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, 1 ] {\displaystyle [0,1]} but does not converges (in R {\displaystyle \mathbb {R} } ) to it. The free prefilter ( R , β ) := { ( r , β ) : r β R } {\displaystyle (\mathbb {R} ,\infty ):=\{(r,\infty ):r\in \mathbb {R} \}} of intervals does not converge (in R {\displaystyle \mathbb {R} } ) to any point. The same is also true of the prefilter [ R , β ) := { [ r , β ) : r β R } {\displaystyle [\mathbb {R} ,\infty ):=\{[r,\infty ):r\in \mathbb {R} \}} because it is equivalent to ( R , β ) {\displaystyle (\mathbb {R} ,\infty )} and equivalent families have the same limits. In fact, if B {\displaystyle {\mathcal {B}}} is any prefilter in any topological space X {\displaystyle X} then for every S β B β X , {\displaystyle S\in {\mathcal {B}}^{\uparrow X},} B β S . {\displaystyle {\mathcal {B}}\to S.} More generally, because the only neighborhood of X {\displaystyle X} is itself (that is, N ( X ) = { X } {\displaystyle {\mathcal {N}}(X)=\{X\}} ), every non-empty family (including every filter subbase) converges to X . {\displaystyle X.} For any point x , {\displaystyle x,} its neighborhood filter N ( x ) β x {\displaystyle {\mathcal {N}}(x)\to x} always converges to x . {\displaystyle x.} More generally, any neighborhood basis at x {\displaystyle x} converges to x . {\displaystyle x.} A point x {\displaystyle x} is always a limit point of the principle ultra prefilter { { x } } {\displaystyle \{\{x\}\}} and of the ultrafilter that it generates. The empty family B = β
{\displaystyle {\mathcal {B}}=\varnothing } does not converge to any point. Basic properties If B {\displaystyle {\mathcal {B}}} converges to a point then the same is true of
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any family finer than B . {\displaystyle {\mathcal {B}}.} This has many important consequences. One consequence is that the limit points of a family B {\displaystyle {\mathcal {B}}} are the same as the limit points of its upward closure: lim X β‘ B = lim X β‘ ( B β X ) . {\displaystyle \operatorname {lim} _{X}{\mathcal {B}}~=~\operatorname {lim} _{X}\left({\mathcal {B}}^{\uparrow X}\right).} In particular, the limit points of a prefilter are the same as the limit points of the filter that it generates. Another consequence is that if a family converges to a point then the same is true of the family's trace/restriction to any given subset of X . {\displaystyle X.} If B {\displaystyle {\mathcal {B}}} is a prefilter and B β B {\displaystyle B\in {\mathcal {B}}} then B {\displaystyle {\mathcal {B}}} converges to a point of X {\displaystyle X} if and only if this is true of the trace B | B . {\displaystyle {\mathcal {B}}{\big \vert }_{B}.} If a filter subbase converges to a point then do the filter and the Ο-system that it generates, although the converse is not guaranteed. For example, the filter subbase { ( β β , 0 ] , [ 0 , β ) } {\displaystyle \{(-\infty ,0],[0,\infty )\}} does not converge to 0 {\displaystyle 0} in X := R {\displaystyle X:=\mathbb {R} } although the (principle ultra) filter that it generates does. Given x β X , {\displaystyle x\in X,} the following are equivalent for a prefilter B : {\displaystyle {\mathcal {B}}:} B {\displaystyle {\mathcal {B}}} converges to x . {\displaystyle x.} B β X {\displaystyle {\mathcal {B}}^{\uparrow X}} converges to x . {\displaystyle x.} There exists a family equivalent to B {\displaystyle {\mathcal {B}}} that converges to x . {\displaystyle x.} Because subordination is transitive, if B β€ C then
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lim X B β lim X C {\displaystyle {\mathcal {B}}\leq {\mathcal {C}}{\text{ then }}\lim {}_{X}{\mathcal {B}}\subseteq \lim {}_{X}{\mathcal {C}}} and moreover, for every x β X , {\displaystyle x\in X,} both { x } {\displaystyle \{x\}} and the maximal/ultrafilter { x } β X {\displaystyle \{x\}^{\uparrow X}} converge to x . {\displaystyle x.} Thus every topological space ( X , Ο ) {\displaystyle (X,\tau )} induces a canonical convergence ΞΎ β X Γ Filters β‘ ( X ) {\displaystyle \xi \subseteq X\times \operatorname {Filters} (X)} defined by ( x , B ) β ΞΎ if and only if x β lim ( X , Ο ) B . {\displaystyle (x,{\mathcal {B}})\in \xi {\text{ if and only if }}x\in \lim {}_{(X,\tau )}{\mathcal {B}}.} At the other extreme, the neighborhood filter N ( x ) {\displaystyle {\mathcal {N}}(x)} is the smallest (that is, coarsest) filter on X {\displaystyle X} that converges to x ; {\displaystyle x;} that is, any filter converging to x {\displaystyle x} must contain N ( x ) {\displaystyle {\mathcal {N}}(x)} as a subset. Said differently, the family of filters that converge to x {\displaystyle x} consists exactly of those filter on X {\displaystyle X} that contain N ( x ) {\displaystyle {\mathcal {N}}(x)} as a subset. Consequently, the finer the topology on X {\displaystyle X} then the fewer prefilters exist that have any limit points in X . {\displaystyle X.} === Cluster points === A family B {\displaystyle {\mathcal {B}}} is said to cluster at a point x {\displaystyle x} of X {\displaystyle X} if it meshes with the neighborhood filter of x ; {\displaystyle x;} that is, if B # N ( x ) . {\displaystyle {\mathcal {B}}\#{\mathcal {N}}(x).} Explicitly, this means that B β© N β β
for every B β B {\displaystyle B\cap N\neq \varnothing
|
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{\text{ for every }}B\in {\mathcal {B}}} and every neighborhood N {\displaystyle N} of x . {\displaystyle x.} In particular, a point x β X {\displaystyle x\in X} is a cluster point or an accumulation point of a family B {\displaystyle {\mathcal {B}}} if B {\displaystyle {\mathcal {B}}} meshes with the neighborhood filter at x : B # N ( x ) . {\displaystyle x:\ {\mathcal {B}}\#{\mathcal {N}}(x).} The set of all cluster points of B {\displaystyle {\mathcal {B}}} is denoted by cl X β‘ B , {\displaystyle \operatorname {cl} _{X}{\mathcal {B}},} where the subscript may be dropped if not needed. In the above definitions, it suffices to check that B {\displaystyle {\mathcal {B}}} meshes with some (or equivalently, meshes with every) neighborhood base in X {\displaystyle X} of x or S . {\displaystyle x{\text{ or }}S.} When B {\displaystyle {\mathcal {B}}} is a prefilter then the definition of " B and N {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {N}}} mesh" can be characterized entirely in terms of the subordination preorder β€ . {\displaystyle \,\leq \,.} Two equivalent families of sets have the exact same limit points and also the same cluster points. No matter the topology, for every x β X , {\displaystyle x\in X,} both { x } {\displaystyle \{x\}} and the principal ultrafilter { x } β X {\displaystyle \{x\}^{\uparrow X}} cluster at x . {\displaystyle x.} If B {\displaystyle {\mathcal {B}}} clusters to a point then the same is true of any family coarser than B . {\displaystyle {\mathcal {B}}.} Consequently, the cluster points of a family B {\displaystyle {\mathcal {B}}} are the same as the cluster points of its upward closure: cl X β‘ B = cl X β‘ ( B β X ) . {\displaystyle \operatorname {cl} _{X}{\mathcal {B}}~=~\operatorname {cl} _{X}\left({\mathcal {B}}^{\uparrow X}\right).} In particular, the
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cluster points of a prefilter are the same as the cluster points of the filter that it generates. Given x β X , {\displaystyle x\in X,} the following are equivalent for a prefilter B on X {\displaystyle {\mathcal {B}}{\text{ on }}X} : B {\displaystyle {\mathcal {B}}} clusters at x . {\displaystyle x.} The family B β X {\displaystyle {\mathcal {B}}^{\uparrow X}} generated by B {\displaystyle {\mathcal {B}}} clusters at x . {\displaystyle x.} There exists a family equivalent to B {\displaystyle {\mathcal {B}}} that clusters at x . {\displaystyle x.} x β β F β B cl X β‘ F . {\displaystyle x\in {\textstyle \bigcap \limits _{F\in {\mathcal {B}}}}\operatorname {cl} _{X}F.} X β N β B β X {\displaystyle X\setminus N\not \in {\mathcal {B}}^{\uparrow X}} for every neighborhood N {\displaystyle N} of x . {\displaystyle x.} If B {\displaystyle {\mathcal {B}}} is a filter on X {\displaystyle X} then x β cl X β‘ B if and only if X β N β B {\displaystyle x\in \operatorname {cl} _{X}{\mathcal {B}}{\text{ if and only if }}X\setminus N\not \in {\mathcal {B}}} for every neighborhood N of x . {\displaystyle N{\text{ of }}x.} There exists a prefilter F {\displaystyle {\mathcal {F}}} subordinate to B {\displaystyle {\mathcal {B}}} (that is, F β₯ B {\displaystyle {\mathcal {F}}\geq {\mathcal {B}}} ) that converges to x . {\displaystyle x.} This is the filter equivalent of " x {\displaystyle x} is a cluster point of a sequence if and only if there exists a subsequence converging to x . {\displaystyle x.} In particular, if x {\displaystyle x} is a cluster point of a prefilter B {\displaystyle {\mathcal {B}}} then B ( β© ) N ( x ) {\displaystyle {\mathcal {B}}(\cap ){\mathcal {N}}(x)} is a prefilter subordinate to B {\displaystyle {\mathcal {B}}} that converges to x . {\displaystyle x.}
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"page_id": 47516955,
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The set cl X β‘ B {\displaystyle \operatorname {cl} _{X}{\mathcal {B}}} of all cluster points of a prefilter B {\displaystyle {\mathcal {B}}} satisfies cl X β‘ B = β B β B cl X β‘ B . {\displaystyle \operatorname {cl} _{X}{\mathcal {B}}=\bigcap _{B\in {\mathcal {B}}}\operatorname {cl} _{X}B.} Consequently, the set cl X β‘ B {\displaystyle \operatorname {cl} _{X}{\mathcal {B}}} of all cluster points of any prefilter B {\displaystyle {\mathcal {B}}} is a closed subset of X . {\displaystyle X.} This also justifies the notation cl X β‘ B {\displaystyle \operatorname {cl} _{X}{\mathcal {B}}} for the set of cluster points. In particular, if K β X {\displaystyle K\subseteq X} is non-empty (so that B := { K } {\displaystyle {\mathcal {B}}:=\{K\}} is a prefilter) then cl X β‘ { K } = cl X β‘ K {\displaystyle \operatorname {cl} _{X}\{K\}=\operatorname {cl} _{X}K} since both sides are equal to β B β B cl X β‘ B . {\displaystyle {\textstyle \bigcap \limits _{B\in {\mathcal {B}}}}\operatorname {cl} _{X}B.} === Properties and relationships === Just like sequences and nets, it is possible for a prefilter on a topological space of infinite cardinality to not have any cluster points or limit points. If x {\displaystyle x} is a limit point of B {\displaystyle {\mathcal {B}}} then x {\displaystyle x} is necessarily a limit point of any family C {\displaystyle {\mathcal {C}}} finer than B {\displaystyle {\mathcal {B}}} (that is, if N ( x ) β€ B and B β€ C {\displaystyle {\mathcal {N}}(x)\leq {\mathcal {B}}{\text{ and }}{\mathcal {B}}\leq {\mathcal {C}}} then N ( x ) β€ C {\displaystyle {\mathcal {N}}(x)\leq {\mathcal {C}}} ). In contrast, if x {\displaystyle x} is a cluster point of B {\displaystyle {\mathcal {B}}} then x {\displaystyle x} is necessarily a cluster point of any family C {\displaystyle {\mathcal {C}}} coarser
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"page_id": 47516955,
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than B {\displaystyle {\mathcal {B}}} (that is, if N ( x ) and B {\displaystyle {\mathcal {N}}(x){\text{ and }}{\mathcal {B}}} mesh and C β€ B {\displaystyle {\mathcal {C}}\leq {\mathcal {B}}} then N ( x ) and C {\displaystyle {\mathcal {N}}(x){\text{ and }}{\mathcal {C}}} mesh). Equivalent families and subordination Any two equivalent families B and C {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}} can be used interchangeably in the definitions of "limit of" and "cluster at" because their equivalency guarantees that N β€ B {\displaystyle {\mathcal {N}}\leq {\mathcal {B}}} if and only if N β€ C , {\displaystyle {\mathcal {N}}\leq {\mathcal {C}},} and also that N # B {\displaystyle {\mathcal {N}}\#{\mathcal {B}}} if and only if N # C . {\displaystyle {\mathcal {N}}\#{\mathcal {C}}.} In essence, the preorder β€ {\displaystyle \,\leq \,} is incapable of distinguishing between equivalent families. Given two prefilters, whether or not they mesh can be characterized entirely in terms of subordination. Thus the two most fundamental concepts related to (pre)filters to Topology (that is, limit and cluster points) can both be defined entirely in terms of the subordination relation. This is why the preorder β€ {\displaystyle \,\leq \,} is of such great importance in applying (pre)filters to Topology. Limit and cluster point relationships and sufficient conditions Every limit point of a non-degenerate family B {\displaystyle {\mathcal {B}}} is also a cluster point; in symbols: lim X β‘ B β cl X β‘ B . {\displaystyle \operatorname {lim} _{X}{\mathcal {B}}~\subseteq ~\operatorname {cl} _{X}{\mathcal {B}}.} This is because if x {\displaystyle x} is a limit point of B {\displaystyle {\mathcal {B}}} then N ( x ) and B {\displaystyle {\mathcal {N}}(x){\text{ and }}{\mathcal {B}}} mesh, which makes x {\displaystyle x} a cluster point of B . {\displaystyle {\mathcal {B}}.} But in general, a cluster point need not be a limit
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"page_id": 47516955,
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point. For instance, every point in any given non-empty subset K β X {\displaystyle K\subseteq X} is a cluster point of the principle prefilter B := { K } {\displaystyle {\mathcal {B}}:=\{K\}} (no matter what topology is on X {\displaystyle X} ) but if X {\displaystyle X} is Hausdorff and K {\displaystyle K} has more than one point then this prefilter has no limit points; the same is true of the filter { K } β X {\displaystyle \{K\}^{\uparrow X}} that this prefilter generates. However, every cluster point of an ultra prefilter is a limit point. Consequently, the limit points of an ultra prefilter B {\displaystyle {\mathcal {B}}} are the same as its cluster points: lim X β‘ B = cl X β‘ B ; {\displaystyle \operatorname {lim} _{X}{\mathcal {B}}=\operatorname {cl} _{X}{\mathcal {B}};} that is to say, a given point is a cluster point of an ultra prefilter B {\displaystyle {\mathcal {B}}} if and only if B {\displaystyle {\mathcal {B}}} converges to that point. Although a cluster point of a filter need not be a limit point, there will always exist a finer filter that does converge to it; in particular, if B {\displaystyle {\mathcal {B}}} clusters at x {\displaystyle x} then B ( β© ) N ( x ) = { B β© N : B β B , N β N ( x ) } {\displaystyle {\mathcal {B}}\,(\cap )\,{\mathcal {N}}(x)=\{B\cap N:B\in {\mathcal {B}},N\in {\mathcal {N}}(x)\}} is a filter subbase whose generated filter converges to x . {\displaystyle x.} If β
β B β β ( X ) and S β₯ B {\displaystyle \varnothing \neq {\mathcal {B}}\subseteq \wp (X){\text{ and }}{\mathcal {S}}\geq {\mathcal {B}}} is a filter subbase such that S β x in X {\displaystyle {\mathcal {S}}\to x{\text{ in }}X} then x β cl X β‘ B .
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"page_id": 47516955,
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{\displaystyle x\in \operatorname {cl} _{X}{\mathcal {B}}.} In particular, any limit point of a filter subbase subordinate to B β β
{\displaystyle {\mathcal {B}}\neq \varnothing } is necessarily also a cluster point of B . {\displaystyle {\mathcal {B}}.} If x {\displaystyle x} is a cluster point of a prefilter B {\displaystyle {\mathcal {B}}} then B ( β© ) N ( x ) {\displaystyle {\mathcal {B}}(\cap ){\mathcal {N}}(x)} is a prefilter subordinate to B {\displaystyle {\mathcal {B}}} that converges to x in X . {\displaystyle x{\text{ in }}X.} If S β X {\displaystyle S\subseteq X} and if B {\displaystyle {\mathcal {B}}} is a prefilter on S {\displaystyle S} then every cluster point of B in X {\displaystyle {\mathcal {B}}{\text{ in }}X} belongs to cl X β‘ S {\displaystyle \operatorname {cl} _{X}S} and any point in cl X β‘ S {\displaystyle \operatorname {cl} _{X}S} is a limit point of a filter on S . {\displaystyle S.} Primitive sets A subset P β X {\displaystyle P\subseteq X} is called primitive if it is the set of limit points of some ultrafilter (or equivalently, some ultra prefilter). That is, if there exists an ultrafilter B on X {\displaystyle {\mathcal {B}}{\text{ on }}X} such that P {\displaystyle P} is equal to lim X β‘ B , {\displaystyle \operatorname {lim} _{X}{\mathcal {B}},} which recall denotes the set of limit points of B in X . {\displaystyle {\mathcal {B}}{\text{ in }}X.} Since limit points are the same as cluster points for ultra prefilters, a subset is primitive if and only if it is equal to the set cl X β‘ B {\displaystyle \operatorname {cl} _{X}{\mathcal {B}}} of cluster points of some ultra prefilter B . {\displaystyle {\mathcal {B}}.} For example, every closed singleton subset is primitive. The image of a primitive subset of X {\displaystyle X} under a
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"page_id": 47516955,
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continuous map f : X β Y {\displaystyle f:X\to Y} is contained in a primitive subset of Y . {\displaystyle Y.} Assume that P , Q β X {\displaystyle P,Q\subseteq X} are two primitive subset of X . {\displaystyle X.} If U {\displaystyle U} is an open subset of X {\displaystyle X} that intersects P {\displaystyle P} then U β B {\displaystyle U\in {\mathcal {B}}} for any ultrafilter B on X {\displaystyle {\mathcal {B}}{\text{ on }}X} such that P = lim X β‘ B . {\displaystyle P=\operatorname {lim} _{X}{\mathcal {B}}.} In addition, if P and Q {\displaystyle P{\text{ and }}Q} are distinct then there exists some S β X {\displaystyle S\subseteq X} and some ultrafilters B P and B Q on X {\displaystyle {\mathcal {B}}_{P}{\text{ and }}{\mathcal {B}}_{Q}{\text{ on }}X} such that P = lim X β‘ B P , Q = lim X β‘ B Q , S β B P , {\displaystyle P=\operatorname {lim} _{X}{\mathcal {B}}_{P},Q=\operatorname {lim} _{X}{\mathcal {B}}_{Q},S\in {\mathcal {B}}_{P},} and X β S β B Q . {\displaystyle X\setminus S\in {\mathcal {B}}_{Q}.} Other results If X {\displaystyle X} is a complete lattice then: The limit inferior of B {\displaystyle B} is the infimum of the set of all cluster points of B . {\displaystyle B.} The limit superior of B {\displaystyle B} is the supremum of the set of all cluster points of B . {\displaystyle B.} B {\displaystyle B} is a convergent prefilter if and only if its limit inferior and limit superior agree; in this case, the value on which they agree is the limit of the prefilter. === Limits of functions defined as limits of prefilters === Suppose f : X β Y {\displaystyle f:X\to Y} is a map from a set into a topological space Y , {\displaystyle Y,} B β β
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"page_id": 47516955,
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( X ) , {\displaystyle {\mathcal {B}}\subseteq \wp (X),} and y β Y . {\displaystyle y\in Y.} If y {\displaystyle y} is a limit point (respectively, a cluster point) of f ( B ) in Y {\displaystyle f({\mathcal {B}}){\text{ in }}Y} then y {\displaystyle y} is called a limit point or limit (respectively, a cluster point) of f {\displaystyle f} with respect to B . {\displaystyle {\mathcal {B}}.} Explicitly, y {\displaystyle y} is a limit of f {\displaystyle f} with respect to B {\displaystyle {\mathcal {B}}} if and only if N ( y ) β€ f ( B ) , {\displaystyle {\mathcal {N}}(y)\leq f({\mathcal {B}}),} which can be written as f ( B ) β y or lim f ( B ) β y in Y {\displaystyle f({\mathcal {B}})\to y{\text{ or }}\lim f({\mathcal {B}})\to y{\text{ in }}Y} (by definition of this notation) and stated as f {\displaystyle f} tend to y {\displaystyle y} along B . {\displaystyle {\mathcal {B}}.} If the limit y {\displaystyle y} is unique then the arrow β {\displaystyle \to } may be replaced with an equals sign = . {\displaystyle =.} The neighborhood filter N ( y ) {\displaystyle {\mathcal {N}}(y)} can be replaced with any family equivalent to it and the same is true of B . {\displaystyle {\mathcal {B}}.} The definition of a convergent net is a special case of the above definition of a limit of a function. Specifically, if x β X and Ο : ( I , β€ ) β X {\displaystyle x\in X{\text{ and }}\chi :(I,\leq )\to X} is a net then Ο β x in X if and only if Ο ( Tails β‘ ( I , β€ ) ) β x in X , {\displaystyle \chi \to x{\text{ in }}X\quad {\text{ if and only if }}\quad \chi (\operatorname
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"page_id": 47516955,
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{Tails} (I,\leq ))\to x{\text{ in }}X,} where the left hand side states that x {\displaystyle x} is a limit of the net Ο {\displaystyle \chi } while the right hand side states that x {\displaystyle x} is a limit of the function Ο {\displaystyle \chi } with respect to B := Tails β‘ ( I , β€ ) {\displaystyle {\mathcal {B}}:=\operatorname {Tails} (I,\leq )} (as just defined above). The table below shows how various types of limits encountered in analysis and topology can be defined in terms of the convergence of images (under f {\displaystyle f} ) of particular prefilters on the domain X . {\displaystyle X.} This shows that prefilters provide a general framework into which many of the various definitions of limits fit. The limits in the left-most column are defined in their usual way with their obvious definitions. Throughout, let f : X β Y {\displaystyle f:X\to Y} be a map between topological spaces, x 0 β X , and y β Y . {\displaystyle x_{0}\in X,{\text{ and }}y\in Y.} If Y {\displaystyle Y} is Hausdorff then all arrows " β y {\displaystyle \to y} " in the table may be replaced with equal signs " = y {\displaystyle =y} " and " lim f ( B ) β y {\displaystyle \lim f({\mathcal {B}})\to y} " may be replaced with " lim f ( B ) = y {\displaystyle \lim f({\mathcal {B}})=y} ". By defining different prefilters, many other notions of limits can be defined; for example, lim | x | β | x 0 | | x | β | x 0 | f ( x ) β y . {\displaystyle \lim _{\stackrel {|x|\to |x_{0}|}{|x|\neq |x_{0}|}}f(x)\to y.} Divergence to infinity Divergence of a real-valued function to infinity can be defined/characterized by using the prefilters ( R
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"page_id": 47516955,
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, β ) := { ( r , β ) : r β R } and ( β β , R ) := { ( β β , r ) : r β R } , {\displaystyle (\mathbb {R} ,\infty ):=\{(r,\infty ):r\in \mathbb {R} \}~~{\text{ and }}~~(-\infty ,\mathbb {R} ):=\{(-\infty ,r):r\in \mathbb {R} \},} where f β β {\displaystyle f\to \infty } along B {\displaystyle {\mathcal {B}}} if and only if ( R , β ) β€ f ( B ) {\displaystyle (\mathbb {R} ,\infty )\leq f({\mathcal {B}})} and similarly, f β β β {\displaystyle f\to -\infty } along B {\displaystyle {\mathcal {B}}} if and only if ( β β , R ) β€ f ( B ) . {\displaystyle (-\infty ,\mathbb {R} )\leq f({\mathcal {B}}).} The family ( R , β ) {\displaystyle (\mathbb {R} ,\infty )} can be replaced by any family equivalent to it, such as [ R , β ) := { [ r , β ) : r β R } {\displaystyle [\mathbb {R} ,\infty ):=\{[r,\infty ):r\in \mathbb {R} \}} for instance (in real analysis, this would correspond to replacing the strict inequality " f ( x ) > r {\displaystyle f(x)>r} " in the definition with " f ( x ) β₯ r {\displaystyle f(x)\geq r} "), and the same is true of B {\displaystyle {\mathcal {B}}} and ( β β , R ) . {\displaystyle (-\infty ,\mathbb {R} ).} So for example, if B := N ( x 0 ) {\displaystyle {\mathcal {B}}\,:=\,{\mathcal {N}}\left(x_{0}\right)} then lim x β x 0 f ( x ) β β {\displaystyle \lim _{x\to x_{0}}f(x)\to \infty } if and only if ( R , β ) β€ f ( B ) {\displaystyle (\mathbb {R} ,\infty )\leq f({\mathcal {B}})} holds. Similarly, lim x β x 0 f ( x
