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| {{Redirect|Isometric embedding|related concepts for [[metric space]]s|isometry}}
| | Rabbits make superb pets and can be our companions for a lot of many years. With the right care and dealing with you can encourage your rabbit to be pleasant, enjoying time on your lap to be stroked or groomed. Most rabbits simply adore hay and will consume it continually all day lengthy, nevertheless some bunnies just will not seem to be to want to take in it. Your rabbit's diet plan should consist of about eighty% hay, so if he is not eating it, it can lead to issues this sort of as overgrown enamel and a sluggish digestive transit. It is critical to inspire him to commence eating his hay as it will help to maintain him fit and nicely.<br>Hello fellas, this guide is all about how to obtain Hay Day for Pc Recreation available for Windows seven/8/XP Computer, please keep on studying to know more. Investigation has proven that high sugar diets fall pH in a horse's system. Already prone and inclined to ulcers, the horse needs all the help he can get. Hay is at a pH of vinegar and can have a substantial sugar material. By soaking the hay for a single hour, you reduce the sugar amounts in the hay and help neutralize the pH. The merged business will not adjust the Hay Day identify, but it will replace the firm's computers and update its application.<br>These eye drops does not handle indicators directly or right away. The drops are utilised to stop the allergy symptoms from happening. The treatment stops cells in your eyes from releasing the chemicals that make them itch and operate. So you have to use them on a day-to-day basis, commencing from ahead of the time of yr you typically get eye allergic reactions. If you are already obtaining signs, it will get a few of weeks of use for you to know if this is likely to aid you. It is worth striving, simply because it is really fully protected. It truly is well worth giving it a two to four 7 days try out.<br>There is a good deal of exciting associated although taking part in mobile game titles at the comfort and ease of one’s home. This is particularly when it arrives to Hay day. This is an Iphone and iPad application that folks use to play and earn some cash. A great deal of benefits are included specially when 1 gets to be used to the game such as obtaining bonuses and a good deal of prizes.In buy to play Hay working day cheats, 1 has to download the software from the Application keep and install it. Downloading and installation normally takes a few minutes given that the software is really modest in measurement.<br>Hay Day is a farming based game in which you have to do all as usual issues which folks do in the farming. In situation you are taking part in it on your Android system then make certain the OS model is at minimum four. and increased in any other case the match will not support your gadget. Hay Working day performs very best on Jelly Bean and KitKat operating products. The only problem with feeding straw is that it does not incorporate significantly protein. Oat straw consists of somewhat more protein than other varieties of straw but it is still not adequate to substitute hay altogether. It is very best fed mixed fifty/50 with hay<br><br>If you beloved this article and you simply would like to acquire more info concerning [http://i-showad.com/xe/?document_srl=1876212 hay day cheats] i implore you to visit our own website. |
| {{Other uses}}
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| In [[mathematics]], an '''embedding''' (or '''imbedding''') is one instance of some [[mathematical structure]] contained within another instance, such as a [[group (mathematics)|group]] that is a [[subgroup]].
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| When some object ''X'' is said to be embedded in another object ''Y'', the embedding is given by some [[injective]] and structure-preserving map {{nowrap|''f'' : ''X'' → ''Y''}}. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which ''X'' and ''Y'' are instances. In the terminology of [[category theory]], a structure-preserving map is called a [[morphism]].
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| The fact that a map {{nowrap|''f'' : ''X'' → ''Y''}} is an embedding is often indicated by the use of a "hooked arrow", thus: <math> f : X \hookrightarrow Y.</math> On the other hand, this notation is sometimes reserved for [[inclusion map]]s.
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| Given ''X'' and ''Y'', several different embeddings of ''X'' in ''Y'' may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the [[natural number]]s in the [[integer]]s, the integers in the [[rational number]]s, the rational numbers in the [[real number]]s, and the real numbers in the [[complex number]]s. In such cases it is common to identify the [[Domain (mathematics)|domain]] ''X'' with its [[image (mathematics)|image]] ''f''(''X'') contained in ''Y'', so that then {{nowrap|''X'' ⊆ ''Y''}}.
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| ==Topology and geometry==
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| ===General topology===
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| In [[general topology]], an embedding is a [[homeomorphism]] onto its image.<ref>{{citation| first = R.W. | last = Sharpe | title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher = Springer-Verlag, New York | year = 1997| isbn = 0-387-94732-9}}, page 16.
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| </ref> More explicitly, an [[injective]] [[continuous function (topology)|continuous]] map ''f'' : ''X'' → ''Y'' between [[topological space]]s ''X'' and ''Y'' is a '''topological embedding''' if ''f'' yields a homeomorphism between ''X'' and ''f''(''X'') (where ''f''(''X'') carries the [[topological subspace|subspace topology]] inherited from ''Y''). Intuitively then, the embedding ''f'' : ''X'' → ''Y'' lets us treat ''X'' as a [[topological subspace|subspace]] of ''Y''. Every embedding is [[injective]] and [[continuous function (topology)|continuous]]. Every map that is injective, continuous and either [[open map|open]] or [[closed map|closed]] is an embedding; however there are also embeddings which are neither open nor closed. The latter happens if the image ''f''(''X'') is neither an [[open set]] nor a [[closed set]] in ''Y''.
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| For a given space X, the existence of an embedding X → Y is a [[topological invariant]] of X. This allows two spaces to be distinguished if one is able to be embedded into a space while the other is not.
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| ===Differential topology===
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| In [[differential topology]]:
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| Let ''M'' and ''N'' be smooth [[manifold]]s and <math>f:M\to N</math> be a smooth map. Then ''f'' is called an [[immersion (mathematics)|immersion]] if its [[pushforward (differential)|derivative]] is everywhere injective. An '''embedding''', or a '''smooth embedding''', is defined to be an injective immersion which is an embedding in the topological sense mentioned above (i.e. [[homeomorphism]] onto its image).<ref>{{citation| first = F.W. | last = Warner | title = Foundations of Differentiable Manifolds and Lie Groups | publisher = Springer-Verlag, New York | year = 1983| isbn = 0-387-90894-3}}, page 22.</ref>
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| In other words, an embedding is [[diffeomorphism|diffeomorphic]] to its image, and in particular the image of an embedding must be a [[submanifold]]. An immersion is a local embedding (i.e. for any point <math>x\in M</math> there is a neighborhood <math>x\in U\subset M</math> such that <math>f:U\to N</math> is an embedding.)
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| When the domain manifold is compact, the notion of a smooth embedding is equivalent to that of an injective immersion.
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| An important case is ''N''='''R'''<sup>n</sup>. The interest here is in how large ''n'' must be, in terms of the dimension ''m'' of ''M''. The [[Whitney embedding theorem]] <ref>Whitney H., ''Differentiable manifolds,'' Ann. of Math. (2), '''37''' (1936), 645-680.</ref> states that ''n'' = 2''m'' is enough, and is the best possible linear bound. For example the [[real projective plane]] of dimension m requires ''n'' = 2m for an embedding. An immersion of this surface is, however, possible in '''R'''<sup>3</sup>, and one example is [[Boy's surface]]—which has self-intersections. The [[Roman surface]] fails to be an immersion as it contains [[cross-cap]]s.
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| An embedding is '''proper''' if it behaves well [[List_of_mathematical_abbreviations|w.r.t.]] [[Topological_manifold#Manifolds_with_boundary|boundaries]]: one requires the map <math>f: X \rightarrow Y</math> to be such that
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| *<math>f(\partial X) = f(X) \cap \partial Y</math>, and
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| *<math>f(X)</math> is [[Transversality (mathematics)|transversal]] to <math>\partial Y</math> in any point of <math>f(\partial X)</math>.
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| The first condition is equivalent to having <math>f(\partial X) \subseteq \partial Y</math> and <math>f(X \setminus \partial X) \subseteq Y \setminus \partial Y</math>. The second condition, roughly speaking, says that f(X) is not tangent to the boundary of Y.
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| ===Riemannian geometry===
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| In [[Riemannian geometry]]:
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| Let (''M,g'') and (''N,h'') be [[Riemannian manifold]]s.
