Singularity theory - Revision history

77.255.61.216: /* Algebraic curve singularities */ formula didn't match graph

2014-12-27T13:06:35Z

Algebraic curve singularities: formula didn't match graph

← Older revision		Revision as of 15:06, 27 December 2014
Line 1:		Line 1:
	~~Hello,~~ I'm ~~Deneen, a 20 year old from Thurn, Austria~~.~~<br>My hobbies include (but are not limited to) Bowling, Exhibition Drill and watching Modern Family~~.<br><br>~~My website~~; [http://~~www.lareinaroja.com/info.php?a%5B%5D=%3Ca+href%3Dhttp%3A%2F%2Fseojay~~.zc.~~bz%2Fyangsamy%2Fguestbook%3Efree+cheats%3C%2Fa%3E Look at this hack~~]		Gabrielle Straub is what you can call me although it's not the a large number of feminine of names. Guam has always been my home. Filing presents been my profession blood pressure levels . time and I'm starting pretty good financially. Fish keeping is what I follow every week. See all that is new on my online shop here: http://prometeu.net<br><br>Stop by my web page :: clash of clans cheat; [http://prometeu.net prometeu.net],

en>Dr Greg: remove link to dab page

2014-02-15T21:32:45Z

remove link to dab page

← Older revision		Revision as of 23:32, 15 February 2014
Line 1:		Line 1:
	~~In [[category theory]]~~, ~~a branch of [[mathematics]]~~, a '''pushout''' (also called a '''fibered coproduct''' or '''fibered sum''' or '''cocartesian square''' or '''amalgamed sum''') is the [[colimit]] of a [[diagram (category theory)\|diagram]] consisting of two [[morphism]]s ''f'' : ''Z'' → ''X'' and ''g'' : ''Z'' → ''Y'' with a common [[Domain of a function\|domain]]: it is the colimit of the [[Span (category theory)\|span]] <math>X \leftarrow Z \rightarrow Y</math>.		Hello, I'm Deneen, a 20 year old from Thurn, Austria.<br>My hobbies include (but are not limited to) Bowling, Exhibition Drill and watching Modern Family.<br><br>My website; [http://www.lareinaroja.com/info.php?a%5B%5D=%3Ca+href%3Dhttp%3A%2F%2Fseojay.zc.bz%2Fyangsamy%2Fguestbook%3Efree+cheats%3C%2Fa%3E Look at this hack]

	~~The pushout is the [[dual (category theory)\|categorical dual]] of the [[pullback (category theory)\|pullback]].~~

	~~==Universal property==~~
	Explicitly, the pushout of the morphisms ''f'' and ''g'' consists of an object ''P'' and two morphisms ''i''<sub>1</sub> : ''X'' → ''P'' and ''i''<sub>2</sub> : ''Y'' → ''P'' for which the following diagram [[commutative diagram\|commutes]]:

	~~:[[Image:Categorical pushout.svg\|125px]]~~

	~~Moreover, the pushout (''P'', ''i''<sub>1</sub>~~, ~~''i''<sub>2</sub>) must be [[universal property\|universal]] with respect to this diagram~~. ~~That is, for any other such set (''Q'', ''j''~~<~~sub~~>~~1</sub>, ''j''<sub>2</sub>) for which the following diagram commutes, there must exist a unique ''u'' : ''P'' → ''Q'' also making the diagram commute:~~

	~~:[[Image:Categorical pushout~~ (~~expanded).svg\|225px]]~~

	~~As with all universal constructions, the pushout, if it exists, is unique up to a unique [[isomorphism]].~~

	~~==Examples of pushouts==~~
	~~Here~~ are some examples of pushouts in familiar categories. Note that in each case, we are only providing a construction of an object in the isomorphism class of pushouts; as mentioned above, there may be other ways to ~~construct it, but they are all equivalent.~~

	1. Suppose that ''X'', ''Y'', and ''Z'' as above are [[Set (mathematics)\|sets]], and that ''f'' : ''Z'' → ''X'' and ''g'' : ''Z'' → ''Y'' are set functions. The pushout of ''f'' and ''g'' is the [[disjoint union]] of ''X'' and ''Y'', where elements sharing a common [[preimage]] (in ''Z'') ~~are identified~~, ~~together with certain morphisms from ''X''~~ and ~~''Y''.~~

	~~2. The construction of [[adjunction space]]s is an example of pushouts in the [[category of topological spaces]]~~. More precisely, if ''Z'' is a [[subspace (topology)\|subspace]] of ''Y'' and ''g'' : ''Z'' → ''Y'' is the [[inclusion map]] we can "glue" ''Y'' to another space ''X'' along ''Z'' using an "attaching map" ''f'' : ''Z'' → ''X''. The result is the adjunction space <~~math~~>~~X \cup_{f} Y~~<~~/math~~> ~~which is just the pushout of ''f'' and ''g''. More generally, all identification spaces may be regarded as pushouts in this way.~~

	~~3. A special case of the above is the [[wedge sum]] or one-point union~~; ~~here we take ''X'' and ''Y'' to be~~ [~~[pointed space]]s and ''Z'' the one-point space. Then the pushout is <math>X \vee Y</math>, the space obtained by gluing the basepoint of ''X'' to the basepoint of ''Y''.~~

	4. In the category of [[abelian group]]s, pushouts can be thought of as "[[direct sum of abelian groups\|direct sum]] with gluing" in the same way we think of adjunction spaces as "[[disjoint union]] with gluing". The zero group is a subgroup of every group, so for any abelian groups ''A'' and ''B'', we have [[homomorphisms]]

	~~:''f'' : 0 → ''A''~~

	~~and~~

	:~~''g'' : 0 → ''B''.~~

	The pushout of these maps is the direct sum of ''A'' and ''B''. Generalizing to the case where ''f'' and ''g'' are arbitrary homomorphisms from a common domain ''Z'', one obtains for the pushout a [[quotient group]] of the direct sum; namely, we [[Modulo (jargon)\|mod out]] by the subgroup consisting of pairs (''f''(''z''),−''g''(''z'')). Thus we have "glued" along the images of ''Z'' under ''f'' and ''g''. A similar trick yields the pushout in the category of ''R''-[[Module (mathematics)\|module]]s for any [[Ring (mathematics)\|ring]] ''R''.

	~~5. In the [[category of groups]], the pushout is called the [[free product with amalgamation]]. It shows up in the [[Seifert–van Kampen theorem]] of [[algebraic topology]] (see below).~~

	6. In '''CRing''', the category of [[commutative rings]] (a full subcategory of the [[category of rings]]), the pushout is given by the [[tensor product]] of rings. In particular, let ''A'', ''B'', and ''C'' be objects (commutative rings with identity) in '''CRing''' and let ''f'' : ''C'' → ''A'' and ''g'' : ''C'' → ''B'' be morphisms (ring [[homomorphisms]]) in '''CRing'''. Then the tensor product,

	~~:<math>A \otimes_{C} B = \left\{\sum_{i \in I} (a_{i},b_{i}) \; \big\| \; a_{i} \in A, b_{i} \in B \right\} \Bigg~~/ ~~\bigg\langle (f(c)a,b) - (a,g(c)b) \; \big\| \; a \in A, b \in B, c \in C \bigg\rangle <~~/~~math>~~

	with the morphisms <math> g': A \rightarrow A \otimes_{C} B </math> and <math> f': B \rightarrow A \otimes_{C} B </math> that satisfy <math> f' \circ g = g' \circ f </math> defines the pushout in '''CRing'''. Since the pushout is the [[colimit]] of a span and the [[pullback]] is the limit of a cospan, we can think of the tensor product of rings and the [[Pullback (category theory)\|fibered product of rings]] (see the examples section) as dual notions to each other.

	~~==Properties==~~
	*Whenever ''A''∪<sub>''C''</sub>''B'' and ''B''∪<sub>''C''</sub>''A'' exist, there is an isomorphism ''A''∪<sub>''C''</sub>''B'' &cong; ''B''∪<sub>''C''</sub>''A''.
	*Whenever the pushout ''A''∪<sub>''A''</sub>''B'' exists, there is an isomorphism ''B'' &cong; ''A''∪<sub>''A''</~~sub>''B'' (this follows from the universal property of the pushout)~~.

	~~==Construction via coproducts and coequalizers==~~
	~~Pushouts are equivalent to coproducts, and coequalizers (if there is an initial object) in the sense that:~~
	* Coproducts are a pushout from the initial object, and the coequalizer of ''f'', ''g'' : ''X'' → ''Y'' is the pushout of [''f'', ''g''] and [1<sub>''X''</sub>, 1<sub>''X''</sub>], so if there are pushouts (and an initial object), then there are coequalizers and coproducts;
	* Pushouts can be constructed from coproducts and coequalizers, as described below (the pushout is the coequalizer of the maps to the coproduct).

	~~All of the above examples may be regarded as special cases of the following very general construction, which works in any category ''C'' satisfying:~~
	* For any objects ''A'' and ''B'' of ''C'', their [[coproduct]] exists in ''C'';
	* For any morphisms ''j'' and ''k'' of ''C'' with the same domain and target, the [[coequalizer]] of ''j'' and ''k'' exists in ''C''.

	In this setup, we obtain the pushout of morphisms ''f'' : ''Z'' → ''X'' and ''g'' : ''Z'' → ''Y'' by first forming the coproduct of the targets ''X'' and ''Y''. We then have two morphisms from ''Z'' to this coproduct. We can either go from ''Z'' to ''X'' via ''f'', then include into the coproduct, or we can go from ''Z'' to ''Y'' via ''g'', then include. The pushout of ''f'' and ''g'' is the coequalizer of these new maps.

