University timetabling by constraint-based reasoning: A case study

This paper presents a case-study which applies constraint-based reasoning to university timetable planning. The timetabling problem is formulated as a constraint satisfaction model and this model is solved using constrained-directed search algorithms with built-in forward checking and constraint propagation. The model and algorithms were tested with real data containing 536 lessons to be scheduled into 45 timeslots and 21 rooms. The solution to the problem was obtained with minimal computing effort and processing time. This study showed that modelling and remodelling could be carried out easily through a proposed parameterised model formulation. The proposed approach has also successfully maximised room utilisation and minimised the number of timeslots required to deliver all the lectures. This finding may facilitate the widespread implementation of automated timetabling systems to larger scale problems and a wider variety of application domains.     

A comparison of ratios and fractions and their roles as tools in proportional reasoning

In an attempt to develop our shared understanding of the relationship between ratios and fractions, we began a phenomenological study to gather evidence from teachers and textbooks and to collect evidence from our own experiences. In this article, we present five possible models for this relationship and a summary of evidence to support each. We also present the model that we developed to represent our shared understanding and provide the results of a study for which we have used our model to help us analyze students’ uses of ratios and fractions in their solutions to proportion-related problems.     

The application of automated reasoning to formal models of combinatorial optimization

Many formalisms have been proposed over the years to capture combinatorial optimization algorithms such as dynamic programming, branch and bound, and greedy. In 1989 Helman [9] presented a common formalism that captures dynamic programming and branch and bound type algorithms. The formalism was later extended to include greedy algorithms. In this paper, we describe the application of automated reasoning techniques to the domain of our model, in particular considering some representational issues and demonstrating that proofs about the model can be obtained by an automated reasoning program. The long-term objective of this research is to develop a methodology for using automated reasoning to establish new results within the theory, including the derivation of new lower bounds and the discovery (and verification) of new combinatorial search strategies.     

Using technology to facilitate reasoning: lifting the fog from linear algebra

This article discusses student difficulties in grasping conceptsfrom linear algebra. Using an example from an interview witha student, we propose changes that might positively impact studentunderstanding of concepts within a problem-solving context.In particular, we illustrate barriers to student understandingand suggest technological interventions to address these barriers.     

DB2 and DB2A: Two useful tools for constructing Hamiltonian circuits

The paper presents a procedure, called DB2A, for constructing a Hamiltonian circuit (HC) in a general directed graph. Application examples and a completely developed example problem are included. An appendix recalls the features of DB2, used here as a subprocedure of DB2A, and finding a Hamiltonian Cycle in undirected graphs.     

Mathematical tools of the kinetic theory of active particles with some reasoning on the modelling progression and heterogeneity

The mathematical approach proposed in this paper refers to the modelling, and related mathematical problems, of large systems of interacting entities whose microscopic state includes not only geometrical and mechanical variables (typically position and velocity), but also peculiar functions or specific activities. The number of the above entities is sufficiently large for describing the overall state of the system using a suitable probability distribution over the microscopic state. The first part of the paper is devoted to the derivation of suitable mathematical structures which can be properly used to model a variety of models in different fields of applied sciences. Then some research perspectives are analyzed, focussed on applications to biological systems.     

A task that elicits reasoning: A dual analysis

This paper reports on the forms of reasoning elicited as fourth grade students in a suburban district and sixth grade students in an urban district worked on similar tasks involving reasoning with the use of Cuisenaire rods. Analysis of the two data sets shows similarities in the reasoning used by both groups of students on specific tasks, and the tendency of a particular task to elicit numerous forms of reasoning in both groups of students. Attributes of that task and ways that those attributes can be replicated in other domains may have implications in the teaching of early reasoning.     

Mathematical formalization and diagrammatic reasoning: the case study of the braid group between 1925 and 1950

The standard historical narrative regarding formalism during the twentieth century indicates the 1920s as a highpoint in the mathematical formalization project. This was marked by Hilbert’s statement that the sign stood at the beginning of pure mathematics [‘Neubegründung der Mathematik. Erste Mitteilung’, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 1 (1922), 157–177]. If one takes the braid group as a case study of research whose official goal was to symbolically formalize braids and weaving patterns, a reconsideration of this strict definition of formalism is nevertheless required. For example, does it reflect what actually occurred in practice in the mathematical research of this period? As this article shows, the research on the braid group between 1926 and 1950, led among others by Artin, Burau, Fröhlich and Bohnenblust, was characterized by a variety of practices and reasoning techniques. These were not only symbolic and deductive, but also diagrammatic and visual. Against the historical narrative of formalism as based on a well-defined chain of graphic signs that has freedom of interpretation, this article presents how these different ways of reasoning—which were not only sign based—functioned together within the research of the braid group; it will be shown how they are simultaneously necessary and complementary for each other.     

