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 the fact that you're self-studying is what your question is _about_.

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10
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2k 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 ...
0
votes
1answer
39 views

Derivation of an integration

Can someone explain to me the difference between the results of $ A$ and $B$, where $$A=\frac{d}{dc} \int_{-\infty}^c xf(x) dx $$ $$B= \frac{d}{dc} \int_c^{+\infty} xf(x) dx $$ You can image $f(x)$ ...
6
votes
2answers
106 views

Concrete examples and computations in differential geometry

I've been studying differential geometry by myself for some time now. I studied a fair amount of the basic general theory and gone through a lot of the exercises from several textbooks. Lately I ...
0
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0answers
37 views

Self-learning SAT

I want to take SAT, but I don't know which textbook is good to learn on Mathematics II. At least more than 1 textbook, or just textbook for each curriculum. I really need suggestion.
2
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0answers
42 views

Efficient way to rigorously learn AI prerequisites

Question: My formal goal is to be able to rigorously understand the mathematical basis for modern statistical learning methods (ML, deep learning). I am told by math people that this involves: linear ...
1
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0answers
22 views

Is there a way to maximize this probability by taking the derivative of the cumulative normal distribution function?

I'm self-studying Brownian motion and encountered the following problem. I understand the author's solution, and it is clear why maximizing the right-hand side of the inequality provides such $t$ ...
3
votes
4answers
206 views

Derivative of a determinant whose entries are functions

Happy New Year, everyone! I do not understand a remark in Adams' Calculus (page 628 $7^{th}$ edition). This remark is about the derivative of a determinant whose entries are functions as quoted below. ...
0
votes
1answer
485 views

How to decide whether PDE is Homogeneous or non-homogeneous.

I am studying second order PDE. And I have seen homogeneous and non-homogeneous PDE. But I cannot decide which one is homogeneous or non-homogeneous. For examples; (1) $(D^3-3D^2D'+4D'^3)u=0$ ...
0
votes
2answers
320 views

Complex form of Fourier Series

So, the last part of the university syllabus in the chapter of Fourier Series is: ...
2
votes
1answer
14 views

Commutative Banach algebra and its maximal ideal space

Let $A:=C^{(n)}([0,1])$ be the set consisting of the n-times continuously differentiable complex-valued functions. Consider $A$ with the norm $$ \|f\|:=\max\limits_{0 \leq t \leq 1} \sum_{k=0}^{n}{\...
45
votes
25answers
4k views

What books should I get to self study beyond Calculus for someone about to start undergrad mathematics?

I am struggling to pick out books when it comes to self studying math beyond Calculus. My situation is as follows. I have taken all math courses at my school (up to Calc BC and AP Stats) and I have ...
2
votes
2answers
32 views

Matrix decomposition in unipotent matrices

Consider the positive definite and symmetric matrix $$A = \begin{pmatrix} 1 & 2 & 0 \\ 2 & 6 & -1 \\ 0 & -1 & 1 \end{pmatrix}$$ Find a decomposition with unipotent $U ...
1
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2answers
29 views

Calculate the spectral norm

Consider the four vectors $v_1, v_2, u_1, u_2 \in \mathbb{C}^2$ with $$v_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \qquad v_2 = \begin{pmatrix} 2 \\ 1 \end{pmatrix}, \qquad u_1 = \begin{pmatrix} ...
1
vote
1answer
469 views

Expected value of division

Let $X,Y$ and $Z$ be three indenependent real valued random variables. Al with finite second momennt and all with mean $0$ and variance $1$. Define $$ W= \frac{X+YZ}{\sqrt{1+Z^2}} $$ Show that ...
1
vote
1answer
43 views

Which one is more complex; $(1-e^a)^{(b^c)}$ or $\frac{e^a \times a^b}{b!}$ [closed]

I know this is a simple question but I cannot find an answer for it. I have two functions, $$1:\ (1-e^a)^{(b^c)}$$ and $$2:\ \frac{e^a \times a^b}{b!}$$ where $a,c$ are real numbers, $b=...
2
votes
0answers
28 views

Specific decomposition of quadratic 2x2 matrix

Consider the matrix $A = \begin{pmatrix} 1 & 1 \\ -1 & 3 \end{pmatrix}$. Prove that there is only one decomposition A = B + C with $B,C \in \mathbb{R}^{2x2}$ that fulfill the following ...
4
votes
1answer
26 views

ODE system solving by sequence of functions

Let $y' = Ay$ where $A = \begin{pmatrix} 0&1 \\ -1& 0 \end{pmatrix}$ and $y( 0 ) = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$. Consider the map $$G: C(\mathbb{R},\mathbb{R}^2) \to C(\mathbb{R},\...
0
votes
0answers
25 views

Constant solutions to the Lotka-Volterra equations

Let $$x' = ax - bxy, \\ y' = -cy + dxy,$$ be the Lotka-Volterra equations with $a,b,c,d > 0$. How to show that for every choice of $a,b,c,d$ there is an initial condition $x(0) = x_0$ and $y(0) = ...
-6
votes
1answer
69 views

Asymptotics of $\lim\inf X_n$, missing part of exam question [closed]

So I was working through an old exam and encountered the hilarious situation that part of the statement of the exercise was illegible. I was wondering if anyone could figure it out for me, so that I ...
0
votes
0answers
13 views

error term in birth-death process

This is from notes: I have three questions I understand that for small $h$, when $h\to 0$, $h^2, h^3,h^4....$ are negligibly small compared to $h$, so I think the euqation 12 should be $(\...
8
votes
1answer
65 views

How math help reduce terms and conditions of someone's dying wish?

Good morning everyone... This is my very first question here, so I apologise in advance for any wrongdoing which I possibly make unintentionally. So here is a little background story. I'm working at a ...
1
vote
1answer
37 views

o(h) term in birth-death process

This is from the note I have two questions. Since $o(h)$ represents the probability of 2 birth and 1 death, 3 birth and 2 death, etc, why it still says $P(|X(t+h)-X(t)|>1)=o(h)$? shouldn't it ...
1
vote
1answer
69 views

Regarding to OCD about Reading Mathematical Books [closed]

Dear Math Stack Exchange advisers, I recently started to develop an OCD-like symptom about reading books in mathematics. Whenever I read previous pages and proceed to next, I always feel under a ...
0
votes
3answers
60 views

Calculus problem (Differential)

First off, no, this is not homework. This comes from self-study and has stymied me. Please explain your answer as thoroughly as you can! Find increment $\Delta y$ and differential $dy$ for the ...
0
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0answers
29 views

Taylor polynomial reminder

Page 277 of Apostol's Calculus 1 has the following theorem: Let $P_{n}$ be a polynomial of degree $n\geq1$. Let f and g be two functions with derivatives of order n and assume that $f(x)=P_{n}+ x^ng(x)...
1
vote
1answer
45 views

Proof verification: $f(x) \le g(x) \implies \lim_{x\to a}f(x) \le \lim_{x\to a}g(x)$

I am self-teaching calculus using Spivak's book, and it's hard for me to know whether my proofs are correct, if they are different from the proofs that Spivak gives. Could you help me to check whether ...
4
votes
3answers
215 views

Self study plan [closed]

I don't want to go to college and I came up with plan in the following order: ...
0
votes
1answer
34 views

uneven probability problem

I was encountered with a probability problem, and here I tried to explain it in a easier way. Now we have a square(w*l), and I am going to randomly put a white point on the surface of this square, ...
0
votes
0answers
27 views

Elementary literature on Group theoretic Power Diophantine Equation

I am looking for an elementary books/pdf notes on group theory related to Power Diophantine Equation. I have read elementary group theory. Please advise some books/pdf notes. Also, it would be ...
1
vote
0answers
79 views

Solutions to Laplace's equation between surfaces of a constant coordinate

In Physics (specifically electrostatics) it is often encountered that one must solve the Dirichlet problem for Laplace's equation \begin{equation} \nabla^2 \varphi(x_1,x_2,x_3) = 0 \end{equation} ...
1
vote
1answer
440 views

Inverse Laplace Transformation of a heaviside function.

I'm working through an example of an inverse laplace transformation: $$\mathscr{L}^{-1}[\frac{e^{-3s}}{s+1}] = u_3(t)e^{-(t-3)}$$ I am having trouble seeing how this works. I know that: $\mathscr{L}...
1
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0answers
32 views

Number-Theory Books to read before studying Analytic Number Theory

S.E friends, Due to my genuine interest to Goldbach's conjecture, I decided to self-study the subject of additive number theory on this upcoming Fall. Before jumping to such fascinating field of ...
1
vote
3answers
72 views

Recommend book Taylor expansions

I've taken up self-study of math and i start using the book called : Mathematical Analysis I Authors: Canuto, Claudio, Tabacco, Anita I would like to start from zero to understand taylor ...
0
votes
2answers
135 views

Physics Book Recommendation Request

General Requirements Physics for Mathematicians Philosophy + Foundations Mathematical Derivation of Theories I want to know if there is a physics book for mathematicians. I attempted to read some ...
1
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3answers
260 views

Introductory Algebra Book Suggestions

General Requirements The algebra book must be no more than 400-500 pages in length and should contain end-of-lesson/chapter exercises. Required Topics linear equations linear inequalities ...
0
votes
2answers
59 views

Advanced Calculus Book for Computer Science Student

I study Computer Science, but our mathematic coures are a little bit to basic. I'm looking for a "advanced Calculus" book for self-study that has a lot of exercises. The book should focus on ...
1
vote
0answers
26 views

Book search on statistics

I am searching a book that Analysis of Failure and Survival Data (Chapman & Hall/CRC Texts in Statistical Science) by Peter Smith. Its link is here. I tried to buy it from Amazon, but it is out ...
2
votes
1answer
44 views

Why Gaussian elimination on the columns changes the column space?

This page on theorem 8.2 states that, Neither of the operations of the gaussian elimination changes the row space of an $m \times n$ matrix after applying the operation. It says later that this is ...
0
votes
1answer
26 views

Testing the independence of two jointly normal variables

Variables $u$ and $v$ are jointly normal, correlated with zero mean. $X$ is a linear combination of $u$ and $v$: \begin{align*} X := \frac{u}{\sqrt{E(u^2)}}-\rho\frac{v}{\sqrt{E(v^2)}} \end{align*} ...
7
votes
4answers
452 views

How to stay motivated when solving diffucult problems? [closed]

From your experience, what is the most effective way to solve competition math problems without getting discouraged/frustrated because you cannot find the solution? For example, I was trying to solve ...
0
votes
3answers
834 views

Contraction Mapping, why constant and weak inequality?

From Wikipedia, a contraction mapping is a function $f: M \rightarrow M$ on a metric space $(M,d)$ such that there exists a nonnegative real number $k<1$ such that for all $x,y\in M$, $$ d \left(f(...
2
votes
2answers
874 views

Introduction into Operations Research

I am a first year graduate student who advisor wants me to learn about operations research and to use stochastic integer programming in my research. He keeps giving me papers to read but they ...
2
votes
0answers
38 views

Book recommendations for someone with a B.S in mathematics - Self study

I have finished my B.S degree in mathematics recently and I would like to continue to study on my own. I'm looking for books in all subjects you can recommend. I want to start each subject from its ...
1
vote
4answers
363 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 x\to0}\dfrac{...
0
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0answers
22 views

Cholesky decomposition: No uniqueness when A is positive semidefinite

I can get as corolary of the following theorem, that Cholesky decomposition is not unique for positive semidefinite matrices? Thank's in advance If $A\in \mathbb{M}_{n\times n}$ is a positive ...
0
votes
2answers
60 views

Meaning of $ E(u|v) = p\,v$

I am having problem understanding the intuition behind this: if $u$ and $v$ are jointly normal (with zero mean), then $E(u|v) = p\,v$, for some parameter $p$ ?
1
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0answers
17 views

Poisson and conditional distribution

Given a source that emits photons let be $X$ the number of photons emitted in $s$ seconds. Let be $p$ the probability that photons reach a detector and let be $Y$ the number of photos that reached the ...
0
votes
0answers
27 views

Reading commutative algebra book

Now I am just a beginner in commutative algebra, so I just want to ask which book I should read step by step. I am reading Step in commutative algebra of Sharp, then I want to read Commutative ring ...
2
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0answers
19 views

Help needed with partial products

I understand that 4 times 6 tens is 240. I am now being asked to add the extra tens to 240. how do I do that? are the extra tens 40. so if I add 240+40=280 is that the answer?