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{
"page_id": 47516955,
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"title": "Filters in topology"
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) β β β {\displaystyle \lim _{x\to x_{0}}f(x)\to -\infty } if and only if ( β β , R ) β€ f ( N ( x 0 ) ) , {\displaystyle (-\infty ,\mathbb {R} )\leq f\left({\mathcal {N}}\left(x_{0}\right)\right),} or equivalently, if and only if ( β β , R ] β€ f ( N ( x 0 ) ) . {\displaystyle (-\infty ,\mathbb {R} ]\leq f\left({\mathcal {N}}\left(x_{0}\right)\right).} More generally, if f {\displaystyle f} is valued in Y = R n or Y = C n {\displaystyle Y=\mathbb {R} ^{n}{\text{ or }}Y=\mathbb {C} ^{n}} (or some other seminormed vector space) and if B β₯ r := { y β Y : | y | β₯ r } = Y β B < r {\displaystyle B_{\geq r}:=\{y\in Y:|y|\geq r\}=Y\setminus B_{<r}} then lim x β x 0 | f ( x ) | β β {\displaystyle \lim _{x\to x_{0}}|f(x)|\to \infty } if and only if B β₯ R β€ f ( N ( x 0 ) ) {\displaystyle B_{\geq \mathbb {R} }\leq f\left({\mathcal {N}}\left(x_{0}\right)\right)} holds, where B β₯ R := { B β₯ r : r β R } . {\displaystyle B_{\geq \mathbb {R} }:=\left\{B_{\geq r}:r\in \mathbb {R} \right\}.} == Filters and nets == This section will describe the relationships between prefilters and nets in great detail because of how important these details are applying filters to topology β particularly in switching from utilizing nets to utilizing filters and vice verse. === Nets to prefilters === In the definitions below, the first statement is the standard definition of a limit point of a net (respectively, a cluster point of a net) and it is gradually reworded until the corresponding filter concept is reached. If f : X β Y {\displaystyle f:X\to Y} is a map and x β {\displaystyle x_{\bullet }} is a net
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{
"page_id": 47516955,
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in X {\displaystyle X} then Tails β‘ ( f ( x β ) ) = f ( Tails β‘ ( x β ) ) . {\displaystyle \operatorname {Tails} \left(f\left(x_{\bullet }\right)\right)=f\left(\operatorname {Tails} \left(x_{\bullet }\right)\right).} === Prefilters to nets === A pointed set is a pair ( S , s ) {\displaystyle (S,s)} consisting of a non-empty set S {\displaystyle S} and an element s β S . {\displaystyle s\in S.} For any family B , {\displaystyle {\mathcal {B}},} let PointedSets β‘ ( B ) := { ( B , b ) : B β B and b β B } . {\displaystyle \operatorname {PointedSets} ({\mathcal {B}}):=\{(B,b)~:~B\in {\mathcal {B}}{\text{ and }}b\in B\}.} Define a canonical preorder β€ {\displaystyle \,\leq \,} on pointed sets by declaring ( R , r ) β€ ( S , s ) if and only if R β S . {\displaystyle (R,r)\leq (S,s)\quad {\text{ if and only if }}\quad R\supseteq S.} There is a canonical map Point B β‘ : PointedSets β‘ ( B ) β X {\displaystyle \operatorname {Point} _{\mathcal {B}}~:~\operatorname {PointedSets} ({\mathcal {B}})\to X} defined by ( B , b ) β¦ b . {\displaystyle (B,b)\mapsto b.} If i 0 = ( B 0 , b 0 ) β PointedSets β‘ ( B ) {\displaystyle i_{0}=\left(B_{0},b_{0}\right)\in \operatorname {PointedSets} ({\mathcal {B}})} then the tail of the assignment Point B {\displaystyle \operatorname {Point} _{\mathcal {B}}} starting at i 0 {\displaystyle i_{0}} is { c : ( C , c ) β PointedSets β‘ ( B ) and ( B 0 , b 0 ) β€ ( C , c ) } = B 0 . {\displaystyle \left\{c~:~(C,c)\in \operatorname {PointedSets} ({\mathcal {B}}){\text{ and }}\left(B_{0},b_{0}\right)\leq (C,c)\right\}=B_{0}.} Although ( PointedSets β‘ ( B ) , β€ ) {\displaystyle (\operatorname {PointedSets} ({\mathcal {B}}),\leq )} is not, in general, a partially ordered set,
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"page_id": 47516955,
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"title": "Filters in topology"
}
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it is a directed set if (and only if) B {\displaystyle {\mathcal {B}}} is a prefilter. So the most immediate choice for the definition of "the net in X {\displaystyle X} induced by a prefilter B {\displaystyle {\mathcal {B}}} " is the assignment ( B , b ) β¦ b {\displaystyle (B,b)\mapsto b} from PointedSets β‘ ( B ) {\displaystyle \operatorname {PointedSets} ({\mathcal {B}})} into X . {\displaystyle X.} If B {\displaystyle {\mathcal {B}}} is a prefilter on X then Net B {\displaystyle X{\text{ then }}\operatorname {Net} _{\mathcal {B}}} is a net in X {\displaystyle X} and the prefilter associated with Net B {\displaystyle \operatorname {Net} _{\mathcal {B}}} is B {\displaystyle {\mathcal {B}}} ; that is: Tails β‘ ( Net B ) = B . {\displaystyle \operatorname {Tails} \left(\operatorname {Net} _{\mathcal {B}}\right)={\mathcal {B}}.} This would not necessarily be true had Net B {\displaystyle \operatorname {Net} _{\mathcal {B}}} been defined on a proper subset of PointedSets β‘ ( B ) . {\displaystyle \operatorname {PointedSets} ({\mathcal {B}}).} If x β {\displaystyle x_{\bullet }} is a net in X {\displaystyle X} then it is not in general true that Net Tails β‘ ( x β ) {\displaystyle \operatorname {Net} _{\operatorname {Tails} \left(x_{\bullet }\right)}} is equal to x β {\displaystyle x_{\bullet }} because, for example, the domain of x β {\displaystyle x_{\bullet }} may be of a completely different cardinality than that of Net Tails β‘ ( x β ) {\displaystyle \operatorname {Net} _{\operatorname {Tails} \left(x_{\bullet }\right)}} (since unlike the domain of Net Tails β‘ ( x β ) , {\displaystyle \operatorname {Net} _{\operatorname {Tails} \left(x_{\bullet }\right)},} the domain of an arbitrary net in X {\displaystyle X} could have any cardinality). Partially ordered net The domain of the canonical net Net B {\displaystyle \operatorname {Net} _{\mathcal {B}}} is in general not partially ordered. However,
|
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"page_id": 47516955,
"source": null,
"title": "Filters in topology"
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in 1955 Bruns and Schmidt discovered a construction (detailed here: Filter (set theory)#Partially ordered net) that allows for the canonical net to have a domain that is both partially ordered and directed; this was independently rediscovered by Albert Wilansky in 1970. Because the tails of this partially ordered net are identical to the tails of Net B {\displaystyle \operatorname {Net} _{\mathcal {B}}} (since both are equal to the prefilter B {\displaystyle {\mathcal {B}}} ), there is typically nothing lost by assuming that the domain of the net associated with a prefilter is both directed and partially ordered. If can further be assumed that the partially ordered domain is also a dense order. === Subordinate filters and subnets === The notion of " B {\displaystyle {\mathcal {B}}} is subordinate to C {\displaystyle {\mathcal {C}}} " (written B β’ C {\displaystyle {\mathcal {B}}\vdash {\mathcal {C}}} ) is for filters and prefilters what " x n β = ( x n i ) i = 1 β {\displaystyle x_{n_{\bullet }}=\left(x_{n_{i}}\right)_{i=1}^{\infty }} is a subsequence of x β = ( x i ) i = 1 β {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }} " is for sequences. For example, if Tails β‘ ( x β ) = { x β₯ i : i β N } {\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)=\left\{x_{\geq i}:i\in \mathbb {N} \right\}} denotes the set of tails of x β {\displaystyle x_{\bullet }} and if Tails β‘ ( x n β ) = { x n β₯ i : i β N } {\displaystyle \operatorname {Tails} \left(x_{n_{\bullet }}\right)=\left\{x_{n_{\geq i}}:i\in \mathbb {N} \right\}} denotes the set of tails of the subsequence x n β {\displaystyle x_{n_{\bullet }}} (where x n β₯ i := { x n j : j β₯ i and j β N } {\displaystyle x_{n_{\geq i}}:=\left\{x_{n_{j}}~:~j\geq i{\text{ and }}j\in \mathbb {N}
|
{
"page_id": 47516955,
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\right\}} ) then Tails β‘ ( x n β ) β’ Tails β‘ ( x β ) {\displaystyle \operatorname {Tails} \left(x_{n_{\bullet }}\right)~\vdash ~\operatorname {Tails} \left(x_{\bullet }\right)} (which by definition means Tails β‘ ( x β ) β€ Tails β‘ ( x n β ) {\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)\leq \operatorname {Tails} \left(x_{n_{\bullet }}\right)} ) is true but Tails β‘ ( x β ) β’ Tails β‘ ( x n β ) {\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)~\vdash ~\operatorname {Tails} \left(x_{n_{\bullet }}\right)} is in general false. If x β = ( x i ) i β I {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}} is a net in a topological space X {\displaystyle X} and if N ( x ) {\displaystyle {\mathcal {N}}(x)} is the neighborhood filter at a point x β X , {\displaystyle x\in X,} then x β β x if and only if N ( x ) β€ Tails β‘ ( x β ) . {\displaystyle x_{\bullet }\to x{\text{ if and only if }}{\mathcal {N}}(x)\leq \operatorname {Tails} \left(x_{\bullet }\right).} If f : X β Y {\displaystyle f:X\to Y} is an surjective open map, x β X , {\displaystyle x\in X,} and C {\displaystyle {\mathcal {C}}} is a prefilter on Y {\displaystyle Y} that converges to f ( x ) , {\displaystyle f(x),} then there exist a prefilter B {\displaystyle {\mathcal {B}}} on X {\displaystyle X} such that B β x {\displaystyle {\mathcal {B}}\to x} and f ( B ) {\displaystyle f({\mathcal {B}})} is equivalent to C {\displaystyle {\mathcal {C}}} (that is, C β€ f ( B ) β€ C {\displaystyle {\mathcal {C}}\leq f({\mathcal {B}})\leq {\mathcal {C}}} ). ==== Subordination analogs of results involving subsequences ==== The following results are the prefilter analogs of statements involving subsequences. The condition " C β₯ B , {\displaystyle {\mathcal {C}}\geq {\mathcal {B}},} " which is also
|
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"page_id": 47516955,
"source": null,
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written C β’ B , {\displaystyle {\mathcal {C}}\vdash {\mathcal {B}},} is the analog of " C {\displaystyle {\mathcal {C}}} is a subsequence of B . {\displaystyle {\mathcal {B}}.} " So "finer than" and "subordinate to" is the prefilter analog of "subsequence of." Some people prefer saying "subordinate to" instead of "finer than" because it is more reminiscent of "subsequence of." ==== Non-equivalence of subnets and subordinate filters ==== Subnets in the sense of Willard and subnets in the sense of Kelley are the most commonly used definitions of "subnet." The first definition of a subnet ("Kelley-subnet") was introduced by John L. Kelley in 1955. Stephen Willard introduced in 1970 his own variant ("Willard-subnet") of Kelley's definition of subnet. AA-subnets were introduced independently by Smiley (1957), Aarnes and Andenaes (1972), and Murdeshwar (1983); AA-subnets were studied in great detail by Aarnes and Andenaes but they are not often used. A subset R β I {\displaystyle R\subseteq I} of a preordered space ( I , β€ ) {\displaystyle (I,\leq )} is frequent or cofinal in I {\displaystyle I} if for every i β I {\displaystyle i\in I} there exists some r β R {\displaystyle r\in R} such that i β€ r . {\displaystyle i\leq r.} If R β I {\displaystyle R\subseteq I} contains a tail of I {\displaystyle I} then R {\displaystyle R} is said to be eventual in I {\displaystyle I} ; explicitly, this means that there exists some i β I {\displaystyle i\in I} such that I β₯ i β R {\displaystyle I_{\geq i}\subseteq R} (that is, j β R {\displaystyle j\in R} for all j β I {\displaystyle j\in I} satisfying i β€ j {\displaystyle i\leq j} ). A subset is eventual if and only if its complement is not frequent (which is termed infrequent). A map h :
|
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"page_id": 47516955,
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A β I {\displaystyle h:A\to I} between two preordered sets is order-preserving if whenever a , b β A {\displaystyle a,b\in A} satisfy a β€ b , {\displaystyle a\leq b,} then h ( a ) β€ h ( b ) . {\displaystyle h(a)\leq h(b).} Kelley did not require the map h {\displaystyle h} to be order preserving while the definition of an AA-subnet does away entirely with any map between the two nets' domains and instead focuses entirely on X {\displaystyle X} β the nets' common codomain. Every Willard-subnet is a Kelley-subnet and both are AA-subnets. In particular, if y β = ( y a ) a β A {\displaystyle y_{\bullet }=\left(y_{a}\right)_{a\in A}} is a Willard-subnet or a Kelley-subnet of x β = ( x i ) i β I {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}} then Tails β‘ ( x β ) β€ Tails β‘ ( y β ) . {\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)\leq \operatorname {Tails} \left(y_{\bullet }\right).} Example: If I = N {\displaystyle I=\mathbb {N} } and x β = ( x i ) i β I {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}} is a constant sequence and if A = { 1 } {\displaystyle A=\{1\}} and s 1 := x 1 {\displaystyle s_{1}:=x_{1}} then ( s a ) a β A {\displaystyle \left(s_{a}\right)_{a\in A}} is an AA-subnet of x β {\displaystyle x_{\bullet }} but it is neither a Willard-subnet nor a Kelley-subnet of x β . {\displaystyle x_{\bullet }.} AA-subnets have a defining characterization that immediately shows that they are fully interchangeable with sub(ordinate)filters. Explicitly, what is meant is that the following statement is true for AA-subnets: If B and F {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}} are prefilters then B β€ F {\displaystyle {\mathcal {B}}\leq {\mathcal {F}}} if and only if Net F {\displaystyle \operatorname {Net} _{\mathcal {F}}} is
|
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"page_id": 47516955,
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an AA-subnet of Net B . {\displaystyle \operatorname {Net} _{\mathcal {B}}.} If "AA-subnet" is replaced by "Willard-subnet" or "Kelley-subnet" then the above statement becomes false. In particular, as this counter-example demonstrates, the problem is that the following statement is in general false: False statement: If B and F {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}} are prefilters such that B β€ F then Net F {\displaystyle {\mathcal {B}}\leq {\mathcal {F}}{\text{ then }}\operatorname {Net} _{\mathcal {F}}} is a Kelley-subnet of Net B . {\displaystyle \operatorname {Net} _{\mathcal {B}}.} Since every Willard-subnet is a Kelley-subnet, this statement thus remains false if the word "Kelley-subnet" is replaced with "Willard-subnet". If "subnet" is defined to mean Willard-subnet or Kelley-subnet then nets and filters are not completely interchangeable because there exists a filterβsub(ordinate)filter relationships that cannot be expressed in terms of a netβsubnet relationship between the two induced nets. In particular, the problem is that Kelley-subnets and Willard-subnets are not fully interchangeable with subordinate filters. If the notion of "subnet" is not used or if "subnet" is defined to mean AA-subnet, then this ceases to be a problem and so it becomes correct to say that nets and filters are interchangeable. Despite the fact that AA-subnets do not have the problem that Willard and Kelley subnets have, they are not widely used or known about. == Topologies and prefilters == Throughout, ( X , Ο ) {\displaystyle (X,\tau )} is a topological space. === Examples of relationships between filters and topologies === Bases and prefilters Let B β β
{\displaystyle {\mathcal {B}}\neq \varnothing } be a family of sets that covers X {\displaystyle X} and define B x = { B β B : x β B } {\displaystyle {\mathcal {B}}_{x}=\{B\in {\mathcal {B}}~:~x\in B\}} for every x β X . {\displaystyle x\in X.} The definition of
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"page_id": 47516955,
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a base for some topology can be immediately reworded as: B {\displaystyle {\mathcal {B}}} is a base for some topology on X {\displaystyle X} if and only if B x {\displaystyle {\mathcal {B}}_{x}} is a filter base for every x β X . {\displaystyle x\in X.} If Ο {\displaystyle \tau } is a topology on X {\displaystyle X} and B β Ο {\displaystyle {\mathcal {B}}\subseteq \tau } then the definitions of B {\displaystyle {\mathcal {B}}} is a basis (resp. subbase) for Ο {\displaystyle \tau } can be reworded as: B {\displaystyle {\mathcal {B}}} is a base (resp. subbase) for Ο {\displaystyle \tau } if and only if for every x β X , B x {\displaystyle x\in X,{\mathcal {B}}_{x}} is a filter base (resp. filter subbase) that generates the neighborhood filter of ( X , Ο ) {\displaystyle (X,\tau )} at x . {\displaystyle x.} Neighborhood filters The archetypical example of a filter is the set of all neighborhoods of a point in a topological space. Any neighborhood basis of a point in (or of a subset of) a topological space is a prefilter. In fact, the definition of a neighborhood base can be equivalently restated as: "a neighborhood base is any prefilter that is equivalent the neighborhood filter." Neighborhood bases at points are examples of prefilters that are fixed but may or may not be principal. If X = R {\displaystyle X=\mathbb {R} } has its usual topology and if x β X , {\displaystyle x\in X,} then any neighborhood filter base B {\displaystyle {\mathcal {B}}} of x {\displaystyle x} is fixed by x {\displaystyle x} (in fact, it is even true that ker β‘ B = { x } {\displaystyle \ker {\mathcal {B}}=\{x\}} ) but B {\displaystyle {\mathcal {B}}} is not principal since { x } β B
|
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"page_id": 47516955,
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. {\displaystyle \{x\}\not \in {\mathcal {B}}.} In contrast, a topological space has the discrete topology if and only if the neighborhood filter of every point is a principal filter generated by exactly one point. This shows that a non-principal filter on an infinite set is not necessarily free. The neighborhood filter of every point x {\displaystyle x} in topological space X {\displaystyle X} is fixed since its kernel contains x {\displaystyle x} (and possibly other points if, for instance, X {\displaystyle X} is not a T1 space). This is also true of any neighborhood basis at x . {\displaystyle x.} For any point x {\displaystyle x} in a T1 space (for example, a Hausdorff space), the kernel of the neighborhood filter of x {\displaystyle x} is equal to the singleton set { x } . {\displaystyle \{x\}.} However, it is possible for a neighborhood filter at a point to be principal but not discrete (that is, not principal at a single point). A neighborhood basis B {\displaystyle {\mathcal {B}}} of a point x {\displaystyle x} in a topological space is principal if and only if the kernel of B {\displaystyle {\mathcal {B}}} is an open set. If in addition the space is T1 then ker β‘ B = { x } {\displaystyle \ker {\mathcal {B}}=\{x\}} so that this basis B {\displaystyle {\mathcal {B}}} is principal if and only if { x } {\displaystyle \{x\}} is an open set. Generating topologies from filters and prefilters Suppose B β β ( X ) {\displaystyle {\mathcal {B}}\subseteq \wp (X)} is not empty (and X β β
{\displaystyle X\neq \varnothing } ). If B {\displaystyle {\mathcal {B}}} is a filter on X {\displaystyle X} then { β
} βͺ B {\displaystyle \{\varnothing \}\cup {\mathcal {B}}} is a topology on X {\displaystyle X} but the
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"page_id": 47516955,
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converse is in general false. This shows that in a sense, filters are almost topologies. Topologies of the form { β
} βͺ B {\displaystyle \{\varnothing \}\cup {\mathcal {B}}} where B {\displaystyle {\mathcal {B}}} is an ultrafilter on X {\displaystyle X} are an even more specialized subclass of such topologies; they have the property that every proper subset β
β S β X {\displaystyle \varnothing \neq S\subseteq X} is either open or closed, but (unlike the discrete topology) never both. These spaces are, in particular, examples of door spaces. If B {\displaystyle {\mathcal {B}}} is a prefilter (resp. filter subbase, Ο-system, proper) on X {\displaystyle X} then the same is true of both { X } βͺ B {\displaystyle \{X\}\cup {\mathcal {B}}} and the set B βͺ {\displaystyle {\mathcal {B}}_{\cup }} of all possible unions of one or more elements of B . {\displaystyle {\mathcal {B}}.} If B {\displaystyle {\mathcal {B}}} is closed under finite intersections then the set Ο B = { β
, X } βͺ B βͺ {\displaystyle \tau _{\mathcal {B}}=\{\varnothing ,X\}\cup {\mathcal {B}}_{\cup }} is a topology on X {\displaystyle X} with both { X } βͺ B βͺ and { X } βͺ B {\displaystyle \{X\}\cup {\mathcal {B}}_{\cup }{\text{ and }}\{X\}\cup {\mathcal {B}}} being bases for it. If the Ο-system B {\displaystyle {\mathcal {B}}} covers X {\displaystyle X} then both B βͺ and B {\displaystyle {\mathcal {B}}_{\cup }{\text{ and }}{\mathcal {B}}} are also bases for Ο B . {\displaystyle \tau _{\mathcal {B}}.} If Ο {\displaystyle \tau } is a topology on X {\displaystyle X} then Ο β { β
} {\displaystyle \tau \setminus \{\varnothing \}} is a prefilter (or equivalently, a Ο-system) if and only if it has the finite intersection property (that is, it is a filter subbase), in which case a subset B
|
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"page_id": 47516955,
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"title": "Filters in topology"
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β Ο {\displaystyle {\mathcal {B}}\subseteq \tau } will be a basis for Ο {\displaystyle \tau } if and only if B β { β
} {\displaystyle {\mathcal {B}}\setminus \{\varnothing \}} is equivalent to Ο β { β
} , {\displaystyle \tau \setminus \{\varnothing \},} in which case B β { β
} {\displaystyle {\mathcal {B}}\setminus \{\varnothing \}} will be a prefilter. === Topological properties and prefilters === Neighborhoods and topologies The neighborhood filter of a nonempty subset S β X {\displaystyle S\subseteq X} in a topological space X {\displaystyle X} is equal to the intersection of all neighborhood filters of all points in S . {\displaystyle S.} A subset S β X {\displaystyle S\subseteq X} is open in X {\displaystyle X} if and only if whenever F {\displaystyle {\mathcal {F}}} is a filter on X {\displaystyle X} and s β S , {\displaystyle s\in S,} then F β s in X implies S β F . {\displaystyle {\mathcal {F}}\to s{\text{ in }}X{\text{ implies }}S\in {\mathcal {F}}.} Suppose Ο and Ο {\displaystyle \sigma {\text{ and }}\tau } are topologies on X . {\displaystyle X.} Then Ο {\displaystyle \tau } is finer than Ο {\displaystyle \sigma } (that is, Ο β Ο {\displaystyle \sigma \subseteq \tau } ) if and only if whenever x β X and B {\displaystyle x\in X{\text{ and }}{\mathcal {B}}} is a filter on X , {\displaystyle X,} if B β x in ( X , Ο ) {\displaystyle {\mathcal {B}}\to x{\text{ in }}(X,\tau )} then B β x in ( X , Ο ) . {\displaystyle {\mathcal {B}}\to x{\text{ in }}(X,\sigma ).} Consequently, Ο = Ο {\displaystyle \sigma =\tau } if and only if for every filter B on X {\displaystyle {\mathcal {B}}{\text{ on }}X} and every x β X , B β x in ( X
|
{
"page_id": 47516955,
