Questions about studying mathematics without formal instruction.

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Example for a set in $\Bbb R^p$ whose interior is $\emptyset$ and closure is $\Bbb R^p$

The following exercise was in the Elements of Real Analysis by Bartle. Give an example of a set $A$ in $\Bbb R^p$ such that $A^{\circ} = \emptyset$ and $A ^ - = \Bbb R^p$. Can such a set be ...
2
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1answer
134 views

Proof Verification: Show sequence is bounded and find limit: $x_1 \gt 1$ and $x_{n + 1} = 2 - \frac{1}{x_n}$

Came across the following exercise in Bartle's Elements of Real Analysis and am a little unsure about my solution. Would be extremely grateful if someone could verify it for me. Let $x_1 \in ...
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6answers
5k views

Why can't you pick socks using coin flips?

I'm teaching myself axiomatic set theory and I'm having some trouble getting my head around the axiom of choice. I (think I) understand what the axiom says, but I don't get why it is so 'contentious', ...
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0answers
39 views

Please check the question: Compute $EX$

Question: A box contains $10$ balls numbered $1,2,\ldots,10$. A random sample of $7$ balls is selected. $X=$ the smallest of the numbers drawn. Compute $E(X)$ $R(X)= \{1, 2, 3, 4\}$ ...
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0answers
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On Lucas Lehmer primality Test

http://primes.utm.edu/notes/proofs/LucasLehmer.html is proof of the Lucas Lehmer Test I read. The part I do not understand is why did he consider the sequence $S_n=S_{n+1}^2-2$. I mean why would ...
1
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0answers
29 views

Continuity of a positive preserving operator between C(X) and C(Y)

I've been struggling with this question in Reed and Simon while I'm prepping for quals. Suppose that $T:C(X)\rightarrow C(Y)$ is a positive operator. Prove that T is continuous and $\Vert ...
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1answer
49 views

circular differentiation

Suppose one starts with a function $f: \mathbb R^2 \rightarrow \mathbb R$ using $\mu, \sigma^2$ as its input, i.e. $f=f(\mu, \sigma^2)$. (Note that here I omitted the specific form of $f$ since I ...
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1answer
48 views

$a, z_1 \gt 0$ and $z_{n + 1} = (a + z_n)^{\frac 1 2}$ then $(z_n)$ is monotone and bounded?

Having trouble with the following exercise in Bartle's Elements of Real Analysis. Let $a, z_1 \gt 0$. Define $z_{n + 1} = (a + z_n)^{\frac 1 2}$ for $n \in \Bbb N$. Show that $(z_n)$ converges. ...
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4answers
1k views

Learning trigonometry on my own.

I have been self teaching myself math beginning with a grade 10 level for a while now and need learn trigonometry from near scratch. I am seeking both books and perhaps lectures on trigonometry and ...
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0answers
24 views

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 ...
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3answers
156 views

Derivatives of sine and cosine at $x=0$ give all values of $\frac{d}{dx}\sin x$ and $\frac{d}{dx}\cos x$?

In video 3 of the video lectures by MIT on Single Variable Calculus presented by David Jerison, the latter says: Remarks: $\dfrac{d}{dx}\cos x\left|\right._{x=0}=\lim\limits_{\Delta ...
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1answer
1k views

What are the prerequisites for stochastic calculus?

I am not a math student, and only kind of picking up something whenever I need it. After emerged in the field of machine learning, probability, measure theory and functional analysis seem to be quite ...
7
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3answers
245 views

Self-study: what fractions of problems to solve?

I am self-studying measure-theoretic probability out of Billingsley's Probability and Measure. So far I have been trying to solve all the exercises. While the exercises are wonderful and I can ...
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2answers
57 views

primitive root of residue modulo p

I was trying to prove that for the set $\{1,2,....,p-1\}$ modulo p there are exactly $\phi(p-1)$ generators.Here p is prime.Also the operation is multiplication. My Try: So I first assumed that if ...
0
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1answer
21 views

Probability question.

How many ways are there to distribute 2 indistinguishable white and 4 indistinquisable black balls into 4 indistinquisable boxes? If the question is asked as "distinct boxes", I can solve. But now, ...
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4answers
89 views

Does $\int_1^\infty \frac{\log x}{x^{3}} \sin x \,dx $ exist?

