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| [[Image:Simplicial complex nonexample.svg|thumb|200px|A geometrical representation of an abstract simplicial complex that is not a valid [[simplicial complex]].]]
| | Christmas lights are an particularly significant part at all people's Christmas decorations. Unfortunately, there are a regarding common mistakes that people make when hanging Christmas lights. Fortunately, there are several simple things you can do to be sure that they are as attractive as suitable.<br><br><br><br>If it becomes clear that your heating costs is usually a bit greater you to be able to be paying, you needs to have someone install some better insulation to your house. While this is really a bit pricey, it a lot less than you might have to order inflated utility bills, period.<br><br>If you are going to be home for a long period of time, have someone pick up your junk mail such as flyers by the doorstep. Purchase also stop newspaper delivery until you're back quarters.<br><br>Once the work is complete, I can accessorize area with casual living patio furniture, weather-resistant fabrics, container gardens, outdoor lighting, plants, fountains and fire sets.<br><br>There is convincing evidence that buyers make their brains up the actual first 1/2 minute or so, when first viewing a property. That's why the front walkway, steps, front door, and entrance are so important. Even if your master bedroom on right away . floor is gorgeous, in the event an buyer is turned off by time they get upstairs, it's probably too newer. Make sure your front walkway is obvious of weeds and debris (and ice in the winter!), do your best to make the entrance clean and tidy (put away extra shoes and boots), consider painting best door (nice and sleek!) - creates a great first impression; consider replacing a rusted mailbox, and so on. If your front screen door is torn, replace the interface.<br><br>Start by tidying moving upward. Clear the yard involving most excess clutter. Put away the frost rags that you used safeguard your plants and bushes from the freezing temperatures and chin-up all the dead plants that were lost throughout the winter 12 weeks. Now is the perfect time start off trimming the hedges and trees with your yard.<br><br>Exercise with him (or her)----Don't just tell your beloved to get some exercise. Go for it and join him. Going how my dad used to walk his little pint-size min pin (miniature pinscher), "Willie Boy", down their mountain road. Tony horton created a send to go outside and walks, as well as talk with my dad. What's more, he got the opportunity to greet the neighbors. Simply did he get physical exercise, nonetheless needed socialization as very well.<br><br>Tree surgical treatment is a associated with art. Sometimes it may be even dangerous when performed on tall trees. That's why tree surgeons are specially trained a cordless the right equipment as a way to avoid accidents as their consequences end up being fatal. Their skills and data gained through years of practice makes it possible to keep our closest environment tidy and clean, prepared to serve the other generations.<br><br>If you loved this information and you would like to receive details relating to [http://www.hedgingplants.com/ hedgingplants.com] assure visit our internet site. |
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| In [[mathematics]], an '''abstract simplicial complex''' is a purely combinatorial description of the geometric notion of a [[simplicial complex]], consisting of a family of [[finite set]]s closed under the operation of taking [[subset]]s.<ref name=Lee>Lee, JM, Introduction to Topological Manifolds, Springer 2011, ISBN 1-4419-7939-5, p153</ref> In the context of [[matroid]]s and [[greedoid]]s, abstract simplicial complexes are also called '''independence systems'''.<ref>{{cite book
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| | author = [[Bernhard Korte|Korte, Bernhard]]; [[László Lovász|Lovász, László]]; Schrader, Rainer
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| | year = 1991
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| | title = Greedoids
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| | publisher = Springer-Verlag
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| | isbn = 3-540-18190-3
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| | page = 9}}</ref>
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| ==Definitions==
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| A nonempty [[family of sets|family]] Δ of [[finite set|finite]] [[subset]]s of a [[universal set]] ''S'' is an '''abstract simplicial complex''' if, for every set ''X'' in Δ, and every subset ''Y'' ⊂ ''X'', ''Y'' also belongs to Δ. Equivalently, it is an abstract simplicial complex if and only if there do not exist two sets ''Y'' ⊂ ''X'' such that ''X'' belongs to Δ but ''Y'' does not.
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| Note that the [[empty set]] belongs to every non-empty abstract simplicial complex, because it is a subset of every other set ''X'' in the complex. The finite sets that belong to Δ are called '''faces''' of the complex, and a face ''Y'' is said to belong to another face ''X'' if ''Y'' ⊂ ''X'', so the definition of an abstract simplicial complex can be restated as saying that every face of a face of a complex Δ is itself a face of Δ. The '''vertex set''' of Δ is defined as ''V''(Δ) = ∪Δ, the union of all faces of Δ. The elements of the vertex set are called the '''vertices''' of the complex. So for every vertex ''v'' of Δ, the set {''v''} is a face of the complex.
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| The maximal faces of Δ (i.e., faces that are not subsets of any other faces) are called '''facets''' of the complex.