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, Ο ) {\displaystyle x\in X,{\mathcal {B}}\to x{\text{ in }}(X,\sigma )} if and only if B β x in ( X , Ο ) . {\displaystyle {\mathcal {B}}\to x{\text{ in }}(X,\tau ).} However, it is possible that Ο β Ο {\displaystyle \sigma \neq \tau } while also for every filter B on X , B {\displaystyle {\mathcal {B}}{\text{ on }}X,{\mathcal {B}}} converges to some point of X in ( X , Ο ) {\displaystyle X{\text{ in }}(X,\sigma )} if and only if B {\displaystyle {\mathcal {B}}} converges to some point of X in ( X , Ο ) . {\displaystyle X{\text{ in }}(X,\tau ).} Closure If B {\displaystyle {\mathcal {B}}} is a prefilter on a subset S β X {\displaystyle S\subseteq X} then every cluster point of B in X {\displaystyle {\mathcal {B}}{\text{ in }}X} belongs to cl X β‘ S . {\displaystyle \operatorname {cl} _{X}S.} If x β X and S β X {\displaystyle x\in X{\text{ and }}S\subseteq X} is a non-empty subset, then the following are equivalent: x β cl X β‘ S {\displaystyle x\in \operatorname {cl} _{X}S} x {\displaystyle x} is a limit point of a prefilter on S . {\displaystyle S.} Explicitly: there exists a prefilter F β β ( S ) on S {\displaystyle {\mathcal {F}}\subseteq \wp (S){\text{ on }}S} such that F β x in X . {\displaystyle {\mathcal {F}}\to x{\text{ in }}X.} x {\displaystyle x} is a limit point of a filter on S . {\displaystyle S.} There exists a prefilter F on X {\displaystyle {\mathcal {F}}{\text{ on }}X} such that S β F and F β x in X . {\displaystyle S\in {\mathcal {F}}{\text{ and }}{\mathcal {F}}\to x{\text{ in }}X.} The prefilter { S } {\displaystyle \{S\}} meshes with the neighborhood filter N ( x ) . {\displaystyle {\mathcal {N}}(x).} Said differently,
|
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"page_id": 47516955,
"source": null,
"title": "Filters in topology"
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x {\displaystyle x} is a cluster point of the prefilter { S } . {\displaystyle \{S\}.} The prefilter { S } {\displaystyle \{S\}} meshes with some (or equivalently, with every) filter base for N ( x ) {\displaystyle {\mathcal {N}}(x)} (that is, with every neighborhood basis at x {\displaystyle x} ). The following are equivalent: x {\displaystyle x} is a limit points of S in X . {\displaystyle S{\text{ in }}X.} There exists a prefilter F β β ( S ) on { S } β { x } {\displaystyle {\mathcal {F}}\subseteq \wp (S){\text{ on }}\{S\}\setminus \{x\}} such that F β x in X . {\displaystyle {\mathcal {F}}\to x{\text{ in }}X.} Closed sets If S β X {\displaystyle S\subseteq X} is not empty then the following are equivalent: S {\displaystyle S} is a closed subset of X . {\displaystyle X.} If x β X and F β β ( S ) {\displaystyle x\in X{\text{ and }}{\mathcal {F}}\subseteq \wp (S)} is a prefilter on S {\displaystyle S} such that F β x in X , {\displaystyle {\mathcal {F}}\to x{\text{ in }}X,} then x β S . {\displaystyle x\in S.} If x β X and F β β ( S ) {\displaystyle x\in X{\text{ and }}{\mathcal {F}}\subseteq \wp (S)} is a prefilter on S {\displaystyle S} such that x {\displaystyle x} is an accumulation points of F in X , {\displaystyle {\mathcal {F}}{\text{ in }}X,} then x β S . {\displaystyle x\in S.} If x β X {\displaystyle x\in X} is such that the neighborhood filter N ( x ) {\displaystyle {\mathcal {N}}(x)} meshes with { S } {\displaystyle \{S\}} then x β S . {\displaystyle x\in S.} Hausdorffness The following are equivalent: X {\displaystyle X} is a Hausdorff space. Every prefilter on X {\displaystyle X} converges to at most one point
|
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"page_id": 47516955,
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in X . {\displaystyle X.} The above statement but with the word "prefilter" replaced by any one of the following: filter, ultra prefilter, ultrafilter. Compactness As discussed in this article, the Ultrafilter Lemma is closely related to many important theorems involving compactness. The following are equivalent: ( X , Ο ) {\displaystyle (X,\tau )} is a compact space. Every ultrafilter on X {\displaystyle X} converges to at least one point in X . {\displaystyle X.} That this condition implies compactness can be proven by using only the ultrafilter lemma. That compactness implies this condition can be proven without the ultrafilter lemma (or even the axiom of choice). The above statement but with the word "ultrafilter" replaced by "ultra prefilter". For every filter C on X {\displaystyle {\mathcal {C}}{\text{ on }}X} there exists a filter F on X {\displaystyle {\mathcal {F}}{\text{ on }}X} such that C β€ F {\displaystyle {\mathcal {C}}\leq {\mathcal {F}}} and F {\displaystyle {\mathcal {F}}} converges to some point of X . {\displaystyle X.} The above statement but with each instance of the word "filter" replaced by: prefilter. Every filter on X {\displaystyle X} has at least one cluster point in X . {\displaystyle X.} That this condition is equivalent to compactness can be proven by using only the ultrafilter lemma. The above statement but with the word "filter" replaced by "prefilter". Alexander subbase theorem: There exists a subbase S for Ο {\displaystyle {\mathcal {S}}{\text{ for }}\tau } such that every cover of X {\displaystyle X} by sets in S {\displaystyle {\mathcal {S}}} has a finite subcover. That this condition is equivalent to compactness can be proven by using only the ultrafilter lemma. If F {\displaystyle {\mathcal {F}}} is the set of all complements of compact subsets of a given topological space X , {\displaystyle X,} then F
|
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"page_id": 47516955,
"source": null,
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{\displaystyle {\mathcal {F}}} is a filter on X {\displaystyle X} if and only if X {\displaystyle X} is not compact. Continuity Let f : X β Y {\displaystyle f:X\to Y} be a map between topological spaces ( X , Ο ) and ( Y , Ο
) . {\displaystyle (X,\tau ){\text{ and }}(Y,\upsilon ).} Given x β X , {\displaystyle x\in X,} the following are equivalent: f : X β Y {\displaystyle f:X\to Y} is continuous at x . {\displaystyle x.} Definition: For every neighborhood V {\displaystyle V} of f ( x ) in Y {\displaystyle f(x){\text{ in }}Y} there exists some neighborhood N {\displaystyle N} of x in X {\displaystyle x{\text{ in }}X} such that f ( N ) β V . {\displaystyle f(N)\subseteq V.} f ( N ( x ) ) β f ( x ) in Y . {\displaystyle f({\mathcal {N}}(x))\to f(x){\text{ in }}Y.} If B {\displaystyle {\mathcal {B}}} is a filter on X {\displaystyle X} such that B β x in X {\displaystyle {\mathcal {B}}\to x{\text{ in }}X} then f ( B ) β f ( x ) in Y . {\displaystyle f({\mathcal {B}})\to f(x){\text{ in }}Y.} The above statement but with the word "filter" replaced by "prefilter". The following are equivalent: f : X β Y {\displaystyle f:X\to Y} is continuous. If x β X and B {\displaystyle x\in X{\text{ and }}{\mathcal {B}}} is a prefilter on X {\displaystyle X} such that B β x in X {\displaystyle {\mathcal {B}}\to x{\text{ in }}X} then f ( B ) β f ( x ) in Y . {\displaystyle f({\mathcal {B}})\to f(x){\text{ in }}Y.} If x β X {\displaystyle x\in X} is a limit point of a prefilter B on X {\displaystyle {\mathcal {B}}{\text{ on }}X} then f ( x ) {\displaystyle f(x)} is a limit point
|
{
"page_id": 47516955,
"source": null,
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of f ( B ) in Y . {\displaystyle f({\mathcal {B}}){\text{ in }}Y.} Any one of the above two statements but with the word "prefilter" replaced by "filter". If B {\displaystyle {\mathcal {B}}} is a prefilter on X , x β X {\displaystyle X,x\in X} is a cluster point of B , and f : X β Y {\displaystyle {\mathcal {B}},{\text{ and }}f:X\to Y} is continuous, then f ( x ) {\displaystyle f(x)} is a cluster point in Y {\displaystyle Y} of the prefilter f ( B ) . {\displaystyle f({\mathcal {B}}).} A subset D {\displaystyle D} of a topological space X {\displaystyle X} is dense in X {\displaystyle X} if and only if for every x β X , {\displaystyle x\in X,} the trace N X ( x ) | D {\displaystyle {\mathcal {N}}_{X}(x){\big \vert }_{D}} of the neighborhood filter N X ( x ) {\displaystyle {\mathcal {N}}_{X}(x)} along D {\displaystyle D} does not contain the empty set (in which case it will be a filter on D {\displaystyle D} ). Suppose f : D β Y {\displaystyle f:D\to Y} is a continuous map into a Hausdorff regular space Y {\displaystyle Y} and that D {\displaystyle D} is a dense subset of a topological space X . {\displaystyle X.} Then f {\displaystyle f} has a continuous extension F : X β Y {\displaystyle F:X\to Y} if and only if for every x β X , {\displaystyle x\in X,} the prefilter f ( N X ( x ) | D ) {\displaystyle f\left({\mathcal {N}}_{X}(x){\big \vert }_{D}\right)} converges to some point in Y . {\displaystyle Y.} Furthermore, this continuous extension will be unique whenever it exists. Products Suppose X β := ( X i ) i β I {\displaystyle X_{\bullet }:=\left(X_{i}\right)_{i\in I}} is a non-empty family of non-empty topological spaces and
|
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"page_id": 47516955,
"source": null,
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}
|
that is a family of prefilters where each B i {\displaystyle {\mathcal {B}}_{i}} is a prefilter on X i . {\displaystyle X_{i}.} Then the product B β {\displaystyle {\mathcal {B}}_{\bullet }} of these prefilters (defined above) is a prefilter on the product space β X β , {\displaystyle {\textstyle \prod }X_{\bullet },} which as usual, is endowed with the product topology. If x β := ( x i ) i β I β β X β , {\displaystyle x_{\bullet }:=\left(x_{i}\right)_{i\in I}\in {\textstyle \prod }X_{\bullet },} then B β β x β in β X β {\displaystyle {\mathcal {B}}_{\bullet }\to x_{\bullet }{\text{ in }}{\textstyle \prod }X_{\bullet }} if and only if B i β x i in X i for every i β I . {\displaystyle {\mathcal {B}}_{i}\to x_{i}{\text{ in }}X_{i}{\text{ for every }}i\in I.} Suppose X and Y {\displaystyle X{\text{ and }}Y} are topological spaces, B {\displaystyle {\mathcal {B}}} is a prefilter on X {\displaystyle X} having x β X {\displaystyle x\in X} as a cluster point, and C {\displaystyle {\mathcal {C}}} is a prefilter on Y {\displaystyle Y} having y β Y {\displaystyle y\in Y} as a cluster point. Then ( x , y ) {\displaystyle (x,y)} is a cluster point of B Γ C {\displaystyle {\mathcal {B}}\times {\mathcal {C}}} in the product space X Γ Y . {\displaystyle X\times Y.} However, if X = Y = Q {\displaystyle X=Y=\mathbb {Q} } then there exist sequences ( x i ) i = 1 β β X and ( y i ) i = 1 β β Y {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }\subseteq X{\text{ and }}\left(y_{i}\right)_{i=1}^{\infty }\subseteq Y} such that both of these sequences have a cluster point in Q {\displaystyle \mathbb {Q} } but the sequence ( x i , y i ) i = 1 β β X Γ Y
|
{
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{\displaystyle \left(x_{i},y_{i}\right)_{i=1}^{\infty }\subseteq X\times Y} does not have a cluster point in X Γ Y . {\displaystyle X\times Y.} Example application: The ultrafilter lemma along with the axioms of ZF imply Tychonoff's theorem for compact Hausdorff spaces: == Examples of applications of prefilters == === Uniformities and Cauchy prefilters === A uniform space is a set X {\displaystyle X} equipped with a filter on X Γ X {\displaystyle X\times X} that has certain properties. A base or fundamental system of entourages is a prefilter on X Γ X {\displaystyle X\times X} whose upward closure is a uniform space. A prefilter B {\displaystyle {\mathcal {B}}} on a uniform space X {\displaystyle X} with uniformity F {\displaystyle {\mathcal {F}}} is called a Cauchy prefilter if for every entourage N β F , {\displaystyle N\in {\mathcal {F}},} there exists some B β B {\displaystyle B\in {\mathcal {B}}} that is N {\displaystyle N} -small, which means that B Γ B β N . {\displaystyle B\times B\subseteq N.} A minimal Cauchy filter is a minimal element (with respect to β€ {\displaystyle \,\leq \,} or equivalently, to β {\displaystyle \,\subseteq } ) of the set of all Cauchy filters on X . {\displaystyle X.} Examples of minimal Cauchy filters include the neighborhood filter N X ( x ) {\displaystyle {\mathcal {N}}_{X}(x)} of any point x β X . {\displaystyle x\in X.} Every convergent filter on a uniform space is Cauchy. Moreover, every cluster point of a Cauchy filter is a limit point. A uniform space ( X , F ) {\displaystyle (X,{\mathcal {F}})} is called complete (resp. sequentially complete) if every Cauchy prefilter (resp. every elementary Cauchy prefilter) on X {\displaystyle X} converges to at least one point of X {\displaystyle X} (replacing all instance of the word "prefilter" with "filter" results in equivalent statement). Every
|
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"page_id": 47516955,
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compact uniform space is complete because any Cauchy filter has a cluster point (by compactness), which is necessarily also a limit point (since the filter is Cauchy). Uniform spaces were the result of attempts to generalize notions such as "uniform continuity" and "uniform convergence" that are present in metric spaces. Every topological vector space, and more generally, every topological group can be made into a uniform space in a canonical way. Every uniformity also generates a canonical induced topology. Filters and prefilters play an important role in the theory of uniform spaces. For example, the completion of a Hausdorff uniform space (even if it is not metrizable) is typically constructed by using minimal Cauchy filters. Nets are less ideal for this construction because their domains are extremely varied (for example, the class of all Cauchy nets is not a set); sequences cannot be used in the general case because the topology might not be metrizable, first-countable, or even sequential. The set of all minimal Cauchy filters on a Hausdorff topological vector space (TVS) X {\displaystyle X} can made into a vector space and topologized in such a way that it becomes a completion of X {\displaystyle X} (with the assignment x β¦ N X ( x ) {\displaystyle x\mapsto {\mathcal {N}}_{X}(x)} becoming a linear topological embedding that identifies X {\displaystyle X} as a dense vector subspace of this completion). More generally, a Cauchy space is a pair ( X , C ) {\displaystyle (X,{\mathfrak {C}})} consisting of a set X {\displaystyle X} together a family C β β ( β ( X ) ) {\displaystyle {\mathfrak {C}}\subseteq \wp (\wp (X))} of (proper) filters, whose members are declared to be "Cauchy filters", having all of the following properties: For each x β X , {\displaystyle x\in X,} the discrete ultrafilter at
|
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"page_id": 47516955,
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x {\displaystyle x} is an element of C . {\displaystyle {\mathfrak {C}}.} If F β C {\displaystyle F\in {\mathfrak {C}}} is a subset of a proper filter G , {\displaystyle G,} then G β C . {\displaystyle G\in {\mathfrak {C}}.} If F , G β C {\displaystyle F,G\in {\mathfrak {C}}} and if each member of F {\displaystyle F} intersects each member of G , {\displaystyle G,} then F β© G β C . {\displaystyle F\cap G\in {\mathfrak {C}}.} The set of all Cauchy filters on a uniform space forms a Cauchy space. Every Cauchy space is also a convergence space. A map f : X β Y {\displaystyle f:X\to Y} between two Cauchy spaces is called Cauchy continuous if the image of every Cauchy filter in X {\displaystyle X} is a Cauchy filter in Y . {\displaystyle Y.} Unlike the category of topological spaces, the category of Cauchy spaces and Cauchy continuous maps is Cartesian closed, and contains the category of proximity spaces. === Topologizing the set of prefilters === Starting with nothing more than a set X , {\displaystyle X,} it is possible to topologize the set P := Prefilters β‘ ( X ) {\displaystyle \mathbb {P} :=\operatorname {Prefilters} (X)} of all filter bases on X {\displaystyle X} with the Stone topology, which is named after Marshall Harvey Stone. To reduce confusion, this article will adhere to the following notational conventions: Lower case letters for elements x β X . {\displaystyle x\in X.} Upper case letters for subsets S β X . {\displaystyle S\subseteq X.} Upper case calligraphy letters for subsets B β β ( X ) {\displaystyle {\mathcal {B}}\subseteq \wp (X)} (or equivalently, for elements B β β ( β ( X ) ) , {\displaystyle {\mathcal {B}}\in \wp (\wp (X)),} such as prefilters). Upper case double-struck letters
|
{
"page_id": 47516955,
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|
for subsets P β β ( β ( X ) ) . {\displaystyle \mathbb {P} \subseteq \wp (\wp (X)).} For every S β X , {\displaystyle S\subseteq X,} let O ( S ) := { B β P : S β B β X } {\displaystyle \mathbb {O} (S):=\left\{{\mathcal {B}}\in \mathbb {P} ~:~S\in {\mathcal {B}}^{\uparrow X}\right\}} where O ( X ) = P and O ( β
) = β
. {\displaystyle \mathbb {O} (X)=\mathbb {P} {\text{ and }}\mathbb {O} (\varnothing )=\varnothing .} These sets will be the basic open subsets of the Stone topology. If R β S β X {\displaystyle R\subseteq S\subseteq X} then { B β β ( β ( X ) ) : R β B β X } β { B β β ( β ( X ) ) : S β B β X } . {\displaystyle \left\{{\mathcal {B}}\in \wp (\wp (X))~:~R\in {\mathcal {B}}^{\uparrow X}\right\}~\subseteq ~\left\{{\mathcal {B}}\in \wp (\wp (X))~:~S\in {\mathcal {B}}^{\uparrow X}\right\}.} From this inclusion, it is possible to deduce all of the subset inclusions displayed below with the exception of O ( R β© S ) β O ( R ) β© O ( S ) . {\displaystyle \mathbb {O} (R\cap S)~\supseteq ~\mathbb {O} (R)\cap \mathbb {O} (S).} For all R β S β X , {\displaystyle R\subseteq S\subseteq X,} O ( R β© S ) = O ( R ) β© O ( S ) β O ( R ) βͺ O ( S ) β O ( R βͺ S ) {\displaystyle \mathbb {O} (R\cap S)~=~\mathbb {O} (R)\cap \mathbb {O} (S)~\subseteq ~\mathbb {O} (R)\cup \mathbb {O} (S)~\subseteq ~\mathbb {O} (R\cup S)} where in particular, the equality O ( R β© S ) = O ( R ) β© O ( S ) {\displaystyle \mathbb {O} (R\cap S)=\mathbb {O} (R)\cap \mathbb
|
{
"page_id": 47516955,
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{O} (S)} shows that the family { O ( S ) : S β X } {\displaystyle \{\mathbb {O} (S)~:~S\subseteq X\}} is a Ο {\displaystyle \pi } -system that forms a basis for a topology on P {\displaystyle \mathbb {P} } called the Stone topology. It is henceforth assumed that P {\displaystyle \mathbb {P} } carries this topology and that any subset of P {\displaystyle \mathbb {P} } carries the induced subspace topology. In contrast to most other general constructions of topologies (for example, the product, quotient, subspace topologies, etc.), this topology on P {\displaystyle \mathbb {P} } was defined without using anything other than the set X ; {\displaystyle X;} there were no preexisting structures or assumptions on X {\displaystyle X} so this topology is completely independent of everything other than X {\displaystyle X} (and its subsets). The following criteria can be used for checking for points of closure and neighborhoods. If B β P and F β P {\displaystyle \mathbb {B} \subseteq \mathbb {P} {\text{ and }}{\mathcal {F}}\in \mathbb {P} } then: Closure in P {\displaystyle \mathbb {P} } : F {\displaystyle \ {\mathcal {F}}} belongs to the closure of B in P {\displaystyle \mathbb {B} {\text{ in }}\mathbb {P} } if and only if F β β B β B B β X . {\displaystyle {\mathcal {F}}\subseteq {\textstyle \bigcup \limits _{{\mathcal {B}}\in \mathbb {B} }}{\mathcal {B}}^{\uparrow X}.} Neighborhoods in P {\displaystyle \mathbb {P} } : B {\displaystyle \ \mathbb {B} } is a neighborhood of F in P {\displaystyle {\mathcal {F}}{\text{ in }}\mathbb {P} } if and only if there exists some F β F {\displaystyle F\in {\mathcal {F}}} such that O ( F ) = { B β P : F β B β X } β B {\displaystyle \mathbb {O} (F)=\left\{{\mathcal {B}}\in \mathbb {P} ~:~F\in
|
{
"page_id": 47516955,
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"title": "Filters in topology"
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{\mathcal {B}}^{\uparrow X}\right\}\subseteq \mathbb {B} } (that is, such that for all B β P , if F β B β X then B β B {\displaystyle {\mathcal {B}}\in \mathbb {P} ,{\text{ if }}F\in {\mathcal {B}}^{\uparrow X}{\text{ then }}{\mathcal {B}}\in \mathbb {B} } ). It will be henceforth assumed that X β β
{\displaystyle X\neq \varnothing } because otherwise P = β
{\displaystyle \mathbb {P} =\varnothing } and the topology is { β
} , {\displaystyle \{\varnothing \},} which is uninteresting. Subspace of ultrafilters The set of ultrafilters on X {\displaystyle X} (with the subspace topology) is a Stone space, meaning that it is compact, Hausdorff, and totally disconnected. If X {\displaystyle X} has the discrete topology then the map Ξ² : X β UltraFilters β‘ ( X ) , {\displaystyle \beta :X\to \operatorname {UltraFilters} (X),} defined by sending x β X {\displaystyle x\in X} to the principal ultrafilter at x , {\displaystyle x,} is a topological embedding whose image is a dense subset of UltraFilters β‘ ( X ) {\displaystyle \operatorname {UltraFilters} (X)} (see the article StoneβΔech compactification for more details). Relationships between topologies on X {\displaystyle X} and the Stone topology on P {\displaystyle \mathbb {P} } Every Ο β Top β‘ ( X ) {\displaystyle \tau \in \operatorname {Top} (X)} induces a canonical map N Ο : X β Filters β‘ ( X ) {\displaystyle {\mathcal {N}}_{\tau }:X\to \operatorname {Filters} (X)} defined by x β¦ N Ο ( x ) , {\displaystyle x\mapsto {\mathcal {N}}_{\tau }(x),} which sends x β X {\displaystyle x\in X} to the neighborhood filter of x in ( X , Ο ) . {\displaystyle x{\text{ in }}(X,\tau ).} If Ο , Ο β Top β‘ ( X ) {\displaystyle \tau ,\sigma \in \operatorname {Top} (X)} then Ο = Ο {\displaystyle \tau =\sigma }
|
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"page_id": 47516955,
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if and only if N Ο = N Ο . {\displaystyle {\mathcal {N}}_{\tau }={\mathcal {N}}_{\sigma }.} Thus every topology Ο β Top β‘ ( X ) {\displaystyle \tau \in \operatorname {Top} (X)} can be identified with the canonical map N Ο β Func β‘ ( X ; P ) , {\displaystyle {\mathcal {N}}_{\tau }\in \operatorname {Func} (X;\mathbb {P} ),} which allows Top β‘ ( X ) {\displaystyle \operatorname {Top} (X)} to be canonically identified as a subset of Func β‘ ( X ; P ) {\displaystyle \operatorname {Func} (X;\mathbb {P} )} (as a side note, it is now possible to place on Func β‘ ( X ; P ) , {\displaystyle \operatorname {Func} (X;\mathbb {P} ),} and thus also on Top β‘ ( X ) , {\displaystyle \operatorname {Top} (X),} the topology of pointwise convergence on X {\displaystyle X} so that it now makes sense to talk about things such as sequences of topologies on X {\displaystyle X} converging pointwise). For every Ο β Top β‘ ( X ) , {\displaystyle \tau \in \operatorname {Top} (X),} the surjection N Ο : ( X , Ο ) β image β‘ N Ο {\displaystyle {\mathcal {N}}_{\tau }:(X,\tau )\to \operatorname {image} {\mathcal {N}}_{\tau }} is always continuous, closed, and open, but it is injective if and only if Ο is T 0 {\displaystyle \tau {\text{ is }}T_{0}} (that is, a Kolmogorov space). In particular, for every T 0 {\displaystyle T_{0}} topology Ο on X , {\displaystyle \tau {\text{ on }}X,} the map N Ο : ( X , Ο ) β P {\displaystyle {\mathcal {N}}_{\tau }:(X,\tau )\to \mathbb {P} } is a topological embedding (said differently, every Kolmogorov space is a topological subspace of the space of prefilters). In addition, if F : X β Filters β‘ ( X ) {\displaystyle {\mathfrak {F}}:X\to
|
{
"page_id": 47516955,
