Bayesian experimental design: Difference between revisions

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[[File:Mapping cone.PNG|thumb|The mapping cone of <math>f: X \to Y</math> is obtained by gluing the cone over ''X'' to ''Y''.]]
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In [[mathematics]], especially [[homotopy theory]], the '''mapping cone''' is a construction <math>C_f</math> of [[topology]], analogous to a [[quotient space]]. It is also called the '''homotopy cofiber,''' and also notated <math>Cf</math>.
 
==Definition==
 
Given a [[continuous function|map]] <math>f\colon X \to Y</math>, the mapping cone <math>C_f</math> is defined to be the quotient topological space of <math>(X \times I) \sqcup Y</math> with respect to the equivalence relation <math>(x, 0) \sim (x',0)\,</math>, <math>(x,1) \sim f(x)\,</math> on ''X''. Here <math>I</math> denotes the unit interval [0,1] with its standard topology. Note that some (like May) use the opposite convention, switching 0 and 1.
 
Visually, one takes the cone on ''X'' (the cylinder <math>X \times I</math> with one end (the 0 end) identified to a point), and glues the other end onto ''Y'' via the map ''f'' (the identification of the 1 end).
 
Coarsely, one is taking the [[quotient space]] by the image of ''X,'' so ''Cf'' "=" ''Y''/''f''(''X''); this is not precisely correct because of point-set issues, but is the philosophy, and is made precise by such results as the [[#Homology of a pair|homology of a pair]] and the notion of an [[n-connected|''n''-connected]] map.
 
The above is the definition for a map of unpointed spaces; for a map of pointed spaces <math>f\colon (X,x_0) \to (Y,y_0),</math> (so <math>f\colon x_0 \mapsto y_0</math>), one also identifies all of <math>{x_0}\times I</math>; formally, <math>(x_0,t) \sim (x_0,t')\,.</math> Thus one end and the "seam" are all identified with <math>y_0.</math>
 
=== Example of circle ===
If <math>X</math> is the [[circle]] ''S''<sup>1</sup>, ''C''<sub>''f''</sub> can be considered as the [[quotient space]] of the [[disjoint union]] of ''Y'' with the [[disk (mathematics)|disk]] ''D''<sup>2</sup> formed by identifying a point ''x'' on the [[boundary (topology)|boundary]] of ''D''<sup>2</sup> to the point ''f(x)'' in ''Y''.
 
Consider, for example, the case where ''Y'' is the disc ''D''<sup>2</sup>, and  
 
:''f'': ''S''<sup>1</sup> &rarr; ''Y'' = ''D''<sup>2</sup>
 
is the standard inclusion of the circle ''S''<sup>1</sup> as the boundary of ''D''<sup>2</sup>. Then the mapping cone ''C''<sub>''f''</sub> is [[homeomorphism|homeomorphic]] to two disks joined on their boundary, which is topologically the sphere ''S''<sup>2</sup>.
 
=== Double mapping cylinder ===
 
The mapping cone is a special case of the double [[mapping cylinder]]. This is basically a cylinder joined on one end to a space ''X''<sub>1</sub> via the [[continuous function|map]]
 
:''f''<sub>1</sub>: ''S''<sup>1</sup> &rarr; ''X''<sub>1</sub>
 
and joined on the other end to a space ''X''<sub>2</sub> via the map
 
:''f''<sub>2</sub>: S<sup>1</sup> &rarr; ''X''<sub>2</sub>.
 
The mapping cone is the degenerate case of the double mapping cylinder (also known as the homotopy pushout), in which one space is a single point.
 
==Applications==
===CW-complexes===
Attaching a cell
 
===Effect on fundamental group===
Given a [[topological space|space]] ''X'' and a loop
 
:<math>\alpha\colon S^1 \to X</math>
 
representing an element of the [[fundamental group]] of ''X'', we can form the mapping cone ''C''<sub>''&alpha;''</sub>. The effect of this is to make the loop ''&alpha;'' [[contractible]] in ''C''<sub>''&alpha;''</sub>, and therefore the [[equivalence class]] of ''&alpha;'' in the fundamental group of ''C''<sub>''&alpha;''</sub> will be simply the [[identity element]].
 
Given a group presentation by generators and relations, one gets a 2-complex with that fundamental group.
 