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| An '''isometric embedding''' is a smooth embedding ''f'' : ''M'' → ''N'' which preserves the [[Riemannian metric|metric]] in the sense that ''g'' is equal to the [[pullback (differential geometry)|pullback]] of ''h'' by ''f'', i.e. ''g'' = ''f''*''h''. Explicitly, for any two tangent vectors
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| :<math>v,w\in T_x(M)</math>
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| we have
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| :<math>g(v,w)=h(df(v),df(w)).\,</math>
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| Analogously, '''isometric immersion''' is an immersion between Riemannian manifolds which preserves the Riemannian metrics.
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| Equivalently, an isometric embedding (immersion) is a smooth embedding (immersion) which preserves length of [[curve]]s (cf. [[Nash embedding theorem]]).<ref>Nash J., ''The embedding problem for Riemannian manifolds,'' Ann. of Math. (2), '''63''' (1956), 20-63.</ref>
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| ==Algebra==
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| In general, for an algebraic category ''C'', an embedding between two ''C''-algebraic structures ''X'' and ''Y'' is a ''C''-morphism ''e:X→Y'' which is injective.
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| ===Field theory===
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| In [[field theory (mathematics)|field theory]], an '''embedding''' of a [[field (mathematics)|field]] ''E'' in a field ''F'' is a [[ring homomorphism]] σ : ''E'' → ''F''.
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| The [[Kernel (algebra)|kernel]] of σ is an [[ideal (ring theory)|ideal]] of ''E'' which cannot be the whole field ''E'', because of the condition σ(1)=1. Furthermore, it is a well-known property of fields that their only ideals are the zero ideal and the whole field itself. Therefore, the kernel is 0, so any embedding of fields is a [[monomorphism]]. Hence, ''E'' is [[isomorphic]] to the subfield σ(''E'') of ''F''. This justifies the name ''embedding'' for an arbitrary homomorphism of fields.
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| ===Universal algebra and model theory===
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| {{further2|[[Substructure]] and [[Elementary equivalence]]}}
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| If σ is a [[signature (logic)|signature]] and <math>A,B</math> are σ-[[structure (mathematical logic)|structures]] (also called σ-algebras in [[universal algebra]] or models in [[model theory]]), then a map <math>h:A \to B</math> is a σ-embedding [[iff]] all the following holds:
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| * <math>h</math> is [[injective]],
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| * for every <math>n</math>-ary function symbol <math>f \in\sigma</math> and <math>a_1,\ldots,a_n \in A^n,</math> we have <math>h(f^A(a_1,\ldots,a_n))=f^B(h(a_1),\ldots,h(a_n))</math>,
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| * for every <math>n</math>-ary relation symbol <math>R \in\sigma</math> and <math>a_1,\ldots,a_n \in A^n,</math> we have <math>A \models R(a_1,\ldots,a_n)</math> iff <math>B \models R(h(a_1),\ldots,h(a_n)).</math>
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| Here <math>A\models R (a_1,\ldots,a_n)</math> is a model theoretical notation equivalent to <math>(a_1,\ldots,a_n)\in R^A</math>. In model theory there is also a stronger notion of [[elementary embedding]].
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| ==Order theory and domain theory==
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| In [[order theory]], an embedding of [[partial order]]s is a function F from X to Y such that:
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| :<math>\forall x_1,x_2\in X: x_1\leq x_2\Leftrightarrow F(x_1)\leq F(x_2)</math>.
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| In [[domain theory]], an additional requirement is:
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| :<math> \forall y\in Y:\{x: F(x)\leq y\}</math> is [[Directed set|directed]].
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| ==Metric spaces==
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| A mapping <math>\phi: X \to Y</math> of [[metric spaces]] is called an ''embedding''
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| (with distortion <math>C>0</math>) if
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| :<math> L d_X(x, y) \leq d_Y(\phi(x), \phi(y)) \leq CLd_X(x,y) </math>
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| for some constant <math>L>0</math>.
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| === Normed spaces ===
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| An important special case is that of [[normed spaces]]; in this case it is natural to consider linear embeddings.
