Questions about the process of studying mathematics without formal instruction.

learn more… | top users | synonyms (1)

0
votes
1answer
36 views

Continuity of a map to a Frechet space

Let $(A,\| \cdot \|)$ be a normed space and $B$ be a Frechet space equipped with a family $\{ p_k \}_{k \in \mathbb{N}}$ of seminorms. Let $\phi: A \to B$ be a linear transformation satisfying the ...
5
votes
3answers
61 views

Area preserving transformation in a higher dimensional space is unitary.

In $\mathbb{R}^3$, a linear operator $Q:\mathbb{R}^3 \to \mathbb{R}^3$ preserves the area of parallelograms: that is, given $x,y\in \mathbb{R}^3$, the area of a parallelogram formed by $x$ and $y$ is ...
0
votes
1answer
17 views

quick question on measurability of random variables and what becoming a deterministic function means.

we stated a theorem in class: if X r.v. is $\sigma(Y)$ measurable then X is a function of Y, where $\sigma(Y)$ signifies the sigma algebra of Y. This is fine. The Professor sometimes states that X ...
0
votes
1answer
30 views

Logarithm with variable base

I am trying to define a function that maps polynomials in the form of $x^{3^n}$ to the value of $n$ in the polynomial, where $n\in{Z}$.* Is is valid to define this function as $log_{x^3}(u)$, where ...
1
vote
0answers
56 views

Soft question — I need books and exercise books that will be working on my fundamental skills.

I need help, urgently. I acquired a book called: Mathematics, Its Content, Method and Meaning. Now the problems is the book doesn't provide me with any exercises. I was searching for a book that would ...
2
votes
2answers
36 views

Probability of returning to a given state after n transitions-Markov chains

Let us denote $f_j^{(n)}$ denote the probability of the first return to state $j $after n transitions. Let $p_{jj}^{(n)}$ be the probability of returning to the state $j$ after $n$ transitions when ...
1
vote
1answer
32 views

Confusion with Bolyai-Gerwien theorem

The Bolyai-Gerwien theorem states: Given two polygons with the same area, it is possible to cut up one polygon into a finite number of smaller polygonal pieces and from those pieces assemble into ...
1
vote
1answer
70 views

Queuing theory-Multiple server (reducing simple recurrence formulas)

The equations given in 6.3 have been reduced which really eases the computation in further studies. But I tried to find the method of reducing these but I could not find a way at all. Any hints will ...
-1
votes
2answers
32 views

Prove this result about power sets [duplicate]

I have to prove this result: If $P$ be the power set, and $B$ and $C$ are two sets, then if $B \subseteq C$ prove that $P(B) \subseteq P(C)$. Now, it seems obvious to me that since all the ...
2
votes
1answer
100 views

What is the metric spaces needed to motivate concepts of general topology?

I intend to start learning some topology on my own. I wonder How much metric spaces I should know in order to motivate the concepts of topology? I know it's possible to learn topology without any ...
0
votes
0answers
53 views

Can ergodic Markov chains be periodic?

I found a statement in one of my notes which said If a state is persistent, aperiodic and not null the it is said to be ergodic Is it necessary that it should be aperiodic? This statement ...
1
vote
1answer
27 views

Markov chains: An issue in classification of states

I recently came across a lemma which goes as follows. Suppose a Markov chain has N states. Let i and j be pair of states. Then j can be reached from i iff there is an integer $ 0 ≤n< N$ such ...
0
votes
2answers
68 views

Find the galois group of the polynomial when a root is given

If $\alpha$ is a root of a polynomial $f(x)=x^3 +x^2-4x+1$ then show that $2 - 2\alpha - \alpha^2$ is also a root of $f(x)$. Use this fact to compute the Galois group of the splitting field of $f(x)$ ...
2
votes
2answers
65 views

Intersection of ideals $I=(2x)$ and $J=(2x^2)$ of $\mathbb{Z}[2x,2x^2,2x^3,\dots]$ is not finitely generated.

