formulasearchengine - User contributions [en]

Spectral space

2013-12-02T10:22:49Z

159.149.107.58: /* Spectral maps */

In [[mathematics]], a '''bicategory''' is a concept in [[category theory]] used to extend the notion of [[Category (mathematics)|category]] to handle the cases where the composition of morphisms is not (strictly) [[associative]], but only associative ''[[up to]]'' an isomorphism. The notion was introduced in 1967 by [[Jean Bénabou]].

Formally, a bicategory '''B''' consists of:
* objects ''a'', ''b''... called ''0-cells'';
* morphisms ''f'', ''g'', ... with fixed source and target objects called ''1-cells'';
* "morphisms between morphisms" ρ, σ... with fixed source and target morphisms (which should have themselves the same source and the same target), called ''2-cells'';
with some more structure:
* given two objects ''a'' and ''b'' there is a category '''B'''(''a'', ''b'') whose objects are the 1-cells and morphisms are the 2-cells, the composition in this category is called ''vertical composition'';
* given three objects ''a'', ''b'' and ''c'', there is a bifunctor <math>*:\mathbf{B}(b,c)\times\mathbf{B}(a,b)\to\mathbf{B}(a,c)</math> called ''horizontal composition''.
The horizontal composition is required to be associative up to a natural isomorphism α between morphisms <math>h*(g*f)</math> and <math>(h*g)*f</math>. Some more coherence axioms, similar to those needed for [[monoidal category|monoidal categories]], are moreover required to hold.

Bicategories may be considered as a weakening of the definition of [[2-categories]]. A similar process for 3-categories leads to [[tricategory|tricategories]], and more generally to [[weak n-category|weak ''n''-categories]] for [[n-category|''n''-categories]].

== References ==
* J. Bénabou. "Introduction to bicategories, part I". In ''Reports of the Midwest Category Seminar'', Lecture Notes in Mathematics 47, pages 1-77. Springer, 1967.

[[Category:Higher category theory]]

Measure-preserving dynamical system

2013-11-12T12:45:10Z

159.149.2.252: Aspect modifications

[[File:Na+H2O.svg|thumb|right|The first solvation shell of a [[sodium]] [[ion]] dissolved in water]]
A '''solvation shell''', in [[physical chemistry]], is a structure used to describe [[solvation]]. A solvation shell is a shell of any [[chemical species]] that acts as a [[solvent]] and surrounds a [[Solution|solute]] species. When the solvent is [[water]] it is often referred to as a '''hydration shell''' or '''hydration sphere'''.

A classic example is when water molecules form a sphere around a metal ion. The [[electronegative]] oxygen atom contained in the water molecule is attracted electrostatically to the positive charge on the metal ion. The result is a solvation shell of water molecules that surround the ion. This shell can be several molecules thick, dependent upon the charge of the ion.

==Solvation number==
The solvation number could be determined from the [[compressibility]] of the solvent and solution.

<math>n_s = (1 - \frac {\beta}{\beta_0}) \frac{n_i}{n_0}</math>

where <math>\beta</math> and <math>\beta_0</math> denote compressibilities of solution and the solvent and <math>n_i</math> and <math>n_0</math> the amounts of solvate and solvent.

==Hydration shells of proteins==

The hydration shell (also sometimes called hydration layer) that forms around proteins is of particular importance in biochemistry. This interaction of the protein surface with the surrounding water is often referred to as protein hydration and is fundamental to the activity of the protein.<ref name="Mapping hydration dynamics">{{cite doi|10.1073/pnas.0707647104}}</ref> The hydration layer around a protein has been found to have dynamics distinct from the bulk water to a distance of 1 nm with effects on the surrounding water network extending beyond 2 nm.<ref>{{cite doi|10.1073/pnas.0709207104}}</ref> The duration of contact of a specific water molecule with the protein surface may be in the subnanosecond range while [[molecular dynamics]] simulations suggest the time water spends in the hydration shell before mixing with the outside bulk water could be in the femtosecond to picosecond range.<ref name="Mapping hydration dynamics" />

With other solvents and solutes, varying steric and kinetic factors can also affect the solvation shell.

