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The Encyclopedia of Integer Sequences
"... This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs ..."
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Cited by 719 (15 self)
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This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs to generate the sequence, references, links to relevant web pages, and other
On Kleene Algebras and Closed Semirings
, 1990
"... Kleene algebras are an important class of algebraic structures that arise in diverse areas of computer science: program logic and semantics, relational algebra, automata theory, and the design and analysis of algorithms. The literature contains several inequivalent definitions of Kleene algebras and ..."
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Cited by 48 (6 self)
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Kleene algebras are an important class of algebraic structures that arise in diverse areas of computer science: program logic and semantics, relational algebra, automata theory, and the design and analysis of algorithms. The literature contains several inequivalent definitions of Kleene algebras and related algebraic structures [2, 14, 15, 5, 6, 1, 10, 7]. In this paper we establish some new relationships among these structures. Our main results are: There is a Kleene algebra in the sense of [6] that is not *continuous. The categories of *continuous Kleene algebras [5, 6], closed semirings [1, 10] and Salgebras [2] are strongly related by adjunctions. The axioms of Kleene algebra in the sense of [6] are not complete for the universal Horn theory of the regular events. This refutes a conjecture of Conway [2, p. 103]. Righthanded Kleene algebras are not necessarily lefthanded Kleene algebras. This verifies a weaker version of a conjecture of Pratt [15].
Covering arrays and intersecting codes
 Journal of Combinatorial Designs
, 1993
"... A tcovering array is a set of k binary vectors of length n with the property that, in any t coordinate positions, all 2t possibilities occur at least once. Such arrays are used for example in circuit testing, and one wishes to minimize k for given values of n and t. The case t = 2 was solved by Rkn ..."
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Cited by 36 (0 self)
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A tcovering array is a set of k binary vectors of length n with the property that, in any t coordinate positions, all 2t possibilities occur at least once. Such arrays are used for example in circuit testing, and one wishes to minimize k for given values of n and t. The case t = 2 was solved by Rknyi, Katona, and Kleitman and Spencer. The present article is concerned with the case t = 3, where important (but unpublished) contributions were made by Busschbach and Roux in the 1980s. One of the principal constructions makes use of intersecting codes (linear codes with the property that any two nonzero codewords meet). This article studies the properties of 3covering arrays and intersecting codes, and gives a table of the best 3covering arrays presently known. For large n the minimal k satisfies 3.21256 < k / log n < 7.56444. 01993
What Are All the Best Sphere Packings in Low Dimensions?
 DISCRETE & COMPUTATIONAL GEOMETRY 9
, 1995
"... We describe what may be all the best packings of nonoverlapping equal spheres in dimensions n < 10, where "best" means both having the highest density and not permitting any local improvement. For example, the best fivedimensional sphere packings are parametrized by the 4colorings of ..."
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Cited by 12 (4 self)
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We describe what may be all the best packings of nonoverlapping equal spheres in dimensions n < 10, where "best" means both having the highest density and not permitting any local improvement. For example, the best fivedimensional sphere packings are parametrized by the 4colorings of the onedimensional integer lattice. We also find what we believe to be the exact numbers of "uniform" packings among these, that is, those in which the automorphism group acts transitively. These assertions depend on certain plausible but as yet unproved postulates. Our work may be regarded as a continuation of Lfiszl6 Fejes T6th's work on solid packings.
Codes over GF(4) and complex lattices
 J
, 1978
"... This paper studies the relationship between errorcorrecting codes over GF(4) and complex lattices (more precisely, H[o]modules in C^n, where w = e*n’l*). The thetafunctions of selfdual lattices are characterized. Two general methods are presented for constructing lattices from codes. Several exa ..."
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Cited by 6 (2 self)
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This paper studies the relationship between errorcorrecting codes over GF(4) and complex lattices (more precisely, H[o]modules in C^n, where w = e*n’l*). The thetafunctions of selfdual lattices are characterized. Two general methods are presented for constructing lattices from codes. Several examples are given, including a new lattice spherepacking in R^36.
GoldbergCoxeter Construction for 3 and 4valent Plane Graphs
, 2004
"... We consider the GoldbergCoxeter construction GC k,l (G 0 ) (a generalization of a simplicial subdivision of the dodecahedron considered in [Gold37] and [Cox71]), which produces a plane graph from any 3 or 4valent plane graph for integer parameters k, l.Azigzag in a plane graph is a circuit of ed ..."
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Cited by 4 (3 self)
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We consider the GoldbergCoxeter construction GC k,l (G 0 ) (a generalization of a simplicial subdivision of the dodecahedron considered in [Gold37] and [Cox71]), which produces a plane graph from any 3 or 4valent plane graph for integer parameters k, l.Azigzag in a plane graph is a circuit of edges, such that any two, but no three, consecutive edges belong to the same face; a central circuit in a 4valent plane graph G is a circuit of edges, such that no two consecutive edges belong to the same face. We study the zigzag (or central circuit) structure of the resulting graph using the algebraic formalism of the moving group,the(k, l)product and a finite index subgroup of SL 2 (Z), whose elements preserve the above structure. We also study the intersection pattern of zigzags (or central circuits) of GC k,l (G 0 ) and consider its projections, obtained by removing all but one zigzags (or central circuits).
