Questions about studying mathematics without formal instruction.

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1answer
48 views

How to think of this problem: $\mathcal{A}_{\mu}$ not necessarily equal to $\mathcal{A}_{\nu}$ when $\mu$ and $\nu$ are finite measures?

How to think of this problem: $\mathcal{A}_{\mu}$ not necessarily equal to $\mathcal{A}_{\nu}$ when $\mu$ and $\nu$ are finite measures on a measurable space $\left(X,\mathcal{A}\right)$? NOTE: I ...
2
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1answer
251 views

Calculus - Finding the minimum vertical distance between graphs

Question:Find the minimum vertical distance between the graphs of $2+\sin x$ and $\cos x$? In order to find out the required distance, what should I do? It seems that there is a problem if I ...
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1answer
49 views

Polar equation and Cartesian equation

For the polar equation, $r \sin \theta = \ln r + \ln (\cos\theta)$ Is that equivalent to $y = \ln x $ ?
2
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0answers
30 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.
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1answer
76 views

Breaking Out of Algorithmic Thinking

I have been taught my whole life to solve math with an algorithm or method. My teachers in school trained us to identify patterns in equations and then use a method to solve it. No one has ever ...
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0answers
23 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}) ...
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0answers
79 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 ...
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1answer
46 views

Reformulations of Inverse Function Theorem

Inverse Function Theorem: Let $U\subset\mathbb{R^n}$ be open, $f:U\longrightarrow\mathbb{R^n}$ be $C^k$ such that for $a\in U,\quad d_a f:\mathbb{R^n}\longrightarrow\mathbb{R^n}$ is invertible. ...
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6answers
111 views

Why is an algebra not a $\sigma$-algebra by induction?

I am studying probability theory by reading Sidney Resnick's "A Probability Path". On page 12 and 13, algebra and $\sigma$-algebra are defined. The only difference between the two is the third ...
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1answer
38 views

Show that the intersection of two probabilities in a certain interval

I am struggeling with the following problem: Suppose that $P(A)= \frac{3}{4}$ and $P(B)= \frac{1}{3}$. Show that $\frac{1}{12} \leq P(A \cap B) \leq \frac{1}{3} $. Basically I try to show this ...
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1answer
45 views

how many ways are there to distribute 10white and 10black balls into 20 distinct boxes so that at most one box is empty

Question: how many ways are there to distribute 10white and 10black balls into 20 distinct boxes so that at most one box is empty. I understand case-1. But I cannot understand a part of answer ...
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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 ...
2
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1answer
61 views

Calculating joint MGF

This is an end-of-chapter question from a Korean textbook, and unfortunately it only has solutions to the even-numbered q's, so I'm seeking for some hints or tips to work out this particular joint ...
1
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2answers
83 views

Can we regard Hausdorff space as a manifold?

Can we regard Hausdorff space as a manifold of class ?(p≥1) And I want to know the relation among the concept Hausdorff space,metric space,vector space,tangent space and manifold. What's the common ...
3
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1answer
137 views

Help proving Cantor Intersection Theorem using Bolzano-Weierstrass Theorem

Came across the following exercise in Bartle's Elements of Real Analysis and am quite unsure about my solution. Would greatly appreciate it if someone could take a look at it. The Bolzano - ...
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1answer
78 views

I don't understand a paragraph about tangent space

I don't understand how author associate the smooth manifolds and linear subspace. TM is a linear subspace,what 's the mean of T?A set of vector? And find the definition on Wikipedia. I still ...
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1answer
313 views

What pure mathematics foundations should an applied mathematician have?

