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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1answer
14 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 $u \in [0,1]^A$ ...
1
vote
1answer
23 views

Under the Borel measure associated to the Cantor function each of the intervals remaining in the construction of the Cantor set has measure $2 ^{-n}$

Let $f$ be a function such that agrees with the cantor function on $[0,1]$, vanishes on $(-\infty,0)$, and is identically $1$ on $(1,+\infty)$ and let $\mu_f$ the Borel measure associated to $f$. Show ...
2
votes
0answers
19 views

Explanation - singling out terms

i am trying to understand the derivation of the mean field equations - and my text books shows the following formulas (as part of a larger derivation). $$L(q_j)=\sum_{x}\prod_{i}q_i(x_i)[\log ...
2
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0answers
23 views

Mathematics of Magic Squares

I have seen many popular accounts of simple magic squares but I would like to find a proper mathematical background to understanding magic squares. What background knowledge do I need. I am a retired ...
6
votes
1answer
96 views

Is “Categories and Sheaves” a good followup to Aluffi's “Algebra: Chapter 0”?

I'm about to finish Aluffi's "algebra: chapter 0" and am a bit confused as to what should be my next move. I've been planning to read Tom Dieck's Algebraic Topology for some time now. I glimpsed at it ...
2
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0answers
65 views

Student working with a researcher [on hold]

I was wondering if it is possible for a student to "work with" a researcher on a regular basis. That is, the researcher would give him articles to read, as well as small problems he feels might be ...
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0answers
79 views

Comments about “Topics in Algebra” by I.N. Herstein and “Abstract Algebra” by Dummit/Foote? [on hold]

Today, I got two gifts from my research mentor: "Topics in Algebra" by I.N. Herstein and "Abstract Algebra" by Dummit/Foote. I am very happy and grateful for his gifts, but I already have been ...
5
votes
2answers
127 views

How to understand mathematics on a deep level? [on hold]

I've been focusing on self studying mathematics for the past couple months, and I'm currently working on discrete mathematics. Here's my attempt at a metaphor to describe my issue. Imagine you have a ...
0
votes
1answer
26 views

Trace $\sigma$-algebra and measurable envelope

I'm stuck on a problem from Cohn's book. Let $(X,\mathscr{A})$ a measurable space, and let $C$ be a subset of $X$. Let $\mathscr{A}_C$ be the trace of $\mathscr{A}$ on $C$, that is all the ...
3
votes
2answers
68 views

Why do we care if a function is uniformly continuous? [duplicate]

There are a lot of question regarding whether a function is or is not uniformly continuous or just continuous and there are a lot of $\epsilon_s$ and $\delta_s$ trying to show whether a function is ...
3
votes
3answers
172 views

How to overcome the temptation to read many books covering the same topics [on hold]

S.E advisers, I am a college sophomore in US with a major in mathematics and an aspiring mathematician in the computational complexity theory. I have been reading some math books on different topics, ...
1
vote
3answers
71 views

What does $\overline{r}m:=rm$ mean?

On this Wikipedia article, it says that you can define an $R$-module $M$ as an $R/Ann_R(M)$-module using the action $\overline{r}m:=rm.$ What does that action actually mean? What is $\overline{r}$?
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votes
2answers
60 views

Probability in a fixed die

I have that transition matrix is ...
1
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0answers
22 views

Decompose finitely generated modules and use Krull-Schmidt theorem [duplicate]

I'm trying to show that if $R$ is an Artinian ring, then for finitely generated modules $M,N,N'$, we have that $M\oplus N\cong M\oplus N'$ implies that $N\cong N'$. I'm supposed to do this by ...
2
votes
2answers
76 views

Probability returning to initial state

Let $P=\begin{bmatrix}0&\frac{1}{2}&\frac{1}{2}\\\frac{1}{2}&0&\frac{1}{2}\\\frac{1}{2}&\frac{1}{2}&0\end{bmatrix}$ and $P^{(n+1)}=P^{(n)}P.$ I know that if you start in any ...
1
vote
1answer
22 views

Conditional probability and disjoint events

If $\cup_{n=1}^\infty B_n=\Omega$ and $P(\Omega)=1$ then $\sum_{n=1}^\infty P(B_n)=1$, now $$P(A)=\sum_{n=1}^\infty P(A|B_n)P(B_n)=p\sum_{i=1}^\infty P(B_n)=p$$ If $X$ and $Y$ are independents ...
1
vote
1answer
23 views

