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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20 views

Encyclopedia of Mathematics?(non-Alphabetical)

Do you know any Encyclopedia of Mathematics which is in non-alphabetical order, like it starts from basic mathematics and then goes up to very advanced level. And what's the difference between say, ...
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0answers
55 views

How to invent mathematics to find solution to real world problems?(without high level mathematical knowledge) [on hold]

Let's say I want to know how waves are formed when a stone is dropped in water, how trees are deformed by the wind, etc, and I want to invent the mathematical equation to predict the behaviour of ...
0
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0answers
75 views

Too stupid for math? [on hold]

I enrolled in an analysis course and I feel like I'm out of my league. I just find myself not being able to follow the lecturer as she's going through the material in class. Also, some of the ...
1
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2answers
29 views

Dickson's Lemma (proof of Prop. 2.23 in Hasset's Intro to Alg Geom)

I'm studying Hasset's book by myself but I had no previous formal algebra training. To prove Dickson's lemma (prop. 2.23, p. 19) he defines the auxiliary monomial ideals $$J_m=\left<x^\alpha \in ...
11
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7answers
794 views

Strategy for reading math books, is it better to prove the theorems yourself or just read them?

Context: I'm self-studying some mid to upper level undergraduate math subjects. For example, right now I'm reading Munkres' Topology book. Usually, the approach I use is to go through the book in ...
1
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0answers
23 views

do Carmo or Pressley as introduction to Differential Geometry?

I'm choosing between these authors for self-studying Differential Geometry. The contents seem pretty similar for both books and I was wondering which book I should choose. The level of these books ...
2
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3answers
71 views

How do mathematician make sense of “outcome” and “events” in probability?

One of the biggest challenge for me to understand probability is to make sense of this concept of outcomes and events. To put it plainly, it just doesn't feel like mathematics anymore when we talk ...
2
votes
1answer
41 views

Looking for a clarification of the Suslin $\mathcal{A}$-Operation with a (finite) example

I have a problem concerning the output of (and the intuition behind) the Suslin $\mathcal{A}$-Operation. More specifically, I really don't see exactly what the output of it really is (even if I can ...
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2answers
41 views

Probability that among 3 random digits two different one

I have been trying to solve the following problem: What is the probability that among 3 random digits, there appear exactly 2 different ones? The formula for no repititions is: ...
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2answers
28 views

Simplifying with Summation

This is a problem out of my statistics book but my issue is simplifying from Step 3 to Step 4 below: Step 1: var X=$\sum\:p_i\:(x_i-E[X])^2$ Step 2: var X=$\sum\:p_i[x_i^2+E[X]^2-2x_iE[X]]$ Step ...
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1answer
16 views

Find $P(\eta_t=m)$, $m=0,1,2,\dots,$

Let $\epsilon_t$, $t=1,2,\dots$ independent random variables with $P(\epsilon_t=1)=p$ and $P(\epsilon_t=-1)=1-p$. If $\eta_0=0,\eta_t=\eta_{t-1}+\epsilon_t$ , $t=1,2,\dots$ where $\eta_t$ is ...
3
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1answer
44 views

Question about Branch Cuts

I'm starting to learn a little complex analysis, and I'm a little confused as to what the purpose of a branch cut is. Is it to make a function continuous, or single valued? For example, the $\sqrt{}$ ...
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0answers
19 views

Properties of Fitted Regression Line

Suppose the following regression model $$Y_i=BX_i+\epsilon_i$$ where $\epsilon_i\sim N(0,\sigma^2)$independents is the random error. Verify which properties of the estimated regression line are ...
2
votes
2answers
179 views

Why Study Homological Algebra?

I'm very interested in learning Homological Algebra. But I'm not sure about the prerequisites for learning this. My current knowledge in algebra consists of Abstract Algebra (Group,Rings,Fields), ...
2
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1answer
37 views

The Differential Geometry of a 2-D Surface

I'm currently self-studying the differential geometry of embedded surfaces. My question is, how am I to chose the appropriate coordinates and derive the covariant basis for the surface I'm interested ...
0
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1answer
36 views

Show that if $\{X_n\}$ is a Markov Chain

Show that, if $\{X_n\}$ is a Markov Chain then $$P(X_n=j\mid X_k=l,X_m=i)=P(X_n=j\mid X_m=i),0\leq k<m<n$$ What I did is $$P(X_n=j\mid ...
-1
votes
1answer
67 views

Recommended courses [closed]

I'm an advanced soon to be 7th grade student and I do a lot of self-learning. I have done Pre-Algebra, Algebra, and am about half way through Algebra 2. I am wondering what I should do next- Trig, ...
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0answers
29 views

