Questions about studying mathematics without formal instruction.

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2
votes
1answer
131 views

Is this an abuse of notation?

Here is a proof says that the differential of Gauss map is self-adjoint. But I seems there is an abuse of notation at (1) in it. Since $dN_p$ is linear, it suffices to verify that $\langle ...
1
vote
2answers
109 views

Exercise in well-ordered sets

Definitions: Given a subset $A$ of a non-empty well-ordered set $X$ we define the supremum as follows: $\text{sup(A)}:=\text{min}\bigg(\big\{\,y\in X \oplus\{ +\infty\}: \forall x\in A \;(x\le y)\, ...
1
vote
2answers
181 views

What does the 'triple line equals' sign with a strikethrough mean?

I understand that the triple equal sign can mean "identical to" or "defined to be equal to", so intuitively, I assume that the triple equal sign with a strikethrough means "is not identical to" or ...
12
votes
5answers
814 views

Self studying math, how can I learn the most?

I am currently studying Pre-Calculus on my own. I have a few texts I am working with but feel like I could learning a lot more than I am. When people typically ask these kind of questions the common ...
0
votes
1answer
213 views

Square Root of Random Variables

Question: Suppose that $\displaystyle \frac{2}{\theta_0}\sum_{i=1}^n y_i\sim\displaystyle\chi_{2n}^2$ and $\displaystyle 2\theta_0\sum_{i=1}^n x_i\sim\displaystyle\chi_{2n}^2$. And these two are ...
0
votes
2answers
69 views

Positive recurrent, zero recurrent and transient states

How do we prove that a state in a Markov chain process is positive recurrent, zero recurrent or transient? For example, if we have a transition matrix $$P=\left(\begin{array}{ccc} \frac{1}{3} ...
2
votes
1answer
83 views

Evaluate $\iint\limits_{\substack{x<u,y<v, \\ x^2+y^2<1}} dxdy$

How can I evaluate the following double integral: $$\iint\limits_{\substack{x<u,y<v, \\ x^2+y^2<1}} dxdy$$ If we didn't have the restrictions $x<u, y<v$ polar coordinates would have ...
3
votes
2answers
82 views

The set of all the partial ordering of $X$ is a partially ordered set.

I have trouble with the following exercise. I don't understand it. Also I think that $x,y\in P$ should be $x,y\in X$. Could someone give an insight of the exercise and also some hint of how to solve ...
3
votes
3answers
64 views

$X$ and $1 - X$ are identically distributed

I have a two-fold goal for this question. First, I'm trying my hand at making hypotheses and proving them as far as I can. I want to understand the limits of proof, not just the techniques. Second, ...
1
vote
1answer
46 views

polynomial interpolation

I have a function, for example $f(x)=\frac{-x^2}{2}+|x|$, which is divided on $[-1,0)$ and $[0,1]$. How do we interpolate this function with a polynomial $p$ in the maximum degree 4 with ...
6
votes
1answer
205 views

Intuition behind proof and verification partially ordered sets

Hi everyone in the book that I read I have trouble to understand the argument of the proof at the below proposition. There is a lot of point which are left as exercises, which is great. One of these ...
2
votes
0answers
32 views

Question on numerical quadrature and precision

I am studying numerical analysis and I came across this question: Let $P_{n+1}(x)\in\Pi_{n+1}$ be orthogonal to $\Pi_n$ relative to a weight function $w(x)\geq 0$ on $[a,b].$ Denote by ...
2
votes
1answer
226 views

Exercise: Every non-empty subset of $X$ contains a minimum and maximum element and the converse

Hi everyone is my second exercise of posets. And find the following, the first part I think is not difficult at all. But for the converse I have some serious troubles to prove it. I have two ...
2
votes
0answers
80 views

Partially ordered set exercise

Hi everyone is my first time working with posets and I'd like to know if the solution of the next exercise is really correct or maybe there is some errors that I cannot see. Thank in advance. ...
2
votes
2answers
97 views

Calculate the following conditional expectation.

Let $\Theta$ and $R$ be two independent random variables, where $R$ has density $f_{R}(r)=re^{-\frac{1}{2}r^2}$ for $r>0$ (zero otherwise) and $\Theta$ is uniform on $(-\pi,\pi)$. Let $X=R ...
0
votes
0answers
212 views

Problem with Spivak calculus.What should I do?

For past two weeks I have been working throught the Spivaks calculus book.Needless to say I am very pleased with his writing style,but I have a slight issue. The issue is namely that in those two ...
2
votes
2answers
117 views

What is the purpose of universal quantifier?

