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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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
43 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 ...
2
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1answer
88 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 ...
2
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0answers
62 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
36 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
81 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. ...
2
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1answer
156 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
23 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
32 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
32 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 ...
2
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1answer
30 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
56 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
50 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
39 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
95 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
29 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
35 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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46 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
48 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
2
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1answer
37 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 ...
4
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2answers
88 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
43 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 ...
0
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1answer
31 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
127 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
50 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
73 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
43 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 ...
3
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2answers
68 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
59 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 ...
0
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1answer
127 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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0answers
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 ...
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1answer
95 views

A little help for a 14 year old.

I am 14 years old and i am really into maths. I can say that i am really good at it(at what we do at school) and i really want to make a step ahead and start exploring a more complicated sector in ...
2
votes
1answer
43 views

Arc length of curve (regular condition)

I have a question regarding the defintion of arc length of a curve in $\mathbb{R}^n$. If $\gamma$ is a regular curve, the define arc length as $S(t)=\int_{t_0}^t|\gamma'(t)|dt$. Since $\gamma$ is ...
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1answer
35 views

Trying to prove a sequence of functions is increasing.

Put $\delta_n = 2^{-n}$.To each positive integer $n$ and each real number $t$ corresponds a unique integer $k = k_n(t)$ that satisfies $k\delta_n \le t < (k+1)\delta_n$. Define $$\psi_n(t)= ...
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1answer
38 views

If $f$ is differentiable and $f'\geq m\geq0$, $|\int_a^b\cos{f(x)}dx|\leq2/m$

Suppose $f:[a,b]\to\mathbb R$ is a differentiable function such that its derivative is monotonically decreasing and $f'(x)\geq m>0$ for all $x\in[a,b]$. Prove that $$|\int_a^b\cos ...
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38 views

Corollary of theorem 1.14 Rudin real and complex, supremum of a sequence of measurable functions.

Theorem 1.14 states: If $f_n:X\rightarrow [-\infty, \infty]$ is measurable, for $n = 1,2,3, ...,$ and $$g = \sup_{n \ge 1} f_n, \ h = \lim_{n \rightarrow \infty} \sup f_n$$ then g and h are ...
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41 views

How to Solve this Implicit Equation

Let $Y$ to be a uniformly distributed random variable. Consider function $z(\gamma)$ defined by the following equation. $$ \int_{\left\{y\in(0,1), -\frac{\log ...
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1answer
18 views

Transformation of Extreme Value Distribution

Let $X$ be a random variable following distribution function (i.e., generalized Pareto distribution) $$ F_{\gamma, \sigma}(x) = 1-\left( 1+\frac{\gamma x}{\sigma} \right)^{-\frac{1}{\gamma}}, $$ ...
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2answers
44 views

Does $\int_0^1 \frac{x \ln x}{1+ x^2}dx$ converge?

$\int_0^1 \frac{x \ln x}{1+ x^2}dx$ converges? Kinda stuck doing this problem. I just need a hint on what to start with. I know that it is an improper integral and I have to use limits but I need to ...
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1answer
21 views

Borel Sigma-Algebra on $\mathbb{R}$

Show that the Borel sigma-algebra on $\mathbb{R}$, denoted $B_R$ is generated the open intervals in $\mathbb{R}$. My attempt: Let $I$ be the collection of all open intervals, let $\sigma I$ be the ...
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0answers
101 views

Every Riemann Integrable Function can be approximated by a Continuous Function

Prove that given any Riemann Integrable function $f$ on $[a,b]$, and given any $\varepsilon>0$ one can find a continuous function $g$ on $[a,b]$ such that $$\int_a^b|f(x)-g(x)|dx<\varepsilon ...
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1answer
258 views

Most efficient way to learn mathematics

So there's a lot of advice on how to learn mathematics most efficiently, and it mostly revolves around doing problems, asking questions, and considering all possible generalizations. However, I was ...