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# Topic: Closure operator

###### In the News (Sun 18 Aug 19)

 Closure (mathematics) - Wikipedia, the free encyclopedia An operation of a different sort is that of finding the limit points of a subset of a topological space (if the space is first countable, it suffices to restrict consideration to the limits of sequences but in general one must consider at least limits of nets). 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. en.wikipedia.org /wiki/Closure_(mathematics)   (978 words)

 Closure (topology) - Wikipedia, the free encyclopedia The closure of an intersection of sets is always a subset of (but need not be equal to) the intersection of the closures of the sets. In a union of finitely many sets, the closure of the union and the union of the closures are equal; the union of zero sets is the empty set, and so this statement contains the earlier statement about the closure of the empty set as a special case. The closure of the union of infinitely many sets need not equal the union of the closures, but it is always a superset of the union of the closures. en.wikipedia.org /wiki/Closure_(topology)   (1176 words)

 Grok/QL operators   (Site not responding. Last check: 2007-10-08) The result of such an operation is a set (or a relation, if the original relation had more than two columns) containing such tuples from the original relation that their entries in the projection column are equal to one of the members of the set being projected. The domain operator can be applied to relations with any number of columns, but is most meaningful for two-column relations, which can be seen as mapping functions having a domain and a range; when applied to a set, the domain operation returns the set itself. The range operator can be applied to relations with any number of columns, but is most meaningful for two-column relations, which can be interpreted as mapping functions with a domain and a range; when applied to a set, the range operation returns the set itself. swag.uwaterloo.ca /~nsynytskyy/grokdoc/operators.html   (2865 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. Briefly, a closure operator CL is said to be uniquely generated if for any closed set there a unique minimal subset for which this is the closure. Closure operators are fairly common, although they frequently have other names, for example ``convexity''. www.cs.virginia.edu /~jlp/closure.overview.html   (792 words)

 Section: 22225 - 22237. Article 1. Subchapter 3. Chapter 6. Title 27. California Code of Regulations. A private operator may combine a financial means test with a guarantee only if, for the purpose of meeting the requirements of the financial means test, the financial statements of the operator are not consolidated with the financial statements of the guarantor. An operator using a trust fund shall maintain documentation of the remaining capacity filled during the past year for each disposal facility covered by the fund for each year of the buildup period and a copy of the trust agreement and statements verifying the current balance of the fund. An operator using the insurance and environmental fund shall maintain the original or a copy of the comprehensive general liability insurance coverage certification and a copy of the environmental liability fund agreement and statements verifying the current balance of the environmental liability fund. www.ciwmb.ca.gov /regulations/title27/ch6sb3a.htm   (3661 words)

 [No title]   (Site not responding. Last check: 2007-10-08) This closure operator (let us assume that the composition is G = P.R, it is, it receives and returns a set of attributes) is used to determine the set of closed sets of attributes. The closure operator plays a central role in FCA, since it produces the set of closed itemsets that are later used to generate a concept lattice, or implication rules or, as it is now the case, functional dependencies. We will prove that this closure operator (which will be the composition of a Galois connection) allows to derive the functional dependencies that hold on a given dataset, in the same way that implication rules were derived: using the generators and the closed itemsets provided by this operator. www.lsi.upc.es /~jbaixer/recerca/dep_abstract.html   (1667 words)

 Campo Band of Kumeyaay (Mission) - Campo Environmental Protection Agency Regulation Title 5 Solid Waste Management   (Site not responding. Last check: 2007-10-08) The operator shall be responsible for accurate characterization of solid waste, including determinations of whether or not solid waste will be compatible with containment features and other solid waste at a solid waste facility and whether or not solid waste is required to be managed as hazardous waste. The operator shall not commence closure of such a discrete unit until CEPA has approved the final closure and post-closure maintenance plans for proposing to implement any one or a combination of individual closure activities shall obtain approval of the final closure and post-closure maintenance plans before proceeding to implement closure or post-closure maintenance activities. The purposes of the preliminary closure plan are to: (A) Allow the operator to prepare an estimate of closure costs; (B) Enable CEPA to assess the reasonableness of the cost estimate; and (C) Allow a registered civil engineer to certify to the accuracy of the cost estimate. www.nplnews.com /toolbox/tribal/58.html   (14011 words)

 closure operators   (Site not responding. Last check: 2007-10-08) Today in class, I said that a topology on X was a collection of subsets of X (called the open sets), containing X and the empty set, which is closed under finite intersection and arbitrary union. That is, even when their primary objective isn't defining the closed sets in some topology as the sets that are fixed by the closure operator in question. Moral: there is always a topology generated by a closure operator, but it may be on a somewhat abstract space. www.math.harvard.edu /~jessica/math101/closure.html   (297 words)

 [No title] This is what makes it possible to speak of the functor C as *the* completion (or closure) operator with respect to the notion of completion embodied by the subcategory C_0. For closure, I think you would do very much the same thing, where you call an object A in X closed if C(A) = A, i.e, C is a closure operator. In my work, the problem was that I was starting with a collection of objects X' (spacetimes) and a sort of completion operator C (addition of the future causal boundary), but for A in X', C(A) was not in X' (a topological or causal space, but not a spacetime). www.math.niu.edu /~rusin/known-math/99/closure_op   (718 words)

