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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11 views

Workshop and courses

I'm a math student (I also know a bit of theoretical informatic and programming) that will in some months ends his 3° year and earn (it's how work in Europe, don't know in America) the "small degree". ...
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2answers
114 views

Are there any books on real analysis that explain what goes on in their proofs for a self studying student?

Are there any books in real analysis that explains what goes on in their proofs? I want to self study real analysis. I read through proofs in each of these real analysis books and I'm not ...
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39 views

How to show that $f(x,y) = |x| + |y|$ is continous at origin

How to show that $f(x,y) = |x| + |y|$ is continous at origin. CLearly it goes to 0 , but how do i prove it? Thanks
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1answer
58 views

A question about Fisher–Neyman factorization theorem

$f_{\theta}(x)$, then $T$ is sufficient for $\theta$ if and only if nonnegative functions $g$ and $h$ can be found such that $f_{\theta} = h(x)g_{\theta}(T(x)) $ The statement is: if $F(t)$ is a ...
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2answers
49 views

Finite sums of integers and similar problems: book request

I recently learned about Faulhaber's formula, which says that for each integer $p \geq 1,$ we can simplify the finite sum $\sum_{k \in \mathbb{N}}[k<n]k^p$ so that it becomes an (integer-valued) ...
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1answer
70 views

Stationary distribution in continuous-time Markov chain

Consider a barbershop with one barber who can cut hair at rate 4 (people per hour), and three waiting chairs. Customers arrive at rate 5 per hour. Customers who arrive to a fully occupied shop ...
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1answer
81 views

Dirac Delta Function as a Measure

I was always told in my college physics classes to not worry too much about the dirac delta function because it can be made rigorous using distributions or measure theory. I've just started learning ...
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1answer
156 views

Bridge between High School Mathematics and University-level Mathematics?

I've graduated from High School and I am going to major in math at a local University. I've finished High School Calculus and I've self-studied very very basic Multivariable Calculus, Linear Algebra, ...
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1answer
20 views

Subtraction of a sub-squence and a sequence explanation

In my assignment I have the following question, True of false: Let $a_{n}$ be a sequence. If $$\lim\limits_{n\to\infty} (a_{2n}-a_{n})=0$$ Then $a_{n}$ is convergent. The statment is false ...
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3answers
91 views

Proving that $\lim \limits_{n\to \infty} \sqrt[n]{n^5-2n+7}=1$

In my assignment I have to prove the following: $$\lim \limits_{n\to \infty} \sqrt[n]{n^5-2n+7}=1$$ I don't know how to start, I believe it has to do with the squeeze theorem, but I can't be sure. ...
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40 views

Dirac Measure is Purely Atomic

In my book, "Probability and Stochastics" by Cinlar, it's stated that for some measurable space $(E,\scr E)$, and fixed $x\in E$, the Dirac measure $\delta_x(A)=\left\{ \begin{array}{lcc} ...
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25 views

Properties of the Kernel from the measurable space $(X,\mathscr{A})$ to $(Y,\mathscr{B})$

Hi everyone this is an exercise from Cohn's book. I'd appreciate if someone can check part (d) and (e) where I have more problems because this concept is completely new for me. Let $(X,\mathscr{A})$ ...
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1answer
44 views

Variance of Transformed Random Vectors

Consider an $n$-dimensional normal random vector $\mathbf X:= (X_1, \dots, X_n)^T$ with mean $\mathbf 0$ and covariance matrix $\mathbf \Sigma$. Now define a new random vector $\mathbf Y:= (a_1X_1, ...
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0answers
29 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 ...
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1answer
81 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 ...
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1answer
97 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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1answer
17 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
50 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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23 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 ...
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1answer
55 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 ...
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1answer
234 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 ...
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74 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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2answers
210 views

How to understand mathematics on a deep level?

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 ...
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1answer
78 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 ...
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2answers
114 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 ...
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3answers
102 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
71 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 ...
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2answers
88 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 ...
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1answer
26 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
39 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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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 ...
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3answers
51 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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0answers
37 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 ...
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1answer
103 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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46 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 ...
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3answers
53 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
51 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
43 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 ...
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30 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 ...
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3answers
46 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
21 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
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3answers
150 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 ...
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1answer
36 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
193 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
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2answers
37 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
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1answer
24 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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33 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 ...
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1answer
92 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) $A$ ...
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19 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 ...