The process of studying mathematics without formal instruction. _Don't_ use this tag just because you were self-studying when you came across the mathematical question you're asking; it is only for when fact that you're self-studying is what your question is _about_.

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-4
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1answer
38 views

Proving that $\mathbb{N}$ (or any countable set) is infinite

I was watching real analysis lectures by Francis Su. In the lecture on countable and uncountable sets, he writes a theorem which is my question: Thm. $\mathbb{N}$ (or any countable set) is ...
0
votes
0answers
36 views

Should you create a dictionary when learning mathematics?

Usually when I need to look up a term, I just consult the internet. But recently I found myself in an unfortunate situation where I was camping high up in the mountains and needed to look up the ...
0
votes
0answers
23 views

Stirling aproximation [duplicate]

I was reading my book of stochastic processes, when suddenly appear the following approach $$n!\sim n^{n+\frac{1}{2}}e^{-n}\sqrt{2\pi}$$ From where this result comes? I looked on Wikipedia but ...
0
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0answers
57 views

Conditional Expected Value of Product of Normal and Log Normal Distribution

Could someone please provide the answer and steps to solve this expression? \begin{eqnarray*} E\left[\left.\left(e^{X}Y+k\right)\right|\left.\left(e^{X}Y+k\right)>0\right]\right. \end{eqnarray*} ...
1
vote
0answers
22 views

Show that $f$ is constant on the convex set $S$

Call a set $S$ convex if whenever $x,y\in S$, then $tx+(1-t)y\in S$ for any $t\in[0,1]$. Suppose that $S$ is an open convex set in $\mathbb R^n$ and suppose that $f:\mathbb R^n\to\mathbb R$. If ...
3
votes
1answer
35 views

What can we conclude about the function $f$?

Let $f$ be a scalar field, $f:\mathbb R^n\to\mathbb R$. Suppose there is an $n$-ball $B(a;r)$ centered at $a$ with radius $r$ and a fixed vector $y\in\mathbb R^n$ such that $f'(x;y)=0$ for every ...
0
votes
1answer
25 views

Matrix Multiplication, Trace and Integration

Let $\omega(x)$ be a $p\times 1$ vector-valued function defined on a random variable $X$ with CDF $F$. Now define $$V:=\int \omega(x)[\omega(x)]^T dF(x).$$ Then define $\gamma$ as follows. $$ \gamma ...
0
votes
2answers
20 views

how do they calculate these following columns

I have these data: I am sorry the data is in Portuguese, and it is an image so I can't convert it to a table but the translate "probably" ( i am not a native speakers for Portuguese language) is: ...
2
votes
1answer
31 views

Probability theory required for learning statistics rigorously

I would like to learn statistics rigorously. The only book that I can find that seems to do statistics rigorously is this book "Theory of statistics" by Schervish (which seems advanced): ...
0
votes
1answer
23 views

Markov Chains, reccurent and transient

Let the Markov Chain consisting of the states $0,1,2,3$ have the transition probability matrix ...
1
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0answers
33 views

How do I make sure that I've learned and mastered a part of the Visual Complex Analysis book?

So I'm reading Visual Complex Analysis by Tristan Needham. It's a beautiful book that's not very hard to understand at all; however, I just don't know if I have sufficiently learned what I'm supposed ...
3
votes
1answer
44 views
+50

Finiteness of the sum of the product of an i.i.d. sequence

Before I go to the statement of my question I just want to say a few words about the personal background of this question. I have recently taken a course on stochastic differential equations without ...
2
votes
2answers
482 views

Geodesics on torus

Describe the geodesics on Torus $$\sigma (u,v)= ((a+b \cos u)\cos v, (a+b\cos u)\sin v, b\sin u)$$ First fundamental form for torus is $$b^2 du^2 +(a+b \cos u)^2dv^2$$ Consider unit-speed ...
5
votes
1answer
258 views

Most efficient way to learn mathematics

So there's a lot of advice on how to learn mathematics most efficiently, and it mostly revolves around doing problems, asking questions, and considering all possible generalizations. However, I was ...
0
votes
1answer
22 views

How to construct a transition matrix?

