23,003 research outputs found
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
Polynomial Interpretations for Higher-Order Rewriting
The termination method of weakly monotonic algebras, which has been defined
for higher-order rewriting in the HRS formalism, offers a lot of power, but has
seen little use in recent years. We adapt and extend this method to the
alternative formalism of algebraic functional systems, where the simply-typed
lambda-calculus is combined with algebraic reduction. Using this theory, we
define higher-order polynomial interpretations, and show how the implementation
challenges of this technique can be tackled. A full implementation is provided
in the termination tool WANDA
Can Computer Algebra be Liberated from its Algebraic Yoke ?
So far, the scope of computer algebra has been needlessly restricted to exact
algebraic methods. Its possible extension to approximate analytical methods is
discussed. The entangled roles of functional analysis and symbolic programming,
especially the functional and transformational paradigms, are put forward. In
the future, algebraic algorithms could constitute the core of extended symbolic
manipulation systems including primitives for symbolic approximations.Comment: 8 pages, 2-column presentation, 2 figure
Exploring the concept of interaction computing through the discrete algebraic analysis of the BelousovâZhabotinsky reaction
Interaction computing (IC) aims to map the properties of integrable low-dimensional non-linear dynamical systems to the discrete domain of finite-state automata in an attempt to reproduce in software the self-organizing and dynamically stable properties of sub-cellular biochemical systems. As the work reported in this paper is still at the early stages of theory development it focuses on the analysis of a particularly simple chemical oscillator, the Belousov-Zhabotinsky (BZ) reaction. After retracing the rationale for IC developed over the past several years from the physical, biological, mathematical, and computer science points of view, the paper presents an elementary discussion of the Krohn-Rhodes decomposition of finite-state automata, including the holonomy decomposition of a simple automaton, and of its interpretation as an abstract positional number system. The method is then applied to the analysis of the algebraic properties of discrete finite-state automata derived from a simplified Petri net model of the BZ reaction. In the simplest possible and symmetrical case the corresponding automaton is, not surprisingly, found to contain exclusively cyclic groups. In a second, asymmetrical case, the decomposition is much more complex and includes five different simple non-abelian groups whose potential relevance arises from their ability to encode functionally complete algebras. The possible computational relevance of these findings is discussed and possible conclusions are drawn
Inferring Algebraic Effects
We present a complete polymorphic effect inference algorithm for an ML-style
language with handlers of not only exceptions, but of any other algebraic
effect such as input & output, mutable references and many others. Our main aim
is to offer the programmer a useful insight into the effectful behaviour of
programs. Handlers help here by cutting down possible effects and the resulting
lengthy output that often plagues precise effect systems. Additionally, we
present a set of methods that further simplify the displayed types, some even
by deliberately hiding inferred information from the programmer
Slow invariant manifolds as curvature of the flow of dynamical systems
Considering trajectory curves, integral of n-dimensional dynamical systems,
within the framework of Differential Geometry as curves in Euclidean n-space,
it will be established in this article that the curvature of the flow, i.e. the
curvature of the trajectory curves of any n-dimensional dynamical system
directly provides its slow manifold analytical equation the invariance of which
will be then proved according to Darboux theory. Thus, it will be stated that
the flow curvature method, which uses neither eigenvectors nor asymptotic
expansions but only involves time derivatives of the velocity vector field,
constitutes a general method simplifying and improving the slow invariant
manifold analytical equation determination of high-dimensional dynamical
systems. Moreover, it will be shown that this method generalizes the Tangent
Linear System Approximation and encompasses the so-called Geometric Singular
Perturbation Theory. Then, slow invariant manifolds analytical equation of
paradigmatic Chua's piecewise linear and cubic models of dimensions three, four
and five will be provided as tutorial examples exemplifying this method as well
as those of high-dimensional dynamical systems
Algebraic thinking of grade 8 students in solving word problems with a spreadsheet
This paper describes and discusses the activity of grade 8 students on two word
problems, using a spreadsheet. We look at particular uses of the spreadsheet, namely
at the studentsâ representations, as ways of eliciting forms of algebraic thinking
involved in solving the problems. We aim to see how the spreadsheet allows the solution of formally impracticable problems at studentsâ level of algebra knowledge,
by making them treatable through the computational logic that is intrinsic to the
operating modes of the spreadsheet. The protocols of the problem solving sessions
provided ways to describe and interpret the relationships that students established
between the variables in the problems and their representations in the spreadsheet
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