Questions about studying mathematics without formal instruction.

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5
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3answers
172 views
+100

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 ...
3
votes
6answers
222 views

I want a good dictionary of mathematics/ geometry

I noticed I a made a mistake in some geometrical terminology and wanted to better my life by buying a new dictionary of mathematics or more specialised Geometry. (okay I am just a shopaholic for ...
0
votes
1answer
12 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 ...
23
votes
10answers
12k views

How to start with mathematics?

I fell in love with mathematics a bit too late when I've already taken decisions regarding my future, career-wise. Now I would like to learn math on my own but I'm a bit confused as where to start. My ...
0
votes
1answer
25 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
42 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
22 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 ...
0
votes
1answer
13 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
0answers
11 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
24 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 ...
1
vote
1answer
59 views

Math self-study in the holidays

In the upcoming holidays, I have got 6 weeks free to learn some new math (I was thinking of calculus and linear algebra). It's useful for my high school math skills (I don't live in America, so my ...
0
votes
0answers
38 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 ...
0
votes
1answer
63 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 ...
2
votes
2answers
55 views

How to calculate Frenet-Serret equations

How to calculate Frenet-Serret equations of the helix $$\gamma : \Bbb R \to \ \Bbb R^3$$ $$\gamma (s) =\left(\cos \left(\frac{s}{\sqrt 2}\right), \sin \left(\frac{s}{\sqrt 2}\right), ...
10
votes
1answer
326 views

Looking for an easy lightning introduction to Hilbert spaces and Banach spaces

I'm co-organizing a reading seminar on Higson and Roe's Analytic K-homology. Most participants are graduate students and faculty, but there are a number of undergraduates who might like to ...
4
votes
1answer
4k views

How can I learn math of 1st to 10th Standard? [closed]

I am 30+ graduate in Arts from India. I'm very poor in math. I can do basic math calculations, i.e. addition, subtraction, multiplication, and division of simple numbers in writing (on paper only) - I ...
36
votes
3answers
2k views

Why learning modern algebraic geometry is so complicated?

Many students - myself included - have a lot of problems in learning scheme theory. I don't think that the obstacle is the extreme abstraction of the subject, on the contrary, this is really the ...
5
votes
4answers
107 views

Evaluate $\int_0^{\infty} x p^xe^{\Large-\frac{x}{a}}\ dx$

I need to evaluate the integral: $$\int_0^{\infty} x p^xe^{\Large-\frac{x}{a}}\ dx$$ for $0<p<1$. Unfortunately I do not know where to begin. I tried integration by parts but got nowhere ...
0
votes
0answers
37 views

Describe the language generated by the grammar $G = \{\Sigma, \Delta\ S, I \} $

I need to describe the language generated by the grammar $G = \{\Sigma, \Delta\ S, I \} $, where $$\Sigma=\{0,1\epsilon \}, \Delta = \{S,X,Y,Z\}$$ and $$I = \{S \to0X|1Y, x \to1Y|1Z, Y \to0X|0Z, Z ...
1
vote
1answer
21 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 ...
151
votes
27answers
14k views

Too old to start math

I'm sorry if this question goes against the meta for posting questions - I attached all the "beware, this is a soft-question" tags I could. This is a question I've been asking myself now for some ...
15
votes
0answers
832 views

Efficient ways to read and learn a new topic

I started reading the book "Topology without tears" by Sidney A Morris and lecture notes on "Elementary Number Theory" by WWL.Chen. To get the maximum out of the book and understand the material ...
0
votes
2answers
46 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
54 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 ...
24
votes
3answers
561 views

Do free online collaborative solution manuals exist?

I'm not a mathematician by training and a rarely come in contact with mathematicians. For this reason I find solution manuals to be incredibly useful - reading them allows me to see how experienced ...
2
votes
8answers
699 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 ...
24
votes
8answers
2k 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
4answers
45 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 ...
1
vote
2answers
28 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
30 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 ...
1
vote
2answers
20 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
166 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
28 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
20 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
43 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
29 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
585 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 ...
2
votes
2answers
29 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.
61
votes
23answers
11k views

Complete course of self-study

I am about $16$ years old and I have just started studying some college mathematics. I may never manage to get into a proper or good university (I do not trust fate) but I want to really study ...
2
votes
1answer
62 views

Decided to finally jump in

I've finally decided to jump into teaching myself math because I am a junior (soon senior) in high school and have been interested in math for the longest time. I am not sure if this question belongs ...
0
votes
1answer
241 views

Equivalent systems of Linear equation

I've just begun to re-learn linear algebra because is so important, the book that I chose is naturally the Hoffman's for a lot of reason. Well, In the first chapter I'm stuck with the following, ...
1
vote
1answer
36 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 ...
0
votes
2answers
56 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: ...
70
votes
8answers
17k views

Learning mathematics as if an absolute beginner?

I dread mathematics, and I believe it's because I have come to associate mathematics with the experience of terrible teachers. All of my math teachers have been grumpy, but one in particular was the ...
1
vote
1answer
399 views

Linear algebra and Multivariable calculus prerequisites for Stochastic Calculus

Which topics are considered "graduate-level" for the following subjects: Linear algebra Multivariable calculus On Internet, it is said that you need "graduate level" Linear algebra and ...
6
votes
0answers
87 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
34 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
50 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
25 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 ...
1
vote
0answers
43 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 ...