Questions about studying mathematics without formal instruction.

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Existence of Non-Trivial, Convex, Open Set in $C_{\mathbb{C}}[0,1]$ Under $L^{0}$ Metric

I've been struggling with this problem for the last four hours. The problem is to show that the space of $\mathbb{C}$-valued continuous functions on $[0,1]$ under the metric ...
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2answers
50 views

Evaluating integral limit in two ways gives different limits

Problem Show that $$\lim\limits_{h \rightarrow 0^{+}} \int_{-1}^{1} \frac{h}{h^{2}+x^{2}} \, dx = \pi.$$ I can do this by evaluating the integral directly and showing that it is equal to ...
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1answer
43 views

Is a butterfly network on 8-inputs planar?

I could prove that a four input butterfly network is planar. For that I simply drew it such that no two edges intersect. But I could not use the same approach for the 8-input butterfly network. So I ...
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1answer
42 views

Proving that Markov Chain Monte Carlo converges

I am trying to understand how the very basic Markov Chain Monte Carlo approach works: We try to approximately calculate the expected value $E_{\pi(x)}[X]$ by drawing sequential samples from a Markov ...
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1answer
35 views

Differentiable functions defined on a regular surface

First, recall a general definition of a differentiable function as follows. Suppose $f :D\subset \mathbb R^3 \rightarrow \mathbb R$. Then $f$ is differentiable at $\mathbf a \in D$ if there is a ...
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1answer
19 views

What test could I use to test $\mu_1$ and $\mu_2$ instead of $\bar{x}$ and $\mu$?

A single sample $t$ test is only against a sample and a population correct? Can it test against a population and itself or two populations? ie $\mu_1$ vs. $\mu_2$....if not what tests would you use ...
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0answers
54 views

Recommended Textbook/Resources

I'm looking for a textbook or resources my younger brother could use. (He is in year 9, equivalent to US high school freshman) He is wanting to advance upon his math, he currently does exercises out ...
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1answer
41 views

Sequence of learning mathematics from basic algebra to calculus.

What would be a step by step sequence of learning mathematics from basic algebra to basic calculus? I pose this question because I am in the process of self-learning mathematics as a preparation for a ...
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2answers
50 views

Galois Theory problem (primitive roots of unity)

If $e_1,e_2......,e_{p-1}$ denote primitive $p^{th}$ roots of unity. Here $p$ is prime. And set the sum of the $n^{th}$ powers of the $e_i$ as $p_n=e^n_1+e^n_2......+e^n_{p-1}$ Now I want to show that ...
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14answers
4k views

A good way to retain mathematical understanding?

What is a good way to remember math concepts/definitions and commit them to long term memory? Background: In my current situation, I'm at an undergraduate institution where I have to take a lot of ...
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3answers
68 views

Effective enumerability and Enumerability

I am reading the book by Peter Smith "An Introduction to Gödel's Theorems". I am struck at the point where he tries to construct an non effective enumerable set by using the diagonalisation idea. ...
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1answer
33 views
+50

Solving the LaPlace PDE by complete induction

I have a problem with an induction step that appears to be bothersome for me to comprehend it. Problem: Show that $f(x_1, \dots ,x_n)=(x_1^2+ \dots + x_n^2)^{1- n/2}$ for $(x_1^2 + \dots + ...
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2answers
21 views

Standard Deviation of Population from Sample.

I guess I am a little confused. I am doing t test statistics in my class. I think I know but I would love some insight. I am trying to get the estimated standard deviation for the population. These ...
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2answers
33 views

Free $A$-module isomorphic to a direct sum of copies of $A$?

Does this proposition hold even if it's not finitely generated? I think it does, since $M$ isomorphic to the direct sum of $M_i$, $M_i$ isomorphic to the direct sum of ...
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2answers
33 views

rank of a matrix with two columns s.t. their dot product is zero

I have function $\sigma(u,v)=(f(u,v),g(u,v),h(u,v))$ s.t. $\sigma_u$ x $\sigma_v\neq(0,0,0)$ (cross-product) also, there is the $3\times 2$ matrix : $$ ...
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1answer
87 views

Prove that the curves of the family $v^3/u^2=k$ are geodesics on a surface

Prove that the curves of the family $v^3/u^2=k$ where $k$ is a constant are geodesics on a surface with the metric $$v^2 \, du^2-2uv \, du+2u^2 \, dv^2$$ where $u,v \gt 0$.
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1answer
41 views

Baby Rudin Chapter 4 Exercise Questions 5 and 6

4.5: If f is continuous on a closed set in $R^1$, prove there exist continuous functions $g$ on $R^1$ such that $g(x)=f(x)$ for all $x \in \mathbb{E}$. 4.6: Suppose $\mathbb{E}$ is compact, and prove ...
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1answer
28 views