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\operatorname {Filters} (X)} is a map such that x β ker β‘ F ( x ) := β F β F ( x ) F for every x β X {\displaystyle x\in \ker {\mathfrak {F}}(x):={\textstyle \bigcap \limits _{F\in {\mathfrak {F}}(x)}}F{\text{ for every }}x\in X} (which is true of F := N Ο , {\displaystyle {\mathfrak {F}}:={\mathcal {N}}_{\tau },} for instance), then for every x β X and F β F ( x ) , {\displaystyle x\in X{\text{ and }}F\in {\mathfrak {F}}(x),} the set F ( F ) = { F ( f ) : f β F } {\displaystyle {\mathfrak {F}}(F)=\{{\mathfrak {F}}(f):f\in F\}} is a neighborhood (in the subspace topology) of F ( x ) in image β‘ F . {\displaystyle {\mathfrak {F}}(x){\text{ in }}\operatorname {image} {\mathfrak {F}}.} == See also == Characterizations of the category of topological spaces Convergence space β Generalization of the notion of convergence that is found in general topology Filtration (mathematics) β Indexed set in mathematics Filtration (probability theory) β Model of information available at a given point of a random process Filtration (abstract algebra) FrΓ©chet filter β frechet filterPages displaying wikidata descriptions as a fallback Generic filter β in set theory, given a collection of dense open subsets of a poset, a filter that meets all sets in that collectionPages displaying wikidata descriptions as a fallback Ideal (set theory) β Non-empty family of sets that is closed under finite unions and subsets StoneβΔech compactification#Construction using ultrafilters β Concept in topology The fundamental theorem of ultraproducts β Mathematical constructionPages displaying short descriptions of redirect targets == Notes == Proofs == Citations == == References == Adams, Colin; Franzosa, Robert (2009). Introduction to Topology: Pure and Applied. New Delhi: Pearson Education. ISBN 978-81-317-2692-1. OCLC 789880519. Arkhangel'skii, Alexander Vladimirovich; Ponomarev, V.I. (1984). Fundamentals of General Topology: Problems
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{
"page_id": 47516955,
"source": null,
"title": "Filters in topology"
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and Exercises. Mathematics and Its Applications. Vol. 13. Dordrecht Boston: D. Reidel. ISBN 978-90-277-1355-1. OCLC 9944489. Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. Vol. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401. Bourbaki, Nicolas (1989) [1966]. General Topology: Chapters 1β4 [Topologie GΓ©nΓ©rale]. ΓlΓ©ments de mathΓ©matique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64241-1. OCLC 18588129. Bourbaki, Nicolas (1989) [1967]. General Topology 2: Chapters 5β10 [Topologie GΓ©nΓ©rale]. ΓlΓ©ments de mathΓ©matique. Vol. 4. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64563-4. OCLC 246032063. Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1β5. ΓlΓ©ments de mathΓ©matique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190. Burris, Stanley; Sankappanavar, Hanamantagouda P. (2012). A Course in Universal Algebra (PDF). Springer-Verlag. ISBN 978-0-9880552-0-9. Archived from the original on 1 April 2022. Cartan, Henri (1937a). "ThΓ©orie des filtres". Comptes rendus hebdomadaires des sΓ©ances de l'AcadΓ©mie des sciences. 205: 595β598. Cartan, Henri (1937b). "Filtres et ultrafiltres". Comptes rendus hebdomadaires des sΓ©ances de l'AcadΓ©mie des sciences. 205: 777β779. Comfort, William Wistar; Negrepontis, Stylianos (1974). The Theory of Ultrafilters. Vol. 211. Berlin Heidelberg New York: Springer-Verlag. ISBN 978-0-387-06604-2. OCLC 1205452. CsΓ‘szΓ‘r, Γkos (1978). General topology. Translated by CsΓ‘szΓ‘r, KlΓ‘ra. Bristol England: Adam Hilger Ltd. ISBN 0-85274-275-4. OCLC 4146011. Dixmier, Jacques (1984). General Topology. Undergraduate Texts in Mathematics. Translated by Berberian, S. K. New York: Springer-Verlag. ISBN 978-0-387-90972-1. OCLC 10277303. Dolecki, Szymon; Mynard, FrΓ©dΓ©ric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917. Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485. Dunford, Nelson; Schwartz, Jacob T. (1988). Linear Operators. Pure and applied mathematics. Vol. 1. New York: Wiley-Interscience. ISBN 978-0-471-60848-6. OCLC 18412261. Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover
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{
"page_id": 47516955,
"source": null,
"title": "Filters in topology"
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Publications. ISBN 978-0-486-68143-6. OCLC 30593138. Howes, Norman R. (23 June 1995). Modern Analysis and Topology. Graduate Texts in Mathematics. New York: Springer-Verlag Science & Business Media. ISBN 978-0-387-97986-1. OCLC 31969970. OL 1272666M. Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. Jech, Thomas (2006). Set Theory: The Third Millennium Edition, Revised and Expanded. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-44085-7. OCLC 50422939. Joshi, K. D. (1983). Introduction to General Topology. New York: John Wiley and Sons Ltd. ISBN 978-0-85226-444-7. OCLC 9218750. Kelley, John L. (1975) [1955]. General Topology. Graduate Texts in Mathematics. Vol. 27 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-90125-1. OCLC 1365153. Kâthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704. MacIver R., David (1 July 2004). "Filters in Analysis and Topology" (PDF). Archived from the original (PDF) on 2007-10-09. (Provides an introductory review of filters in topology and in metric spaces.) Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250. Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365. Schubert, Horst (1968). Topology. London: Macdonald & Co. ISBN 978-0-356-02077-8. OCLC 463753. Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. Wilansky, Albert (2013). Modern Methods
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{
"page_id": 47516955,
"source": null,
"title": "Filters in topology"
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in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. Wilansky, Albert (17 October 2008) [1970]. Topology for Analysis. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-46903-4. OCLC 227923899. Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.
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Transcription activator-like effector nucleases (TALEN) are restriction enzymes that can be engineered to cut specific sequences of DNA. They are made by fusing a TAL effector DNA-binding domain to a DNA cleavage domain (a nuclease which cuts DNA strands). Transcription activator-like effectors (TALEs) can be engineered to bind to practically any desired DNA sequence, so when combined with a nuclease, DNA can be cut at specific locations. The restriction enzymes can be introduced into cells, for use in gene editing or for genome editing in situ, a technique known as genome editing with engineered nucleases. Alongside zinc finger nucleases and CRISPR/Cas9, TALEN is a prominent tool in the field of genome editing. == TALE DNA-binding domain == TAL effectors are proteins that are secreted by Xanthomonas bacteria via their type III secretion system when they infect plants. The DNA binding domain contains a repeated highly conserved 33β34 amino acid sequence with divergent 12th and 13th amino acids. These two positions, referred to as the Repeat Variable Diresidue (RVD), are highly variable and show a strong correlation with specific nucleotide recognition. This straightforward relationship between amino acid sequence and DNA recognition has allowed for the engineering of specific DNA-binding domains by selecting a combination of repeat segments containing the appropriate RVDs. Notably, slight changes in the RVD and the incorporation of "nonconventional" RVD sequences can improve targeting specificity. == DNA cleavage domain == The non-specific DNA cleavage domain from the end of the FokI endonuclease can be used to construct hybrid nucleases that are active in a yeast assay. These reagents are also active in plant cells and in animal cells. Initial TALEN studies used the wild-type FokI cleavage domain, but some subsequent TALEN studies also used FokI cleavage domain variants with mutations designed to improve cleavage specificity and cleavage activity.
|
{
"page_id": 31001884,
"source": null,
"title": "Transcription activator-like effector nuclease"
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|
The FokI domain functions as a dimer, requiring two constructs with unique DNA binding domains for sites in the target genome with proper orientation and spacing. Both the number of amino acid residues between the TALE DNA binding domain and the FokI cleavage domain and the number of bases between the two individual TALEN binding sites appear to be important parameters for achieving high levels of activity. == Engineering TALEN constructs == The simple relationship between amino acid sequence and DNA recognition of the TALE binding domain allows for the efficient engineering of proteins. In this case, artificial gene synthesis is problematic because of improper annealing of the repetitive sequence found in the TALE binding domain. One solution to this is to use a publicly available software program (DNAWorks) to calculate oligonucleotides suitable for assembly in a two step PCR oligonucleotide assembly followed by whole gene amplification. A number of modular assembly schemes for generating engineered TALE constructs have also been reported. Both methods offer a systematic approach to engineering DNA binding domains that is conceptually similar to the modular assembly method for generating zinc finger DNA recognition domains. == Transfection == Once the TALEN constructs have been assembled, they are inserted into plasmids; the target cells are then transfected with the plasmids, and the gene products are expressed and enter the nucleus to access the genome. Alternatively, TALEN constructs can be delivered to the cells as mRNAs, which removes the possibility of genomic integration of the TALEN-expressing protein. Using an mRNA vector can also dramatically increase the level of homology directed repair (HDR) and the success of introgression during gene editing. == Genome editing == === Mechanisms === TALEN can be used to edit genomes by inducing double-strand breaks (DSB), which cells respond to with repair mechanisms. Non-homologous end
|
{
"page_id": 31001884,
"source": null,
"title": "Transcription activator-like effector nuclease"
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joining (NHEJ) directly ligates DNA from either side of a double-strand break where there is very little or no sequence overlap for annealing. This repair mechanism induces errors in the genome via indels (insertion or deletion), or chromosomal rearrangement; any such errors may render the gene products coded at that location non-functional. Because this activity can vary depending on the species, cell type, target gene, and nuclease used, it should be monitored when designing new systems. A simple heteroduplex cleavage assay can be run which detects any difference between two alleles amplified by PCR. Cleavage products can be visualized on simple agarose gels or slab gel systems. Alternatively, DNA can be introduced into a genome through NHEJ in the presence of exogenous double-stranded DNA fragments. Homology directed repair can also introduce foreign DNA at the DSB as the transfected double-stranded sequences are used as templates for the repair enzymes. === Applications === TALEN has been used to efficiently modify plant genomes, creating economically important food crops with favorable nutritional qualities. They have also been harnessed to develop tools for the production of biofuels. In addition, it has been used to engineer stably modified human embryonic stem cell and induced pluripotent stem cell (IPSCs) clones and human erythroid cell lines, to generate knockout C. elegans, knockout rats, knockout mice, and knockout zebrafish. Moreover, the method can be used to generate knockin organisms. Wu et al.obtained a Sp110 knockin cattle using Talen nickases to induce increased resistance of tuberculosis. This approach has also been used to generate knockin rats by TALEN mRNA microinjection in one-cell embryos. TALEN has also been utilized experimentally to correct the genetic errors that underlie disease. For example, it has been used in vitro to correct the genetic defects that cause disorders such as sickle cell disease, xeroderma
|
{
"page_id": 31001884,
"source": null,
"title": "Transcription activator-like effector nuclease"
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pigmentosum, and epidermolysis bullosa. Recently, it was shown that TALEN can be used as tools to harness the immune system to fight cancers; TALEN-mediated targeting can generate T cells that are resistant to chemotherapeutic drugs and show anti-tumor activity. In theory, the genome-wide specificity of engineered TALEN fusions allows for correction of errors at individual genetic loci via homology-directed repair from a correct exogenous template. In reality, however, the in situ application of TALEN is currently limited by the lack of an efficient delivery mechanism, unknown immunogenic factors, and uncertainty in the specificity of TALEN binding. Another emerging application of TALEN is its ability to combine with other genome engineering tools, such as meganucleases. The DNA binding region of a TAL effector can be combined with the cleavage domain of a meganuclease to create a hybrid architecture combining the ease of engineering and highly specific DNA binding activity of a TAL effector with the low site frequency and specificity of a meganuclease. In comparison to other genome editing techniques, TALEN falls in the middle in terms of difficulty and cost. Unlike ZFNs, TALEN recognizes single nucleotides. It's far more straightforward to engineer interactions between TALEN DNA binding domains and their target nucleotides than it is to create interactions with ZFNs and their target nucleotide triplets. On the other hand, CRISPR relies on ribonucleotide complex formation instead of protein/DNA recognition. gRNAs have occasionally limitations regarding feasibility due to lack of PAM sites in the target sequence and even though they can be cheaply produced, the current development lead to a remarkable decrease of cost for TALENs, so that they are in a similar price and time range like CRISPR based genome editing. == TAL effector nuclease precision == The off-target activity of an active nuclease may lead to unwanted double-strand breaks
|
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"page_id": 31001884,
"source": null,
"title": "Transcription activator-like effector nuclease"
}
|
and may consequently yield chromosomal rearrangements and/or cell death. Studies have been carried out to compare the relative nuclease-associated toxicity of available technologies. Based on these studies and the maximal theoretical distance between DNA binding and nuclease activity, TALEN constructs are believed to have the greatest precision of the currently available technologies. == See also == Genome editing with engineered nucleases Zinc finger nuclease Meganuclease CRISPR == References == == External links == E-TALEN.org A comprehensive tool for TALEN design PDB Molecule of the Month An entry in the Protein Database's monthly structural highlight
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{
"page_id": 31001884,
"source": null,
"title": "Transcription activator-like effector nuclease"
}
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In generalized blockmodeling, the blockmodeling is done by "the translation of an equivalence type into a set of permitted block types", which differs from the conventional blockmodeling, which is using the indirect approach. It's a special instance of the direct blockmodeling approach. Generalized blockmodeling was introduced in 1994 by Patrick Doreian, Vladimir Batagelj and AnuΕ‘ka Ferligoj. == Definition == Generalized blockmodeling approach is a direct one, "where the optimal partition(s) is (are) identified based on minimal values of a compatible criterion function defined by the difference between empirical blocks and corresponding ideal blocks". At the same time, the much broader set of block types is introduced (while in conventional blockmodeling only certain types are used). The conventional blockmodeling is inductive due to nonspecification of neither the clusters or the location of block types, while in generalized blockmodeling the blockmodel is specified with more detail than just the permition of certain block types (e.g., prespecification). Further, it's possible to define departures from the permitted (ideal) blocktype, using criterion function.: 16β17 Using local optimization procedure, firstly the initial clustering (with specified number of clusters is done, based on random creation. How the clusters are neighboring to each other, is based on two transformations: 1) a vertex is moved from one to another cluster or 2) a pair of vertices is interchanged between two different clusters. This process of transformation steps is repeated many times, until only the best fitting partitions (with the minimized value of the criterion function) are kept as blockmodels for the future exploration of the network. Different types of generalized blockmodeling are: generalized binary blockmodeling, generalized valued blockmodeling and generalized homogeneity blockmodeling. == Benefits == According to Patrick Doreian, the benefits of generalized blockmodeling, are as follows: usage of explicit criterion function, compatible with a given type of equivalence,
|
{
"page_id": 68488477,
"source": null,
"title": "Generalized blockmodeling"
}
|
results to in-built measure of fit, which is integral to the establishment of the blockmodels (in conventional blockmodeling, there is no compelling and coherent measures of fit); partitions, based on generalized blockmodeling, regularly outperform and never perform less well than the partitions, based on conventional approach; with generalized blockmodeling it's possible to specify new types of blockmodels; this potentially unlimited set of new block types also results in permittion of inclusion of substantively driven blockmodels; in generalized blockmodeling, the specification of the block types and the location of some of them in the blockmodel is possible; researcher can speficy which (pair of) vertices must be (not) clustered together; this approach also allows the imposition of penalties, resulting into identification of empirical null blocks without inconsistencies with a corresponding ideal null block. == Problems == According to Doreian, the benefits of generalized blockmodeling, are as follows: unknown sensitivity to particular data features, examination of boundary problems, computationally burdensome, which results in a constraint regarding practical network size (generalized blockmodeling is thus primarily used to analyse smaller networks (below 100 units)), identifying structure from incomplete network information, most of generalized blockmodeling is based on binary networks, but there is also development in the field of valued networks, criterion function is minimized for a specified blockmodel, with results in issues of evaluating statistically, based on the structural data alone, problems regarding three dimensional network data, problems regarding the evolution of fundamental network structure. == Book == The book with the same title, Generalized blockmodeling, written by Patrick Doreian, Vladimir Batagelj and AnuΕ‘ka Ferligoj, was in 2007 awarded the Harrison White Outstanding Book Award by the Mathematical Sociology Section of American Sociological Association. == See also == Generative model == References == == Selected bibliography == Patrick Doreian, Vladimir Batagelj, AnuΕ‘ka Ferligoj, Mark Granovetter
|
{
"page_id": 68488477,
"source": null,
"title": "Generalized blockmodeling"
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(Series Editor), Generalized Blockmodeling (Structural Analysis in the Social Sciences), Cambridge University Press 2004 (ISBN 0-521-84085-6)
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{
"page_id": 68488477,
"source": null,
"title": "Generalized blockmodeling"
}
|
Fluctuating asymmetry (FA), is a form of biological asymmetry, along with anti-symmetry and direction asymmetry. Fluctuating asymmetry refers to small, random deviations away from perfect bilateral symmetry. This deviation from perfection is thought to reflect the genetic and environmental pressures experienced throughout development, with greater pressures resulting in higher levels of asymmetry. Examples of FA in the human body include unequal sizes (asymmetry) of bilateral features in the face and body, such as left and right eyes, ears, wrists, breasts, testicles, and thighs. Research has exposed multiple factors that are associated with FA. As measuring FA can indicate developmental stability, it can also suggest the genetic fitness of an individual. This can further have an effect on mate attraction and sexual selection, as less asymmetry reflects greater developmental stability and subsequent fitness. Human physical health is also associated with FA. For example, young men with greater FA report more medical conditions than those with lower levels of FA. Multiple other factors can be linked to FA, such as intelligence and personality traits. == Measurement == Fluctuating asymmetry (FA) can be measured by the equation: Mean FA = mean absolute value of left sides - mean absolute value of right sides. The closer the mean value is to zero, the lower the levels of FA, indicating more symmetrical features. By taking many measurements of multiple traits per individual, this increases the accuracy in determining that individual's developmental stability. However, these traits must be chosen carefully, as different traits are affected by different selection pressures. This equation can further be used to study the distribution of asymmetries at population levels, to distinguish between traits that show FA, directional asymmetry, and anti-symmetry. The distribution of FA around a mean point of zero suggests that FA is not an adaptive trait, where symmetry is
|
{
"page_id": 1969440,
"source": null,
"title": "Fluctuating asymmetry"
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ideal. Directional asymmetry of traits can be distinguished by showing significantly biased measurements towards traits being larger on either the left or right sides, for example, human testicles (where the right is more commonly larger), or handedness (85% are right handed, 15% are left handed). Anti-symmetry can be distinguished by the bimodal distributions, due to some adaptive functions. == Causes == Fluctuating asymmetry (FA) is often considered to be the product of developmental stress and instability, caused by both genetic and environmental stressors. The notion that FA is a result of genetic and environmental factors is supported by Waddington's notion of canalisation, which implies that FA is a measure of the genome's ability to successfully buffer development to achieve a normal phenotype under imperfect environmental conditions. Various factors causing developmental instability and FA include infections, mutations, and toxins. === Genetic factors === Research on twins suggests that there are genetic influences on FA, and increased levels of mutations and perturbations is also linked to greater asymmetry. FA may also result from a lack of genetic immunity to diseases, as those with higher FA show less effective immune responses. This is further supported by evidence showing an association between FA and the number of respiratory infections experienced by an individual, such that those with higher levels of FA experience more infections. Increased prevalence of parasites and diseases in an organism is also seen more in individuals with greater levels of FA. However, the research in this field is predominantly correlational, so caution must be taken when inferring causation. For example, rather than a lack of immunity causing FA, FA may weaken the immune responses of an organism, or there may be another factor involved. There is some speculation that inbreeding contributes towards FA. One study on ants demonstrated that, although inbred