A characterization of dynamic reasoning: Reasoning with time as parameter

Students incorporate and use the implicit and explicit parameter time to support their mathematical reasoning and deepen their understandings as they participate in a differential equations class during instruction on solutions to systems of differential equations. Therefore, dynamic reasoning is defined as developing and using conceptualizations about time as a parameter that implicitly or explicitly coordinates with other quantities to understand and solve problems. Students participate in the following types of mathematical activity related to dynamic reasoning: making time an explicit quantity, using the metaphor of time as “unidimensional space”, using time to reason both quantitatively and qualitatively, using three-dimensional visualization of time related functions, fusing context and representation of time related functions, and using the fictive motion metaphor for function. The purpose of this article is to present a characterization of dynamic reasoning and promote more explicit attention to this type of reasoning by teachers in K-16 mathematics in order to improve student understanding in time related areas of mathematics.     

Using dynamic tools to develop an understanding of the fundamental ideas of calculus

Although dynamic geometry software has been extensively used for teaching calculus concepts, few studies have documented how these dynamic tools may be used for teaching the rigorous foundations of the calculus. In this paper, we describe lesson sequences utilizing dynamic tools for teaching the epsilon-delta definition of the limit and the fundamental theorem of calculus. The lessons were designed on the basis of observed student difficulties and the existing scholarly literature. We show how a combination of dynamic tools and guide questions allows students to construct their understanding of these calculus ideas.     

Analysis and visualization tools in CFD, part II: A case study in grid adaptivity

The objective of the present paper is to show, in the context of a configurable resolution and analysis environment, how grid adaptivity methodology can be made configurable and rapidly applied to various problems. In order to reach this objective, our error estimation and grid adaptation tools were configured to integrate Zienkiewicz and Zhu's local error estimator. The environment was then applied to the simulation of incompressible isothermal flow and natural convection problems.     

Algebraic and analytical tools for the study of the period function

In this paper we consider analytic planar differential systems having a first integral of the form H(x,y)=A(x)+B(x)y+C(x)y²$H(x,y)=A(x)+B(x)y+C(x)y^2$ and an integrating factor κ(x)$\kappa (x)$ not depending on y. Our aim is to provide tools to study the period function of the centers of this type of differential system and to this end we prove three results. Theorem A gives a characterization of isochronicity, a criterion to bound the number of critical periods and a necessary condition for the period function to be monotone. Theorem B is intended for being applied in combination with Theorem A in an algebraic setting that we shall specify. Finally, Theorem C is devoted to study the number of critical periods bifurcating from the period annulus of an isochrone perturbed linearly inside a family of centers. Four different applications are given to illustrate these results.     

Using fuzzy set theory in a scheduling problem: A case study

This paper deals with a real scheduling problem where it seems interesting to use fuzzy sets The question of knowing how and when it is possible to use fuzzy sets (rather than probabilities for instance) is discussed in great detail for the studied case. Fuzzy concepts are shown to be very useful and easy to work with in this decision-aid problem.     

Derivation of the hypergeometric distribution: an alternative reasoning approach

This note presents an alternative approach to the reasoning process and derivation of the hypergeometric probability mass function (pmf), and contrasts it with a binomial model. It utilizes the essential concept of sampling without replacement directly in the development of the mass function.     

A New Computer Construction of the Monster Using 2-Local Subgroups

A construction of the Monster simple group is described implicitlyas 196882x196882 matrices over the field of 3 elements.     

Sentences in the language of reasoning

In the present paper we investigate a few grammatical aspects of texts in which a reasoning has been expressed. We are mainly interested in text units (called sentences) that are constituent parts of a reasoning.

Firstly we discuss the full conceptual framework reflected in the reasoning: its under lying form. Next we divide sentences into two main classes: fundamental and informative sentences. Each of these main classes will thereupon be subdivided. In classifying we are guided by the manner in which sentences function in a reasoning. The nature of each class is elucidated by a discussion and examples.

We subsequently deal with compound sentences and sentence groups. Finally we analyse two texts according to the grammatical classification principles explained in the paper.