	~~==Application: The Seifert–van Kampen theorem~~==
	~~Returning to topology, the [[Seifert–van Kampen theorem]] answers the following question~~. ~~Suppose we have a path-connected space ''X'', covered by path-connected open subspaces ''A'' and ''B'' whose intersection is also path-connected~~. (Assume also that the basepoint * lies in the intersection of ''A'' and ''B''.) If we know the [[fundamental group]]s of ''A'', ''B'', and their intersection ''D'', can we recover the fundamental group of ''X''? The answer is yes, provided we also know the induced homomorphisms
	~~<math>\pi_1(D,) \to \pi_1(A,)</math>~~
	~~and~~
	~~<math>\pi_1(D,) \to \pi_1(B,).</math>~~
	The theorem then says that the fundamental group of ''X'' is the pushout of these two induced maps. Of course, ''X'' is the pushout of the two inclusion maps of ''D'' into ''A'' and ''B''. Thus we may interpret the theorem as confirming that the fundamental group functor preserves pushouts of inclusions. We might expect this to be simplest when ''D'' is [[simply connected]], since then both homomorphisms above have trivial domain. Indeed this is the case, since then the pushout (of groups) reduces to the [[free product]], which is the coproduct in the category of groups. In a most general case we will be speaking of a [[free product with amalgamation]].

	~~There is a detailed exposition of~~ this~~, in a slightly more general setting ([[covering space\|covering]] [[groupoid]]s) in the book by J. P. May listed in the references.~~

	~~==References==~~
	*[[J.P. May\|May, J. P.]] ''A concise course in algebraic topology.'' University of Chicago Press, 1999.
	*:An introduction to categorical approaches to algebraic topology: the focus is on the algebra, and assumes a topological background.

	~~== External links ==~~

	*[http://nlab.mathforge.org/nlab/show/pushout pushout in nLab ]

	~~[[Category:Limits (category theory)]~~]

en>R'n'B: Help needed: which Folding?

2014-01-31T19:16:41Z

Help needed: which Folding?

Show changes

en>AnomieBOT: Dating maintenance tags: {{Lede}}

2012-09-02T01:38:19Z

Dating maintenance tags: {{Lede}}

New page

Absolutely nothing can be more interesting than shopping for pairs of sweet and beautiful Ugg boots for your cute kids. Ugg boots have occur in their unique epitome of style and has brought enormous little ones Uggs collection of boots at your destination. In several shapes and dimensions these lovable pairs are in excellent demand and maintain your youngsters joyful all over the working day. This integrated sheepskin boots are special in layouts and are offered in several shades to enrich your child seems to be. Most at ease footwear with gentle good sole is finest for the chilled wintertime year when it turns into difficult to preserve your tender types heat. This cozy and dampness resistance boots are of fantastic use. With an simplicity to put on and take off they come in strings and laces which not only brighten up their appears to be but give a mark of soothe in severe frost. Ugg boots for little ones can be effortlessly accessed at your close. Even, if you can simply avail these spectacular boots through on the web retailers. Though facilitating payment procedure, these stores will help you to obtain large array of collections at sensible prices. And the greatest profit of them is that they allow you know the real quality of boots with integrated logos which mark their originality by all implies. These real twin-confronted sheepskin higher, comfortable sheepskin included foam insoles, and a gentle flexible outsole gives an greatest convenience encounter to your children while strolling, functioning, skipping, standing or sitting down. Youngsters Uggs are extraordinary footwear's which just about every promising dad and mom look for for their caring birds. With magnificent seems to be they match outfit of your young children and make them look smarter and eye-catching. These authentic Ugg Australian boots comes with various forms and their effortless online availability at retailers make them more interesting to identify few Ugg Bailey button boots in chestnut, black, chocolate, sand, pink, gray and blue. Crafted and manufactured in Australia for many years and many years they are of top quality high-quality and magnificent fashion. Young children do need these stupendous boots for enormous security but make absolutely sure that right before you invest in child Ugg boots for your child they ought to be comfy, sturdy and flexible in mother nature. Uggs has not only marked its dawn with affordable children Uggs boots but also with substantial assortment of Little ones Uggs slippers. Their kinds in slippers selection have supplied a blend of placate and kindness to the growing buds. Availability of neat, economical slippers and scuffs have raised sale of Uggs in all aspects. Youngsters Ugg boots, slippers and scuff have get attention of the purchasers and lifted its value. Bagged for quality and truly worth, young children Uggs sale has immensely increased and touched the sky.Just before you immediately choose for these great pairs of boots you need to check out their authenticity by all means. Even if you make on-line purchase there are specific parameter that will need to be cross checked. On-line buy gives you an immense facility to avail these boots by conserving your important time and provided you wide info on its a variety of new and current collections. But make confident that you go for the appropriate supplier or manufacturer. This well known Ugg Australian Brand is created by Deckers Corporation and is designed in New Zealand, Australia and far more lately in China. With a substantial middle G on the brand as noticed on the Workplace web page you can test its legitimacy. Do track down their formal web-sites or you can also go for Uggs boot authorized sellers. If you come across your self at confusion state and want to check out that a retailer is legit or not, electronic mail Deckers at and check with no matter if the retailer is accredited to market UGG Australian boots or not. This would quite nicely resolve your challenge and assures you with the investment decision to be accomplished. If you beloved this report and you would like to obtain far more details concerning [http://tinyurl.com/k7shbtq uggs on sale] kindly stop by the web page.