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Topic: Closure mathematics

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	Closure (mathematics)
		For example, the closure of a subset of a group is the subgroup generated by that set.
		The closure is idempotent: the closure of the closure equals the closure.
		In linear algebra, the linear span of a set X of vectors is the closure of that set; it is the smallest subset of the vector space that includes X is closed under the operation of linear combination.
	www.algebra.com /algebra/college/abstract/Closure_(mathematics).wikipedia (1012 words)

	NationMaster - Encyclopedia: Closure (mathematics)
		In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.
		In combinatorial mathematics, a matroid is a structure that captures the essence of a notion of independence (hence independence structure) that generalizes linear independence in vector spaces.
		In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.
	www.nationmaster.com /encyclopedia/Closure-%28mathematics%29 (2699 words)

	NationMaster - Encyclopedia: Kuratowski closure axioms
		In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure.
		In topology, a praclosure operator, or ÄŒech closure operator is a map between subsets of a set, similar to a closure operator, except that it is not required to be idempotent.
		Mathematical axioms In topology, a praclosure operator, preclosure operator, or ÄŒech closure operator is a map between subsets of a set, similar to a closure operator, except that it is not required to be idempotent.
	www.nationmaster.com /encyclopedia/Kuratowski-closure-axioms (748 words)

	Closure (mathematics) - Biocrawler
		In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property.
		In linear algebra, the linear span of a set X of vectors is the closure of that set; it is the smallest subset of the vector space that includes X and is a subspace.
		In algebra, the closure of a set S under a binary operation is the smallest set C(S) that includes S and is closed under the binary operation.
	www.biocrawler.com /encyclopedia/Closure_%28mathematics%29 (365 words)

	Closure (mathematics) Summary
		"Closure" is a property which a set either has or lacks with respect to a given operation.
		Closure can be associated with operations on single numbers as well as operations between two numbers.
		Although closure is usually thought of as a property of sets of ordinary numbers, the concept can be applied to other kinds of mathematical elements.
	www.bookrags.com /Closure_(mathematics) (1481 words)

	MathCog Idiocy: What is closure?
		In mathematics, there is a concept for closure that is common across most (possibly all) branches of mathematics.
		· The closure is idempotent: the closure of the closure equals the closure.
		· In matroid theory, the closure of X is the largest superset of X that has the same rank as X. In set theory, the transitive closure of a binary relation.
	mathcogidiocy.blogspot.com /2006/03/what-is-closure.html (1430 words)

	RELATIONAL CLOSURE:
		This concept of algebraic closure of a transformation system illustrates some of the important features of the concept we are looking for, but it is not general enough for the task of modelling complex systems by picking out all the relevant distinctions.
		This means that the complement of a closure (in the sense of complete absence of the missing elements, not in the sense of incomplete presence) can in general also be interpreted as a closure.
		Fourth, certain types of closure may be seen as generalizations or specializations of other types of closure, in the sense that a more general closure is characterized by less strict requirements, and hence is less distinction-reducing or redundancy-generating.
	pespmc1.vub.ac.be /papers/RelClosure.html (3936 words)

	Mathematics
		Mathematics 3 and 8 cover the basic calculus of functions of a single variable, as well as vector geometry and calculus of scalar-valued functions of several variables.
		Mathematics 17, "An Introduction to Mathematics Beyond Calculus", is a course designed for students interested in learning about some of the aspects of mathematics not usually encountered in the first years of mathematical studies.
		Prerequisite: Mathematics 8, or Mathematics 3 and 6.
	www.dartmouth.edu /~reg/courses/desc/math.html (7962 words)

	Closure Medical -- Recommendations and Resources (Site not responding. Last check: )
		In mathematics, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed.
		An equivalent definition is that ''R'' is integrally closed in ''S'' iff the integral closure of ''R'' in ''S'' is equal to ''R'' (in general the integral closure is a superset of ''R'').
		The integral closure of Z in the complex numbers C is the set of all algebraic integers.
	www.becomingapediatrician.com /health/33/closure-medical.html (1330 words)

	Closure operator - ExampleProblems.com
		The name comes from the fact that forming the closure of subsets of a topological space has these properties if the set of all subsets is ordered by inclusion ⊆.
		Given a closure operator C, a closed element of P is an element x that is a fixed point of C, or equivalently, that is in the image of C.
		The closure operators on the partially ordered set P are then nothing but the monads on the category P.
	www.exampleproblems.com /wiki/index.php?title=Closure_operator&printable=yes (583 words)

	Institute of Mathematics and Computer Science University of Latvia - Department of Mathematics
		In the field of mathematical modeling main efforts are devoted to elaboration of mathematical models, difference schemes and programs for describing electro-kinetic processes, flows of gas and liquid matter in non-homogeneous media with rapidly varying coefficients, and phase transitions.Transport kinetic models are based on nonlinear Fokker-Planck,diffusion-absortion, advection-diffusion, Schrodinger and master equations.
		On the weak closure of sets of feasible states for linear elliptic equations in the scalar case.
		On strong closure of the graphs associated with families of elliptic operators.
	www.lumii.lv /math.htm (629 words)

