Domain (ring theory): Difference between revisions

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In mathematics and mathematical economics, '''correspondence''' is a term with several related but not identical meanings.
 
* In general [[mathematics]], a '''correspondence''' is an ordered triple (''X'',''Y'',''R''), where ''R'' is a [[binary relation|relation]] from ''X'' to ''Y'', i.e. any [[subset]] of the [[Cartesian product]] ''X''×''Y''.<ref>{{cite book |title=Encyclopedic dictionary of Mathematics |publisher=MIT |pages=1330–1331 |year=2000 |isbn=0-262-59020-4 |url=http://books.google.co.uk/books?id=azS2ktxrz3EC&pg=PA1331#v=onepage&f=false }}</ref>
 
{{anchor|algebraic geometry}}
* In [[algebraic geometry]], a '''correspondence''' between [[algebraic variety|algebraic varieties]] ''V'' and ''W'' is in the same fashion a subset ''R'' of ''V''&times;''W'', which is in addition required to be closed in the [[Zariski topology]]. It therefore means any relation that is defined by algebraic equations. There are some important examples, even when ''V'' and ''W'' are [[algebraic curve]]s: for example the [[Hecke operator]]s of [[modular form]] theory may be considered as correspondences of [[modular curve]]s.
 
:However, the definition of a correspondence in algebraic geometry is not completely standard. For instance, Fulton, in his book on [[Intersection theory]],<ref>{{Citation | last1=Fulton | first1=William | author1-link=William Fulton (mathematician) | title=Intersection theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] | isbn=978-0-387-98549-7 | mr=1644323  | year=1998 | volume=2}}</ref> uses the definition above. In literature, however, a correspondence from a variety ''X'' to a variety ''Y'' is often taken to be a subset ''Z'' of ''X''&times;''Y'' such that ''Z'' is finite and surjective over each component of ''X''. Note the asymmetry in this latter definition; which talks about a correspondence from ''X'' to ''Y'' rather than a correspondence between ''X'' and ''Y''. The typical example of the latter kind of correspondence is the graph of a function f:''X''&rarr;''Y''. Correspondences also play an important role in the construction of [[motive (algebraic geometry)|motives]].<ref>{{Citation | last1=Mazza | first1=Carlo | last2=Voevodsky | first2=Vladimir | author2-link=Vladimir Voevodsky | last3=Weibel | first3=Charles | title=Lecture notes on motivic cohomology | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Clay Mathematics Monographs | isbn=978-0-8218-3847-1 | mr=2242284  | year=2006 | volume=2}}</ref>
 
* In [[category theory]], a '''correspondence''' from <math> C </math> to <math> D </math> is a functor <math>C^{op}\times D\to\mathbf{Set}</math>. It is the "opposite" of a [[profunctor]].
 
* '''''One-to-one correspondence''''' is an alternate name for a [[bijection]].
 
* In von Neumann algebra theory, a '''correspondence''' is a synonym for a [[von Neumann algebra]] bimodule.
 
* In [[economics]], a '''correspondence''' between two sets ''A'' and ''B'' is a [[map (mathematics)|map]] f:''A''→''P''(''B'') from the elements of the set ''A'' to the [[power set]]  of ''B''. This is similar to a correspondence as defined in general mathematics (i.e., a [[Relation (mathematics)|relation]],) except that the range is over sets instead of elements.  However, there is usually the additional property that for all ''a'' in ''A'', ''f''(''a'') is not empty. In other words, each element in ''A'' maps to a non-empty subset of ''B''; or in terms of a relation ''R'' as subset of ''A''&times;''B'', ''R'' projects to ''A'' [[surjective]]ly. A correspondence with this additional property is thought of as the generalization of a [[function (mathematics)|function]], rather than as a special case of a relation, and is referred to in other contexts as a [[multivalued function]].
 
:An example of a correspondence in this sense is the [[best response]] correspondence in [[game theory]], which gives the optimal action for a player as a function of the strategies of all other players. If there is always a unique best action given what the other players are doing, then this is a function.  If for some opponent's strategy, there is a set of best responses that are equally good, then this is a correspondence.
 
==References==
<references />
 
[[Category:Mathematical terminology]]

Latest revision as of 19:08, 12 January 2015

Richard Gallo, chair of the dermatology department at the University of California, sensitive people could be irritated by or have an allergic reaction to toothpaste. Removing lipids from the surface of the skin prevents clogged pores and the subsequent development of pimples. So, the advantage in using homemade acne face mask on your skin will be safer for you and your health. Just sprinkle a little baking soda in the palm of your hand, dampen your toothbrush and press the bristles into the baking soda, add toothpaste and brush your teeth as usual. Very low levels of melanin can cause problems too; and that's why it works towards striking just the right balance.

The tasteless water could very well help in improving your complexion. Carbamide peroxide is the most commonly used in gels and can be found in some more effective whitening toothpastes as well, while hydrogen peroxide is typically used in whitening strips and whitening pens. Proper dental hygiene can go a long way, but sometimes it isn't enough to get the white smile that is so desirable to complement a glowing summer complexion. Your dreams to a dazzling smile can now be easily fulfilled with the help of a home kit that promises to whiten teeth. Laser technology is probably the latest invention in the field of skin whitening.

Research thoroughly all products associated with skin lightening by reading up on each product including what's in it, how it actually works to solve your skin problems, adverse effects as well as the maintaining aspects and other qualities of the products. Using layers and selection tools may seem more like work than fun to some, but if you really want to experience the full power of Paint - Shop Pro, it's a good idea to learn as much about these features as you can. To minimize that look, apply a moisturizer to your face that is formulated for your skin type. According to these colors and to the type of skin you have certain products are recommended and certain are prohibited. They work simply by ramping way up mobile or portable return costs.

It is a more expensive treatment that is usually offered only by offices that specialize in cosmetic dentistry, but is a great option for quick results. Adverse reactions include postinflammatory hyperpigmentation, erythema, pruritus, milia formation, loss of skin texture, and scarring. These pour once empties gives way for the fresh air to go inside cells of skin. As delicious as they are, these food items are deemed to be the enemies of your skin. Other individuals choose laser treatments or surgical treatment which might provide instantaneous color change but are expensive.

Once you have identified the allergy you may have and start to treat it, you may find that your circles will begin to fade. After makeing a paste with these ingredients, apply it on your skin. Some over the counter face whitening creams can cause damage to your face if misused. Skin bleaching is the practice of applying specific chemical substances or natural recipes to make the skin tone lighter or rather make it even. Exposure to Ultraviolet radiation is also cause of skin tanning.

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