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A Nation of Cheaters
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Poets.org Poems: To Speak of Woe That Is in Marriage
a Woman Loves a Man by David Lehman Poems about Cheating and Infidelity Good Night by Wilhelm Müller What Goes On by Stephen Dunn Related Prose Poems for Breakups and Divorce Adopt a Poet | Add to Notebook | E-mail to Friend | Print "To Speak of Woe That Is in Marriage&q...
Margie
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What�s Wrong with Cheating?
Faculty Resources Faculty Expectations Confronting Cheating Report an Incident Exclusion From Class Faculty use of student work Other Resources & Information Cell Phone Policy Legal Aid Constitution Day Student Resources Student Resources Academic Integrity Academic Integrity Video...
NYTimes Topic: Cheating
Cheating - News - Times Topics - The New York Times Log In Register Now Help Home Page Today's Paper Video Most Popular Times Topics Search All NYTimes.com Tuesday, November 13, 2012 Times Topics World U.S. N.Y. / Region Business Technology Science Health Sports Opinion Arts Style Tra...
 Riemannian Geometry (PDF)
from above that (TM,M, pi) together with the maximal bundle atlas B̂ defined by B is a differen- tiable vector bundle. Definition 4.8. Let M be a differentiable manifold, then a section X : M → TM of the tangent bundle is called a vector field. The set of smooth vector fields X : M →...
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from above that (TM,M, pi) together with the maximal bundle atlas B̂ defined by B is a differen- tiable vector bundle. Definition 4.8. Let M be a differentiable manifold, then a section X : M → TM of the tangent bundle is called a vector field. The set of smooth vector fields X : M → TM is denoted by C∞(TM). Example 4.9. We have seen earlier that the 3-sphere S3 in H ∼= C2 carries a group structure · given by (z, w) · (α, β) = (zα− wβ̄, zβ + wᾱ). This makes (S3, ·) into a Lie group with neutral element e = (1
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www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=19
→ p · q̄ and a real valued norm given by |p|2 = p · p̄. Then the 3-dimensional unit sphere <span class="highlight">S3</span> in H ∼= R4 with the restricted multiplication forms a compact Lie subgroup (<span class="highlight">S3</span>, ·) of (H∗, ·). They are both non-abelian. We shall now introduce some of the classical real and complex matrix Lie groups. As a reference on this topic we recommend the wonderful book: A. W. Knapp, Lie Groups Beyond an Introduction, Birkhäuser (2002). Example 2.31. Let Nil3 be the subset of R3×3 given by Nil3 = { 1 x z0 1 y 0 0 1
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www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=22
R3 and the Riemann sphere Ĉ are diffeomorphic. Exercise 2.8. Find a proof of Proposition 2.24. Exercise 2.9. Let the spheres S1, <span class="highlight">S3</span> and the Lie groups SO(n), O(n), SU(n), U(n) be equipped with their standard differentiable structures introduced above. Use Proposition 2.24 to prove the fol- lowing diffeomorphisms S1 ∼= SO(2), <span class="highlight">S3</span> ∼= SU(2), SO(n)×O(1) ∼= O(n), SU(n)×U(1) ∼= U(n). Exercise 2.10. Find a proof of Corollary 2.28. Exercise 2.11. Let (G, ∗) and (H, ·) be two Lie groups. Prove that the product
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www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=32
embedding if and only if k = ±1. Example 3.23. Let q ∈ <span class="highlight">S3</span> be a quaternion of unit length and φq : S 1 → <span class="highlight">S3</span> be the map defined by φq : z 7→ qz. For w ∈ S1 let γw : R → S1 be the curve given by γw(t) = weit. Then γw(0) = w, γ̇w(0) = iw and φq(γw(t)) = qwe it. By differentiating we yield dφq(γ̇w(0)) = d dt (φq(γw(t)))|t=0 = d dt (qweit)|t=0 = qiw. Then |dφq(γ̇w(0))| = |qwi| = |q||w| = 1 6= 0 implies that the differen- tial dφq is injective. It is easily checked that the immersion φq is an embedding. In the next
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pi : Rn → Rm given by pi : (x1, . . . , xn) 7→ (x1, . . . , xm). Its differential dpix at a point x is surjective since dpix(v1, . . . , vn) = (v1, . . . , vm). This means that the projection is a submersion. An important sub- mersion between spheres is given by the following. Example 3.30. Let <span class="highlight">S3</span> and S2 be the unit spheres in C2 and C× R ∼= R3, respectively. The Hopf map φ : <span class="highlight">S3</span> → S2 is given by φ : (x, y) 7→ (2xȳ, |x|2 − |y|2). For p ∈ <span class="highlight">S3</span> the Hopf circle Cp through p is given by Cp = {eiθ(x, y)| θ ∈ R
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www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=36
ψk : z 7→ zk. For which k ∈ N0 are φk, ψk immersions, submersions or embeddings. Exercise 3.7. Prove that the map φ : Rm → Cm given by φ : (x1, . . . , xm) 7→ (eix1 , . . . , eixm) is a parametrization of the m-dimensional torus Tm in Cm. Exercise 3.8. Find a proof for Theorem 3.26. Exercise 3.9. Prove that the Hopf-map φ : <span class="highlight">S3</span> → S2 with φ : (x, y) 7→ (2xȳ, |x|2 − |y|2) is a submersion.