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| The '''dimension of a face''' ''X'' in Δ is defined as dim(''X'') = |''X''| - 1: faces consisting of a single element are zero-dimensional, faces consisting of two elements are one-dimensional, etc. The '''dimension of the complex''' dim(Δ) is defined as the largest dimension of any of its faces, or infinity if there is no finite bound on the dimension of the faces.
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| The complex Δ is said to be '''finite''' if it has finitely many faces, or equivalently if its vertex set is finite. Also, Δ is said to be '''pure''' if it is finite-dimensional (but not necessarily finite) and every facet has the same dimension. In other words, Δ is pure if dim(Δ) is finite and every face is contained in a facet of dimension dim(Δ).
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| One-dimensional abstract simplicial complexes are mathematically equivalent to [[simple graph|simple]] [[undirected graph]]s: the vertex set of the complex can be viewed as the vertex set of a graph, and the two-element facets of the complex correspond to undirected edges of a graph. In this view, one-element facets of a complex correspond to isolated vertices that do not have any incident edges.
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| A '''subcomplex''' of Δ is a simplicial complex ''L'' such that every face of ''L'' belongs to Δ; that is, ''L'' ⊂ Δ and ''L'' is a simplicial complex. A subcomplex that consists of all of the subsets of a single face of Δ is often called a '''simplex''' of Δ. (However, some authors use the term "simplex" for a face or, rather ambiguously, for both a face and the subcomplex associated with a face, by analogy with the non-abstract (geometric) [[simplicial complex]] terminology. To avoid ambiguity, we do not use in this article the term "simplex" for a face in the context of abstract complexes.)
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| The '''[[d-skeleton]]''' of Δ is the subcomplex of Δ consisting of all of the faces of Δ that have dimension at most ''d''. In particular, the [[skeleton (topology)|1-skeleton]] is called the '''underlying graph''' of Δ. The 0-skeleton of Δ can be identified with its vertex set, although formally it is not quite the same thing (the vertex set is a single set of all of the vertices, while the 0-skeleton is a family of single-element sets).
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| The '''link''' of a face ''Y'' in Δ, often denoted Δ/''Y'' or lk<sub>Δ</sub>(''Y''), is the subcomplex of Δ defined by
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| :<math> \Delta/Y := \{ X\in \Delta \mid X\cap Y = \varnothing,\, X\cup Y \in \Delta \}. </math>
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| Note that the link of the empty set is Δ itself.
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| Given two abstract simplicial complexes, Δ and Γ, a '''simplicial map''' is a function ƒ that maps the vertices of Δ to the vertices of Γ and that has the property that for any face ''X'' of Δ, the image set ƒ(''X'') is a face of Γ.
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| ==Geometric realization==
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| We can associate to an abstract simplicial complex ''K'' a topological space |''K''|, called its '''geometric realization''', which is a [[simplicial complex]]. The construction goes as follows.
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| First, define |''K''| as a subset of ''[0,1]^S'' consisting of functions ''t:S → [0,1]'' satisfying the two conditions:
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| :<math>\sum_{s\in S}t_s=1</math>
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| :<math>\{s\in S:t_s>0\}\in\Delta</math>
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| Now think of ''[0,1]^S'' as the direct limit of ''[0,1]^A'' where ''A'' ranges over finite subsets of ''S'', and give ''[0,1]^S'' the [[final topology|induced topology]]. Now give |''K''| the subspace topology.
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| Alternatively, let <math>\mathcal{K}</math> denote the [[category (mathematics)|category]] whose objects are the faces of ''K'' and whose [[morphism]]s are inclusions. Next choose a [[total order]] on the vertex set of ''K'' and define a functor ''F'' from <math>\mathcal{K}</math> to the category of [[topological space]]s as follows. For any face ''X'' ∈ ''K'' of dimension ''n'', let ''F''(''X'') = Δ<sup>''n''</sup> be the standard ''n''-simplex. The order on the vertex set then specifies a unique bijection between the elements of ''X'' and vertices of Δ<sup>''n''</sup>, ordered in the usual way ''e''<sub>0</sub> < ''e''<sub>1</sub> < ... < ''e''<sub>''n''</sub>. If ''Y'' ⊂ ''X'' is a face of dimension ''m'' < ''n'', then this bijection specifies a unique ''m''-dimensional face of Δ<sup>''n''</sup>. Define ''F''(''Y'') → ''F''(''X'') to be the unique [[affine transformation|affine linear]] embedding of Δ<sup>''m''</sup> as that distinguished face of Δ<sup>''n''</sup>, such that the map on vertices is order preserving.
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| We can then define the geometric realization |''K''| as the [[colimit]] of the functor ''F''. More specifically |''K''| is the [[quotient space]] of the [[disjoint union]]
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| :<math>\coprod_{X \in K}{F(X)}</math>
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| by the equivalence relation which identifies a point ''y'' ∈ ''F''(''Y'') with its image under the map ''F''(''Y'') → ''F''(''X''), for every inclusion ''Y'' ⊂ ''X''.
