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In [[mathematics]], '''differential rings''', '''differential fields''', and '''differential algebras''' are [[ring (mathematics)|rings]], [[field (mathematics)|fields]], and [[algebra over a field|algebras]] equipped with a  [[derivation (abstract algebra)|derivation]], which is a [[Unary operation|unary]] function that is [[linear]] and satisfies the [[Product rule|Leibniz product rule]].  A natural example of a differential field is the field of [[rational function]]s ''C''(''t'') in one variable, over the [[complex number]]s, where the derivation is differentiation with respect to ''t''.
== 'goods in what place ==


'''Differential algebra''' refers also to the area of mathematics consisting in the study of these algebraic objects and their use for an algebraic study of the differential equations. Differential algebra has essentially been introduced by [[Joseph Ritt]].
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==Differential ring==
== good leadership.' ==
A ''differential ring'' is a ring ''R'' equipped with one or more ''derivations'', that is additive [[homomorphisms]]


:<math>\partial:R \to R\,</math>
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such that each derivation ∂ satisfies the [[product rule|Leibniz product rule]]
== drug trafficking and sent her to the camp ==


:<math>\partial(r_1 r_2)=(\partial r_1) r_2 + r_1 (\partial r_2),\,</math>
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for every <math>r_1, r_2 \in R</math>.  Note that the ring could be noncommutative, so the somewhat standard ''d(xy) = xdy + ydx'' form of the product rule in commutative settings may be false.  If <math>M:R \times R \to R</math> is multiplication on the ring, the product rule is the identity
== Lee Dong-laws coauthored two ==


:<math>\partial \circ M =  
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M \circ (\partial \times \operatorname{id}) +
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M \circ (\operatorname{id} \times \partial). </math>
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where <math>f\times g</math> means the function which maps a pair <math>(x,y)</math> to the pair <math>(f(x),g(y))</math>.
== did not give you the backing of the brothers ==


==Differential field==
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A differential field is a field ''K'', together with a derivation.  The theory of differential fields, ''DF'', is given by the usual field axioms along with two extra axioms involving the derivation.  As above, the derivation must obey the [[product rule]], or Leibniz rule over the elements of the field, to be worthy of being called a derivation. That is, for any two elements ''u'', ''v'' of the field, one has
== mother and daughter there is dependency ==


:<math>\partial(uv) = u \,\partial v + v\, \partial u</math>
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since multiplication on the field is commutative. The derivation must also be distributive over addition in the field:
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:<math>\partial (u + v) = \partial u + \partial v\ .</math>
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If ''K'' is a differential field then ''the field of constants'' <math> k = \{u \in K : \partial(u) = 0\}.</math>
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==Differential algebra==
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A differential algebra over a field ''K'' is a ''K''-algebra ''A'' wherein the derivation(s) commutes with the field. That is, for all <math>k \in K</math> and <math>x \in A</math> one has
 
 
</ul>
:<math>\partial (kx) = k \partial x</math>
 
In [[index-free notation]], if <math>\eta \colon K\to A</math> is the [[ring morphism]] defining scalar multiplication on the algebra, one has
 
:<math>\partial \circ M \circ (\eta \times \operatorname{Id}) =
M \circ (\eta \times \partial)</math>
 
As above, the derivation must obey the Leibniz rule over the algebra multiplication, and must be linear over addition. Thus, for all <math>a,b \in K</math> and <math>x,y \in A</math> one has
 
:<math>\partial (xy) = (\partial x) y + x(\partial y)</math>
 
and
 
:<math>\partial (ax+by) = a\,\partial x + b\,\partial y.</math>
 
== Derivation on a Lie algebra ==
A derivation on a [[Lie algebra]] <math>\mathfrak{g}</math> is a linear map <math>D \colon \mathfrak{g} \to \mathfrak{g}</math> satisfying the Leibniz rule:
 
:<math>D([a,b]) = [a,D(b)] + [D(a),b]</math>
 
For any <math>a \in \mathfrak{g}</math>, ad(''a'') is a derivation on <math>\mathfrak{g}</math>, which follows from the [[Jacobi identity]]. Any such derivation is called an '''inner derivation'''.
 
== Examples ==
 
If <math>A</math> is [[unital algebra|unital]], then ∂(1) = 0 since ∂(1) = ∂(1 &times; 1) = ∂(1) + ∂(1).  For example, in a differential field of characteristic zero the rationals are always a subfield of the constant field.
 
Any field pure can be interpreted as a constant differential field. 
 
