Questions about the process of studying mathematics without formal instruction.

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3
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2answers
327 views

Any good, undergraduate level introductions to Functional Analysis?

In my lower division math classes, my instructors referenced functional analysis as essentially the extension of linear algebra to infinite dimensional vector spaces along with some real analysis. As ...
1
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1answer
38 views

Why is the measure of this set 0?

Williams has the following note in his book Probability with Martingales: Lemmma 5.2b simply states that I don't see why $\mu(\{L\neq U\})=0$. I tried doing a proof by contradiction (If ...
3
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1answer
68 views

proposition 1.10 ii) A&M Introduction of commutative algebra

I am working through Introduction of commutative algebra and am having trouble with the following question: (I'll use f instead of the map,since I don't know how to input it.) Q1: Why there exist ...
1
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1answer
30 views

is there a counterexample of this map isn't surjective?

The ring A is a commutative ring with identity. I think ii) is true if they are not coprime. because for every (x+a1,....,x+an) we can find a x such that f(x)= (x+a1,....,x+an). Could you please ...
3
votes
1answer
105 views

Showing to be Unit-sphere

First and second fundamental forms are both $du^2 +\cos^2 u dv^2$ I want to show that the surface is a part of the unit sphere. What I did is following; $E=L=1$ $F=M=0$ $N=G=\cos^2 u$ ...
1
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1answer
41 views

Smallest Algebra Containing Singletons

$\Omega:=\mathbb N$. What is the smallest algebra containing all singleton $\{\omega\}$, i.e. $\{1\}, \{2\}$, and so on. Any hint, please?
1
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3answers
51 views

Definition of limit of function

I'm reading Calculus: Basic Concepts for High School Students and am trying to digest the definition of 'limit of function'. There are two details that I am struggling to fully accept: If you are ...
1
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0answers
19 views

Are they $\pi$ systems?

I am not sure whether the following two systems are closed under finite intersections. $\{(a,b):-\infty<a<b<\infty\}$: I do not think it is if I consider $(0,1)\cap(1,2)=\emptyset\notin ...
1
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3answers
67 views

Why is $[0,1]$ not homeomorphic to $[0,1]^2$?

Why is $[0,1]$ not homeomorphic to $[0,1]^2$? It seems that the easiest way to show this is to find some inconsistency between the open set structures of the two. It is clear that the two share the ...
1
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1answer
38 views

Determine the set $A$ = {$m\in Z|mR52$} and give its cardinality $|A|$.

Given a relation R on $Z^+$ defined as: $mRn$ if and only if $m|n$, I need to determine the set $A$ = {$m\in Z|mR52$} and give its cardinality $|A|$. I know that $mR52$ = $m|52$ and that $52 = mk$ ...
0
votes
1answer
49 views

Tool to reteach Algebra?

I was never a very good math student and over the years, I simply forgot a lot of math. But sometimes it annoys me to no end, because I work in a mathematically related field (applied statistics) and ...
3
votes
1answer
89 views

Book/Books leading up to the the axiom of choice?

I am familiar with the axioms of ZF set theory and some basic uses of them to completely formally construct more complex objects such as natural numbers etc. However I have pretty much no background ...
4
votes
1answer
83 views

Don't understand a proposition and its proof

Proposition 5.1 from Commutative Algebra by Atiyah and Macdonald: $x∈B$ is integral over $A$,then $A[x]$ is a finitely generated $A$-module. The elements in $A[x]$ are the set of all the sum. If ...
3
votes
0answers
95 views

WKB and asymptotic behavior of second order differential equation

I want to study the large $x$ solution to a Riccati equation. After listening to the lectures on Mathematical Physics by Carl Bender, I have fallen in love with asymptotic analysis. But, by no means ...
0
votes
2answers
54 views

Initial value problem of $y' = \sqrt{|y|}(y+1)$

i'm trying to determinate the solution of the intial value problem $$y' = \sqrt{|y|}(y+1)$$ my solution was as follow applying substitution as follow let $u^{2} = y$ and $dy = 2u\ du$ $$2 \int ...
1
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2answers
73 views

Calculus of several variables

I have problem in solving the definite integral of the following function: $$f(x, y, z) = \int_0^z \int_0^y \int_0^x e^{-x-y-z} dx dy dz, $$ All I know is the calculus for single variable. But with ...
4
votes
1answer
186 views

Learning to understand proofs faster?