I'm studying mathematics, with some statistics also, and I've always chosen applied courses. I'm getting to the point where I'm studying 3rd year undergraduate to graduate level material. My first ...
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1answer
45 views

How to prove m is maximal iff A/m is a field?

m is a maximal ideal of a commutative ring A. then m is maximal iff A/m is a field. Use Lattice theorem we get there is a bijection between m and an ideal of A/M. A/M is a field =>the only odeals in ...
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1answer
109 views

help and verification of 3 short exercises

I'm reading an old book and find the following three question. I'd like to know two things: if my attempts are correct and also it would be great if someone could give suggestions in more detail. ...
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1answer
88 views

How to prove there exists a bijection between the ideals of $A/a$ and the ideals of $A$ containing $a$

original proposition is there is a one-to-one order-presserving correspondence between the ideals of $A$ which contain $a$ and the ideals of $A/a$. I think one-to-one correspondence mean ...
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1answer
27 views

Proving the Boolean expressions

Are these two Boolean expressions the same? *$co$ is the carry out while $ci$ is the carry in.
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2answers
1k views

Show that these two numbers have the same number of digits

I want to show that for $n>0$, $2^n$ and $2^n + 1$ have the same number of digits. What I did was I found that the formula for the number of digits of a number $x$ is $\left ...
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2answers
71 views

How to prove “ideal $I$ is prime iff $A/I$ is a integral domain ”?

$A$ is a commutative ring with identity. $I$ is a ideal of $A$. then ideal $I$ is prime iff $A/I$ is a integral domain. here is what I thought $(\Rightarrow)$ We want to prove $A/I$ is a integral ...
2
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2answers
43 views

Qualification of a Universal Quantification

Let us say I have a predicate, $P(n)$, and I want to say that it holds for every integer greater than $2$ (an example would be $P(n) = 2n>2+n$). Let us furthermore say that the UOD (universe of ...
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1answer
43 views

Correct Way to Write a Statement in First-Order Logic

I am teaching myself set theory. I am at a point where the set of rationals, $\mathbb{Q}$, has been defined, along with its ordering relation, $<_\mathbb{Q}$. Now, working towards a definition of a ...
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1answer
231 views

Cambridge Press pure maths book contents

I am revisiting maths after studying it at school some time ago, I'm planning on using the Cambridge Press books as they worked well for me before. However I am confused as to what is covered in ...
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2answers
66 views

Question about some algebra theorem

1. order-presserving = monotonic But we haven't define order structure on the ring. 2. I try to prove x is a unit <=> (x)=A ,and fail. That's what I think: => we want to prove (x)=A. ...
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3answers
71 views

How to find the sum of series $\sum_{i=1}^{\infty}\frac{i}{2^i}$? [duplicate]

I am learning about series of numbers at the moment. In the book there is an exercise in which I need to find the sum of : $$\sum_{i=1}^{\infty}\frac{i}{2^i}$$ I know it is equal to $2$. But how do I ...
5
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4answers
920 views

Self-teaching myself math from pre-calc and beyond.

Going to be starting grade 12 (pre-calculus) shortly and looking to get ahead. I would like to try some more rigorous stuff on my own and have a couple questions. Ideally I would like to be prepared ...
4
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8answers
1k views

Studying Math, All Over Again

I am 16 and never really paid attention to math. For me it was just one more obstacle in passing the exams. Now that I study computers, mostly on my own, I find that there is indeed a need for me to ...
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1answer
19 views

Question regarding dimension of linear transformation.

I saw in an exercise that if $T$ is a linear transformation $T: V\rightarrow W$ and $T_2: W\rightarrow Z$ and $T_1: X\rightarrow V$ are invertible then the rank of the composition doesn't change. So, ...
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1answer
58 views

How is an open set defined without referring to any distance function in a topology?

I am currently studying general topology. The definition given by Royden looks very confusing to me. It says that the elements of a topology is called open sets without actually defining what exactly ...
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2answers
200 views

$A \times B$ is an open set in $\Bbb R^2 \implies A$ and $B$ are both open in $\Bbb R$; $A,B \neq \emptyset$

I am studying Analysis on my own. Reading The Elements of Real Analysis by Bartle. Came across the above problem and I came up with the following solution but am very unsure about it. Would be very ...
0
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1answer
46 views

