Articles of self learning

How to show the curves are conics.

Solve the equation $$\frac{dx}{x^2+b^2} =\frac{dy}{xy-bz}=\frac{dz}{xz+by}$$ How to show the curves are conics. First of all , I need to find its integral curves. I tried to solve. But I get strange results. So I cannot write there. Sorry. Please show me way. Thanks I found one solution. $$\frac{xdy+bdz}{x^2y-bxz+bxz+b^2z}=\frac{dx}{x^2+b^2}$$ Then I get $bz=c_1$

Equivalent Cauchy sequences.

Hi everyone I’m having a bad time with two questions in the Analysis book of Terry Tao. I finally finished one of the exercises and I’m wondering if the next reasoning is correct or maybe needs some changes: Definitions: Two sequence are equivalence $\iff$ $(\forall \varepsilon \in \mathbb{Q}^+\,) ( \, \exists N\in \mathbb{N}\,) \text{ s.t. […]

Prove that the empirical measure is a measurable fucntion

This problem came from Schervish, Theory of Statistics, Sec. 1.4 Prob. 24. Suppose that $X_1, \ldots, X_n$ are exchangeable and take values in the Borel space $(\mathcal{X}, \mathcal{B})$. Prove that the empirical probability measure $P_n$ is a measurable function from the $n$-fold product space $(\mathcal{X}^n,\mathcal{B}^n)$ to $(\mathcal{P}, \mathcal{C_p})$, where $\mathcal{P}$ is the sets of all […]

$L^1$ is complete in its metric

Theorem: The vector space $L^1$ is complete in its metric. The following proof is from Princeton Lectures in Analysis book $3$ page $70$. Some of my questions about the proof of this theorem are as follows. First assume a Cauchy sequence $(f_n)\in L^1$, then we try to extract a subsequence $\left(f_{n_k}\right)$ of $(f_n)$ which converges […]

Minimizing quadratic objective function on the unit $\ell_1$ sphere

I would like to solve the following optimization problem using a quadratic programming solver $$\begin{array}{ll} \text{minimize} & \dfrac{1}{2} x^T Q x + f^T x\\ \text{subject to} & \displaystyle\sum_{i=1}^{n} |x| = 1\end{array}$$ How can I re-write the problem using linear constraints? Note: I have read other similar questions. However, they define the absolute value differently.

Find a conformal map from the disc to the first quadrant.

Find a conformal mapping of the disk $x^2+(y-1)^2\lt 1$ onto the first quadrant $x, y \gt 0$ I did something, which may be false or not, I cannot exactly say anything. I used the composition of a conformal map, which is conformal. Firstly, let’s get a conformal map from the disk to the unit disk, […]

Sequential Criterion for Functional Limits

How can I use the sequential criterion for functional limits to show that the following limit exist and compute the limit: $$\lim_{x \rightarrow 0} \sqrt{|x|}\cos\left(1/x\right) \ \text{for} \ x \in \mathbb{R} \backslash \{0\}$$ My attempt: I have a hunch the limit is 0, but I can’t really use the sequential criterion to show it is […]

Is $m\mathbb{Z}$ not isomorphic to $n\mathbb{Z}$ when $m\neq n$?

Exercise from “Abstarct Algebra: An Introduction” by T.W.Hungerford. For each positive integer $k,$ let $k\mathbb{Z}$ denote the ring of all integer multiples of $k$. Prove that if $m\neq n$, then $m\mathbb{Z}$ is not isomorphic to $n\mathbb{Z}$. I do understand that I should find some property $P$ that should be preserved by an isomorphism, however, just […]

What are some easy to understand applications of Banach Contraction Principle?

I know that Banach contraction principle guarantees a unique solution to problems of the form $$f(x) = x$$ But for the life of me I cannot understand why this problem is important at all. I don’t remember encountering such a question in calculus or ODE (at least not the most important problems) Further more all […]

Geodesic equations and christoffel symbols

I want to learn explicitly proof of the proposition 9.2.3. Which books or lecture notes I can find? Please give me a suggestion. Thank you:)