Questions about the process of studying mathematics without formal instruction.

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1answer
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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 ...
4
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2answers
276 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
56 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
225 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
96 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
75 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
223 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
26 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
59 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
50 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
67 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
68 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
116 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 ...
1
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0answers
18 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
122 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
68 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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1answer
38 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
54 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
27 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
102 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
35 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
55 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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0answers
25 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
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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
49 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
46 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
59 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 ...
2
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1answer
509 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
114 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 ...
2
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1answer
71 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
85 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
308 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
84 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 ...
0
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1answer
48 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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2answers
83 views

Calculus: L′ Hopital's Rule

$\displaystyle\lim_{x\to\frac{\pi}{2}}(\sin x)^{\tan x}$ $\displaystyle\lim_{x\to0}x^2\ln x$ $\displaystyle\lim_{x\to1^+}x^{\frac{1}{1-x}}$ Do I have to apply l'Hôpital's Rule to evaluate these ...
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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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1answer
126 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
110 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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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 ...
2
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2answers
80 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
47 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 ...
2
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1answer
44 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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2answers
67 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
73 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 ...
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1answer
23 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, ...
1
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1answer
62 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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1answer
60 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
85 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? ...
4
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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 ...
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2answers
29 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} + ...