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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Is small triangle is similar to big triangle

A triangle, $ABC$, has point, $M_1$, $M_2$, $M_3$, where $M_1$ is the mid point of Line $AB$, $M_2$ is the mid point of Line $BC$, and $M_3$ is the mid point of Line $AC$. A smaller triangle, $...
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2answers
73 views

Proof in ruin player problem

Let $M_i$ the average number of matches until the player, or lose all, or wins the capital $N$ as it began with the capital $i$. Show that $$M_i=i(N-1);p=\frac{1}{2}$$ $$M_i=\frac{i}{1-2p}-\frac{...
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1answer
20 views

Image of matrix $\int_0^t e^{sA}BB^T e^{sA^T} ds$

Let $A \in R^{n \times n}$ and $B \in R^{n \times m}$. Define $$Q_t = \int_{0}^t e^{sA}BB^T e^{sA^T} ds$$ Suppose that $x \in \text{Im } Q_t$, ie, $\exists \eta \in R^n$ such that $$x = Q_t \eta$$ ...
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0answers
19 views

If state is reachable in time T_1, then it is reachable in time $T > T_1$

Consider a Linear Time System with the admissble control set $$U = \left\{ u: R \rightarrow R^m \;|\;\text{u is integrable in any finite interval} \right\} $$. Show that, if starting on $x_0=0$ we ...
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0answers
35 views

Integrating product of logs

I am failing to integrate $$ \int \log {\bigg(\frac{a}{x}\frac{x-c}{a-c}\bigg)^{s-1}} \log{\bigg(\frac{b}{x}\bigg)}\bigg(\frac{c}{x}\frac{1}{x-c}\bigg) dx $$ for a positive integer $s$, and real ...
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1answer
38 views

Integrate product of x^s and log(x)

Let $$ f(x) = 1 - (1-c\log (F(x)))^s\\ g(x) = \log(\frac{a}{x}) $$ for some positive integers $s$, $c$, and some real-valued function $F(x)$. $\log$ denotes the natural logarithm. One can think ...
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0answers
26 views

The Need for Tangent to a curve at a point and its definition

I was understanding derivative function when I thought that why "concept of tangent", was invented.If it was so because of influence of Physics - instantaneous velocity and other stuff then why ...
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1answer
20 views

Work with this implicitely defined function

I want to check whether the following transformation is correct: $$ \sum_{s=1}^\infty (1-x)^s \exp(-\lambda)\frac{\lambda^s}{s!} = \exp(-\lambda)\sum_{s=1}^\infty \frac{X^s}{s!}\\ = \exp(-\lambda)\...
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1answer
28 views

Sample size and confidence interval

We want to produce a $0.90$ confidence interval for the proportion of vegetarian recipes at one cookbook. We will use simple random sampling without replacement to select a sample of $2311$ ...
0
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1answer
58 views

On the mathematical convention used to talk about biconditional proofs

Given $P \Longleftrightarrow Q$, the following does apply: $P \Rightarrow Q$ is equivalent to: $P$ is a sufficient condition for $Q$, $Q$ is a necessary condition for $P$. $Q \Rightarrow P$ is ...
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0answers
98 views

Proof that if $A$ is open, then int$(\bar{A})=A$

I am sure this is really a trivial result, but I would like to check my proving skills (plus I find it is sort of related to nowhere dense sets, that are quite difficult for me to digest. ...
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0answers
35 views

Tangent Vectors as Infinitesimal Displacements

I'm reading Wald's General Relativity, and I'm stuck on something that is stated very early on in the book. For an abstract manifold $M$, he goes through the usual definition of a tangent vector at $p\...
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1answer
21 views

Finding the stationary points and their types

I am trying to calculate the local maximum and minimum of $f(x, y) = x$$3$$y$$2$$(2 − x − y)$, however I seem to keep running into issues. My steps were to find the partial derivative for $f$$x$$(x,...
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1answer
23 views

ODE using integrating factor

So I have started learning ODEs for the first time. I need to find the general solution of the differential equation $$x \frac{dy}{ dx} + 2y = 3x$$ where the solution satisfying the initial condition ...
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2answers
34 views

Find a tangent plane

I am asked to find a tangent plane of $f(x,y) = e^{x\ln y}$ at the point (2,1). When I ask wolfram alpha this, I am given the line $z=2y-1$. I don't intuatively understand this, shouldn't there be a ...
1
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1answer
86 views

