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Topic: Steiner triple system

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	Dictionary of Combinatorics -- S
		A Steiner Triple System of order v is a collection of triples or 3-subsets of a set X of size v such that each pair of elements of X occurs in exactly one triple.
		In other words a Steiner Triple System is a 2-design with parameters (v, 3, 1, (v-1)/2, v(v-1)/6).
		Kirkman showed that this is a sufficient condition that a Steiner Triple System exist.
	www.southernct.edu /~fields/comb_dic/S.html (533 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Since $248$ is a triple, we have that 4 occurs only as a row or an element, and since $145$ is a triple, we have that 4 occurs only as a column or an element.
		Since $49A$ is a triple, we have that 9 occurs only as a row or a column, and since $579$ is a triple, we have that 9 occurs only as a row or an element.
		Since $678$ is a triple, we have that $6$ occurs only as a column or an element, and since $136$ is a triple, we have that $6$ occurs only as a row or a column.
	www.eskimo.com /~rwb/papers/trades.tex (5616 words)

	[No title]
		As we can see all the elements {1,,12} are partition into triples, every element appears in a block once, and the sum of 1+5=6, 2+8=10, 4+7=11, and 3+9=12 so we have a Skolem triple system of order 4 and hence yield a cyclic STS(25).
		Note that, if k=1, we omit those triples defined for 1£r£k-1; the remaining triples are {1,11,12}, {3,5,8}, {2,7,9}, and {4,6,10} which form a Skolem triple system of order 4 different from the one in example 8a.
		Note that, if k=1, we omit those triples defined for 1£r£k-1; the remaining triples are {1,14,15}, {3,6,9}, {5,8,13}, {2,10,12}, and {4,7,11} which form a Skolem triple system of order 5 different from the one in example 8b.
	courses.csusm.edu /math540ak/STS.doc (2133 words)

	Introduction and ToC: Linearly derived Steiner triple systems (Site not responding. Last check: 2007-10-21)
		To find a ``linear extension'' of a Steiner triple system one restricts one's attention to those 4-subsets of the underlying set of the system that are symmetric differences of two triples --- that is, those 4-subsets that are the support of the binary sum of the incidence vectors of two triples.
		There are Steiner triple systems whose graphs have chromatic number two; they are linearly derived and seem to form an interesting class of systems, which, as far as we know, have never been considered.
		Just as the geometric binary systems are characterized as those systems with the maximum number of quadrilaterals, the Hall triple systems are characterized as those systems with the maximum number of mitres.
	www.lehigh.edu /efa0/public/www-data/lindertoc.html (902 words)

	Steiner system (Site not responding. Last check: 2007-10-21)
		A Steiner S(l, m, n) system is an n -element set S together with a set of m -element subsets of S (which wewill call the special m -element subsets) with the property that each l -element subset of S iscontained in exactly one of the special m -element subsets.
		A Steiner S(2,3, n) system is often called a Steiner triple system, and its special 3-element subsetsare called triples.
		If S is a Steiner triple system, we can define a multiplication on it by setting aa = a for all a in S, and ab = c if { a, b, c } isa triple.
	www.therfcc.org /RFCC/steiner-system-283954.html (415 words)

	Steiner Triple System (Site not responding. Last check: 2007-10-21)
		is called a Steiner triple system and is a special case of a
		Steiner triple systems of order 19 (Stinson and Ferch 1985; Colbourn and Dinitz 1996, p.
		Colbourn, C. and Dinitz, J. (Eds.) ``Steiner Triple Systems.'' §4.5 in
	www.math.sdu.edu.cn /mathency/math/s/s720.htm (121 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		On 2-ranks of Steiner triple systems Our main result is an existence and uniqueness theorem for Steiner triple systems which associates to every such system a binary code --- called the ``carrier'' --- which depends {\it only\/} on the order of the system and its 2-rank.
		When the Steiner triple system is of 2-rank less than the number of points of the system, the carrier organizes all the information necessary to construct directly {\it all systems of the given order and $2$-rank\/} from Steiner triple systems of a specified smaller order.
		We also discuss Steiner quadruple systems and prove an analogous existence and uniqueness theorem; in this case the binary code (corresponding to the carrier in the triple system case) is the dual of the code obtained from a first-order Reed-Muller code by repeating it a certain specified number of times.
	www.combinatorics.org /Volume_2/Abstracts/v2i1r9.tex (260 words)