I want to determine if the following indefinite integral exists: $$\int_{1}^{\infty} \frac{\log x}{x^{3}} \sin x dx.$$ I tried to solve the integral then calculate the limit $$ \lim_{\lambda \to ...
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1answer
96 views

How to prove the supremum of this set is 1

Let $B = \{n/(n+m) : m, n \in \mathbb{N}\}$, find the supremum and prove it is indeed the supremum. I claim that $\sup(B) = 1$ and I prove it using the following definition: So to prove $(i)$, ...
0
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1answer
82 views

A different type of Convergence of Fourier Series

I have just started studying fourier series. All the convergences I have seen considered the partial sums to be $\sum\limits_{i=-n}^n a_n Sin(n\theta)$. But in all practical systems the harmonics ...
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1answer
53 views

A question about outer measure.

When two sets are positively separated we know that $\mu(A \cup B)=\mu(A)+\mu(B)$. My question is what happens when their intersection is null. Will the above equation be invalid? My Try:It has to be ...
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1answer
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How to prove that two curves are not path homotopic

I have a unit circle around origin.And another unit circle around $(2,0)$. Consider the domain $R^2 / \{(0,0)\}$. I am able to clearly see that both are not homotopic but i am unable to prove it ...
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2answers
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Uniqueness of Topology and Basis

In measure theory, we know there is a (unique) minimal $\sigma$-algebra generated by a generator. I am wondering whether this applies to topology and its basis. There are two directions to consider ...
0
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1answer
61 views

Periodic absolute value function

Define $$h(x)=|x|$$ on the interval $[-1,1]$ and extend the defintion of $h$ to all of $\mathbb{R}$ by requiring that $h(x+2)=h(x)$. Now define the function: $$h_n (x)=\frac{1}{2^n} h(2^n x)$$ ...
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0answers
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Trimmed mean is translation invariant Statistics.

I want to know why trimmed mean is a translation invariant statistics? I know that if $\mu$ be a measure on $(\mathbb R^n, \mathcal B)$ then it is called translation invariant iff $\forall x \in ...
4
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1answer
144 views

indefinite integral $\int\sin\sqrt[3]{x}~dx$

I want to determinate the integral $\int\sin\sqrt[3]{x}~dx$ . I tried to use integration by partitions and integration by substitution but I came to no result. I know the result which is shown here ...
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1answer
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$F,G \in \text{End} (V)$ share the same eigenvalues for $F \circ G$ and $G \circ F$

Problem: Let $V$ be a finite dimensional Vector Space over a field $\mathbb{F}$ and $F,G \in \text{End}(V) $ Show that $F \circ G$ and $G \circ F$ have the same Eigenvalues $\lambda$ My ...
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0answers
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How to deform a curve in specific manner

I am wondering whether we can deform a path in specific ways continuously i mean if there is a closed piece wise $C^1$ smooth path which has to be deformed to another piece wise $C^1$ smooth path. Let ...
3
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2answers
85 views

Linear Algebra Self Study

I'm currently a high school student with a love for math. I have taken Plane and Coordinate Geometry, both Algebra I and II, Trigonometry, and am halfway done with Calc A. I want to major in quantum ...
0
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0answers
151 views

Set of infinitely differentiable functions compactly supported in a domain of $\mathbb{R}^n$ not dense in $L^\infty$

How does one show that the set of infinitely differentiable functions compactly supported in a domain $\Omega\subset\mathbb{R}^n$ is not dense in $L^\infty(\Omega)$? Thanks!
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0answers
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Conditional probability question

Please check the conditional probabilty question I posted. I solved this. But I am not sure. Thank you:)
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0answers
71 views

Which proof can I skip? [closed]

Suppose that a student is doing self study from some book: "Introduction to subject $X$" because he wants to learn the first things about the subject $X$. Now, I know that the first rule is "do lots ...
0
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0answers
73 views

Learning about the gamma function.

I have just started learning about the gamma function but the books I have are not sufficient to give me a complete picture of it. Can you guys suggest some online resources/free books where I can ...
1
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0answers
38 views

How can I finish formulating this problem?

