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On Transitive Permutation Groups
 LMS J. Comput. Math
, 1996
"... We assign names and new generators to the transitive groups of degree up to 15, reflecting their structure. ..."
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We assign names and new generators to the transitive groups of degree up to 15, reflecting their structure.
Explicit Construction of Families of LDPC Codes with No 4Cycles
 IEEE Trans. Inform. Theory
, 2003
"... LDPC codes are serious contenders to Turbo codes in terms of decoding performance. ..."
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LDPC codes are serious contenders to Turbo codes in terms of decoding performance.
Counting cases in marching cubes: Toward a generic algorithm for producing substitopes
 Proceedings of the 14th IEEE Visualization 2003 (VIS'03) 8
, 2003
"... Distinct cases of colorings for a square, assigning one color to each vertex. Top row: seven cases result from using four colors (fluid, bone, tissue, lesion) when applying Separating Surfaces to a square. Bottom row: thirteen cases result from using three colors (+ – =) when applying Marching Cubes ..."
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Distinct cases of colorings for a square, assigning one color to each vertex. Top row: seven cases result from using four colors (fluid, bone, tissue, lesion) when applying Separating Surfaces to a square. Bottom row: thirteen cases result from using three colors (+ – =) when applying Marching Cubes to a square. This paper describes a technique for counting the cases that arise in a family of visualization techniques. This family includes Marching Cubes, Sweeping Simplices, Contour Meshing, Interval Volumes, and Separating Surfaces. Counting the cases is the first step toward developing a generic visualization algorithm to produce substitopes (geometric substitutions of polytopes). To count the cases, we observe that discrete “color ” values are assigned to the vertices of a polytope. A group of symmetries acts on the colorings to produce equivalence classes called orbits, each of which corresponds to a single case. Using a software system (“GAP”) for computational group theory, we calculate the cases that arise in the family of visualization techniques. These casecounts are organized into a table that provides a taxonomy of members of the family; numbers in the table are derived from actual lists of cases, which are computed by our methods. The calculation confirms previously reported casecounts for large dimensions that are too large to check by hand, and predicts the number of cases that will arise in algorithms that have not yet been invented.
Simple groups in computational group theory
 Proceedings of the International Congress of Mathematicians
, 1998
"... ..."
A Type System for Computer Algebra
 Journal of Symbolic Computation
, 1994
"... ing RationalFun from Rational yields a higher order type operator that, given a specification, forms the type of objects that satisfy it. Philip Santas DeclareDomain := (Fun: Type?Category) +? (((Rep: Type) +? with(Rep,Fun(Rep))) SomeRep) The type of Rational objects can now be expressed by applyi ..."
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ing RationalFun from Rational yields a higher order type operator that, given a specification, forms the type of objects that satisfy it. Philip Santas DeclareDomain := (Fun: Type?Category) +? (((Rep: Type) +? with(Rep,Fun(Rep))) SomeRep) The type of Rational objects can now be expressed by applying the DeclareDomain constructor to the specification RationalFun: Rational := DeclareDomain(RationalFun) or the shortcut: Rational : RationalFun In order to give proper treatment to the interaction between representations and subtyping, it is necessary to separate Rational into the specifications of its functions and the operators which capture the common structure of all object types. This separation is also important for the semantical construction of categories and the definition of the internal structures of the types. 2.1. Multiple Representations Rationals are created using the function box, which captures the semantics of dynamic objects in object oriented programming.A rational...
Involutory decomposition of groups into twisted subgroups and subgroups
 J. Group Theory
, 2000
"... Gyrogroups are generalized groups modelled on the Einstein groupoid of all relativistically admissible velocities with their Einstein’s velocity addition as a binary operation. Einstein’s gyrogroup fails to form a group since it is nonassociative. The breakdown of associativity in the Einstein addit ..."
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Gyrogroups are generalized groups modelled on the Einstein groupoid of all relativistically admissible velocities with their Einstein’s velocity addition as a binary operation. Einstein’s gyrogroup fails to form a group since it is nonassociative. The breakdown of associativity in the Einstein addition does not result in loss of mathematical regularity owing to the presence of the relativistic effect known as the Thomas precession which, by abstraction, becomes an automorphism called the Thomas gyration. The Thomas gyration turns out to be the missing link that gives rise to analogies shared by gyrogroups and groups. In particular, it gives rise to the gyroassociative and the gyrocommuttive laws that Einstein’s addition possesses, in full analogy with the associative and the commutative laws that vector addition possesses in a vector space. The existence of striking analogies shared by gyrogroups
Some applications of Magma in designs and codes: oval designs, hermitian unitals and generalized ReedMuller codes
"... We describe three applications of Magma to problems in the area of designs and the associated codes: . Steiner systems, Hadamard designs and symmetric designs arising from a oval in an even order plane, leading in the classical case to bent functions and differenceset designs; . the hermitian unita ..."
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We describe three applications of Magma to problems in the area of designs and the associated codes: . Steiner systems, Hadamard designs and symmetric designs arising from a oval in an even order plane, leading in the classical case to bent functions and differenceset designs; . the hermitian unital as a 2(q 3 + 1, q + 1, 1) design, and the code over F p where p divides q + 1; . a basis of minimumweight vectors for the code over F p of the design of points and hyperplanes of the affine geometry AG d (F p ), where p is a prime.
Catalan Monoids, Monoids of Local Endomorphisms, and their Presentations
, 1996
"... The Catalan monoid and partial Catalan monoid of a directed graph are introduced. ..."
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The Catalan monoid and partial Catalan monoid of a directed graph are introduced.
Presentations of finite simple groups: a computational approach
"... All nonabelian finite simple groups of Lie type of rank n over a field of size q, with the possible exception of the Ree groups 2 G2(q), have presentations with at most 49 relations and bitlength O(log n + log q). Moreover, An and Sn have presentations with 3 generators, 7 relations and bitlength ..."
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All nonabelian finite simple groups of Lie type of rank n over a field of size q, with the possible exception of the Ree groups 2 G2(q), have presentations with at most 49 relations and bitlength O(log n + log q). Moreover, An and Sn have presentations with 3 generators, 7 relations and bitlength O(log n), while SL(n, q) has a presentation with 6 generators, 25 relations and