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 fact that you're self-studying is what your question is _about_.

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6
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2answers
230 views

Prerequisits for Gauss-Green theorem

Consider the following theorem from the appendix C from Evans PDE book: I know about integration in $\mathbb{R}^n$ but not about how to make sense of the integrals on the right-hand side. As my ...
1
vote
0answers
16 views

Scheduling problem

Consider the following setting: $N$ jobs, each has a starting time, which is assumed to be a natural number and all N numbers are distinct, e.g., the 1st job has starting time at 5, the 2nd is 6, the ...
0
votes
0answers
17 views

Reference material on Alternating Minimization Algorithm

I am looking for some good reference material (book/paper) for learning Alternating Minimization Algorithm. Any recommendation from optimization experts will be much appreciated. Thank you.
2
votes
2answers
29 views

Question about an exercise from Feller

The following is an exercise from the classical textbook of Feller on probability theory. Four girls take turns at washing dishes. Out of the total of four breakages, three were caused by the ...
0
votes
4answers
72 views

Is $f\colon\mathbb{Z}\to\mathbb{Z}, f(x)=x^2$ injective? Surjective?

I would say no: $\text{Suppose } f(a)=f(b) \text{ then } a^2=b^2 \implies \pm a = \pm b \implies -a=b$. Or simply by counterexample: $f(-1)=f(1)$ Further, I would say it does not map $\mathbb{Z}$ ...
2
votes
1answer
69 views

Infinite horizon cost function

The following quote is from Bertsekas's Dynamic Programming and Optimal Control. I'm only looking for a nudge in the right direction as to how to interpret the following equations, particularly ...
6
votes
3answers
338 views

Equivalence Relation between Derivative Being Odd and Function Being Even

In the exercise, I am required to prove that $f'$ is odd $\iff$ $f$ is even Moving from right to left was pretty trivial, however, I couldn't move from left to right. Note that we can only use very ...
1
vote
2answers
22 views

If $A_n \downarrow A.$ then $A_1 - An \uparrow A_1 - A$? Set theory.

Let $A_1, A_2 , \dots$ be subsets of a set $\Omega$. If $ A_1 \subset A_2 \subset \dots$ and $\bigcup_{n = 1}^{\infty} A_n = A $ then we write $A_n \uparrow A.$ $ A_1 \supset A_2 \supset \dots$ and ...
0
votes
1answer
19 views

liminf and limsup in probability

Consider a sequence of random variable $(X_n)$. Prove the following inequality: $$ \mathbb P\left(\liminf\{X_n \leq x\}\right) \leq \liminf\mathbb P\left(\{X_n \leq x\}\right)\leq \limsup\mathbb ...
1
vote
1answer
37 views

Set Inclusion Properties

Consider a sequence of random variables $(X_n)$ converge almost surely to $X$. Define set $N:=\{\omega: X_n \to X \}^C$. Then it is claimed that we would have the following set inclusion properties. ...
2
votes
2answers
46 views

$4$ random digits, $2$ different ones.

I have been trying to solve the following problem What is the probability that among $4$ random digits, there appear exactly $2$ different ones? Two different digits means that there should be ...
2
votes
1answer
99 views

Proof that this specific function is measurable

Bounty Edit: Considering the nature of the problem at hand (i.e. proving that a specific function is measurable), I think this can be an easy but relevant problem. In particular, it is relevant to ...
2
votes
2answers
40 views

Prove cyclic group with one generator can have atmost 2 elements

Prove cyclic group with one generator can have atmost 2 elements . Attempt Consider a cyclic group generated by $a \neq e$ ie G = .So G is also generated by <$a^{-1}$> .Now Since it is given ...
0
votes
0answers
36 views

How to construct a two sided confidence interval?

A machine produces metal rods used in an automobile suspension system. A random sample of 15 rods is selected and the diameter is measured. The resulting data are shown below. 5.21 5.28 5.29 5.27 ...
0
votes
1answer
22 views

Consequence of linear combination in matrix .

If a column of a matrix is linear combination of another column, what are the consequences ? Several terminology coming into my mind to relate with this such as Rank of the matrix ; Determinant ...
1
vote
1answer
48 views

Showing that Determinant is a Volume Multiplier

I want to show using the change of change of variables theorem for (Riemann) integration that the determinant of a linear transformation $T$ is a scaling factor for the volume of a space. If $1_A$ is ...
1
vote
0answers
46 views

Convergence of Types Theorem

(Convergence of Types Theorem) Suppose that $F_n(u_nx+v_n) \Rightarrow F(x)$ and $F_n(a_nx+b_n) \Rightarrow G(x)$, where $u_n>0, a_n>0$ and $F$ an $G$ are non-degenerate. Then there exist ...
3
votes
1answer
68 views

$P( A\triangle B ) = 0 \Rightarrow P(A)=P(B)=P(A\cup B) + P(A \cap B) $

I want to prove the following statement; $$P( A\triangle B ) = 0 \Rightarrow P(A)=P(B)=P(A\cup B) \color{blue}{=} P(A \cap B) $$ What I did is that $$P(A\triangle B) ...
0
votes
3answers
57 views

$ P( A \mathrel{\triangle} B ) \le P(A \mathrel{\triangle} C) + P(B\mathrel{\triangle} C)$

Show that $$ P( A \mathrel{\triangle} B ) \le P(A \mathrel{\triangle} C) + P(B\mathrel{\triangle}C)$$ where $\mathrel{\triangle}$ indicates the symmetric difference I cannot write my idea, because ...
0
votes
1answer
28 views

Definition of Multiple .

