Questions about studying mathematics without formal instruction.

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16 views

Manipulation of $C^1$ funcitons

I think I read somewhere that composition of $C^1$ functions is also $C^1$, but I could not find the reference now. Also, is the difference of two $C^1$ functions still a $C^1$ function, please? And ...
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0answers
36 views

X is a compact topological space?

Let $X$ be a topological space. Then, when it is said that $K \subset X$ is compact, it is clear to me that every open cover of $K$ contains finite subcover of $K$. But what do we mean when it is ...
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32 views

triangle inequality on a given metric

$X$ be set consisting of all sequences $(x_1,x_2, \dots)$ s.t $x_i \in \mathbb R$ and $\sum x_i^2$ converges I need to prove triangle inequality for the metric on $X$ given by, $d(x,y) = [ ...
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1answer
40 views

Map induced by localization on categories

I have been doing some reading in Hartshorne's Algebraic Geometry on derived functors and subsequent results in cohomology. Given $A$ an abelian category of groups, I have seen that the map ...
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2answers
100 views

There is no Pythagorean triple in which the hypotenuse and one leg are the legs of another Pythagorean triple.

According to Wikipedia, There are no Pythagorean triples in which the hypotenuse and one leg are the legs of another Pythagorean triple. I cannot find the proof in the citation provided. I am ...
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1answer
29 views

How many positive integers n can we make with the digits 3, 3, 4, 5, 5, 6, 7, if the number n > 4, 000, 000?

According to my study guide the answer to the exercise, How many positive integers, (n), can we make with the digits 3, 3, 4, 5, 5, 6, 7, if the number n > 4, 000, 000, : The total of numbers n > ...
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4answers
120 views

$\hom(V,W)$ is canonic isomorph to $\hom(W^*, V^*)$

Introduction My Semester just started and we have a new Professor for Linear Algebra II (replacing our former Professor). Apparently we are behind our schedule and thus we only had a brief ...
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1answer
37 views

How many different 5 characters words are there with only one letter a?

I just need to clarify my answer to this exercise. This is a permutations exercise. If we define a word to be a string of 5 letters of the English alphabet, regardless of meaning, then mnnnw is a ...
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2answers
66 views

From Engineering-Style to Proper Mathematics

I currently have an engineering-style education in mathematics. We covered quite a lot of material (e.g. real and complex analysis, some probability theory and graph theory), but more often than not ...
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27 views

Prove convergence of a serie little with Direct comparison test

I have the following Serie $$\sum_{k=1}^{\infty} \log(1+\frac{1}{k^2})$$ this serie should converge but when i apply the Direct comparison test it diverges $$|\sum_{k=1}^{\infty} ...
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1answer
40 views

If a continuous function$f: (X, \mathscr T_X) \to (Y, \mathscr T_Y)$ is injective (Given $Y$ is Hausdorff), show that X is hausdorff

$(X, \mathscr T_X)$ and $(Y, \mathscr T_Y)$ be 2 topological spaces. $Y$ be Hausdorff. $f$ be a continuous function, $f: X \to Y$. To show that if $f$ is injective $\implies$ $X$ is Hausdorff. ...
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2answers
48 views

Solution Verification: Convergence of Norms $\implies$ Convergence of Sequence

I found the following exercise in Bartle's Elements of Real Analysis. I'm learning on my own and have a couple of doubts would love it if someone could take a look. ($\left|{\left|{x}\right|}\right|$ ...
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2answers
66 views

Analysis/Inequality question about proving an infinite product greater than 0

This is from David Williams' book Probability using Martingales. I'm self-studying. Question Prove that if $$0\leq p_n < 1 \quad\text{ and }\quad S:=\sum p_n < \infty$$ then $$\prod (1-p_n) ...
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2answers
69 views

Evaluating $\lim_{n\to\infty}\left(\frac{1-i}{4}\right)^n$

It's been a while since I have taken Calculus II so my experiences on sequences and series has gone down the drain. I'm trying to find the limit of the sequence ...
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3answers
63 views

Counter Example about Continuous Functions

(James Munkres page 104 Theorem 18.1) Let $X$ and $Y$ be topological spaces; let $f: X \rightarrow Y$. If $f$ is continuous, then for every subset $A$ of $X$, one has $f(\overline{A})\subset ...
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1answer
32 views

Sub-Basis of a Topology James Munkres

James Munkres defines a subbasis $\mathcal S$ for a topology on a set $X$ as a collection of subsets of $X$ whose union equals $X$. Then the topology generated by the subbasis $\mathcal S$ is defined ...
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1answer
46 views

Question about construction of an algebraic closure

$A=K[x]$,$\mathfrak{m}$ is a maximal ideal containing a principle ideal of $A$. every element of $K[x]/\mathfrak{m}$ can be described by form $f+\mathfrak{m}$. every element of $K$ is the ...
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2answers
163 views

Careers in Mathematics?