I'm a software engineer with a very limited background in maths, and I'm trying to teach myself to think more mathematically as I try to learn more about maths. I'm currently trying to formulate a ...
3
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2answers
212 views

Is there another Analysis book that is based on the Cartesian space $\Bbb R^p$

I am in the middle of a slightly ambitious attempt to learn Analysis on my own. I skimmed through Rudin(Baby), Chapman Pugh, William Wade, Stephen Abbott and Strichartz and ended up preferring the ...
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3answers
785 views

Careers in Math

I am a highschool freshman, and I really like to have goals for my life, one of the big ones is my career of choice. Previously, I have always wanted to be a programmer, and I have written a lot of ...
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2answers
245 views

Careers in Mathematics?

I am a college freshman, and I really like to have goals for my life, one of the big ones is my career of choice. Previously, I have always wanted to be a programmer, and I have written a lot of code. ...
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2answers
1k views

Why is cross product only defined in 3 and 7 dimensions? [duplicate]

Why $3$ and $7$? I know from some reading that Hurwitz's Theorem explains this, but can someone help me build some intuition behind this or perhaps provide a simpler explanation? It still seems ...
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2answers
83 views

Time managment while self-studying rigorous mathematical textbooks

Lately I have been introducing myself to Calculus.But I have an issue with time. Namely I have found that on average I need about 2-3 hours per page of material.This of course includes finishing all ...
5
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1answer
204 views

When should I start learning Set Theory?

I started to learn a few disciplines on my own over the break after my first year in college and one of them was Real Analysis. In the process I came across many issues in Analysis texts concerning ...
0
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1answer
46 views

Pre requisites of linear algebra

I want to learn abstract linear algebra. Do I require the knowledge of discrete mathematics before I start? I have the impression that abstract maths and their proofs can be understood easily by the ...
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5answers
1k views

What's the best way to measure mathematical ability?

Very soft question I admit, but it's something that's been bothering me for a while. I've been thinking that being self taught has the problem of accreditation. You can't evaluate a mathematician ...
134
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6answers
5k views

Best Sets of Lecture Notes and Articles

Let me start by apologizing if there is another thread on math.se that subsumes this. I was updating my answer to the question here during which I made the claim that "I spend a lot of time sifting ...
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1answer
57 views

Tools or Resources for pictures and visualizations

The popularity of books like Visual Group Theory and Visual Complex Analysis validates the importance of pictures and visualization for complex subjects. Unfortunately, I'm not aware of similar books ...
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4answers
3k views

Books, Video lectures, other resources to Teach Yourself Analysis

So my limited mathematics education has been especially ignorant of analysis. In this vein, I'd like to teach myself some of the introductory basics. I'm intrigued by sources that might contain ...
3
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5answers
258 views

Number of groups of a given order

In general, for what $n$ do there exist two groups of order $n$? How about three groups of order $n$? I know that if $n$ is prime, there only exists one group of order $n$, by Lagrange's Theorem, but ...
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2answers
80 views

Analysis/Inequality question about proving an infinite product greater than 0

This is from David Williams' book Probability using Martingales. I'm self-studying. Question Prove that if $$0\leq p_n < 1 \quad\text{ and }\quad S:=\sum p_n < \infty$$ then $$\prod (1-p_n) ...
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1answer
80 views

How to calculate the tensor product?

This question might be stupid for you.I ask because I have no clue about it. I don't really understand what is tensor product,although I know its definition. I have search what is tensor product,so I ...
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2answers
178 views

Geodesics on torus

Describe the geodesics on Torus $$\sigma (u,v)= ((a+b \cos u)\cos v, (a+b\cos u)\sin v, b\sin u)$$ First fundamental form for torus is $$b^2 du^2 +(a+b \cos u)^2dv^2$$ Consider unit-speed ...
3
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2answers
67 views

Geodesics on spheroid

Describe the geodesics A Spheroid obtained by rotating the ellipse $\frac{x^2}{p^2}+\frac{z^2}{q^2}=1$ around the z-axis where $p, q\gt 0$ Please explain this question explicitly. Thank you:)
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2answers
51 views

Find the number of real solutions of the system of equations.

Find the number of real solutions of the system of equations. $$x^2-y^2=z\\ y^2-z^2=x\\z^2-x^2=y$$