Questions about the process of studying mathematics without formal instruction.

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2
votes
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 ...
3
votes
0answers
70 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 ...
1
vote
1answer
23 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) := ...
1
vote
1answer
27 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) = ...
4
votes
1answer
98 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 ...
0
votes
0answers
40 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 ...
1
vote
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 ...
0
votes
2answers
21 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 ...
0
votes
1answer
34 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 ...
0
votes
1answer
22 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
votes
1answer
33 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
votes
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.
3
votes
2answers
47 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 ...
0
votes
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 ...
1
vote
1answer
37 views

How to solve this implicit equation involving integral

Consider the following equation with respect to $\alpha$ defined through a probability density function $f(x) = \exp[x-\exp(x)]$. $$ \int_{-\infty}^\infty (x-2) e^{\alpha(x-2)}e^{x-e^x}dx = 0. $$ ...
2
votes
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
votes
1answer
26 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 ...
3
votes
1answer
112 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. ...
1
vote
1answer
45 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 ...
0
votes
2answers
50 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 ...
1
vote
2answers
40 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
votes
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 ...
0
votes
2answers
51 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
votes
1answer
115 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 ...
1
vote
0answers
25 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 ...
1
vote
1answer
94 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
35 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 ...
1
vote
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)= ...
0
votes
0answers
25 views

Are Random variables in delay embedded phase space independent?

Consider a smooth manifold $M=R^d$ embedded in a higher dimensional space $R^D$ using Takens Attractor reconstruction. Let, the Random Variable $X \in R^d$ have a Gaussian pdf and the random variable ...
1
vote
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 ...
0
votes
2answers
33 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 ...
0
votes
0answers
40 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 ...
2
votes
1answer
16 views

Transformation of Extreme Value Distribution

Let $X$ be a random variable following distribution function (i.e., generalized Pareto distribution) $$ F_{\gamma, \sigma}(x) = 1-\left( 1+\frac{\gamma x}{\sigma} \right)^{-\frac{1}{\gamma}}, $$ ...
2
votes
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 ...
1
vote
1answer
18 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 ...
2
votes
0answers
73 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 ...
6
votes
0answers
162 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 ...
2
votes
0answers
114 views

Geometric meaning of $\nabla_{[i}(x^i \nabla_{j]}x^j)$ and $(\nabla_{[i}x^i )\nabla_{j]}x^j$

While teaching myself tensor calculus I have come up with [this] http://mathematica.stackexchange.com/a/71613/12306 {The proof of the 2-D hairy ball theorem). When trying to generalize this proof ...
0
votes
2answers
46 views

Awodey's first UMP example

I am reading Awodey's "Category Theory" by myself and got stuck in a simple passage. He writes: If $g:A^\ast\rightarrow N$ satisfies $g(a)=f(a)$ for all $a\in A$ then, for all ...
0
votes
0answers
51 views

Understanding a measure theory statement from Wikipedia

I was reading the Measure (mathematics) page on wikipedia: http://en.wikipedia.org/wiki/Measure_%28mathematics%29 I was confused with one of their sections at the end, "Additivity". They were ...
6
votes
1answer
120 views

Help with diagram chasing

Given the diagram $\require{AMScd}$ \begin{CD} 0 @>>> A @>f>> B @>g>> C @>>> 0 \\ @. @V\alpha VV \#@V\beta V V\# @VV\gamma V @. \\ 0 @>>> {A'} ...
0
votes
1answer
19 views

Find $\sup$ and $\inf$ of $A$ and justify

$$A=\left \{x\in \Bbb R :x<\dfrac{2}{x} \right \}.$$ a) $\sup A = -1$ since $\max A=-1;$ b) $\inf A$ does not exist since $A$ is not bounded below. Is this the only justifications? Can anyone ...
1
vote
0answers
23 views

Homework : Anti log expression

I have this expression $x(r) = y(a)r^a$ where $r$ is a random variable and I want to express the expression in terms of $r$. The objective is to substitute the variable $r$ into the pdf of $r$, ...
2
votes
0answers
20 views

Integration of unknown derivative

I am unable to solve this integral, have forgotten basics and so need help. Shall be very thankful If a way out is provided: $\int_0^R \ln[p'(t)]dN(t) - \int_0^R p'(t) dt$ If $p(t)$ was known then I ...
1
vote
1answer
28 views

Points defining plane - starting step? [closed]

If the points $P, Q, R$, not all lying on the same straight line, have position vectors $a, b, c$ respectively, show that $(a \times b) + (b \times c) + (c \times a)$ is a vector perpendicular to the ...
1
vote
1answer
28 views

An Example of Taylor Series

Consider a positive function $f(x)$ and suppose that we would like to approximate its value around some point $x_0$. One way to do so is to use two-term Taylor series expansion as follows. $$ f(x) ...
2
votes
1answer
72 views

What are the steps to this functional derivative problem?

I'm trying to derive equations from Matthew Beal's Thesis, Variational Algorithms for Approximate Bayesian Inference pg.55, but I'm stuck on one of the equations (well I'm stuck on a lot of equations ...
0
votes
1answer
33 views

Find the extremals of a functional of the form $\int^{x_1}_{x_0}F(y',z')dx$

I was working on Problem 3 in Ch. 2 of Gelfand & Fomin's Calculus of Variations, which reads: Find the extremals of a functional of the form $$\int^{x_1}_{x_0}F(y',z')dx$$ given that ...
0
votes
0answers
33 views

Proving Frobenius Theorem for Eigen Values

In my mulitivariable calculus class to justify second derivative test my professor used a theorem he called the frobenius theorem. But when I searched on wiki all I could find was Perron Frobenius ...
0
votes
0answers
35 views

Calculating Power of a Paired T Test

$ 239$ subjects had their cholesterol measured, and then were put on high-fiber diets. After a month on the high-fiber diet, the cholesterol was measured again. The mean LDL cholesterol level before ...