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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12 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
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1answer
59 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
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1answer
43 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
46 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 ...
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2answers
35 views

How are surface area and volume related? [on hold]

Is there even going to be a time when the surface area and the volume of a cube are the same numerical value? The cube is a $3cm \times 3cm\times 3cm$. With an area of $54\ cm^2$. And a volume of ...
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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 $u \in [0,1]^A$ ...
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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 ...
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0answers
20 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 ...
4
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1answer
36 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
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1answer
113 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
69 views

Student working with a researcher [closed]

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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84 views

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

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
140 views

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

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
32 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
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2answers
71 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
182 views

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

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, ...
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3answers
74 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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2answers
63 views

Probability in a fixed die

I have that transition matrix is ...
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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
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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
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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 ...
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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 ...
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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
48 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 ...
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4answers
347 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
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0answers
26 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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1answer
58 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
42 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
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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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1answer
38 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
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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 ...
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2answers
17 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
66 views

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

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 ...
6
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1answer
145 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 ...
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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
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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
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1answer
32 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
32 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
69 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
67 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 ...