H.: Intricate Isohedral Tilings of 3D Euclidean Space, Bridges Conference
"... Various methods to create intricate tilings of 3D space are presented. They include modulated extrusions of 2D Escher tilings, freeform deformations of the fundamental domain of various 3D symmetry groups, highly symmetrical polyhedral toroids of genus 1, highergenus cage structures derived from t ..."
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Cited by 3 (2 self)
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Various methods to create intricate tilings of 3D space are presented. They include modulated extrusions of 2D Escher tilings, freeform deformations of the fundamental domain of various 3D symmetry groups, highly symmetrical polyhedral toroids of genus 1, highergenus cage structures derived from the cubic lattice as well as from the diamond and triamond lattices, and finally interlinked tiles with the connectivity of simple knots.
Recent progress in sphere packing
"... Most of this article will be about comparatively minor progress in the sphere packing problem in higher dimensions, but it is a pleasure to record that at last the original 3dimensional density problem has been de nitively solved. The attempted proof by Hsiang [Hs93] unfortunately turned out to be ..."
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Cited by 2 (0 self)
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Most of this article will be about comparatively minor progress in the sphere packing problem in higher dimensions, but it is a pleasure to record that at last the original 3dimensional density problem has been de nitively solved. The attempted proof by Hsiang [Hs93] unfortunately turned out to be incomplete.
Nets in groups, minimum length gadic representations, and minimal additive complements, arXiv: 0812.0560
, 2008
"... Abstract. The number theoretic analogue of a net in metric geometry suggests new problems and results in combinatorial and additive number theory. For example, for a fixed integer g ≥ 2, the study of hnets in the additive group of integers with respect to the generating set Ag = {0} ∪ {±g i: i = 0 ..."
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Cited by 1 (1 self)
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Abstract. The number theoretic analogue of a net in metric geometry suggests new problems and results in combinatorial and additive number theory. For example, for a fixed integer g ≥ 2, the study of hnets in the additive group of integers with respect to the generating set Ag = {0} ∪ {±g i: i = 0, 1,2,...} requires a knowledge of the word lengths of integers with respect to Ag. A gadic representation of an integer is described that algorithmically produces a representation of shortest length. Additive complements and additive asymptotic complements are also discussed, together with their associated minimality problems. 1. Nets in metric spaces Let (X, d) be a metric space. For z ∈ X and r ≥ 0, the sphere with center z and radius r is the set Sz(r) = {x ∈ X: d(x, z) = r}. The open ball Bz(r) of radius r and center z and the closed ball Bz(r) of radius r and center z are, respectively, Bz(r) = {x ∈ X: d(x, z) ≤ r} = ⋃ Sz(r ′) and r ′ <r Bz(r) = {x ∈ X: d(x, z) ≤ r} = ⋃ r ′ ≤r Sz(r ′). An rnet in (X, d) is a subset C of X such that, for all x ∈ X, there exists z ∈ C with d(x, z) ≤ r. Equivalently, C is an rnet in X if and only if X = ⋃ Bz(r). z∈C Note that X is the unique 0net in X. The set C is a net in X if C is an rnet for some r ≥ 0. The set C in X is called rseparated if d(z, z ′) ≥ r for all z, z ′ ∈ C with z ̸ = z ′. By Zorn’s lemma, every metric space contains a maximal rseparated set, and a maximal rseparated set is an rnet in X. A minimal rnet in a metric space (X, d)
Library of Congress Catalog Card Number 85600641 For sale by the Superintendent of Documents
, 1986
"... Medicare coverage of physician services for elderly and disabled beneficiaries improves their financial access to medical care. But Medicare’s payment methods have also fueled increases in expenditures for physician services, which are now one of the most rapidly growing parts of the Federal budget. ..."
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Medicare coverage of physician services for elderly and disabled beneficiaries improves their financial access to medical care. But Medicare’s payment methods have also fueled increases in expenditures for physician services, which are now one of the most rapidly growing parts of the Federal budget. The method of customary, prevailing, and reasonable charge payment is inherently inflationary and contains incentives for providers to use additional and more expensive services. To curtail continuing increases in expenditures for physician services, the Deficit Reduction Act of 1984 (Public Law 98369) froze physician charges to Medicare beneficiaries for 15 months beginning July 1, 1984. That act also mandated OTA to examine alternative methods of paying for physician services in order to guide payment reform. The House Energy and Commerce Committee, the House Ways and Means Committee, and the Senate Finance Committee have jurisdiction over physician services under Medicare and that section of the act. The Senate Special Committee on Aging also requested OTA to study the effect of physician payment methods on the use of medical technology. In preparing this report, OTA staff drew on the expertise of members of the advisory