I'm giving my first steps in stochastic processes but I'm having some difficulties. See the following example Suppose that whether or not it rains today depends on previous weather conditions ...
0
votes
0answers
41 views

How to effectively learn from and use Ramanujan's notebooks? [duplicate]

I will come back and elaborate on the question if necessary (I must be off for a while...). But I'll try being specific. I have all four of Ramanujan's notebooks, with their respective Errata ...
0
votes
2answers
31 views

Find the density of their average

If $f_{X,Y,Z}(x,y,z)=e^{-(x+y+z)}I_{[0,\infty]}(x)I_{[0,\infty]}(y)I_{[0,\infty]}(z)$ find the density of their average $\frac{X+Y+Z}{3}$ I'm a little lost on how to solve this exercise, ...
4
votes
1answer
100 views

Is Alfred Tarski's Introduction to Logic still helpful for self study?

I am trying to setup a self study path to enhance my knowledge of mathematical logic. I haven't taken a logic course for a few years and my confidence on mathematical proofs is unnerving. I am ...
2
votes
2answers
51 views

Disjoint events

Let $A$ and $B$ two disjoint events such that $P(A)=0.3$ and $P(B)=0.5$. Find the probability that i)$A$ or $B$ ocurrs ii)$A$ occur but not $B$ iii)repeat $i)$ and $ii)$ with $A$ ...
1
vote
1answer
37 views

Principle of well ordering

Every non-empty set $A\subset\mathbb{N}$ have a smallest element, i.e. an element $n_0\in A$ such that $n_0\leq n$ $\forall n\in\mathbb{A}$ Proof: Let $I_n=\{p\in\mathbb{N};p\leq n\}$ the set ...
0
votes
2answers
34 views

Reference request for this topics

I need a good reference to learn these topics Markov Chains in discrete time.    1.1. Classification of states, recurrence notions of transience.    1.2. Stationary measure.    1.3. ...
0
votes
4answers
48 views

Logic, writing proof

i)Suppose that $x$ and $y$ are real numbers. Prove that if $x\neq 0$, then if $y=\frac{3x^2+2y}{x^2+2}$ then $y=3$ ii)Suppose that $x$ and $y$ are real numbers. Prove that if $x^2y=2x+y$, ...
0
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0answers
49 views

How to prove it and how to solve it

Tomorrow I will begin my studies, real analysis, however I have some difficulties in making statements so I thought before starting the study in real analysis, learn how to do demonstrations properly. ...
0
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0answers
26 views

Integral gaussian hypergeometric function

How can we define integral with interval $[b,\infty)$ $$ \begin{align} C(b,\alpha) & = \int_b^\infty \frac{1}{1+w^{\alpha/2}}\,\mathrm{d}w \\[8pt] & = 2\pi/\alpha \csc(2\pi/\alpha)-b_2 F_1 ...
34
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6answers
27k views

Good books for self-studying algebra?

I have a few weeks off from school soon, and I was hoping to self-study a bit of algebra. I don't think this question has been asked on here before, but does anyone have any suggestions for algebra ...
0
votes
0answers
22 views

non-linearity and non-convexity

I am taking a course on linear regression online and it talks about the sum of square difference cost function and one of the points it makes is that the cost function is always convex i.e. it has ...
0
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0answers
51 views

How to obtain the genus of the Riemann surface corresponding to an algebraic curve

I am trying to read about the genus of an algebraic curve. I have been told that there is a connection between topological genus and genus defined for an algebraic curve. Since an algebraic curve ...
3
votes
4answers
117 views

Why memorize trig identities?

I want to be a mathematician or computer scientist. I'm going to be a junior in high school, and I skipped precalc/trig to go straight to AP Calc since I've studied a lot of analysis and stuff on my ...
1
vote
2answers
58 views

What kind of geometry is useful to study for mathematical competitions?

I'm bad in geometry but I would like to be better. What kind of geometry is useful to learn olympiad level geometry? I mean, can projective geometry solve more problems than geometry with complex ...
11
votes
4answers
475 views

The best balance in studying Mathematics? [closed]

I'm a student studying Mathematics at a university level. I've completed Single Variable Calculus, Differential Equations, Multivariable Caculus, Real/Complex Analysis, and Linear Algebra and I've ...
1
vote
1answer
54 views

Learning Galois theory - required subtopics that are prerequisite?

This is not a reference request, that is, I have access to many textbooks I am happy with. What I don't know is, what are the things I need to know to get started? My idea on the path of knowledge ...
181
votes
7answers
9k views

Best Sets of Lecture Notes and Articles

Let me start by apologizing if there is another thread on math.se that subsumes this. I was updating my answer to the question here during which I made the claim that "I spend a lot of time sifting ...
1
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0answers
17 views

Random variable with respect to the event space [closed]

Having the probability space $(S, \mathcal F, \Pr(\cdot))$ and a very large $\mathcal F$, like the power set, how do we define a function $X(\cdot): S \rightarrow \mathbb{R}$ which is not a random ...
0
votes
1answer
502 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
12 views

Variance of Inhomogenous Poisson process in a given window

Consider some variable $X\sim \operatorname{Poi}(\lambda(t))$ to be Poisson-distributed with some parameter $\lambda$ dependent on time, where we know how the random variable $\lambda$ is distributed. ...
1
vote
3answers
243 views

Derivatives of sine and cosine at $x=0$ give all values of $\frac{d}{dx}\sin x$ and $\frac{d}{dx}\cos x$?

In video 3 of the video lectures by MIT on Single Variable Calculus presented by David Jerison, the latter says: Remarks: $\dfrac{d}{dx}\cos x\left|\right._{x=0}=\lim\limits_{\Delta ...
1
vote
2answers
236 views

Third and average price auction

Third price auction: the winner is the highest bidder but this time instead of paying the second highest bid, he would pay the third highest bid. -assume there are at least 3 bidders. - Average price ...
3
votes
0answers
67 views

Teaching to Learn [closed]

I am interested in using teaching as a way of learning, but I am uncertain as how to best start. At the moment, I am only a sophomore in university and am relatively new to studying math. Currently,my ...
8
votes
4answers
616 views

Mathematics is not a spectator's sport? [closed]

The title is a sentence by John M. Lee, from his book "Introduction to Topological Manifolds". Indeed, I was wondering if one can learn mathematics in a passive way, just reading the books and ...
3
votes
3answers
748 views

Bartle vs Rudin, which one is better for real analysis? [closed]

I'm in high school and I want to study real analysis, and I can choose between The elements of real analysis by Robert G. Bartle and Principles of mathematical analysis by Walter Rudin, so, from the ...
7
votes
4answers
150 views

How do I decide what problems and how many problems to do when I try to self study?

I am a math major at a relatively small college with barely any choice of classes to choose from so I have to supplement my studying with a lot of self studying. I usually have no problem getting ...
0
votes
1answer
358 views

Book on discrete mathematics for self study

I am searching for book on discrete mathematics which is suitable for self study. This mean I want it to have exercises with answers (It would be ideal if it had solutions). I have already read ...
1
vote
1answer
73 views

Seeking your recommendation on Abstract Algebra textbooks

S.E advisers, I am a college sophomore with major in mathematics. I wrote this email to seek your recommendation on the good, starting textbook for the abstract algebra. I want to start studying ...
2
votes
1answer
77 views

Looking for Advice Self Study Analysis [closed]

$\space$ This summer I have some spare time and I was wanting to dedicate some time to self studying some more math. The reasons are many, but mostly because I am wanting to be best prepared for my ...
8
votes
1answer
163 views

How to begin self study of Mathematics?

I'm aware that this question has been asked several times, but I have specific questions hence why I'm asking again. I began to appreciate the beauty of mathematics when I glossed over the ...
0
votes
1answer
419 views

Question about the mathematics in actuarial studies

I tried Google but there isn't much information on this and I would really like some insights into actuarial studies, the mathematics involved and how it compars to the mathematics in a bachelor of ...
1
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0answers
43 views

Kline, Lang, Silverman…which author provides thorough and rigorous Calculus up Multivariate & Advanced Calculus?

I know this question may appear subjective, but it is not, let me explain; note to mods --> please don't close this if it is considered too subjective, please alert me and I'll try to reword it. I ...
4
votes
1answer
257 views

Are Specific Facts about the Riemann Integral Logically Required?

This question is somewhat in the spirit of this one in that I am trying to understand the most efficient path to the major integral theorems (Fubini, change of variables, etc). My question is this: ...
65
votes
6answers
5k views

Why can't you pick socks using coin flips?

I'm teaching myself axiomatic set theory and I'm having some trouble getting my head around the axiom of choice. I (think I) understand what the axiom says, but I don't get why it is so 'contentious', ...
3
votes
2answers
200 views

Is there any proof for this formula $\lim_{n \to ∞} \prod_{k=1}^n \left (1+\dfrac {kx}{n^2} \right) =e^{x⁄2}$

Some times ago, In a mathematical problem book I sow that this formula. I don't no whether it is true or not. But now I'm try to prove it. I have no idea how to begin it. Any hint or reference would ...