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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2
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39 views

Between Oksendal and Karatzas books

I finished Oksendal "SDE , an introduction with applications" , did most of the exercises (not all with success, but I read the solutions for those); my next project would be to read the Karatzas ...
0
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0answers
21 views

Is every Complex Square Matrix similar to its transpose? [duplicate]

I am aware that every complex square matrix is similar to its transpose but I am having a hard time proving this. Should I try to use the previously asked question listed at $A matrix is similar to ...
2
votes
1answer
56 views

If $B(t)$ is Brownian motion then prove $W(t)$ defined as follows is also Brownian motion

Let $B(t)$ be standard Brownian motion on $[0.1]$ Define $W(t)$ as follows $W(t) = B(t) - \int_0^t \frac{B(1)-B(s)}{1-s} \, ds$ Prove $W(t)$ is also Brownian motion So I'm not sure how to deal ...
4
votes
1answer
94 views

Proof that $(t_1, \dots, t_r) \mapsto \sum^{r}_{i=1} | t_i - \alpha_i|^p$ is continuous - Problem with Inequality

Bounty Edit: I already edited the question after some important comments. The questions I have are highlighted below the supposed proof. Any feedback or answer is most welcome. Thus, I just found a ...
0
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1answer
128 views

Who is a mathematician? [closed]

My first question in Math SE. Basically the question itself, who is a mathematician? Is it someone who solves problems on his leisure time or as a part of a job or even as a hobby? who researches ...
0
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1answer
50 views

Two questions on finding the equation of a parabola word problem- Klein's Calculus: An Intuitive and Physical Approach

I am solving the following word problem "A high voltage cable is supported by two towers 2800 feet apart and 348 feet high. The cable hangs in approximately the shape of a parabola, and the lowest ...
0
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0answers
39 views

Find the set of undominated strategies in Cournot duopoly

Consider a version of the Cournot doupoly game, where firms 1 and 2 simultaneously and independently select quantities to produce in a market. The quantity selected by firm $i$ is denoted $q_i$ and ...
0
votes
1answer
58 views

Unique Positive Definite Square Root of a Positive Definite Matrix

If $A$ be an $n\times n$ positive definite matrix, then there exists a unique positive definite matrix $B$ such that $B^2=A$. My question is how to get this $B$. What is the name of the algorithm for ...
0
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0answers
151 views

What is the expected amount of time until the chain is in state 4?

Consider the continuous-time markov chain with state space {1,2,3,4} and infinitesimal generator ...
1
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1answer
67 views

Topics of analysis beyond in $\mathbb{R}$ and beyond of functions of single real variable?

Could someone list the topics of analysis beyond in $\mathbb{R}$ and beyond of functions of single real variable that a new math graduate student should be familiar with? Also could someone list the ...
1
vote
2answers
101 views

Where to go after Halmos' *Naive Set Theory*

I'm in the process of finishing Halmos' Naive Set Theory, and I found the subject fascinating, so I would like to carry on reading about Set Theory when I'm done. From what I've been able to gather ...
1
vote
1answer
56 views

Bayesian Inference and Disease Testing

I've been working my way through an introduction to Bayesian Inference in a Statistical Physics textbook (Tobochnik and Gould, 2010 - available online, excellent book). I've run across a problem that ...
1
vote
2answers
148 views

I'd like to teach myself Algebra 2 through Calculus BC, where could I start?

Im a junior in high school, since I was young I've had a profound interest in Math, and I'd always looked forward to high school mathematics. Little did I know, Special education had taken away what ...
0
votes
1answer
18 views

Linear Algebra Expression

I have found the rank of M, the basis for the null space and evaluated M$\begin{pmatrix} 1\\ -2\\ -3\\ -4\end{pmatrix}$. But, I am having some trouble answering the last part of the question. Could ...
0
votes
1answer
164 views

Sum of positive infinity and negative infinity

Consider the following function of $\tau$: $$ h(\tau) := C_1 \ln\left(1-\frac{a}{\tau}\right) - C_2 \ln\left(1-\frac{b}{\tau}\right), $$ where $a > b>0$ and ...
0
votes
1answer
45 views

Is this an acceptable way to find an eigenvalue?