There are many books, written by highly decorated academics, which feature proofs that I can hardly comprehend in an acceptable amount of time. Roughly each week, it happens that I find myself having ...
3
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3answers
147 views

Integrating a function with an infinite number of discontinuities

I would appreciate some help with the following exercise: Let $$f(x)=\begin{cases} 1 & \text{if}\ x= 1/n\ \text{for some}\ n \in \mathbb{N} \\ 0 & \text{otherwise} \end{cases}$$ Show that ...
1
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0answers
38 views

Show a subset A is compact if and only if the image of the map T(A) is compact

The question is as follows: Let $\{v_1,v_2,\dots,v_n\}$ be a set of linearly independent vectors of an $n$-dimensional normed linear vector space $V$. Define the map $T: R^n \to V$ by $T(a) = ...
0
votes
1answer
102 views

Abstract Algebra: Ring Homomorphism injective

Reference: Ring Homomorphism $\phi:f\to S$ is injective Referring this I have a doubt which I needed to clear. Below is my answer and query. We know $\phi:f\to S$ be ring homomorphism, where $f$ is a ...
12
votes
1answer
173 views

What's the motivation of the definition of primary ideals?

$$xy\in\mathfrak q\:\Rightarrow\:\text{either $x\in\mathfrak q$ or $y^n\in\mathfrak q$ for some $n\gt0$}.$$ Primary ideals can be regard as the generalization of prime ideals and radical. But ...
2
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1answer
90 views

Question about some details of a proof

i) Why it's a unit can prove this proposition ii)see picture
0
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1answer
41 views

Regarding jointly multivariate normal X1,X2…X5

So I have a question from statistical inference that I need some help with: $X_1,X_2,...,X_5$ are jointly multivariate normal with means = $\mu_i$, variances = $\sigma^2_i$, correlation = $\rho$ ...
1
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2answers
68 views

Example- $l_p$ norm space

$$||x||= {[{\sum _{i=1}^\infty |x_i|^p}]}^{1/p}$$ Is a norm on $l_p$ space :- space of all sequences made of scalars from $\mathfrak C$(filed of complex numbers). To prove that above is norm on $l_p$ ...
3
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0answers
47 views

Where does this series converge?

Let $ \{r_1, r_2 ,r_3,... \}$ be an enumeration of $\mathbb{Q}$. For each $r_n \in \mathbb{Q}$ define: $$u_n(x)=\begin{cases} 1/{2^n} & x>r_n \\ 0 & x \leq r_n \end{cases} $$ and let $$h ...
4
votes
1answer
144 views

Geodesic eqautions and length of a curve in geodesic coordinate system.

About geodesic coordinates: Let S be regular surface. $p\in S$ $\gamma$ be unit speed geodesic on $S$ with parameter $v$ and $\gamma (0)=p$ $\tilde \gamma^v$ be unit speed geodesic s.t. ...
1
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1answer
66 views

A function that is differentiable at a single point

Consider the function $$f(x)=\begin{cases} x^2 & x \in \mathbf{Q}, \\ 0 & x \notin \mathbf{Q} \end{cases} $$ $f$ is continuous only at $0$ and now I need to show that at this point it is ...
2
votes
4answers
83 views

Are different constructions of an algebraic structure always isomorphic?

Any two complete ordered fields are isomorphic (as proved, e.g., in Spivak's Calculus; see also this question). While I understand this proof, I cannot yet appreciate why it is necessary. Given any ...
1
vote
2answers
154 views

Determining constant in a CDF

I have a question that I literally have no idea how to begin, I was hoping someone could help me: It says $X_1,X_2,\ldots,X_n$ is a sample from a distribution It says that the Cumulative ...
1
vote
1answer
393 views

Cubic root formula derivation

I'm trying to understand the derivation for the cubic root formula. The text I am studying from describes the following steps: $$x^3 + ax^2 + bx + c = 0$$ Reduce this to a depressed form by ...
0
votes
2answers
76 views

What is this distribution???

Let $X_1, X_2, \ldots, X_n$ be a random sample from a population with $E(X_i) = \mu$ for all $i \in \{1,\ldots, n \}$. Define $ Y_i = \begin{cases} 1 & \mbox{ if } X_i < \mu \\ 0 ...
1
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1answer
64 views

Abbott's Exercise $6.2.14$ : Convergent subsequences for bounded sequences of functions

I have been trying to solve the following exercise from Abbott's "Understanding Analysis". I understand that $(a)$ comes from an application of the Bolzano Weierstrass Theorem as we assume that ...
0
votes
1answer
22 views

Covering of a universal space and covering of a subset

It seems to me that the definition of covering depends on the set we are referring to. For example, if $X$ is the universal topological space, then a collection $\mathcal A$ of subsets of $X$ is said ...
0
votes
3answers
185 views

Converse of order limit theorem

Part of the Order Limit Theorem states that: Assume $(u_n) \rightarrow l$ and $(v_n) \rightarrow m$, then if $v_n \le u_n$ for all $n \ge N \in \mathbb{N}$, then $m \le l$. I can prove this. ...
1
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2answers
48 views

Showing that two Matrices are not similar over GL$_n(\mathbb{F}_2)$

Problem: Show that $$ \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$ Are not similar over GL$_n ( \mathbb{F}_2)$ Note: We write ...
3
votes
3answers
582 views

Is it possible to learn abstract algebra no precalculus or calculus?

