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
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
votes
1answer
121 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
votes
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 ...
1
vote
0answers
30 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 ...
0
votes
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 ...
-2
votes
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 ...
0
votes
3answers
107 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
votes
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 ...
2
votes
0answers
79 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
votes
1answer
94 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. $$\int_0^{\infty}[I_{(0,2)}(z)] \frac{(n-1)(y-z)^{n-2}}{y^{n-1}}dy$$ $0<z<y$ ...
0
votes
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, ...
1
vote
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
votes
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 $$x = Q_t \eta$$ ...
1
vote
0answers
18 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 ...
2
votes
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 ...
0
votes
1answer
36 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 ...
1
vote
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 ...
0
votes
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!}\\ = ...
1
vote
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
votes
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 ...
1
vote
0answers
93 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. ...
0
votes
0answers
34 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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
2answers
32 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
vote
1answer
66 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) = ...
0
votes
0answers
69 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
vote
4answers
205 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
votes
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...
1
vote
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 ...
0
votes
1answer
84 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
votes
1answer
35 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) = ...
0
votes
0answers
35 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
votes
0answers
25 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 ...
1
vote
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 ...
3
votes
2answers
87 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
votes
3answers
50 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, ...
1
vote
2answers
45 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 ...
12
votes
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
votes
3answers
95 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
47 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 ...
1
vote
2answers
72 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: ...
1
vote
2answers
54 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 ...
0
votes
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
votes
1answer
85 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
votes
2answers
331 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 ...
2
votes
1answer
45 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
vote
1answer
71 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
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, ...