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| | Airplanes, airports, stability, baggage, misplaced baggage, almost missed your plane, they ran out of snacks, your headphones don't work, the film acquired trapped so they can't show it, discouraged, calm, now irate, anticipation, anticipation, anticipation...<br><br>Although Strawberry had some legal difficulties ahead of coming to L.A. things received worse on the left coastline. Drug and alcoholic beverages abuse would price him chunks of actively playing time and what after was a promising career turned into unfortunate tabloid fodder.<br><br>Liking oneself unconditionally is the key to joy and achievement. Of course, you want to "love" yourself unconditionally but you need to initial "like" oneself as you are and increase from there into loving Belinda Broido oneself.<br><br>Louis Gehrig was soon after Babe Ruth in the batting line up and his RBI quantities ended up constantly extremely [http://kasian.com/node/1713 Belinda Broido] higher. On his initial working day with the group he didn't provide his possess bat so when the crew manager led him to the batting cages he chose one particular from the fence line, the bat he chose was Babe Ruth's bat (his favourite bat) surprisingly he didn't demand from customers that he return the bat but rather said hello to him.<br><br>Sports is constantly a a single of the most popular nursery themes for infant boys. If you are going with a generic sporting activities concept, you can choose can enhance with all of the athletics this kind of as basketball, baseball, football, soccer, and hockey. You can also chose to do only one particular sport as the concept for the infant's place. and just adhere with that one activity only throughout the total area. One more concept for a sports activities themed nursery is to pick your favourite athletics group this kind of as the Belinda Broido, Boston Crimson Sox, Cleveland Cavaliers, or Dallas Cowboys. Beautify the nursery in the crew's colours and locate any things that have the groups symbol on it to decorate with.<br><br>The final results will be announced on January nine, live on MLB Community. In buy to achieve election to the Hall, a participant requirements to protected Belinda Broido seventy five per cent of the votes cast.<br><br>David Wells (1st calendar year) - The male acknowledged as "Boomer" invested two independent two-12 months stints in pinstripes, winning a World Championship in 1998. Wells is greatest known for tossing a ideal game as a member of the Yankees on Could seventeen, 1998. His 21-calendar year occupation finished with 239 victories and three all-star sport appearances.<br><br>Recently, Doug Drabek reflected on his career with Houston Astros Examiner Stephen Goff via a phone job interview, as well as his greatest influences, his son, Kyle's, baseball journey in large faculty and the minors, and what to expect from the youthful Drabek as he climbs the ladder to the huge leagues.<br><br>A: He's obtained what it will take to be a effective significant league pitcher. He'll make it to the massive leagues and possibly be added to the forty-male roster for 2010. It's critical he goes to Spring Coaching and gains valuable encounter with the big club. If he doesn't earn a place on the active roster, at minimum Kyle will start off the time in Triple-A. I consider he'll locate himself with the Phillies sometime in 2010 or as a September call-up. |
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| In [[mathematics]], a '''bialgebra''' over a [[Field (mathematics)|field]] ''K'' is a [[vector space]] over ''K'' which is both a [[unital algebra|unital]] [[associative algebra]] and a [[coalgebra]], such that the algebraic- and coalgebraic structure satisfy certain compatibility relations. Specifically, the [[comultiplication]] and the [[counit]] are both unital algebra [[homomorphisms]], or equivalently, that the multiplication and the unit of the algebra both be [[Coalgebra#Further concepts and facts|coalgebra morphisms]]. These statements are equivalent in that they are expressed by ''the same [[commutative diagram]]s''. A bialgebra homomorphism is a [[linear map]] that is both an algebra and a coalgebra homomorphism.
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| As reflected in the symmetry of the commutative diagrams, the definition of bialgebra is [[Dual (category theory)|self-dual]], so if one can define a [[Dual space|dual]] of ''B'' (which is always possible if ''B'' is finite-dimensional), then it is automatically a bialgebra.