 PA Bulletin, Doc. No. 99-744c (1) An owner or operator, or the agent of the owner or operator, may not accept hazardous waste for treatment, storage or disposal unless it is accompanied by a manifest approved by the Department, unless a manifest is not required by 40 CFR 262.20(e) (relating to the manifest general requirements). Applicant--An owner or operator of a hazardous waste treatment, storage or disposal facility which is attempting to demonstrate the capability to self-insure all or part of its liabilities to third persons for personal injury and property damage from sudden or nonsudden pollution occurrences, or both. An owner or operator of a new facility shall submit the bond to the Department at least 60 days before the date that hazardous waste is first received for treatment, storage or disposal. www.pabulletin.com /secure/data/vol29/29-18/744c.html   (5389 words)

 Waste Tire Facility Permits - Title 14 CCR, Division 7, Chapter 6 The operator shall adjust the closure cost estimate for inflation within 60 days of the anniversary date of the establishment of the financial mechanism for closure costs. After receiving the documentation for closure activities, the Board or its designee shall determine whether the closure expenditures are in accordance with the closure plan or otherwise justified. (b) An operator shall be deemed to be without the financial assurances in the event of bankruptcy of its provider, or in the event of a suspension or revocation of the authority of the provider to issue such coverage. www.ciwmb.ca.gov /Regulations/Title14/ch6a9.htm   (3419 words)

 The Meta-Evaluator   (Site not responding. Last check: 2007-10-08) Extraction of the expression, operator and operator definition of the current continuation closure of the level are just as in the standard evaluator, as is the automatic conversion of unknown operators into procedure calls. The Platypus operator (or Lisp macro, in the meta-evaluator) The shadow operators are implemented in a similar way to those in Platypus89, but the C code that defines them must be pre-processed before compilation, to insert some of the standard helper code for taking apart levels and closures, and for evaluating arguments for operators that simply want all their arguments evaluated. www.cb1.com /~john/thesis/chapters/platypus89.html   (7852 words)

 On the pullback stability of a quotient map with respect to a closure operator   (Site not responding. Last check: 2007-10-08) There are well-known characterizations of the hereditary quotient maps in the category of topological spaces, (that is, of quotient maps stable under pullback along embeddings), as well as of universal quotient maps (that is, of quotient maps stable under pullback). In this paper hereditary and stable quotient maps are characterized in the broader context given by a category equipped with a closure operator. To this end, we derive explicit formulae and conditions for the closure in the codomain of such a quotient map in terms of the closure in its domain. www.tac.mta.ca /tac/volumes/8/n6/8-06abs.html   (154 words)

 SUPPORTING STATEMENT FOR   (Site not responding. Last check: 2007-10-08) Section 264.12(b) requires the owner or operator of a facility receiving off-site waste to send a one-time notice to the generator stating that he or she has the appropriate permits, and will accept, the waste the generator is shipping. Sections 264.12(c) and 265.12(b) require owners and operators transferring ownership of a facility during its operating life, or of a disposal facility during the post-closure care period, to notify the new owner or operator in writing of the requirements of 40 CFR Parts 264 and 270. In addition, owners or operators of tank systems or drip pads that intend to remove or decontaminate the hazardous waste at partial or final closure are required to submit a contingent post-closure plan, as specified in sections 264.197(c)(2) and 265.197(c)(2), and 264.575(c)(1)(ii) and 265.445(c)(1)(ii), respectively. www.epa.gov /opperid1/icr1999/1571.06/1571ss06.htm   (11586 words)

 Gabriele Castellini, Eraldo Giuli   (Site not responding. Last check: 2007-10-08) Abstract:A notion of hereditarity of a closure operator with respect to a class of monomorphisms is introduced. Let \$C\$ be a regular closure operator induced by a subcategory \$\Cal A\$. It is shown that, if every object of \$\Cal A\$ is a subobject of an \$\Cal A\$-object which is injective with respect to a given class of monomorphisms, then the closure operator \$C\$ is hereditary with respect to that class of monomorphisms. www.univie.ac.at /EMIS/journals/CMUC/cmuc9201/abs/caste.htm   (82 words)

 PA Bulletin, Doc. No. 99-744e (e) The Department will not issue a closure or postclosure certification for a facility causing adverse effects on the public health, safety or welfare or the environment, creating a public nuisance, or is in violation of this article, the act or the statutes set forth in section 505(a) of the act (35 P. § 6018.505(a)). (2) The owner or operator abandons the facility without providing closure or postclosure care, or otherwise fails to properly close the facility in accordance with this article, the act, the statutes in section 505(a) of the act, the terms and conditions of the permit or orders of the Department. An owner or operator of a waste oil-fired space heater that burns waste oil fuel under § 266a.41(b)(2) is exempt from this notification requirement. www.pabulletin.com /secure/data/vol29/29-18/744e.html   (5029 words)