	"Lack of Closure in Mathematics Teaching andLearning: Understanding Primary Teachers'Beliefs and Understandings in ...
		Although lack of closure is perceived to be at odds with some beliefs about the value of open-ended mathematics teaching and learning, the idea seems nevertheless to be an appealing one.
		She seems to refer to a lack of closure in what’s it got to do with teaching Maths in the primary school and is typical in referring positively to the ideas being challenging.
		The CAME concept of lack of closure in terms of children’s thinking is distinct from their ideas of openness and closure and is at odds with more traditional task-orientated ideas of differentiation.
	www.leeds.ac.uk /educol/documents/000000928.htm (6238 words)

	Mathematics - Apronus.com
		Provenmath - mathematical project which states the foundations of mathematics and derives mathematical knowledge from these foundations.
		Recommended Math Books - list of mathematical books recommended by Apronus; displays the tables of contents of the recommended books, so that you can get a good idea about what the books are about.
		In principle, it can be used to write mathematical content of any complexity...
	www.apronus.com /math/math.htm (406 words)

	Closure Spaces
		The concept of uniquely generated closure spaces has begun to be studied as a common thread emerging in computer applications, in graphs, and in discrete geometries.
		The importance of uniquely generated closure spaces lies in the fact that in discrete systems they play a role that is in many respects analogous to the vector spaces of classical mathematics.
		Algorithmic closure, in particular that of greedy algorithms is found in [KLS91] which introduces the term ``greedoid'', a special kind of antimatroid.
	www.cs.virginia.edu /~jlp/closure.overview.html (842 words)

	Closure
		For the closure of a set in topology and related branches, see Topological closure.
		For the algebraic closure of a field, see Algebraic closure.
		Closure is also a property of binary operations.
	www.ebroadcast.com.au /lookup/encyclopedia/cl/Clausure.html (92 words)

	Closure - Wikipedia, the free encyclopedia
		Closure (mathematics), the smallest object that both includes the object as a subset and possesses some given property
		Closure (topology), the set of all points intuitively "close to" a given set
		Closure (psychology), the state of experiencing an emotional conclusion to a difficult life event
	en.wikipedia.org /wiki/Closure (213 words)

	Amazon.com: Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being: Books: George ... (Site not responding. Last check: )
		Their basic premise, that all of mathematics is derived from the metaphors we use to maneuver in the world around us, is easy enough to grasp, but following the reasoning requires a willingness to approach complex mathematical and linguistic concepts--a combination that is sure to alienate a fair number of readers.
		By attacking the transcendental nature of mathematics, and elaborating the grounding of mathematical thought in the metaphorical mapping of the mind, many important implications arise ranging from the meaning of mathematics, the way mathematics is practiced and proofs are formulated, to the way mathematics should be taught.
		The first is with their technique of "mathematical idea analysis", in which they state that a particular metaphor is being applied in some area of mathematics (between two mathematical domains or between a mathematical domain and some conception of the real world), and then provide an explicit mapping between concepts in the two domains.
	www.amazon.com /Where-Mathematics-Comes-Embodied-Brings/dp/0465037712 (3651 words)

	AMS Website News 2004 (Site not responding. Last check: )
		She was recently awarded the Canadian Mathematical Society's Krieger-Nelson Prize, has served as a Vice President of the Society for Industrial and Applied Mathematics, and is incoming President of the Association for Women in Mathematics.
		The Grand Prize winner of the 2003 Biographies of Contemporary Women in Mathematics essay contest of the Association for Women in Mathematics is Esther Feldblum of Maimonides School in Sharon, MA, for her essay, "Dr. Harpreet Chowdhary: The Mathematician as Executive." Read this and the prizewinning essays in the Grades 6-8, 9-12, and College catgeories.
		The American Institute of Mathematics has announced that Jacob Lurie is the recipient of the 2004 AIM Five-Year Fellowship, which is awarded each year to an outstanding new Ph.D. student in an area of pure mathematics.
	www.ams.org /dynamic_archive/home-news-2004.html (4141 words)

	Internet Math Lesson Plans (Site not responding. Last check: )
		The pitch of a musical note is determined by the frequency or vibrations per second, of the sound waves emanating from the instrument.
		The closure will be given by random volunteers whichwill be eager to summarize their learned knowledge.
		Mathematics teacher at the 21st Century Preparatory School.
	www.utc.edu /~thecmath/lesson_plans/DeborahTaylor/thec98dtaylor.html (220 words)

	Karen E. Smith (Site not responding. Last check: )
		Although she always loved mathematics and wanted to be a mathematician from a young age, she did not realize that one could have a career as a mathematician until college, when her freshman calculus teacher, Professor Charles Fefferman, suggested it.
		After teaching high school mathematics for a year, she looked into the possibilities of graduate school and learned that one could actually get full support to work on a Ph.D. At this point, she decided to make a big change, and went off to the midwest for graduate school.
		The Ruth Lyttle Satter Prize in Mathematics is awarded to Karen E. Smith of the University of Michigan for her outstanding work in commutative algebra, which has established her as a world leader in the study of tight closure, an important tool in the subject introduced by Hochster and Huneke.
	www.agnesscott.edu /lriddle/women/smithk.htm (471 words)

