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Wikipedia OT game
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| 31 Jan 2017 11:18 PM |
| Go click random page,band copy/paste the contents of the page that you get |
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| 31 Jan 2017 11:21 PM |
Ina Jang (born 1982, South Korea) is a photographer based out of Brooklyn, New York.[2] She received her BFA in photography in 2010 from the School of the Visual Arts in New York City. In 2012, she completed the school's MPS program in fashion photography.[3][4]
Contents [hide] 1 Photographs 2 Exhibitions 2.1 Solo exhibitions 2.2 Group exhibitions 3 Awards 4 References Photographs[edit] In an interview with Unseen, Jang describes her approach to photography as "playful, light-hearted and dreamy."[5] In the same interview, she describes how many of her photographs are inspired by ideas which began in the form of drawings. One sees this close connection between drawing and photography in her photographs' compositions, which emphasize the images' two-dimensionality.[5]
Exhibitions[edit] Solo exhibitions[edit] 2011
SO, TOO & VERY, Curated by Ja#####o########nagnorisis Fine Arts, NYC (USA) 2009
WORLD, School of Visual Arts, NYC (USA) XOXO, Anagnorisis Fine Arts, NYC (USA) 2008
BY INA, School of Visual Arts, NYC (USA) Group exhibitions[edit] 2012
ICONS OF TOMORROW, Christophe Guye Galerie, Zurich (Switzerland) FLASH FORWARD 2011, Magenta Foundation, Portland (Canada) PHOTO ASSIGNMENT EXHIBITION, Festival International de Mode et de Photographie, Hyeres (France) GIRLCORE, Orange Dot Gallery, London (UK) 2011
FLASH FORWARD 2011, Magenta Foundation, Portland (Canada) FOAM TALENT 2011 Foam Fotografiemuseum Amsterdam, International Photography Magazine, Amsterdam (The Netherlands) TOKYO PHOTO 2011, Danziger Gallery, Tokyo Midtown Hall, Tokyo (Japan) GROUP EXHIBITION, Festival International de Mode et de Photographie, Hyeres (France) New Visual Artists 2011, Print Magazine, NYC (USA) 2010
KiptonART RISING g 2011, KiptonART, NYC (USA) FUTURE PERFECT, Curated by Stephen Frailey, KiptonART, NYC (USA) MENTORS AT NEW YORK PHOTO FESTIVAL, NYPH’10, NYC (USA) ANOTHER ART SHOW, Submergedart Gallery, Newark (USA) MENTORS, Visual Arts Gallery, NYC (USA) GROUP SHOW 34, Humble iGavel Emering Artist Auction, Daniel Cooney Fine Art, Online 2009
BASICALLY HUMAN, Curated by Stephen Frailey, The Empty Quarter, Dubai (UAE); Visual Arts Gallery, NYC (USA) SURFACE LIFE, Visual Arts Gallery, NYC (USA) 2008
OUR LADY _ist Gallery, NYC (USA)[3] Awards[edit] Jang was one of 15 artists from over 800 submissions selected to be featured in Foam's Issue #28/Talent. Additional awards and nominations include:
2012
Foam Paul Huf Award from Foam Fotografiemuseum Amsterdam, Nominee 2011
PDN 30, Nominee Foam Talent 2011 Flash Forward 2011, Winner Festival d’Hyeres 2011, Finalist 20 Under 30 New Visual Artists Foam Paul Huf Award, Nominee 2010
PDN 30, Nominee KiptonART Rising 2011, Winner Creativity 40th Annual Print, Platinum Creativity International Awards The Tierney Fellowship, Nominee[3] |
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| 31 Jan 2017 11:23 PM |
Jacobson density theorem From Wikipedia, the free encyclopedia In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring R.[1] The theorem can be applied to show that any primitive ring can be viewed as a "dense" subring of the ring of linear transformations of a vector space.[2][3] This theorem first appeared in the literature in 1945, in the famous paper "Structure Theory of Simple Rings Without Finiteness Assumptions" by Nathan Jacobson.[4] This can be viewed as a kind of generalization of the Artin-Wedderburn theorem's conclusion about the structure of simple Artinian rings. Contents [hide] 1 Motivation and formal statement 2 Proof 2.1 Proof of the Jacobson density theorem 3 Topological characterization 4 Consequences 5 Relations to other results 6 Notes 7 References 8 External links Motivation and formal statement[edit] Let R be a ring and let U be a simple right R-module. If u is a non-zero element of U, u • R = U (where u • R is the cyclic submodule of U generated by u). Therefore, if u, v are non-zero elements of U, there is an element of R that induces an endomorphism of U transforming u to v. The natural question now is whether this can be generalized to arbitrary (finite) tuples of elements. More precisely, find necessary and sufficient conditions on the tuple (x1, ..., xn) and (y1, ..., yn) separately, so that there is an element of R with the property that xi • r = yi for all i. If D is the set of all R-module endomorphisms of U, then Schur's lemma asserts that D is a division ring, and the Jacobson density theorem answers the question on tuples in the affirmative, provided that the xi are linearly independent over D. With the above in mind, the theorem may be stated this way: The Jacobson Density Theorem. Let U be a simple right R-module, D = End(UR), and X ⊂ U a finite and D-linearly independent set. If A is a D-linear transformation on U then there exists r ∈ R such that A(x) = x • r for all x in X.