|
{
"page_id": 1969440,
"source": null,
"title": "Fluctuating asymmetry"
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|
individuals show more asymmetry in observed bilateral traits, the differences were not significant. Furthermore, ant colonies created by an inbreeding queen do not show significantly higher FA than those produced by a non-inbreeding queen. === Environmental factors === Multiple sources provide information on environmental factors that are correlated with FA. A meta-analysis of related studies suggests that FA is an appropriate marker of environmental stress during development. Some evidence suggests that poverty and lack of food during development may contribute to greater levels of FA. Infectious diseases can also lead to FA, as studies have repeatedly shown that those with higher FA report more infections. Alternatively, this association between levels of FA and infections may be due to a lack of immunity to diseases, as mentioned earlier (see 'Genetic factors'). Fluctuating asymmetry in human males is also seen to positively correlate with levels of oxidative stress. This process occurs when an organism creates excess reactive oxygen species (ROS) compared to ROS-neutralising antioxidants. Oxidative stress may mediate the association seen between high FA and infection amounts during development. Toxins and poisons are considered to increase FA. Pregnancy sickness is argued to be an adaptation for avoiding toxins during foetal development. Research has reported that when a mother has no sickness or a sickness that extends beyond week 12 of gestation, the offspring shows higher FA (as demonstrated by measuring thigh circumferences). This suggests that when a mother fails to expel environmental toxins, this creates stress and developmental instability for the foetus, later leading to increased asymmetry in that individual. Greater exposure to pollution may also be a fundamental cause of FA. Research on skull characteristics of Baltic grey seals (Halichoerus grypus) demonstrated that those born after 1960 (marking an increase in environmental pollution) had increased levels of asymmetry. Also, shrews (Crocidura
|
{
"page_id": 1969440,
"source": null,
"title": "Fluctuating asymmetry"
}
|
russula) from more polluted areas show higher levels of asymmetry. Radioactive contamination may also increase FA levels, as mice (Apodemus flavicollis) living closer to the failed Chernobyl reactor show greater asymmetry. == Developmental stability == Developmental stability is achieved when an organism is able to withstand genetic and environmental stress, to display the bilaterally symmetrical traits determined by its developmentally programmed phenotype. To measure an individual's developmental stability, the FA measurements of 10 traits are added together, including ear width, elbows, ankles, wrists, feet, length of ears and fingers. This is achieved by: (L - R)trait 1 + (L - R)trait 2 + ......(L - R)trait 10. This provides a good overall measure of body FA, as every individual has some features that are not perfectly symmetrical. Common environmental pressures leading to lower developmental stability include exposure to toxins, poison and infectious diseases, low food quality and malnutrition. Genetic pressures include spontaneous new mutations, and "bad genes" (genes that once had adaptive functions, but are being removed through evolutionary selection). A large fluctuating asymmetry (FA) and a low developmental stability suggests that an organism is unable to develop according to the ideal state of bilateral symmetry. The energy required for bilateral symmetry development is extremely high, making fully perfect bilateral symmetry functionally nonexistent in natural organic creatures. Energy is invested into survival in spite of the genetic and environmental pressures, before making bilaterally symmetrical traits. Research has also revealed links between FA and depression, genetic or environmental stress and measures of mate quality for sexual selection. == Health == === Susceptibility to diseases === Research has linked higher levels of fluctuating asymmetry (FA) to poorer outcomes in some domains of physical health in humans. For example, one study found that individuals with higher levels of FA report a higher number
|
{
"page_id": 1969440,
"source": null,
"title": "Fluctuating asymmetry"
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|
of medical conditions than those with lower levels of FA. However, they did not experience worse outcomes in areas such as systolic blood pressure or cholesterol levels. Higher levels of FA have also been linked to higher body mass index (BMI) in women, and lower BMI in men. Research has shown that both men and women with higher levels of FA, both facial and bodily, report a higher number of respiratory infections and a higher number of days ill, compared to men and women with lower levels of FA. In men, higher levels of FA have been linked to lower levels of physical attractiveness and higher levels of oxidative stress, regardless of smoking or levels of toxin exposure. There is no gender difference in the susceptibility of diseases depending on body FA. A large-scale review of the human and non-human literature by MΓΈller found that higher levels of fluctuating asymmetry were linked to increased vulnerability to parasites, and also to lower levels of immunity to disease. A large-scale longitudinal study in Britain found that facial FA was not associated with poorer health over the course of childhood, which was interpreted as suggesting smaller effects of FA in Western societies with generally low levels of FA A review of the relationship between various attractiveness features and health in Western societies produced similar results, finding that symmetry was not related to health in either sex, but was related to attractiveness in males. === Health-risk behaviours === It has been suggested that individuals with lower levels of FA may engage in more biologically costly behaviours such as recreational drug use and risky body modifications such as piercings and tattoos. These ideas have been proposed in the context of Zahavi's handicap principle, which argues that highly costly behaviours or traits serve as signals of
|
{
"page_id": 1969440,
"source": null,
"title": "Fluctuating asymmetry"
}
|
an organism's genetic quality. The relationship between FA and behaviours with high health risks has received mixed support. Individuals with body piercings and tattoos (which increase risk of blood-borne infections) have been shown to have lower levels of FA, but individuals with lower FA do not engage in any more recreational drug use than those with higher FA levels. === Mental health in humans === Higher levels of FA have been linked to higher levels of some mental health difficulties. For instance, it has been shown that, among university students, higher FA is associated with higher levels of schizotypy. Depression scores have been found to be higher in men, but not women, with higher levels of FA. One study by Shackelford and Larsen found that men and women with higher facial asymmetry reported more physiological complaints than those with lower facial asymmetry, and that both men and women with higher asymmetry experienced higher levels of psychological distress overall. For example, men with higher facial asymmetry experienced higher levels of depression compared to men with lower facial asymmetry. Fluctuating asymmetry has also been studied in relation to psychopathy. One study looking at offenders and non-offenders found that, although offenders had higher levels of FA overall, psychopathic offenders had lower levels of FA compared to offenders who did not meet the criteria for psychopathy. Additionally, offenders with the highest levels of psychopathy were found to have similar levels of FA to non-offenders. === Other health issues in humans === Research has also linked FA to conditions such as lower back pain, although the evidence is mixed. While one study found no notable link between pelvic asymmetry and lower back pain, other studies have found pelvic asymmetry (as well as FA in other traits not directly related to pelvic function) to be higher
|
{
"page_id": 1969440,
"source": null,
"title": "Fluctuating asymmetry"
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|
in patients experiencing lower back pain, and higher levels of FA have also been linked to congenital spinal problems. Studies have also shown increased levels of FA of ear length in individuals with cleft lip and/or non-syndromic cleft palate syndrome. === Physical fitness in humans === In addition to general health and susceptibility to disease, research has also studied the link between FA and physical fitness. Research has found that lower levels of lower-body FA is associated with faster running speeds in Jamaican sprinters, and individuals with greater body asymmetry have been shown to move more asymmetrically while running, although do not experience higher metabolic costs than more symmetrical individuals. It has also been shown that children with lower levels of lower-body FA have faster sprinting speeds and are more willing to sprint when followed up in adulthood. === Health in non-human populations === The relationship between FA, health and susceptibility to disease has also been studied in non-human animals. For example, studies have found that higher levels of facial asymmetry are associated with poorer overall health in female rhesus macaques (Macaca mulatta), and that higher FA is also linked to more health issues in chimpanzees (Pan troglodytes). The link between FA and health has also been investigated in non-primates. In three gazelle species (Gazella cuvieri; Gazella dama; Gazella dorcas), for instance, FA has been linked to a range of blood parameters associated with health in mammals, although the specific relevance of these blood parameters for these gazelle species was not examined. It has also been found that, among Iberian red deer (Cervus elaphus hispanicus), higher FA was slightly negatively related to both antler size and overall body mass (traits thought to indicate overall condition). Antlers more involved in fighting were found to be more symmetrical than those not involved,
|
{
"page_id": 1969440,
"source": null,
"title": "Fluctuating asymmetry"
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|
and antler asymmetry at reproductive age was lower than in development or at post-reproductive age. FA and health outcomes have been examined within insect populations. For instance, it has been found that Mediterranean field crickets (Gryllus bimaculatus) with higher levels of FA in three hind-limb traits have lower encapsulation rates, but do not differ from low-FA crickets in lytic activity (both are measures of immunocompetence). While research on the relationship between FA and longevity is sparse in humans, some studies using non-human populations have suggested an association between the symmetry of an organism and its lifespan. For instance, it has been found that flies whose wing veins showed more bilateral symmetry live longer than less symmetrical flies. This difference was greatest for male flies. == In sexual selection == === Mate attraction === Symmetry has been shown to affect physical attractiveness. Those with lower levels of fluctuating asymmetry (FA) are often rated as more attractive. Various studies have supported this. The relationship between FA and mate attraction has been studied in both males and females. As FA reflects developmental stability and quality, it has been suggested that we prefer those as more attractive/with low FA because it signals traits such as health and intelligence. Research has shown that the female partners of men with lower levels of FA experience a higher number of copulatory orgasms, compared to the female partners of males with higher levels of FA. Other studies have also found that the voices of men and women with low fluctuating asymmetry are rated as more attractive, suggesting that voice may be indicative of developmental stability. Research has shown attractiveness ratings of men's scent are negatively correlated with FA, but FA is unrelated to attractiveness ratings for women's scent, and women's preferences for the scent of more symmetric men
|
{
"page_id": 1969440,
"source": null,
"title": "Fluctuating asymmetry"
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appears limited to the most fertile phases of the menstrual cycle. However, research has failed to find changes in women's preferences for low FA across the menstrual cycle when assessing pictures of faces, as opposed to scents. Facial symmetry has been positively correlated with higher occurrences of mating. Also, one study used 3-D scans of male and female bodies, and showed videos of these scans to a group of individuals who rated the bodies on attractiveness. It was found that, for both males and females, lower levels of FA were associated with higher attractiveness ratings. It was also found that sex-typical joint configurations were rated as more attractive and linked to lower FA in men, but not women. Men with higher FA have been shown to have higher levels of oxidative stress and lower levels of attractiveness. Research has also provided evidence that FA is linked to extra-pair copulation, as women have been shown to prefer men with lower levels of FA as extra-pair partners. However, the literature is mixed regarding the relationship between attractiveness and FA. For example, in one study, altering images of faces to in a way that reduced asymmetry led to observers rating such faces as less, rather than more, attractive. Research by Van Dongen also found FA to be unrelated to attractiveness, physical strength and level of masculinity in both men and women. === Sexual selection in non-human animals === Many non-human animals have been shown to be able to distinguish between potential partners, based upon levels of FA. As with humans, lower levels of FA are seen in the most reproductively successful members of species. For instance, FA of male forewing length seem to have an important role in successful mating for many insect species, such as dark-wing damselflies and Japanese scorpionflies. In the
|
{
"page_id": 1969440,
"source": null,
"title": "Fluctuating asymmetry"
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dark-winged damselfly (Calopteryx maculate), successfully mating male flies showed significantly lower levels of FA in their forewings than unsuccessful males, while for Japanese scorpionflies, FA levels are a good predictor for the outcome of fights between males in that more symmetrical males won significantly more fights. Other animals also show similar patterns, for example, many species of butterfly, males with lower levels of FA tended to live longer and flew more actively, allowing them to have more reproductive success. Also, female swallows have been shown to prefer longer, and more symmetrical tails as a cue for mate choice. Therefore, the males with longer and more symmetrical tails show higher levels of reproductive success with more attractive females. In red deer, sexual selection has affected antler development, in that larger and more symmetrical antlers are favoured in males at prime mating age. However, some evidence for the effects of sexual selection of FA levels have been inconsistent, suggesting that the relationship between FA and sexual selection may be more complex than originally thought. For instance, in the lekking black grouse and red junglefowl, no correlations were found between FA and mating success. Furthermore, when manipulating paradise whydahs' tails to be more and less symmetrical, females showed no preferences for more symmetrical tails (but they did show preferences for longer tails). == Other associated factors == === Intelligence === Through research, fluctuating asymmetry (FA) has been found to have a negative correlation to measurements of human traits such as working memory and intelligence, such that individuals showing greater asymmetry have lower IQ scores. As FA links with both intelligence and facial attractiveness, it is possible that our perceptions of attractiveness have evolved based upon developmental quality, which includes traits such as intelligence and health. However, some literature shows no such correlations between
|
{
"page_id": 1969440,
"source": null,
"title": "Fluctuating asymmetry"
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FA and intelligence. A meta-analysis of the research covering this topic demonstrated that whilst published studies largely report negative correlations, unpublished studies often find no association between FA and intelligence. === Personality === Research into FA suggests that there may be some correlation to specific personality factors, in particular, the Big Five personality traits. From a general view, one would expect someone who is more symmetrical (usually meaning greater attractiveness), to be high on agreeableness, conscientiousness, extraversion and openness, and low on neuroticism. One of the most consistent findings reported is that low FA is positively associated with measures of extraversion, suggesting that more symmetrical people tend to be more extraverted than less symmetrical individuals, particularly when specifying to symmetry within the face. A correlation has also been reported between FA and human social dominance. However, research is proving less consistent with other personality factors, with some finding some weak correlations between low FA and conscientiousness and openness to experience, and others finding no significant differences between those with high or low FA. === Antisocial behaviours === Some studies suggest a link between FA and aggression, but the evidence is mixed. In humans, criminal offenders show greater FA than nonoffenders. However, other studies report that human males with higher FA show less physical aggression and less anger. Females show no association between FA and physical aggression, but some research has suggested that older female adolescents with higher facial FA are less hostile. The type of aggression being studied may account for the mixed evidence that is seen here. For example, one study found that females with higher FA demonstrated higher levels of reactive aggression in response to high levels of provocation, whereas high FA males showed more reactive aggression under low levels of provocation. Research is also mixed in other
|
{
"page_id": 1969440,
"source": null,
"title": "Fluctuating asymmetry"
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animals. In Japanese scorpionflies (Panorpa nipponensis and Panorpa ochraceopennis), FA differences between members of the same sex competing for food determines the outcome of interspecific contests and aggression better than body size or ownership of food. Furthermore, cannibalistic laying hens (Gallus gallus domesticus) demonstrate more asymmetry than normal hens. However, this link between FA and aggression in hens is questionable, as victimised hens also showed greater asymmetry. Furthermore, when prenatally injecting hen eggs with excess serotonin (5-HT), the hens later exhibited more FA at 18 weeks of age, but displayed less aggressive behaviours. It is suggested that the stress introduced during early embryonic stages via certain factors (such as excess serotonin) may create developmental instability, causing phenotypic and behavioural variations (such as increased or decreased aggression). === Aging === In old age, facial symmetry has been associated with better cognitive aging, as lower levels of FA have been associated with higher intelligence and more efficient information processing in older men. However, it has been found that risk of mortality cannot be predicted accurately from levels of FA in photographs of older adults. === Other factors === Additionally, FA has been shown to predict atypical asymmetry of the brain. Research has also shown that growth rates after birth positively correlate with FA. For example, increased FA has been found in people who were obese. == See also == Bilateria Facial symmetry Physical attractiveness Symmetry in biology Minor physical anomalies == References ==
|
{
"page_id": 1969440,
"source": null,
"title": "Fluctuating asymmetry"
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The iliococcygeal raphe is a raphe representing the midline location where the levatores ani converge. == See also == Anococcygeal body == References ==
|
{
"page_id": 24644898,
"source": null,
"title": "Iliococcygeal raphe"
}
|
Vat Green 1 is an organic compound that is used as a vat dye. It is a derivative of benzanthrone. It is a dark green solid. Vat Green 1 can dye viscose, silk, wool, paper, and soap. == References ==
|
{
"page_id": 42208547,
"source": null,
"title": "Vat Green 1"
}
|
This list of sequenced animal genomes contains animal species for which complete genome sequences have been assembled, annotated and published. Substantially complete draft genomes are included, but not partial genome sequences or organelle-only sequences. For all kingdoms, see the list of sequenced genomes. == Porifera (Sponges) == Amphimedon queenslandica, a sponge (2009) Stylissa carteri (2016) Ephydatia muelleri (2020) Xestospongia testudinaria (2016) == Ctenophora == Mnemiopsis leidyi (Ctenophora), (order Lobata) (2012/2013) Hormiphora californensis (Ctenophora) (2021) Pleurobrachia bachei (Ctenophora) (2014) Bolinopsis microptera(Ctenophora) (2022) == Placozoa == == Cnidaria == Hydra vulgaris, (previously Hydra magnipapillata), a model hydrozoan (2010) Nematostella vectensis, a model sea anemone (starlet sea anemone) (2007) Aiptasia pallida, a sea anemone (2015) Renilla muelleri, an octocoral (2017, 2019) Stylophora pistillata, a coral (2017) Aurelia aurita, moon jellyfish (2019) Clytia hemisphaerica, Hydrozoan jellyfish (2019) Myxobolus honghuensis (2022) Nemopilema nomurai, Nomura jellyfish (2019) Rhopilema esculentum, Flame jellyfish (2020) Cassiopea xamachana (Scyphozoa) (2019) Alatina alata (Cubozoa) (2019) Calvadosia cruxmelitensis (Staurozoa) (2019) Dendronephthya gigantea, an octocoral (2019) Acropora acuminata (2020) Acropora awi (2020) Acropora cytherea, Table coral (2020) Acropora digitifera, a coral (2011) Acropora echinata (2020) Acropora florida, branching staghorn coral(2020) Acropora gemmifera (2021) Acropora hyacinthus, Brush coral (2020) Acropora intermedia, Noble Staghorn Coral (2020) Acropora microphthalma (2020) Acropora muricata, Staghorn coral (2020) Acropora nasta, branching staghorn coral (2020) Acropora pulchra (2025) Acropora selago, Green Selago Acropora (2020) Acropora tenuis, Purple Tipped Acropora (2020) Acropora yongei ,Yonge's staghorn coral (2020) Corallium rubrum, Precious coral (2024) Astreopora myriophthalma, Porous star coral (2020) Lophelia pertusa, Deepwater White Coral (2023) Montipora cactus (2020) Montipora capitata, Rice coral (2022) Montipora efflorescens, Velvet coral (2020) Orbicella faveolata, mountainous star coral (2016) Paragorgia papillata, Bubble-gum coral (2025) Pocillopora acuta, Hosoeda Hanayasai coral (2022) Pocillopora damicornis, cauliflower coral (2018) Pocillopora meandrina, Cauliflower coral (2022) Porites astreoides, Mustard hill coral (2022) Porites