==See also==
*[[Water model]]
*[[Poisson-Boltzmann equation]]

==References==
<references />

==External links==
* [http://www1.lsbu.ac.uk/water/protein.html London South Bank University pages on protein hydration]
* [http://jcp.aip.org/resource/1/jcpsa6/v36/i12/p3382_s1?isAuthorized=no Journal of Chemical Physics]
{{Chemical solutions}}

{{DEFAULTSORT:Solvation Shell}}
[[Category:Solutions]]

{{physical-chemistry-stub}}

Grothendieck local duality

2013-06-21T13:37:46Z

159.149.2.252:

{{Infobox enzyme
| Name = Dimethylglycine N-methyltransferase
| EC_number = 2.1.1.161
| CAS_number =
| IUBMB_EC_number = 2/1/1/161
| GO_code =
| image =
| width =
| caption =
}}
'''Dimethylglycine N-methyltransferase''' ({{EC number|2.1.1.161}}, ''BsmB'', ''DMT'') is an [[enzyme]] with system name ''S-adenosyl-L-methionine:N,N-dimethylglycine N-methyltransferase (betaine-forming)''.<ref>{{cite journal | title = Identification of glycine betaine as compatible solute in Synechococcus sp. WH8102 and characterization of its N-methyltransferase genes involved in betaine synthesis |author = Lu, W.D., Chi, Z.M. and Su, C.D. |journal = Arch. Microbiol. |date = 2006 |volume = 186 |pages = 495–506 |pmid = 17019606 | doi = 10.1007/s00203-006-0167-8 | issue = 6}}</ref> This enzyme [[catalysis|catalyses]] the following [[chemical reaction]]

: [[S-adenosyl-L-methionine]] + [[N,N-dimethylglycine]] <math>\rightleftharpoons</math> [[S-adenosyl-L-homocysteine]] + [[betaine]]

This enzyme is purified from the marine [[cyanobacterium]] ''[[Synechococcus]] sp. WH8102''.

== References ==
{{reflist}}

== External links ==
* {{MeshName|Dimethylglycine+N-methyltransferase}}

[[Category:EC 2.1.1]]

Wick product

2012-11-19T15:04:42Z

159.149.107.42:

In mathematics, '''Wedderburn's little theorem''' states that every [[finite set|finite]] [[domain (ring theory)|domain]] is a [[field (mathematics)|field]]. In other words, for [[finite ring]]s, there is no distinction between domains, [[skew-field]]s and fields. The theorem is essentially equivalent to saying that the [[Brauer group]] of a finite field is trivial. In fact, this characterization immediately yields a proof of the theorem as follows: let ''k'' be a finite field. Since the [[Herbrand quotient]] vanishes by finiteness, <math>\operatorname{Br}(k) = H^2(k^{\text{al}}/k)</math> coincides with <math>H^1(k^{\text{al}}/k)</math>, which in turn vanishes by Hilbert 90.

The [[Artin–Zorn theorem]] generalizes the theorem to [[alternative ring]]s: every finite simple alternative ring is a field.<ref>{{cite book | last=Shult | first=Ernest E. | title=Points and lines. Characterizing the classical geometries | series=Universitext | location=Berlin | publisher=[[Springer-Verlag]] | year=2011 | isbn=978-3-642-15626-7 | zbl=1213.51001 | page=123 }}</ref>

== History ==
The original proof was given by [[Joseph Wedderburn]] in 1905,<ref name="Lam-2001-p204">Lam (2001), {{Google books quote|id=f15FyZuZ3-4C|page=204|text=little|p. 204}}</ref> who went on to prove it two other ways. Another proof was given by [[Leonard Eugene Dickson]] shortly after Wedderburn's original proof, and Dickson acknowledged Wedderburn's priority. However, as noted in {{Harv|Parshall|1983}}, Wedderburn's first proof was incorrect – it had a gap – and his subsequent proofs appeared only after he had read Dickson's correct proof. On this basis, Parshall argues that Dickson should be credited with the first correct proof.

