Questions about studying mathematics without formal instruction.

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2
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3answers
180 views

Algebra Textbook

Perhaps this questions was asked already, but I browsed through other threads and couldn't find exactly what I am looking for. I am looking for an Algebra Textbook (high-school/undergrad level) that ...
1
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2answers
56 views

Please checking to find an arc-length reparametrization

Find an arc-length reparametrization of $$c(t)=\langle \cos t+t\sin t, \sin t-t\cos t\rangle$$ for $t\in [\pi, 3\pi/2]$ solution trial: $$c'(t)=\langle -\sin t+\sin t+t\cos t, \cos t-\cos t+t\sin ...
6
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2answers
105 views

Independence of sigma-algebras

Good day to everyone. While solving some problem of studying character I obtained some statement to prove, which is like following (this is my internal interest to prove it rigorously). Assume that ...
1
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0answers
21 views

Determine the direction of given parametrization.

I saw an example, which I posted below. First of all, I understand how to show paramtrized curve but I dont understand how to determine the direction of the parametrization. For example, how can ...
1
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1answer
57 views

Interspersing of integers by reals

Hi everyone I'm asking to myself if the next argument is sound. I know it is costumary to use the well-ordering principle to prove it, but at this point in the book I have not seen yet. Proposition: ...
0
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1answer
31 views

Question on the convergence of $|x|^n\rightarrow 0$ when $|x|<1$.

My question is about calculating the $N$ for a convergent sequence. I want to prove that $|x|^n \rightarrow 0$ when $|x|<1$ using the $\epsilon, N$ definition. So I know that $|x^n-0| \leq ...
1
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1answer
179 views

Venn diagram question

Here is my question. A math examination has three questions. Twenty-six students took the examination, and every student answered at least one question. Six students did not answer the first ...
4
votes
2answers
125 views

If $f^{-1}(G)$ is open in $X$ for every open set $G$ in $Y$, then $f$ is continuous. Question on proof.

Let $X,Y$ be metric spaces and $f:X\rightarrow Y$. If $f^{-1}(G)$ is open in $X$ for every open set $G$ in $Y$, then $f$ is continuous. The text I am using proves this proposition like so: Suppose ...
3
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1answer
188 views

Show convergence of a sequence of continuous functions $f_n$ to a continuous function $f$ does not imply convergence of corresponding integrals.

Let $f_n\in C([0,1])$ be a sequence of functions converging uniformly to a function $f$. Show that $$\lim_{n\rightarrow\infty}\int_0^1f_n(x)dx = \int_0^1 f(x)dx.$$ Give a counterexample to show that ...
0
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1answer
41 views

Calculate the Burning Time for a Lamp

If you have a lamp with burning time 4000 hours. If the time goes forward until the lamp will be destroyed the exponential distribution is 3675 hours, what is the probability of a lamp to be working ...
1
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1answer
200 views

Questions about coercive functions and its implications

Given this definition: A function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is $coercive$ if $$\lim_{||x||\rightarrow\infty}f(x) = \infty.$$ Explicitly, this means that for any $M>0$ there is an ...
3
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1answer
69 views

$F = \{f\in C^1([0,1])| \hspace{2mm} \|f\|\leq M, \|f'\|\leq N\}$. Showing it is precompact and not closed.

I have an example in my book: Let $C([0,1])$ denote the space of all continuous functions $f$ on $[0.1]$ with continuous derivative $f'$. For constants $M>0$ and $N>0$, we define the subset $F$ ...
1
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2answers
481 views

Proof of the limit laws (Analysis)

Hi everyone I'd like to know if my arguments for the next proof are sound or needs some changes to be correct. I hope they are not a little flaws. Proposition (limit laws): Let $(a_n)_{n=m}^\infty$ ...
0
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3answers
69 views

Exercise: product of transposition

How would I go about computing $$(1 2 3)\cdot(12)(34)$$ I know the definitions but I do not know how to apply them here. This is rather strange and odd-looking to me. I know I have to construct a ...
0
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1answer
49 views

Verification of my proof of some elementary exercises in Analysis

Hi everyone after lost the fight with the exercise in my last question I decide to continue with the book; I hope someday answer it. I finished some stack of questions and I appreciate is somebody ...
1
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3answers
67 views