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from above that (TM,M, pi) together with the maximal bundle atlas B̂ defined by B is a differen- tiable vector bundle. Definition 4.8. Let M be a differentiable manifold, then a section X : M → TM of the tangent bundle is called a vector field. The set of smooth vector fields X : M → TM is denoted by C∞(TM). Example 4.9. We have seen earlier that the 3-sphere <span class="highlight">S3</span> in H ∼= C2 carries a group structure · given by (z, w) · (α, β) = (zα− wβ̄, zβ + wᾱ). This makes (<span class="highlight">S3</span>, ·) into a Lie group with neutral element e = (1
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www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=83
(− � , � )→ O(n) is a geodesic if and only if γt · γ̈ = γ̈t · γ. Exercise 7.3. Find a proof for Proposition 7.23. Exercise 7.4. Find a proof for Corollary 7.24. Exercise 7.5. For the real parameter θ ∈ (0, pi/2) define the 2- dimensional torus T 2θ by T 2θ = {(cos θeiα, sin θeiβ) ∈ <span class="highlight">S3</span>| α, β ∈ R}. Determine for which θ ∈ (0, pi/2) the torus T 2θ is a minimal submanifold of the 3-dimensional sphere <span class="highlight">S3</span> = {(z1, z2) ∈ C2| |z1|2 + |z2|2 = 1}. Exercise 7.6. Find a proof for Corollary 7.27. Exercise 7.7. Determine the totally
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zkw̄k and let Tm = {z ∈ Cm| |z1| = ... = |zm| = 1} be the m-dimensional torus in Cm with the induced metric. Find an isometric immersion φ : Rm → Tm, determine all geodesics on Tm and prove that the torus is flat. Exercise 8.6. Find a proof for Proposition 8.17. Exercise 8.7. Let the Lie group <span class="highlight">S3</span> ∼= SU(2) be equipped with the metric g(Z,W ) = 1 2 Re(trace(Z̄tW )). (i) Find an orthonormal basis for TeSU(2). (ii) Prove that (SU(2), g) has constant sectional curvature +1. Exercise 8.8. Let Sm be the unit sphere in
ReadWriteThink: I Love You Rebus Poem Handout
Every loves a
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Every <span class="highlight">loves</span> a
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Every <span class="highlight">loves</span> a
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Every <span class="highlight">loves</span> a
English Bread Assizes from Reigns of Henry II to Edward II
Strangely, though, only certain people appear to have been fined for the infraction, and Alice Salvage's widow was fined twice as much as anyone else. Are these people, perhaps, repeat offenders? Or had they made more money from their cheating? Another observation worth noting is proof of...
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Strangely, though, only certain people appear to have been fined for the infraction, and Alice Salvage's widow was fined twice as much as anyone else. Are these people, perhaps, repeat offenders? Or had they made more money from their cheating? Another observation worth noting is proof of an old stereotype about the greedy, cheating miller -- sure enough, here we find the miller's wife among the guilty; worse yet, she stands among the fined. 2 Maitland, F.W., ed
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http://www.engr.psu.edu/mtah/articles/pdf/bread_assizes.pdf#page=3
www.engr.psu.edu/mtah/articles/pdf/bread_assizes.pdf#page=3
Strangely, though, only certain people appear to have been fined for the infraction, and Alice Salvage's widow was fined twice as much as anyone else. Are these people, perhaps, repeat offenders? Or had they made more money from their <span class="highlight">cheating</span>? Another observation worth noting is proof of an old stereotype about the greedy, <span class="highlight">cheating</span> miller -- sure enough, here we find the miller's <span class="highlight">wife</span> among the guilty; worse yet, she stands among the fined. 2 Maitland, F.W., ed
The High Cost of Fame for Singer/Actress Brandy
Campbell, who has struggled with addiction and was publicly humiliated when shocking videos of her hit the Internet. Take a first look Latest episode: "Fix My Cheating Husband" Oprah's Favorite Things 2012 Sunday at 9/8c Oprah's Favorite Things returns to TV...
Kids Reading List: Oprah's Book Club
Campbell, who has struggled with addiction and was publicly humiliated when shocking videos of her hit the Internet. Take a first look Latest episode: "Fix My Cheating Husband" Oprah's Favorite Things 2012 Sunday at 9/8c Oprah's Favorite Things returns to TV...
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