An Approach to Providing Mathematical Annotation in Plots

Abstract

A simple method for providing mathematical annotation of plots produced with the R environment is described. Although the implementation is specific to R, a similar method could be used in any environment which uses an expression-based command interface and provides a basic quoting mechanism.     

A Priori knowledge contextualised and Benacerraf’s dilemma

In this article, I discuss Hawthorne’s contextualist solution to Benacerraf’s dilemma. He wants to find a satisfactory epistemology to go with realist ontology, namely with causally inaccessible mathematical and modal entities. I claim that he is unsuccessful. The contextualist theories of knowledge attributions were primarily developed as a response to the skeptical argument based on the deductive closure principle. Hawthorne uses the same strategy in his attempt to solve the epistemologist puzzle facing the proponents of mathematical and modal realism, but this problem is of a different nature than the skeptical one. The contextualist theory of knowledge attributions cannot help us with the question about the nature of mathematical and modal reality and how they can be known. I further argue that Hawthorne’s account does not say anything about a priori status of mathematical and modal knowledge. Later, Hawthorne adds to his account an implausible claim that in some contexts a gettierized belief counts as knowledge.     

Constraint propagation,relational arithmetic in AI systems and mathematical programs

This paper explores the interrelationships between methods developed in mathematical programming to discover the structure of constraint (feasibility) sets and constraint propagation over networks used by some AI systems to perform inferences about quantities. It is shown that some constraint set problems in mathematical programming are equivalent to inferencing problems for constraint networks with interval labels. This makes the inference and query capabilities associated with AI systems that use logic programming, directly accessible to mathematical programming systems. On the other hand, traditional and newer methods which mathematical programming uses to obtain information about its associated feasibility set can be used to determine the propagation of constraints in a network of nodes of an AI system. When viewed from this point of view, AI problems can access additional mathematical programming analytical tools including new ways to incorporate qualitative data into constraint sets via interval and fuzzy arithmetic.This work was partially supported by the Industrial Consortium to Develop an Intelligent Mathematical Programming System — Amoco Oil Company, General Research Corporation, Ketron Management Science, Shell Oil Company, MathPro, and US West Advanced Technologies.     

Evaluation of German parties and coalitions by methods of the mathematical theory of democracy

The paper continues author’s studies on applications of the mathematical theory of democracy. Five German leading parties and their coalitions are evaluated with indices of popularity and universality to show how well they represent the electorate. The evaluation goes beyond the information on election results and uses an operational representation of party manifestos.     

The Structure of Cities

We give an account of recent work, where the spatial structure of cities is analysed as an equilibrium problem between the various uses of land. The mathematical theory of optimal transportation plays a crucial role in the proof.     

AgentGeom: a multiagent system for pedagogical support in geometric proof problems

This paper aims, first, to describe the fundamental characteristics and workings of the AgentGeom artificial tutorial system, which is designed to help students develop knowledge and skills related to problem solving, mathematical proof in geometry, and the use of mathematical language. Following this, we indicate the manner in which a secondary school student can appropriate these abilities through interactions with the system. Our system uses strategic messages of the agent tutor in an argumentative process that collaborates with a student in the construction of a proof.     

Effective metastability of Halpern iterates in CAT(0) spaces

This paper provides an effective uniform rate of metastability (in the sense of Tao) on the strong convergence of Halpern iterations of nonexpansive mappings in CAT(0) spaces. The extraction of this rate from an ineffective proof due to Saejung is an instance of the general proof mining program which uses tools from mathematical logic to uncover hidden computational content from proofs. This methodology is applied here for the first time to a proof that uses Banach limits and hence makes a substantial reference to the axiom of choice.     

