Questions about the process of studying mathematics without formal instruction.

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1answer
36 views

How to evaluate: $\int_{0}^{1-z} y^{j-1} (1-z-y)^{n-k} dy$

How can I compute the integral: $$\int_{0}^{1-z} y^{j-1} (1-z-y)^{n-k} dy\quad\text{where}\ z \in (0,1) $$ Had it not been for $z$ , the integral would look like an incomplete beta function but what ...
3
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1answer
299 views

An Increasing Function Discontinuous at All Rational Numbers

Let $q: \mathbb{N}\rightarrow \mathbb{Q}$ be a bijective map and let $g: \mathbb{Q}\rightarrow \mathbb{R}$ define $g(q(n))=2^{-n}$. Show that $\sum_{r\in \mathbb{Q}}g(r)$ is absolutely convergent. ...
1
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1answer
32 views

Find the point of inflection

Will there be an inflection point if there is no solution for $x$ when $f ''(x) = 0$? For example, $$ f(x)=\frac{x^2-x+1}{x-1} $$ with domain $\mathbb{R}-\{1\}$ Also, is that when $x$ is smaller than ...
2
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1answer
30 views

Example of an equivlance relation that is transitive

I am just tying to figure our this example but am having difficulty understand the math being used. The example state: Let R be a relation on the set $\mathbb{Z}$ defined as $(m,n)\in$ R if and only ...
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2answers
42 views

Trying to understand an example of an equivlance relation that is symmetric

I am just tying to figure our this example but am having difficulty understand the math being used. The example state: Let R be a relation on the set $\mathbb{Z}$ defined as (m,n)$\in$ R if and only ...
0
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2answers
60 views

How to integrate absolute function

I have this absolute e-function, but I don't know how to calculate the integration $$ \int_{-2}^{2} e^{\frac{1}{2}j\omega |x|}dx $$ Any idea?
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1answer
401 views

How to prove the Jordan's Inequality?

Can anyone tell me how to prove it by using the concepts related to limit?
0
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0answers
21 views

Inverse Fourier Transform of $S_Y(f)$

I have this power spectral density $$ S_Y(f) =\frac{N_0}{4 \pi ^{2} f^{2}}\left [ 1- \cos(2\pi f T) \right ] $$ Can any one help me how to find the Inverse Fourier transform?
3
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1answer
92 views

Exponentiation of Real Numbers?

I'm looking to learn Real Analysis on my own. Am reading Elements of Real Analysis by Bartle. I came across this project which defines the powers of real numbers i.e. exponentiation. Firstly I am ...
6
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2answers
596 views

What can I do with measure theory that I can't with probability and statistics

I've studied mathematics and statistics at undergraduate level and am pretty happy with the main concepts. However, I've come across measure theory several times, and I know it is a basis for ...
3
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2answers
74 views

one-to-one and onto functions help

I am trying to understand this exercise. Define $S : \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ by the rule: For all integers $n$, $S(n) =$ the sum of the positive divisors of $n$. a. Is $S$ one-to-one? ...
6
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2answers
195 views

What all maths do I need to know to become good at machine learning.

I am a computer science engineer and I took a couple of maths classes in my first year they were on Fourier series(not transform) partial differential equations, vector calculus, infinite series ...
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3answers
87 views

Calculus: Limit and continuity

Would anyone mind telling me how to solve these two questions? I know it sounds silly but I really have no idea.
0
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1answer
38 views

PMF of Two Random Variables

X and Y are independent and geometrically distributed random variables with $$ P(X = m) = p(1-p)^{m}, m=0,1,2... $$ $$ P(Y = n) = p(1-p)^{n}, n=0,1,2... $$ To find the probability mass function ...
0
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2answers
80 views

one-to-one and onto function question

I am trying to understand this exercise: Let S be the set of all strings of 0's and 1's, and define f: S -> $Z^{nonneg}$ by f(s) = the length of s, for all string in S. a. Is f one-to-one? The ...
10
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2answers
578 views

Modern research into Grassman's “theory of forms”?

I quote from Petsche's Hermann Graßmann: Biography (emphasis mine): The mathematical part of the book begins with the conception of the “General Theory of Forms”. Starting with a perspective on ...
1
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0answers
108 views

$(x^r)^s=x^{rs}$ for the real case

Hi everyone I'd like to know if the following is correct and if someone knows a better way to do it. Definition Let $x>0$ be a real, and $\alpha$ be a real number. We define the quantity ...
3
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1answer
50 views

Determining quadratic residues quickly

Let's say that I'm looking for all quadratic residues of a number. THe example from my book is 31. So I can just evaluate $i^2\equiv{a}\pmod{31}$, for $i=1..15$. While not a terribly difficult ...
0
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1answer
65 views

Most powerful size $\alpha$ test

Someone can help me to check this answer? How to find the Most Powerful Test size $\alpha$ and Power of Test, Since I have $H_0 : X \thicksim f_{\theta 0}= (1/\sqrt(2\pi) \exp^{(-x^2/2)}$ and $H_1 : ...
10
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2answers
193 views

Suggestions for a learning roadmap for universal algebra?

I think a useful combination of resources for universal algebra would ideally, when taken together: Provide ample motivation behind the various developments in the field. Either provide powerful ...
1
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2answers
171 views

Continuity of $f(x)=x^p$ when $p$ is a real number and $x\in (0,\infty)$

Here is my final answer. Definition Let $x>0$ be a real, and $\alpha$ be a real number. We define the quantity $x^{\alpha}$, by the formula $\text{lim}_{n\rightarrow\infty} x^{q_n}$ where $(q_n)$ ...
1
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1answer
113 views

Leibniz rule in a Double Integral

I have been trying to evaluate the following double integral: $$\frac{\partial}{\partial \theta_1 \partial \theta_2} \int_{\theta_1-\theta_2}^{\theta_1+\theta_2} \int_{\theta_1 -\theta_2}^{x} u(y,x) ...
0
votes
1answer
108 views

Self teaching “Curiculum”

This will probably be flagged but here goes anyway. I guess I am a glutton for punishment, but, I have been trying to self teach advanced math ever since I realized that if I want to read anything ...
0
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1answer
35 views

Analysis of first order differential equation

I'm working through a question where the differential equation is $$ y^2(y'^2 -1)(3y'^2 +1) = c, \;\;\; y(0) = 0 $$ and the answer proceeds with two cases (1) $c=0 \implies y(x)=0 \vee y(x) = \pm ...
0
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1answer
70 views

Confusion in an order topology being Hausdorff and $T_1$.

There is a theorem which says that every order topology is Hausdorff. Also every Hausdorff follows $T_1$ Axiom. So suppose $X = \{1,2\}$. Now $1 < 2$. A basis for the topological space $X$ is ...
1
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1answer
28 views

Continuity on finite interval

Problem: $f$ is a continuous function on $[a,b]$, and $f(x)>0$ for all $x$ in $[a,b]$. Prove that there is an $\alpha>0$ such that $f(x)>\alpha$ for all $x$. Please help in the steps of this ...
0
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1answer
92 views

Limit of function using sequences

One easy question about limit of function at a point. In the book what I read says: Let $X\subset \mathbb{R}, $ $f: X\rightarrow \mathbb{R}$, $E\subset X$, $x_0$ be an adherent point of $E$ and $L\in ...
0
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1answer
150 views

Issue with proof: Cauchy Completeness of Real Numbers

Having trouble understanding a cardinality-related argument when proving that all Cauchy sequences of reals numbers converge to a real limit. Came across it on CC Pugh's Real Mathematical Analysis ...
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0answers
43 views

If $x^m=y^n=e$ and $(m,n)=1$, then $o(xy)=mn$ [duplicate]

If $G$ is a group with $xy=yx$ for $x,y\in G$, and $x^m=y^n=e$, then: (1) $o(xy)|mn$, where $o(xy)$ is the order of $xy$, and (2) if $(m,n)=1$ then $o(xy)=mn$. Well, I've already proved (1), but ...
2
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0answers
113 views

The Heine-Borel Theorem for the real line

Hi everyone I'd like to know if the following argument is correct and also I'm very interested in a constructive approach for (2)$\Rightarrow$(1) (a link or a hint it will sufficient for me) I was ...
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0answers
42 views

Resources for self-learning “relational” abstract algebra? [please see body of post for details]

I have been studying Grassman and Clifford algebras a bit, and it is fascinating to see how, for example, the rules defining the inner product operator are enough to the capture something of the ...
3
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0answers
70 views

Equality of nested radicals with different operations [duplicate]

I was playing around on Maple with some nested radicals and I notices that $$\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}}=\sqrt{2\sqrt{2\sqrt{2\sqrt{2\cdots}}}}=2$$ I thought my mind was playing tricks ...
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2answers
57 views

For 2nd order linear homogenous ODE, what is the effect of changing the value of the IC for $\mathrm{d}y/\mathrm{d}x$?

Given a linear homogenous second-order ODE $$f_2(x)\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + f_1(x)\frac{\mathrm{d}y}{\mathrm{d}x} + f_0(x)y = 0$$ with initial conditions $$y(x_0)=0$$ ...
0
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0answers
60 views

Proof verification and suggestion to elude the AC (equivalent definition of adherent points).

