Questions about the process of studying mathematics without formal instruction.

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13
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0answers
304 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 ...
7
votes
0answers
983 views

Learning higher-mathematics on your own

I was hoping someone had an opinion on how to learn higher-mathematics (specific fields that could be of use to me) outside of a classroom setting. I graduated with an M.S. in Computer science about ...
6
votes
0answers
126 views

Finitely additive function on an infinite set, s.t., $m(A)=0$ for any finite set and $m(X)=1$ (constructive approach)

Other exercise which I found in Dudley's Analysis book: Show that there is a measure on a infinite set $X$, defined on $2^X$ s.t. is finitely additive, $m(A)=0$ for any finite set and $m(X)=1$. ...
6
votes
0answers
165 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
votes
0answers
161 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 ...
6
votes
0answers
176 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 ...
5
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0answers
139 views

Analysis or (abstract) algebra first?

Which one would you recommend? I only know calculus and linear algebra when it comes to university-level mathematics. Is one required to understand the other?
5
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0answers
95 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
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0answers
64 views

How can I retain the mathematics that I've supposedly learnt?

So my question simply is "What is the best method to make sure you retain what you have learnt?" Okay so I've tried learning mathematics up to where I should be at in the past. Every time though I ...
4
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0answers
75 views

How to select good exercises?

I'm studying on Rudin "Principles of of Mathematical Analysis" which I begin to find as a good and complete reference. I wonder how many exercises shall I do at the end of each chapter ? In case of ...
4
votes
0answers
91 views

Compact family of Lip functions under the sup norm metric, proof verification.

Hi everyone I'd like to know if the following is correct, I'd appreciate your opinion and also any suggestion to improve my argument. Thanks in advance for your time. If $(K,d)$ is a compact ...
4
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0answers
32 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 ...
4
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0answers
91 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
379 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
121 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
345 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
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0answers
70 views

Geometry textbook question

I have just started the textbook Geometry: its elements and structure by Alfred Posamentier. The first set of questions refers to the following diagram: The very first question is "What is the ...
3
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0answers
44 views

Finding dominating integrable function

Hi everyone I'm not completely familiar with this kind of argument and I'd appreciate if someone can help me to see if the argument is correct and also any suggestion to improve it. Thanks in advance. ...
3
votes
0answers
69 views

Periodic curve on unit sphere and torsion

Define $S^2 \subset \mathbb{R^3}$ be the unit sphere. Suppose that $\alpha :\mathbb{R} \to S^2$ is a differentiable curve parametrized by arc-length. a) Show that $\kappa(s)$, the curvature of ...
3
votes
0answers
59 views

How to approach real analaysis

I'm just starting first year in university in Europe and here there there is no Calculus, instead you jump right into Analysis. The trouble is, for some time I self-studied through US style books and ...
3
votes
0answers
219 views

Which is the best transitional mathematics book for self-teaching among the ones listed?

What is Mathematics, An Elementary Approach to Ideas and Methods - Courant Robbins Stewart How to Solve It, A New Aspect of Mathematical Solving - Polya Introductory Mathematics, Algebra and Analysis ...
3
votes
0answers
210 views

Isolation and self-study

A little background: I am currently a sophomore (studying mathematics) at an unknown university in the Middle East. My mother is European so it does not make sense to study mathematics in the Middle ...
3
votes
0answers
82 views

Cauchy-Euler Equation of order $n$

What I wish to prove is that for a Cauchy-Euler equation of order $n$, the substitution $x=e^{t}$ transforms it into a linear differential equation with constant coefficients. To put it as a theorem: ...
3
votes
0answers
88 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
45 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
57 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
52 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
112 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
354 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 ...
3
votes
0answers
112 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
131 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
239 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
18 views

Learning from Alternative Sources

I have a very general question about people's experiences with learning math. I can think of a couple of times where I had the following situation. I was seeking to learning about topic A. However, ...
2
votes
0answers
72 views

Background & Advice for a self-learner of Descriptive Set Theory

A rather straight to the point soft-question: What kind of background should have somebody who wants to study properly descriptive set theory? More specifically, how much analysis should she/he ...
2
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0answers
35 views

Proof of Heisenberg Uncertainty Principle Exercise

I'm not very knowledgeable in QM, and I know many physics books derive the uncertainty principle using commutators, but as an exercise in my PDE book (by Asmar), I should be able to derive it from one ...
2
votes
0answers
51 views

Directional derivative (Vector)

Given $f:\mathbb{R}^2 \to \mathbb{R}^2$ is a map $f(x,y)=(u(x,y),v(x,y))$ and $\alpha=(\alpha_1,\alpha_2)$ is a point, then how does one show that $f$ is differentiable (or not) in the direction ...
2
votes
0answers
30 views

Continued fraction approximation to a function and its derivative

I am recently working on fitting a model with incomplete beta function. In order to put it into my optimization algorithm, I must find out the derivatives of the incomplete beta function $B_p(x,y)$ ...
2
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0answers
62 views

Convergence of Expectations (cont'd)

The question is related to this question. Suppose $\{X_n\}$ is a sequence of indep. random variables with zero expectation. Consider their sum $S_n$ which have the following properties: $S_n$ ...
2
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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 ...
2
votes
0answers
74 views

measure theory exercise (verification)

Hi I found the following exercise in the Dudley's book and I'd like to see if my answer is correct; the last part is what I'm not entirely sure, since I'm not completely familiar with this kind of ...
2
votes
0answers
77 views

self-study hints

A question to those who took rigirous courses like math 25 (Harvard), MATH 295-396 Michigan and etc Being not able to collectively discuss problem sets from the course, as those who involved in ...
2
votes
0answers
134 views

A Compact Real Analysis book for a graduate student, who is short of time.

I am a Phd student in Computer Science and I want to focus on Machine Learning, especially on statistical methods. My problem is, I always keep hitting the wall when it comes to studying underlying ...
2
votes
0answers
92 views

In what order should I study?

I would like to study the basic fundamentals of mathematics from the beginning and move on from there as my understanding of the subject is lacking. In what order should I study? Arithmetic ...
2
votes
0answers
87 views

Rotate the unit circle by a fixed angle, what does happen is $\alpha/\pi$ is rational? and irrational?

Hi everyone I´d like if someone could say me if the following is correct. Thanks in advance Rotate the unit circle by a fixed angle $\alpha$, say $R: C \rightarrow C$; $(1,\theta)\mapsto ...
2
votes
0answers
132 views

Most Suitable Book after Kline's Calculus?

I've been working through Morris Kline's Calculus: An Intuitive and Physical Approach and it's an absolutely excellent book for self-studying applied single-variable (and some multi-variable) calculus ...
2
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0answers
74 views

Continuity of the right-hand derivative of a Convex function (help with the proof)

Hi everyone I have some trouble with one point in the following proof. Let $f$ be a convex function (strict convex function) on a real interval. If $f'_-(a)=f'_+(a)$ where $f'_-$ and $f'_+$ are ...
2
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0answers
44 views

Revise high school material

Can you suggest me a comprehensive book to revise high school mathematics (up to besic calculus)? It should be extremely clear and complete and "scientific" (not like most high school books). Thank ...
2
votes
0answers
53 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
votes
0answers
35 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
votes
0answers
69 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}} . . . ...