Abstract Algebra: Automorphism and irreducible

I done how to prove this the question is A. Show that there is an automorphism of the ring of rational number $\mathbb{Q}(i)[x] $that has order of $4$. I suppose i can say that if $\varphi$ is an ...
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1answer
28 views

Abstract Algebra: Irreducible polynomials and automorphism

While I was studying and referring few examples I came up with few question. Hope you guys can help me solving my confusion. Irreducible polynomials problem 1 in this how to argue that why the ...
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1answer
62 views

The meaning of a symbol in the proposition

The question is what's the meaning of the symbol $\phi$? If it just a mapping,what's the mean of the equation? I guess it's $\phi(x)$, $x$ is the element of $M$. Then $\phi(x)$ is the element of ...
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1answer
22 views

For $f: \mathbb{R}^n \to \mathbb{R}$ homogenous, show that $\sum_{i=1}^n x_i \frac{\partial f}{\partial x_i}(x_1, \dots ,x_n)= kf(x_1, \dots , x_n)$

Definition: A function $f: \mathbb{R}^n \to \mathbb{R}$ is said to be homogenous of degree $k$ if $\forall t \in \mathbb{R}$ and $(x_1, \dots , x_n) \in \mathbb{R}^n$ the equations $f(tx_1, \dots , ...
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1answer
57 views

Problem about Galois Extension.

$Gal(\Bbb K/\Bbb F)\cong S$. Where $\Bbb K$ is extension of $\Bbb F$. And $S= G_1\times G_2\times\cdots\times G_K$ is a solvable group. Where each $G_i$ are groups of prime power order and $o(S)= n.$ ...
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0answers
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Where can I find proof - There're infinitely many primes $p$ such that $p(mod\ N)\not\in H$ - Name?

Origin - http://math.uga.edu/~pete/4400FULL.pdf - on p120, Theorem 122 Fix a positive integer $N>2$, and let $H$ be a proper subgroup of $U(N)=(Z/NZ)^{\times}$. There are infinitely many ...
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1answer
43 views

The differential is NOT the Jacobi Matrix?

In the book Analysis II by C.T. Michaels the differential is introduced as the Jacobi-Matrix. In class we had the following definition: Definition: Let $U \subset \mathbb{R}^m$ be open, $f: U \to ...
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1answer
27 views

Finite field and Automorphism

Problem 1. Let S be a finite field of characteristics 2 and the map be define as $\eta$: S$\longrightarrow$S x$\longmapsto$x$^p$ Show that $\eta$ is automorphism, i.e., S is isomorphism ...
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0answers
26 views

Question of a proposition about direct product

I try to prove it's injective, surjective and homomorphism. define f(x)=(x+a1,x+a2,....,x+an),it's homomorphism. it's injective <=> the intersection of ai=0 I don't know how to prove the ...
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2answers
83 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 ...
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1answer
33 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 ...
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1answer
46 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 ...
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1answer
28 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 ...
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0answers
23 views

Nonrejection region- equivalently the area determined by confidence interval for $H_0: \hat \beta=\beta^*$

For $H_0: \hat \beta=\beta^*$ I want to prove that the non-rejection region in level of significance approach Will be eqaul to the area determined by upper and lower bounds in confidence interval ...
3
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1answer
99 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; $E=L=1$ $F=M=0$ $N=G=\cos^2 u$ ...
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1answer
27 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?
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3answers
34 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 ...
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0answers
15 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 ...
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3answers
44 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 ...
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1answer
11 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 $52 = mk$ ...
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0answers
41 views

Using Picard-Lindelöf Theorem to elegantly demonstrate uniqueness of an IVP

I am trying to keep this question clean and short, therefore I won't write down the entire theorem of Picard-Lindelöf here. Problem: $$y'=1+y^2 =:F(y), \ y(0)=0 $$ Find a solution on a ...
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1answer
28 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 ...
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1answer
57 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 ...
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1answer
70 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 ...
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0answers
38 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 ...
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2answers
20 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 ...
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2answers
56 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 ...
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1answer
56 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 ...
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1answer
66 views

Where should somebody who isn't very good at math, start?

I'm 18 years old and I want to learn math. I'm wondering how far can somebody in math go, if they have an average IQ, but math interests them. Also where should one start?
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3answers
59 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 ...
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0answers
30 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) = ...
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1answer
58 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 ...
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1answer
113 views

What's the motivation of definition of primary?

Primary ideal can be regard as the generalization of prime ideal and radical. But Why it's defined like that?It's not symmetry. Why not define like that: