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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What is the importance that an assumption needs to state whether a space is Banach space?

I am self studying functional analysis and I don't not see the utility of authors trying make it clear that a space $X$ is a Banach space before proceeding with a definition. For example, going ...
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28 views

Outer measure exclusion of zero set

I've just started self-studying measure theory by reading Pugh's Mathematical Analysis. He shows that the exclusion of a zero set does not change the outer measure: $m^*(E\setminus Z)=m^*(E)$, but ...
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2answers
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Matrix Differentiation

Consider a differentiable function $f: \mathbb R \to \mathbb R$ and two $p\times 1$ vectors $x$ and $\theta$. Then define a new function as follows. $$ f\left( x^T\theta \right)x. $$ Now we want to ...
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35 views

Doubly stochastic matrix proof

A transition matrix $P$ is said to be doubly stochastic if the sum over each column equals one, that is $\sum_i P_{ij}=1\space\forall i$. If such a chain is irreducible and aperiodic and ...
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35 views

Proof that state can be reached

Prove that if the number of states in a Markov Chain is $M$, and if state $j$ can be reached from state $i$, then it can be reached in $M$ steps or less. To me it just seems the definition of ...
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30 views

If $R$ is a simple Artinian ring, then when is a finitely generated module free?

Here's an exercise from my book, which only gives a brief solution which leaves me very confused. Let $R$ be a simple Artinian ring, say $R=K_r$. Show that there is only one simple right ...
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2answers
54 views

Notation - “' sign” in summation

I am studying logistic regression - and i saw the following symbol: c' (in sum symbol of the denominator). What does this mean? I thought it might be all other elements of the vector except the one ...
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1answer
41 views

What is the best way to motivate yourself to review what you have already learned? [closed]

I went through a course in complex variables recently with great zest and enthusiasm and thought that I had mastered the material. Two month after school I turned my interest to one of the chapter ...
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50 views

equivalent form of almost sure convergence

Consider random variables $X_1, X_2, \dots$ and $X$ on $(\Omega, \mathcal F, \mathbb P)$. We say that $X_n$ converges to $X$ almost surely if $$\mathbb P\left(\lim_{n \to \infty} X_n =X\right)=1.$$ It ...
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1answer
19 views

Convexity proven as max of linear functions

i am studying convexity, and stumbled upon the statement and example below. Am i right to understand that the function in the example is convex because maximizing the equation on the right hand size ...
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1answer
33 views

Introduction to Lebesgue Integration for Statistical Use

I am studying statistics at the graduate level and have a moderate background in real analysis however I unfortunately have no experience with Lebesgue integration. Does anyone have some recommended ...
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1answer
33 views

Seeking the Recommendation on Complexity Theory books

S.E advisers, I am a rising college junior in US with a major in mathematics and an aspiring applied mathematician in the fields of theoretical computing. I just recently got a research project on ...
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1answer
32 views

If $P^r$ has all positive entries, then so does $P^n$

Let $P$ be the transition probability matrix of a Markov Chain. Argue that it for some positive integer r, $P^r$ has all positive entries, then so does $P^n$, for all integers $n\geq r$ I ...
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1answer
41 views

Vector spaces - $\mathbb{R}^n$ and $\mathbb{R}^m$

I stumbled on the following text on Wikipedia: Suppose the random column vectors X, Y live in $\mathbb{R}^n$ and $\mathbb{R}^m$ respectively, and the vector $(X, Y)$ in $\mathbb{R}^{n+m}$ has a ...
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0answers
65 views

Understanding the formula

Let $P$ the transition probability matrix and $\mu$ the row vector of initial distribution. $$P_\mu(X_n=j)=\sum_j\mu(i)p^n(i,j)=\mu p^n(j)$$ I don't want to make a proof of that, I want to ...
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1answer
28 views

Expectation in markov chain

A Markov Chain {$X_n,n\geq0$} with states $0,1,2$, has the transition probability matrix ...
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0answers
25 views

Looking for resources on these topics from Linear Algebra

I am looking for Characteristic roots and characteristic vectors of a linear transformation or of a matrix, Algebraic and Geometric multiplicity of a characteristic value, Cayley-Hamilton theorem, ...
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1answer
36 views

Is {$X_n,n\geq 0$} a markov chain?