|
{
"page_id": 35917093,
"source": null,
"title": "List of sequenced animal genomes"
}
|
compressa, Finger coral (2022) == Hemichordata == === Order Enteropneusta (Acorn Worms) === == Echinodermata == Acanthaster planci, starfish (2014) Apostichopus japonicus, sea cucumber (2017) Arbacia lixula, black sea urchin (2024) Astropecten irregularis, sand sea star (2024) Australostichopus mollis, Australian sea cucumber (2016) Chiridota hydrothermica, deep sea cucumber (2024) Diadema setosum, Long-spined sea urchin (2024) Echinometra lucunter, rock boring urchin (2023) Ophionereis fasciata, mottled brittlestar (2016) Patiriella regularis, the New Zealand common cushion star (2016) Plazaster borealis, Octopus starfish (2022) Strongylocentrotus purpuratus, a sea urchin and model deuterostome (2006) == Cephalocordata (Lancelets) == == Tunicates == === Appendicularia === ==== Order Copelata (Larvaceans) ==== Oikopleura dioica, a larvacean (2001). === Acopa === ==== Order Stolidobranchia ==== ==== Thaliacea ==== ===== Order Pyrosomida (Pyrosomes) ===== ===== Order Salpida (Salps) ===== ===== Order Doliolida ===== ==== Enterogona ==== ===== Order Phlebobranchia ===== Ciona intestinalis, a tunicate (2002) Ciona savignyi, a tunicate (2007) ===== Order Aplousobranchia ===== == Vertebrates == === Cartilaginous fish === ==== Holocephali ==== ===== Order Chimaeriformes (Chimeras) ===== ==== Selachimorpha (True Sharks) ==== Superorder Galeomorphi Order Carcharhiniformes (Ground Sharks) Scyliorhinus torazame, Cloudy catshark (2018) Order Lamniformes (Mackerel Sharks) Carcharodon carcharias, Great white shark (2018) Order Orectolobiformes (Carpet Sharks) Chiloscyllium plagiosum, Whitespotted bamboo shark (2020) Chiloscyllium punctatum, Brownbanded bamboo shark (2018) Rhincodon typus, Whale shark (2017) ==== Batomorphi (Rays) ==== Order Myliobatiformes Potamotrygon leopoldi, Xingu river ray (2023 draft) Order Rajiformes Leucoraja erinacea, Little skate (2023) === Ray-Finned Fish === ==== Cladistia ==== Order Polypteriformes Polypterus senegalus, Senegal bichir, (2021) ==== Chondrostei ==== Order Acipenseriformes Polyodon spathula, American paddlefish, (2021 draft) ==== Holostei ==== Order Amiiformes Amia calva, Bowfin, (2021 draft) Order Lepisosteiformes Atractosteus spatula, Aligator gar, (2024) Lepisosteus oculatus, Spotted gar ==== Teleostei ==== Order Anabantiformes Betta splendens, Siamese fighting fish (2018) Helostoma temminkii, Kissing gourami (2020) Order Anguilliformes
|
{
"page_id": 35917093,
"source": null,
"title": "List of sequenced animal genomes"
}
|
Anguilla anguilla, European Eel (2012) Anguilla japonica, Japanese Eel (2022) Order Atheriniformes Atherinopsis californiensis, Jack silverside (2023) Order Beloniformes Oryzias latipes, medaka (2007) Order Callionymiformes Callionymus lyra, common dragonet (2020) Order Carangiformes Caranx ignobilis, Giant trevally (2022) Caranx melampygus, Bluefin trevally (2021) Pseudocaranx georgianus, New Zealand trevally (2021) Order Centrarchiformes Oplegnathus fasciatus, barred knifejaw (2019) Order Characiformes Astyanax jordani, Mexican cavefish (2014) Astyanax mexicanus, Mexican tetra (2021) Colossoma macropomum, Tambaqui (2021) Hasemania nana, Silvertip tetra (2013) Hyphessobrycon heterorhabdus, Flag tetra (2023) Petitella bleheri, Firehead tetra (2015) Psalidodon paranae, (2016) Order Cichliformes Oreochromis niloticus, Nile tilapia (2019) Metriaclima zeb, Lake Malawi cichlid (2019) Order Clupeiformes Clupea harengus, Atlantic herring (2020) Coilia nasus, Japanese grenadier anchovy (2020) Sardina pilchardus, European pilchard (2019) Order Cypriniformes Anabarilius grahami, Kanglang fish (2018) Danio rerio, zebrafish (2007) Leuciscus baicalensis, Siberian dace (2014) Megalobrama amblycephala, Wuchang bream (2017) Metzia formosae, (2015) Opsarius caudiocellatus, (2022) Oxygymnocypris stewartii, (2019) Pseudobrama simoni (2020) Rhodeus ocellatus, Rosy bitterling (2020) Triplophysa bleekeri, Tibetan stone loach (2020) Order Cyprinodontiformes Fundulus catenatus, Northern studfish (2020) Fundulus olivaceus, Blackspotted topminnow (2020) Fundulus nottii, Bayou topminnow (2020) Fundulus xenicus, Diamond killifish (2020) Gambusia affinis, western mosquitofish (2020) Heterandria formosa, least killifish (2019) Micropoecilia picta, swamp guppy (2021) Xiphophorus maculatus, platyfish (2013) Nothobranchius furzeri, turquoise killifish (2015) Order Esociformes Esox lucius, northern pike (2014) Order Gadiformes Gadus macrocephalus, Pacific cod (2022) Gadus morhua, Atlantic cod (2011) Order Gasterosteiformes Gasterosteus aculeatus, three-spined stickleback (2006, 2012) Order Gobiiformes Oxyeleotris marmorata, marble goby (2020) Periophthalmus modestus, shuttles hoppfish or shuttles mudskipper (2022) Order Gymnotiformes Electrophorus electricus, electric eel (2014) Order Lampriformes Lampris incognitus, Smalleye Pacific Opah (2021) Order Osmeriformes Neosalanx tangkahkeii, Chinese icefish (2015) Protosalanx hyalocranius, clearhead icefish (2017) Order Osteoglossiformes Heterotis niloticus, African arowana (2020) Paramormyrops kingsleyae, mormyrid electric fish (2017) Scleropages formosus, Asian arowana (2016) Order Perciformes Centropyge
|
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bicolor, bicolor angelfish (2021) Chaetodon trifasciatus, melon butterflyfish (2020) Channa argus, northern snakehead (2017) Channa maculata, blotched snakehead (2021) Chelmon rostratus, copperband butterflyfish (2020) Dissostichus mawsoni, Antarctic toothfish (2019) Eleginops maclovinus, Patagonian robalo (2019) Epinephelus moara, kelp grouper (2021) Larimichthys crocea, large yellow croaker (2014) Lutjanus campechanus, Northern red snapper (2020) Naso vlamingii, bignose unicornfish (2020) Parachaenichthys charcoti, Antarctic dragonfish (2017) Seriola dumerili, Greater amberjack (2017) Sillago sinica, chinese sillago (2018) Siniperca knerii, Big-Eye Mandarin Fish (2020) Sparus aurata, gilt-head bream (2018) Order Salmoniformes Salmo salar, Atlantic salmon (2016) Oncorhynchus mykiss, rainbow trout (2014) Oncorhynchus tshawytscha, Chinook salmon (2018) Salvelinus namaycush, Lake Trout (2021) Order Scorpaeniformes Sebastes schlegelii, Black rockfish (2018) Order Siluriformes Clarias batrachus, walking catfish (2018) Ictalurus punctatus, channel catfish (2016) Pangasianodon hypophthalmus, Iridescent shark catfish (2021) Silurus glanis, Wels catfish (2020) Order Spariformes Datnioides pulcher, Siamese tigerfish (2020) Datnioides undecimradiatus, Mekong tiger perch (2020) Order Syngnathiformes Syngnathus scovelli, Gulf pipefish (2016, 2023) Order Tetraodontiformes Diodon holocanthus, Long-spine porcupinefish (2020) Mola mola, ocean sunfish (2016) Takifugu rubripes, a puffer fish (2002) Tetraodon nigroviridis, a puffer fish (2004) === Lobe-Finned Fish (Excluding Tetrapods) === ==== Coelacanths (Actinistia) ==== ===== Order Coelacanthiformes ===== ==== Lungfish (Dipnoi) ==== ===== Order Ceratodontiformes ===== === Amphibians === ==== Frogs (Anura) ==== Taudactylus pleione, Kroombit tinker frog (2023) Leptobrachium leishanense, Leishan Moustache toad (2019) Limnodynastes dumerilii dumerilii, Eastern banjo frog (2020) Nanorana parkeri, High Himalaya frog (2015) Oophaga pumilio, Strawberry poison-dart frog (2018) Platyplectrum ornatum, Ornate burrowing frog (2021) Pyxicephalus adspersus, African bullfrog (2018) Rana [Lithobates] catesbeiana, North American bullfrog (2017) Rana kukunoris, Plateau brown frog (2023) Rhinella marina, Cane toad (2018) Vibrissaphora ailaonica, Moustache toad (2019) Xenopus tropicalis, western clawed frog (2010) ==== Salamanders (Urodela) ==== Clade Salamandroidea Family Salamandridae Pleurodeles waltl, Iberian ribbed newt, (2025) Family Ambystomatidae (Tiger Salamanders) Ambystoma mexicanum, Axolotl
|
{
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(2018) ==== Caecillians ==== Family Dermophiidae Geotrypetes seraphini, Gaboon caecillian, (2023) Family Siphonopidae Microcaecilia unicolor, a caecillian, (2023) === Birds === ==== Ratites (Palaeognathae) ==== ===== Order Struthioniformes ===== ===== Order Rheiformes (Rheas) ===== ===== β Order Dinornithiformes (Moas) ===== ===== Order Tinamiformes (Tinamous) ===== ===== Order Apterygiformes (Kiwis) ===== ===== Order Casuariiformes ===== ==== Fowl (Galloanserae) ==== ===== Order Anseriformes (Waterfowl) ===== ===== Order Galliformes (Landfowl) ===== ==== Neoaves ==== ===== Mirandornithes ===== ====== Order Phoenicopteriformes (Flamingos) ====== ====== Order Podicipediformes (Grebes) ====== ===== Columbaves ===== ====== Order Columbiformes ====== ====== Order Mesitornithiformes (Mesites) ====== ====== Order Pterocliformes (Sandgrouse) ====== ====== Order Musophagiformes (Turacos) ====== ====== Order Otidiformes (Bustards) ====== ====== Order Cuculiformes (Cuckoos) ====== ===== Gruae ===== ====== Order Opisthocomiformes ====== ====== Order Gruiformes ====== ====== Order Charadriiformes ====== ===== Strisores ===== ====== Order Caprimulgiformes (Nightjars) ====== ====== Order Apodiformes ====== ===== Phaethoquornithes ===== Order Phaethontiformes ====== Order Eurypygiformes ====== ====== Order Gaviiformes (Loons) ====== ====== Order Procellariiformes (Petrels) ====== ====== Order Sphenisciformes (Penguins) ====== Genus Aptenodytes (Great Penguins) Aptenodytes forsteri, Emperor penguin (2014) Aptenodytes patagonicus, King penguin (2019) Genus Eudyptes (Crested Penguins) Eudyptes chrysocome, Western rockhopper penguin (2019) Eudyptes chrysolophus chrysolophus, Macaroni penguin (2019) Eudyptes chrysolophus schlegeli, Royal penguin (2019) Eudyptes filholi, Eastern rockhopper penguin (2019) Eudyptes moseleyi, Northern rockhopper penguin (2019) Eudyptes pachyrhynchus, Fiordland penguin (2019) Eudyptes robustus, Snares penguin (2019) Eudyptes sclateri, Erect-crested penguin (2019) Genus Eudyptula (Little Penguins) Eudyptula minor albosignata, White-flippered penguin (2019) Eudyptula minor minor, Little blue penguin (2019) Eudyptula novaehollandiae, Fairy penguin (2019) Genus Megadyptes (Hoiho Penguins) Megadyptes antipodes antipodes, Yellow-eyed penguin (2019) Pygoscelis (Brush-tailed Penguins) Pygoscelis adeliae, AdΓ©lie penguin (2014) Pygoscelis antarctica, Chinstrap penguin (2019) Pygoscelis papua, Gentoo penguin (2019) Genus Spheniscus (Banded Penguins) Spheniscus demersus, African penguin (2019) Spheniscus humboldti, Humboldt penguin (2019) Spheniscus magellanicus, Magellanic penguin (2019) Spheniscus
|
{
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mendiculus, GalΓ‘pagos penguin (2019) ====== Order Ciconiiformes (Storks) ====== ====== Order Suliformes ====== ====== Order Pelecaniformes ====== ===== Afroaves ===== ====== Order Strigiformes (Owls) ====== ====== Order Accipitriformes ====== Add 87 hawk genomes found here: https://pmc.ncbi.nlm.nih.gov/articles/PMC9851080/ ====== Order Coliiformes (Mousebirds) ====== ====== Order Leptosomiformes ====== ====== Order Trogoniformes (Trogons) ====== ====== Order Bucerotiformes ====== ====== Order Coraciiformes ====== ====== Order Piciformes ====== ===== Australaves ===== ====== Order Cariamiformes ====== ====== Order Falconiformes (Falcons) ====== ====== Order Psittaciformes (Parrots) ====== ====== Order Passeriformes (Passerines) ====== === Crocodilians === === Turtles === ==== Crytodira (Hidden-Neck Turtles) ==== ===== Trionychia (Softshell Turtles) ===== ===== Testudinoidea ===== ===== Chelonioidea (Sea Turtles) ===== ===== Chelydroidea ===== ==== Pleurodira ==== === Rhynchocephalia === ==== Neosphenodontia ==== === Squamates === ==== Gekkota (Gekkos) ==== Clade Gekkomorpha Family Eublepharidae Eublepharis macularius, Leopard gecko (2016) ==== Scinciformata ==== Clade Cordylomorpha Family Cordylidae Hemicordylus capensis, Cape cliff lizard, (2023) ==== Laterata ==== Clade Teiformata Family Teiidae Salvator merianae, Argentine black and white tegu, (2018) Clade Lacertiformata Family Lacertidae Zootoca vivipara, Viviparous lizard (2020) ==== Toxicofera ==== ===== Anguimorpha ===== Clade Paleoanguimorpha Family Shinisauridae Shinisaurus crocodilurus, Chinese crocodile lizard, (2017) Clade Neoanguimorpha Family Helodermatidae Heloderma charlesbogerti, Guatemalan beaded lizard, (2022) Family Anguidae Dopasia gracilis, Burmese glass lizard, (2015) ===== Iguania ===== Clade Acrodonta Family Agamidae Pogona vitticeps, Central bearded dragon, (2015) Clade Pleurodonta Family Dactyloidae Anolis carolinensis, Carolina anole, (2011) Family Phrynosomatidae Phrynosoma platyrhinos, Dessert horned lizard, (2021) Phrynosoma cornutum, Texas horned lizard, (2021) Sceloporus undulatus, Eastern fence lizard (2021) ===== Serpentes (Snakes) ===== Clade Scolecophidia (Blindsnakes) Family Typhlopidae Anilios bituberculatus, Prong-snouted blind snake (2021) Indotyphlops braminus, Brahminy blindsnake, (2022) Clade Booidea Family Pythonidae Morelia viridis, Green Tree Python (2022) Python bivittatus, Burmese python (2013) Python regius, Ball python (2020) Simalia boeleni, Boelen's Python (2022) Family Boidae Boa constrictor, Boa constrictor
|
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(2019) Charina bottae, Rubber boa, (2022) Clade Caenophidia Family Viperidae Azemiops feae, Fea's viper (2022) Bothrops jararaca, Jararaca lancehead, (2021) Crotalus adamanteus, Eastern diamondback rattlesnake (2021) Crotalus mitchellii pyrrhus, southwestern speckled rattlesnake (2014) Crotalus oreganus helleri, southern Pacific rattlesnake (2023) Crotalus tigris, Tiger rattlesnake (2021) Crotalus viridis, Great Plains rattlesnake (2018) Daboia siamensis, Eastern Russell's viper (2022) Deinagkistrodon acutus, Five-pacer viper (2016) Protobothrops flavoviridis, Okinawa Habu (2018) Protobothrops mucrosquamatus, Taiwanese Habu (2017, 2024) Trimeresurus albolabris, White-lipped tree pit viper (2024) Family Homalopsidae Myanophis thanlyinesis, (No common name), (2021) Family Colubridae Ahaetulla prasina, Asian vine snake (2023) Arizona elegans occidentalis, California glossy snake (2022) Chrysopelea ornata, Ornate Flying Snake (2023) Diadophis punctatus, ring-necked snake (2023) Dolichophis caspius, Caspian whipsnake (2020) Elaphe carinata, King ratsnake (2024) Pantherophis guttatus, corn snake (2014) Pantherophis obsoletus, Leucistic Texas Rat Snake (2021) Ptyas mucosa, Oriental rat snake (2024) Thamnophis sirtalis, Common garter snake (2018) Thermophis baileyi, Tibetan hot-spring snake (2018) Family Elapidae Bungarus multicinctus, Many-banded krait (2022) Emydocephalus ijimae, Ijima's turtle-headed sea snake, (2019) Hydrophis curtus, Shaw's Sea Snake (2020) Hydrophis cyanocinctus, blue-banded sea snakes (2021) Hydrophis melanocephalus, slender-necked sea snake, (2019) Laticauda colubrina, yellow-lipped sea krait, (2019) Laticauda laticaudata, blue-lipped sea krait, (2019) Naja atra, Chinese cobra (2024) Naja naja, Indian cobra (2020) Notechis scutatus, mainland tiger snake (2022) Ophiophagus hannah, king cobra (2013) Pseudonaja textilis, Eastern brown snake (2022) === Mammals === ==== Monotremes ==== ==== Marsupials ==== Order Didelphimorphia Family Didelphidae (opossums) Monodelphis domestica, gray short-tailed opossum (2007) Order Dasyuromorphia Family Dasyuridae Antechinus stuartii, brown antechinus (2020) Sarcophilus harrisii, Tasmanian devil () Sminthopsis crassicaudata, fat-tailed dunnart (ongoing) Dasyurus hallucatus, northern quoll (ongoing) Family Myrmecobiidae Myrmecobius fasciatus, numbat (ongoing) β Family Thylacinidae β Thylacinus cynocephalus, thylacine () Order Peramelemorphia Family Peramelidae Perameles gunnii, eastern barred bandicoot (ongoing) Family Thylacomyidae Macrotis lagotis, greater bilby (ongoing)
|
{
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Order Notoryctemorphia, Family Notoryctidae Notoryctes typhlops, southern marsupial mole (ongoing) Order Diprotodontia Family Macropodidae Macropus eugenii, tammar wallaby (2011) Petrogale penicillata, brush-tailed rock-wallaby (ongoing) Family Potoroidae Bettongia gaimardi, eastern bettong (ongoing) Bettongia penicillata ogilbyi, woylie (2021) Family Petauridae Gymnobelideus leadbeateri, Leadbeater's possum (ongoing) Family Burramyidae Burramys parvus, mountain pygmy possum (ongoing) Family Vombatidae Vombatus ursinus, common wombat (ongoing) Family Phascolarctidae Phascolarctos cinereus, koala (2013 draft) ==== Placentals ==== ===== Afrotheria ===== ====== Order Hyracoidea ====== Family Procaviidae Procavia capensis, Rock hyrax, (2011) ====== Order Proboscidea ====== Family Elephantidae (Elephants) Elephas maximus, Asian elephant, (2015, 2024) β Mammuthus primigenius, Wooly mammoth, (2015) Loxodonta africana, African bush elephant, (2009, 2024) Loxodonta cyclotis, African forest elephant, (2018) ====== Order Sirenia (Sea Cows) ====== Family Trichechidae Trichechus manatus, West Indian manatee, (2015) Family Dugongidae Dugong dugon, Dugong, (2024) ===== Euarchontoglires ===== Order Lagomorpha Family Leporidae Oryctolagus cuniculus, European rabbit (2010) Order Primates Family Callitrichidae Callithrix jacchus, Common marmoset (2010, whole genome 2014) Family Cercopithecidae Macaca mulatta, rhesus macaque (2007 & Chinese rhesus macaque Macaca mulatta lasiota in 2011) Macaca fascicularis, Cynomolgus or crab-eating macaque (2011) Papio anubis, olive baboon (2020) Papio cynocephalus, yellow baboon (2016) Rhinopithecus roxellana, golden snub-nosed monkey (2019) Family Galagidae Otolemur garnettii, small-eared galago, or bushbaby () Family Hominidae Subfamily Ponginae Pongo pygmaeus/Pongo abelii, orangutan (Borneo/Sumatra) (2011) Subfamily Homininae Gorilla gorilla, western gorilla (2012) Homo sapiens, modern human (draft 2001, whole genome 2022) β Homo neanderthalensis, Neanderthal (draft 2010) Pan troglodytes, chimpanzee (2005) Pan paniscus, bonobo (2012) Order Rodentia Family Caviidae Hydrochoerus hydrochaeris, capybara (2018) Family Cricetidae Microtus montanus, Montane vole (2021) Microtus richardsoni, North American Water Vole (2021) Peromyscus leucopus, white-footed mouse (2019) Family Heteromyidae Perognathus longimembris pacificus, Pacific Pocket Mouse Family Muridae Mastomys coucha, Southern multimammate mouse (2019) Mus musculus Strain: C57BL/6J, House mouse (2002) Rattus norvegicus, Brown rat
|
{
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(2004) ===== Laurasiatheria ===== Order Artiodactyla (even-toed ungulates) Family Antilocapridae Antilocapra americana, pronghorn (2019) Family Balaenidae Balaena mysticetus, bowhead whale (2015) Eubalaena glacialis, North Atlantic right whale (2018) Family Balaenopteridae Balaenoptera acutorostrata, common minke whale (2014) Balaenoptera borealis, sei whale (2018) Balaenoptera musculus, blue whale (2018) Balaenoptera physalus, fin whale (2014) Megaptera novaeangliae, humpback whale (2018) Family Bovidae Ammotragus lervia, Barbary sheep (2019) Antidorcas marsupialis, Springbox (2019) Bison bonasus, European bison (2017) Bos grunniens, yak 2012 () Bos primigenius indicus, zebu or Brahman cattle (2012) Bos primigenius taurus, cow 2009 () Bubalus bubalis, river buffalo (2017) Capra ibex, Goats (2019) Cephalophus harveyi, Harvey's duiker (2019) Connochaetes taurinus, blue wildebeest (2019) Damaliscus lunatus, common tsessebe (2019) Gazella thomsoni, Thomson's gazelle (2019) Hippotragus niger, Sable Antelope (2019) Kobus ellipsiprymnus, Waterbuck (2019) Litocranius walleri, Gerenuk (2019) Oreotragus oreotragus, Klipspringer (2019) Oryx gazella, Gemsbok (2019) Ourebia ourebi, Oribi (2019) Ovis ammon, Argali (2019) Ovis ammon polii, marco polo sheep (2017) Nanger granti, Grant's gazelle (2019) Neotragus moschatus, Suni (2019) Neotragus pygmaeus, Royal antelope (2019) Philantomba maxwellii, Maxwell's duiker (2019) Procapra przewalskii, Przewalski's gazelle (2019) Pseudois nayaur, Bharal (2019) Pseudoryx nghetinhensis, Saola (2025) Raphicerus campestris, Steenbox (2019) Redunca redunca, Bohor reedbuck (2019) Syncerus caffer, African buffalo (2019) Sylvicapra grimmia, common duiker (2019) Tragelaphus, Spiral-horned bovine (2019) Tragelaphus buxtoni, Mountain nyala (2019) Tragelaphus strepsiceros, Greater kudu (2019) Tragelaphus imberbis, Lesser kudu (2019) Tragelaphus spekii, Sitatunga (2019) Tragelaphus scriptus, Bushbuck (2019) Taurotragus oryx, Common eland (2019) Family Camelidae Camelus ferus, Wild Bactrian camel (2007) Family Cervidae Cervus albirostris, Tharold's deer (2019) Elaphurus davidianus, Père David's deer (2018) Muntiacus crinifrons, hairy-fronted muntjac (2019) Muntiacus muntjak, Indian muntjac (2019) Muntiacus reevesi, Reeves's muntjac (2019) Odocoileus hemionus, mule deer (2021) Rangifer tarandus, Reindeer (2017) Family Delphinidae Tursiops truncatus, bottlenosed dolphin (2012) Neophocaena phocaenoides, finless porpoise (2014) Orcinus orca, killer whale
|
{
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(2015) Sousa chinensis, Indo-Pacific humpback dolphin (2019) Family Eschrichtiidae Eschrichtius robustus, gray whale (2018) Family Giraffidae Giraffa camelopardalis, Giraffe (2019) Giraffa camelopardalis tippelskirchi, Masai giraffe (2019) Okapia johnstoni, Okapi (2019) Family Monodontidae Delphinapterus, beluga whale (2017) Family Moschidae Moschus berezovskii, forest musk deer (2018) Moschus chrysogaster, Alpine musk deer (2019) Family Phocoenidae Neophocaena asiaeorientalis sunameri, East Asian finless porpoise Family Physeteridae Physeter macrocephalus, sperm whale (2019) Family Suidae Sus scrofa, pig (2012) Family Tragulidae Tragulus javanicus, Java mouse-deer (2019) Order Carnivora Family Felidae Acinonyx jubatus, cheetah (2015) Felis catus, cat (2007) Panthera leo, lion (2013) Panthera pardus, Amur leopard (2016) Panthera tigris tigris, Siberian tiger (2013) Panthera tigris tigris, Bengal tiger (2013) Panthera uncia, snow leopard (2013) Prionailurus bengalensis, leopard cat (2016) Family Canidae Canis familiaris, dog (2005) Canis lupus lupus, wolf (2017). Lycaon pictus, african wild dog (2018) Family Ursidae Ailuropoda melanoleuca, giant panda (2010) Ursus arctos ssp. horribilis, Grizzly bear (2018) Ursus americanus, American black bear (2019) Ursus maritimus, Polar bear (2014) Family Odobenidae Odobenus rosmarus, walrus (2015) Family Mustelidae Enhydra lutris kenyoni, sea otter (2017) Mustela erminea, stoat (2018) Mustela furo, ferret (2014) Pteronura brasiliensis, giant otter (2019) Order Chiroptera Family Megadermatidae Megaderma lyra, greater false vampire bat (2013) Family Mormoopidae Pteronotus parnellii, Parnell's mustached bat (2013) Family Pteropodidae Pteropus vampyrus, fruit bat (2012) Eidolon helvum, Old World fruit bat (2013) Family Rhinolophidae Rhinolophus ferrumequinum, greater horseshoe bat (2013) Family Vespertilionidae Myotis lucifugus, little brown bat (2010) Family Phyllostomidae Leptonycteris yerbabuenae, long nosed bat (2020) Leptonycteris nivalis, greater long nosed bat (2020) Musonycteris harrisoni, banana bat (2020) Artibeus jamaicensis, Jamaican fruit bat (2020) Macrotus waterhousii, Waterhouse's leaf-nosed bat (2020 Order Erinaceomorpha, Family Erinaceidae Erinaceus europaeus, western European hedgehog () Order Eulipotyphla, Family Solenodontidae Solenodon parodoxus, Hispaniolan solenodon (2018) Order Perissodactyla (odd-toed ungulates) Family Equidae Equus caballus, horse
