Questions about the process of studying mathematics without formal instruction.

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35 views

How to do this Integration (in Orthonormal Family for Continuous Functions)

Here is a common example in the discussion on orthonormal family. Let $\mathcal L = C[a, b]$. For $k \in \mathbb Z$, let $e_k \in \mathcal L$ be defined by $$e_k(\xi) := \frac{1}{\sqrt{b-a}} ...
0
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0answers
40 views

Learning pipeline for developing own optical flow algorithms

I am really sorry if this question is outside of this resource or too silly I am bachelor of computer science and a programmer in small company. And i am faced with the task of developing own custom ...
1
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1answer
25 views

Convolution Operator and Integration Operator

I have some questions about the following two operators. A convolution operator $T$. If $k \in \mathcal L^1(\mathbb R)$, then $$f(x) \mapsto \int_{-\infty}^\infty k(x-y)f(y) dy: \mathcal L^2(\mathbb ...
2
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1answer
57 views

Closed Subspaces of Hilbert Spaces

I read the following statements. But I do not know how to show it or any example to support it. Could anyone provide some explanation and examples, please? Thank you! The subspace $C^\infty$ ...
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0answers
29 views

Basis for Finite Dimensional Hilbert Spaces

Verify that a Hilbert space orthonormal basis in a finite dimensional Hilbert space is the same as an orthonormal basis in the sense of linear algebra. Here is what I know. Hilbert space ...
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0answers
26 views

Maximum of squared point to plane distance?

I am looking at the squared distance from a hyperplane that passes throught the point $\mathbf{a}$ to the point $t$ which I understand is given by the formula $$\frac{\left[\mathbf{k} \left( ...
3
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3answers
82 views

$f$ an analytic function on $\mathbb{C}$ which takes values in $\mathbb{C}\backslash(-\infty,0]$ implies constant

Let $f$ be an analytic function on $\mathbb{C}$ which takes values in $\mathbb{C}\backslash(-\infty,0]$, i.e. takes values in the complement of the nonpositive part of the real axis. Show that $f$ is ...
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1answer
33 views

Eigenvalue of Compact Operators

To prove that the set of eigenvectors of a compact linear operator on a normed space $X$ is countable, I read "it suffices to show that for every real $k > 0$ the set of all eigenvalues whose ...
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1answer
59 views

A Question on Compact Operators on Hilbert Space

I read this question which I have no idea how to start. Could anyone provide me with some detailed answer, please? Thanks. Suppose that a linear operator $F$ from a Hilbert space $\mathcal H$ to ...
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2answers
46 views

To discuss differentiability of function at origin and my attempt

P1: $F = |x| + |y| when x,y is not equal to 0, = 0 when x = y = 0 P2 :Discuss the differentiability at origin of $F = y sin(1/x)$ :
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0answers
40 views

Finite Sums and Riemann-Stieltjes integral

Show that every finite sum $\sum_{k=1}^{n}a_k$ can be written as a Riemann-Stieltjes integral. My thoughts As far as I understand step functions provide a bridging link between Riemann-Stieltjes ...
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0answers
53 views

Can any one recommend a way to “quickly” learn a subject?

I would love to read a well written book on a subject - provided that I have the time. But sometimes we do not need to become experts on a particular field but still need the basics. For example, a ...
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0answers
26 views

Sum converging a.s. (cont'd)

Let $X_k$ be independent random variables with zero expectation s.t. $\sum_{k=1}^nX_k\rightarrow_{a.s.} X$ and $X$ has a nondegenerate distribution. Is there a way to estimate $E[(\sum_{k=1}^\infty ...
2
votes
3answers
73 views

limits multivariable calculus. where am i wrong with my attempt?

P : $\lim_{(x,y) \to (0,0)} f(x,y)$ where $$f(x,y) = y\sin\frac1x + \frac{xy}{x^{2} + y^{2}}$$ Text book says Limit doesnot exist . So where i am wrong with my proof below ? EDITED ATTEMPT : Or ...
6
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1answer
127 views

Real Analysis : Self Studying vs Doing a Course

I am an engineering graduate student. Recently I got interested in studying Maths. So, I have started self-studying Real Analysis(let's call it RA) using a few books. I will also be using problem ...
1
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1answer
55 views

Sum converging a.s.

