Questions about the process of studying mathematics without formal instruction.

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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
147 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
43 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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35 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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43 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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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 ...
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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 ...
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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 ...
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41 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
40 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
789 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
64 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
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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
61 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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51 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
243 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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0answers
51 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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0answers
23 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
37 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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0answers
8 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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7 views

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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107 views

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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3answers
44 views

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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58 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
45 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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12 views

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
19 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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25 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 ...
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62 views

What minimum subset of fields of mathematics is needed to understand homomorphic encryption?

Without the luxury of full undergraduate training in mathematics, if one worked part time could the community list the smallest set of mathematical fields needed to understand homomorphic encryption? ...
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1answer
45 views

Are my calculations using Neymann Pearson lemma correct?

I read this post, but I need to use N-P lemma to verify hypothesis doing it really step by step, so please help me. $X_1,X_2,\ldots,X_{30}\sim N(\mu, 1)$, so $\sigma=1$ (I assume that) and $n=30$. ...
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59 views

The canonical height of a point on an elliptic curve

I am struggling with exercise 3.3 in Silverman-Tate Rational Points on Elliptic Curves. Here is the paraphrased problem with necessary background: Let $C:y^2 = x^3 + a x + b$ be a nonsingular cubic ...
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18 views

Limit of generalized mean

I just want someone to check my proof because I feel that it might have a mistake. I'm not really sure, but I feel like it wasn't meant to be solved this way, and that I might have messed up ...
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1answer
24 views

How is the Binomial coefficient simplified to a falling factorial?

I'm learning how to take the derivative of the binomial coefficient and found a blog post that was quite useful. However I am unclear as to how the first step bellow was simplified to the second step ...
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1answer
85 views

Dual Pairs, topology of weak convergence and weak* topology

Edit for Bounty: I decided to put a bounty on this question because I would really like to get it properly. Thus, I would like to get feedbacks on my basic questions, and a detailed answer on my ...
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1answer
38 views

limit supremum and infimum question

Question: Show that $\lim sup A_n -\lim inf A_n = \limsup(A_n A^c_{n+1}) =lim sup (A^c_n A_{n+1})$ the thing I understand from this queston is the following; $$\bigcap_{n=1}^\infty ...
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2answers
41 views

Topology proof: dense sets and no trivial intersection

I was wondering if this proof of this basic topological result concerning the closure works. Proposition: Let $A \subseteq (X,\tau)$. Then, $A$ is dense in $X$ if and only if every non-empty open ...
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1answer
73 views

Dual Spaces and Topological Vector Spaces

I have a question regarding dual spaces. Before, let me write that this all issue looks really problematic to me, and I already touched it quickly in another question. However, in that occasion, the ...
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0answers
48 views

Open sets in the topology of weak convergence

I do have various questions regarding the topic of probability measures on polish spaces in general, thus I am trying to divide them in “small” subquestions. Hence, this is my first question on this ...
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1answer
33 views

Proof that the discrete metric $d$ is complete in $\mathbb{N}$

This is an attempt of a proof of a rather basic result. Proposition: The discrete metric $d$ is complete in $\mathbb{N}$. Proof: Let $x_n$ be an arbitrary sequence in $\mathbb{N}$ endowed ...
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65 views

What are the prerequisites for learning abstract algebra?

Well, I want to learn abstract algebra. So, I get across this http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra. But I'm not sure whether I've understanding of prerequisites ...
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1answer
102 views

A Better Approximation of $e$

So, I'm trying to self-learn Analysis, and I don't have any solutions, so I hope you don't mind if I put my answer here for you guys to help me check it, as it seems I haven't solved it correctly. ...
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1answer
72 views

Exponential Tilting

Consider a random variable $Y$ with density function $f_Y(y)$ and moment generating function $m_Y(t)$ and cumulant generating function $\kappa_Y(t)$. Then a random variable $X$ derived from $Y$ by the ...
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1answer
18 views

Expansion of cumulant transform

Verify the following expansion for a cumulant generating function of a random variable $X$. \begin{align} \kappa(t) & = \mu t + \frac{1}{2}\sigma^2t^2+\frac{1}{6}\rho_3\sigma^3t^3 + ...
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2answers
31 views

How to take this exponentials

Given an expansion of a cumulant function as follows: $$ \kappa(t) = \frac{t^2}{2} + \frac{\rho_3 t^3}{6\sqrt{n}} + \frac{\rho_4t^4}{24n} +O\left(\frac{1}{n\sqrt{n}}\right), (*) $$ where ...
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1answer
29 views

Expected value for random walk

A point starts at the origin and can randomly go up, down, left, right (equally likely). The question asks to write the expression of the point's position in terms of $x_1$ -units up, $x_2$ -units ...
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1answer
28 views

For finite dimensional $F$-Vectorspaces $V$ it is true that $\forall U \subset V: U = \bigcap_{\lambda \in U^0}\ker ( \lambda)$

In E. Oeljeklaus & ‎R. Remmert Linear Algebra they proof this little lemma: Lemma: Let $V$ be a finite dimensional $F$-Vectorspace over a field $F$ and $U \subset V$, then $$U = ...
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Help with a Proof of exercise 7.3 Apostol on two different definitions of the Riemann integral.

This is a follow-up to this question. Here I ask to check my work and improve the final part that I feel is missing some important steps: So to prove what is asked for in the link above (I am not ...
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1answer
45 views

Inner Product of a Dual Space

I feel like this has to have been asked before, but my searches turned up nothing. I was tutoring a student today and they asked me what is a good intuition for an adjoint. I still am not sure I know ...
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2answers
30 views

Exponential Families defined by Radon-Nikodym Theorem

Let $X \in \mathbb R^d$ be a random vector on space $(\Omega, \mathcal F, \mathbb P)$ and its Laplace transform $\varphi(\theta) := \int e^{\theta\cdot X(\omega)}\mathbb P(d\omega)$ exists for a row ...