Given that a throw with 10 dice produced at least one ace, what is probability p of two or more aces

So, the question is: Given that a throw with 10 dice produced at least one ace, what is probability p of two or more aces? It is conditional probability problem. The formula is $$P(A|B) = \frac{P(A\...
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0answers
82 views

Uniqueness of Maximal Integral Curve in Manifold

(*) I know that for an open set $V\subseteq\mathbb{R}^n$ containing the point $p$ and a smooth function $f:V\to\mathbb{R}^n$, differential equation $\frac{dy}{dt}=f(y), y(0)=p$ has a unique solution $...
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4answers
234 views

Is there better alternative to Princeton Companion to Mathematics, because I can't make sense of it? [closed]

I am currently reading first chapter of this book, and here are few quotes from the book and I can't make sense of these,they are - 1.) Historically, the abstract structures emerged as ...
2
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2answers
81 views

How can we prove that $\frac{a}{b }\times\frac{c}{d} =\frac{ac}{bd}$

I am slowly reading calculus by michael spivak and it is one of the problems in first chapter. however I cant prove it please help me with it...
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1answer
25 views

Problem understanding how a linear equation is simplified

Using this paper as a reference (Section IV.C, page 4318), We have the following objective function which we wish to minimize with respect to $D \in \mathbb R^{n \times K}$ ($X \in \mathbb R^{K \times ...
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1answer
92 views

Decimal expansion of $x\in [0,1]$

This is an exercise from Royden Real Analysis: Let $p$ be a natural number greater than 1, and $x$ a real number, $0 \leq x \leq 1$. Show that there is a sequence $\{a_n\}$ of integers with $0 \leq ...
2
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1answer
36 views

Birhdays: find the probabilities for the various configurations of the birthdays of 22 people

Let S,D,T,Q stand for simple,double,triple and quadruple, respectively: So, for example: the probabilities of 22 simple birthdays(22 person have birthdays in different days) are $ P(22S) = (365!/343!)/...
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0answers
36 views

Tangent Spaces of Distinct Points are Disjoint?

I'm reading Tu's "An Introduction to Manifolds", and he defines the tangent bundle on $M$ as the disjoint union $TM:=\bigcup_{p\in M}\{p\}\times T_pM$, but he remarks that for $p\neq q$, we already ...
2
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0answers
26 views

What is the purpose of continuous and differentiable dependence

In learning Gronwall's inequality you also get to learn about continuous an differentiable dependence. I know the theorems but I have no idea about their application. What is the big idea of ...
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1answer
40 views

Probability that in bridge game the Players N,E,S,W have a,b,c,d spades respectively.

There are 52 cards in bridge and 13 cards of each suit. The formula for numerator is: $${13\choose a}{39 \choose 13-a}{13-a\choose b}{26+a\choose 13-b}{13-a-b\choose c}{13+a+b\choose 13-c}$$ But i ...
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2answers
98 views

Applying the definition of Lebesgue Integral to specific functions

I am fairly sure this question will sound rather naive, but I do have a problem with applying the Lebesgue Integral. Actually this question can be divide in two sub-question, related to two examples I ...
2
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3answers
53 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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2answers
46 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 k[...
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7answers
1k 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 ...
2
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3answers
96 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
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1answer
49 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
75 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
55 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 3:...
0
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1answer
22 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 ...
4
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1answer
96 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 $\sqrt{}$ ...
2
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2answers
367 views

Prerequisites and references for homological algebra

I'm very interested in learning Homological Algebra, but I'm not sure about the prerequisites for learning it. My current knowledge in algebra consists of Abstract Algebra (groups, rings, and fields),...
2
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1answer
46 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 ...
1
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1answer
77 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 X_k=l,X_m=i)=\frac{P(X_n=j,X_k=l,X_m=i)}{P(X_k=l,X_m=i)...
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1answer
70 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, Pre-...
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0answers
68 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 $\int_\Omega Xd\mathbb{P}$, ...
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1answer
44 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 ...
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0answers
42 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 $|x^...
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4answers
153 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 ...
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1answer
40 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
74 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 ...
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1answer
60 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 $$E[...
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2answers
25 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
42 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
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1answer
201 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
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3answers
72 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 ...