54,335 research outputs found
Tribonacci and Tribonacci-Lucas Sedenions
The sedenions form a 16-dimensional Cayley-Dickson algebra. In this paper, we
introduce the Tribonacci and Tribonacci-Lucas sedenions. Furthermore, we
present some properties of these sedenions and derive relationships between
them.Comment: 17 pages, 1 figur
Categorisation of Mental Computation Strategies to Support Teaching and to Encourage Classroom Dialogue
Mental strategies are a desired focus for computational instruction in schools and have been the focus of many syllabus documents and research papers. Teachers though, have been slow to adopt such changes in their classroom planning. A possible block to adoption of this approach is their lack of knowledge about possible computation strategies and a lack of a clear organisation of a school program for this end. This paper discusses a framework for the categorisation of mental computation strategies that can support teachers to make the pedagogical shift to use of mental strategies by providing a framework for the development of school and classroom programs and provide a common language for teachers and students to discuss strategies in use
Can a computer be "pushed" to perform faster-than-light?
We propose to "boost" the speed of communication and computation by immersing
the computing environment into a medium whose index of refraction is smaller
than one, thereby trespassing the speed-of-light barrier.Comment: 7 pages, 1 figure, presented at the UC10 Hypercomputation Workshop
"HyperNet 10" at The University of Tokyo on June 22, 201
Type classes for efficient exact real arithmetic in Coq
Floating point operations are fast, but require continuous effort on the part
of the user in order to ensure that the results are correct. This burden can be
shifted away from the user by providing a library of exact analysis in which
the computer handles the error estimates. Previously, we [Krebbers/Spitters
2011] provided a fast implementation of the exact real numbers in the Coq proof
assistant. Our implementation improved on an earlier implementation by O'Connor
by using type classes to describe an abstract specification of the underlying
dense set from which the real numbers are built. In particular, we used dyadic
rationals built from Coq's machine integers to obtain a 100 times speed up of
the basic operations already. This article is a substantially expanded version
of [Krebbers/Spitters 2011] in which the implementation is extended in the
various ways. First, we implement and verify the sine and cosine function.
Secondly, we create an additional implementation of the dense set based on
Coq's fast rational numbers. Thirdly, we extend the hierarchy to capture order
on undecidable structures, while it was limited to decidable structures before.
This hierarchy, based on type classes, allows us to share theory on the
naturals, integers, rationals, dyadics, and reals in a convenient way. Finally,
we obtain another dramatic speed-up by avoiding evaluation of termination
proofs at runtime.Comment: arXiv admin note: text overlap with arXiv:1105.275
Computing Invariants of Simplicial Manifolds
This is a survey of known algorithms in algebraic topology with a focus on
finite simplicial complexes and, in particular, simplicial manifolds. Wherever
possible an elementary approach is chosen. This way the text may also serve as
a condensed but very basic introduction to the algebraic topology of simplicial
manifolds.
This text will appear as a chapter in the forthcoming book "Triangulated
Manifolds with Few Vertices" by Frank H. Lutz.Comment: 13 pages, 3 figure
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
- …