Let $X_k$ be independent random variable s.t. $\sum_{k=1}^nX_k\rightarrow_{a.s.} X$. So, $$X=\sum_{k=1}^\infty (X_k^+ -X_k^-)$$ Is it true that $X=\sum_{k=1}^\infty (X_k^+)-\sum_{k=1}^\infty ...
1
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2answers
26 views

On subgroups of isometries and their respective fixed-points

I am working on the following problemset: Let $G < \DeclareMathOperator{\Iso}{Iso}\Iso(E)$ be a finite subgroup of the isometries in the euclidean plane. Denote: $$G = \{g_1, \ldots , g_n \} ...
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1answer
56 views

Proving Euler Summation by Parts Without Using Integration by Parts

Assume $f$ has continuous derivative $f'$ on [a,b]. Prove the following summation formula, without using partial integration: \begin{equation} \sum_{a< x \le ...
2
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1answer
47 views

Exercise on measure theory, (verification and suggestion)

Hi everyone I'd like to know if the following is correct and also I'd appreciate any suggestion to improve the argument. Thanks in advance For every positive integer $n$, let $f_n:{\bf{R}}\to ...
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0answers
31 views

Proof check - if a set is a $\sigma$-algebra or not.

From Rudin Real and Complex Analysis Theorem 1.12. Suppose $M$ is a $\sigma$-algebra in $X$, and $Y$ is a topological space. let $f$ map $X$ into $Y$. If $\Omega$ is the collection of all ...
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0answers
8 views

On inverse images of a function from a $\sigma$-algebra into a topological space.

Let $f$ map $X$ into $Y$. Call the $\sigma$-algebra $X$ and let it be the set $\{\{1,2\},\{1\},\{2\},\{\emptyset\}\}$. Call the topological space Y and let it be the set of all open sets of $R$. ...
2
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1answer
88 views

Prove that $\frac{1}{a^3(b+c)}+\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)}\ge \frac32$

$a,b,c$ are positive reals with $abc = 1$. Prove that $$\frac{1}{a^3(b+c)}+\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)}\ge \frac32$$ I try to use AM $\ge$ HM. ...
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0answers
23 views

Riemann-Stieltjes Integrals with $n$ discontinuities (Proof Review)

First of all is the proof legitimate? If so, then are there other methods to achieve the same result that are neater than mine? Let $\alpha : [a,b] \to \mathbb{R}$ be a step function with ...
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2answers
78 views

Borel-Cantelli (proof and application)

Hi I was reading the second volume of the Tao's Analysis book and in one exercise he's asking for a proof of Borel-Cantelli If we have a sequence $s_n\in \Omega$ of measurable sets s.t. ...
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0answers
30 views

Convergence equivalent random sequences

Suppose we have a sequence of independent random vars $X_n$ and consider a sequence of truncated random variables $Y_n=X_n1_{X_n\le n}$ s.t. $E[Y_n]=0$. We know that $X_n$'s and $Y_n$'s are ...
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1answer
49 views

Riemann-Stieltjes Integral and the Step Function

Let $a < c< b$ and let $\alpha (x)$ be defined as $\alpha (x) =\begin{cases} 0 & a \le x \le c \\ 1 &c<x \le b \end{cases}$. Show that $f \in \mathcal{R}(\alpha)$ if and only if ...
0
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1answer
347 views

James R. Munkres' TOPOLOGY, 2nd edition: How to check my work?

I'm trying to learn, or revise, some topology from James R. Munkres' TOPOLOGY, 2nd edition. I'm working alone; that is, I'm self-learning. It is quite fun. But the problem is how do I check if I've ...
0
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2answers
43 views

Proof that every open set in the plane is a countable union of rectangles.

I came across this statement in the first chapter or Rudin Real and Complex Analysis. Rudin states that every open set in the plane is a countable union of rectangles. Looking for a proof I ...
1
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1answer
62 views

Norm of the Resolvent of a Self-Adjoint Operator

Let $\mathcal H$ be a Hilbert space and $\mathcal L$ is a self- adjoint operator with a discrete spectrum $\{\lambda_{j}\}$. I read that it is well known that for, $\lambda \notin \sigma(\mathcal ...
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2answers
71 views

Spectrum of Self-Adjoint Operators

This is an exercise (5-i) from here. It has two parts as follows. For a self-adjoint operator $A$. Show that $A \geq k I, \ k \in \mathbb R$ if and only if $\lambda \geq k$ for all $\lambda$ ...
2
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1answer
72 views

Most efficient mental way to convert Decimal to Hexadecimal

My question is as follows: What is the most efficient mental way to convert Decimal to Hexadecimal? I've heard of many methods. Some people divide the decimal by 16 and find the remainder. Others ...
3
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1answer
76 views

Nash's Axiomatic Bargaining: Source of problems sets and practice questions.

From where can I practice questions related to the following topic: Nash's Axiomatic Bargaining. Any form of book reference or a link to some online problem set would be highly appreciated.
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1answer
27 views

On loss of generality proving the Cauchy Schwartz inequality.

In Rudins Real and Complex Analysis The Schwartz inequality: $|(x,y)| \le ||x|| \ ||y||$ is proven in the following manner: Put $A = ||x||^2, B=|(x,y)|$ and $C = ||y||^2$. There is a complex number ...
2
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2answers
224 views

Vladimir Zorich vs Rudin/Pugh/Abbott

There have seen various comparisons between books on Analysis. I was surprised to find out that Zorich's book on Analysis was not compared anywhere. Can anyone give a comparison between Zorich and ...
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1answer
96 views

In what order should I learn linear algebra and multivariable calculus?