Interpretation - exponential loss formula

i stumbled upon the following formula while studying the concept of exponential loss. I try to understand this formula. I read it as follows: The function that minimizes the expected exponential loss ...
0
votes
0answers
11 views

minimization of adaptive basis functions

i am self-studying the topic of boosting, and trying to understand the following argument.i am failing to see the connection between 16.39 and 16.40 - why is this function of $\phi_m$ the optimal ...
3
votes
3answers
47 views

Find $p_{ij}^{(n)}$ for the transition matrix

Let $$P=\begin{bmatrix}\frac{1}{3}&0&\frac{2}{3}\\\frac{1}{3}&\frac{2}{3}&0\\\frac{1}{3}&\frac{1}{3}&\frac{1}{3}\end{bmatrix}$$ find ...
8
votes
4answers
338 views

How To Develop A Higher Mathematical Aptitude? [closed]

First off I must say I'm pretty blown away by the vast majority of the people in this forum. I do aspire to reach the knowledge of mathematics as shown on the site, but honestly it's a little daunting ...
2
votes
0answers
25 views

number of vertices in a solid

Determine the number of vertices in a solid made up of $x$ triangles, $y$ squares and $z$ pentagons. Without using the Euler's formula $v-e+f=2$ and without counting up all vertices by hand I am not ...
1
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0answers
44 views

Expected time to failure

A machine needs two types of components in order to function. We have a stockpile of $n$ type-$1$ components and $m$ type-$2$ components. Type-$1$ components last for an exponential time with ...
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0answers
39 views

Help in partial derivative during maximization for estimation problem

The joint pdf is: $$P((\mathbf{X,y}) |y_n, \theta) = \frac{1}{\sqrt{2 \pi \sigma^2_c}} \exp \big(\frac{-(c_0)^2}{2 \sigma^2_c} \big) \prod_{n=1}^{N-1} \frac{1}{\sqrt{2 \pi \sigma^2_w}} \exp ...
1
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3answers
52 views

$\lim_{n \to \infty} (\sqrt{9n^2 + 2n + 1} - 3n) = \frac{1}{3}$ [duplicate]

Prove $$\lim_{n \to \infty} (\sqrt{9n^2 + 2n + 1} - 3n) = \frac{1}{3}$$ I got this problem in Harro Heuser's "Lehrbuch der Analysis Teil 1". It is surely smaller than 1 because $\sqrt{9n^2 + 2n + 1} ...
3
votes
1answer
35 views

Dynamic programming recursion

In a book by Wayne Winston for operations research I found this question. Here's how I did it: Let $t$ be the no.of subjects to pass and let h be the no.of hours she has in hand for studying. ...
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votes
0answers
34 views

Getting Started [closed]

I am new to code and need a good program for a basic knowledge. I have started looking at programs but am in need of advice of something that does not need a masters or doctorate in the subject and i ...
1
vote
1answer
36 views

Finding a maximal complete subspace of Riemann Integrable functions on $[0,1]$

I know that the space of Riemann Integrable functions on $[0,1]$ is not complete under the norm $|f|= \int f$. So I was wondering as to what would be a maximal complete subspace of Riemann Integrable ...
0
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0answers
25 views

Increasing Matrix

Consider real matrix-vector multiplication. I am just curious whether there exists a $p\times p$ matrix $A$ such that if $x$ is a $p\times 1$ real vector whose entry is in ascending order, i.e., $x_1 ...
2
votes
3answers
31 views

Independent Poisson process

Suppose that $\{N_1(t),t\geq0\}$ and $\{N_2(t),t\geq0\}$ are independent Poisson Process with rates $\lambda_1$ and $\lambda_2$. Show that $\{N_1(t)+N_2(t),t\geq0\}$ is a Poisson process with ...
1
vote
2answers
16 views

Steps to Simplify

I am struggling to see how the following problem is simplified. Can someone include any steps that may have been skipped? Original Equation= $\frac{T(p-b)}{(p-b+q-a)}$ Simplified Equation= ...
2
votes
3answers
61 views

Brief book on calculus to read before studying the analysis [on hold]

S.E advisers, I am going to start studying the analysis texts (Rudin-PMA, Apostol-MA, Pugh-RMA) on the first week of August. I have a good proof skills through working on Artin's Algebra and ...
1
vote
1answer
24 views

Expectation and Poisson process

Let {$N(t),t\geq0$} be a Poisson process with rate $\lambda$. Calculate $E[N(t).N(t+s)]$ I know that $N(t)\sim Poisson(\lambda t)$ and $N(t+s)\sim Poisson(\lambda(t+s))$ I can assume that ...
7
votes
1answer
143 views

Reading mathematics at the graduate level - does every single proof matter?

I would be interested in hearing from PhD students (and upwards) specializing in pure maths (in particular, the more algebraic aspects). My question is this: When reading to learn mathematics at ...
2
votes
2answers
32 views

Poisson Process proof that

For a Poisson process show, for $s<t$ that $$P(N(s)=k\mid N(t)=n)={n\choose k}\left(\frac{s}{t}\right)^k\left(1-\frac{s}{t}\right)^{n-k},\space > k=0,1,\dots,n$$ I tried a few things but ...
2
votes
1answer
22 views

Notation - matrix calculatoin

i am self-studying linear algebra, and came across the following statement about singular value decomposition (X = $USV^T$). I am somewhat confused on how to interpret the (|) and (--) symbols. Should ...
0
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0answers
22 views

Derivation - EM algorithm

I am self-studying the EM algorithm from Kevin Murphy's book (http://www.cs.ubc.ca/~murphyk/MLbook/index.html), and have a question that i am struggling with: Moving from step 11.24 and 11.25 - why ...
0
votes
1answer
55 views

Equivalent conditions of Lebesgue measurable sets

Hi I'd appreciate if someone can check the following exercise any suggestions are welcome. Thanks ;) Let $A$ a subset of ${\bf{R}}^d$ show that the following conditions are equivalent: (i) $A$ ...
0
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0answers
10 views

Singular value decomposition - unique determination

i am self-studying SVD - and stumbled upon the Wikipedia page (https://en.wikipedia.org/wiki/Singular_value_decomposition) on the statement that a common convention is to order the singular values in ...
0
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0answers
10 views

low-rank matrix decomposition

i am self-studying factor analysis - and the text states that the covariance matrix is approximated with a low-rank matrix decomposition. i am trying to understand what this means in practice - and ...
1
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2answers
50 views

Expectation and waiting time

There are three jobs that need to be processed, with the processing time of job $i$ being exponential with rate $\mu_i$. There are two processors available, so processing on two of the jobs can ...
2
votes
1answer
31 views

Failure time and exponential distribution

One hundred items are simultaneously put on a life test. Suppose the lifetimes of the individual items are independent exponential random variables with mean $200$ hours. The test will end when ...
0
votes
1answer
31 views

Bounded Matrix-Vector Multiplication

Given a $p\times p$ square matrix $A$. Can I say that the 2 norm of their product is always bounded for any $p \times 1$ vector, please? That is, $$ \| Ax \| <\infty, \forall x\in\mathbb R^p. $$ ...
2
votes
1answer
52 views

Expectation with exponential random variable

If $X_i$, $i=1,2,3$ are independent exponential random variable with rates $\lambda_i$, find $$E[\max(X_i) \mid X_1<X_2<X_3]$$ I really did not understand this exercise, because if ...
3
votes
1answer
68 views

What are some easy to understand applications of Banach Contraction Principle?

I know that Banach contraction principle guarantees a unique solution to problems of the form $$f(x) = x$$ But for the life of me I cannot understand why this problem is important at all. I don't ...
2
votes
0answers
64 views

Inquiry about My Self-Study Plan for Real Analysis (associated with my undergraduate research) [closed]

S.E advisers, I am a college sophomore in US with a major in mathematics and an aspiring mathematician in the computation theory and cryptography. I recently got an undergraduate research in the ...
1
vote
1answer
40 views

Probability with exponential random variable

Machine $1$ is currently working. Machine $2$ will be put in use at time $t$ from now. If the lifetime of machine $i$ is exponential with rate $\lambda_i=1,2$, what is the probability that ...
2
votes
1answer
13 views

log normalizer - exponential family

i am studying the exponential family- and read that, for $p(x|\mu)=h(x)exp(\eta^T t(x)-a(\eta))$, that $a(\eta)$ is the log normalizer, which ensures that the probability distribution integrates to ...
1
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0answers
17 views

Characterize the $\mu^*$- measurable sets where $\mu^∗ = \lambda^* \circ \text{proj}_1 $ and $ \lambda^*$ is the Lebesgue outer measure

Hi I'm working with Cohn's book and I have other problem with the necessity condition, I'd appreciate any help. Let $\lambda^*$ the Lebesgue outer measure on $\bf{R}$, and let $\pi$ be the ...
0
votes
2answers
46 views

Splitting the summation sign

i am trying to understand the second step in the formula per below - and how the summation sign $\sum_{k=1}^K$ splits into the terms 1-$\sum_{k=1}^{K-1}$ terms. Any help much appreciated
0
votes
1answer
19 views

Indicator function - exponential form of Bernouilli [closed]

i am struggling to understand the derivation per below, which aims to explain how $Ber(x|\mu$) can be written as exp[$\phi$(x)$^t$$\theta$]. i would have interpreted that the indicator function ...