The field  '''Q'''(''t'') has a unique structure as a differential field, determined by setting  ∂(''t'') = 1:  the field axioms along with the axioms for derivations  ensure that the derivation is differentiation with respect to ''t''.  For example, by commutativity of multiplication and the Leibniz law one has that ∂(''u''<sup>2</sup>) = ''u'' ∂(''u'') + ∂(''u'')''u''= 2''u''∂(''u'').
 
The differential field '''Q'''(''t'') fails to have a solution to the differential equation
 
:<math> \partial(u) = u </math>
 
but expands to a larger differential field including the function ''e''<sup>''t''</sup> which does have a solution to this equation.
A differential field with solutions to all systems of differential equations is called a [[differentially closed field]]. Such fields exist, although they do not appear as natural algebraic or geometric objects. All differential fields (of bounded cardinality) embed into a large differentially closed field.  Differential fields are the objects of study in [[differential Galois theory]].
 
Naturally occurring examples of derivations are [[partial derivative]]s, [[Lie derivative]]s, the [[Pincherle derivative]], and the [[commutator]] with respect to an element of the algebra. All these examples are tightly related, with the concept of derivation as the major unifying theme.
 
==Ring of pseudo-differential operators==
Differential rings and differential algebras are often studied by means of the ring of [[pseudo-differential operator]]s on them.
 
This is the ring
 
:<math>R((\xi^{-1})) = \left\{ \sum_{n<\infty} r_n \xi^n | r_n \in R \right\}.</math>
 
Multiplication on this ring is defined as
 
:<math>(r\xi^m)(s\xi^n) =  
\sum_{k=0}^m r (\partial^k s) {m \choose k} \xi^{m+n-k}.</math>
 
Here <math>{m \choose k}</math> is the [[binomial coefficient]]. Note the identities
 
:<math>\xi^{-1} r = \sum_{n=0}^\infty (-1)^n (\partial^n r) \xi^{-1-n}</math>
 
which makes use of the identity
 
:<math>{-1 \choose n} = (-1)^n</math>
 
and
 
:<math>r \xi^{-1} = \sum_{n=0}^\infty \xi^{-1-n} (\partial^n r).</math>
 
==See also==
 
*[[Differential Galois theory]]
*[[Kähler differential]]
*[[Differentially closed field]]
* A [[D-module]] is an algebraic structure with several differential operators acting on it.
* A [[differential graded algebra]] is a differential algebra with an additional grading.
*[[Arithmetic derivative]]
*[[Differential calculus over commutative algebras]]
*[[Difference algebra]]
*[[Differential algebraic geometry]]
*[[Picard–Vessiot theory]]
 
==References==
 
* Buium, ''Differential Algebra and Diophantine Geometry'', Hermann (1994).
* I. Kaplansky, ''Differential Algebra'', Hermann (1957).
* [[E. Kolchin]], ''Differential Algebra and Algebraic Groups,'' 1973
* D. Marker, Model theory of differential fields, ''Model theory of fields'', Lecture notes in Logic 5, D. Marker, M. Messmer and A. Pillay, Springer Verlag (1996).
* A. Magid, ''Lectures on Differential Galois Theory,''  American Math. Soc., 1994
 
==External links==
* [http://www.math.uic.edu/~marker/ David Marker's home page] has several online surveys  discussing differential fields.
 
[[Category:Differential algebra| ]]

Latest revision as of 14:01, 22 November 2014

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mother and daughter there is dependency

However, an oil rose heart fear, like a child who skip class, mischief, fighting and smashing glass man was stabbed to kate spade マザーズバッグ all sorts of bad things there dad, he always tens of meters away from a place outside hesitate undecided, back a Breaking butt bigger than bags, cocked his head to worry about.

now gone as the burden of school bags, that can be a psychological burden I did not eliminate the crime only to find today, perhaps too many childhood provoke angry father's sake, do not want to look after his father always kind of frantic, furious look, others can not understand this lack of feeling of single-parent families, more than the crime of outsiders is many ケイトスペード マザーズバッグ years later, to see Dad pains one ハンドバッグ ケイトスペード point a hair is not easy to make money, to see him around to ask for things that make life difficult, slowly understood.

if so to say, in fact, Yaya is also to be understood here is accompanied by father and son, mother and daughter ケイトスペードの財布 there is dependency, always worried about the 財布 ケイトスペード kind of profound feelings of feeling, because an outsider intervention and disappear. He was worried as he has such a stepmother aunt took his love, like, maybe even more worried after Yaya dad took her a profiteer 相关的主题文章: