Homework set 8 - Solutions Math 407 – Renato Feres
نویسنده
چکیده
• Integration of forms. I review the basics about integrating an n-form over an n-dimensional manifold. Let M be a smooth n-manifold. We need M to be orientable. There are several equivalent characterizations of orientability; for example, M is orientable if there exists a nowhere vanishing continuous n-form ω. Given such an n-form we can select from a smooth atlas of M a subatlas for which the Jacobian determinant of coordinate changes are always positive. This subatlas can be defined by selecting only coordinatesφ−1 = (x1, . . . , xn) so that ω(∂/∂x1, . . . ,∂/∂xn) > 0. With the slight abuse of notation of thinking of ∂/∂x j as vectors either tangent to M or on Rn , this is the same as saying that φ∗ω= ρd x1 ∧·· ·∧d xn where ρ is a positive real-valued function. Let {(Uα,φα) :α ∈ A} be an oriented atlas of smooth parametrizations of M , where Uα ⊂Rn and φα : Uα ⊂Rn →Vα =φα(Uα) ⊂ M is a diffeomorphism for eachα. We assume that {Vα :α ∈ A} is locally finite. This means that for each p ∈ M there is an open neighborhood of p that intersects at most finitely many Vα. Let {ηα :α ∈ A} be a smooth partition of unity subordinate to the open cover {Vα :α ∈ A}. By definition, {ηα :α ∈ A} satisfies the properties: 1. ηα : M → [0,1] has compact support contained in Vα for each α.
منابع مشابه
Notes for Math 450 Elements of Stochastic Calculus
These notes supplement the paper by Higham and provide more information on the basic ideas of stochastic calculus and stochastic differential equations. You will need some of this material for homework assignment 12 in addition to Higham’s paper. There are many places where you can find this theory developed in greater detail and better than here. See, for example, [CKO]. Also note that the inf...
متن کاملNotes for Math 450 Lecture Notes 2 Renato Feres 1 Probability Spaces
We first explain the basic concept of a probability space, (Ω,F , P ). This may be interpreted as an experiment with random outcomes. The set Ω is the collection of all possible outcomes of the experiment; F is a family of subsets of Ω called events; and P is a function that associates to an event its probability. These objects must satisfy certain logical requirements, which are detailed below...
متن کاملHomework set 7 - Solutions
The Student’s t Test. A very common problem in statistics is to estimate the population mean of a probability distribution given a sample, the subject of sections 7.1 and 7.2 of the statistics textbook. As explained there, this relates directly to the so-called t test. For large samples we may assume, due to the central limit theorem, that the sample mean is normally distributed; one often cons...
متن کاملHomework set 9 - Due 4 / 19 / 2013
The purpose of this homework assignment is to introduce some of the R-functions that can be used to perform the kind of data analysis we studied in chapters 7, 8, and 9. Not all the tests listed below will be needed for this problems set. I’ve added them for reference; themain tests for inference on proportions you are going to use are prop.testand chisq.test. Since the main point of this assig...
متن کامل