	Encyclopaedia of Design Theory: Steiner triple systems (Site not responding. Last check: 2007-10-21)
		A Steiner triple system consists of a set X of n points, and a collection B of subsets of X called blocks or triples, such that each block contains exactly 3 points, and any two points lie together in exactly one block.
		A binary representation: the points are all triples of binary digits except 000, and the blocks are all sets of three such triples with sum zero (for example, {110,101,011}).
		A Kirkman triple system is a resolvable Steiner triple system; that is, one whose blocks can be partitioned into "parallel classes" so that each point lies on a unique block of each parallel class.
	www.designtheory.com /library/encyc/sts/g (536 words)

	Introduction and ToC: On 2-ranks of Steiner triple systems (Site not responding. Last check: 2007-10-21)
		Thus the systems of full 2-rank are seen as the building blocks since, from the binary point of view, Steiner triple systems of full 2-rank must be viewed as unintelligible and hence taken as given facts of life.
		In particular, then, we show that the binary code of a Steiner triple system is completely determined by its 2-rank; this explains why Tonchev and Weishaar found only five codes (one for each dimension between eleven and fifteen) among the eighty Steiner triple systems on fifteen points.
		The construction of all triple systems of deficient 2-rank is treated in Section 4 and Section 5 discusses some particularly easy cases of the construction.
	www.lehigh.edu /efa0/public/www-data/onSTStoc.html (1200 words)

	Steiner triple systems (Site not responding. Last check: 2007-10-21)
		A Steiner triple system (of order n) STS(n) is a 2-(n,3,1) design, that is, a Steiner system S(2,3,n), in other words, a collection of 3-subsets of an n-set such that any pair of elements of the n-set is contained in a unique one among these 3-sets.
		Up to isomorphism, there are 2 Steiner triple systems of order 13, and 80 Steiner triple systems of order 15.
		The block graph of a Steiner triple system is the graph with these 3-sets as vertices, where two 3-sets are adjacent when they have nonempty intersection.
	www.win.tue.nl /~aeb/drg/graphs/STS.html (214 words)

	Finite Geometries?
		Steiner is known for a variety of contributions to geometry, including work on isoperimetric problems (what region in the plane has the largest area with a fixed perimeter?) and projective geometry (geometry where all lines meet).
		Given that numbers and number systems are at the very heart of mathematics, it is surprising that the development of finite arithmetics systems by the mathematics community came so late.
		It was Evariste Galois (1811-1832) who in conjunction with his efforts to show that polynomial equations of the fifth degree (quintics) could not be solved by providing formulas for their roots (as is true for polynomial equations of degree 1 to 4) first discovered the idea of finite arithmetics.
	www.ams.org /featurecolumn/archive/finitegeometries.html (5303 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		A dexagon triple is a configuration consisting of 6 triangles whose "inside edges" form a copy of K_4.
		A dexagon triple system is a pair (X,D), where D is a collection of edge disjoint dexagons which partitions 3K_n (= every pair of vertices are joined by 3 edges).
		We show that a necessary and sufficient condition of the existence of a perfect dexagon triple system is n \equiv 1 (mod 12).
	www.maths.uq.edu.au /cdmc/sem2aug04 (360 words)

	Problem
		By "extending" a STS to an embedding we mean an embedding in which all blocks of the triple system form facial cycles.
		However, it is known [5] that there is a pair of Steiner triple systems with n = 15 that cannot be embedded together on an orientable surface.
		But perhaps if n is large enough, arbitrary pairs of Steiner triple systems (satisfying obvious necessary parity conditions) could still be embedded on an orientable surface.
	www.fmf.uni-lj.si /~mohar/Problems/P0211KnFlexibility.html (552 words)