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| One of the basic questions that can be asked about a finite-dimensional [[normed space]] <math>(X, \| \cdot \|)</math> is, ''what is the maximal dimension <math>k</math> such that the [[Hilbert space]] <math>\ell_2^k</math> can be linearly embedded into <math>X</math> with constant distortion?''
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| The answer is given by [[Dvoretzky's theorem]].
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| ==Category theory==
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| In [[category theory]], there is no satisfactory and generally accepted definition of embeddings that is applicable in all categories. One would expect that all isomorphisms and all compositions of embeddings are embeddings, and that all embeddings are monomorphisms. Other typical requirements are: any [[monomorphism|extremal monomorphism]] is an embedding and embeddings are stable under [[Pullback (category theory)|pullback]]s.
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| Ideally the class of all embedded [[subobject]]s of a given object, up to isomorphism, should also be [[small class|small]], and thus an [[ordered set]]. In this case, the category is said to be well powered with respect to the class of embeddings. This allows to define new local structures on the category (such as a [[closure operator]]).
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| In a [[concrete category]], an '''embedding''' is a morphism ''ƒ'': ''A'' → ''B'' which is an [[injective function]] from the underlying set of ''A'' to the underlying set of ''B'' and is also an '''initial morphism''' in the following sense: | |
| If ''g'' is a function from the underlying set of an object ''C'' to the underlying set of ''A'', and if its composition with ''ƒ'' is a morphism ''ƒg'': ''C'' → ''B'', then ''g'' itself is a morphism.
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| A [[factorization system]] for a category also gives rise to a notion of embedding. If (''E'', ''M'') is a factorization system, then the morphisms in ''M'' may be regarded as the embeddings, especially when the category is well powered with respect to ''M''. Concrete theories often have a factorization system in which ''M'' consists of the embeddings in the previous sense. This is the case of the majority of the examples given in this article.
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| As usual in category theory, there is a [[dual (category theory)|dual]] concept, known as quotient. All the preceding properties can be dualized.
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| An embedding can also refer to an [[Subcategory#Embeddings|embedding functor]].
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| ==See also==
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| *[[Cover (algebra)|Cover]]
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| *[[Immersion (mathematics)|Immersion]]
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| *[[Submanifold]]
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| *[[Subspace (topology)|Subspace]]
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| *[[Closed immersion]]
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| *[[Johnson–Lindenstrauss lemma]]
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| *[[Dimension reduction]]
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| ==Notes==
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| {{reflist}}
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| == References ==
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| * {{citation| first = R.W. | last = Sharpe | title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher = Springer-Verlag, New York | year = 1997| isbn = 0-387-94732-9}}.
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| * {{citation| first = F.W. | last = Warner | title = Foundations of Differentiable Manifolds and Lie Groups | publisher = Springer-Verlag, New York | year = 1983| isbn = 0-387-90894-3}}.
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| == External links ==
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| *{{cite book|last=Adámek|first=Jiří|coauthors=Horst Herrlich, George Strecker|title=Abstract and Concrete Categories (The Joy of Cats)|url=http://katmat.math.uni-bremen.de/acc/|year=2006}}
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| * [http://www.map.mpim-bonn.mpg.de/Embedding Embedding of manifolds] on the Manifold Atlas
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| [[Category:Abstract algebra]]
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| [[Category:Category theory]]
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| [[Category:General topology]]
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| [[Category:Differential topology]]
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| [[Category:Functions and mappings]]
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| [[Category:Maps of manifolds]]
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| [[Category:Model theory]]
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| [[Category:Order theory]]
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Hello fellas, this guide is all about how to obtain Hay Day for Pc Recreation available for Windows seven/8/XP Computer, please keep on studying to know more. Investigation has proven that high sugar diets fall pH in a horse's system. Already prone and inclined to ulcers, the horse needs all the help he can get. Hay is at a pH of vinegar and can have a substantial sugar material. By soaking the hay for a single hour, you reduce the sugar amounts in the hay and help neutralize the pH. The merged business will not adjust the Hay Day identify, but it will replace the firm's computers and update its application.
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