https://metaphor.ethz.ch/x/2018/hs/401-1511-00L/literatur/hilfsaetze.pdf
... Definition N. Let φ(x) = Ax+ b be an isometry of Rn, where A ∈ Rnxn is an orthogonal matrix and b ∈ Rn be an ... isometry of Rn. Define (a) φ is orientation-preserving ⇐⇒ det(A) > 0⇐⇒ det(A) = 1. (b) φ is orientation ... Definition N. Let φ(x) = Ax+ b be an isometry of Rn, where A ∈ Rnxn is an orthogonal matrix and b ∈ Rn be an ... isometry of Rn. Define (a) φ is orientation-preserving ⇐⇒ det(A) > 0⇐⇒ det(A) = 1. (b) φ is orientation ...
FAI HS2022 Problem Set 2
... ) Prove that (c0)∗ ∼= ℓ1, i.e. show that there exists a surjective isometry I : ℓ1 → (c0)∗. (b) Prove that ... (ℓ1)∗ ∼= ℓ∞, i.e. show that there exists a surjective isometry I˜ : ℓ∞ → (ℓ1)∗. (c) Prove that there ... ) Prove that (c0)∗ ∼= ℓ1, i.e. show that there exists a surjective isometry I : ℓ1 → (c0)∗. (b) Prove that ... (ℓ1)∗ ∼= ℓ∞, i.e. show that there exists a surjective isometry I˜ : ℓ∞ → (ℓ1)∗. (c) Prove that there ...
https://metaphor.ethz.ch/x/2022/fs/401-3532-08L/ex/ex_9.pdf
... )|y−en|−2 give an isometry between the two previous Riemannian manifolds b) Show that, for the second ... isometry between Rn with metric g(w,w) = ( w · x|x| )2 + (|w|2 − (w · x|x|)2)sinh2 |x||x|2 (1) andM ... )|y−en|−2 give an isometry between the two previous Riemannian manifolds b) Show that, for the second ... isometry between Rn with metric g(w,w) = ( w · x|x| )2 + (|w|2 − (w · x|x|)2)sinh2 |x||x|2 (1) andM ...
https://metaphor.ethz.ch/x/2020/fs/401-3532-08L/ex/ex_4.pdf
... induced distance functions d and d¯, respectively. Further, let f : (M,d) → (M¯, d¯) be an isometry of ... isometry by proving ‖F (X)‖ = ‖X‖. d) Prove that F is linear and conclude that f is smooth in a ... induced distance functions d and d¯, respectively. Further, let f : (M,d) → (M¯, d¯) be an isometry of ... isometry by proving ‖F (X)‖ = ‖X‖. d) Prove that F is linear and conclude that f is smooth in a ...
https://metaphor.ethz.ch/x/2018/hs/401-1511-00L/literatur/fixedpoin...
... establish a general principle: An isometry ψ carries the center of gravity of a figure to the center of ... . (Proof elsewhere) Every isometry ψ of Rn has the form ψ(x) = Ax+ b, where A is an orthogonal matrix and b ... establish a general principle: An isometry ψ carries the center of gravity of a figure to the center of ... . (Proof elsewhere) Every isometry ψ of Rn has the form ψ(x) = Ax+ b, where A is an orthogonal matrix and b ...
Serie 13
... → M the isometry with φp(p) = p and dφp(p) = −1. Prove that the map σ : M ×M →M, (p, q) 7→ φp(q) is ... symmetric space. For this verify that the map φp : Sn → Sn given by φp(x) = −x + 2〈p, x〉p is an isometry ... → M the isometry with φp(p) = p and dφp(p) = −1. Prove that the map σ : M ×M →M, (p, q) 7→ φp(q) is ... symmetric space. For this verify that the map φp : Sn → Sn given by φp(x) = −x + 2〈p, x〉p is an isometry ...
https://metaphor.ethz.ch/x/2018/hs/401-3461-00L/ex/sheet02.pdf
... of a metric space (X, d) is a pair consisting of a complete metric space (X∗, d∗) and an isometry ι ... exists and is equal to ‖x‖∞. 6. (Non-affine isometry). Find a non-linear isometry ϕ : (R, | · |) → (R2 ... of a metric space (X, d) is a pair consisting of a complete metric space (X∗, d∗) and an isometry ι ... exists and is equal to ‖x‖∞. 6. (Non-affine isometry). Find a non-linear isometry ϕ : (R, | · |) → (R2 ...
https://metaphor.ethz.ch/x/2022/fs/401-3532-08L/ex/ex_10.pdf
... Hadamard manifolds Let (M, g) be a two dimensional Hadamard manifold. For fixed point p ∈M and isometry H ... exercise 2. We call cY the center of Y . (b) Let γ be an isometry of M . Then γ is elliptic if and only if ... Hadamard manifolds Let (M, g) be a two dimensional Hadamard manifold. For fixed point p ∈M and isometry H ... exercise 2. We call cY the center of Y . (b) Let γ be an isometry of M . Then γ is elliptic if and only if ...
http://edu.itp.phys.ethz.ch/hs14/QIT/Exercise6.pdf
... [X] 2 ≥ 14~2. Exercise 6.4 Stinespring Isometry (Extra question) Show the existence of the ... Stinespring isometry directly from the Choi state. ... [X] 2 ≥ 14~2. Exercise 6.4 Stinespring Isometry (Extra question) Show the existence of the ... Stinespring isometry directly from the Choi state. ...
Mathematics of Machine Learning Homework 2
... is one, if and only if 퐴 is a scaled isometry. 1 As always, 핊푛−1 ∶= {푥 ∈ ℝ푛| ‖푥‖2 = 1} is the ... is one, if and only if 퐴 is a scaled isometry. 1 As always, 핊푛−1 ∶= {푥 ∈ ℝ푛| ‖푥‖2 = 1} is the ...