Expectation in measure theory

I'm reading a book on measure-theoretic probability, and the author defines the expectation of a random variable $X$ on a probability space $(\Omega,\scr H,\mathbb{P})$ as $\int_\Omega Xd\mathbb{P}$, ...
1
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1answer
29 views

Decreasing sequence of sets: Power of natural numbers

Let $P(N)$ be the set of all the possible subsets of natural numbers (power set of $N$). Suppose that we have a decreasing sequence of sets $S_n$, ie $S_{n+1} \subseteq S_n\;,\in P(N)$ such that they ...
2
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0answers
37 views

Help with Definition of Limits (Finding a delta given epsilon)

The problem says: Find a $\delta$ such that $|f(x)-l| < \epsilon$ for all x satisfying $0 < |x-a| < \delta$ when $f(x) = x^4; l = a^4$. What I did so far was $|x^4-a^4| < \epsilon$ so ...
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4answers
98 views

IF $\lim_{n\to\infty}a_{n}=l$, Then prove that $\lim_{n\to\infty}\frac{a_{1}+a_2+\cdot..+a_n}{n}=l$

Given $a_n$ be a sequence and IF $\lim_{n\to\infty}a_{n}=l$, Then prove that $\lim_{n\to\infty}\frac{a_{1}+a_2+\cdot..+a_n}{n}=l$ I do not know how to do this. Can someone help me with this? Thanks ...
1
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1answer
29 views

General question about eigenvalue, eigenvectors.

I have the following question : $A$ is a $n \times n$ matrix, and this is the characteristic polynom $$p(x)=(x+3)^2(x-1)(x-5)$$ Then I can conclude that $n=4$ since the number of the roots is $4$, ...
2
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4answers
31 views

Eigenvector proof for repeated eigenvalues

I am stuck trying to solve the following problem: In diagonalizing a symmetric matrix $S$, we find that two of the eigenvalues ($\lambda_1$ and $\lambda_2$) are equal but the third ($\lambda_3$) is ...
1
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1answer
30 views

Sum of random error in regression

If you know that $\sum_{i=1}^n e_i=0$.What can you say about $\sum_{i=1}^n\epsilon_i=0$? Where $e_i=Y_i-\hat{Y_i}$ and $\epsilon_i=Y_i-E[Y_i]$. I know that $$Y_i=B_0+B_1X_i+\epsilon_i$$ and ...
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2answers
19 views

Commenting results in a ratio scale

Consider the following plot: Is it mathematically correct if I say blue is 50% lower than red. Because from the plot it ...
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0answers
39 views

Help in understanding a probability calculation from a paper

Paper: A Framework for Investigating the Performance of Chaotic-Map Truly Random Number Generators download link = http://arxiv.org/pdf/1211.1234.pdf explains a method to determine if a random ...
2
votes
1answer
38 views

Distributions, PDFs, and Random Variables in Measure Theory

I'm currently reading a book on measure-theoretic probability theory, and I'm having trouble seeing how the familiar objects distributions, pdfs/pmfs, and random variables from my calc-based prob/stat ...
2
votes
3answers
54 views

Are there any significant differences between studying functional analysis from a normed space perspective versus a metric space perspective?

Does it matter if functional analysis was introduced from a normed space versus a metric space formulation? Are all major theorems from functional analysis (such as Banach contraction mapping, Hahn ...
1
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0answers
24 views

Pre-College Algebra Book

I am looking for a high school/ pre-college level Algebra book that is self contained for self-study. Nothing special, I don't want a book about number theory, but a book in preparation of high school ...
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4answers
86 views

To compute improper integral $\int_3^{5}\frac{x^{2}\, dx}{\sqrt{x-3}{\sqrt{5-x}}}$

I am given improper integral as $$\int_3^{5}\frac{x^{2}}{\sqrt{x-3}{\sqrt{5-x}}}dx$$ DOUBT I see that problem is at both the end points, so i need to split up the integral. But problem seems to me ...
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2answers
38 views

Verifying a Proof for Spivak's Calculus Question (Chapter 2 Problem 9)

It says "Prove that if a set $A$ of natural numbers contains $n_0$ and contains $k+1$ whenever it contains $k$, then A contains all natural numbers $\ge n_0$". Am I allowed to construct another set ...
0
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0answers
12 views

Unit Impulse response vs Impulse response in ODE

I'm was watching MIT OCW lectures for Differential Equations and in lecture 23, the professor goes over impulse inputs where impulse is $\int_a^b{f(t)dt}$ where $f(t)$ can vary or be constant. He ...
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0answers
9 views

Workshop and courses

I'm a math student (I also know a bit of theoretical informatic and programming) that will in some months ends his 3° year and earn (it's how work in Europe, don't know in America) the "small degree". ...
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2answers
54 views

Are there any books on real analysis that explain what goes on in their proofs for a self studying student?

