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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103 views

Proof that if $\phi \in \mathbb{R}^X$ is continuous, then $\{ x \mid \phi(x) \geq \alpha \}$ is closed.

Recently, having realized I did not properly internalize it (shame on me!), I went back to the definition of continuity in metric spaces and I found a proposition for which I was looking for a proof. ...
3
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2answers
61 views

what is a curve ? Is the concept of derivative limited to curves only?

I am trying to understand derivative and I want to know intuitive and rigorous definitions for a curve and if derivative is lmited only to curves or not..
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1answer
39 views

Support of random variable is a closed set

Let the support $S$ of a distribution function $F$ be $$S = \left\{x: F(x+\epsilon)-F(x-\epsilon) > 0, \forall \epsilon>0\right\}$$ I want to show that this is a closed set. In case anyone ...
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3answers
100 views

Advanced complex function theory book recommendation

I would like to have some recommendations in order to self study the above topic. I have already studied some complex function theory, covering some of the more 'classical' theorems (the Bloch-Landau ...
2
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2answers
88 views

Let $X_t$ and $Y_t$ Poisson Process

Let $X_t$ and $Y_t$ be two independent Poisson Process with rate parameters $\lambda_1$ and $\lambda_2$ respectively, measuring the number of customers arriving in stores 1 and 2. a)What ...
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1answer
306 views

Fair coin tosses until until two consecutive heads or two consecutive tails appear

A fair coin is tossed repeatedly and independently until two consecutive heads or two consecutive tails appear. Find the PMF, the expected value, and the variance of the number of tosses. Let X be a ...
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4answers
82 views

What does $a\mid p$ mean?

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0answers
14 views

Polar plane spiral repitions

I'm just starting out teaching my self about the polar plane using tools like Desmos and have been wondering: When graphing an equation in the polar plane, does it extend forever? All the tools ...
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1answer
22 views

Rank of a product of a positive definite and a rank $2$ matrix.

If we assume that $\mathbf{U}$ is a matrix $n\times2$, with rank $2$ (two independent columns) and $\mathbf{A}$ is a positive definite matrix of order $n$, what would be the best way to see that the ...
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6answers
2k views

How do I teach university level mathematics to myself? [closed]

So here I go, I have enrolled myself in maths major this year but due to less marks in SSC I couldn't secure admission in a good university so I have to take admission wherever I could get with my ...
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1answer
29 views

How does this function behave in terms of $\gamma$ when we take $\lim_{b \to \infty}$

$$\mu_n = \frac{(b^{n-\gamma+1}-a^{n-\gamma+1})(-\gamma+1)}{(b^{-\gamma+1}-a^{-\gamma+1})(b-\gamma+1)}$$ I'm interested in how this behaves as $\gamma$ changes. We can assume $\gamma > 1$. I've ...
3
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0answers
106 views

What's the right way to read Princeton companion to mathematics?(for non-mathematician)

I am thinking of ways,how one(who is not a mathematician, but wants to know what's going on in the field of mathematics) can properly read Princeton companion to mathematics to make sense of it. I ...
3
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1answer
77 views

Being ready to tackle the math courses in my CS program

Here's my (long) story cut short. I was awful at math in high school. I did 4 years in the service and now I'm going to start college in just a few weeks. I am really nervous because I will have 5 ...
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0answers
148 views

Solutions for “What is Mathematics?” by Richard Courant

I am currently reading through Courant's "What is Mathematics?" Most of the time I am not taking the exercises too seriously, given that I am reading this for pleasure, and that often my solutions are ...
1
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2answers
59 views

Question about Quantifiers

I've been reading Velleman's How to Prove It and it says that the $\forall x$ in$\forall x P(x) \rightarrow Q(x)$ only applies to $P(x)$ unless there is a parenthesis $\forall x (P(x) \rightarrow ...
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2answers
36 views

Problematic exercise on alternate way of expressing random variables

I found this exercise on a book on probability theory, and I find it problematic. Let $(X, \Sigma, p)$ be a probability space, $\mathcal{A} \subseteq \Sigma$ a finite partition of $X$; and $\phi ...
4
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3answers
97 views

What is the equivalent of musical ear training with regards to studying mathematics