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| {{Algebraic structures |Algebra}}
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| == Formal definition ==
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| '''(''B'', ∇, η, Δ, ε)''' is a '''bialgebra''' over ''K'' if it has the following properties:
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| * ''B'' is a vector space over ''K'';
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| * there are ''K''-[[linear map]]s (multiplication) ∇: ''B'' ⊗ ''B'' → ''B'' (equivalent to ''K''-[[multilinear map]] ∇: ''B'' × ''B'' → ''B'') and (unit) η: ''K'' → ''B'', such that (''B'', ∇, η) is a unital associative [[Algebra over a field|algebra]];
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| * there are ''K''-linear maps (comultiplication) Δ: ''B'' → ''B'' ⊗ ''B'' and (counit) ε: ''B'' → ''K'', such that (''B'', Δ, ε) is a (counital coassociative) [[coalgebra]];
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| * compatibility conditions expressed by the following [[commutative diagram]]s:
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| # Multiplication ∇ and comultiplication Δ <ref>Dăscălescu, Năstăsescu & Raianu (2001), {{Google books quote|id=pBJ6sbPHA0IC|page=147|text=is a morphism of coalgebras|p. 147 & 148}}</ref>
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| #::[[Image:Bialgebra2.svg|500px|Bialgebra commutative diagrams]]
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| #: where τ: ''B'' ⊗ ''B'' → ''B'' ⊗ ''B'' is the [[linear map]] defined by τ(''x'' ⊗ ''y'') = ''y'' ⊗ ''x'' for all ''x'' and ''y'' in ''B'',
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| # Multiplication ∇ and counit ε
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| #::[[Image:Bialgebra3.svg|310px|Bialgebra commutative diagrams]]
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| # Comultiplication Δ and unit η <ref>Dăscălescu, Năstăsescu & Raianu (2001), {{Google books quote|id=pBJ6sbPHA0IC|page=148|text=is a morphism of coalgebras|p. 148}}</ref>
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| #::[[Image:Bialgebra4a.svg|310px|Bialgebra commutative diagrams]]
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| # Unit η and counit ε
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| #::[[Image:Bialgebra1.svg|125px|Bialgebra commutative diagrams]]
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| ==Coassociativity and counit==
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| The [[multilinear map|''K''-linear map]] Δ: ''B'' → ''B'' ⊗ ''B'' is [[coalgebra|coassociative]] if <math>(\mathrm{id}_B \otimes \Delta) \circ \Delta = (\Delta \otimes \mathrm{id}_B) \circ \Delta</math>.
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| The ''K''-linear map ε: ''B'' → ''K'' is a counit if <math>(\mathrm{id}_B \otimes \epsilon) \circ \Delta = \mathrm{id}_B = (\epsilon \otimes \mathrm{id}_B) \circ \Delta</math>.
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| Coassociativy and counit are expressed by the [[commutative diagram|commutativity]] of the following two diagrams with ''B'' in place of ''C'' (they are the duals of the diagrams expressing associativity and unit of an algebra):
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| [[Image:coalg.png|center|800px]]
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| == Compatibility conditions ==
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| The four commutative diagrams can be read either as "comultiplication and counit are [[homomorphism]]s of algebras" or, equivalently, "multiplication and unit are [[homomorphism]]s of coalgebras".
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| These statements are meaningful once we explain the natural structures of algebra and coalgebra in all the vector spaces involved besides ''B'': (''K'', ∇<sub>0</sub>, η<sub>0</sub>) is a unital associative algebra in an obvious way and (''B'' ⊗ ''B'', ∇<sub>2</sub>, η<sub>2</sub>) is a unital associative algebra with unit and multiplication
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| :<math>\eta_2 := (\eta \otimes \eta) : K \otimes K \equiv K \to (B \otimes B) </math>
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| :<math>\nabla_2 := (\nabla \otimes \nabla) \circ (id \otimes \tau \otimes id) : (B \otimes B) \otimes (B \otimes B) \to (B \otimes B) </math>,
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| so that <math>\nabla_2 ( (x_1 \otimes x_2) \otimes (y_1 \otimes y_2) ) = \nabla(x_1 \otimes y_1) \otimes \nabla(x_2 \otimes y_2) </math> or, omitting ∇ and writing multiplication as juxtaposition, <math>(x_1 \otimes x_2)(y_1 \otimes y_2) = x_1 y_1 \otimes x_2 y_2 </math>; | |
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| similarly, (''K'', Δ<sub>0</sub>, ε<sub>0</sub>) is a coalgebra in an obvious way and ''B'' ⊗ ''B'' is a coalgebra with counit and comultiplication
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| :<math>\epsilon_2 := (\epsilon \otimes \epsilon) : (B \otimes B) \to K \otimes K \equiv K</math>
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| :<math>\Delta_2 := (id \otimes \tau \otimes id) \circ (\Delta \otimes \Delta) : (B \otimes B) \to (B \otimes B) \otimes (B \otimes B)</math>.