 Computing Papers on Closure   (Site not responding. Last check: 2007-10-08) Language concatenation behaves as a multiplicative operator is a zero and { } is an identity. T he other was tosee whether of English sentences a the automatic syntactic module region operate s or not, according to the ambiguous in each sentence. A regular graph constraint is given by a graph G. A heap H satis es the constraint iff there exists a graph homomorphism from H to G. Regular graph constraints allow specifying properties of graphs in some given class of graphs C. computing.breinestorm.net /Closure   (2774 words)

 On the Closure Operator and the Closure System of Many Sorted Sets - Korni, On (ResearchIndex) On the closure operator and the closure system of many sorted sets. 5 the many sorted closure operator and the many sorted closure.. 1 the closure operator and the closure (context) - Madras, many et al. citeseer.ist.psu.edu /370654.html   (623 words)

 Mathematical Structures: Closure algebras   (Site not responding. Last check: 2007-10-08) A closure algebra is a modal algebra A = (A,∨,0, ∧,1,¬,◊) such that ◊ is closure operator: x ≤ ◊x and ◊◊x = ◊x. The operator ◊ is the possibility operator, and the necessity operator □ is defined as □x = ¬◊¬x. math.chapman.edu /cgi-bin/structures.pl?Closure_algebras   (102 words)

 Amazon.com: Categorical Closure Operators: Books: Gabriele Castellini   (Site not responding. Last check: 2007-10-08) This book presents the general theory of categorical closure operators together with examples and applications to the most common categories, such as topological spaces, fuzzy topological spaces, groups, and abelian groups. The main aim of the theory is to develop a categorical characterization of the classical basic concepts in topology via the newly introduced concept of categorical closure operators. "Categorical Closure Operators" is self-contained and can be considered as a graduate level text for topics courses in category theory, algebra, and topology. www.amazon.com /exec/obidos/tg/detail/-/0817642501?v=glance   (660 words)

 US PATENT SUBCLASS 49 / 52-- .~ Closure operator extends through grille   (Site not responding. Last check: 2007-10-08) (under subclass 50) Device provided with a mechanical means movable relative to the closure for imparting movement to the closure which means extends through the openwork panel. 356, for a single lever push rod operator for a movable closure. 290, and 322+ for an operator extending through a register grille to impart movement to a pivoted valve combined with ventilating structure, e.g., air duct, air directing structure. www.patentec.com /data/class/defs/49/52.html   (90 words)

 On the Closure Operator and the Closure System of Many Sorted Sets - Korni, On (ResearchIndex) Abstract: this paper definitions of many sorted closure system and many sorted closure operator are introduced. In this article closure system is absolutely multiplicative subset family of many sorted sets and in [10] is many sorted absolutely multiplicative subset family of many sorted sets. Analogously, closure operator is function between many sorted sets and in [10] is many sorted function from a many sorted set into a many... citeseer.ist.psu.edu /465619.html   (685 words)

 A Characterization Theorem for the Canonical Basis of a Closure Operator A Characterization Theorem for the Canonical Basis of a Closure Operator The purpose of this paper is to provide a characterization result for the canonical basis of an arbitrary closure operator. This theorem strengthens the result of Burosch, Demetrovics and Katona, who propose in [1] a characterization of the generating system of a closure operator defined by the quasi-closed sets of the closure operator. ideas.repec.org /p/fth/pariem/98.44.html   (197 words)

 Recombination Closure Sets The intuitive motivation for this development is that it might be interesting to know which part of the type set can be covered by recombining a given set of types to start with. All this holds for the general recombination operators defined in section 3. It may be worthwhile to consider recombination closure under the string recombination operators studied above. www.cbc.yale.edu /old/cce/papers/Reco/node11.html   (296 words)

 AMCA: F-net convergence with respect to a closure operator by Josef Slapal   (Site not responding. Last check: 2007-10-08) In a category with a closure operator, we introduce the notion of a neighborhood of a point. The neighborhoods are then discussed and used for defining a net convergence structure on the category with respect to the closure operator. The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. at.yorku.ca /c/a/j/p/67.htm   (187 words)

 Gap-Definability as a Closure Property - Fenner, Fortnow, Li (ResearchIndex)   (Site not responding. Last check: 2007-10-08) Gap-definability and the gap-closure operator were defined in [FFK91]. Few complexity classes were known at that time to be gapdefinable. In this paper, we give simple characterizations of both gapdefinability and the gap-closure operator, and we show that many complexity classes are gap-definable, including P #P, P #P[1], PSPACE, EXP, NEXP, MP, and BP \Delta \PhiP. sherry.ifi.unizh.ch /57466.html   (502 words)

 PlanetMath: closure axioms , and such that the following (Kuratowski's closure axioms) hold for any subsets The following theorem due to Kuratowski says that a closure operator characterizes a unique topology on This is version 4 of closure axioms, born on 2002-12-09, modified 2003-07-27. planetmath.org /encyclopedia/ClosureAxioms.html   (58 words)

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