VIEWS: 66 PAGES: 13 CATEGORY: Fitness POSTED ON: 2/14/2011
Warm-up exercises (Warm-up), also known as ready to exercise, named after the former due to physiological responses, and the latter is a general concept. Warm-up exercises, the activities of some combination of the body; in the main physical activity prior to the lesser activity in advance of physical activity for more intense physical activity then prepare for the purpose of enhancing the efficiency of subsequent intense exercise, intense exercise Security, while meeting the physical and psychological human needs. Before exercise, the body's functional ability and efficiency can not be reached at the beginning of the highest level, and thus need to adjust the movement warm-up state.
A Warm-up Exercise from Calculus Sturm's Theorem in Ordinary Dierential Equation Rauch Comparison Theorem and Its Application On the Rauch Comparison Theorem and Its Applications Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar April 30, 2009 Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications A Warm-up Exercise from Calculus Sturm's Theorem in Ordinary Dierential Equation Rauch Comparison Theorem and Its Application A Warm-up Exercise from Calculus Exercise 1 1 2 sin 2t 3 sin 3t on [0, 3 ]. π Show that ≥ Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications A Warm-up Exercise from Calculus Sturm's Theorem in Ordinary Dierential Equation Rauch Comparison Theorem and Its Application A Warm-up Exercise from Calculus 0.5 1 sin2 t 2 0.4 1 0.3 sin3 t 3 0.2 0.1 t 0.2 0.4 0.6 0.8 1.0 tt Figure: Graphs of 1 sin 2t and 1 sin 3t 2 3 Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications A Warm-up Exercise from Calculus Sturm's Theorem in Ordinary Dierential Equation Rauch Comparison Theorem and Its Application Sturm's Theorem Theorem (Sturm) Let x1 (t ) and x2 (t ) be solutions to equations x1 (t ) + p1 (t )x1 (t ) = 0 (1) and x2 (t ) + p2 (t )x2 (t ) = 0 (2) respectively with initial conditions x1 (0) = x2 (0) = 0 and x1 (0) = x2 (0) = 1, where p1 (t ) and p2 (t ) are continuous on [0, T ]. Suppose p1 (t ) ≤ p2 (t ) on [0, T ] and x2 (t ) > 0 on (0, T ]. Then x1 (t ) ≥ x2 (t ) on [0, T ]. Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications A Warm-up Exercise from Calculus Sturm's Theorem in Ordinary Dierential Equation Rauch Comparison Theorem and Its Application Sturm's Theorem x1 t x2 t 0 T t Figure: Comparison of the Solutions to Two ODEs Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications A Warm-up Exercise from Calculus Some Ingredients Sturm's Theorem in Ordinary Dierential Equation Rauch Comparison Theorem Rauch Comparison Theorem and Its Application A Tool for the Proof Application of Rauch Comparison Theorem Jacobi Field Denition (Jacobi Field) A vector eld J along a geodesic g in a Riemmannian manifold is said to be a Jacobi eld if it satises the Jacobi equation D2 J (t ) + R (J (t ), γ (t ))γ (t ) = 0, (3) dt 2 where R is the Riemann curvature tensor. Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications A Warm-up Exercise from Calculus Some Ingredients Sturm's Theorem in Ordinary Dierential Equation Rauch Comparison Theorem Rauch Comparison Theorem and Its Application A Tool for the Proof Application of Rauch Comparison Theorem Conjugate Point Denition (Conjugate Point) A point q is said to be a conjugate point to another point p along a geodesic g in a Riemannian manifold if there exists a non-zero Jacobi eld along g that vanishes at p and q. Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications A Warm-up Exercise from Calculus Some Ingredients Sturm's Theorem in Ordinary Dierential Equation Rauch Comparison Theorem Rauch Comparison Theorem and Its Application A Tool for the Proof Application of Rauch Comparison Theorem The Rauch Comparison Theorem Theorem (Rauch, 1951) Let M1 and M2 be Riemannian manifolds, γ1 : [0, T ] → M1 and γ2 : [0, T ] → M2 be normalized geodesic segments such that γ2 (0) has no conjugate points along γ2 , and J1 , J2 be normal Jacobi elds along γ1 and γ2 such that J1 (0) = J2 (0) = 0 and |J1 (0)| = |J2 (0)| . Suppose that the sectional curvatures of M1 and M2 satisfy K1 ≤ K2 for all 2-planes containing γ1 and γ2 on each manifold. Then |J1 (t )| ≥ |J2 (t )| for all t ∈ [0, T ]. Remark The condition normal Jacobi elds along γ1 and γ2 can be replaced by the Jacobi elds satisfying J1 (0), γ1 (0) = J2 (0), γ2 (0) , that we can see in the proof. Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications A Warm-up Exercise from Calculus Some Ingredients Sturm's Theorem in Ordinary Dierential Equation Rauch Comparison Theorem Rauch Comparison Theorem and Its Application A Tool for the Proof Application of Rauch Comparison Theorem The Rauch Comparison Theorem Theorem (Rauch, 1951) Let M1 and M2 be Riemannian manifolds, γ1 : [0, T ] → M1 and γ2 : [0, T ] → M2 be normalized geodesic segments such that γ2 (0) has no conjugate points along γ2 , and J1 , J2 be normal Jacobi elds along γ1 and γ2 such that J1 (0) = J2 (0) = 0 and |J1 (0)| = |J2 (0)| . Suppose that the sectional curvatures of M1 and M2 satisfy K1 ≤ K2 for all 2-planes containing γ1 and γ2 on each manifold. Then |J1 (t )| ≥ |J2 (t )| for all t ∈ [0, T ]. Remark The condition normal Jacobi elds along γ1 and γ2 can be replaced by the Jacobi elds satisfying J1 (0), γ1 (0) = J2 (0), γ2 (0) , that we can see in the proof. Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications A Warm-up Exercise from Calculus Some Ingredients Sturm's Theorem in Ordinary Dierential Equation Rauch Comparison Theorem Rauch Comparison Theorem and Its Application A Tool for the Proof Application of Rauch Comparison Theorem Index Form Denition (Index Form) The index form of a piecewise dierentiable vector eld V along a geodesic γ on a Riemannian manifold M is dened as ˆt It (V , V ) := ( V ,V − R (γ , V )γ , V )dt . (4) 0 Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications A Warm-up Exercise from Calculus Some Ingredients Sturm's Theorem in Ordinary Dierential Equation Rauch Comparison Theorem Rauch Comparison Theorem and Its Application A Tool for the Proof Application of Rauch Comparison Theorem Index Lemma Lemma (Index Lemma) Let J be a Jacobi eld along a geodesic γ : [0, T ] → M, which has no conjugate point to γ(0) in the interval (0, T ], with J , γ = 0, and V be a piecewise dierentiable vector eld along γ , with V , γ = 0. Suppose that J (0) = V (0) = 0 and J (t0 ) = V (t0 ), t0 ∈ (0, T ]. Then It0 (J , J ) ≤ It0 (V , V ). Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications A Warm-up Exercise from Calculus Some Ingredients Sturm's Theorem in Ordinary Dierential Equation Rauch Comparison Theorem Rauch Comparison Theorem and Its Application A Tool for the Proof Application of Rauch Comparison Theorem Sphere Theorem Theorem (Berger-Klingenberg, 1960) Any compact, simply connected, and strictly 1 -pinched manifold 4 M n , that is the sectional curvature K of M n satisfying 1 0 < Kmax < K ≤ Kmax , is homeomorphic to S . n 4 Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications A Warm-up Exercise from Calculus Some Ingredients Sturm's Theorem in Ordinary Dierential Equation Rauch Comparison Theorem Rauch Comparison Theorem and Its Application A Tool for the Proof Application of Rauch Comparison Theorem Dierentiable Sphere Theorem Theorem (Brendle and Schoen, 2009) Any compact, simply connected and strictly 1 -strictly pinched 4 manifold M n is dieomorphic to S n . Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications