Questions about studying mathematics without formal instruction.

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13
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0answers
695 views

Efficient ways to read and learn a new topic

I started reading the book "Topology without tears" by Sidney A Morris and lecture notes on "Elementary Number Theory" by WWL.Chen. To get the maximum out of the book and understand the material ...
10
votes
0answers
934 views

Which is better strategy to learn and read books, traditionally one by one OR re-read carefully on perfect books

(Just focus on how to learn and master the stuff pretty well, not involve the aspect of courses or exam) Because recently I always feel that the time and energy are pretty limited, I want to try ...
7
votes
0answers
201 views

Modern research into Grassman's “theory of forms”?

I quote from Petsche's Hermann Graßmann: Biography (emphasis mine): The mathematical part of the book begins with the conception of the “General Theory of Forms”. Starting with a perspective on ...
7
votes
0answers
186 views

How much time is reasonable to complete baby Rudin?

I've been teaching myself math for more than a year. My current aim is towards algebraic topology and differential geometry. Apart from a messy (by which i mean some rigorous and some not) ...
6
votes
0answers
103 views

What is the best way to go about learning math?

I know this is a very subjective question, but after struggling on my own for a while I figured I might as well ask it. I did all the normal math classes in college (LinAlg, MultiVariable Calc, ...
6
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0answers
136 views

Advice on learning mathematics

I know it is hard to ask for the perfect method for doing mathematics, but I hope there are some paths that are more preferable over others. I am a engineering student who has switched to mathematics ...
5
votes
0answers
139 views

The high road to learn algebraic geometry

Suppose that a student has a basic knowledge in commmutative algebra at the level of the Atiyah-MacDonald. What do you think about the following steps to learn algebraic geometry? step 1: read ...
4
votes
0answers
62 views

Transitioning to Higher Level Mathematics

I am just finishing grade 12 pre-calculus at my school and have strong interest in math. The problem is, it seems some important elements of higher level math are not in my schools curriculum that are ...
4
votes
0answers
86 views

Prove that a series is $O(t^a)$.

Consider the series $$ u(t,x) = \sum_{i \geq 1} {u_i(x) t_1^i } + \sum_{i+2j \geq k+2, j\geq 1} {\varphi_{i,j,k}(x) t_1^i t_2^j y^k} $$ where $t \in \tilde{\mathbb{C} \setminus \{ 0 \}}$, $x$ is ...
4
votes
0answers
209 views

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 ...
4
votes
0answers
114 views

Where do I go from Linear algebra past Calc III to try to learn complex physics (relativity and quantum group theory)?

I'm mainly a programmer, but I have a love for Mathematics that's been, well, insatiable. I've had my eye on learning Quantum Groups and Relativity, but I want to stay in something I can do with ...
4
votes
0answers
188 views

How does one study with many textbooks?

Suppose you wanted a good understanding of a subject, and so you start reading a recommended textbook. But it turns out that the textbook you have has a different perspective/omits certain topics, ...
3
votes
0answers
38 views

WKB and asymptotic behavior of second order differential equation

I want to study the large $x$ solution to a Riccati equation. After listening to the lectures on Mathematical Physics by Carl Bender, I have fallen in love with asymptotic analysis. But, by no means ...
3
votes
0answers
39 views

Where does this series converge?

Let $ \{r_1, r_2 ,r_3,... \}$ be an enumeration of $\mathbb{Q}$. For each $r_n \in \mathbb{Q}$ define: $$u_n(x)=\begin{cases} 1/{2^n} & x>r_n \\ 0 & x \leq r_n \end{cases} $$ and let $$h ...
3
votes
0answers
36 views

Prime divisor of the form $2kp+1$ that divides $2^p-1$

The book that I'm reading (Elementary Number Theory by Underwood Dudley) gives a Theorem: If $p$ and $q$ are odd primes and $q|a^p-1$, then either $q|a-1$ or $q=2kp+1$, for some integer $k$. Then it ...
3
votes
0answers
46 views

Totient Function problem

Suppose we know that $(m,n)=2$. Show that this implies that $\phi{(mn)}=2\phi{(m)}\phi{(n)}$. My attempt: So let $m=p_1^{r_1}p_2^{r_2}...p_k^{r_k}, n=p_1^{s_1}p_2^{s_2}...p_k^{s_k}$. Then ...
3
votes
0answers
94 views

Examples of Talagrand's inequality

I am trying to understand Talagrand's inequality and when it gives better results than Markov/Chebyshev/Chernoff. However I find the formal definition hard to understand. Are there any nice simple ...
3
votes
0answers
68 views

Interchange of finite sum with convergence sequences.

Hi everyone I'm wondering if the following proof is correct (to be honest at the beginning I have some troubles to understand what the sequences of the sums of convergent sequences has to be, but I ...
3
votes
0answers
97 views

Where to start?

I want to learn Mathematics but I don't know where to start. Sometimes I really get frustrated as I am a Software Engineering graduate (currently working) and I feel like I don't know anything about ...
3
votes
0answers
180 views

Convex analysis books and self study.

I have taken some courses in Convex optimization. Now I would like to know a little bit more about the pure mathematical side. Is there any good books in convex analysis? I have read and worked with ...
3
votes
0answers
215 views

Are Specific Facts about the Riemann Integral Logically Required?

This question is somewhat in the spirit of this one in that I am trying to understand the most efficient path to the major integral theorems (Fubini, change of variables, etc). My question is this: ...
2
votes
0answers
22 views

Where can I find proof - There're infinitely many primes $p$ such that $p(mod\ N)\not\in H$ - Name?

Origin - http://math.uga.edu/~pete/4400FULL.pdf - on p120, Theorem 122 Fix a positive integer $N>2$, and let $H$ be a proper subgroup of $U(N)=(Z/NZ)^{\times}$. There are infinitely many ...
2
votes
0answers
42 views

How to count Dynkin system for finite sets?

For a set of finite elements, is there a good way to list all of its Dynkin systems, please? I understand that all $\sigma$-algebras of a set are also Dynkin systems. Therefore, we should as many ...
2
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0answers
29 views

Calculus: Reduction formula

For this question, I can find out $I3$, but I have no idea how to find the reduction formula. Please advise me.
2
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0answers
60 views

Proof involving prime factorization

I'm beginning some self-study in Number Theory and have come across a problem that I'm not really sure how to solve. Here's the problem: Prove that, if, $$ a=q_{1}^{e_{1}}q_{2}^{e_{2}} . . . ...
2
votes
0answers
72 views

Difference between Metric Space and Topological Space

I am reading Chapter 11 of Real Analysis written by Royden and Patrick (4th). It says "The concept of uniform convergence of a sequence of functions is a metric concept. The concept of pointwise ...
2
votes
0answers
109 views

Unordered summation

I have a question regard unordered sumation. Definitions: Let $X$ be a set (possible uncountable) and let $f: X\rightarrow \mathbb{R}$ be a function. We say that $\sum _{x\in X}f(x)$ converges ...
2
votes
0answers
72 views

The Heine-Borel Theorem for the real line

Hi everyone I'd like to know if the following argument is correct and also I'm very interested in a constructive approach for (2)$\Rightarrow$(1) (a link or a hint it will sufficient for me) I was ...
2
votes
0answers
28 views

Question on numerical quadrature and precision

I am studying numerical analysis and I came across this question: Let $P_{n+1}(x)\in\Pi_{n+1}$ be orthogonal to $\Pi_n$ relative to a weight function $w(x)\geq 0$ on $[a,b].$ Denote by ...
2
votes
0answers
51 views

Partially ordered set exercise

Hi everyone is my first time working with posets and I'd like to know if the solution of the next exercise is really correct or maybe there is some errors that I cannot see. Thank in advance. ...
2
votes
0answers
44 views

Set Theory - Well Order (Lexiographical combination)

Question: Prove constructively that if $(A_{1},\prec_{1})$ and $(A_{2},\prec_{2})$ are two well-ordered sets then their lexicographical combination $(A_{1} \times A_{2},<_{1,2})$ is also well ...
2
votes
0answers
47 views

Counter-Example for $f$ unbounded, $f$ continuous at $s$, and $\alpha(x)=I(x-s)$ such that $\int_a^b f \,d\alpha \neq f(s)$ (Rudin Thm 6.15)

I'm reading Theorem 6.15 in Baby Rudin, which states that: Theorem 6.15: If $a<s<b$, $f$ is bounded on $[a,b]$, $f$ is continuous at $s$, and $\alpha(x)=I(x-s)$, then $$\int_a^b f \,d\alpha = ...
2
votes
0answers
91 views

Absolute Convergent Series Laws (Exercise)

Me again, I have troubles to understand the concept of absolute convergence in the general setting when the set can be uncountable. In the book what I read says: Definition: Let $X$ be a set ...
2
votes
0answers
140 views

Fubini's Theorem for Infinite series

In the book what I've read, there is one point where the author suggest to begin the proof of the Fubini's Theorem for infinite sum in the case when is non-negative after this try to generalize. But ...
2
votes
0answers
34 views

Why can the group of isomorphism classes of line bundles be identified with $H^1(C,\mathbb O_C^*)$?

This is a reference request to the fact in the title. Is there a book at most as advanced as Hartshorn's which explains this result?
2
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0answers
21 views

How to find required surface.

Hi I am studying this question. I understand all parts expect for the part I posted below. Please explain it. Thank you so much. I am new learner of PDE.
2
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0answers
63 views

Progress upwards?

I'm looking for a somewhat sequential list of books to learn math beyond calculus at home. I've taken calc 3 and am going further but I'm having a lot of trouble nailing down a real order of things ...
2
votes
0answers
251 views

Schaum's Outlines for self-study

I wish to learn Physics via self-study. I realize that I have a long way to go as far as Mathematics prerequisites: I will need to begin with Precalculus(I took that course many years ago, so I am ...
2
votes
0answers
222 views

Isomorphism of groups with certain property, p-groups

I'm doing exercises from Hungerford's book "Abstract Algebra: An Introduction". The exercise is in section 8.2, numbered 22. I would like someone to check my proof, as I have reasonable doubts that ...
1
vote
0answers
19 views

Existence of Non-Trivial, Convex, Open Set in $C_{\mathbb{C}}[0,1]$ Under $L^{0}$ Metric

I've been struggling with this problem for the last four hours. The problem is to show that the space of $\mathbb{C}$-valued continuous functions on $[0,1]$ under the metric ...
1
vote
0answers
15 views

Are they $\pi$ systems?

I am not sure whether the following two systems are closed under finite intersections. $\{(a,b):-\infty<a<b<\infty\}$: I do not think it is if I consider $(0,1)\cap(1,2)=\emptyset\notin ...
1
vote
0answers
30 views

Show a subset A is compact if and only if the image of the map T(A) is compact

The question is as follows: Let $\{v_1,v_2,\dots,v_n\}$ be a set of linearly independent vectors of an $n$-dimensional normed linear vector space $V$. Define the map $T: R^n \to V$ by $T(a) = ...
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 ...
1
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0answers
21 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 ...
1
vote
0answers
36 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 ...
1
vote
0answers
22 views

how to find matrix nonsingular P and Q such that $PAQ $ is a normal form?

Given matrix $A$: $$\pmatrix{1&1&1\\2&2&2\\-1&1&-3\\1&2&0}$$ how to find matrix nonsingular P and Q such that $PAQ $ is a normal form? Thanks!
1
vote
0answers
41 views

What does “for sufficiently large $n$” mean?

The question essentially is, In a general/common context what does "for sufficiently large $n$" mean? My initial grasp of the phrase went, "for an $n$ that is greater than a required bound". ...
1
vote
0answers
12 views

Lower bound for non-negative definite matrix

I wonder if the following inequality is true, which I can not prove: $$ e^T A^{-1} \mathrm{diag}(A) \geq 1 $$ where $A$ is non-negative definite matrix, $\mathrm{diag}(A)$ is the vector of diagonal ...
1
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0answers
21 views

Fourier Transform of $sin(5t - \frac{\pi}{4})U(t+8)$

I have this function $$ sin(5t - \frac{\pi}{4})U(t+8) $$ I know the Fourier Transform of $sin(5t - \frac{\pi}{4})$, which is $$ \frac{e^{-\frac{\pi^2}{2}fj}}{2j}\left [\delta (f-\frac{5}{2 \pi}) ...
1
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0answers
77 views

$(x^r)^s=x^{rs}$ for the real case

Hi everyone I'd like to know if the following is correct and if someone knows a better way to do it. Definition Let $x>0$ be a real, and $\alpha$ be a real number. We define the quantity ...