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| Then, diagrams 1 and 3 say that Δ: ''B'' → ''B'' ⊗ ''B'' is a homomorphism of unital (associative) algebras (''B'', ∇, η) and (''B'' ⊗ ''B'', ∇<sub>2</sub>, η<sub>2</sub>)
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| :<math>\Delta \circ \nabla = \nabla_2 \circ (\Delta \otimes \Delta) : (B \otimes B) \to (B \otimes B)</math>, or simply Δ(''xy'') = Δ(''x'') Δ(''y''),
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| :<math>\Delta \circ \eta = \eta_2 : K \to (B \otimes B)</math>, or simply Δ(1<sub>''B''</sub>) = 1<sub>''B'' ⊗ ''B''</sub>;
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| diagrams 2 and 4 say that ε: ''B'' → ''K'' is a homomorphism of unital (associative) algebras (''B'', ∇, η) and (''K'', ∇<sub>0</sub>, η<sub>0</sub>):
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| :<math>\epsilon \circ \nabla = \nabla_0 \circ (\epsilon \otimes \epsilon) : (B \otimes B) \to K</math>, or simply ε(''xy'') = ε(''x'') ε(''y'')
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| :<math>\epsilon \circ \eta = \eta_0 : K \to K</math>, or simply ε(1<sub>''B''</sub>) = 1<sub>''K''</sub>.
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| Equivalently, diagrams 1 and 2 say that ∇: ''B'' ⊗ ''B'' → ''B'' is a homomorphism of (counital coassociative) coalgebras (''B'' ⊗ ''B'', Δ<sub>2</sub>, ε<sub>2</sub>) and (''B'', Δ, ε):
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| :<math> \nabla \otimes \nabla \circ \Delta_2 = \Delta \circ \nabla : (B \otimes B) \to (B \otimes B),</math>
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| :<math>\epsilon \circ \nabla = \nabla_0 \circ \epsilon_2 : (B \otimes B) \to K</math>;
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| diagrams 3 and 4 say that η: ''K'' → ''B'' is a homomorphism of (counital coassociative) coalgebras (''K'', Δ<sub>0</sub>, ε<sub>0</sub>) and (''B'', Δ, ε):
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| :<math>\Delta \circ \eta = \eta_2 \circ \Delta_0: K \to (B \otimes B),</math>
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| :<math>\epsilon \circ \eta = \eta_0 \circ \epsilon_0 : K \to K</math>.
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| ==Examples==
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| A simple example of a bialgebra is the set of functions from a [[group (mathematics)|group]] ''G'' to <math>\mathbb R</math>, which we may represent as a vector space <math>\mathbb R^G</math> consisting of linear combinations of standard basis vectors '''e'''<sub>''g''</sub> for each ''g'' ∈ ''G'', which may represent a [[probability distribution]] over ''G'' in the case of vectors whose coefficients are all non-negative and sum to 1. An example of suitable comultiplication operators and counits which yield a counital coalgebra are
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| :<math>\Delta(\mathbf e_g) = \mathbf e_g \otimes \mathbf e_g \,,</math>
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| which represents making a copy of a [[random variable]] (which we extend to all <math>\mathbb R^G</math> by linearity), and
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| :<math>\varepsilon(\mathbf e_g) = 1 \,,</math>
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| (again extended linearly to all of <math> \mathbb R^G</math>) which represents "tracing out" a random variable — ''i.e.,'' forgetting the value of a random variable (represented by a single tensor factor) to obtain a [[marginal distribution]] on the remaining variables (the remaining tensor factors).
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| Given the interpretation of (Δ,ε) in terms of probability distributions as above, the bialgebra consistency conditions amount to constraints on (∇,η) as follows:
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| 4. η is an operator preparing a normalized probability distribution which is independent of all other random variables;
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| 3. The product ∇ maps a probability distribution on two variables to a probability distribution on one variable;
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| 2. Copying a random variable in the distribution given by η is equivalent to having two independent random variables in the distribution η;
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| 1. Taking the product of two random variables, and preparing a copy of the resulting random variable, has the same distribution as preparing copies of each random variable independently of one another, and multiplying them together in pairs.