Cyclic Groups in Artin's Algebra

I'm currently working through Michael Artin's Algebra in my spare time, and I seem to be stuck on an easy question in section 2.4: Cyclic groups. Let a and b be elements of a group G. Assume that a ...
3
votes
1answer
134 views

Help with a lemma of the nth root (without the binomial formula)

I have no idea of how to solve it. I would appreciate if someone gives me a hint, please. Definitions Let $\,x^{1/n}:= sup\{\, y \in \mathbb{R}: y\ge0 \text{ and } y^n\le x\, \}$ Lemma: Let ...
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0answers
47 views

Please help me about conjugate gradient method

As I know, the error function of neural network is the sum of difference between actual output and the target value. But in conjugate gradient method, they use quadratic function: $E(w) = \frac{1}{2} ...
5
votes
2answers
122 views

Looking for explanation of bound on Dirichlet's L-Function

I am reading Stein and Shakarchi's Fourier Analysis text and the proof Dirichlet's theorem and I am looking for clarification on how he derives the following for large $s$, $\lim_{s\to\infty}$ and ...
0
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1answer
50 views

A question about continuity at a point

I have a question about continuity of a function defined in terms of limits. For simplicity consider only a function $f: \mathbb{R} \rightarrow \mathbb{R}$. I have seen a sufficient condition for ...
0
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1answer
103 views

A basis question of least upper bound property.

A set $A$ is said to have least upper bound property if every subset $A_0 \subset A$ has a least upper bound. $\mathbb{R}$ has least upper bound property is well known. Now consider the subset $A = ...
2
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1answer
36 views

Theory of periodic solution

I want to study the theory of the periodic solution of ordinary differential equations, but I don't know how to study. Could you give me some references?
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2answers
50 views

$u= \frac {1}{2i} -\frac 3j$ , find a vector perpendicular to u [closed]

Find the interior angle of the triangle ABC give the point $A(3,4), B(-1,-7), C(-8,-2)$. I am a beginner.
2
votes
4answers
165 views

Relationship between O and o notation

In big-O notation, $f(x) = O(g(x))$ as $x\rightarrow \pm\infty$ if $$\exists C, \delta>0: \forall |x| \geq \delta: |f(x)| < C |g(x)|$$ and, for the case I'm more interested in here, $f(x) = ...
4
votes
2answers
291 views

Lie bracket is a connection?

In Road to Reality, section 14.6 on Lie derivative Penrose writes: Now $\epsilon^2 [j,h]$ corresponds to an $O(\epsilon^2)$ gap in the ‘parallelogram’ whose initial sides are $e_j$ and $e_h$ at ...
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2answers
84 views

In a metric space are the terms $bounded$ and $totally$ $bounded$ interchangeable?

The question is: Is there a theorem that says in a metric space a set is bounded if and only if it is totally bounded?
0
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1answer
89 views

Combinatorial Game Theory Prerequisites

I am planning to self-study Combinatorial Game Theory. I have gathered some useful references from here. Reference for combinatorial game theory. I plan to make a study about a local combinatorial ...
5
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6answers
165 views

When learning maths, should I review past topics?

So I've been learning maths from online video lectures and books for some time now and, as is probably quite natural, I have forgotten some of the stuff I've already read through. That'd be mostly ...
2
votes
1answer
81 views

Density of smooth functions in fractional Sobolev space

I am reading a paper on the analysis of numerical methods, and am confused about a statement made. I am working in fractional Hilbert spaces, but I don't think that this has much bearing on the answer ...
2
votes
2answers
114 views

Lipschitz Condition…

I am independently studying Numerical analysis and came across a question for which I am stuck at. Assume that $g(x)$ is differentiable. Show that if $|g'(x)|<1$ over $[x_0-p, x_0+p]$, then $g(x)$ ...
4
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1answer
407 views

How to get PDF from characteristic function

I would appreciate if anybody could explain to me with a simple example how to find PDF of a random variable from its characteristic function. Thank you.
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0answers
16 views

Bound of Degree of representation

Let $G$ be a finite group of order $n$. Let $\rho\colon G\rightarrow GL_m(V)$ be a representation with $m>n$ Show that $\rho$ is irreducible. I have been trying to construct a subrepresentation ...
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2answers
140 views

Operator Theory References and Topics

I wish to do a reading course in Operator Theory. Thus, I am looking for some references in the area. Right now, I have the following two sources available: Unbounded Self-Adjoint Operators ...
19
votes
2answers
536 views

History of the theory of equations: John Colson

This is an EDIT version of my original question: Recently I've been interested in the history of the Theory of Equations. The thing is that I learned about this mathematician named John Colson, he ...
1
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1answer
157 views

Proof of the trichotomy of real numbers using Cauchy sequences.