I have a matrix M where $$ M = \begin{pmatrix} -2 & 2 & 2 \\ 2 & 1 & 2 \\ -3 & -6 & -7 \\ \end{pmatrix} $$ and it has an eigenvector of ...
0
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0answers
38 views

Extension of Scalars, Tensor Prodcuts Vs Cartesian Products

My goal is to in a sense, create an additional set of "scalars" from the field of complex numbers, and the ring of integers. That will also maintain the property of being bilinear under bilinear forms ...
0
votes
1answer
19 views

Getting the rational function with given characteristics

The curve C has an equation $$y = \frac{ax^2+bx+c}{x+d},$$ where $a$, $b$, $c$, and $d$ are constants. The curve cuts the $y$-axis at $(0,-2)$ and has asymptotes $x=2$ and $y = x + 1$. From a ...
1
vote
2answers
60 views

Product of two sums over the same interval

I have some terms of an expression as sums but I would like to simplify the solution to an easier and less complicated one. What I have is $$ X = \sum_{k=0}^\infty \left(\frac{z}{5}\right)^k ...
0
votes
1answer
20 views

Self study-Common expectation and variance for sum of independent random variables

I am doing a problem that reads Suppose $X_1, X_2..., X_n$ are independent random variables with common expectation $\mu$ and variance $\sigma^2$. Let $S_n$=$X_1+X_2+...+X_n$. Find the expectation ...
1
vote
1answer
51 views

Proving the limit of $\frac{n!}{10^{n}}$ using definitions

$\cdot \lim \limits_{n \to \infty} \frac{n!}{10^n} = \frac{10!}{10^{10}} * (\frac{n!}{10^n})$ for all $n \ge 11$ So we must find $N(M)$ such that $\lvert \frac{10!}{10^{10}} * (\frac{n!}{10^n}) ...
3
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2answers
40 views

Proving the nested interval theorem

Theorem: Let $\{I_n\}_{n \in \mathbb N}$ be a collection of closed intervals with the following properties: $I_n$ is closed $\forall \,n$, say $I_n = [a_n,b_n]$; $I_{n+1} \subseteq ...
2
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0answers
51 views

Some advice on self studying [closed]

I'm currently studying Mechanical Engineering and of course doing that I've run into proof based mathematics. I would love to do classes for it but I don't want to kill my GPA for a non major course. ...
0
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0answers
61 views

Bolzano-Weierstrass Theorem proof question

Since $[a_n,b_n] \subset [x,y] \forall n$, we know that $Q = \{a_n\}^\infty_{n=1}$is bounded Let $t= \sup Q$ (which will be the accumulation point) Let $P$ be any neighborhood of $t$, so that there ...
0
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2answers
35 views

Solving a question on trigonometric series

I have stated the sum of the sum of the series (by geometric series) which is $$S_n= \frac{z(1-z^n)}{1-z}$$ I am trying to prove the second part of the question. However, I am unable to reach to ...
0
votes
1answer
22 views

This is the first half of proving Bolzano-Weierstrass theorem

Just making sure I'm on the right track so far Every bounded infinite set of real numbers has at least one accumulation point Pf: Let S be a bounded set. Since S is bounded, there are real numbers ...
5
votes
2answers
182 views

Is it acceptable to use reduced row echelon to show basis?

I am asked to show that {a, b, c} forms a basis for $\Bbb R^3$. I'm just wondering if it is acceptable to use reduced row echelon to show it since it is not shown that way in the marking scheme? ...
1
vote
1answer
45 views

Proof of sets A and B involving set theory, showing: $(B^c - AB)^c = B$

Use set algebra rules to show why the complement of $(B^c - AB)^c = B$ => Let x be an object Assume $x\in (B^c -AB)^c $ or $x \notin (B^c - AB)$ So then $x \in B $ but $x\notin AB$, therefore $x ...
1
vote
1answer
60 views

proving if $a \le b$ then $\sup(A) \le \inf(B)$

Let $A$ and $B$ be bounded sets of real numbers such that $a\le b$ for all $a \in A$ and for all $b \in B$. Show that $\sup(A) \le \inf(B)$ Pf: Assume A and B are bounded sets. This means they have a ...
0
votes
1answer
21 views

Proof using triangle inequality

Fix a real number x and a positive number $\epsilon$. If $\lvert x-1\rvert \le \epsilon$, show that $\lvert 2-x\rvert \ge 1 - \epsilon$ Pf: Fix $x \in R$ and let $\epsilon>0$, We know ...
0
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0answers
68 views

Proving that $\sup(-A) = -\inf(A)$ [duplicate]

Let $A$ be a bounded set of real numbers. Define $-A =$ {$x: -x \in A$}. Show that $-A$ is bounded and that $\sup(-A) = -\inf(A)$. Pf: A is bounded so $\exists x,y \in A $ such that $inf(A) \le y ...
3
votes
2answers
44 views

Proving the limit of a sequence using definitions

This is a review that my professor posted and I want to make sure I'm on the right path as I study $\cdot \lim \limits_{n \to \infty} n - \sqrt{2n^2+1} = n ...
3
votes
3answers
130 views

mathematical rigore for an engineer! [closed]

I recently bought a used copy of "Mathematical Analysis" by Apostol for \$1.0 and "Probability and Measure Theory" by Robert Ash for \$3.0 (well another \$3.99 for shipping)! When I read the first few ...
3
votes
1answer
61 views

$\lim\limits_{A \to \infty} (1 - \frac{c}{A})^{N-1}$, $N \sim \mathrm{Poisson}(\lambda A)$

Given $N \sim \mathrm{Poisson}(\lambda A)$, what will be $(1 - \frac{c}{A})^{N-1}$, where $c$ is a constant, when $A \to \infty$? Will it converge into a distribution or a constant? And what is it? ...
0
votes
2answers
110 views

What are the minimum and maximum prerequisites to study Stochastic Processes?

