper introduces the need for ‘intermediate-level’ and ‘value-based’ modelling and emphasises the natural role of system dynamics in supporting a methodology to fulfil the need. It describes the development of an approach and its application in the oil industry to coordinate the response of people and tools within operational, financial and commercial functions across the value chain to address a variety of problems and issues.     

Chains and Trees: ‘Strong’–‘Weak’ Order in Job Scheduling

We present a summary of recent NP-hardness and polynomial time solvability results for the distinction between strong and weak precedence for chains and trees in scheduling. We distinguish between chains and proper trees which are not chains, and demonstrate that the strong-weak precedence distinction for chains is not inclusive with regards to NP-hardness, and conjecture that the same holds for strong-weak tree precedence. The objective is to show that different interpretations for chain and tree order relations in scheduling might have far reaching computational implications.     

Determinants and Symmetries in ‘Yetter—Drinfeld’ Categories

A Yetter—Drinfeld category over a Hopf algebra H with a bijective antipode, is equipped with a braiding which may be symmetric for some of its subcategories (e.g. when H is a triangular Hopf algebra). We prove that under an additional condition (which we term the u-condition) such symmetric subcategories completely resemble the category of vector spaces over a field k, with the ordinary flip map. Consequently, when Char k=0, one can define well behaving exterior algebras and non-commutative determinant functions.     

An improvement on the Tracy and Chen model ‘A generalized model for weight restrictions in DEA’

Recently Tracy and Chen presented a parametric DEA model (PDEA) to assess relative efficiency in the presence of a generalized form of linear weight restrictions. This paper proposes a modification to the PDEA model that avoids the need to resort to searching algorithms to estimate efficiency, and assures that the correct efficiency scores are obtained in a single stage using mathematical programming solvers. The results of this model and the results of Tracy and Chen's PDEA model are compared using the examples reported in their paper. The results confirm the superiority of the model proposed in this paper.     

Achieving ‘Desirable and Feasible’ Change: An Application of Soft Systems Methodology

Rational intervention in human affairs, if it is to constitute not only action but also research, so that future interventions may be made more effective, needs a well-defined methodological framework. Soft systems methodology (S.S.M.) provides one such framework. S.S.M. is doubly systemic: it is itself a learning system, and within that system it uses systems models, models of human activity systems. It accepts that such models are not models of parts of the real world, only models of ways of perceiving the real world, that is to say, models relevant to debate about ‘reality’ (one man's ‘terrorism’ is another's ‘freedom fighting’).S.S.M. is here illustrated by means of an account of a systems study of change in the information function of a sophisticated science-based company. The study is described from the point of view of a professional analyst who was helping a team of three managers carry out the study. The course of the study is described, and a separate commentary relates its activity to the seven stages of S.S.M. The study involved three cycles round stages 2–3–4–5 of S.S.M., the stages in which models of relevant human activity systems are built and compared with the real world in order to construct a rich debate about changes whose introduction in the problem situation world be both (systemically) desirable and (culturally) feasible.     

An Extension of Yuan’s Lemma and Its Applications in Optimization

We prove an extension of Yuan’s lemma to more than two matrices, as long as the set of matrices has rank at most 2. This is used to generalize the main result of Baccari and Trad (SIAM J Optim 15(2):394–408, 2005), where the classical necessary second-order optimality condition is proved, under the assumption that the set of Lagrange multipliers is a bounded line segment. We prove the result under the more general assumption that the Hessian of the Lagrangian, evaluated at the vertices of the Lagrange multiplier set, is a matrix set with at most rank 2. We apply the results to prove the classical second-order optimality condition to problems with quadratic constraints and without constant rank of the Jacobian matrix.     

Existence of a solution ‘in the large’ for the 3D large-scale ocean dynamics equations

For the 3D system of equations describing large-scale ocean dynamics in the Cartesian coordinate system existence and uniqueness of a solution on an arbitrary time interval $[0,T]$ is proved and the norm $6{\stackrel{?}{\mathbf{u}}}_{x}6$ is shown to be continuous in time on $[0,T]$. To cite this article: G.M. Kobelkov, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

The concept ‘model’ and its potential role in mathematics education

Under the general term ‘mathematical models’ is now subsumed a sufficiently diversified collection of distinct types of models to merit categorization and an attempt at a unified definition. In this study a broad ‘working definition’ for the concept ‘model’ is proposed; basic types of models are identified and illustrated, and the impact of conscious instruction of models on mathematics education is analysed.     

Consistencies and inconsistencies in students' solutions to algebraic ‘single-value’ inequalities

This paper describes students’ solutions to a commonly taught and not commonly taught inequality. The findings showed students’ difficulties. Participants implicitly and explicitly exhibited two intuitive beliefs: inequalities must result in inequalities and solving inequalities and equations are the same process. Following the analysis of students’ written solutions, individual interviews were conducted that gave a better insight into their reasoning and provided some ideas for teaching. The concluding section of the paper offers relevant educational implications.