Questions about studying mathematics without formal instruction.

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2
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1answer
43 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 ...
0
votes
0answers
29 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 ...
0
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0answers
7 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
votes
1answer
87 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. ...
1
vote
0answers
20 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 ...
3
votes
1answer
58 views

I love maths, but my school is limited in its teachings. [closed]

I'm not sure how well this question will be received but, I feel there is no harm in asking. I'm 17 soon to be 18. I was in my applied maths course, but not anymore, not due to difficulty, but I can ...
1
vote
2answers
77 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. ...
1
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0answers
29 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 ...
1
vote
1answer
48 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
votes
1answer
125 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
votes
2answers
39 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
vote
1answer
56 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 ...
1
vote
2answers
68 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
votes
1answer
46 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
votes
1answer
75 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.
0
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1answer
26 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
votes
2answers
149 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 ...
0
votes
1answer
76 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 ...
6
votes
2answers
46 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 $a+b>c,b+c>a,a+c>b$ ...
1
vote
0answers
78 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 ...
0
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0answers
20 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 ...
1
vote
1answer
32 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
vote
1answer
17 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
votes
2answers
55 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 ...
0
votes
0answers
32 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 ...
0
votes
1answer
26 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
votes
1answer
16 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
votes
1answer
61 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
votes
1answer
29 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' ...
0
votes
1answer
20 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
votes
1answer
52 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
265 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
votes
1answer
27 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 ...
0
votes
2answers
18 views

Converges of a sequences defined through a continued fraction

Consider the following sequence $(b_n)_{n \geq 1}$ recursively given through the continued fraction $b_1 = \frac{1}{1}$, $b_2 = \frac{1}{1+ \frac{1}{2}}$, $ \dots , b_n = \frac{1}{1+b_{n-1}}$ ...
0
votes
2answers
51 views

Find the equation of the parabola given the tangent to a point and another point.

I have a problem with derivatives, I've been trying to solve but I was not able to do it. A parabola is tangent to the line $3x-y+6 = 0$ in the point $(0,6)$ and goes through the point $(1,0)$. ...
1
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0answers
27 views

(soft question) Kreyszig's Functional Analysis or Rudin's Real and complex analysis for an introduction into more advanced analysis?

I realise they are quite different in their approach and material covered, but they share the central stuff like normed/Banach/Hilbert spaces, Hahn-Banach theorem etc. Not really understanding what ...
1
vote
1answer
59 views

Categories and the direct image functor

I've just started learning about sheaves and yesterday I had a look at MacLane/Moerdijk's book "Sheaves in Geometry and Logic". I've discovered something in this book on which I'd like some ...
7
votes
3answers
90 views

How do I decide what problems and how many problems to do when I try to self study?

I am a math major at a relatively small college with barely any choice of classes to choose from so I have to supplement my studying with a lot of self studying. I usually have no problem getting ...
0
votes
1answer
24 views

A Question on the Scaling Invariance of Brownian Motion

I read the following paragraph. Let $B_t, \ t \in [0, \infty)$ be a standard linear Brownian motion. For each $q > 4$, define the following sequence of sets. $$ \Omega_k := \left\{\omega \in ...
1
vote
1answer
36 views

How to show this inequality in the law of iterated logarithm

Let $\psi(t) = \sqrt{2t\log\log t}$ and $q>4$. Then we have for large $k$, $$ \psi\left(q^k - q^{k-1}\right) \geq \psi\left(q^k\right) \left(1-\frac{1}{q}\right). $$ To prove this, I can tell by ...
2
votes
1answer
17 views

Zero divisors and inverstible elements

I have learned about $X_n = \mathbb{Z} / n\mathbb{Z}$. I understand that a zero divisor is an element $x\neq 0$ in $X_n$ such that $xy = 0$ for some $y\neq 0$. I understand that an element $x$ in ...
0
votes
0answers
26 views

On Nonlinear Autonomous system of two equations if the eigenvalues of the Jacobian matrix are 0.

Suppose we have a non-linear autonomous system of two equations: $$\begin{cases} x'(t) = F(x,y) \\ y'(t) = G(x,y) \end{cases} $$ and we obtain a fixed point for this equation, but the eigenvalues of ...
0
votes
3answers
97 views

I want to study higher mathematics. Where do I start?

Since last year, I've been interested in higher mathematics and don't really know where to start and don't know what knowledge I still need to obtain, before being able to understand the concepts. ...
0
votes
0answers
30 views

Total Variation of a Signed Measure

In Royden's Real Analysis the total variation $|\mu|$ of a signed measure $\mu$ is defined to be $$|\mu|(E) := \sup\sum_{k=1}^n |\mu(E_k)|,$$ where the supremum is taken over all finite disjoint ...
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vote
2answers
37 views

A Question on Bolzano-Weierstrass Theorem

The following are two equivalent statements about the Bolzano-Weierstrass theorem. Every bounded real sequence has a convergent subsequence. A subset of $\mathbb R$ is compact if and only if it is ...
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0answers
23 views

On a cauchy problem (differential equation system) .

I am approaching the theory of these kind of problems but I am missing an example. I am tasked to solve: $$X'= \left( \begin{array}{ccc} 1 & 4 \\ 1 & 1 \\ \end{array} \right)X + \left( ...
0
votes
1answer
27 views

Convergence of $\sum \frac{i^n}{\log(n)}$

Study the convergence of $$\sum_{n \geq 2 } \frac{i^n}{\log(n)} $$ where $\log$ of course denotes the 'natural logarithm' and $i \in \mathbb{C}$ Oddly enough I managed to show Abel's Test for ...
1
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0answers
27 views

Limit of a random walk

Suppose we have a simple random walk: $X_n\pm1$ with equal probabilities. For any finite $n$, $E[\sum_{k=1}^nX_k]=0$. Does it imply that $E[\lim_{n\rightarrow \infty}\sum_{k=1}^nX_k]=0?$ Thank's! ...
-1
votes
1answer
28 views

How to derive this inequality

I learnt that for a standard normal random varialbe $Z$ and positive $x$, we have $$\mathbb P (Z > x) \geq \frac{x}{x^2+1} \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} = \frac{1}{x+\frac{1}{x}} ...
1
vote
0answers
44 views

Textbooks to complete concurrently - Self learning empowerment

A user is completing some year challenge that takes them through $9$ textbooks and they are alternating in author. Algebra - Cohn Analysis - Rudin Topology - Lee Repeat three times. I would like ...