|
{
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(2009 2018) == Arthropods == === Insects === Order Blattodea Blattella germanica, German cockroach (2018) Periplaneta americana, American cockroach (2018) Zootermopsis nevadensis, a dampwood termite (2014 Cryptotermes secundus, a drywood termite(2018) Macrotermes natalensis, a higher termite (2014 Order Coleoptera Dendroctonus ponderosae Hopkins, beetle (mountain pine beetle) (2013) Aquatica lateralis, Japanese aquatic firefly "Heike-botaru" (firefly) (2018) Photinus pyralis, Big Dipper firefly (2018) Protaetia brevitarsis, White-spotted flower chafer (2019) Tribolium castaneum Strain:GA-2, beetle (red flour beetle) (2008) Allomyrina dichotoma, Japanese rhinoceros beetle (2022) Order Collembola Family Isotomidae Desoria tigrina, (2021) Family Sminthurididae Sminthurides aquaticus, (2021) Order Diptera Family Calliphoridae Aldrichina grahami, Forensic blowfly (2020) Family Chironomidae Dasypogon diadema, Hunting Robber fly (2019) Parochlus steinend, Antarctic winged midge (2017) Proctacanthus coquilletti, Assassin fly (2017) Family Culicidae (mosquitoes) Aedes aegypti Strain:LVPib12, mosquito (vector of dengue fever, etc.) (2007) Aedes albopictus (2015) Anopheles darlingi Anopheles gambiae Strain: PEST, mosquito (vector of malaria) (2002) Anopheles gambiae Strain: M, mosquito (vector of malaria) (2010) Anopheles gambiae Strain: S, mosquito (vector of malaria) (2010) Anopheles sinensis, mosquito (vector of vivax malaria, lymphatic filariasis and Setaria infections), (2014) Anopheles stephensii Anopheles arabiensis (2015) Anopheles quadriannulatus (2015) Anopheles merus (2015) Anopheles melas (2015) Anopheles christyi (2015) Anopheles epiroticus (2015) Anopheles maculatus (2015) Anopheles culicifacies (2015) Anopheles minimus (2015) Anopheles funestus (2015, 2019) Anopheles dirus (2015) Anopheles farauti (2015) Anopheles atroparvus (2015) Anopheles sinensis (2015) Anopheles albimanus (2015) Culex quinquefasciatus, mosquito (vector of West Nile virus, filariasis etc.) (2010) Family Drosophilidae (fruit flies) Drosophila albomicans, fruit fly (2012) Drosophila ananassae, fruit fly (2007) Drosophila biarmipes, fruit fly (2011) Drosophila bipectinata, fruit fly (2011) Drosophila erecta, fruit fly (2007) Drosophila elegans, fruit fly (2011) Drosophila eugracilis, fruit fly (2011) Drosophila ficusphila, fruit fly (2011) Drosophila grimshawi, fruit fly (2007) Drosophila kikkawai, fruit fly (2011) Drosophila melanogaster, fruit fly (model organism) (2000) Drosophila
|
{
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"title": "List of sequenced animal genomes"
}
|
mojavensis, fruit fly (2007) Drosophila neotestacea, fruit fly (transcriptome 2014) Drosophila persimilis, fruit fly (2007) Drosophila pseudoobscura, fruit fly (2005) Drosophila rhopaloa, fruit fly (2011) Drosophila santomea, fruit fly () Drosophila sechellia, fruit fly (2007) Drosophila simulans, fruit fly (2007) Drosophila takahashi, fruit fly (2011) Drosophila virilis, fruit fly (2007) Drosophila willistoni, fruit fly (2007) Drosophila yakuba, fruit fly (2007) Family Phoridae Megaselia abdita, scuttle fly (transcriptome 2013) Family Psychodidae (drain flies) Clogmia albipunctata, moth midge (transcriptome 2013) Family Sarcophagidae (flesh flies) Sarcophaga Bullata, Flesh fly (2019) Family Syrphidae (hoverflies) Episyrphus balteatus, hoverfly (transcriptome 2011) Order Hemiptera Acyrthosiphon pisum, aphid (pea aphid) (2010) Ericerus pela, Chinese wax scale insect (2019) Laodelphax striatellus, small brown planthopper (2017) Lycorma delicatula, spotted lanternfly (2019) Rhodnius prolixus, kissing-bug (2015) Rhopalosiphum maidis, Corn leaf aphid (2019) Sitobion miscanthi, Indian grain aphid (2019) Triatoma rubrofasciata, assassin bug (2019) Order Hymenoptera Acromyrmex echinatior colony Ae372, ant (Panamanian leafcutter) (2011) Apis mellifera, bee (honey bee), (model for eusocial behavior) (2006) Atta cephalotes, ant (leaf-cutter ant) (2011) Camponotus floridanus, ant (2010) Cerapachys biroi, ant (clonal raider ant)(2014) Euglossa dilemma, Green orchid bee (2017) Harpegnathos saltator, ant (2010) Lasius niger, ant (black garden ant)(2017) Linepithema humile, ant (Argentine ant) (2011) Nasonia giraulti, wasp (parasitoid wasp) (2010) Nasonia longicornis, wasp (parasitoid wasp) (2010) Nasonia vitripennis, wasp (parasitoid wasp; model organism) (2010) Nomia Melanderi, Alkali bee (2019) Pogonomyrmex barbatus, ant (red harvester ant) (2011) Solenopsis invicta, ant (fire ant) (2011) Order Lepidoptera Abrostola tripartita Hufnagel, Spectacle (2021) Achalarus lyciades, Hoary Edge Skipper (2017) Ahamus jianchuanensis, Jianchuan ghost moth (2024) Antharaea yamamai, Japanese oak silk moth (2019) Arctia plantaginis, Wood tiger moth (2020) Bicyclus anynana, squinting bush brown (2017) Bombyx mori Strain:p50T, moth (domestic silk worm) (2004) Calycopis cecrops, Red-Banded Groundstreak (2016) Calycopis isobeon, Dusky-Blue Groundstreak (2016) Coenonympha arcania, Pearly Heath (2024) Cydia
|
{
"page_id": 35917093,
"source": null,
"title": "List of sequenced animal genomes"
}
|
pomonella, codling moth (2019) Danaus plexippus, monarch butterfly) (2011) Heliconius melpomene, butterfly (2012) Keiferia lycopersicella, Tomato pinworm (2024) Melitaea cinxia, Glanville fritillary butterfly (2014) Megathymus ursus violae, bear giant skipper butterfly (2018) Morpho helenor, Common blue morpho (2023) Morpho achilles, Blue-banded morpho (2023) Morpho deidamia (2023) Papilio bianor, Chinese peacock butterfly (2019) Phthorimaea absoluta, Tomato leafminer (2024) Pieris rapae, small cabbage white butterfly (2016) Plodia interpunctella, Indianmeal moth (2022) Plutella xylostella, moth (diamondback moth) (2013) Scrobipalpa atriplicella, Goosefoot groundling moth (2024) Spodoptera frugiperda, Fall armyworm (2017) Thitarodes armoricanus, Himalaya ghost moth (2024) Thitarodes xiaojinensis, Xiaojin ghost moth (2024) Troides aeacus, Golden birdwing (2024) Eudocima phalonia, fruit-piercing moth (2017) Order Orthoptera Locusta migratoria, migratory locust (2014) Schistocerca gregaria, desert locust (2020) Gryllus bimaculatus, two-spotted cricket (2021) Order Phthiraptera Pediculus humanus, louse (sucking louse; parasite) (2010) Psocoptera Liposcelis brunnea, booklouse (2022) Order Raphidioptera Venustoraphidia nigricollis, black-necked snakefly (2023) Order Trichoptera Eubasilissa regina, purple caddisfly (2022,) Stenopsyche tienmushanensisi, Caddisfly (2018) === Crustaceans === Acartia tonsa dana, cosmopolitan calanoid copepod (2019) Cherax quadricarinatus, Red claw crayfish (2020) Daphnia pulex, water flea (2007) Eulimnadia texana, Clam Shrimp (2018) Macrobrachium nipponense, oriental river prawn (2021) Neocaridina denticulata, shrimp (2014) Parhyale hawaiensis, amphipod (2016) Pollicipes pollicipes, Gooseneck barnacle (2022) Portunus trituberculatus, swimming crab (2020) Procambarus virginalis, marbled crayfish (2018) Sphaeroma terebrans, a wood-boring isopod (2019) Tigriopus kingsejongensis, antarctic-endemic copepod (2017) === Chelicerates === Order Xiphosura: Limulus polyphemus, Atlantic horseshoe crab (2014) Carcinoscorpius rotundicauda, mangrove horseshoe crab (2021) Tachypleus tridentatus, tri-spine horseshoe crab (2021) Order Ixodida: Ixodes scapularis, deer tick (2016) Order Mesostigmata: Tropilaelaps mercedesae, honeybee mite (2017) Order Trombidiformes: Tetranychus urticae, spider mite (2011) Order Scorpiones: Mesobuthus martensii, Chinese scorpion (2013) Order Araneae: Acanthoscurria geniculata, Brazilian whiteknee tarantula (2014) Argiope bruennichi, European wasp spider (2021) Dysdera silvatica, Canary Island nocturnal endemic woodlouse spider (2019) Latrodectus elegans, Black
|
{
"page_id": 35917093,
"source": null,
"title": "List of sequenced animal genomes"
}
|
widow spider (2022) Nephila clavipes, (golden silk orb-weaver) (2017) Parasteatoda tepidariorum, (common house spider) (2017) Stegodyphus mimosarum, African social velvet spider (2014) Uloborus diversus, Cribellate orb-weaving spider, (2023) === Myriapods === Strigamia maritima, centipede Trigoniulus corallinus, millipede == Onychophora (Velvet Worms) == == Tardigrades == Hypsibius dujardini, water bear (2015) == Nematodes == Ancylostoma ceylanicum, zoonotic hookworm infecting both humans and other mammals (2015) Aplectana chamaeleonis, amphibian parasite (2023) Ascaris suum, pig-infecting giant roundworm, closely related to human-infecting giant roundworm Ascaris lumbricoides (2011) Brugia malayi (Strain:TRS), human-infecting filarial parasite (2007) Bursaphelenchus xylophilus, infects pine trees (2011) Caenorhabditis angaria (Strain:PS1010) (2010) Caenorhabditis brenneri, a gonochoristic (male-female obligate) species more closely related to C. briggsae than C. elegans Caenorhabditis briggsae (2003) Caenorhabditis elegans (Strain:Bristol N2), model organism (1998) Caenorhabditis remanei, a gonochoristic (male-female obligate) species more closely related to C. briggsae than C. elegans Dirofilaria immitis, dog-infecting filarial parasite (2012) Globodera pallida, plant pathogen (2014) Haemonchus contortus, blood-feeding parasite infecting sheep and goats (2013) Heterodera glycines, soybean cyst nematode (2019) Heterorhabditis bacteriophora, (2013) Loa loa, human-infecting filarial parasite (2013) Meloidogyne hapla, northern root-knot nematode (plant pathogen) (2008) Meloidogyne incognita, southern root-knot nematode (plant pathogen) (2008) Necator americanus, human-infecting hookworm (2014) Onchocerca volvulus, human-infecting filarial parasite Pristionchus pacificus, model invertebrate (2008) Romanomermis culicivorax, entomopathogenic nematode that invades larvae of various mosquito species (2013) Trichuris suis, pig-infecting whipworm (2014) Trichuris muris, mouse-infecting whipworm (2014) Trichuris trichiura, human-infecting whipworm (2014) Wuchereria bancrofti, human-infecting filarial parasite == Nematomorpha == == Priapulida == == Kinorhyncha == == Loricifera == == Molluscs == === Polyplacophora (Chitons) === === Caudofoveata === === Cephalopods === Architeuthis dux, giant squid (2020) Euprymna scolopes, Hawaiian bobtail squid (2019) Hapalochlaena maculosa, Southern blue-ringed octopus (2020) Octopus bimaculoides, California two-spot octopus (2015) Octopus minor, common long-arm octopus (2018) Octopus vulgaris, common octopus (2019) ===
|
{
"page_id": 35917093,
"source": null,
"title": "List of sequenced animal genomes"
}
|
Bivalves === Argopecten purpuratus, peruvian scallop (2018) Bathymodiolus platifrons, seep mussel (2017) Chlamys farreri, Zhikong scallop (2017) Crassostrea angulata, Portuguese oyster (2023) Crassostrea gigas, Pacific oyster (2012) Dreissena rostriformis, Quagga mussel (2019) Limnoperna fortunei, invasive golden mussel (2017) Margaritifera margaritifera, European freshwater pearl mussel (2023) Modiolus philippinarum, shallow water mussel (2017) Mytilus galloprovincialis, Mediterranean mussel (2016) Panopea generosa, Pacific geoduck (2023) Patinopecten yessoensis, Yesso scallop (2017) Pecten maximus, Great scallop (2020) Pinctada fucata, Pearl oyster (2012) Ruditapes philippinarum, Manila clam (2017) Saccostrea glomerata, Sydney rock oyster (2018) Scapharca broughtonii, Blood clam (2019) Tridacna crocea, Giant clam (2023) Venustaconcha ellipsiformis, freshwater mussel (2018) === Gastropods === Achatina fulica, giant African snail (2019) Biomphalaria glabrata, a medically important air-breathing freshwater snail in the family Planorbidae (2017) Biomphalaria straminea, Ramshorn snail (2022) Candidula unifasciata, Land snail (2021) Conus ventricosus, Mediterranean cone snail (2021) Elysia chlorotica, a solar-powered sea slug (2019) Haliotis discus hannai, pacific abalone (2017) Kalloconus canariensis, Canary Island cone shell (2023) Kelletia kelletii, Kelletβs whelk (2023) Lottia gigantea, owl limpet (2013) Plakobranchus ocellatus, Kleptoplastic sea slug (2021) Pomacea canaliculata, golden apple snail (2018) === Scaphopods === ==== Dentaliida ==== ==== Gadilida ==== == Platyhelminthes == Clonorchis sinensis, liver fluke (human pathogen) (draft 2011) Echinococcus granulosus, tapeworm (dog pathogen) (2013, 2013) Echinococcus multilocularis, tapeworm (2013) Hymenolepis microstoma, tapeworm (2013) Schistosoma haematobium, schistosome (human pathogen) (2012 2019) Schistosoma japonicum, schistosome (human pathogen) (2009) Schistosoma mansoni, schistosome (human pathogen) (2009, 2012) Schmidtea mediterranea, planarian (model organism) (2006) Taenia solium, tapeworm (2013) == Annelids == Capitella teleta, polychaete (2007, 2013) Helobdella robusta, leech (2007, 2013) Eisenia fetida, earthworm (2015, 2016) Paraescarpia echinospica, deep-sea tubeworm (2021,) Hirudinaria manillensis, Asian Buffalo leech (2023) Hirudo nipponia, Japanese blood-sucking leech (2023) Whitmania pigra, Asian freshwater leech (2023) == Bryozoa == Bugula neritina, bryozoan (2020,) == Brachiopoda == Lingula anatina,
|
{
"page_id": 35917093,
"source": null,
"title": "List of sequenced animal genomes"
}
|
brachiopod (2015,) == Rotifera == Adineta vaga, rotifer (2013,) == See also == List of sequenced bacterial genomes List of sequenced archaeal genomes List of sequenced eukaryotic genomes List of sequenced fungi genomes List of sequenced plant genomes List of sequenced protist genomes List of sequenced plastomes == References ==
|
{
"page_id": 35917093,
"source": null,
"title": "List of sequenced animal genomes"
}
|
In statistical mechanics, the random-subcube model (RSM) is an exactly solvable model that reproduces key properties of hard constraint satisfaction problems (CSPs) and optimization problems, such as geometrical organization of solutions, the effects of frozen variables, and the limitations of various algorithms like decimation schemes. The RSM consists of a set of N binary variables, where solutions are defined as points in a hypercube. The model introduces clusters, which are random subcubes of the hypercube, representing groups of solutions sharing specific characteristics. As the density of constraints increases, the solution space undergoes a series of phase transitions similar to those observed in CSPs like random k-satisfiability (k-SAT) and random k-coloring (k-COL). These transitions include clustering, condensation, and ultimately the unsatisfiable phase where no solutions exist. The RSM is equivalent to these real CSPs in the limit of large constraint size. Notably, it reproduces the cluster size distribution and freezing properties of k-SAT and k-COL in the large-k limit. This is similar to how the random energy model is the large-p limit of the p-spin glass model. == Setup == === Subcubes === There are N {\displaystyle N} particles. Each particle can be in one of two states β 1 , + 1 {\displaystyle -1,+1} . The state space { β 1 , + 1 } N {\displaystyle \{-1,+1\}^{N}} has 2 N {\displaystyle 2^{N}} states. Not all are available. Only those satisfying the constraints are allowed. Each constraint is a subset A i {\displaystyle A_{i}} of the state space. Each A i {\displaystyle A_{i}} is a "subcube", structured like A i = β j β 1 : N A i j {\displaystyle A_{i}=\prod _{j\in 1:N}A_{ij}} where each A i j {\displaystyle A_{ij}} can be one of { β 1 } , { + 1 } , { β 1 , +
|
{
"page_id": 77073703,
"source": null,
"title": "Random subcube model"
}
|
1 } {\displaystyle \{-1\},\{+1\},\{-1,+1\}} . The available states is the union of these subsets: S = βͺ i A i {\displaystyle S=\cup _{i}A_{i}} === Random subcube model === Each random subcube model is defined by two parameters Ξ± , p β ( 0 , 1 ) {\displaystyle \alpha ,p\in (0,1)} . To generate a random subcube A i {\displaystyle A_{i}} , sample its components A i j {\displaystyle A_{ij}} IID according to P r ( A i j = { β 1 } ) = p / 2 P r ( A i j = { + 1 } ) = p / 2 P r ( A i j = { β 1 , + 1 } ) = 1 β p {\displaystyle {\begin{aligned}Pr(A_{ij}&=\{-1\})&=p/2\\Pr(A_{ij}&=\{+1\})&=p/2\\Pr(A_{ij}&=\{-1,+1\})&=1-p\end{aligned}}} Now sample 2 ( 1 β Ξ± ) N {\displaystyle 2^{(1-\alpha )N}} random subcubes, and union them together. === Entropies === The entropy density of the r {\displaystyle r} -th cluster in bits is s r := 1 N log 2 β‘ | A r | {\displaystyle s_{r}:={\frac {1}{N}}\log _{2}|A_{r}|} The entropy density of the system in bits is s := 1 N log 2 β‘ | βͺ r A r | {\displaystyle s:={\frac {1}{N}}\log _{2}|\cup _{r}A_{r}|} == Phase structure == === Cluster sizes and numbers === Let n ( s ) {\displaystyle n(s)} be the number of clusters with entropy density s {\displaystyle s} , then it is binomially distributed, thus E [ n ( s ) ] = 2 ( 1 β Ξ± ) N P β 2 N Ξ£ ( s ) + o ( N ) V a r [ n ( s ) ] = 2 ( 1 β Ξ± ) N P ( 1 β P ) V a r [ n ( s ) ] E [ n
|
{
"page_id": 77073703,
"source": null,
"title": "Random subcube model"
}
|
( s ) ] 2 β 2 β N Ξ£ ( s ) {\displaystyle {\begin{aligned}E[n(s)]&=2^{(1-\alpha )N}P\to 2^{N\Sigma (s)+o(N)}\\Var[n(s)]&=2^{(1-\alpha )N}P(1-P)\\{\frac {Var[n(s)]}{E[n(s)]^{2}}}&\to 2^{-N\Sigma (s)}\end{aligned}}} where P := ( N s N ) p ( 1 β s ) N ( 1 β p ) s N , Ξ£ ( s ) := 1 β Ξ± β D K L ( s β 1 β p ) D K L ( s β 1 β p ) := s log 2 β‘ s 1 β p + ( 1 β s ) log 2 β‘ 1 β s p {\displaystyle {\begin{aligned}P&:={\binom {N}{sN}}p^{(1-s)N}(1-p)^{sN},\\\Sigma (s)&:=1-\alpha -D_{KL}(s\|1-p)\\D_{KL}(s\|1-p)&:=s\log _{2}{\frac {s}{1-p}}+(1-s)\log _{2}{\frac {1-s}{p}}\end{aligned}}} By the Chebyshev inequality, if Ξ£ > 0 {\displaystyle \Sigma >0} , then n ( s ) {\displaystyle n(s)} concentrates to its mean value. Otherwise, since E [ n ( s ) ] β 0 {\displaystyle E[n(s)]\to 0} , n ( s ) {\displaystyle n(s)} also concentrates to 0 {\displaystyle 0} by the Markov inequality. Thus, n ( s ) β { 2 N Ξ£ ( s ) + o ( N ) if Ξ£ ( s ) > 0 0 if Ξ£ ( s ) < 0 {\displaystyle n(s)\to {\begin{cases}2^{N\Sigma (s)+o(N)}\quad &{\text{if }}\Sigma (s)>0\\0\quad &{\text{if }}\Sigma (s)<0\end{cases}}} almost surely as N β β {\displaystyle N\to \infty } . When Ξ£ = 0 {\displaystyle \Sigma =0} exactly, the two forces exactly balance each other out, and n ( s ) {\displaystyle n(s)} does not collapse, but instead converges in distribution to the Poisson distribution P o i s s o n ( 1 ) {\displaystyle Poisson(1)} by the law of small numbers. === Liquid phase === For each state, the number of clusters it is in is also binomially distributed, with expectation 2 ( 1 β Ξ± ) N ( 1 β p / 2
|
{
"page_id": 77073703,
"source": null,
"title": "Random subcube model"
}
|
) N = 2 N ( log 2 β‘ ( 2 β p ) β Ξ± ) {\displaystyle 2^{(1-\alpha )N}(1-p/2)^{N}=2^{N(\log _{2}(2-p)-\alpha )}} So if Ξ± < log 2 β‘ ( 2 β p ) {\displaystyle \alpha <\log _{2}(2-p)} , then it concentrates to 2 N ( log 2 β‘ ( 2 β p ) β Ξ± ) {\displaystyle 2^{N(\log _{2}(2-p)-\alpha )}} , and so each state is in an exponential number of clusters. Indeed, in that case, the probability that all states are allowed is [ 1 β [ 1 β ( 1 β p / 2 ) N ] 2 ( 1 β Ξ± ) N ] 2 N βΌ e β e β 2 N ( log 2 β‘ ( 2 β p ) β Ξ± ) + N ln β‘ 2 β 1 {\displaystyle [1-[1-(1-p/2)^{N}]^{2^{(1-\alpha )N}}]^{2^{N}}\sim e^{-e^{-2^{N(\log _{2}(2-p)-\alpha )}+N\ln 2}}\to 1} Thus almost surely, all states are allowed, and the entropy density is 1 bit per particle. === Clustered phase === If Ξ± > Ξ± d := log 2 β‘ ( 2 β p ) {\displaystyle \alpha >\alpha _{d}:=\log _{2}(2-p)} , then it concentrates to zero exponentially, and so most states are not in any cluster. Those that do are exponentially unlikely to be in more than one. Thus, we find that almost all states are in zero clusters, and of those in at least one cluster, almost all are in just one cluster. The state space is thus roughly speaking the disjoint union of the clusters. Almost surely, there are n ( s ) = 2 N Ξ£ ( s ) {\displaystyle n(s)=2^{N\Sigma (s)}} clusters of size 2 N s {\displaystyle 2^{Ns}} , therefore, the state space is dominated by clusters with optimal entropy density s β = arg β‘ max s ( Ξ£ ( s
|
{
"page_id": 77073703,
"source": null,
"title": "Random subcube model"
}
|
) + s ) {\displaystyle s^{*}=\arg \max _{s}(\Sigma (s)+s)} . Thus, in the clustered phase, the state space is almost entirely partitioned among 2 N Ξ£ ( s β ) {\displaystyle 2^{N\Sigma (s^{*})}} clusters of size 2 N s β {\displaystyle 2^{Ns^{*}}} each. Roughly, the state space looks like exponentially many equally-sized clusters. === Condensation phase === Another phase transition occurs when Ξ£ ( s β ) = 0 {\displaystyle \Sigma (s^{*})=0} , that is, Ξ± = Ξ± c := p ( 2 β p ) + log 2 β‘ ( 2 β p ) {\displaystyle \alpha =\alpha _{c}:={\frac {p}{(2-p)}}+\log _{2}(2-p)} When Ξ± > Ξ± c {\displaystyle \alpha >\alpha _{c}} , the optimal entropy density becomes unreachable, as there almost surely exists zero clusters with entropy density s β {\displaystyle s^{*}} . Instead, the state space is dominated by clusters with entropy close to s c {\displaystyle s_{c}} , the larger solution to Ξ£ ( s c ) = 0 {\displaystyle \Sigma (s_{c})=0} . Near s c {\displaystyle s_{c}} , the contribution of clusters with entropy density s = s c β Ξ΄ {\displaystyle s=s_{c}-\delta } to the total state space is 2 N s β size of clusters Γ 2 N Ξ£ ( s ) β number of clusters = 2 N ( s + Ξ£ ( s ) ) = 2 N ( s c β Ξ΄ β Ξ£ β² ( s c ) Ξ΄ ) {\displaystyle \underbrace {2^{Ns}} _{\text{size of clusters}}\times \underbrace {2^{N\Sigma (s)}} _{\text{number of clusters}}=2^{N(s+\Sigma (s))}=2^{N(s_{c}-\delta -\Sigma '(s_{c})\delta )}} At large N {\displaystyle N} , the possible entropy densities are s c , s c β 1 / N , s c β 2 / N , β¦ {\displaystyle s_{c},s_{c}-1/N,s_{c}-2/N,\dots } . The contribution of each is 2 N s c , 2 N s
|
{
"page_id": 77073703,
"source": null,
"title": "Random subcube model"
}
|
c 2 β ( 1 + Ξ£ β² ( s c ) ) , 2 N s c 2 β 2 ( 1 + Ξ£ β² ( s c ) ) , β¦ {\displaystyle 2^{Ns_{c}},2^{Ns_{c}}2^{-(1+\Sigma '(s_{c}))},2^{Ns_{c}}2^{-2(1+\Sigma '(s_{c}))},\dots } We can tabulate them as follows: Thus, we see that for any Ο΅ > 0 {\displaystyle \epsilon >0} , at N β β {\displaystyle N\to \infty } limit, over 1 β Ο΅ {\displaystyle 1-\epsilon } of the total state space is covered by only a finite number of clusters. The state space looks partitioned into clusters with exponentially decaying sizes. This is the condensation phase. === Unsatisfiable phase === When Ξ± > 1 {\displaystyle \alpha >1} , the number of clusters is zero, so there are no states. == Extensions == The RSM can be extended to include energy landscapes, allowing for the study of glassy behavior, temperature chaos, and the dynamic transition. == See also == Random energy model == References == Mora, Thierry; ZdeborovΓ‘, Lenka (2008-06-01). "Random Subcubes as a Toy Model for Constraint Satisfaction Problems". Journal of Statistical Physics. 131 (6): 1121β1138. arXiv:0710.3804. Bibcode:2008JSP...131.1121M. doi:10.1007/s10955-008-9543-x. ISSN 1572-9613. ZdeborovΓ‘, Lenka (2008-06-25). "Statistical Physics of Hard Optimization Problems". Acta Physica Slovaca. 59 (3): 169β303. arXiv:0806.4112. Bibcode:2008PhDT.......107Z. MΓ©zard, Marc; Montanari, Andrea (2009-01-22). Information, Physics, and Computation. Oxford University Press. ISBN 978-0-19-154719-5.
