Questions about studying mathematics without formal instruction.

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how to find matrix nonsingular P and Q such that $PAQ $ is a normal form?

Given matrix $A$: $$\pmatrix{1&1&1\\2&2&2\\-1&1&-3\\1&2&0}$$ how to find matrix nonsingular P and Q such that $PAQ $ is a normal form? Thanks!
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2answers
69 views

Proof Verification: $\lim (a^n + b^n)^{\frac 1 n} = b$ for $ 0 \lt a \le b$

Found the following exercise in Bartle's Elements of Real Analysis in the section on combinations of sequences. Am unsure about my solution and would really appreciate it if someone could verify it. ...
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2answers
35 views

Calculus problem (Differential)

First off, no, this is not homework. This comes from self-study and has stymied me. Please explain your answer as thoroughly as you can! Find increment $\Delta y$ and differential $dy$ for the ...
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2answers
107 views

Geodesic equations and christoffel symbols

I want to learn explicitly proof of the proposition 9.2.3. Which books or lecture notes I can find? Please give me a suggestion. Thank you:)
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0answers
26 views

Minimising line integral over a scalar field part 2

This is a continuation of this question whose general point is summarised below Say we want to find a path $y=y(x)$ in the scalar field $S(x,y)$ that finds the extrema of of its line integral. ...
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1answer
65 views

Affine variety example.

I know how to show the statement. But I cannot find an example (the part I underlined by a yellow pen) please help me for finding an example. Note: For example, can I consider the following ...
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41 views

What does “for sufficiently large $n$” mean?

The question essentially is, In a general/common context what does "for sufficiently large $n$" mean? My initial grasp of the phrase went, "for an $n$ that is greater than a required bound". ...
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0answers
42 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 ...
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1answer
27 views

Why we cannot in general define the product of two submodules

why we can define product of two ideals of a commutative ring,but we can't in general define the product of two submodules? ...
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1answer
32 views

How to prove M/Ker(f) & Im(f) are isomomorphic

f:M->N us an A-module homomorphism. A is a commutative ring. How to prove M/Ker(f) and Im(f) are isomomorphic I can't prove this statement. But if A is a field,it's not hard to be proved. For ...
2
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1answer
71 views

Doubts in definition of continuity in a topological space

EDITED Let $(X, \mathscr T_X)$ and $(Y, \mathscr T_Y)$ be two topological spaces. Let there be a function $f: X \to Y$. Then $f$ is said to be continuous if for every $E \in \mathscr T_Y$ the ...
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1answer
69 views

Differential geometry question.

Please explain how to solve this question. Thank you:) And sorry for hand-writing.
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1answer
28 views

understanding Continuity in topological space

Let $(X, \mathscr T_X)$ and $(Y, \mathscr T_Y)$ be topological spaces and $f: X \to Y$. $f$ is continuous iff $f^{-1} (E) \in \mathscr T_X $ for every $E \in \mathscr T_Y$. My doubt is: I dont know ...
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3answers
48 views

Understanding the definition of compactness.

$(X, \mathscr T )$ be a topological space and $A \subset X$. $\{ U_i \mid i \in I \}$ is said to be an open cover of $A$ if $A \subset \cup_{i \in I} U_i$. $A$ is said to be compact if there exists ...
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1answer
26 views

Showing if $H,K$ are subgroups of $G$, the $\varphi(hak)=a^{-1}hak$ is bijective.

Here's my problem...I have to ultimately show that $\varphi(hak)=\varphi(hbk)\Rightarrow hak=hbk$ and I'm having issues... I am told, both $H,K$ are subgroups of $G$ and $a\in G$. The goal is to show ...
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1answer
62 views

Help proving exercise on sequences in Bartle's Elements

Self learning Analysis and found the following exercise in Bartle's Elements of Real Analysis: Let $X = (x_n)$ be a sequence of strictly positive real numbers such that $\lim \left({\frac {x_{n ...
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1answer
31 views

to show $\sum_{i=1}^{\infty} |x_i y_i|$ converges

$X$ consists of sets of the form $(x_1, x_2, x_3, \dots)$ where $x_i \in \mathbb R$. Suppose $\sum_{i=1}^{\infty} x_i ^2$ converges. Show that : $\sum_{i=1}^{\infty} |x_i y_i$| converges. where $x,y ...
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1answer
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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1answer
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
102 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
125 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
39 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
67 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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0answers
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} ...
2
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1answer
41 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
64 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
164 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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1answer
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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1answer
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 ...
2
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2answers
100 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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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.