I'm trying to proof the next proposition but I have stuck in one point of the proof. I would really appreciate some help, thanks in advance. Definition: We shall say that a Cauchy sequences is ...
0
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2answers
7k views

Proof that when $x + y$ are irrational then $x$ and $y$ are irrational

I want to prove by contrapositive that: Proof that if $x + y$ are irrational then $x$ and $y$ are irrational. $x,y \in \mathbb{R}$ I did the following: Negation of the statement: $x + y$ are ...
1
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1answer
45 views

Underlying Connections (The Tower of Hanoi)

It is well established that the least number of steps of solving the Hanoi tower problem for $n$ disks is $2^{n}-1$. Now consider the following problem: Player 1 picks a natural number between 1 and ...
1
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1answer
344 views

Equivalent Cauchy sequences.

Hi everyone I'm having a bad time with two questions in the Analysis book of Terry Tao. I finally finished one of the exercises and I'm wondering if the next reasoning is correct or maybe needs some ...
3
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2answers
177 views

Good books to learn Riemann integration

I am looking for a good text book to learn Riemann integration. Please suggest books with theories and proofs comprehensively explained.
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3answers
140 views

Change of variable problems in probability

If $X$ is a continuous random variable with positive values,find it's Cumulative distribution function (CDF) and it's probability density function (PDF) for $$Y = \sqrt{X}$$ This was a detable ...
0
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3answers
302 views

Given a continuous random variable $X$ and the pdf find…

There is a pattern excercise: First pattern: We have $X$ and the probability density function is usually: $$ f(x) = \begin{cases} {ax+bx^2} & {0\lt x \lt 1} \\ 0 & \text{else} ...
0
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3answers
107 views

How many typos are in the document?

We assume that $1\%$ of characters in a document are typos. We want to find the probability of having at most 2 typos inside an 100 characters document. We want to find it in two ways: a)Precisely ...
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2answers
82 views

Find the probability for defective diskettes.

It is known that diskettes produced by a certain company will be defective with probability $0.0014$, independently of each other.* --Find exactly-- and --compute approximately (?)-- the probability ...
2
votes
1answer
113 views

What is a Math Learning Graph?

I went through may topics on learning maths effectively, learning math at later years (I am 30, so I read it to get motivated) One thing I missed is a natural flow of topics that one must learn to ...
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2answers
245 views

Probability about students and exams.

In exams, $80\%$ of the students go prepared and $20\%$ go unprepared. From the prepared ones, $90\%$ passes the exams. From the unprepared one, only $10\%$ passes the exams. First question: If a ...
3
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1answer
163 views

What Is The Difference Between The Maths That Physicists Use And The Maths On A Typical Mathematics Degree

Physicists are widely respected for using and sometimes even inventing mathematics yet physicists study Physics which is a subject in its own right. So surely someone studying physics spends less ...
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2answers
75 views

Interspersing of integers by rationals

I'm wondering if the next argument is sound or maybe need some adjustments; Proposition (Interspersing of integers by rationals): Let $x\in \mathbb{Q}$. Then there exists an integer such that $n\le ...
2
votes
1answer
46 views

Why $O_i$'s are blocks in this group action?

I was thinking about the problem from Dummit-Foote (Art. 4.1 Question 9) N.B. The definition of block as given in Ex. 7: Let $G$ be a transitive permutation group on the finite set $A$. A ...
3
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0answers
109 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 ...
0
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1answer
81 views

Help to understand a proof of Tao's book (analysis)

Hi I have troubles to understand a proof that is in the the notes and in the book of Terry Tao of Analysis. I Proposition in question is: The problems that I have it's to understand some tricky ...