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Some Primality Testing Algorithms
 Notices of the AMS
, 1993
"... We describe the primality testing algorithms in use in some popular computer algebra systems, and give some examples where they break down in practice. 1 Introduction In recent years, fast primality testing algorithms have been a popular subject of research and some of the modern methods are now i ..."
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We describe the primality testing algorithms in use in some popular computer algebra systems, and give some examples where they break down in practice. 1 Introduction In recent years, fast primality testing algorithms have been a popular subject of research and some of the modern methods are now incorporated in computer algebra systems (CAS) as standard. In this review I give some details of the implementations of these algorithms and a number of examples where the algorithms prove inadequate. The algebra systems reviewed are Mathematica, Maple V, Axiom and Pari/GP. The versions we were able to use were Mathematica 2.1 for Sparc, copyright dates 19881992; Maple V Release 2, copyright dates 19811993; Axiom Release 1.2 (version of February 18, 1993); Pari/GP 1.37.3 (Sparc version, dated November 23, 1992). The tests were performed on Sparc workstations. Primality testing is a large and growing area of research. For further reading and comprehensive bibliographies, the interested re...
Media Theory: Representations and Examples
, 2008
"... In this paper we develop a representational approach to media theory. We construct representations of media by well graded families of sets and partial cubes and establish the uniqueness of these representations. Two particular examples of media are also described in detail. ..."
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In this paper we develop a representational approach to media theory. We construct representations of media by well graded families of sets and partial cubes and establish the uniqueness of these representations. Two particular examples of media are also described in detail.
Computing Science Digital Diffraction
"... Note: This document is available in other formats. Some years ago I visited M. F. Perutz, the Cambridge biochemist who deciphered the structure of the hemoglobin molecule. Professor Perutz showed me a series of artifacts from his 20year struggle to unravel the twists and folds of the oxygencarryin ..."
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Note: This document is available in other formats. Some years ago I visited M. F. Perutz, the Cambridge biochemist who deciphered the structure of the hemoglobin molecule. Professor Perutz showed me a series of artifacts from his 20year struggle to unravel the twists and folds of the oxygencarrying protein. There was an xray diffraction film whose symmetrical array of dots looked like a lace doily, a contour map of electron density drawn on stacked sheets of transparent plastic, and a huge molecular model supported by a forest of brass rods. Looking at these objects, I could understand in a general way how an xray diffraction pattern reveals the structure of a crystallized molecule. The diffraction pattern is the Fourier transform of the crystal lattice, representing in "frequency space " the positions of the atoms in ordinary space. The lattice and the diffraction pattern have a reciprocal relationship: Widely separated dots on the xray film correspond to closely spaced planes of atoms in the crystal, and nearby dots on the film are generated by widely spaced atoms. That much I understood. What was lacking was morespecific knowledge of how to translate from crystal lattice to diffraction pattern and back again. I wanted to look at the xray film and see the geometry of the crystal. What I wanted most was a chance to experiment and explore. I wanted to nudge an atom in the crystal, and see how that displacement altered the diffraction pattern.
ILLUSTRATING MATHEMATICS USING 3D PRINTERS
"... Abstract. 3D printing technology can help to visualize proofs in mathematics. In this document we aim to illustrate how 3D printing can help to visualize concepts and mathematical proofs. As already known to educators in ancient Greece, models allow to bring mathematics closer to the public. The new ..."
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Abstract. 3D printing technology can help to visualize proofs in mathematics. In this document we aim to illustrate how 3D printing can help to visualize concepts and mathematical proofs. As already known to educators in ancient Greece, models allow to bring mathematics closer to the public. The new 3D printing technology makes the realization of such tools more accessible than ever. This is an updated version of a paper included in [62]. 1. Visualization Visualization has always been an important ingredient for communicating mathematics [80]. Figures and models have helped to express ideas even before formal mathematical language was able to describe the structures. Numbers have been recorded as marks on bones, represented with pebbles, then painted onto stone, inscribed into clay, woven into talking knots, written onto papyrus or paper, then printed on paper or displayed on computer screens. While figures extend language and pictures allow to visualize concepts, realizing objects in space has kept its value. Already in ancient Greece, wooden models of Apollonian cones were used to teach conic sections. Early research in mathematics was often visual: figures on Babylonian Clay tablets illustrate Pythagorean triples, the Moscow mathematical papyrus features a picture which helps to derive the volume formula for a frustum. AlKhwarizmi drew figures to solve the quadratic equation. Visualization is not only illustrative, educational or heuristic, it has practical value: Pythagorean triangles realized by ropes helped measuring and dividing up of land in Babylonia.