Questions about the process of studying mathematics without formal instruction.

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1answer
20 views

Find the curve which together with $\gamma$ encloses the greatest area.

I'm working through Gelfand & Fomin's Calculus of Variations by myself, and could use the guidance of someone familiar with the subject. The problem I'm on now is the following: "Given two points ...
3
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0answers
17 views

Prove that two functionals with identical differentials differ by a constant.

I am self-studying Calculus of Variations and am struggling to prove results about the variation of a functional that are analogous to results in elementary analysis about differentials/derivatives. ...
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2answers
17 views

On the horizontal integration of the Lebesgue integral

I'm studying Lebesgue integral and its difference with respect to the Riemann one. I'm reading that the key difference (at least graphically speaking) is that the first slices the function ...
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1answer
29 views

Strategies for linear systems

Consider I have the following equations. Is there a faster way for me to solve the system without going through a series of substitutions? $$-20a+13b+13c=0$$ $$10a-26b+13c=0$$ $$10a-13b-16c=0$$ ...
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2answers
56 views

What is a good book for reviewing high school math, and preparing for university?

I'm signing up for University soon (Compsci program) as a mature student. It's been a long time since I've done any math, and I went as far as grade 11 in high school. So, I'm looking for a book that ...
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2answers
33 views

If $x_n$ $\rightarrow $ 1 Then show that sequence $\frac {4+ (x_n)^{2}}{x_n}$ approaches to limit 5

If $x_n$ $\rightarrow $ 1 Then show that sequence $\frac {4+ (x_n)^{2}}{x_n}$ approaches to limit 5 I have tried to find epsilon proof ,But i am not successful .Can anyone help me with this ...
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4answers
120 views

Complex number, series

Show that $$\frac{1}{z^2}=1+\sum_{n=1}^\infty (n+1)(z+1)^n$$ when $|z+1|<1$ I'm having problems to resolve this type of exercise since my book has virtually no exercises of this type, these ...
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1answer
27 views

Proof that limit of sequence is unique

I am learning real analysis on my own from this book http://books.google.co.in/books?id=TZ-NAgAAQBAJ&printsec=frontcover#v=onepage&q&f=false On page 33 , i do not get proof of that limit ...
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1answer
29 views

Complex number, series representation

Show that for any finite value of $z$ $$e^z=e+e\sum_{n=1}^\infty \frac{(z-1)^n}{n!}$$ For $z=1$ $$f(z)=f(z_0)+\sum f^{(n)}(z_0)\frac{(z-z_0)^n}{n!}$$ equality is checked, but I do not know how to ...
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1answer
17 views

Topology of weak convergence, linear functionals and probabilistic intuition

One very basic question regarding the topology of weak convergence. We know that given the following: $X$ metrizable topological space, $\mathcal{B} (X)$ Borel $\sigma$-algebra, $\Delta (X)$ ...
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28 views

$\sum_{n=1}^{\infty}\frac{a_n}{10^n}$ where $\{a_n\}_{n=1}^{\infty}$ is a sequence in the ten digits {0, 1, 2 , 3 , 4 , 5 , 6 , 7 , 8 ,9}

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence in the ten digits {0, 1, 2 , 3 , 4 , 5 , 6 , 7 , 8 ,9} And consider the sum $\sum_{n=1}^{\infty}\frac{a_n}{10^n}$ $\in$ $[0,1]$ What characteristics of ...
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3answers
77 views

Let $R$ be a finite ring (with unity) and $S$, $T$ be subrings of $R$. Is $S \cup T$ a subring of R?

Let $R$ be a finite ring (with unity) and $S$, $T$ be subrings of $R$. Is $S \cup T$ a subring of R? (Counterexamples are easy to find to me when $R$ is an infinite ring or a finite rng.) P.S. I am ...
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1answer
27 views

column space of a matrix

If $A\in M_{m\times n}\mathbb{(R)}$, show that $\mathcal{R}(AA^t)=\mathcal{R}(A)$ and $\mathcal{R}(A^tA)=\mathcal{R}(A^t)$ where $\mathcal{R}$ denotes the column space of matrix. How can I prove it ...
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0answers
41 views

How to fill my mathematical gaps?

To do the story short, I became interested in mathematics in a serious way like two years ago, I'm currently in graduate school, but the problem is that my mathematical background is not as good as ...
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1answer
60 views

Basic concerns about dependent function types

I am working my way through the introductory material of Homotopy Type Theory and by the end of section $1.5$ it is clear that I did not have the earlier material down as clearly as I had thought. I ...
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1answer
46 views

Long term probability in Markov Chains

I was practicing some questions on transition probability matrices and I came up with this question. You have 3 coins: A (Heads probability 0.2),B (Heads probability 0.4), C (Heads probability ...
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1answer
13 views

Visualizing a probability measures through a probability density functions

I found a previous question with a very nice answer, but still there is something that is not completely clear to me. We start from a space $(X, \Sigma)$, endowed with a $\sigma$-algebra, and we let ...
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0answers
27 views

how to prove this function is a probability measure in $U_B$

Let $(\Omega, U, P)$ be a probability space. and $B\in U$, $P(B)\gt 0$ $U_B =\{A: A=B\cap C, C\in U\}$ its class in $\Omega$ is a $\sigma$-algebra and $P_B : U_B \to \Bbb R$ $A \to ...
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2answers
22 views

Transition probability matrix

In the article here it had this question. A walker moves on two positions a and b. She begins at a at time 0, and is at a next time as well. Subsequently, if she is at the same position for two ...
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2answers
203 views

Application of Picard-Lindelöf to determine uniqueness of a solution to an IVP

I am still struggling quite a lot with the Picard-Lindelöf-Theorem (also known as the Cauchy-Lipschitz-Theorem). Problem: Consider the following IVP with $\alpha \neq 1$ $$\begin{cases} y'&= ...
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1answer
43 views

Moments of a random variable in terms of its cumulative distribution function

Consider a random variable $X$ with distribution function $F(x)$. Calculate the $r$th moment of $X$, $\mathbb E X^r$. I read that the desired moment can be calculated as follows. $$ \begin{align} ...
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0answers
50 views

How to think/see point-set topology abstractly?

I've started learning point-set topology this semester. I've learned basic material about: topology on a set topological space open sets closed sets clopen sets closure neighborhoods interior point ...
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2answers
97 views

Are basic trigonometry functions ( sine, cosine, tangent ) intuitive or memorized?

First, I'm really sorry for this somewhat vague and possibly just silly question. I also apologize if the following context runs a bit long. But please trust me that I'm asking with total sincerity ...
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2answers
45 views

Understanding proof that sequence $x_n =( -1)^{n}$ is not convergent

To Understanding proof that sequence $x_n =( -1)^{n}$ is not convergent . Here goes the proof :- Firstly i assume that sequence converges to x .i.e -1 and 1 lies between $x$-$\epsilon$ and ...
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5answers
63 views

To show sequence $a_n = n$ is not convergent [on hold]

Here is proof that sequence n is not convergent. It is from the book A Basic Course in Real Analysis by Ajit Kumar, S. Kumaresan, page 31. I have not understood last three lines of the proof. Can ...
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3answers
113 views

Please help collecting examples of finite/infinite rings satisfying different conditions about units/zero divisors (Added question 4)

0) Every nonzero element of a finite ring is either a zero divisor or a unit. This is proved in Every nonzero element in a finite ring is either a unit or a zero divisor 1) If a ring R satisfies the ...
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1answer
54 views

Characterize all finite unital rings with only zero divisors

Is it true that for every finite (for simplicity, commutative) ring $R$ in which every element not equal to $1$ is a zero divisor, is isomorphic to the zero ring or $\mathbb{Z}/2\mathbb{Z}$, ...
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1answer
12 views

Regression project in octave/matlab

I'm trying to establish a polynomial model to adjust the variation of the dollar throughout the year. Suppose hypothetically that I have the following data ...
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0answers
28 views

What does $\mathbb P(\overline{\mathbf X} = \mathbf x)$ mean

I am reading Peter Hall's "the bootstrap and edgeworth expansion". In Theorem 2.3 on page 57, it claims that if the characteristic function $\chi$ of a $d$-dimension random variable $\mathbf X$ ...
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9answers
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Calculus book recommendations (for complete beginner)

Well I have not started calculus yet but I am really keen to. I would love if you suggest some books. Points to be noted: I really don't like the way textbooks are written so please no "textbooks" ...
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1answer
13 views

Complete separable metric space X represented represented as union of closed sets

I have a problem concerning a statement I found in volume 2 of the classic reference book on measure theory by Bogachev. More precisely, I have a problem concerning theorem 6.1.13. I the proof the ...
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1answer
50 views

Is self study of proof-based mathematics difficult?

I heard from a renowned Mathematician that self study of proof based Mathematics is extremely difficult as there is not only right and wrong but also degree's of correctness. So without a teacher ...
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0answers
83 views

Help in understanding the prove whether a map is one-to-one and disconnected for homeomorphism from the book

I need to prove that amp is a homeomorphism. I am following the basics from the book For the proof I have taken the help of the book "An introduction to dynamical system" Download link ...
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0answers
59 views

Which version of MIT Single Variable Calculus is most complete? [closed]

MIT's OpenCourseWare offers nine versions of the single variable calculus course. five with course number 18.01..., and four versions under the course number heading "Supplemental." 1) Which of ...
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1answer
31 views

Proving that a subspace of $L^2$ is closed.

Suppose $Z$ is a random variable on a probability space $(\Omega, F, P)$. $M(Z)$ is the subspace of $L^2$ consisting of all random variables in $L^2$ which can be written in the form $\phi(Z)$ for ...
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6answers
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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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2answers
58 views

Can the given transformation possible for given determinant?

In forth step $(x-1)(x-2)$ is obtained by applying transformation R$1 \frac{1}{(x+1)}$ and R$2 \frac{1}{(x+2)}$. But we get value of $x = -1$ or $ x = -2$ so $\frac{1}{(x+1)}$ and ...
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1answer
24 views

Solving the equations .

Say , I have two equations : $$y_1=a+bx_{1}+e_1$$ $$y_2=a+bx_{2}+e_2$$ Say , $a=.5$ , $b=2.1$ , $x_1=2$ , $x_2=2.2$ . Now if $e_1=e_2$ , I have to find the relationship between $y_1$ and $y_2$ . ...
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3answers
74 views

What would be an effective way to learn group theory on my own?

I've read the basics of this branch and I found it extremely interesing, and I would really love to learn more about it. I want to study as much as I can on my own, as my course doesn't have group ...
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0answers
57 views

Good starting point for learning noncommutative geometry?

Currently, I am attempting to learn noncommutative geometry. My interests mostly lie on the boundaries of pure mathematics and theoretical physics, so I am not only interested in the mathematical ...
2
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0answers
27 views

Double Integral of an Exponential Function with an Absolute Value in the Numerator of the Exponent

This is a question related to statistics, but my major concern relates to the setup and evaluation of integrals. So I decided this question was better suited for Mathematics Exchange than CV. I know ...
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1answer
21 views

A proof related to diameter of a simplex S

Question: Prove that the diameter $\mathcal p(S)$ of a simplex $\mathcal S$ equals the greatest Eucledian distance between two vectors in the simplex. My opinion: We all know what every vector in the ...
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1answer
23 views

Show that a positive definite (not necessarily symmetric) matrix induces a hyperellipse

Consider $A\in M_n(\mathbb{R})$ a positive definite matrix and a matrix $B\in M_{n \times p}(\mathbb{R})$, with $n\geq p$ and $rank(B)=p$. i) Show that $C=B^TAB$ is positive definite. ii) Show that ...
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1answer
44 views

How to get this inequality

Let $c>0$, $n \in \mathbb N$ and $q>1$. How to get the following approximating inequality when $n$ is large, please? To be more specific, I cannot see how to get rid of the square root. $$ ...
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0answers
5 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. ...
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0answers
88 views

Are “Transition Books” (Spivak/Apostol/Courant) really necessary?

Why do so many people recommend Spivak, Apostol, and Courant calculus textbooks, especially as a preparation toward the advanced courses like analysis and abstract algebra? Are they really necessary? ...
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2answers
36 views

An orthogonal projection matrix in $ \Bbb{R}^{3} $.

Consider the vector space $\mathbb{R^3}$ with usual inner product. Find the orthogonal projection matrix on the xy plane. I've found sometimes the orthogonal projection of a vector in a given ...
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3answers
32 views

Linear algebra, inner product and matrix

Let $A\in M_{m \times n}(\mathbb{R})$, $x\in \mathbb{R}^n$ and $b,y\in \mathbb{R}^m$. Show that if $Ax=b$ and $A^ty=0_{\mathbb{R}^m}$, then $\langle b,y\rangle=0$. Also make a geometric ...
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1answer
470 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 ...
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15answers
7k views

A good way to retain mathematical understanding?

What is a good way to remember math concepts/definitions and commit them to long term memory? Background: In my current situation, I'm at an undergraduate institution where I have to take a lot of ...