I am a college freshman, and I really like to have goals for my life, one of the big ones is my career of choice. Previously, I have always wanted to be a programmer, and I have written a lot of code. ...
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44 views

A conditional probabilty question.

Question: $8$ identical balls are randomly distributed into $8$ boxes. Given first box and second box are not both empty, find the probability that first box is not empty? $A:=$ B1 is not ...
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3answers
190 views

Proving the Obvious

Is it normal that I have the hardest time when I'm trying to prove statements that are blatantly obvious on a visual and/or intuitive level? For instance, how does one go about formally proving the ...
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1answer
56 views

Question about construction of an algebraic closure of a field

In Constructing algebraic closures by Keith Conrad, the author writes: Let $K$ be a field. We want to construct an algebraic closure of $K$, i.e., an algebraic extension of $K$ which is ...
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5answers
49 views

Given any real numbers $x \gt 0$ and $r \gt 1$ there exists $m \in \Bbb N$ such that $\frac 1 {r^m} \lt x$

I know it can be proven that given any real number $x \gt 0$ there exists $m \in \Bbb N$ such that $\frac 1 {2^m} \lt x$. I tried to generalise it but am surprisingly not getting anywhere. I ...
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2answers
190 views

Is there another Analysis book that is based on the Cartesian space $\Bbb R^p$

I am in the middle of a slightly ambitious attempt to learn Analysis on my own. I skimmed through Rudin(Baby), Chapman Pugh, William Wade, Stephen Abbott and Strichartz and ended up preferring the ...
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1answer
22 views

Does this opetation and structure have a name?

A is a commutative ring ,a is an ideal of A. then we can get a structure A/a called quotient ring by operation of quotient. question is :if we have a ring A/a and a set a . How to get A? This ...
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40 views

how to prove a=r(a)<=>a is a intersection of prime ideals.if a is an ideal≠(1) in commutative ring A.

how to prove a=r(a)<=>a is a intersection of prime ideals.if a is an ideal≠(1) in commutative ring A.r mean radical. I can't prove it,here is what I did. a is a intersection of prime ideals mean ...
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1answer
39 views

Bolzano-Weirstrass

This is derived of other question where my proof of the Bolzano thm is as follows Proof: Suppose that $(a_n)$ is a bounded sequence then we have to show that it has a convergent subsequence. Since ...
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1answer
53 views

Modern Algebra: Groups

Is this the way to solve the question? Question a). Find the center of the group $S_3 \times \mathbb Z/6\mathbb Z$ Ans: $S_3$is the order of 6 element therefore, {1,(12),(23),(13),(123),(132)} and ...
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2answers
62 views

Two probability questions.

I have two questions. (1) solution(1): Sample size $=|S|=12^{20}$ $11^{20}\rightarrow$ guarantee that one box is empty. $10^{20}\rightarrow$ guarantee that two boxes are empty. ...
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1answer
50 views

Chain rule (proof verification)

Hi everyone I'm asking two thinks is this proof correct (my other idea was using limit of sequences)? and are there a simpler alternative than this using Newton's approximation? If someone could help ...
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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 ...
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2answers
97 views

Is $m\mathbb{Z}$ not isomorphic to $n\mathbb{Z}$ when $m\neq n$?

Exercise from "Abstarct Algebra: An Introduction" by T.W.Hungerford. For each positive integer $k,$ let $k\mathbb{Z}$ denote the ring of all integer multiples of $k$. Prove that if $m\neq n$, then ...
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1answer
52 views

Question about some details of a proof of Chinese Remainder Theorem

In the proof of 3rd proposition I can prove the intersection of all ideals is the kernel of the map, but why does it imply this proposition is true?
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2answers
16 views

Special function example

Given: $0< f(x) < x^2$ for all $x \in R$ Give an example of a function whcih satisfies the hypothesis, but is not continuous at $x \neq 0$, and explain whether the function is continuous at 0.
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33 views

A question about a detail of proof

proposition: x∈The Jacobson radical <=> 1-xy is a unit in commutative ring A for all y∈A I have proved (=>) I don't figure out a detail of the proof of (<=). Here is the proof on book: ...
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1answer
43 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 ...
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2answers
20 views

Question about method in an algebra precalculus exercise

($a$ real number) So if $\frac{1}{4}<a<\frac{1}{3}$ prove that $\frac{10}9<R(a)<\frac{11}{6}$ where R(x)=$(2x-1)(x+1)(x-3)=2x^3-5x^2-4x+3$ So my idea was to do the same operations in ...
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3answers
60 views

How to prove a simple proposition about local rings and maximal ideals

(The word ring shall mean a commutative ring with an identity element in this question.) Actually, there is a proof about this proposition, but I don't get it, even the first step. Proposition: ...
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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.
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1answer
33 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 $ ?
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1answer
55 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
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}) ...
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64 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
42 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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1answer
24 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
41 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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1answer
79 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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6answers
94 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
44 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 ...
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2answers
72 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 ...
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1answer
94 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 - ...