	Anstice (Site not responding. Last check: 2007-10-21)
		During his time at Wigginton, Anstice became interested in the mathematical work of another rector, Kirkman, who had written on the subject of Steiner triple systems (as they are now called).
		In one of his papers Kirkman gave an elegant construction of a resolvable Steiner triple system on 15 elements (the famous Kirkman 15 schoolgirls problem), making use of what are now known as a Room square of order 8 and the Fano plane.
		Infinite families of cyclic Steiner triple systems and Room squares are constructed [ in the papers ].
	www.gap-system.org /~history/Mathematicians/Anstice.html (689 words)

	Clearing up the market cycle... best Steiner Triple System (Site not responding. Last check: 2007-10-21)
		Steine r Triple System -- from MathWorld Steiner Triple System -- from MathWorld Let X be a set of v\geq 3 elements together with a set B of 3-subset (triples) of X such that every 2-subset of X occurs in exactly one triple of B. Then B is...
		Bicoloring Steiner Triple Systems Bicoloring Steiner Triple Systems A Steiner triple system has a bicoloring with m color classes if the points are partitioned into m subsets and the three points in every block are contained in exactly two of...
		A Steiner S(2,3,n) system is often called a Steiner triple system, and its special 3...
	ascot.pl /th/Fourier5/Steiner-Triple-System.htm (637 words)

	Darryn Bryant's Talk (Site not responding. Last check: 2007-10-21)
		A Steiner triple system of order v is a partition of the edges of the complete graph on v vertices into triangles.
		A partial Steiner triple system of order v is a partition of some subset of the edges of the complete graph of order v into triangles.
		Not every partial Steiner triple system of order v can be completed to a Steiner triple system of order v, but it is known that any partial Steiner triple system can be completed to a Steiner triple system of larger order.
	www.cs.rhul.ac.uk /~simonb/darryn.html (198 words)

	glossary
		Steiner System - Steiner System S(v,k,t,=b) is also a t-design with usually l=1.
		Steiner Triple System A Steiner Triple System of order v is a collection of triples or 3-subsets of a set X of size v such that each pair of elements of X occurs in exactly one triple.
		In other words a Steiner Triple System is a 2-design with parameters (v, 3, 1, (v-1)/2, v(v-1)/6).Since the design parameters must be integers, it is necessary that v = 6n + 1 or v = 6n + 3.
	members.tripod.com /~TheLotteryInstitute/glossary.htm (2977 words)

	PlanetMath: incidence structure
		Again, there may be several non-isomorphic systems with the same values of the parameters, and no systems at all for certain combinations of values.
		(often, the system is so symmetric that it makes no difference which point you choose).
		If any of the divisibility conditions on the way there do not hold, there cannot exist a Steiner system with the original parameters either.
	planetmath.org /encyclopedia/Block2.html (982 words)

	DCI 2000 Research Program Abstracts - Week 1
		A Steiner triple system of order v, STS(v), is a partition of the edges of the complete graph on v vertices into triangles (called blocks).
		Unfortunately many software systems are "fragile" in the sense that they are unreliable, suffer from security and performance lapses, and are difficult to maintain and upgrade to satisfy new demands.
		We consider colourings of Steiner triple systems and Steiner systems S(2,4,v) in which blocks have prescribed colour patterns, as a refinement of classical weak colourings.
	www.dimacs.rutgers.edu /dci/2000/abstractswk1.html (3602 words)

	Quantum continuity
		Colbourn and Rosa write: = = "Any Steiner triple system may be provided with = a structure of a Steiner quasigroup by setting = = z=x*y = = when {x,y,z} is a block of the design or when = = x=y=z" = = Another important concept is a factorization.
		Colbourn and Rosa write: "Any Steiner triple system may be provided with a structure of a Steiner quasigroup by setting z=x*y when {x,y,z} is a block of the design or when x=y=z" Another important concept is a factorization.
		The important theorem that tells us when a triple is necessarily unquantized is stated as "A TS(v,k) having a maximum PPC [partial parallel class] of floor(v/3) exists for every admissible pair (v,k) except for (6,2) and (7,1)." The "floor" function is the "greatest integer less than" function.
	www.pych-one.com /new-715748-4765.html (12668 words)