I took AP calculus in high school and I really enjoyed it, but when I got to my university I was upset that I couldn't take Calculus II as it didn't fit in my schedule. I feel kind of behind now that ...
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2answers
48 views

How to prove $\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b} <2$ [duplicate]

Prove the inequality for a triangle with sides $a,b,c$ we have $$\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b} <2$$ Trial: Since $a,b,c$ are sides of a triangle I know $a+b>c,b+c>a,a+c>b$ ...
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0answers
81 views

How to deal with a lack of ability to apply ideas in math?

I am currently studying Theoretical Computer Science, but as a Computer Science student who does not have a formal background in mathematics, past A Level (High School), I find that the ideas I learn ...
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0answers
21 views

fractal dimension estimation for an image

Apparently it is possible to estimate the fractal dimension of an image by using the following method: from: Fractal-based texture analysis, K L Chan, 16-20 Nov 1992, pp. 102 - 106 vol.1, in ...
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1answer
37 views

Determine all the generators of $\mathbb{Z}_{25}^{\times}$

Determine all the generators of $\mathbb{Z}_{25}^{\times}$. Is there some way that I can use the fact that $\mathbb{Z}_{25}^{\times}$ is cyclic generated by $3$?
1
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1answer
18 views

Show that zero sequences satisfy the following equation

I am working on the following problem and got puzzled: $\\$ Show that every zero sequence $(a_n), a_n \neq 0$ satisfies the equation: $$ \lim_{n \rightarrow \infty} \frac{\sqrt {1 + a_n} - 1}{a_n} = ...
0
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2answers
56 views

Optimizing a box

I'm learning the use of derivatives and I have found a problem: Supposing we want to build a box of $4000\, \textrm{cm}^3$ of volume without top and a square base. Which are the measures so we ...
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0answers
35 views

Inequality of gamma distribution

Let $X_{\alpha} \sim \text{Gammma}(\alpha,1),\alpha >0$ with distribution function $G_{\alpha}$ and $X_{\beta} <_{(c)} X_{\alpha}\,\forall ~0<\alpha<\beta<\infty$. Then show ...
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1answer
37 views

Existence of a unique maximizer of a strict quasi-concave function defined over a convex set

Set $S \subset \mathbb R^2$ is compact and convex. A typical element of $S$ is $s=(s_1,s_2) \in S$. Also, $d \in \mathbb R^2$ is a fixed element such that there exists $s \in S$ such that $s \gt d$. ...
0
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1answer
17 views

If $s_1 +s_2 \gt 1$ and $(t_1,t_2)$ be a convex combination of this with $(0.5,0.5)$ then show that $t_1t_2 \gt 0.25$

Let $(s_1,s_2)$ be such that $s_1 + s_2 \gt 1$. Let $(t_1,t_2)=((1-\epsilon )(0.5) + \epsilon s_1 , (1- \epsilon)(0.5) +\epsilon s_2)$, where $0< \epsilon \lt1$. I need to show that for $\epsilon ...
2
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1answer
72 views

A Counter Example about Closed Graph Theorem

This is an example I read around closed graph theorem. Let $Y=C[0, 1]$ and $X$ be its subset $C^\infty[0, 1]$. Equip both with uniform norm. Define $D: X \to Y$ by $f \mapsto f'$. Suppose $(f_n, f_n') ...
0
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1answer
30 views

A mapping defined by an equivalence relation $\sim$ that is compatible with a semigroup

I am working on the following: Problem: Let $(H, \cdot)$ be a semigroup and $\sim$ an equivalence relation on $H$. We say that $\sim$ is compatible with $(H, \cdot)$ $$: \iff \forall a,a',b,b' ...
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1answer
22 views

question on a stopping time problem.

I borrowed some lecture notes on stochastic calculus, which contained the following exercise: Let $(X_n)_{n>0}$ be a sequence of random variables with $X_n: \Omega \to [0,\infty)$. We set $S_n= ...
3
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1answer
62 views

Self-learning Book recommendation for topics in ring-theory

I failed badly in my Internal examination in ring theory , and at any cost want to improve upon my grades in the final eamination,with a month and a half to go .... I haven't yet covered the below ...
13
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0answers
300 views

IMO programs of different nations?

We have a good team in the IMO, and this year I can, and probably will, be part of it. Since we as a country do not have a public training programme, I have to consult the training programms of ...
0
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1answer
28 views

Continuity and Differentiability of a series of functions

Consider the function $f(x)=\sum_{n=1}^{\infty} 2^{-n}g(2^{2^{n}}x)$ where \begin{equation} g(x)=\begin{cases} 1+x &-2 \le x \le 0 \\ 1-x &0 \le x \le 2 \end{cases} \end{equation} where ...