Questions about the process of studying mathematics without formal instruction.

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0
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19 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
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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
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1answer
45 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
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0answers
26 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
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1answer
61 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) ...
-1
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3answers
51 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
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1answer
27 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 ...
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3answers
68 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
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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
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1answer
30 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
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1answer
32 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
85 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
47 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
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0answers
19 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
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1answer
33 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
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1answer
76 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 ...
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1answer
12 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 ...
6
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1answer
154 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
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1answer
20 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
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1answer
49 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
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0answers
25 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
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4answers
213 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 ...
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2answers
38 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 ...
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0answers
26 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 ...
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3answers
86 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
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1answer
32 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
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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
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1answer
33 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 : ...
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3answers
321 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
78 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 ...
0
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0answers
27 views

Similar matrices for k^th power

If $A^k$ and $B^k$ are similar matrices then A and B are similar matrices. I started with $A^k = PB^kP^{-1}$ and after some arranging I got $A^k=(PBP^{-1})^k$ and I am not sure if I can conclude from ...
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4answers
54 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
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1answer
24 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 \\ ...
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0answers
34 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
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1answer
50 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
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2answers
98 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
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1answer
46 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
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1answer
43 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 ...
0
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2answers
25 views

Prove that if G is Abelian group ,then for all a,b in G $ (a.b)^{n} = a^{n}b^{n}$

This question have already been asked on this site ,But i couldnot understand details so i ask it again .Also what i have done is that first for n=1 its trivial ,for n=2 , we have $a(ba)b=a(ab)b $ ...
0
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3answers
45 views

If G is a group such that $(a.b)^{2}=a^{2}.b^{2}$ for all a and b,Then show that G is abelian

This is problem from I.N Herstein Page 35 Q3 .How should i start doing this ?Hints ? Thanks
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4answers
110 views

What books should a high school calculus student read to learn more about truly beautiful mathematics? [closed]

I love mathematics! Unfortunately, I don't know as much about it as I would like to. I honestly spend a large portion of my free time reading further in my Calculus textbook, and it's very ...
1
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1answer
78 views

Multivariable/Vector Calculus Textbook Recommendation Please!

S.E friends, I am a college sophomore with a major in mathematics. I am trying to self-study multivariable and vector calculus (they means the same, right?) and prepare for Summer course on ...
4
votes
1answer
146 views

Compactness, continuity and the discrete topology

Assume that $X, Y$ are compact metric spaces, and that there is a map $$ \mu : X \to \Delta (X \times Y)$$ such that $\mu$ is continuous, where $\Delta (\Omega)$ denotes the set of probability ...
1
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1answer
42 views

The mean value theorem in $\mathbb{R}^n$ and its application to show that functions are independent of a variable

I am currently reading through several multivariable calculus books to understand the proofs better (most of which go back to introduce functions in $\mathbb{R}$ for which the results are already ...
0
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1answer
34 views

Looking for an online course

My friend and I are interesting in doing an online math course together. He has the basic high school math up to Calculus AB and will be doing BC while we are doing the course. I, however have done ...
0
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1answer
41 views

What to read after Shreve's “Stochastic calculus for finance 2”?

I am finishing the last pages of Shreve's Stochastic calculus for finance 2, and I was wondering what would be the best book to follow. I would like to go on with a book introducing more technical ...
0
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2answers
40 views

mathematical induction to establish inequality

Studying for a test in discrete mathematics and I cannot seem to grasp the explanations in the textbook regarding these questions. Using mathematical induction, prove that $$2^n > n^2, \text{for ...
1
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0answers
50 views

Compactness & Continuity - Looking for feedbacks on a specific setting

I am trying to get the implications of the following general setting concerning compact spaces and continuous maps. Any feedback would be greatly appreciated, because I have some difficulties in ...
2
votes
1answer
29 views

Application of the chain rule for curves

Problem: Let $f: \mathbb{R}^3 \to \mathbb{R}$ be a differentiable function such that $$y \frac{\partial f}{\partial x}(x,y,z) -x \frac{\partial f}{\partial y}(x,y,z) + \frac{\partial f}{\partial ...
0
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0answers
38 views

How to acquire Mathematical Reasoning & Proof Skills

Dear Math Stack Exchange advisers, I am going to start self-studying the introductory analysis soon by using the textbooks called "Understanding Analysis" by Abbott and "Mathematical Analysis" by ...