|
{
"page_id": 77073703,
"source": null,
"title": "Random subcube model"
}
|
Tire-derived fuel (TDF) is composed of shredded scrap tires. Tires may be mixed with coal or other fuels, such as wood or chemical wastes, to be burned in concrete kilns, power plants, or paper mills. An EPA test program concluded that, with the exception of zinc emissions, potential emissions from TDF are not expected to be very much different from other conventional fossil fuels, as long as combustion occurs in a well-designed, well-operated and well-maintained combustion device. In the United States in 2017, about 43% of scrap tires (1,736,340 tons or 106 million tires) were burnt as tire-derived fuel. Cement manufacturing was the largest user of TDF, at 46%, pulp and paper manufacturing used 29% and electric utilities used 25%. Another 25% of scrap tires were used to make ground rubber, 17% were disposed of in landfills and 16% had other uses. == Theory == Historically, there has not been any volume use for scrap tires other than burning that has been able to keep up with the volume of waste generated yearly. Tires produce the same energy as petroleum and approximately 25% more energy than coal. Burning tires is lower on the hierarchy of reducing waste than recycling, but it is better than placing the tire waste in a landfill or dump, where there is a possibility for uncontrolled tire fires or the harboring of disease vectors such as mosquitoes. Tire Derived Fuel is an interim solution to the scrap tire waste problem. Advances in tire recycling technology might one day provide a solution other than burning by reusing tire derived material in high volume applications. == Characteristics == Tire derived fuel is usually consumed in the form of shredded or chipped material with most of the metal wire from the tire's steel belts removed. The analytical properties of
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{
"page_id": 18877738,
"source": null,
"title": "Tire-derived fuel"
}
|
this refined material are published in TDF Produced From Scrap Tires with 96+% Wire Removed. Tires are typically composed of about 1 to 1.5% Zinc oxide, which is a well known component used in the manufacture of tires and is also toxic to aquatic and plant life. The chlorine content in tires is due primarily to the chlorinated butyl rubber liner that slows the leak rate of air. The Rubber Manufacturers Association (RMA) is a very good source for compositional data and other information on tires. The use of TDF for heat production is controversial due to the possibility for toxin production. Reportedly, polychlorinated dibenzodioxins and furans are produced during the combustion process and there is supportive evidence to suggest that this is true under some incineration conditions. Other toxins such as NOx, SOx and heavy metals are also produced, though whether these levels of toxins are higher or lower than conventional coal and oil fired incinerators is not clear. == Environmental impact == While environmental controversy surrounding use of this fuel is wide and varied, the greatest supported evidence of toxicity comes from the presence of dioxins and furans in the flue gases. Zinc has also been found to dissolve into storm water, from shredded rubber, at acutely toxic levels for aquatic life and plants. A study of dioxin and furan content of stack gasses at a variety of cement mills, paper mills, boilers, and power plants conducted in the 1990s shows a wide and inconsistent variation in dioxin and furan output when fueled partially by TDF as compared to the same facilities powered by only coal. Some facilities added as little as 4% TDF and experienced as much as a 4,140% increase in dioxin and furan emissions. Other facilities added as much as 30% TDF and experienced dioxin
|
{
"page_id": 18877738,
"source": null,
"title": "Tire-derived fuel"
}
|
and furan emissions increases of only as much as 58%. Still other facilities used as much as 8% TDF and experienced a decrease of as much as 83% of dioxin and furan emissions. One facility conducted four tests with two tests resulting in decreased emissions and two resulting in increased emissions. Another facility also conducted four tests and had widely varying increases in emissions. A 2004 study showed that huge polyaromatic emissions are generated from combustion of tire rubber, at a minimum, 2 orders of magnitude higher than coal alone. The study concludes with, "atmospheric contamination dramatically increases when tire rubber is used as the fuel. Other different combustion variables compared to the ones used for coal combustion should be used to avoid atmospheric contamination by toxic, mutagenic, and carcinogenic pollutants, as well as hot-gas cleaning systems and COx capture systems." == References ==
|
{
"page_id": 18877738,
"source": null,
"title": "Tire-derived fuel"
}
|
A hair dryer (the handheld type also referred to as a blow dryer) is an electromechanical device that blows ambient air in hot or warm settings for styling or drying hair. Hair dryers enable better control over the shape and style of hair, by accelerating and controlling the formation of temporary hydrogen bonds within each strand. These bonds are powerful, but are temporary and extremely vulnerable to humidity. They disappear with a single washing of the hair. Hairstyles using hair dryers usually have volume and discipline, which can be further improved with styling products, hairbrushes, and combs during drying to add tension, hold and lift. Hair dryers were invented in the late 19th century. The first model was created in 1911 by Gabriel Kazanjian. Handheld, household hair dryers first appeared in 1920. Hair dryers are used in beauty salons by professional stylists, as well as by consumers at home. == History == In 1888 the first hair dryer was invented by French stylist Alexandre Godefroy. His invention was a large, seated version that consisted of a bonnet that attached to the chimney pipe of a gas stove. Godefroy invented it for use in his hair salon in France, and it was not portable or handheld. It could only be used by having the person sit underneath it. Armenian American inventor Gabriel Kazanjian was the first to patent a hair dryer in the United States, in 1911. Around 1920, hair dryers began to go on the market in handheld form. This was due to innovations by National Stamping and Electricworks under the white cross brand, and later U.S. Racine Universal Motor Company and the Hamilton Beach Co., which allowed the dryer to be small enough to be held by hand. Even in the 1920s, the new dryers were often heavy, weighing
|
{
"page_id": 986413,
"source": null,
"title": "Hair dryer"
}
|
in at approximately 2 pounds (0.9 kg), and were difficult to use. They also had many instances of overheating and electrocution. Hair dryers were only capable of using 100 watts, which increased the amount of time needed to dry hair (the average dryer today can use up to 2000 watts of heat). Since the 1920s, development of the hair dryer has mainly focused on improving the wattage and superficial exterior and material changes. In fact, the mechanism of the dryer has not had any significant changes since its inception. One of the more important changes for the hair dryer is to be made of plastic, so that it is more lightweight. This really caught on in the 1960s with the introduction of better electrical motors and the improvement of plastics. Another important change happened in 1954 when GEC changed the design of the dryer to move the motor inside the casing. The bonnet dryer was introduced to consumers in 1951. This type worked by having the dryer, usually in a small portable box, connected to a tube that went into a bonnet with holes in it that could be placed on top of a person's head. This worked by giving an even amount of heat to the whole head at once. The 1950s also saw the introduction of the rigid-hood hair dryer which is the type most frequently seen in salons. It had a hard plastic helmet that wraps around the person's head. This dryer works similarly to the bonnet dryer of the 1950s but at a much higher wattage. In the 1970s, the U.S. Consumer Product Safety Commission set up guidelines that hair dryers had to meet to be considered safe to manufacture. Since 1991 the CPSC has mandated that all dryers must use a ground fault circuit interrupter
|
{
"page_id": 986413,
"source": null,
"title": "Hair dryer"
}
|
so that it cannot electrocute a person if it gets wet. By 2000, deaths by blowdryers had dropped to fewer than four people a year, a stark difference to the hundreds of cases of electrocution accidents during the mid-20th century. == Function == Most hair dryers consist of electric heating coils and a fan that blows the air (usually powered by a universal motor). The heating element in most dryers is a bare, coiled nichrome wire that is wrapped around mica insulators. Nichrome is used due to its high resistivity, and low tendency to corrode when heated. A survey of stores in 2007 showed that most hair dryers had ceramic heating elements (like ceramic heaters) because of their "instant heat" capability. This means that it takes less time for the dryers to heat up and for the hair to dry. Many of these dryers have "cool shot" buttons that turn off the heater and blow room-temperature air while the button is pressed. This function helps to maintain the hairstyle by setting it. The colder air reduces frizz and can help to promote shine in the hair. Many feature "ionic" operation, to reduce the build-up of static electricity in the hair, though the efficacy of ionic technology is of some debate. Manufacturers claim this makes the hair "smoother". Hair dryers are available with attachments, such as diffusers, airflow concentrators, and comb nozzles. A diffuser is an attachment that is used on hair that is fine, colored, permed or naturally curly. It diffuses the jet of air, so that the hair is not blown around while it dries. The hair dries more slowly, at a cooler temperature, and with less physical disturbance. This makes it so that the hair is less likely to frizz and it gives the hair more volume. An
|
{
"page_id": 986413,
"source": null,
"title": "Hair dryer"
}
|
airflow concentrator does the opposite of a diffuser. It makes the end of the hair dryer narrower and thus helps to concentrate the heat into one spot to make it dry rapidly. The comb nozzle attachment is the same as the airflow concentrator, but it ends with comb-like teeth so that the user can dry the hair using the dryer without a brush or comb. Hair dryers have been cited as an effective treatment for head lice. == Types == Today there are two major types of hair dryers: the handheld and the rigid-hood dryer. A hood dryer has a hard plastic dome that fits over a person's head to dry their hair. Hot air is blown out through tiny openings around the inside of the dome so the hair is dried evenly. Hood dryers are mainly found in hair salons. === Hair dryer brush === A hair dryer brush (also called "hot air brush" and "round brush hair dryer" and "hair styler") has the shape of a brush and it is used as a volumizer too. There are two types of round brush hair dryers β rotating and static. Rotating round brush hair dryers have barrels that rotate automatically while static round brush hair dryers don't. == Cultural references == The British historical drama television series Downton Abbey made note of the invention of the portable hair dryer when a character purchased one in Series 6 Episode 9, set in the year 1925. == Gallery == == See also == Curling iron Heat gun == Notes == == References == == External links == Media related to Hairdryers at Wikimedia Commons
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{
"page_id": 986413,
"source": null,
"title": "Hair dryer"
}
|
Sven Anders FlodstrΓΆm (born 1 October 1944) is a Swedish professor of materials physics at the Royal Institute of Technology. FlodstrΓΆm was born in SΓΆderhamn, Sweden. He studied engineering physics and electrical engineering in LinkΓΆping. In 1975, he was awarded a Ph.D. in physics in LinkΓΆping with the thesis "Electronic structure of clean and oxygen covered aluminium and magnesium surfaces studied by photoelectron spectroscopy". FlodstrΓΆm was also one of the initiators of the synchrotron facility MAX-Lab in Lund, where he served as a coordinator until 1985. In 1985 he was appointed professor of materials physics at the Royal Institute of Technology in Stockholm. He was previously the rector of LinkΓΆping University from 1996 to 1999 and of the Royal Institute of Technology from 1999 to 2007 and University Chancellor of Sweden and head of the Swedish National Agency for Higher Education from 1 August 2007 to 30 June 2010. Since November 2012, Anders FlodstrΓΆm is the Chief Education Officer of EIT Digital and a member of the Management Committee of EIT Digital. == References ==
|
{
"page_id": 22875437,
"source": null,
"title": "Anders FlodstrΓΆm"
}
|
Zeta potential is the electrical potential at the slipping plane. This plane is the interface which separates mobile fluid from fluid that remains attached to the surface. Zeta potential is a scientific term for electrokinetic potential in colloidal dispersions. In the colloidal chemistry literature, it is usually denoted using the Greek letter zeta (ΞΆ), hence ΞΆ-potential. The usual units are volts (V) or, more commonly, millivolts (mV). From a theoretical viewpoint, the zeta potential is the electric potential in the interfacial double layer (DL) at the location of the slipping plane relative to a point in the bulk fluid away from the interface. In other words, zeta potential is the potential difference between the dispersion medium and the stationary layer of fluid attached to the dispersed particle. The zeta potential is caused by the net electrical charge contained within the region bounded by the slipping plane, and also depends on the location of that plane. Thus, it is widely used for quantification of the magnitude of the charge. However, zeta potential is not equal to the Stern potential or electric surface potential in the double layer, because these are defined at different locations. Such assumptions of equality should be applied with caution. Nevertheless, zeta potential is often the only available path for characterization of double-layer properties. The zeta potential is an important and readily measurable indicator of the stability of colloidal dispersions. The magnitude of the zeta potential indicates the degree of electrostatic repulsion between adjacent, similarly charged particles in a dispersion. For molecules and particles that are small enough, a high zeta potential will confer stability, i.e., the solution or dispersion will resist aggregation. When the potential is small, attractive forces may exceed this repulsion and the dispersion may break and flocculate. So, colloids with high zeta potential (negative
|
{
"page_id": 1183025,
"source": null,
"title": "Zeta potential"
}
|
or positive) are electrically stabilized while colloids with low zeta potentials tend to coagulate or flocculate as outlined in the table. Zeta potential can also be used for the pKa estimation of complex polymers that is otherwise difficult to measure accurately using conventional methods. This can help studying the ionisation behaviour of various synthetic and natural polymers under various conditions and can help in establishing standardised dissolution-pH thresholds for pH responsive polymers. == Measurement == Some new instrumentations techniques exist that allow zeta potential to be measured. The Zeta Potential Analyzer can measure solid, fibers, or powdered material. The motor found in the instrument creates an oscillating flow of electrolyte solution through the sample. Several sensors in the instrument monitor other factors, so the software attached is able to do calculations to find the zeta potential. Temperature, pH, conductivity, pressure, and streaming potential are all measured in the instrument for this reason. Zeta potential can also be calculated using theoretical models, and an experimentally-determined electrophoretic mobility or dynamic electrophoretic mobility. Electrokinetic phenomena and electroacoustic phenomena are the usual sources of data for calculation of zeta potential. (See Zeta potential titration.) === Electrokinetic phenomena === Electrophoresis is used for estimating zeta potential of particulates, whereas streaming potential/current is used for porous bodies and flat surfaces. In practice, the zeta potential of dispersion is measured by applying an electric field across the dispersion. Particles within the dispersion with a zeta potential will migrate toward the electrode of opposite charge with a velocity proportional to the magnitude of the zeta potential. This velocity is measured using the technique of the laser Doppler anemometer. The frequency shift or phase shift of an incident laser beam caused by these moving particles is measured as the particle mobility, and this mobility is converted to the zeta
|
{
"page_id": 1183025,
"source": null,
"title": "Zeta potential"
}
|
potential by inputting the dispersant viscosity and dielectric permittivity, and the application of the Smoluchowski theories. ==== Electrophoresis ==== Electrophoretic mobility is proportional to electrophoretic velocity, which is the measurable parameter. There are several theories that link electrophoretic mobility with zeta potential. They are briefly described in the article on electrophoresis and in details in many books on colloid and interface science. There is an IUPAC Technical Report prepared by a group of world experts on the electrokinetic phenomena. From the instrumental viewpoint, there are three different experimental techniques: microelectrophoresis, electrophoretic light scattering, and tunable resistive pulse sensing. Microelectrophoresis has the advantage of yielding an image of the moving particles. On the other hand, it is complicated by electro-osmosis at the walls of the sample cell. Electrophoretic light scattering is based on dynamic light scattering. It allows measurement in an open cell which eliminates the problem of electro-osmotic flow except for the case of a capillary cell. And, it can be used to characterize very small particles, but at the price of the lost ability to display images of moving particles. Tunable resistive pulse sensing (TRPS) is an impedance-based measurement technique that measures the zeta potential of individual particles based on the duration of the resistive pulse signal. The translocation duration of nanoparticles is measured as a function of voltage and applied pressure. From the inverse translocation time versus voltage-dependent electrophoretic mobility, and thus zeta potentials are calculated. The main advantage of the TRPS method is that it allows for simultaneous size and surface charge measurements on a particle-by-particle basis, enabling the analysis of a wide spectrum of synthetic and biological nano/microparticles and their mixtures. All these measuring techniques may require dilution of the sample. Sometimes this dilution might affect properties of the sample and change zeta potential. There is
|
{
"page_id": 1183025,
"source": null,
"title": "Zeta potential"
}
|
only one justified way to perform this dilution β by using equilibrium supernatant. In this case, the interfacial equilibrium between the surface and the bulk liquid would be maintained and zeta potential would be the same for all volume fractions of particles in the suspension. When the diluent is known (as is the case for a chemical formulation), additional diluent can be prepared. If the diluent is unknown, equilibrium supernatant is readily obtained by centrifugation. ==== Streaming potential, streaming current ==== The streaming potential is an electric potential that develops during the flow of liquid through a capillary. In nature, a streaming potential may occur at a significant magnitude in areas with volcanic activities. The streaming potential is also the primary electrokinetic phenomenon for the assessment of the zeta potential at the solid material-water interface. A corresponding solid sample is arranged in such a way to form a capillary flow channel. Materials with a flat surface are mounted as duplicate samples that are aligned as parallel plates. The sample surfaces are separated by a small distance to form a capillary flow channel. Materials with an irregular shape, such as fibers or granular media, are mounted as a porous plug to provide a pore network, which serves as capillaries for the streaming potential measurement. Upon the application of pressure on a test solution, liquid starts to flow and to generate an electric potential. This streaming potential is related to the pressure gradient between the ends of either a single flow channel (for samples with a flat surface) or the porous plug (for fibers and granular media) to calculate the surface zeta potential. Alternatively to the streaming potential, the measurement of streaming current offers another approach to the surface zeta potential. Most commonly, the classical equations derived by Maryan Smoluchowski are used
|
{
"page_id": 1183025,
"source": null,
"title": "Zeta potential"
}
|
to convert streaming potential or streaming current results into the surface zeta potential. Applications of the streaming potential and streaming current method for the surface zeta potential determination consist of the characterization of surface charge of polymer membranes, biomaterials and medical devices, and minerals. === Electroacoustic phenomena === There are two electroacoustic effects that are widely used for characterizing zeta potential: colloid vibration current and electric sonic amplitude. There are commercially available instruments that exploit these effects for measuring dynamic electrophoretic mobility, which depends on zeta potential. Electroacoustic techniques have the advantage of being able to perform measurements in intact samples, without dilution. Published and well-verified theories allow such measurements at volume fractions up to 50%. Calculation of zeta potential from the dynamic electrophoretic mobility requires information on the densities for particles and liquid. In addition, for larger particles exceeding roughly 300 nm in size information on the particle size required as well. == Calculation == The most known and widely used theory for calculating zeta potential from experimental data is that developed by Marian Smoluchowski in 1903. This theory was originally developed for electrophoresis; however, an extension to electroacoustics is now also available. Smoluchowski's theory is powerful because it is valid for dispersed particles of any shape and any concentration. However, it has its limitations: Detailed theoretical analysis proved that Smoluchowski's theory is valid only for a sufficiently thin double layer, when the Debye length, 1 / ΞΊ {\displaystyle 1/\kappa } , is much smaller than the particle radius, a {\displaystyle a} : ΞΊ β
a β« 1 {\displaystyle {\kappa }\cdot a\gg 1} The model of the "thin double layer" offers tremendous simplifications not only for electrophoresis theory but for many other electrokinetic and electroacoustic theories. This model is valid for most aqueous systems because the Debye length is
|
{
"page_id": 1183025,
"source": null,
"title": "Zeta potential"
}
|
typically only a few nanometers in water. The model breaks only for nano-colloids in a solution with ionic strength approaching that of pure water. Smoluchowski's theory neglects the contribution of surface conductivity. This is expressed in modern theories as the condition of a small Dukhin number: D u βͺ 1 {\displaystyle Du\ll 1} The development of electrophoretic and electroacoustic theories with a wider range of validity was a purpose of many studies during the 20th century. There are several analytical theories that incorporate surface conductivity and eliminate the restriction of the small Dukhin number for both the electrokinetic and electroacoustic applications. Early pioneering work in that direction dates back to Overbeek and Booth. Modern, rigorous electrokinetic theories that are valid for any zeta potential, and often any ΞΊ a {\displaystyle \kappa a} , stem mostly from Soviet Ukrainian (Dukhin, Shilov, and others) and Australian (O'Brien, White, Hunter, and others) schools. Historically, the first one was DukhinβSemenikhin theory. A similar theory was created ten years later by O'Brien and Hunter. Assuming a thin double layer, these theories would yield results that are very close to the numerical solution provided by O'Brien and White. There are also general electroacoustic theories that are valid for any values of Debye length and Dukhin number. === Henry's equation === When ΞΊa is between large values where simple analytical models are available, and low values where numerical calculations are valid, Henry's equation can be used when the zeta potential is low. For a nonconducting sphere, Henry's equation is u e = 2 Ξ΅ r s Ξ΅ 0 3 Ξ· ΞΆ f 1 ( ΞΊ a ) {\displaystyle u_{e}={\frac {2\varepsilon _{rs}\varepsilon _{0}}{3\eta }}\zeta f_{1}(\kappa a)} , where f1 is the Henry function, one of a collection of functions which vary smoothly from 1.0 to 1.5 as ΞΊa
|
{
"page_id": 1183025,
"source": null,
"title": "Zeta potential"
}
|
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