See above. I am trying to re-teach myself mathematics in a different manner than is formally taught (i.e., set theory, number theory, mathematical logic, abstract algebra, discrete math and then ...
0
votes
1answer
40 views

Is norm of a differentiable function continuous?

The following argument is from my class notes. "Suppose $\gamma(s): I\rightarrow \mathbb R^3$ is a regular curve. Fix $t_0$ and define $s(t) = \int_{t_0}^t \|\gamma'(t)\|dt$. That is, $s(t)$ is the ...
0
votes
1answer
52 views

Calculating mean vector of a multivariate distribution

I have a question concerning calculating the mean vector (vector of expected values) of a general multivariate distribution. I try to obtain the mean vector by doing a vector integration and I ...
2
votes
1answer
72 views

Convergence in $L_p$, Vitali's theorem, and convergence in measure

Working a measure theory question for practice from Bartle. Assume that $(X,\mathbb{X},\mu)$ is a finite measure space $f_{n}\rightarrow f$ in $L_{p}(X,\mathbb{X},\mu)$ $\varphi$ is a real-valued ...
2
votes
1answer
85 views

Which of these things is not like the others?

What's in a name? Well quite a lot, if you're confused enough. I have an engineering-style mathematics education, based on good old hand waving and learning bits and pieces from all over the place. I ...
4
votes
1answer
282 views

Using the Parseval Identity to compute $ \sum_{n=1}^{+ \infty} \frac{1}{(4n^2-1)^2}$

Parseval's Identity: For continuous $f: [- \pi , \pi] \to \mathbb{R}$ $$ \sum_{n=- \infty}^{+ \infty} |c_n|^2 = \frac{1}{2 \pi} \int_{ - \pi}^{ \pi} |f(x)|^2dx, \text{ where } c_n = ...
0
votes
1answer
48 views

Could you please explain the detail of the proof

Proposition: Proof: Question: Why it's isomorphism?
3
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4answers
314 views

Which book among these would you recommend for first year calculus?

I'm struggling a bit with functions(limits, squeeze theorem, etc). I have done some research and found a list of books on calculus but I'm not sure which one would be better suited for me, so I would ...
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2answers
66 views

Why are these two lemmas worth including and proving?

The following two lemmas are from Stein and Shakarchi (2005). (p4 Lemma 1.1) If a rectangle is the almost disjoint union of finitely many other rectangles, say $R=\cup_{k=1}^n R_k$, then $|R| = ...
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1answer
149 views

Better way to prove this sequence problem relating $\lim x_{n+1}/x_n$ and $\lim\sqrt[n]{x_n}$

Came across the following exercise in Bartle's Elements of Real Analysis. Let $X= (x_n)$ be a sequence of strictly positive real numbers, let $\lim \left({ \dfrac{x_{n + 1}}{x_n}}\right) = L$, ...
3
votes
3answers
154 views

How to show a collection of sets is countable

In the proof for "Every open subset $\mathcal O$ of $\mathbb R$ can be written uniquely as a countable union of disjoint open intervals" Stein and Shakarchi (2005 p6) argue that (after having defined ...
2
votes
2answers
219 views

find the coefficient of the given term when the expression is expanded by the binomial theorem

I am just trying to understand why the term is $\binom{15}8$(3p$^2$ - 2q)$^7$. I need to find the coefficient in $p^{16}q^7$ in $(3p^2 - 2q)^{15}$ So, I know that $n = 15$ and I have $a^{n - k}b^k$ ...
3
votes
1answer
53 views

Isomorphism of Grassmannians

I want to prove that two CW complexes $\mathrm{Gr}_{n}(\mathbb{R}^{n+k})$ and $\mathrm{Gr}_{k}(\mathbb{R}^{n+k})$ are isomorphic to one another. I'm pretty sure I can just show that the number of ...
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2answers
51 views

Where am I going wrong? Converging sequence

I am struck with a dilemma concerning the following exercise in Bartle's Elements of Real Analysis. Determine the convergence or the divergence of the sequence $(x_n)$ given by $$ x_n = ...
0
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1answer
60 views

Could you please show some hints about the proof

I even don't know how to start.