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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483 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) $(D^3-3D^2D'+4D'^3)u=0$ ...
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2answers
320 views

Complex form of Fourier Series

So, the last part of the university syllabus in the chapter of Fourier Series is: ...
1
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1answer
12 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}{\...
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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
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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 ...
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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
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1answer
468 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
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1answer
42 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
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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
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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},\...
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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) = ...
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1answer
68 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 ...
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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 $(\...
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0answers
31 views

What books should I study? [closed]

I will study in college the following courses: Analysis; Differential Equations; Vector Calculus and Applications; Linear Algebra; Numerics and Optimisation; Systems, Series and Transforms; ...
8
votes
1answer
64 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
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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
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1answer
68 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 ...
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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 ...
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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)...
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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
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3answers
213 views

Self study plan [closed]

I don't want to go to college and I came up with plan in the following order: ...
0
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1answer
33 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, ...
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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 ...
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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} ...
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1answer
438 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}...
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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 ...
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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 ...
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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 ...
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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
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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
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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
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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 ...
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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
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4answers
451 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
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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
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2answers
869 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
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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
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4answers
360 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{...
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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
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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
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0answers
26 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 ...
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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?
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3answers
1k views

start mathematics from scratch

I'm sorry if this question goes against the site. I have completed my engineering in computer science; we were taught to get degrees only in our university :(. I love mathematics but time passed ...
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1answer
14 views

Riesz map between Sobolev space and its dual

I'm asked to find the Riesz's map: $$R:H^1_0(\Omega) \rightarrow H^{-1}(\Omega) $$ $$R: u \mapsto F_u \quad \text{s.t.} \quad <F_u,v>_*= (u_F,v)_{H_0^1} \ \ \forall v \in H_0^1 $$ I chose $(u,...
233
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7answers
17k 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 ...
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1answer
2k 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, ...
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2answers
24 views

volume of surface of revolution around y axis

Can anyone help walk me through this problem style? I have a lot of homework problems like this and I really want to understand how to do these problems. Find the volume of the solid generated by ...
1
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1answer
46 views

Are minimizing a function and root finding the same?

What is the relationship between minimizing a function and finding a root of an equation? Are the the same? I know in both problem we have similar algorithms, such as gradient decent, or newton's ...