Consider a process {$X_n,n=0,1,\dots$}, which takes on the values $0,1,2$. Suppose $$P(X_{n+1}=j|X_n=i,X_{n-1}=i_{n-1},\dots,X_0=i_0)$$ $$=P_{ij}^I,\text{when n is even}$$ ...
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1answer
22 views

Is the function $G$ right continuous?

Suppose $S\subset\mathbb R$ such that $S\cap(x,\infty)\neq\phi$ for every $x\in\mathbb R$ and suppose that $g:S\to\mathbb R$ is a non-decreasing bounded function. Define $G:\mathbb R\to\mathbb R$ ...
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2answers
78 views

Question from self-studying Halmos' Finite Dimensional Vector Spaces

For section 1 on Fields, there is a question 2c: 2. a) Is the set of all positive integers a field? b) What about the set of all integers? c) Can the answers to both these question be changed by ...
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1answer
35 views

To show function is Riemann integrable

Given function v(x) = 0 , x=0 v(x)= 1 , x belonging to (0,1] To show that v is Riemann integrable in [0,1] Hints on how to ...
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130 views

Which should I study first: Logic or set theory?

I'm an undergraduate student in a college of sciences and technics studying maths, physics, computing and some chimestry so we studied elementary materials in logic and set theory. As I am interested ...
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To prove $| S(f,P,T) - S(g,P,T) | \leq M(b-a)$ ( Riemann Integration)

To prove $| S(f,P,T) - S(g,P,T) | \leq M(b-a)$ Question : Let $[a,b] \subseteq R$ be a non degenerative closed bounded interval and let $f,g :[a,b] \rightarrow R$ be functions .Suppose that there is ...
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A reference to study Boundary conditions of diffusion processes

I am trying to learn about Wentzell Boundary condition and (A,L) diffusion in the sense of Watanabe's paper (On the existence and uniqueness of diffusion processes with Wentzell's boundary condition ...
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88 views

What is the best way to study Probability? [closed]

Nowadays, I am studying probability. I want to be an actuary and the first exam that I have to pass is P exam. I just want to know what is the best way and if you can recommend any books please let me ...
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3answers
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What does the quotient group $(A+B)/B$ actually mean?

I understand that $A+B$ is the set containing all elements of the form $a+b$, wit $a\in A, b\in B$. When you do the quotient group, that's like forming equivalence classes modulo $B$. All elements ...
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43 views

Proof the statement

Given a finite aperiodic irreducible Markov Chain, prove that for some $n$ all terms of $P^n$ are positive. I'm little lost in how to prove that, but I know that: $i)$ If a Markov Chain is ...
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1answer
18 views

Stochastic matrix proof

Every stochastic $n\times n$ matrix corresponds to a Markov chain for which it is the one-step transition matrix. However, not every stochastic matrix $n\times n$ is the two-step transition ...
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1answer
30 views

Periodicity of states in Markov Chain

Determine the classes and the periodicity of the various states for a Markov Chain with transition probability matrix ...
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1answer
57 views

To show that function is Riemann Integrable on $[0,1]$

I am having difficulty in understanding this proof. Firstly why is the set $E$ defined? Why is specific value of $\delta$ chosen and about possibility of tags counting twice. Can anyone help me with ...
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2answers
50 views

Understanding definition of Riemann Integral

I have read this definition first time today. As far as I can understand it, it seems to me that difference between Riemann sums and a number L can be made small by changing norm of partition. Does ...
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2answers
21 views

Maximising directional derivative of a polynomial in 3 variables

I am a beginner in multivariable calculus and had started reading Apostol. I have solved all the exercises in the portions I have covered, except one problem. Find values of the constants $a,b,c$ ...
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1answer
47 views

To show function is not surjective

I am trouble understanding first three lines of proof . For onto we have to show that there exists some element in P(A) which doesnot have its preimage .But how did they have shown here
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3answers
44 views

Question involving chess master (combinatorics)

I am having hard time with this question .I have not understood what is point and why is sequence $a_1 + 21$ , $a_2 + 21 $... has been taken in second picture .Please help me understand the question ...
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1answer
70 views

Brushing up on Calc

I am an engineering major. I have taken calc 1-3 and did very well in each course. However, I feel that I did well mostly by learning the tricks and feel that I need a better intuitive understanding ...
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0answers
49 views

What is the vector product $(x\wedge y)\wedge z$?

