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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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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13 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
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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0answers
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 ...
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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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54 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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63 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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20 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
30 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
92 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
61 views

limit supremum and infimum question

Question: Show that $\limsup A_n -\liminf A_n = \limsup(A_n A^c_{n+1}) =\limsup (A^c_n A_{n+1})$ the thing I understand from this queston is the following; $$\bigcap_{n=1}^\infty ...
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2answers
42 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
79 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
56 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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1answer
69 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
105 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
111 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
20 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
29 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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0answers
50 views

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
47 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
34 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 ...
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0answers
82 views

gallian's contemporary abstract Algebra exercises

i have just finished doing A first course in abstract algebra by John Fraleigh (group theory only) and now i am looking forward to apply all the stuff i learnt by doing problems. i have on my mind ...
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1answer
27 views

Borel Sets and Borel Functions

I read the following argument in Billingsley (1995). Consider a function $F(x)$ and define four associated quantities at $x$: $$ D^F(x) := \limsup_{h\downarrow 0} \frac{F(x+h)-F(x)}{h}; \\ D_F(x) := ...
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1answer
31 views

An application of the Dominated Convergence Theorem

I read the following argument. Consider an integrable function $f$ satisfying that: $$ F(x) - F(a) = \int_a^x f(t) dt, $$ for $a \leq x \leq b$ and some function $F$. Then $$ F(x+h) - F(x) = ...
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1answer
102 views

Making progress in learning hard math?

I'm not a mathematician, but a computer scientist, but the following problem has puzzled me and I'm wondering how people counter it. In the last 5-6 years, I've probably read the first 5 sections of ...
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42 views

On question 7.3 from Apostol regarding two definitions of Riemann integral.

The question states: Let $f(x) = \alpha(x) = 0$ for $a \le x < c$, $f(x) = \alpha(x) = 1$ for $c < x \le b, f(c) = 0, \alpha(c) = 1$. Show that $\int^a_bf $ exists according to definition 1 but ...
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0answers
32 views

Showing Two Random Variables are Indendent Based on their PDF

Let $X_1,...X_n$ be a random sample from population with pdf $F(x|\theta)=\alpha \theta^{-\alpha}x^{\alpha -1}$ where $ 0 < x < \theta$ Show that $\frac{X_k}{X_{k+1}}$ and ...
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2answers
23 views

If $\varphi: V \to W$ is a $F$-linear mapping, then for every $U \subset V$ it is true that $\dim_F(\varphi(U)) \leq \dim_F(U)$

Problem: Let $V,W$ be finite dimensional $F$-Vectorspaces where $F$ denotes a Field. Let $\varphi: V \to W$ be a $F$-linear mapping. Show that for every $U \subset V$ $$\dim_F(\varphi(U)) \leq ...
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1answer
41 views

Box-Muller Independence Proof by Change of Variables (Help finding the Inverses)

Let $X_1=\cos(2 \pi U_1)\sqrt{-2 \log(U_2)}$ and $X_2=\sin(2 \pi U_1)\sqrt{-2 \log(U_2)}$ wher $U_1$ and $U_2$ are iid uniform (0,1). Prove that $X_1$ and $X_2$ are independent N(0,1) random ...
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1answer
23 views

Help in derivative with summation

I have forgotten how to handle the derivative $\frac{\partial}{\partial x}[(n-1) \sum_{y_i} \log x({y_i})]$ where $x$ is a function of a vector $\mathbf{y_i}$. How do I evaluate this? Thank you
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1answer
34 views

Open neighbourhoods, Polish spaces, and basis for the Baire space

This is a follow-up of a question I asked yesterday answered by GEdgar. I think I see now GEdgar’s answer, but I am not sure about an issue related to it. Thus, I will write my general understanding ...
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2answers
86 views

Prove $\lim\limits_{n \to \infty} \sup \left ( \frac{(2n - 1)^{2n - 1}}{2^{2n} (2n)!)} \right ) ^ {\frac 1 n} = \frac {e^2} 4$

This is a problem in Heuer (2009) "Lerbuch der Analysis Teil 1" on page 366. I assume that the proof should use $e = \sum\limits_{k = 0}^{\infty} \frac 1 {k!}$, but I cannot come further.
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2answers
48 views

Open neighbourhoods & polish spaces - typo in Marker's notes?

A very (very!) easy question that merges together the very basic concepts of two fields that I find always problematic for my understanding, namely topology and descriptive set theory. Everything ...
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1answer
31 views

Suggestion or help to choose a book for theory of ODE

I have recently have a course in theory of ordinary differential equations , I have learned main ideas and now I want to study theory of ode from a book for filling gaps. I have choosed three book byt ...
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1answer
41 views

How to solve this implicit equation involving integral

Consider the following equation with respect to $\alpha$ defined through a probability density function $f(x) = \exp[x-\exp(x)]$. $$ \int_{-\infty}^\infty (x-2) e^{\alpha(x-2)}e^{x-e^x}dx = 0. $$ ...
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4answers
58 views

On the intuition behind a conditional probability problem.

This is a very similar question to this one. But notice the subtle difference that the event that I define $B$ is that I am dealt at least an ace. Suppose I get dealt 2 random cards from a standard ...
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1answer
30 views

Saddle Point by the Cauchy-Riemann Equations

Consider the following complex integral $$ \int_{\mathcal P} e^{v\cdot w(z)} \xi(z) dz, $$ where $v$ is large and positive and the integration path $\mathcal P$ satisfies the following two ...
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1answer
121 views

Guided study of Lie theory

I'm willing to become a mathematical physicist and, as such, it is mandatory that I become better acquainted with some essential mathematical theories, Lie groups and algebras being one of them. ...
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1answer
46 views

Initial segments of trees

I am fairly sure this question will look quite trivial, but I have some trouble getting the reason behind the use of the symbole $\subset$ when we deal with sequences and initial segments of ...
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2answers
58 views

Continuous, bijective - yet not a homeomorphism

I'm going through the earlier chapters in books and making sure I can do everything (and addressed many short-comings, like compactness) but I've come across something I can't do. In "Introduction to ...
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2answers
42 views

Which rational functions $\mathbb{P}^1\rightarrow k$ are regular at the point at infinity?

I am trying to solve a problem in § 1.7 of Shafarevich's "Basic Algebraic Geometry 1": "Let $k$ be an algebraically closed field. Which rational functions $\mathbb{P}^1\rightarrow k$ are regular at ...
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2answers
64 views

What happens if we consider an algebra instead of $\sigma$-algebra in probability theory?

I understand the difference between algebra (of sets) and $\sigma$-algebra. But which are the implications if we use algebra instead of $\sigma$-algebra in probability theory? If it exists, could you ...
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2answers
56 views

What mathematics should I study to understand Neural Nets / Machine Learning?

I am strongly fascinated by neural nets, and perhaps other forms of machine learning. There are so many (potential) applications: teaching a robot with shaft encoders to drive along different ...
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1answer
121 views

How to re-learn math: books or websites?

To re-learn math, both websites and books provide visual content (text and some of them shows illustrations). So are websites an alternative to books (content quality-wise)? My goal is to re-learn ...
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26 views

Projection matrix?

If $\mathbf{X}_{n \times K}= \begin{bmatrix} \mathbf{x}'_1 \\ \cdots \\ \mathbf{x}'_n \end{bmatrix}$, then can we say that ...