The universal generalization rule of predicate logic says that whenever formula M(x) is valid for its free variable x, we can ...
0
votes
2answers
128 views

Pigeonhole Principle and maximum length of the repeating section

The question I have is, when 5 / 20483 is written as a decimal, what is the maximum length of the repeating section of the representation? I believe I need to divide 5 by 20483 which is equal to ...
2
votes
2answers
62 views

system of congruences proof

I've checked a lot of the congruency posts and haven't seen this one yet, so I'm going to ask it. If there is a related one, I'd be happy to see it. Let $x \equiv r\pmod{m}, x \equiv s\pmod{(m+1)}$. ...
0
votes
1answer
77 views

Inverse Fourier Transform of the output, Y(f)

A linear system is defined by the differential equation: $$ y''(t) + 4y'(t) + 25y(t)= x(t) $$ The transfer function of this system is: $$ H(f) = \frac{Y(f)}{X(f)}= \frac{1}{(2\pi fj)^{2}+ 4(2\pi ...
4
votes
5answers
1k views

Math is too hard for me. How can I make it easier?

I am trying to study, and I keep finding that math is hard (any kind), and it doesn't get easier(only harder). I am trying to learn these things all in progression (asynchronously): 1.Math for all ...
0
votes
1answer
39 views

Show that the image of Gaussian map of a generalized cone is a curve on $S^2$ and deduce that the cone has zero Gaussian curvature.

Show that the image of Gaussian map of a generalized cone is a curve on $S^2$ and deduce that the cone has zero Gaussian curvature. I dont have enough idea. Please explain the question clearly. ...
2
votes
1answer
27 views

Show that the set $(\mathbb{I} \cap (-\infty, 0]) \cup (\mathbb{Q} \cap [0, \infty))$ is neither a $F_{\sigma}$ set nor a $G_{\delta}$ set.

Show that the set $(\mathbb{I} \cap (-\infty, 0]) \cup (\mathbb{Q} \cap [0, \infty))$ is neither a $F_{\sigma}$ set nor a $G_{\delta}$ set. Note: $\mathbb{I}$ is the set of irrational numbers. ...
1
vote
1answer
82 views

Laplace Transform and Fourier Transform of a function

I have this transfer function: $$ h(t)= -\frac{1}{16}te^{-2t} $$ and the Laplace Transform is: $$ H(s) = \frac{-\frac{1}{16}}{(s+2)^{2}} $$ I know that to find the Fourier Transform, I would ...
0
votes
6answers
88 views

Factoring $s^2+4s+13$

I was looking at an example, and it was factored as follow: $$ s^{2}+4s+13 = (s+2)^{2}+9 $$ How can we do that?
0
votes
1answer
128 views

Soft question: Learning theory and solving problems in self-study

When trying to learn a new subject in mathematics from a book I usually find myself mostly learning the theory directly presented there(reading through all the theorems, proofs, definitions etc.) not ...
2
votes
0answers
53 views

Set Theory - Well Order (Lexiographical combination)

Question: Prove constructively that if $(A_{1},\prec_{1})$ and $(A_{2},\prec_{2})$ are two well-ordered sets then their lexicographical combination $(A_{1} \times A_{2},<_{1,2})$ is also well ...
0
votes
2answers
54 views

Finite intersections and unions of $F_{\sigma}$ and $G_{\delta}$ sets

Using the following definition: A set $A \subseteq \mathbb{R}$ is called an $F_{\sigma}$ set if it can be written as the countable union of closed sets. A set $B \subseteq \mathbb{R}$ is called a ...
1
vote
1answer
57 views

show $\int g\log (g/f)$ is $0$ only if $g=f$ almost everywhere

Question: Suppose that $f$ and $g$ are two probability density functions, show that $\int g\log (g/f)$ is always non-negative and equals to $0$ $\it only\ if$ $\ g=f$ almost everywhere. I have ...
2
votes
3answers
109 views

Gamma function can be extended as a meromorphic function

Let $$\Gamma (z)= \int_{0}^{\infty} e^{-t}t^{z-1}dz$$ for $\Re z\gt 0$ Can be extended as a meromorphic function to the entire complex plane with simple poles at non positive integers. How can I ...
2
votes
1answer
63 views

Derive a perturbation of period $2\pi$, to order $\epsilon$

I have the following problem: In the equation $\ddot{x}+\Omega^2x+\epsilon f(x) = \Gamma \cos t$, $\Omega$ is not close to an odd integer, and $f(x)$ is an odd function of x, with expansion, $$f(a\cos ...
0
votes
1answer
26 views

Find an integral expression for $\Gamma'(z)$ for $\Re z\gt 0$

Find an integral expression for $\Gamma'(z)$ for $\Re z\gt 0$ I know the result. But I dont know how to show this step by step.
-1
votes
1answer
45 views

Learning way and Resource for Complete math Subject. [closed]

I want to learn [self learning] Mathematics from basic.What is the order [like 1) arithmetic,2) Geometry,Etc..] to learn the maths? and what is the best resource to that particular subject?
2
votes
4answers
70 views

What does countable union mean?

The book I am reading contains the following two definitions: Two sets $A$ and $B$ have the same cardinality if there exists $f: A \rightarrow B$ that is one to one and onto. In this case, we ...
14
votes
7answers
510 views

Introduction to ring theory?

