29,341 research outputs found
The vector algebra war: a historical perspective
There are a wide variety of different vector formalisms currently utilized in
engineering and physics. For example, Gibbs' three-vectors, Minkowski
four-vectors, complex spinors in quantum mechanics, quaternions used to
describe rigid body rotations and vectors defined in Clifford geometric
algebra. With such a range of vector formalisms in use, it thus appears that
there is as yet no general agreement on a vector formalism suitable for science
as a whole. This is surprising, in that, one of the primary goals of nineteenth
century science was to suitably describe vectors in three-dimensional space.
This situation has also had the unfortunate consequence of fragmenting
knowledge across many disciplines, and requiring a significant amount of time
and effort in learning the various formalisms. We thus historically review the
development of our various vector systems and conclude that Clifford's
multivectors best fulfills the goal of describing vectorial quantities in three
dimensions and providing a unified vector system for science.Comment: 8 pages, 1 figure, 1 tabl
Four Departures in Mathematics and Physics
Much of Mathematics, and therefore Physics as well, have been limited by four
rather consequential restrictions. Two of them are ancient taboos, one is an
ancient and no longer felt as such bondage, and the fourth is a surprising
omission in Algebra. The paper brings to the attention of those interested
these four restrictions, as well as the fact that each of them has by now ways,
even if hardly yet known ones, to overcome them
Tarski's influence on computer science
The influence of Alfred Tarski on computer science was indirect but
significant in a number of directions and was in certain respects fundamental.
Here surveyed is the work of Tarski on the decision procedure for algebra and
geometry, the method of elimination of quantifiers, the semantics of formal
languages, modeltheoretic preservation theorems, and algebraic logic; various
connections of each with computer science are taken up
Reductionism and the Universal Calculus
In the seminal essay, "On the unreasonable effectiveness of mathematics in
the physical sciences," physicist Eugene Wigner poses a fundamental
philosophical question concerning the relationship between a physical system
and our capacity to model its behavior with the symbolic language of
mathematics. In this essay, I examine an ambitious 16th and 17th-century
intellectual agenda from the perspective of Wigner's question, namely, what
historian Paolo Rossi calls "the quest to create a universal language." While
many elite thinkers pursued related ideas, the most inspiring and forceful was
Gottfried Leibniz's effort to create a "universal calculus," a pictorial
language which would transparently represent the entirety of human knowledge,
as well as an associated symbolic calculus with which to model the behavior of
physical systems and derive new truths. I suggest that a deeper understanding
of why the efforts of Leibniz and others failed could shed light on Wigner's
original question. I argue that the notion of reductionism is crucial to
characterizing the failure of Leibniz's agenda, but that a decisive argument
for the why the promises of this effort did not materialize is still lacking.Comment: 11 pages, 1 figur
Understanding what you are doing: A new angle on CAS?
Powerful Computer Algebra Systems (CAS) are often used only with reluctance in early undergraduate mathematics teaching, partly because of concerns that they may not encourage students to understand what they are doing. In this exploratory study, a version of a CAS that has been designed for secondary school students was used, with a view to considering the value of this sort of student learning support for first year undergraduate students enrolled in degree programs other than mathematics. Workshops were designed to help students understand aspects of elementary symbolic manipulation, through the use of the Algebra mode of an algebraic calculator, the Casio Algebra FX 2.0. The Algebra mode of this calculator allows a user to undertake elementary algebraic manipulation, routinely providing all intermediate results, in contrast to more powerful CAS software, which usually provides simplified results only. The students were volunteers from an introductory level unit, designed to provide a bridge between school and university studies of mathematics and with a focus on algebra and calculus. The two structured workshop sessions focussed respectively on the solution of linear equations and on relationships between factorising and expanding; attention focussed on using the calculators as personal learning devices. Following the workshops, structured interviews were used to systematically record student reactions to the experience. As a result of the study, the paper offers advice on the merits of using algebraic calculators in this sort of way
Gopakumar-Vafa invariants do not determine flops
Two 3-fold flops are exhibited, both of which have precisely one flopping
curve. One of the two flops is new, and is distinct from all known algebraic
D4-flops. It is shown that the two flops are neither algebraically nor
analytically isomorphic, yet their curve-counting Gopakumar-Vafa invariants are
the same. We further show that the contraction algebras associated to both are
not isomorphic, so the flops are distinguished at this level. This shows that
the contraction algebra is a finer invariant than various curve-counting
theories, and it also provides more evidence for the proposed analytic
classification of 3-fold flops via contraction algebras.Comment: 10 pages, final versio
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