===Homology of a pair===
The mapping cone lets one interpret the homology of a pair as the reduced homology of the quotient:
 
If ''E'' is a [[homology theory]], and <math>i\colon A \to X</math> is a [[cofibration]], then <math>E_*(X,A) = E_*(X/A,*) = \tilde E_*(X/A)</math>, which follows by applying excision to the mapping cone.<ref>
[http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf Peter May "A Concise Course in Algebraic Topology", section 14.2]
</ref>
 
==Relation to homotopy (homology) equivalences==
 
A map <math> f\colon X\rightarrow Y</math> between simply-connected CW complexes is a [[homotopy equivalence]] if and only if its mapping cone is contractible.
 
More generally, a map is called [[n-connected|''n''-connected]] (as a map) if its mapping cone is ''n''-connected (as a space), plus a little more.  
See A. Hatcher [http://www.math.cornell.edu/~hatcher/AT/ATpage.html Algebraic Topology].{{PN|date=February 2013}}
 
Let <math> \mathbb{}H_*</math> be a fixed [[homology theory]]. The map <math> f:X\rightarrow Y</math> induces isomorphisms on <math>\mathbb{}H_*</math>, if and only if the map <math> \{\cdot\}\hookrightarrow C_f</math> induces an isomorphism on <math> \mathbb{}H_*</math>, i.e. <math> \mathbb{}H_*(C_f,pt)=0</math>.
 
==See also==
* [[Mapping cone (homological algebra)]]
 
==References==
<References />
 
[[Category:Algebraic topology]]

Latest revision as of 10:35, 4 October 2014

Panic attacks are very scary at occasions as a result of the symptoms are so intently related to coronary heart assaults. how to lose weight When you might have an nervousness dysfunction any little feeling will scare you and make you think something is gravely fallacious. When you've got a slight feeling of pain somewhere you will over exaggerate it and make it seem to be a cause to worry.

Pulmonary embolism happens when a blood clot detaches from the wall of a deep vein, strikes by way of the bloodstream, and becomes lodged within the pulmonary artery close to or low carb diet inside the lungs. This reduces the volume of fresh blood returning to the left aspect of the center and may produce sudden chest ache, speedy heart fee (tachycardia), fast shallow breathing, and coughing up of vivid crimson spit. Hypoxia means diminished availability of oxygen to the physique tissues. It's a symptom of several potential underlying issues, resembling altitude sickness or a pulmonary disorder. Signs can embody problem respiration (dyspnea), speedy pulse, fainting and chest ache (angina pectoris). So you could be wondering what the brief story above has to do with anxiousness

Weight loss program - A wholesome balanced eating regimen is vitally necessary while pregnant as you are not only feeding your self, but in addition your unborn child. Your food regimen should comprise loads of fruit and greens as they contain important elements required by the infant throughout pregnancy. These foods additionally contain chemicals which are helpful in controlling anxiousness attacks throughout being pregnant. Drug therapies for anxiety problems work finest together with cognitive behavioral therapy or some other type of psychotherapy. Antidepressants There's one key issue that makes the difference between those who fully remove panic assaults from their lives and those who do not. The key ingredient isn't medicine, lifestyle modifications, or leisure workouts. Concern of dropping one's mind

Panic attacks usually reach maximum intensity within one to a few minutes as soon as they begin. They then diminish step by step over the subsequent thirty minutes typically taking on to a few hours to totally dissipate. It is not uncommon for an individual to go to an emergency room the first time they have a panic attack because they are 'shocked' at what is occurring. The concern of the unknown of what's happening makes that first panic assault even more scary. Subsequent attacks might occur a number of times a month or hardly ever and will be as severe as the initial attack Nevertheless when an individual a minimum of turns into educated about panic assaults the subsequent time they have one they can do some encouraging self discuss

As ridiculous as this will likely sound, one efficient solution to stop anxiousness is to avoid reaching a conclusion. What does that mean? Attempt to look at a state of affairs and acknowledge, “yes this has occurred,” and initially maintain it as impersonal as you possibly can. This may be difficult when the media many times blows issues manner out of proportion and presents the news in a manner that detox diet - simpledietburnfat.com - contributes to those overwhelming emotions you experience. However if you happen to take a look at the scenario and ask to yourself, “how does it get any higher than this” or “what else is possible,” that in itself can hold you out of the turmoil. It may also allow different options to come into place.

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