[5] Proof[edit] In the Jacobson density theorem, the right R-module U is simultaneously viewed as a left D-module where D = End(UR), in the natural way: g • u = g(u). It can be verified that this is indeed a left module structure on U.[6] As noted before, Schur's lemma proves D is a division ring if U is simple, and so U is a vector space over D. The proof also relies on the following theorem proven in (Isaacs 1993) p. 185: Theorem. Let U be a simple right R-module, D = End(UR), and X ⊂ U a finite set. Write I = annR(X) for the annihilator of X in R. Let u be in U with u • I = 0. Then u is in XD; the D-span of X. Proof of the Jacobson density theorem[edit] We use induction on |X|. If X is empty, then the theorem is vacuously true and the base case for induction is verified. Assume X is non-empty, let x be an element of X and write Y = X \{x}. If A is any D-linear transformation on U, by the induction hypothesis there exists s ∈ R such that A(y) = y • s for all y in Y. Write I = annR(Y). It is easily seen that x • I is a submodule of U. If x • I = 0, then the previous theorem implies that x would be in the D-span of Y, contradicting the D-linear independence of X, therefore x • I ≠ 0. Since U is simple, we have: x • I = U. Since A(x) − x • s # # # # # I, there exists i in I such that x • i = A(x) − x • s. Define r = s + i and observe that for all y in Y we have: {\displaystyle {\begin{aligned}y\cdot r&=y\cdot (s+i)\\&=y\cdot s+y\cdot i\\&=y\cdot s&&({\text{since }}i\in {\text{ann}}_{R}(Y))\\&=A(y)\end{aligned}}} {\begin{aligned}y\cdot r&=y\cdot (s+i)\\&=y\cdot s+y\cdot i\\&=y\cdot s&&({\text{since }}i\in {\text{ann}}_{R}(Y))\\&=A(y)\end{aligned}} Now we do the same calculation for x: {\displaystyle {\begin{aligned}x\cdot r&=x\cdot (s+i)\\&=x\cdot ######## i\\&=x\cdot s+\left(A(x)-x\cdot s\right)\\&=A(x)\end{aligned}}} {\begin{aligned}x\cdot r&=x\cdot (s+i)\\&=x\cdot s+####################+\left(A(x)-x\cdot s\right)\\&=A(x)\end{aligned}} Therefore, A(#### # # r for all z in X, as desired. This completes the inductive step of the proof. It follows now from mathematical induction that the theorem is true for finite sets X of any size. Topological characterization[edit] A ring R is said to act densely on a simple right R-module U if it satisfies the conclusion of the Jacobson density theorem.[7] There is a topological reason for describing R as "dense". Firstly, R can be identified with a subring of End(DU) by identifying each element of R with the D linear transformation it induces by right multiplication. If U is given the discrete topology, and if UU is given the product topology, and End(DU) is viewed as a subspace of UU and is given the subspace topology, then R acts densely on U if and only if R is dense set in End(DU) with this topology.[8] Consequences[edit] The Jacobson density theorem has various important consequences in the structure theory of rings.[9] Notably, the Artin–Wedderburn theorem's conclusion about the structure of simple right Artinian rings is recovered. The Jacobson density theorem also characterizes right or left primitive rings as dense subrings of the ring of D-linear transformations on some D-vector space U, where D is a division ring.[3] Relations to other results[edit] This result is related to the Von Neumann bicommutant theorem, which states that, for a *-algebra A of operators on a Hilbert space H, the double commutant A′′ can be approximated by A on any given finite set of vectors. See also the Kaplansky density theorem in the von Neumann algebra setting. Notes[edit] Jump up ^ Isaacs, p. 184 Jump up ^ Such rings of linear transformations are also known as full linear rings. ^ Jump up to: a b Isaacs, Corollary 13.16, p. 187 Jump up ^ Jacobson, Nathan "Structure Theory of Simple Rings Without Finiteness Assumptions" Jump up ^ Isaacs, Theorem 13.14, p. 185 Jump up ^ Incidentally it is also a D-R bimodule structure. Jump up ^ Herstein, Definition, p. 40 Jump up ^ It turns out this topology is the same as the compact-open topology in this case. Herstein, p. 41 uses this description. Jump up ^ Herstein, p. 41 References[edit] I.N. Herstein (1968). Noncommutative rings (1st ed.). The Mathematical Association of America. ISBN 0-88385-015-X. I. Ma######s######1993). Algebra, a graduate course (1st ed.). Brooks/Cole Publishing Company. ISBN 0-##############Jacobson, N. (1945), "Structure theory of simple rings without finiteness assumptions", Trans. Amer. Math. Soc., 57: 228–245, doi:10.1090/s0002-9947-1945-0011680-8, ISSN 0002-9947, MR 0011680 |
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