Are there any books in real analysis that explains what goes on in their proofs? I want to self study real analysis. I read through proofs in each of these real analysis books and I'm not ...
1
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2answers
36 views

How to show that $f(x,y) = |x| + |y|$ is continous at origin

How to show that $f(x,y) = |x| + |y|$ is continous at origin. CLearly it goes to 0 , but how do i prove it? Thanks
0
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1answer
17 views

A question about Fisher–Neyman factorization theorem

$f_{\theta}(x)$, then $T$ is sufficient for $\theta$ if and only if nonnegative functions $g$ and $h$ can be found such that $f_{\theta} = h(x)g_{\theta}(T(x)) $ The statement is: if $F(t)$ is a ...
1
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2answers
40 views

Finite sums of integers and similar problems: book request

I recently learned about Faulhaber's formula, which says that for each integer $p \geq 1,$ we can simplify the finite sum $\sum_{k \in \mathbb{N}}[k<n]k^p$ so that it becomes an (integer-valued) ...
1
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1answer
33 views

Stationary distribution in continuous-time Markov chain

Consider a barbershop with one barber who can cut hair at rate 4 (people per hour), and three waiting chairs. Customers arrive at rate 5 per hour. Customers who arrive to a fully occupied shop ...
2
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1answer
21 views

Dirac Delta Function as a Measure

I was always told in my college physics classes to not worry too much about the dirac delta function because it can be made rigorous using distributions or measure theory. I've just started learning ...
4
votes
1answer
112 views

Bridge between High School Mathematics and University-level Mathematics?

I've graduated from High School and I am going to major in math at a local University. I've finished High School Calculus and I've self-studied very very basic Multivariable Calculus, Linear Algebra, ...
0
votes
1answer
14 views

Subtraction of a sub-squence and a sequence explanation

In my assignment I have the following question, True of false: Let $a_{n}$ be a sequence. If $$\lim\limits_{n\to\infty} (a_{2n}-a_{n})=0$$ Then $a_{n}$ is convergent. The statment is false ...
1
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3answers
86 views

Proving that $\lim \limits_{n\to \infty} \sqrt[n]{n^5-2n+7}=1$

In my assignment I have to prove the following: $$\lim \limits_{n\to \infty} \sqrt[n]{n^5-2n+7}=1$$ I don't know how to start, I believe it has to do with the squeeze theorem, but I can't be sure. ...
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0answers
21 views

Dirac Measure is Purely Atomic

In my book, "Probability and Stochastics" by Cinlar, it's stated that for some measurable space $(E,\scr E)$, and fixed $x\in E$, the Dirac measure $\delta_x(A)=\left\{ \begin{array}{lcc} ...
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0answers
20 views

Properties of the Kernel from the measurable space $(X,\mathscr{A})$ to $(Y,\mathscr{B})$

Hi everyone this is an exercise from Cohn's book. I'd appreciate if someone can check part (d) and (e) where I have more problems because this concept is completely new for me. Let $(X,\mathscr{A})$ ...
0
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1answer
14 views

Variance of Transformed Random Vectors

Consider an $n$-dimensional normal random vector $\mathbf X:= (X_1, \dots, X_n)^T$ with mean $\mathbf 0$ and covariance matrix $\mathbf \Sigma$. Now define a new random vector $\mathbf Y:= (a_1X_1, ...
0
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0answers
24 views

Graph Laplacians - self-study

I am self-studying graph laplacians in Kevin Murphy’s book “A probabilistic perspective on machine learning”. I understand that we introduce the vector f to proof that the matrix is positive ...
6
votes
1answer
67 views

Norm $\Vert \cdot \Vert$ on the symmetric group $S_n$

If we define a real valued function $\Vert \cdot \Vert$ on the $n^{th}$ order symmetric group $S_n$ satisfying following conditions $$\begin{align} & \|x\|=0\iff x=\omega\,\,\,(\text{identity ...
5
votes
1answer
56 views

A good, self-study statistical computing book

I'm looking for a book an introductory statistical computing that has proofs for the methods as well as examples. I'd like proofs that are about the same level as (or lower than) proofs in Statistical ...
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0answers
54 views

Learning Math for Computer Science

Apologies if this has been already asked. I have gone through a lot of different questions but they don't adapt to my personal situation. I have a 2 years diploma in software development and I am ...
0
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1answer
15 views

Preference relations and the existence of extensions of functions representing them

In a book I found the following question: Let $\succsim$ be a complete preference relation on a nonempty set $X$, and let $\varnothing \neq B \subseteq A \subseteq X$. If $u \in [0,1]^A$ ...