35,198 research outputs found
Formalization of the fundamental group in untyped set theory using auto2
We present a new framework for formalizing mathematics in untyped set theory
using auto2. Using this framework, we formalize in Isabelle/FOL the entire
chain of development from the axioms of set theory to the definition of the
fundamental group for an arbitrary topological space. The auto2 prover is used
as the sole automation tool, and enables succinct proof scripts throughout the
project.Comment: 17 pages, accepted for ITP 201
The history of the concept of function and some educational implications
Several fields of mathematics deal directly or indirectly with functions: mathematical
analysis considers functions of one, two, or n variables, studying their properties as well as
those of their derivatives; the theories of differential and integral equations aim at solving
equations in which the unknowns are functions; functional analysis works with spaces made
up of functions; and numerical analysis studies the processes of controlling the errors in the
evaluation of all different kinds of functions. Other fields of mathematics deal with concepts
that constitute generalizations or outgrowths of the notion of function; for example, algebra
considers operations and relations, and mathematical logic studies recursive functions.
It has long been argued that functions should constitute a fundamental concept in secondary
school mathematics (Klein, 1908/1945) and the most recent curriculum orientations clearly
emphasize the importance of functions (National Council of Teachers of Mathematics, 1989).
Depending on the dominant mathematical viewpoint, the notion of function can be regarded
in a number of different ways, each with different educational implications.
This paper reviews some of the more salient aspects of the history of the concept of
function,1 looks at its relationship with other sciences, and discusses its use in the study of
real world situations. Finally, the problem of a didactical approach is considered, giving
special attention to the nature of the working concept underlying the activities of students and
the role of different forms of representation
Procedural embodiment and magic in linear equations
How do students think about algebra? Here we consider a theoretical framework which builds from natural human functioning in terms of embodiment â perceiving the world, acting on it and reflecting on the effect of the actions â to shift to the use of symbolism to solve linear equations. In the main, the students involved in this study do not encapsulate algebraic expressions from process to object, they do not solve âevaluation equationsâ such as by âundoingâ the operations on the left, they do not find such equations easier to solve than , and they do not use general principles of âdo the same thing to both sides.â Instead they build their own ways of working based on the embodied actions they perform on the symbols, mentally picking them up and moving them around, with the added âmagicâ of rules such as âchange sides, change signs.â We consider the need for a theoretical framework that includes both embodiment and process-object encapsulation of symbolism and the need for communication of theoretical insights to address the practical problems of teachers and students
Anisotropic eddy-viscosity concept for strongly detached unsteady flows
The accurate prediction of the flow physics around bodies at high Reynolds number is a challenge in aerodynamics nowadays. In the context of turbulent flow modeling, recent advances like large eddy simulation (LES) and hybrid methods [detached eddy simulation (DES)] have considerably improved the physical relevance of the numerical simulation. However, the LES approach is still limited to the low-Reynolds-number range concerning wall flows. The unsteady Reynolds-averaged NavierâStokes (URANS) approach remains a widespread and robust methodology for complex flow computation, especially in the near-wall region. Complex statistical models like second-order closure schemes [differential Reynolds stress modeling (DRSM)] improve the prediction of these properties and can provide an efficient simulationofturbulent stresses. Fromacomputational pointofview, the main drawbacks of such approaches are a higher cost, especially in unsteady 3-D flows and above all, numerical instabilities
A bibliography on formal methods for system specification, design and validation
Literature on the specification, design, verification, testing, and evaluation of avionics systems was surveyed, providing 655 citations. Journal papers, conference papers, and technical reports are included. Manual and computer-based methods were employed. Keywords used in the online search are listed
Encouraging versatile thinking in algebra using the computer
In this article we formulate and analyse some of the obstacles to understanding the notion of a variable, and the use and meaning of algebraic notation, and report empirical evidence to support the hypothesis that an approach using the computer will be more successful in overcoming these obstacles. The computer approach is formulated within a wider framework ofversatile thinking in which global, holistic processing complements local, sequential processing. This is done through a combination of programming in BASIC, physical activities which simulate computer storage and manipulation of variables, and specific software which evaluates expressions in standard mathematical notation. The software is designed to enable the user to explore examples and non-examples of a concept, in this case equivalent and non-equivalent expressions. We call such a piece of software ageneric organizer because if offers examples and non-examples which may be seen not just in specific terms, but as typical, or generic, examples of the algebraic processes, assisting the pupil in the difficult task of abstracting the more general concept which they represent. Empirical evidence from several related studies shows that such an approach significantly improves the understanding of higher order concepts in algebra, and that any initial loss in manipulative facility through lack of practice is more than made up at a later stage
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