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| If ''K'' is finite, then we can describe |''K''| more simply. Choose an embedding of the vertex set of ''K'' as an [[affinely independent]] subset of some Euclidean space '''R'''<sup>''N''</sup> of sufficiently high dimension ''N''. Then any face ''X'' ∈ ''K'' can be identified with the geometric simplex in '''R'''<sup>''N''</sup> spanned by the corresponding embedded vertices. Take |''K''| to be the union of all such simplices.
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| If ''K'' is the standard combinatorial ''n''-simplex, then clearly |''K''| can be naturally identified with Δ<sup>''n''</sup>.
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| ==Examples==
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| *As an example, let ''V'' be a finite subset of ''S'' of cardinality ''n''+1 and let ''K'' be the [[power set]] of ''V''. Then ''K'' is called a '''combinatorial''' ''n''-'''simplex''' with vertex set ''V''. If ''V'' = ''S'' = {0, 1, 2, ..., ''n''}, ''K'' is called the '''standard''' combinatorial ''n''-simplex.
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| *The [[clique complex]] of an undirected graph has a simplex for each [[clique]] (complete subgraph) of the given graph. Clique complexes form the prototypical example of [[flag complex]]es, complexes with the property that every set of elements that pairwise belong to simplexes of the complex is itself a simplex.
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| * In the theory of [[partially ordered set]]s ("posets"), the '''order complex''' of a poset is the set of all finite chains. Its [[Homology (mathematics)|homology]] groups and other topological invariants contain important information about the poset.
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| * The [[Vietoris–Rips complex]] is defined from any metric space ''M'' and distance δ by forming a simplex for every finite subset of ''M'' with diameter at most δ. It has applications in [[homology theory]], [[hyperbolic group]]s, [[image processing]], and [[mobile ad hoc network]]ing. It is another example of a flag complex.
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| ==Enumeration==
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| The number of abstract simplicial complexes on ''n'' elements is the nth [[Dedekind number]]. These numbers grow very rapidly, and are known only for ''n'' ≤ 8; they are
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| :2, 3, 6, 20, 168, 7581, 7828354, 2414682040998, 56130437228687557907788 {{OEIS|id=A000372}}.
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| ==See also==
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| * [[Kruskal–Katona theorem]]
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:Abstract Simplicial Complex}}
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| [[Category:Algebraic topology]]
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| [[Category:Set families]]
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| [[Category:Simplicial sets]]
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Christmas lights are an particularly significant part at all people's Christmas decorations. Unfortunately, there are a regarding common mistakes that people make when hanging Christmas lights. Fortunately, there are several simple things you can do to be sure that they are as attractive as suitable.
If it becomes clear that your heating costs is usually a bit greater you to be able to be paying, you needs to have someone install some better insulation to your house. While this is really a bit pricey, it a lot less than you might have to order inflated utility bills, period.
If you are going to be home for a long period of time, have someone pick up your junk mail such as flyers by the doorstep. Purchase also stop newspaper delivery until you're back quarters.
Once the work is complete, I can accessorize area with casual living patio furniture, weather-resistant fabrics, container gardens, outdoor lighting, plants, fountains and fire sets.
There is convincing evidence that buyers make their brains up the actual first 1/2 minute or so, when first viewing a property. That's why the front walkway, steps, front door, and entrance are so important. Even if your master bedroom on right away . floor is gorgeous, in the event an buyer is turned off by time they get upstairs, it's probably too newer. Make sure your front walkway is obvious of weeds and debris (and ice in the winter!), do your best to make the entrance clean and tidy (put away extra shoes and boots), consider painting best door (nice and sleek!) - creates a great first impression; consider replacing a rusted mailbox, and so on. If your front screen door is torn, replace the interface.
Start by tidying moving upward. Clear the yard involving most excess clutter. Put away the frost rags that you used safeguard your plants and bushes from the freezing temperatures and chin-up all the dead plants that were lost throughout the winter 12 weeks. Now is the perfect time start off trimming the hedges and trees with your yard.
Exercise with him (or her)----Don't just tell your beloved to get some exercise. Go for it and join him. Going how my dad used to walk his little pint-size min pin (miniature pinscher), "Willie Boy", down their mountain road. Tony horton created a send to go outside and walks, as well as talk with my dad. What's more, he got the opportunity to greet the neighbors. Simply did he get physical exercise, nonetheless needed socialization as very well.
Tree surgical treatment is a associated with art. Sometimes it may be even dangerous when performed on tall trees. That's why tree surgeons are specially trained a cordless the right equipment as a way to avoid accidents as their consequences end up being fatal. Their skills and data gained through years of practice makes it possible to keep our closest environment tidy and clean, prepared to serve the other generations.
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