	ARCC Workshop: Integral Closure, Multiplier Ideals and Cores (Site not responding. Last check: )
		Loosely speaking, the integral closure of an ideal I is an ideal contained in the radical of I that shares a number of finer properties with I.
		Determining the integral closure of I is a difficult task, which essentially amounts to finding solutions in the ring itself of special polynomial equations whose coefficients belong to higher and higher powers of I.
		The aspects intimately connected to the integral closure that we are planning to focus on are: computation of the integral closure and its complexity; multiplicities and equisingularity theory; cores of ideals and Briancon-Skoda type theorems; multiplier ideals and test ideals; multiplier ideals and jet schemes.
	www.aimath.org /ARCC/workshops/integralclosure.html (387 words)

	Science Structure Dialogue Response
		The drive for closure makes us seek certainty but it is in the nature of the world and of mathematics that we cannot achieve ultimate certainty (although practical, useful certainty is obtainable in many things).
		Using the language of mathematics, Gödel has shown (self-referentially) that we cannot obtain closure in mathematics in the sense of a set of rules that is both consistent and complete.
		These characteristics of mathematics and science -- no matter how subjectively disturbing they may be to some -- may be viewed optimistically as assuring us that there will never be an end to interesting questions to explore.
	www.dcn.davis.ca.us /~sander/sci_struc/scistr1.html (677 words)

	I P M - Bulletin Board - Events (Site not responding. Last check: )
		The theory of tight closure was started by Hochster and Huneke in the early 1980s, and since then it has proved to be a powerful theory.
		This minicourse will start with the basics of tight closure, proceed to the early results which proved the power of the theory, and end up with the recent developments on the primary decompositions of ideals.
		Tight closure is defined for modules over rings containing fields.
	www.ipm.ac.ir /ipm/Activities/ViewEventInfo.jsp?ETID=168 (254 words)

	Internet Math Lesson Plans (Site not responding. Last check: )
		The pitch of a musical note is determined by the frequency or vibrations per second, of the sound waves emanating from the instrument.
		The closure will be given by random volunteers whichwill be eager to summarize their learned knowledge.
		Mathematics teacher at the 21st Century Preparatory School.
	oneweb.utc.edu /~thecmath/lesson_plans/DeborahTaylor/thec98dtaylor.html (220 words)

	Chapter 9
		The logical closure of mathematics has so far not been possible, tough it has been looked for by a large number of mathematicians, starting with those of the school of logics (Leibnitz, De Morgan, Boole, Frege, Russell, Whitehead) and continuing with those of the school of formalism (Hilbert, von Neumann, etc.)
		The problem of the logical closure of mathematics is known as the problem of consistence of formal systems.
		They proved that the problem of consistence regarding the great chapters of mathematics can be reduced to that of the consistence of the natural numbers arithmetic, which in turn can be put in correspondence with the set theory.
	www.racai.ro /books/doe/chap9-2.html (735 words)

	Set Closure -- from Wolfram MathWorld
		The term "closure" is also used to refer to a "closed" version of a given set.
		The closure of a set can be defined in several equivalent ways, including
		In topologies where the T2-separation axiom is assumed, the closure of a finite set
	mathworld.wolfram.com /SetClosure.html (189 words)

	[No title]
		Closures are also useful in real-life applications (business and social settings, transportation, etc).
		¡:=i¦"ª¦ óÒ¨Closures of Relations ¡$ª ¨Definition: Let R be a relation on a set A. R may or may not have some property P, such as reflexivity, symmetry, or transitivity.
		Solution:¡@¯ "" "ª® ó¨Closures of Relations¡(ª ¨ÐExample 4 (again): Find the reflexive, symmetric and transitive closures of the relation: {(a,a), (b,b), (c,c), (a,c), (a,d), (b,d), (c,a), (d,a)} on the set S = {a,b,c,d}.
	www.cs.umb.edu /~khsuyan/note/Session21.ppt (651 words)

	Irena Swanson
		These are the notes from the mini course I gave at IPM (Institute for Studies in Theoretical Physics and Mathematics) in Tehran, Iran, in January 2002.
		The previous paper is superseded by the much better paper: Associated primes of local cohomology modules and of Frobenius powers (with Anurag Singh), International Mathematics Research Notices 33 (2004) 1703-1733.
		On free integral extensions generated by one element (with Orlando Villamayor), to be published in 'Commutative Algebra with a focus on geometric and homological aspects', Proceedings of Sevilla, June 18-21, 2003 and Lisbon, June 23-27, 2003.
	www.reed.edu /~iswanson/papers.html (1165 words)

	Fuzzy Closure Operators with Truth Stressers -- Belohlávek et al. 13 (5): 503 -- Logic Journal of IGPL
		We study closure operators and closure structures in a fuzzy
		A is included in B then the closure of A is included in the
		Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues.
	jigpal.oxfordjournals.org /cgi/content/short/13/5/503?rss=1 (230 words)

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