A proof problem about congruence relation

Given a ring R and a two-sided ideal I in R, we may define an equivalence relation ~ on R as follows: a ~ b if and only if a − b is in I. Using the ideal properties, it is not difficult to check ...
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1answer
121 views

Prove that $\limsup_{n \to \infty} \left(a_n + b_n \right) \leq \limsup_{n \to \infty} a_n +\limsup_{n \to \infty} b_n$

I want to prove that for two sequences, say $a_k$ and $b_k:$ $$ \limsup_{n \to \infty} \left(a_n + b_n \right) \leq \limsup_{n \to \infty} a_n +\limsup_{n \to \infty} b_n$$ If we let $M_n =\sup \{ ...
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1answer
64 views

Checking if systems of linear equations are equivalent

I'm going through Hoffman and Kunze's Linear Algebra on my own and can't for the life of me figure out part of Exercise 3 of Section 1.2: Are the following two systems of linear equations equivalent? ...
2
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2answers
28 views

Prove $ab + ab\overline{c} + bcd = b(a+c)(a+d)$

Do I need to use absorbtion law to prove them? $ab + ab\overline{c} + bcd = b(a+c)(a+d)$ $ab + cd = (a+c)(a+d)(b+c)(b+d)$. For 1), I simplified $ab+ ab\overline{c} + bcd$ into $b(a\overline{c} + ...
2
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1answer
76 views

Bridge the gap to university mathematics [closed]

Can anyone suggest some good books to help an high school student to "bridge the gap" to university math? I've heard of http://www.amazon.com/How-Study-as-Mathematics-Major/dp/0199661316 and ...
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2answers
43 views

Regular or normal topological space

How do we define an open neighborhood around a closed set? My question is with respect to a normal or regular topological space where we use the concept of an open neighborhood around a closed set.
2
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0answers
62 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
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0answers
81 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 ...
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1answer
26 views

Prove the following property of holomorphic functions.

Let $\rho(x)$ be a holomorphic function on a disk $D \subseteq \mathbb{C}$ with the property that $\rho(x) \notin \mathbb{N^*} = \{1,2,\dots\}$ on $D$. Prove the following: There exists an $R$ ...
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0answers
124 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 ...
4
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1answer
56 views

Show that $T\neq{T^*}$

Let $V=P_2(\mathbb{R}), T\in \mathcal{L}(P_2(\mathbb{R})),$ where $T(p)=(a_1x)$. Make $V$ an inner product space by defining $$\langle p,q\rangle=\int_0^1{p(x)q(x)\,dx}$$ So I calculate $$\langle ...
3
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1answer
182 views

A subset of $\Bbb R^p$ is open iff it is the union of a countable collection of open balls

I am studying analysis on my own and need some help verifying the solution to the above exercise found in Bartle's Elements of Real Analysis. I know there are other posts answering the same question ...
2
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3answers
1k views

What is the equal sign with 3 lines mean in Wilson's theorem?

I'm reading up on Wilson's Theorem, and see a symbol I don't know... what does an equal sign with three lines mean? I'm looking at the example table and I still can't infer what they are trying to ...
3
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0answers
56 views

What else can I do to learn math better? [closed]

I feel like I am not learning math well or efficiently enough. I read the textbook, do the exercises but I still don't get the marks I would like. For the time I put in, it seems like I should be ...
2
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1answer
156 views

An Increasing Function Discontinuous at All Rational Numbers

Let $q: \mathbb{N}\rightarrow \mathbb{Q}$ be a bijective map and let $g: \mathbb{Q}\rightarrow \mathbb{R}$ define $g(q(n))=2^{-n}$. Show that $\sum_{r\in \mathbb{Q}}g(r)$ is absolutely convergent. ...
2
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1answer
46 views

Reference request to study Borel summation

Could someone recommend sources to learn about Borel summation procedure? Books, articles or reviews? I have a background in basic analysis.
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1answer
176 views

Book on discrete mathematics for self study

I am searching for book on discrete mathematics which is suitable for self study. This mean I want it to have exercises with answers (It would be ideal if it had solutions). I have already read ...