Consider the subring $\mathbb{Z}[2x,2x^2,2x^3,\dots]\subset \mathbb{Z}[x]$. Then show that the intersection of ideals $I=(2x)$ and $J=(2x^2)$ of $\mathbb{Z}[2x,2x^2,2x^3,\dots]$ i.e., $I\cap ...
2
votes
7answers
749 views

What are the most important functions every mathematician should know? [closed]

I am an undergrad in math and was wondering, what are for you the most important functions every mathematician should know? At the moment I think ...
1
vote
4answers
52 views

A simple conditional probability problem

Assume that two fair dice are rolled one at a time. Given that the sum of the two numbers that occured was at least $7$, compute the probability that it was equal to $7$. I tried computing the ...
0
votes
2answers
740 views

Proof that the subset relation is reflexive and transitive

I'm teaching myself set theory, and I'm not sure how detailed I should be when asked to prove things. Here is my proof that $A\subseteq A$ (the subset relation is reflexive): $A \subseteq B$ iff ...
1
vote
1answer
59 views

Optimization with both equality and inequality constraints

I need to minimize the following quantity: $$\min x_1^{-1/n}- \left(1-x_2 \right)^{-1/n}$$ subject to: $1-x_1-x_2=\gamma$ and $0<x_1+x_2<1$ $\gamma$ being a constant. Had it been two ...
4
votes
2answers
115 views

How to find the minimum value of $|5^{4m+3}-n^2 |$

How can I find the minimum value of $|5^{4m+3}-n^2 |$ for positive integers n,m. I solve this home work problem, but it is a very long process. So I need a short answer.
28
votes
8answers
3k views

Complex analysis is more “real” than real analysis

In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself ...
1
vote
2answers
34 views

law of total probability and conditiona probability exercise.

Exercise: Let $X$ be an uniform discrete r.v. with four possible values: 1, 2, 3, 4. Let $Y$ be an exponential variable whose parameter is the value taken by $X$. So, if $X = 3$, $Y$ is Exp (3). ...
4
votes
3answers
182 views

Show that $\lim_{x \rightarrow 1} \frac{x^4-2x+1}{x-1} + \sqrt{x} =3$

Show that $\lim_{x \rightarrow 1} \frac{x^4-2x+1}{x-1} + \sqrt{x} =3$ from the definition (using $\epsilon-\delta$) Why can't I do something like this? We want: $|\frac{x^4-2x+1}{x-1} + ...
0
votes
1answer
55 views

Gauss-Jordan Method

I keep getting the wrong set of solutions can someone help me. I know that when using the Gauss-Jordan method, the rules that I must follow can be applied in a variety of different procedures then why ...
0
votes
2answers
26 views

Inverse function of borel sets when function is a constant.

Following a simple proof my professor explained in class I am having problems with a specific step: The proof is of probabilistic nature and we are trying to prove that If $X$ (random variable) is ...
0
votes
0answers
56 views

Difficulty parsing combinatorics exercise

I am working through the wonderful book Proofs and Confirmations by David Bressoud. In the section 2.2, I came across the following exercise, which has me scratching my head. (2.2.8) ~ Let ...
0
votes
0answers
39 views

Showing the modified Dirichlet function is discontinuous

Show, using the $\epsilon-\delta$ definition of continuity, that the modified Dirichlet function, i.e., $f(x) = x$ if $x$ is rational and $f(x) = 0$ if $x$ is irrational, is discontinuous at ...
7
votes
0answers
1k views

Learning higher-mathematics on your own

I was hoping someone had an opinion on how to learn higher-mathematics (specific fields that could be of use to me) outside of a classroom setting. I graduated with an M.S. in Computer science about ...
3
votes
2answers
63 views

Prove $n(A-B)=n(A)-n(A \cap B)$

Prove that: $n(A-B)=n(A)-n(A \cap B)$ This is an example from my book in which first step is like this:$$n(A)=n(A-B)+n(A \cap B) $$ But how did they get it.
1
vote
1answer
94 views

The logical consequence of an empty set of premises.

I am studying propositional logic by self-study, using a dutch book. I hope I am translating the terms to the correct English term. If my words are confusing, please please just let me know instead of ...
6
votes
0answers
107 views

Who wants to learn set theory? [closed]

So set theory is something I really want to learn. I found this document that I really like, except the fact that it doesn't prove all of it's theorems in with a lot of detail (a lot of times they say ...
1
vote
2answers
66 views

Find out the value of $d$

If the mean deviation of number $1,\ 1+d,\ 1+2d,\ 1+3d,\ldots,1+100d$ from their mean deviation $255$ then $d$ equals to ? This was the question asked in AIEEE 2009. MY EFFORTS: ...
1
vote
2answers
64 views

How to get the number of ways of getting a five card hand that is a straight flush from a standard deck of cards

I do not get the result at this page, ex. 13-7: Suppose that Aces can be either high or low; that is, that {Ace, 2, 3, 4, 5} is a straight, and so is {10, Jack, Queen, King, Ace}. The number of ...
0
votes
1answer
148 views

Starting Calculus with a weak foundation in Pre-Calculus

I am struggling in Pre-Calc mathematics, and I want to know is it ok if I start Calculus I with a weak foundation in Pre-calculus mathematics? I understand the general gist of limits, function ...
0
votes
0answers
29 views

Proving that the $[g,x]^n=e$ if $G$ is nilpotent of degree $n$

This is an article from wikipedia which I saw wondering as to how to prove it. The question is If $G$ is nilpotent of degree $n$ then $[g,x]^n=e$ for all $x \in G$, where $[g,x]=g^{-1}x^{-1}gx$. I ...
3
votes
1answer
202 views

Most Suitable Book after Kline's Calculus?