Definition of multiple is : In mathematics, a multiple is the product of any quantity and an integer. In other words, for the quantities a and b, we say that b is a multiple of a if b = na for ...
1
vote
3answers
119 views

If M,N are finite dimensional vector spaces with same dimension ,then if M is subset of N ,then M=N

If M,N are finite dimensiona;l vector spaces with same dimension then if M is subset of N ,then M=N I think i need to show that both vector spaces are spanned by same bases in order to do this or to ...
2
votes
0answers
61 views

Can anyone check if this correct?

Convert to spherical coordinates and evaluate:$$\iiint_{E}z(x^2+y^2+z^2)^{-3/2}dV$$ where E is the region satisfying the following inequalities:$$x^2+y^2+z^2\le16,z\ge 2$$ This is what i have done so ...
2
votes
1answer
38 views

Second moments from survival function

Let X be a non-negative continuous random variable with probability density function f(x). Let $$G(t) = \int_{t}^{\infty} f(x)dx$$ Show that$$E(X^{2}) = 2\int_{0}^{\infty} tG(t)dt$$ My thoughts: I ...
0
votes
1answer
33 views

limt of the function as $\mu\rightarrow\infty$ or $\mu\rightarrow-\infty$ .

$\lim_{\mu\rightarrow\infty}\frac{\exp(\bar x-\mu)^2}{(\bar x-\mu)}=? $ Also, $\lim_{\mu\rightarrow-\infty}\frac{\exp(\bar x-\mu)^2}{(\bar x-\mu)}=? $ I know, ...
3
votes
2answers
481 views

Is $g(x)=\log x$ convex function?

The graph of convex function is : In a book it is written that $g(x)=\log x$ is strictly convex function. So i searched for graph of $g(x)=\log x$ and found that Though it has been said that ...
2
votes
2answers
88 views

To prove any two basis of Finite Dimensional Vector Space have same number of elements

To prove any two basis of Finite Dimensional Vector Space have same number of elements If i take bases as $S_!$ = {$\alpha_!$ ,$\alpha_2$ ,....$\alpha_n$ } $S_2$ = {$\beta_!$,$\beta_2$ .... ...
2
votes
0answers
22 views

Uniform probability bound - checking my understanding

Let x and y be two independent random variables. What is the difference between (1) $P_x[\forall y, f(x,y) < \epsilon] >1- \delta$ (uniform bound), and (2) $\forall y, ...
0
votes
1answer
41 views

gluing together continuous functions

HI I was checking and old question here and I have troubles to proof the following: Proposition: Let $X$ be a space with subspaces $Y,A,B$ such that $X \backslash Y= A \sqcup B$ (disjoint union). Let ...
2
votes
1answer
77 views

To prove set is a group

Given a non empty set together with associative binary operation $*$ on $G$ such that $a*x=b$ and $y*a=b$ have solutions in $G$ for all $a,b$ in $G$ To prove it is a group Hints to get started ...
0
votes
1answer
26 views

Sample Points Calculation.

A coin is tossed three times. There are three sample points that i can get one head and two tail. I can count the sample points after writing the sample space, as ...
7
votes
1answer
208 views

Mathematical introduction to machine learning

At first glance, this is once again a reference request for "How to start machine learning". However, my mathematical background is relatively strong and I am looking for an introduction to machine ...
1
vote
1answer
25 views

To find right cosets of H in G where G=<a> and H=$<a^{2}>$ ,where o(G)=10

To find right cosets of H in G where G= and H=$<a^{2}>$ ,where o(G)=10 Since order of $G =10$ , so $a^{10}=e$ .We have $G= { a,a^{2},a^{3},a^{4},a^{5},a^{6},a^{7},a^{8},a^{9},e}$ and $H = ...
1
vote
1answer
79 views

If G has no non trivial subgroups ,then Show that G must be of prime order

If G has no non trivial subgroups ,then Show that G must be of prime order .This question is from Herstein Page 46 Question 3 . Attempt :- Let G has prime order(say p) .So by Lagrange theorem ...
2
votes
0answers
59 views

On the importance of the Riesz–Markov–Kakutani representation theorem.