I've been teaching myself algebra these couple of months. I already went through the basics of group (Lagrange, action, class equation, Cauchy and Sylow theorems etc.) And I already have some linear ...
7
votes
1answer
100 views

Find $\sum_{k=1}^{\infty}\frac{1}{z_k^2}$

Let $z_1, z_2,\dots, z_k,\dots$ be all the roots of $e^z=z$. Let $C_N$ be the square in the plane centered at the origin with siden parallel to the axis and each of length $2\pi N$. Assume that ...
0
votes
1answer
67 views

Why can't the interval construction argument used to show $\mathbb{R}$ is uncountable be used for other infinite sets?

I read the following proof as to why the set of real numbers is uncountable. Assume that $\mathbb{R}$ is countable. Then we can enumerate $\mathbb{R} = \{x_1, x_2, x_3, \cdots\}$ and be sure that ...
1
vote
1answer
53 views

Zeta function in complex analysis.

Show that $$\frac{\zeta'(z)}{\zeta(z)}=-\sum_{n=2}^{\infty}\frac{f(z)}{n^z}$$ for $\Re z\gt 1$ Where $f(z)= \ln p$ if $n=p^m$ for some prime $p$ and some $m\in \Bbb N^+$ Or $f(z)=0$ otherwise. ...
1
vote
1answer
170 views

How to linearize a nonlinear ODE around its equilibrium?

I am studying for a comprehensive exam in non-linear ODE's and I have this in my book: $$\ddot{\xi}+c\bigg[x_1+\xi-\dfrac{\lambda}{a-x_1-\xi}\bigg] = 0$$ then it goes straight to ...
2
votes
1answer
56 views

Will a bounded sequence in $\mathbb{R}$ necessarily induce a compact subset of $\mathbb{R}$?

Question: We know that a bounded sequence in $\mathbb{R}$ must have a convergent sub-sequence from Bolzano-Weierstrass theorem. However, will a bounded sequence $(x_n)$ in $\mathbb{R}$ necessarily ...
4
votes
2answers
216 views

Exercise Real Analysis

I'm having trouble to understand the following exercise I would appreciate any help? Let $A,B,C$ be sets such that $A\subseteq B\subseteq C$ and let $f:C\rightarrow A$ be an injective map. Define ...
1
vote
1answer
45 views

Boundary Value Problem (Separation of Variables)

Solve the boundary value problem \begin{cases} u_{t}-2u_{xx}=0 \\ u_{x}(0,t)=u_{x}(\pi,t)=0, \quad x\in[0,\pi], t\geq0 \\ u(x,0)=\cos^{2}(x) \end{cases} My Attempt Let $u(x,t)=X(x)T(t)$. Then ...
1
vote
1answer
66 views

Why does $\operatorname{rank}(\mathbf{X})$ equal $\operatorname{rank}(\mathbf{X^TX})$? What is $\dim(\mathbf{X^TX})$?

Why does $\operatorname{rank}(\mathbf{X})$ equal $\operatorname{rank}(\mathbf{X^TX})$? Is this true in general, please? And what is $\dim(\mathbf{X^TX})$, please? Does it equal to $\dim(\mathbf{X})$ ...
2
votes
3answers
105 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. ...
3
votes
1answer
89 views

Branch points of global analytic functions (Ahlfors)

In Ahlfors' complex analysis text, page 298 he discusses the case of a global analytic function $\mathbf{f}$ which can be continued analytically along all arcs in some punctured disk $\{0<|z|< ...
2
votes
1answer
132 views

Cantor-Bernstein Theorem proof

I am self studying real analysis and I am doing an exercise which is proving the Cantor-Bernstein Theorem. Question: Assume there exists a 1-1 function $f:X \rightarrow Y$ and another 1-1 function ...
1
vote
1answer
68 views

Rearrangement Thm

Hi everyone: In the book what I've read of analysis in the proof of the Riemann rearrangement thm there is gaps that I need to fill. There is no real problem in almost everything but for the last one ...
10
votes
1answer
363 views

Preparing for “differential forms in algebraic topology”?

I'd very much like to read "differential forms in algebraic topology". Apart from background in calculus and linear algbra I've thoroughly went through the first 5 chapters of Munkres. I'm thinking ...
1
vote
2answers
88 views

Infinite product convergence

Prove that $$\prod_n\left(1+\frac{i}{n}\right)$$ diverges. But $$\prod_n\left\vert 1+\frac{i}{n}\right\vert$$ converges. I know the theorem $\prod (1+z_k) $ converges $\iff$ $\sum\log (1+z_k)$ ...
1
vote
1answer
68 views

Where am I wrong applying chain rule here?

Why is $$\dfrac{\partial}{\partial x_i}f(tx_1, \dots, tx_n)=\dfrac{\partial f}{\partial(tx_i)}(tx_1,\dots,tx_n)\cdot\dfrac{\partial(tx_i)}{\partial x_i}=\dfrac{\partial f}{\partial ...