A simplified version of the proof was later given by [[Ernst Witt]].<ref name="Lam-2001-p204"/> Witt's proof is sketched below. Alternatively, the theorem is a consequence of the [[Skolem–Noether theorem]] by the following argument.<ref>Theorem 4.1 in Ch. IV of Milne, class field theory, http://www.jmilne.org/math/CourseNotes/cft.html</ref> Let ''D'' be a finite division algebra with center ''k''. Let [''D'' : ''k''] = ''n''2 and ''q'' denote the cardinarity of ''k''. Every maximal subfield of ''D'' has ''qn'' elements; so they are isomorphic and thus are conjugate by Skolem–Noether. But a finite group (the multiplicative group of ''D'' in our case) cannot be a union of conjugates of a proper subgroup; hence, ''n'' = 1.

== Sketch of proof ==
Let ''A'' be a finite domain. For each nonzero ''x'' in ''A'', the two maps
:<math>a \mapsto ax, a \mapsto xa: A \to A</math>
are injective by the cancellation property, and thus, surjective by counting. It follows from the elementary group theory <ref>e.g., Exercise 1.9 in Milne, group theory, http://www.jmilne.org/math/CourseNotes/GT.pdf</ref> that the nonzero elements of ''A'' form a group under multiplication. Thus, ''A'' is a [[Division_ring|skew-field]]. Since the [[Center_(group_theory)|center]] ''Z''(''A'') of ''A'' is a field, ''A'' is a vector space over ''Z''(''A'') with finite dimension ''n''. Our objective is then to show ''n'' = 1. If ''q'' is the order of ''Z''(''A''), then A has order ''qn''. For each ''x'' in ''A'' that is not in the center, the [[Centralizer and normalizer|centralizer]] ''Zx'' of ''x'' has order ''qd'' where ''d'' divides ''n'' and is less than ''n''. Viewing ''Z''(''A'')*, ''Z*x'' and ''A*'' as groups under multiplication, we can write the [[Conjugacy_class#Conjugacy_class_equation|class equation]]
:<math>q^n - 1 = q - 1 + \sum {q^n - 1 \over q^d - 1}</math>
where the sum is taken over all representatives ''x'' that is not in ''Z''(''A'') and ''d'' are the numbers discussed above. ''qn''−1 and ''qd''−1 both admit factorization in terms of [[cyclotomic polynomials]]
:<math>\Phi_f(q)</math>.

From the polynomial identities
:<math>x^n-1 = \prod_{m|n} \Phi_m(x)</math> and <math>x^d-1 = \prod_{m|d} \Phi_m(x)</math>,

we set ''x'' = ''q'' to see that
:<math>\Phi_n(q)</math> divides both ''qn''−1 and <math>{q^n - 1 \over q^d - 1}</math>,

so by the above class equation <math>\Phi_n(q)</math> must divide ''q''−1, and therefore
:<math>|\Phi_n(q)| \leq q-1</math>.

To see that this forces ''n'' to be 1, we will show
:<math>|\Phi_n(q)| > q-1</math>

for ''n'' > 1 using factorization over the complex numbers. In the polynomial identity
:<math>\Phi_n(x) = \prod (x - \zeta)</math>,

where ζ runs over the primitive ''n''-th roots of unity, set ''x'' to be ''q'' and then take absolute values
:<math>|\Phi_n(q)| = \prod |q - \zeta|</math>.

For ''n'' > 1,
:<math>|q-\zeta| > |q-1|</math>

by looking the location of ''q'', 1, and ζ in the complex plane. Thus
:<math>|\Phi_n(q)| > q-1</math>.

==Notes==
<references />

== References ==
* {{cite journal | last = Parshall | first = K. H. | year = 1983 | title = In pursuit of the finite division algebra theorem and beyond: Joseph H M Wedderburn, Leonard Dickson, and Oswald Veblen | journal = Archives of International History of Science | volume = 33 | pages = 274–99|authorlink=Karen Parshall}}
* {{cite book |last1=Lam |first1=Tsit-Yuen |authorlink1= |last2= |first2= |authorlink2= |title=A first course in noncommutative rings |url= |edition=2 |series=Graduate texts in mathematics |volume=131 |year=2001 |publisher=Springer |location= |isbn=0-387-95183-0 |id= }}

== External links ==
*[http://planetmath.org/?op=getobj&from=objects&id=3627 Proof of Wedderburn's Theorem at Planet Math]

[[Category:Ring theory]]
[[Category:Theorems in abstract algebra]]

NNPDF

2012-06-18T11:20:29Z

159.149.45.88:

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