When one aspires to be a professional musician, it is made clear that ear training is a very valuable skill that must be cultivated on a daily basis. The student is advised to put in the time and ...
3
votes
1answer
111 views

Measure from a “distribution function” and integrate with respect the associated measure

Hi this is an exercise from other old exam. I'd like to know if my attempt is correct. I'd really appreciate your help or your suggestions to improve my arguments and also recommendation to similar ...
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1answer
30 views

a question related least common multiple [closed]

$ 1, 3, x, 15, y, 75$ These above six numbers are divisors of a number $z$, and these numbers are ordered from the least one to the highest one. How to find numbers $x, y, z$?
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1answer
105 views

textbook for self studying geometry

Looking for recommended readings in geometry for self study. I am planning to get this book for self study. Geometry: A High School Course
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1answer
44 views

Explicit construction of a sigma algebra that makes a simple function measurable

This is a question from an old exam. I'd like to see if my answer is correct. I'd appreciate any suggestion. Thanks :) Let $f: (\mathbb{R},\mathscr{S}) \to (\mathbb{R}, \mathscr{B}(\mathbb{R}))$ ...
0
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1answer
140 views

Probability of hiting the target

If the probability of hitting a target is 1/5, and ten shots are fired independently, what is the probability that the target is hit at least twice? What is the conditional probability that the target ...
34
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9answers
3k views

How to self learn mathematics if you can't buy the books? [closed]

I'm trying to learn more mathematics, especially number theory and abstract algebra, with my goal being category theory, however, whenever I look for book suggestions here, most guys will link to ...
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0answers
31 views

Condtions to make a semiring a base for a topological space

In the book "Principles of Real Analysis" by Aliprantis and Burkinshaw I found the following exercise: Let $\mathcal{S}$ be a semiring of subsets of a nonempty set $X$. What additional ...
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1answer
15 views

Combinatorics problems: termination at rth step

In Feller's book of probability exixt such formulas: a)Placing balls untill for the first time a ball is placed into a cell already occupied: The probability of the process termitating at the rth ...
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2answers
69 views

Span of two vectors in $\mathbb{R}^2$ [closed]

The span of two vectors in $\mathbb{R}^2$ neither of which is zero vector, and which are not parallel, is- a point. line in $\mathbb{R}^2$ not running through origin. line in $\mathbb{R}^2$ running ...
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3answers
108 views

Easiest (most forgiving?) way to learn category theory?

I'm not a maths student, but I've read a bit on category theory and I'd love to learn about it. Is there any book that's simple enough for a busy student to pick up and learn the fundamentals? What ...
0
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1answer
35 views

Find the sum of this series $\sum_{k=1}^{\infty} \frac{(2-x)^k}{2^k*k}$ How to find the integration $C$?

Hello I have the following series : $$\sum_{k=1}^{\infty} \frac{(2-x)^k}{2^k*k}$$ I found that the series convergents for $0<x \leq 4$ I managed to reach that $$f(x)=-ln(x)+C$$ But for some ...
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0answers
80 views

Rationale behind construction of measure theory from semirings

I am studying a book (Aliprantis & Burkinshaw, "Principles of Real Analysis") that, in order to introduce the concept of measure, starts from semiring. In particular the authors state that: ...
0
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1answer
96 views

Reference request for Stochastic Processes in general

I'm studying stochastic processes through the book "Introduction to Stochastic Processes, Gregory F Lawler". Is there any significant difference between "Stochastic processes, Sheldon Ross" and ...
2
votes
1answer
164 views

How to evaluate the integral $\int_0^{\infty}[I_{(0,2)}(z)]\frac{(n-1)(y-z)^{n-2}}{y^{n-1}}dy$

I am doing a statistical calculation from a statistical exercise but get stuck at the following integral. $$\int_0^{\infty}[I_{(0,2)}(z)] \frac{(n-1)(y-z)^{n-2}}{y^{n-1}}dy$$ $0<z<y$ ...
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1answer
37 views

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 $$M_i=i(N-1);p=\frac{1}{2}$$ ...
2
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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
37 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!}\\ = ...
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1answer
26 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
95 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 ...
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1answer
19 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 ...
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1answer
22 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
33 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 ...
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1answer
80 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) = ...
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0answers
70 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 ...
1
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4answers
209 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
80 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...