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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3
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1answer
72 views

self-teach: In what order should I structure my studies?

I'm a 19 year old, who will be 20 in may. I didn't go to the greatest high school, and I didn't get the proper education to prepare me for college. I need to make a study plan for myself to truly ...
0
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0answers
80 views

Proof that Poisson process interarrival time $T(N+1)-T(N)$ with $T(N)<t<T(N+1)$ is Gamma$(2,\lambda)$

Suppose a Poisson process $N(t)\sim\text{Poisson}(\lambda t)$. Let $T(N)$ be the time of the last arrival before time $t$ and $T(N+1)$ be the time of the first arrival after time $t$. From ...
1
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1answer
102 views

Given undergraduate Algebra background, which introductory Homological Algebra textbook?

I have read the answer for graduate-level Algebra background and all answers in stackexchange and mathoverflow discussing Homological Algebra textbooks. But none of them directly answers my question, ...
0
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0answers
25 views

Let W = X1/(X1+X2), how to prove **W** has a beta distribution?

I am confused when I come across this question, could anyone help? Thanks! Let X1 and X2 have independent gamma distributions with parameters α, θ and β and θ respectively. Now we let W = X1/(X1+X2), ...
1
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2answers
123 views

How do mathematicians come up with beautiful equations [closed]

In Linear regression for example, we can find weights as following: $\hat{\beta}=(X^{T}X)^{-1}X^{T}y$ how someone invented this? I mean how do they transform a problem to such an equation. And ...
3
votes
3answers
132 views

Effective Methods of Studying in different areas of Math

I apologize if this question isn't appropriate for this site, but I am looking for advice that I think other math students might be better able to give me. I am an undergraduate math major entering ...
0
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0answers
60 views

Proving a degree sequence satisfy Chvatal’s criterion

How can I prove that a degree sequence satisfy Chvatal’s criterion? I know that i must prove that sequence A is Hamiltonian if and only if A' is hamiltonian but i am lost on where i should start. for ...
0
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0answers
24 views

how to construct non-Hamiltonian graphs

I have been asked to construct a hamiltonian graph and a non-hamiltonian graph using the same degree sequence. I have had no problem constructing the hamiltonian graph however I am finding it ...
1
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1answer
58 views

Min problem by using Lagrange method

$$\min x^2+y^2 $$ $$\text{s.t.}\ \ (x-2)^2+(y-3)^2\le 4 \ \ \ \text{and} \ \ \ x^2=4y$$ Please explicitly solve this question by using Lagrange multiplier method. I accept $(x-2)^2+(y-3)^2=4$ ...
5
votes
1answer
179 views

Set all measurable real functions on $[0,1]$ with metric $\int_{0}^1 \min \{1,|f(t)−g(t)|\}dt$ is Fréchet without nonzero continuous linear functional

Bounty Edit: In the following, all the questions will be highlighted by a bold number and a text written in italics. I found the following statement in a book, and I am really struggling to see why ...
1
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1answer
49 views

Maximization problem on an ellipsoid [closed]

for three variables, $$\max f(x,y,z)= xyz \\ \text{s.t.} \ \ (\frac{x}{a})^2+(\frac{y}{b})^2+(\frac{z}{c})^2=1$$ where $a,b,c$ are constant how to solve the maximization optimization problem? ...
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2answers
33 views

How to write a discrete dynamical system into first order system

I need guidance on how to solve this here. $$x_{n+1} + 3x_n - 4x_{n-1} = (\sqrt{2})^n cos \left(\frac{n\pi}{6}\right)$$ I am required to transform the above equation into a first order finite ...
0
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0answers
26 views

Marginalizing multivariate-normal distribution canonical form

Regarding the problem of margenalization of canonical forms of multivariate gaussian distribution it was mentioned in probabilistic graphical models text book that $$\int{C(X,Y;k,h,g)}dY$$ is ...
2
votes
2answers
49 views

Expectation Functional in Lebesgue and Riemann Terms – Looking for a clarification

Here there is a really central problem I am having self-studying probability theory, that concerns the relation between the definition of expectation in Lebesgue terms and in Riemann terms. I will ...
0
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1answer
64 views

Negative Mean Square Error

For simple random sampling, I have calculated somemean square errors for ratio-type estimators such as Isaki estimator, and Prasad Singh estimator. But, Mean Square Errors i obtained are negative. ...
3
votes
0answers
55 views

Is It Worth It Working Out Every Practice Problem In Math? (Without a calculator)

I'm bouncing back between trig, algebra, and calc books. I've noticed that most of the problems at some point seem to distill into very tedious arithmetic. It is nice to have the prowess of ...
0
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0answers
67 views

Suitable reference for learning symplectic geometry

I am interested in studying symplectic geometry by myself and I'm looking for a good text to use as a reference in the way. I am a bit lost because I've found a lot of notes and books on the subject ...
4
votes
3answers
870 views

Does the phrase “If you don't use it, you lose it” apply to mathematics? [closed]

I'm asking this because I ran into the following particular situation: I took some calc courses over 2013, where I learned, amongst other things, to integrate some pretty nasty functions, and this ...
1
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1answer
38 views

writing a piecewise regression model as a linear model

lets write the following piecewise regression model $$y= \alpha_0 + \alpha_1 x +\epsilon ;\ \ x\le x_0 $$ $$ y=\beta_0 +\beta_1 x + \epsilon \ \ x\gt x_0$$ according to the variable $x_0$ is ...
1
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0answers
44 views

Self learning complex systems modeling

I am currently a senior studying mathematics. My program is heavily focused on pure maths, however, my interests lie in the area of applied mathematics. Specifically, I am interested in the study of ...
2
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0answers
36 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
55 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
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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
votes
1answer
126 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
48 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
37 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
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1answer
56 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 ...
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0answers
143 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 ...
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2answers
98 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
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1answer
55 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
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2answers
121 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
149 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 ...
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0answers
34 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
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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 ...
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2answers
59 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
19 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
votes
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
50 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. ...
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0answers
60 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 ...
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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
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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
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2answers
181 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
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1answer
44 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
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1answer
59 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
19 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 ...