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| A pair (∇,η) which satisfy these constraints are the [[convolution]] operator
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| :<math>\nabla\bigl(\mathbf e_g \otimes \mathbf e_h\bigr) = \mathbf e_{gh} \,,</math>
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| again extended to all <math>\mathbb R^G \otimes \mathbb R^G</math> by linearity; this produces a normalized probability distribution from a distribution on two random variables, and has as a unit the delta-distribution <math> \eta = \mathbf e_{i} \;,</math> where ''i'' ∈ ''G'' denotes the identity element of the group ''G''.
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| Other examples of bialgebras include the [[Hopf algebra]]s.<ref>Dăscălescu, Năstăsescu & Raianu (2001), {{Google books quote|id=pBJ6sbPHA0IC|page=151|text=Hopf|p. 151}}</ref> Similar structures with different compatibility between the product and comultiplication, or different types of multiplication and comultiplication, include [[Lie bialgebra]]s and [[Frobenius algebra]]s. Additional examples are given in the article on [[coalgebra]]s.
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| ==See also==
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| *[[Quasi-bialgebra]]
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| *[[Frobenius algebra]]
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| *[[Hopf algebra]]
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| == Notes ==
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| <references/>
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| == References ==
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| * {{Citation| last1=Dăscălescu| first1=Sorin| last2=Năstăsescu| first2=Constantin| last3=Raianu| first3=Șerban| year=2001| title=Hopf Algebras| subtitle=An introduction| edition=1st| volume = 235| series=Pure and Applied Mathematics | publisher=Marcel Dekker| isbn = 0-8247-0481-9}}.
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| [[Category:Bialgebras]]
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| [[Category:Coalgebras]]
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| [[Category:Monoidal categories]]
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Airplanes, airports, stability, baggage, misplaced baggage, almost missed your plane, they ran out of snacks, your headphones don't work, the film acquired trapped so they can't show it, discouraged, calm, now irate, anticipation, anticipation, anticipation...
Although Strawberry had some legal difficulties ahead of coming to L.A. things received worse on the left coastline. Drug and alcoholic beverages abuse would price him chunks of actively playing time and what after was a promising career turned into unfortunate tabloid fodder.
Liking oneself unconditionally is the key to joy and achievement. Of course, you want to "love" yourself unconditionally but you need to initial "like" oneself as you are and increase from there into loving Belinda Broido oneself.
Louis Gehrig was soon after Babe Ruth in the batting line up and his RBI quantities ended up constantly extremely Belinda Broido higher. On his initial working day with the group he didn't provide his possess bat so when the crew manager led him to the batting cages he chose one particular from the fence line, the bat he chose was Babe Ruth's bat (his favourite bat) surprisingly he didn't demand from customers that he return the bat but rather said hello to him.
Sports is constantly a a single of the most popular nursery themes for infant boys. If you are going with a generic sporting activities concept, you can choose can enhance with all of the athletics this kind of as basketball, baseball, football, soccer, and hockey. You can also chose to do only one particular sport as the concept for the infant's place. and just adhere with that one activity only throughout the total area. One more concept for a sports activities themed nursery is to pick your favourite athletics group this kind of as the Belinda Broido, Boston Crimson Sox, Cleveland Cavaliers, or Dallas Cowboys. Beautify the nursery in the crew's colours and locate any things that have the groups symbol on it to decorate with.
The final results will be announced on January nine, live on MLB Community. In buy to achieve election to the Hall, a participant requirements to protected Belinda Broido seventy five per cent of the votes cast.
David Wells (1st calendar year) - The male acknowledged as "Boomer" invested two independent two-12 months stints in pinstripes, winning a World Championship in 1998. Wells is greatest known for tossing a ideal game as a member of the Yankees on Could seventeen, 1998. His 21-calendar year occupation finished with 239 victories and three all-star sport appearances.
Recently, Doug Drabek reflected on his career with Houston Astros Examiner Stephen Goff via a phone job interview, as well as his greatest influences, his son, Kyle's, baseball journey in large faculty and the minors, and what to expect from the youthful Drabek as he climbs the ladder to the huge leagues.
A: He's obtained what it will take to be a effective significant league pitcher. He'll make it to the massive leagues and possibly be added to the forty-male roster for 2010. It's critical he goes to Spring Coaching and gains valuable encounter with the big club. If he doesn't earn a place on the active roster, at minimum Kyle will start off the time in Triple-A. I consider he'll locate himself with the Phillies sometime in 2010 or as a September call-up.