I've been working through Morris Kline's Calculus: An Intuitive and Physical Approach and it's an absolutely excellent book for self-studying applied single-variable (and some multi-variable) calculus ...
1
vote
2answers
70 views

Find the limit of $a_n = e^ne^{-e^n}$

Consider the sequence $(a_n)$ where $a_n = e^ne^{-e^n}$, what is $\lim_{n \rightarrow \infty} (a_n)$? I have a feeling it's 0 because $\displaystyle a_n = \frac{e^n}{e^{e^n}}$ and $e^n$ grows ...
5
votes
0answers
215 views

Analysis or (abstract) algebra first?

Which one would you recommend? I only know calculus and linear algebra when it comes to university-level mathematics. Is one required to understand the other?
0
votes
1answer
78 views

What is purpose of these paragraph?

The following paragraphs are from my study notes on probability theory. It is a section within the independence discussion. But to me, they seem to appear here out of blue. I do not understand what ...
8
votes
1answer
133 views

Follow up to Pinter's abstract algebra

I wanted to learn abstract algebra this summer so I bought Pinter's A book of Abstract Algebra. I was planning on reading it over the course of the summer, but just finished the last problem of its ...
1
vote
0answers
31 views

Are there online-platforms where to find people for joint learning and discussions?

After quite some time in academia I ended up in a nice company but the math to use is not really demanding. Hence I am still reading and working a bit on some university level math. It's roughly at a ...
4
votes
3answers
802 views

How is math used in computer graphics? [closed]

I'm doing a research paper on the mathematics of computer graphics and animation (3D) and I do not know where to start. What mathematical equations and concepts are used for computer graphics and ...
1
vote
1answer
13 views

Finding out number of observation

There are $n$ scores $X_1,X_2,X_3,....,X_n$ and their sum is $80$ and sum of their squares is $400$ then which among them is the probable value of $n$ A)$10$ B)$9$ C)$15$ ...
1
vote
2answers
187 views

How wrong is it? - A “proof” of the FTC that I came up with in high school by hand-waving.

In high school calculus, I was first taught that the area under a curve $f(x)$ between $x=a$ and $x=b$ is given by: $$ A = \lim_{\delta x \rightarrow 0} \sum \limits_{a}^{b} f(x) \delta x $$ Then ...
11
votes
3answers
627 views

Book series like AMS' Student Mathematical Library?

I had the joy of discovering AMS' Student Mathematical Library book series today, and I have been pleasantly surprised by how enticing some of the titles seem: exciting and expositionary, a perfect ...
0
votes
1answer
73 views

Convergence in Probability and in Quadratic Mean for a sequence of random variables

I have been trying to determine whether a sequence of random variables, $X_1,X_2,\ldots,X_n$, such that $$P\left(X_n= \frac{1}{n}\right)=1-\frac{1}{n^2}\\ \text{and}\\ ...
2
votes
1answer
120 views

Does math have to be learned linearly?

I am asking because often times one doesn't know where to start in math. "Just learn what you need" is very vague and unspecific ... for example, assume I'm a beginner at Algebra and was considering ...
3
votes
2answers
145 views

Zero divisors in ring of real valued functions.

I'm working though Pinter's A book of Abstract Algebra and would like a quick verification on a simple problem. Exercise 17.B2 asks Describe the divisors of zero in $\mathcal{F}(\mathbb{R})$. ...
8
votes
3answers
228 views

Beginning of Romance

I am a 17-year old student in India, in the standard 12th grade. Recently, I found the fascination in mathematics, and I am eager to dig in further. Currently, the only textbooks I have are the ones ...
0
votes
2answers
43 views

A few questions on Minimal Polynomials

I have been trying to see the properties of Minimal Polynomials. So from the examples I guessed the following properties but I am not sure whether they are true. The properties are: 1) If $\alpha$ is ...
1
vote
0answers
33 views

Wedge product of Lie algebra valued differential forms [duplicate]

Let $\mathfrak{g}$ be the Lie algebra of a matrix Lie group. Furthermore, let us consider the following $\mathfrak{g}$-valued $p$-form and $\mathfrak{g}$-valued $q$-form: \begin{equation} ...