Questions about the process of studying mathematics without formal instruction.

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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 ...
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41 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 ...
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4answers
236 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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41 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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27 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 ...
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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 $i< j$ ...
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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
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1answer
35 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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329 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 ...
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1answer
85 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 ...
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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 ...
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1answer
31 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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39 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 ...
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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 ...
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1answer
112 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 ...
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1answer
52 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 ...
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1answer
44 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 ...
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2answers
27 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 $ ...
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3answers
46 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
123 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 ...
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1answer
111 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 ...
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1answer
152 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 ...
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1answer
44 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 ...
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1answer
37 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 ...
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1answer
50 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 ...
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2answers
43 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 ...
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51 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 ...
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1answer
30 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 ...
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44 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 ...
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2answers
42 views

Problem in Primitive Pythagorean Triples (PPT)

I'm new to number theory. So now I'm starting my journey of 'number theory' by reading this book. I'm currently in chapter 2 which is Pythagorean Triples. I don't understand. It says there are ...
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4answers
794 views

Forgotten old results break my motivation

I'll begin graduate school next year and I am very impatient to learn new things such as theories, ways of thinking and so on (I enjoyed reading about category theory on my own and I find Galois ...
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1answer
70 views

Next book in learning Abstract Algebra

I have just finished the book "C C Pinter - A Book of Abstract Algebra". My aim is to reach to the level of the book "T W Hungerford - Algebra". Hungerford's book is not only too advanced to study ...
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3answers
59 views

Initial value problem for a linear system.

Consider the linear system $$ \frac{dY}{dt} = \begin{pmatrix} 1 & -1 \\ 1 & 3 \\ \end{pmatrix} Y $$ (a) Show that the function $$ Y(t) = \begin{pmatrix} te^{2t} \\ -(t + 1)e^{2t}\\ ...
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1answer
76 views

Regarding to Real Analysis Textbooks

S.E. users, Which one is better for the real analysis, "Mathematical Analysis" by Tom Apostol or "Undergraduate Analysis" by Serge Lang? It is my first time with real analysis, but I will be ...
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54 views

Question about branches of functions (complex power)

I have the following question, I really appreciate if someone can help me to clarify ideas and I apologize if is a stupid question: This is from Conway's complex analysis book: Let $f: G \to ...
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1answer
253 views

How to Self-Study Mathematical Methods?

Edit: Ok, user Chinny84 made comment that truly helps narrow the focus of my question. Basically, I'm asking for a self-study course of Mathematical Methods. Thanks to his recommendation I ...
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57 views

What is the best way to master my algebra skills without taking an algebra class?

I was in advanced math my entire life. I got through all the math I needed for my original degree. 8 years later here I am changing degrees and I need more math. I just took calculus I and I passed ...
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1answer
31 views

Proving that a subspace of $L^2$ is closed.

Suppose $Z$ is a random variable on a probability space $(\Omega, F, P)$. $M(Z)$ is the subspace of $L^2$ consisting of all random variables in $L^2$ which can be written in the form $\phi(Z)$ for ...
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2answers
38 views

On integration of a simple random variable in measure theory.

Suppose we have a simple Random variable $X$ defined on a probability space $(\Omega, F, P)$. A random variable is simple if $X(\Omega) = \{ \alpha_1, \ldots , \alpha_n \}$. We define the integral of ...
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9 views

How to find the points on an Ellipsoid such that the normal has equal angles with the coord. axis?

I have seen that one could find the points of an Ellipsoid: $$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1$$ But i can see the way to reach them.
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Question on specific optimal control problem

I try to understand the problem described in pages 38-40 of this book (Lectures on Macroeconomics, Blanchard and Fischer,1989). Given $\lambda_t = \mu_t e^{\theta t} $ and this Hamiltonian: $$H_t = ...
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In balancing effort and advancement in what concerns learning

What's new besides showing modern advancements in modern mathematics as well as eloquently written notes also contains some good advice to young people like me in this room (career advice:!). in ...
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Proving that the generator of $U$ is normal if $\forall u \in U, g\in G$ $gug^{-1} \in U$

This is from Herstein. $4.$ $\;a)$ Given a group $G$ and a subset $U$ denote by $\hat U$ the smallest subgroup of $G$ which contains $U$ (the subgroup generated by $U$). Prove there is a ...
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1answer
59 views

how to calculate sum of a series? (me or Wangenmakers is wrong)

Wagenmakers in his critical article about p-values wrote that: $$\sum_{i=12}^{\infty} {{n-1} \choose {2}} \cdot \left(\frac{1}{2}\right)^n \approx .033$$ How could he do his calculations if the ...
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1answer
46 views

Definition of continuity in practice

In general I have a problem to recognise if a function is continuous or not. I simply don't know where I should start to actually see it. Here there is an example of my problem that I found in a ...
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66 views

Why does = change to $\leq$ and then to = in this proof of |a+b| = |a|+|b|?

From Spivak's Calculus. This proof is motivated by the observation that |a| = $\sqrt {a^2}$. $\sqrt x$ denotes the positive square root of x; this symbol is defined only when x $\geq 0$. We may ...
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models to experiment with game theory?

I want to learn game theory from a practical point of view. Does anyone know if there exist programs that illustrates the utility of game theory? Or books that contains MATLAB simulation of games? ...
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1answer
20 views

Can I substitute $\beta A \alpha^{1-\gamma}$ with $c^\gamma$?

I reach a point where in the book the author substitutes $\beta A \alpha^{1-\gamma}$ with $c^\gamma$ to simplify the rest of notation, where $\beta, \gamma \in (0,1)$ and $\alpha, A$ two other ...
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28 views

Big O in Stochastic Sense

I understand that if for a real-valued random variable $X$ we have $X = O_p(1)$, then it means that for any $\epsilon>0$, there exists a positive real number $M>0$ such that ...