Suppose, I never studied random variables. This is the syllabus: Lecture contents Review of important notions of probability theory (4h). A few remarks on stochastic processes : ...
5
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0answers
215 views

How can we prove $e^{\pi}-\pi\simeq 20$ geometrically? [closed]

Using a calculator we can easily check that $$\color{Green}{e^{\pi}-\pi}=19.999\cdots\color{Green}{\approx 20}$$ This article and this one provides some details about this almost near identity, but no ...
1
vote
1answer
142 views

Expected mean squared error and MSR

In a small-scale regression study, five observations on $Y$ were obtained corresponding to $X = 1,4,10, 11$, and $14$. Assume that $\sigma=0.6,B_0=5,B_1=3$ a. What are the expected values ...
1
vote
0answers
57 views

Undergrad looking for study material/advice for applied mathematics.

I am an undergraduate math student (junior) who is looking to get a masters degree in Applied Math. I like pure math, but I want to use my education to get a great-paying job. Here are a few questions ...
0
votes
0answers
73 views

Prove that OLS estimator of the intercept has minimum variance

Let $$y_i=B_0+B_1X_i+\epsilon_i$$ where $\epsilon_i\sim > N(0,\sigma^2)$. Find the least squares estimator of $B_0$ and show that it is unbiased and has minimum variance. I will not write in ...
1
vote
1answer
73 views

New to Abstract Algebra, need guidance [closed]

I am new to Abstract Algebra. How should I begin learning it. On the face if it, it doesn't look easy to me.I have bought the book by Prof. Gallian. Is there any other book or any videos which I can ...
2
votes
2answers
57 views

Convergence of fixed point iteration when $g'(p)=1$.

I am dealing with a function $f(x)=e^{-\frac{1}{x^2}}$, which has a root $p$ of infinite multiplicity at 0. I am struggling with the convergence rate of the resulting standard Newton fixed point ...
-1
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1answer
50 views

Book recommendations for learning financial/business mathematics.

Does anyone know a book which covers topics on: Simple interest Compound interest Equations of equivalent values Nominal rate, effective rate and equivalent rate Annuities ...
2
votes
0answers
41 views

What is the link/ relationship and difference between probability measure, Bernoulli measure, Lebesgue measure, Borel measure and Hausdorff measure

I am having difficulty in understanding what is the difference between probability measure and Bernoulli measure. Is the latter used when the random variable has a Bernoulli distribution? What is its ...
3
votes
1answer
55 views

Most Expeditious Way of Adding Consecutive Composite Numbers

In helping my 10-year old son with a homework problem that he was trying to solve by rote, I found myself resorting to arithmetic series. In short, I was calculating the arithmetic series from 12 to ...
0
votes
1answer
25 views

Jar has 10 dimes/12 nickels. After draw1, The coin that is drawn first is added back + 1 more of the same type. Find P(Dime1|Dime2)

To clarify, Find the probability that the first coin was a dime given the 2nd was also a dime. I'm very sorry for the extended title, but it says to be very specific. I think the tree diagram ...
0
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0answers
48 views

Joint and marginal pmf

A class of n students takes a test consisting of m questions. Suppose that student i submitted answers to the first m questions The grader randomly picks one answer, call it (I, J), where I is the ...
1
vote
1answer
47 views

Natural Isomorphism between $T_1^1(V)$ and End$(V)$

I'm a little stuck on showing that there is a natural isomorphism between the $\mathbb{R}$ vector space of $(1,1)$ tensors, and the $\mathbb{R}$ space of of linear maps $T:V\to V$. The hint is define ...
0
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1answer
41 views

Proof with stationary distribution

Let $\pi(k)$ the stationary distribution of the Markov Chain. Show that if $$p_{ij}^{(n)}\geq\varepsilon$$ for some $i,j,n,\varepsilon$ then $$\pi(j)\geq \varepsilon \pi(i)$$ I'm litle lost ...
0
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0answers
46 views

Markov Chain in urn with replacement

Consider a green ball and a yellow distributed in two urnas.Em each step, a ball is selected at random, then if that ball is green it changes of urn with probability $1/4$, and if the ball is ...
1
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0answers
43 views

Reversibility and stationary probability

Let $X_n$ a markov chain with transition matrix given by $$\begin{bmatrix}0.7&0.3&0\\0.2&0.7&0.1\\0.4&0.1&0.5\end{bmatrix}$$ i) Find the stationary probability ...