An investigation of pedagogical techniques in Descartes' La géométrie

Much research has been conducted about the philosophy and mathematical writings of René Descartes, but that which focuses on pedagogy does so in a holistic manner. The present study uses a systematic approach to identify pedagogical techniques within each sentence of Descartes' La géométrie. Next, the study provides an analysis of La géométrie based on the techniques identified, their frequencies, and patterns of use within the text. The results of this analysis indicate that Descartes placed a high value on the use of demonstration, particularly in conjunction with deductive reasoning and multiple representations; that Descartes believed his method of approaching mathematical problems was superior to other methods; and that Descartes was in fact concerned with whether his readers understood his ideas or not.     

Differential games with random noise

A differential game is considered in which an opponent is a random noise. An ally uses ε-strategies, defined in the classical theory of differential games. The mathematical expectation of possible harm caused by the opponent's actions is minimized. Bibliography: 5 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 80, 1996, pp. 78–89.     

The Language of Graphics

Abstract

We describe a system, called the Graphics Production Library (GPL), that implements a language for quantitative graphics. The structure of this system differs from existing statistical graphics, visualization, and mapping systems. Instead of treating a graphics display as a viewer for underlying data, GPL treats data as an accessory to viewing a graph. GPL is based on the mathematical definition of the graph of a function and uses that definition to organize data linked to the graph.     

Using mathematical programming to solve large ranking problems

This paper introduces a mathematical programming model that overcomes the major methodological problem of a large ranking task: respondent fatigue and deteriorated decision quality caused by an excessive number of objects to be ranked. The model was applied to the problem of ranking Marketing and International Business journals. There are more than 200 such journals, making direct ranking or rating very difficult, if not impossible. The result shows that the mathematical programming model uses very little information and yet can produce rankings that are in agreement with results obtained from direct ranking studies.     

The use of total cost of ownership for strategic procurement: a company-wide management information system

We present a general company-wide management information system for defining procurement strategies. We believe that existing practices for determining purchasing strategies can be improved and a new approach developed. The system uses total cost of ownership information. We argue that mathematical programming models should be used for exploiting this information, when evaluating the firm's strategic procurement options. As an example, we show how we have successfully applied our approach to develop a decision support system at Usinor, a European multinational steel company.     

Qualities of examples in learning and teaching

In this paper, we theorise about the different kinds of relationship between examples and the classes of mathematical objects that they exemplify as they arise in mathematical activity and teaching. We ground our theorising in direct experience of creating a polynomial that fits certain constraints to develop our understanding of engagement with examples. We then relate insights about exemplification arising from this experience to a sequence of lessons. Through these cases, we indicate the variety of fluent uses of examples made by mathematicians and experienced teachers. Following Thompson’s concept of “didactic object” (Symbolizing, modeling, and tool use in mathematics education. Kluwer, Dordrecht, The Netherlands, pp 191–212, 2002), we talk about “didacticising” an example and observe that the nature of students’ engagement is important, as well as the teacher’s intentions and actions (Thompson avoids using a verb with the root “didact”. We use the verb “didacticise” but without implying any connection to particular theoretical approaches which use the same verb.). The qualities of examples depend as much on human agency, such as pedagogical intent or mathematical curiosity or what is noticed, as on their mathematical relation to generalities.     

仿射几何在轴测投影中的应用

本文利用仿射几何学建立起一套仿射理论，进而阐明轴测投的数学原理。     

Optimal vaccination and control strategies against dengue

This article focuses on the mathematical modelling of a disease outbreak of dengue fever. A cost‐efficient fighting strategy, which simultaneously uses vaccination, application of insecticides to adult and aquatic mosquitoes, and an approach to decrease the number of man‐made breeding places for the mosquitoes, is computed using optimal control. Vaccination includes a paediatric vaccination and an imperfect random mass vaccination with waning immunity.     

Sometimes binary is better

This note uses material involving perfect numbers and Zeno's paradoxes to show that although most students prefer to use base 10 when working with mathematical concepts there are times when the binary system is best.     

复振幅法在再生放大器增益研究中的应用

在激光原理教学中,针对再生放大器的增益问题,应用复振幅法,并参考多光束干涉的处理方法,给出了增益表达式的详细推导过程,可促进学生对该问题的理解和掌握,并可作为现有教材的一点补充.