Hi everyone I'd like to know if the following is correct and, more importantly, if there is some way to escape of the axiom of choice (as the hint the book says "use AC"). Definition: Let $X\subset ...
0
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1answer
30 views

Induction with two indexes

I want to prove that if $G$ is a group and $a\in G$, $n,m\in \Bbb Z$, then $a^na^m=a^{n+m}$. I think, that it's easier to prove the case when $n,m\in \Bbb N$. I found this question: Induction (over 2 ...
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2answers
109 views

Elementary properties of closure

Hi everyone I'd like to know if the following is correct. I really appreciate any suggestion. (Honestly the only one that matters me is the second property the others are easy, I think) Thanks. ...
2
votes
1answer
128 views

Proving that $G$ is a group if $a*x=b$ and $y*a=b$ have solutions.

(Reference : Fraleigh, A first course in abstract algebra) Prove that a nonempty set $G$, together with an associative binary operation * on $G$ such that $a*x=b$ and $y*a=b$ have solutions in G, ...
2
votes
1answer
318 views

Expected value of sample median given the sample mean.

Let $Y$ denote the median and let $\bar{X}$ denote the mean of a random sample of size $n=2k+1$ from a distribution that is $N(\mu,\sigma^2)$. How can I compute $E(Y|\bar{X}=\bar{x})$? Intuitively, ...
3
votes
1answer
62 views

Figuring out deficiencies in math education.

I'm mostly self-taught and while I know (and use) many advanced mathematical topics, I often enough find holes in my understanding of lower level math. Is there an exam (or series of exams) I could ...
0
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1answer
40 views

problem with isometry that doesn't preserves rays, i just don't understand

I try to teach myself a bit of non-euclidean geometry And I am a bit stumped by the following remark in George E. Martins "The foundations of Geometry and the Non-Euclidean Plane" page 217. extra ...
8
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1answer
289 views

“Easy” (maybe not) question about dual spaces (Lineal Algebra).

Hi everyone is my first time reading about dual spaces and in one part of the notes that I read, says: The dual of the quotient space $V/U$ is naturally a subspace of $V$, namely the annihilators of ...
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2answers
41 views

A question on conditional probability

Question: Let $X$ and $Y$ be two random variables. The relationship between the two is as follows. If $Y$ is less than or equal to $1$, then $X$ is equal to $Y$; If $Y$ is more than $1$, then $X$ ...
3
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3answers
922 views

Difference between continuity and uniform continuity

I understand the geometric differences between continuity and uniform continuity, but I don't quite see how the differences between those two are apparent from their definitions. For example, my book ...
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2answers
35 views

Absolute value of the difference between two sequences

Consider two sequences $(a_n)$ and $(b_n)$ both contained in $\mathbb{R}$ and assume that these two sequences satisfy $|a_n - b_n| \rightarrow 0$, then this does imply that both $a_n$ and $b_n$ are ...
2
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1answer
75 views

Source for Differential Manifolds/Geometry Questions?

I'm looking for a good source for a large collection of Differential Manifolds/Geometry questions covering a subset of the following topics: inverse function theorem, local coordinates, induced ...
0
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1answer
656 views

Brouwer's Fixed Point theorem proof for 2-dimension

I am trying to find a elementary proof of the Brouwer's fixed point theorem only using basics of point set topology and real analysis. In the one of the textbooks I read, they were proving Brouwer's ...
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2answers
55 views

The mean of a geometric random variable

The mean of a geometric random variable is $$ \frac{1-p}{p} $$ What would be the mean of x for this density function? $$ f_{x}(x) = \sum_{k=0}^{\infty} p(1-p)^{k} \delta (x-k) $$ I got confused ...
1
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1answer
101 views

Group with an even number of elements.

If $G$ is a group such that $|G|=2n$. Prove that there's an odd number of elements of order 2, and then there's an element which is its own inverse, besides of the identity. If we consider all the ...
3
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0answers
57 views

Prime divisor of the form $2kp+1$ that divides $2^p-1$

The book that I'm reading (Elementary Number Theory by Underwood Dudley) gives a Theorem: If $p$ and $q$ are odd primes and $q|a^p-1$, then either $q|a-1$ or $q=2kp+1$, for some integer $k$. Then it ...
7
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1answer
418 views

What pure mathematics foundations should an applied mathematician have?

I'm studying mathematics, with some statistics also, and I've always chosen applied courses. I'm getting to the point where I'm studying 3rd year undergraduate to graduate level material. My first ...