Questions about studying mathematics without formal instruction.

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2
votes
2answers
57 views

find the coefficient of the given term when the expression is expanded by the binomial theorem

I am just trying to understand why the term is $\binom{15}8$(3p$^2$ - 2q)$^7$. I need to find the coefficient in $p^{16}q^7$ in $(3p^2 - 2q)^{15}$ So, I know that $n = 15$ and I have $a^{n - k}b^k$ ...
0
votes
0answers
37 views

Intuition analysis-deconstruction-reconstruction.

The following question is a refinement of this question, which caused a lot of people to give answers that were missing the point entirely, probably because the question was not clear. Being human, ...
3
votes
1answer
44 views

Isomorphism of Grassmannians

I want to prove that two CW complexes $\mathrm{Gr}_{n}(\mathbb{R}^{n+k})$ and $\mathrm{Gr}_{k}(\mathbb{R}^{n+k})$ are isomorphic to one another. I'm pretty sure I can just show that the number of ...
1
vote
2answers
43 views

Where am I going wrong? Converging sequence

I am struck with a dilemma concerning the following exercise in Bartle's Elements of Real Analysis. Determine the convergence or the divergence of the sequence $(x_n)$ given by $$ x_n = ...
0
votes
1answer
59 views
1
vote
2answers
115 views

Proof Verification - Every sequence in $\Bbb R$ contains a monotone sub-sequence

Came across the following exercise in Bartle's Elements of Real Analysis. This is the solution I came up with. Would be grateful if someone could verify it for me and maybe suggest better/alternate ...
0
votes
0answers
38 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 $E(X)$ $R(X)= \{1, 2, 3, 4\}$ ...
0
votes
1answer
104 views

Books (and supporting material) that are useful in deconstructing one's intuition?

I recently came across the following problem from Paul Zeitz's book The Art and Craft of Problem Solving. Given the image below, can you find a way to connect corresponding blocks (i.e. A to A, B to ...
1
vote
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 ...
59
votes
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', ...
0
votes
0answers
43 views

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 ...
0
votes
1answer
46 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 ...
1
vote
1answer
37 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. ...
1
vote
0answers
22 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 ...
0
votes
3answers
133 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 ...
1
vote
2answers
53 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 ...
2
votes
1answer
100 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 ...
0
votes
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, ...
2
votes
1answer
88 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 $(i)$, ...
1
vote
4answers
85 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 ...
0
votes
1answer
47 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: $$h_n (x)=\frac{1}{2^n} h(2^n x)$$ ...
0
votes
1answer
47 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 ...
0
votes
0answers
16 views

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 ...
1
vote
2answers
53 views

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 ...
5
votes
1answer
74 views

$F,G \in \text{End} (V)$ 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 ...
4
votes
1answer
111 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 ...
0
votes
0answers
42 views

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
votes
2answers
69 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
votes
1answer
119 views

Show that $L^1$ is strictly contained in $(L^\infty)^*$

How does one show that $L^1$ is strictly contained in $(L^\infty)^*$? Here, $(L^\infty)^*$ is the space of linear continuous functionals on $L^\infty$.
0
votes
0answers
136 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!
0
votes
0answers
40 views

Conditional probability question

Please check the conditional probabilty question I posted. I solved this. But I am not sure. Thank you:)
2
votes
0answers
70 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
votes
0answers
67 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
vote
0answers
37 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 ...
2
votes
1answer
86 views

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 ...
5
votes
3answers
588 views

Why are we interested in irreducible representation but not faithful representation?

I am reading some materials of representation theory (of a group). The motivation of representation theory is to represent (by a homomorphism $h: G \to GL(V)$, from the group $G$ to a vector space ...
10
votes
2answers
826 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 ...
0
votes
1answer
44 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 ...
1
vote
2answers
76 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 ...
3
votes
1answer
53 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 ...
7
votes
3answers
229 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 ...
5
votes
1answer
155 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 ...
3
votes
5answers
243 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 ...
1
vote
1answer
62 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 ...
0
votes
2answers
143 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 ...
1
vote
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. $$x^2-y^2=z\\ y^2-z^2=x\\z^2-x^2=y$$
3
votes
2answers
63 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:)
0
votes
1answer
81 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 ...
3
votes
1answer
152 views

How do you get good at reading research papers with lots of proofs?

tl;dr To get good at math proofs (and thinking math), do you have to first memorize all the different proof tricks, or is there a way to learn as you go? I am a software developer and have recently ...
0
votes
1answer
33 views

Why these two propositions have different requirements

Proposition 2.18 is similar to 2.19. Why we need $N$ flat in 2.19? What's the difference between 2.18 and 2.19?