I am following big Rudin and I have arrived at the representation theorem. Before doing the full long proof I would like to know what results are based on this theorem that for completeness I state ...
0
votes
4answers
279 views

To prove every element of G has finite order where

Let G be a group such that intersection of all its subgroups which are different from e is a subgroup different from e . To prove every element of G has finite order Hints to get started Thanks ...
0
votes
2answers
44 views

To prove $o(HK) = o(H)o(K)/o(H\cap K)$

Given that $H$ and $K$ are finite subgroups of $G$ of order $o(H)$ and $o(K)$, prove that $$o(HK) = \frac{o(H)\,o(K)}{o(H\cap K)}$$ I have proved for specific case when $H$ and $K$ have only ...
1
vote
0answers
28 views

How to show that a function is continuous in the topology of weak convergence

Let $\Omega$ be compact, and let $\omega^* \in \Omega$ be arbitrary. Let $\Delta (\Omega)$ denote the set of all probability measures over $\Omega$, and endow $\Delta ( \Omega)$ with the topology of ...
2
votes
3answers
95 views

Showing $(\mathbb{Q},+)$ is not isomorphic to $(\mathbb{R},+)$

To show: $(\mathbb{Q},+)$ is not isomorphic to $(\mathbb{R},+)$ Now, the equation $x^{2} =3$ has a solution in $\mathbb{R}$, but not in $\mathbb{Q}$. Hence they are not isomorphic to each other. Is ...
0
votes
1answer
33 views

how to get the second equation (related to summation)

$$V(Y) = \sum_{i=1}^N\sum_{j=1}^N [\frac{N^2}{n^2}] (Y_i-Y_j)^2 \frac{n(N-n)}{N(N-1)} $$ for $i< j$ Equation(2.5) $$=(\frac{(N-n)}{n(N-1)})\sum_{i=1}^N \sum_{j=1}^N (Y_i-Y_j)^2 $$ for $i< j$ ...
0
votes
0answers
13 views

Help simplifying this sum $f(x) =\sum_{n=1}^{\infty} \frac{2x}{n} e^{-x^2/n} 2^{-n}$, $ x \ge 0$

I am stuck on this sum $f(x) = \sum_{n=1}^{\infty} \frac{2x}{n} e^{-x^2/n} 2^{-n}$ $ x \ge 0$ Any tips on how to get started? Thanks for any help
1
vote
1answer
37 views

Are the following Stopping Times?

I've been working through the following list of stopping time questions. I am have problems with the final two (e and f). I appreciate any assistance offered. $\textbf{Question:}$ Let $S,T : ...
7
votes
3answers
334 views

How do i visualize Cosets of a group

The Lemma asserted in Herstein as given by $[a] = Ha$ seems very non intuitive to me. How do I think in order that this thing makes sense to me? LEMMA 2.4.4 For all $a$ in $G$ , $$Ha = \{ x \in ...
3
votes
1answer
92 views

What is the prerequisite knowledge for Navier–Stokes Existence and Smoothness problem?

I am highly interested in the Millennium Problem of Navier–Stokes Existence and Smoothness (also here) and my aim is to reach some level of knowledge to do research on it. The problem seems simple to ...
1
vote
4answers
55 views

Prove $R$ is an equivalence relation.

I think I'm on the right track. Set $S = N \times N$, and for any two members $(a,b),(c,d)$ of $S$, define $(a,b) \simeq (c,d)$ provided that $ad = bc$. Prove that $\simeq$ is an equivalence ...
1
vote
1answer
37 views

Determinant of an almost-diagonal matrix

I would like to compute the determinant of the $(k+1)\times (k+1)$ matrix below $$J=\begin{vmatrix} y_{k+1}& 0 & \ldots & 0 & y_1 \\ 0& y_{k+1}& \ldots& 0& y_2 \\ ...
1
vote
0answers
42 views

Suppose a finite set G is closed under associative product and that both cancellation laws hold in G.Prove G must be a group

Suppose a finite set G is closed under associative product and that both cancellation laws hold in G.Prove G must be a group I somehow need to prove identity and inverse ,closure holds to prove that ...
1
vote
1answer
52 views

Group of all $2\times2$ matrices where $a$, $b$, $c$, and $d$ are integers modulo $p$, Herstein Q$26$ Page $37$ [duplicate]

Let $G$ be group of all square matrices of order $2$ $$\begin{bmatrix} a & b\\ c & d \end{bmatrix}$$ such that $a$, $b$, $c$, and $d$ are integers modulo a prime number $p$, such that ...
4
votes
1answer
137 views

Quality of Videos Lectures and Lectures vs Textbooks

I am a student trying to learn different subjects by watching video lectures and reading on my own time. I was wondering if the lectures from ICTP and nptelhrd are a great use of my time. I tried ICTP ...
0
votes
1answer
66 views

To prove in a Group Left identity and left inverse implies right identity and right inverse

Let G be the nonempty set closed under an associative product,which in addition satisfies : A. There exists an e in G such that a.e=a for all a in G B.Give a in G ,there exists an element y(a) in G ...
0
votes
1answer
45 views

Non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i,$ for two consecutive integers.

Given an example of a non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i, \forall a, b \in G$ for two consecutive integers. This is question 5 from Herstein Page 35. I have proved that ...