	Good Math, Bad Math : Woo Math: Steiner and Theosophical Math
		OTOH, if I'm remembering my history right, Steiner didn't exactly invent Steiner triple systems (from which the general systems are generalized); the question of existence was posed in the British Isles, solved by Kirkman, and then later posed on the continent by Steiner (and solved by someeone else).
		Mostly they were just nice, kind parents who thought the school system too cold and uncaring, and who saw in Steiner pedagogics a potential way out, probably tempted by some of the genuinely good common-sense ideas in there (for instance that much of the day is devoted to the week's current topic).
		Steiner's work is littered with that kind of ultra-literalism, where abstract concepts are taken as absolute literal properties of the concrete universe.
	scienceblogs.com /goodmath/2006/11/woo_math_steiner_and_theosophi.php (8930 words)

	DCI 2000 Research Program Abstracts - Week 1
		A Steiner triple system of order v is a partition of the edges of the complete graph on v vertices into triangles.
		A partial Steiner triple system of order v is a partition of some subset of the edges of the complete graph of order v into triangles.
		Not every partial Steiner triple system of order v can be completed to a Steiner triple system of order v, but it is known that any partial Steiner triple system can be completed to a Steiner triple system of larger order.
	dimacs.rutgers.edu /dci/2000/abstractswk1.html (3591 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Abstract: Our main result is an existence and uniqueness theorem for Steiner triple systems which associates to every such system a binary code --- called the ``carrier'' --- which depends {\it only\/} on the order of the system and its 2-rank.
		When the Steiner triple system is of 2-rank less than the number of points of the system, the carrier organizes all the information necessary to construct directly {\it all systems of the given order and $2$-rank\/} from Steiner triple systems of a specified smaller order.
		We also discuss Steiner quadruple systems and prove an analogous existence and uniqueness theorem; in this case the binary code (corresponding to the carrier in the triple system case) is the dual of the code obtained from a first-order Reed-Muller code by repeating it a certain specified number of times.
	optnet.itwm.fhg.de /opt-net/documents/v95w17n4 (282 words)

	Graph Theory Glossary - sp-sz (Site not responding. Last check: 2007-10-21)
		For a given variant of the Steiner tree problem, the maximum possible ratio of the length of a minimum spanning tree of a set of terminals to the length of an optimal Steiner tree of the same set of terminals.
		The Steiner tree of some subset of the vertices of a graph G is a minimum-weight connected subgraph of G that includes all the vertices.
		A Steiner triple system (STS) is a collection of points and triples of these points such that every pair occurs in exactly one triple.
	www.cc.ioc.ee /jus/gtglossary/gtglos_sp_sz.htm (3177 words)

	JAKOB STEINER
		Steiner was one of the greatest of all geometers.
		Steiner's Ellipse Problem (that the ellipse circumscribing triangle ABC and having minimal area is the Steiner circum-ellipse, and that the ellipse inscribed in ABC and having maximal area is the Steiner in-ellipse; the two ellipses have as center the centroid of ABC)
		Steiner's Circle Problem (that of all plane surfaces having equal perimeter, the circle has the greatest area; and of all plane surfaces with equal area, the circle has the least perimeter)
	faculty.evansville.edu /ck6/bstud/steiner.html (776 words)

	Triple systems (Site not responding. Last check: 2007-10-21)
		A Steiner triple system is a non empty
		The usual counting formulas for Steiner systems imply that there are exactly
		The number of isomorphism types of the small Steiner triple systems is well known.
	www.math.uni-kiel.de /geometrie/klein/math/geometry/striple.html (112 words)

	[No title]
		Subject: Steiner Sysrems Date: Wed, 12 Jan 2000 19:13:53 +0200 Newsgroups: sci.math Summary: [missing] Hello, Steiner systems represent special solutions to an extremal set problem — satisfying a condition containing the words “exactly ones” is an extreme case for both “at most one” or “at least one”.
		wrote: > > > this is impossible for N > 4 because of Tit's > > inequality on the parameters of a Steiner system.
		A Steiner system is "trivial" if k = v (all points in one block) or if k = t (the blocks are simply all the subsets of size t).
	www.math.niu.edu /~rusin/known-math/00_incoming/steiner (1055 words)

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