10,880 research outputs found
Finite difference methods fengshui: alignment through a mathematics of arrays
Numerous scientific-computational domains make use of array data. The core computing of the numerical methods and the algorithms involved is related to multi-dimensional array manipulation. Memory layout and the access patterns of that data are crucial to the optimal performance of the array-based computations. As we move towards exascale computing, writing portable code for efficient data parallel computations is increasingly requiring an abstract productive working environment. To that end, we present the design of a framework for optimizing scientific array-based computations, building a case study for a Partial Differential Equations solver. By embedding the Mathematics of Arrays formalism in the Magnolia programming language, we assemble a software stack capable of abstracting the continuous high-level application layer from the discrete formulation of the collective array-based numerical methods and algorithms and the final detailed low-level code. The case study lays the groundwork for achieving optimized memory layout and efficient computations while preserving a stable abstraction layer independent of underlying algorithms and changes in the architecture.Peer ReviewedPostprint (author's final draft
Algebra of Observables for Identical Particles in One Dimension
The algebra of observables for identical particles on a line is formulated
starting from postulated basic commutation relations. A realization of this
algebra in the Calogero model was previously known. New realizations are
presented here in terms of differentiation operators and in terms of
SU(N)-invariant observables of the Hermitian matrix models. Some particular
structure properties of the algebra are briefly discussed.Comment: 13 pages, Latex, uses epsf, 1 eps figure include
PROCRUSTES: A computer algebra package for post-Newtonian calculations in General Relativity
We report on a package of routines for the computer algebra system Maple
which supports the explicit determination of the geometric quantities, field
equations, equations of motion, and conserved quantities of General Relativity
in the post-Newtonian approximation. The package structure is modular and
allows for an easy modification by the user. The set of routines can be used to
verify hand calculations or to generate the input for further numerical
investigations.Comment: 20 pages, 3 figures. The latest version of the package can be
obtained from http://www.thp.uni-koeln.de/~dp/procrustes.htm
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 students’ 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 students’ 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
On Formal Specification of Maple Programs
This paper is an example-based demonstration of our initial results on the
formal specification of programs written in the computer algebra language
MiniMaple (a substantial subset of Maple with slight extensions). The main goal
of this work is to define a verification framework for MiniMaple. Formal
specification of MiniMaple programs is rather complex task as it supports
non-standard types of objects, e.g. symbols and unevaluated expressions, and
additional functions and predicates, e.g. runtime type tests etc. We have used
the specification language to specify various computer algebra concepts
respective objects of the Maple package DifferenceDifferential developed at our
institute
Detecting Similarity of Rational Plane Curves
A novel and deterministic algorithm is presented to detect whether two given
rational plane curves are related by means of a similarity, which is a central
question in Pattern Recognition. As a by-product it finds all such
similarities, and the particular case of equal curves yields all symmetries. A
complete theoretical description of the method is provided, and the method has
been implemented and tested in the Sage system for curves of moderate degrees.Comment: 22 page
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