Here's an exercise from my book (exercise 10, chapter 2.1) Show that the three-dimensional vector space $V=R^3$ forms an associative algebra with respect to the operation $x\uparrow ...
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0answers
26 views

Show that $f$ is constant on the convex set $S$

Call a set $S$ convex if whenever $x,y\in S$, then $tx+(1-t)y\in S$ for any $t\in[0,1]$. Suppose that $S$ is an open convex set in $\mathbb R^n$ and suppose that $f:\mathbb R^n\to\mathbb R$. If ...
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1answer
39 views

What can we conclude about the function $f$?

Let $f$ be a scalar field, $f:\mathbb R^n\to\mathbb R$. Suppose there is an $n$-ball $B(a;r)$ centered at $a$ with radius $r$ and a fixed vector $y\in\mathbb R^n$ such that $f'(x;y)=0$ for every ...
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1answer
30 views

Matrix Multiplication, Trace and Integration

Let $\omega(x)$ be a $p\times 1$ vector-valued function defined on a random variable $X$ with CDF $F$. Now define $$V:=\int \omega(x)[\omega(x)]^T dF(x).$$ Then define $\gamma$ as follows. $$ \gamma ...
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3answers
38 views

how do they calculate these following columns

I have these data: I am sorry the data is in Portuguese, and it is an image so I can't convert it to a table but the translate "probably" ( i am not a native speakers for Portuguese language) is: ...
4
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1answer
74 views

Probability theory required for learning statistics rigorously

I would like to learn statistics rigorously. The only book that I can find that seems to do statistics rigorously is this book "Theory of statistics" by Schervish (which seems advanced): ...
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1answer
28 views

Markov Chains, reccurent and transient

Let the Markov Chain consisting of the states $0,1,2,3$ have the transition probability matrix ...
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0answers
55 views

How do I make sure that I've learned and mastered a part of the Visual Complex Analysis book?

So I'm reading Visual Complex Analysis by Tristan Needham. It's a beautiful book that's not very hard to understand at all; however, I just don't know if I have sufficiently learned what I'm supposed ...
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1answer
32 views

How to construct a transition matrix?

I'm giving my first steps in stochastic processes but I'm having some difficulties. See the following example Suppose that whether or not it rains today depends on previous weather conditions ...
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48 views

How to effectively learn from and use Ramanujan's notebooks? [duplicate]

I will come back and elaborate on the question if necessary (I must be off for a while...). But I'll try being specific. I have all four of Ramanujan's notebooks, with their respective Errata ...
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2answers
32 views

Find the density of their average

If $f_{X,Y,Z}(x,y,z)=e^{-(x+y+z)}I_{[0,\infty]}(x)I_{[0,\infty]}(y)I_{[0,\infty]}(z)$ find the density of their average $\frac{X+Y+Z}{3}$ I'm a little lost on how to solve this exercise, ...
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2answers
56 views

Disjoint events

Let $A$ and $B$ two disjoint events such that $P(A)=0.3$ and $P(B)=0.5$. Find the probability that i)$A$ or $B$ ocurrs ii)$A$ occur but not $B$ iii)repeat $i)$ and $ii)$ with $A$ ...
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1answer
129 views

Is Alfred Tarski's Introduction to Logic still helpful for self study?

I am trying to setup a self study path to enhance my knowledge of mathematical logic. I haven't taken a logic course for a few years and my confidence on mathematical proofs is unnerving. I am ...
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1answer
46 views

Principle of well ordering

Every non-empty set $A\subset\mathbb{N}$ have a smallest element, i.e. an element $n_0\in A$ such that $n_0\leq n$ $\forall n\in\mathbb{A}$ Proof: Let $I_n=\{p\in\mathbb{N};p\leq n\}$ the set ...
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94 views

Reference request for this topics

I'm a second year undergraduate statistic student, I need a good reference to learn these topics Markov Chains in discrete time.    1.1. Classification of states, recurrence notions of transience. ...