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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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. ...
1
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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 ...
0
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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 ...
3
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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 ...
0
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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 ...
0
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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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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 ...
1
vote
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
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1answer
41 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
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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)= ...
0
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0answers
28 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
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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 ...
0
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2answers
35 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
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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
17 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
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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
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0answers
84 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
192 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
116 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
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2answers
47 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 ...
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0answers
56 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
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1answer
122 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
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1answer
20 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 ...
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0answers
25 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
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0answers
21 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
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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
73 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
36 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
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0answers
38 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
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0answers
36 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 ...
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2answers
89 views

Strange behaviors of finitely additive probabilities

Watching a lecture on youtube I heard the lecturer stating that in general finitely additive probabilities behaves strangely. For example, it is possible that every open interval around a point $x$ ...
0
votes
1answer
25 views

How to determine a function whose minima falls on a specified curve?

I have a family of curves given by $g(x,y)=C_0 yx^{-n}$. How can I determine the function $f(x,y)$ for the family of curves that satisfies the condition that the local minima $\frac{\partial ...
2
votes
1answer
47 views

Learning from Alternative Sources

I have a very general question about people's experiences with learning math. I can think of a couple of times where I had the following situation. I was seeking to learning about topic A. However, ...
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0answers
14 views

Showing there is a cartesian coordinate system on EG.

I'm pretty sure I should just show there is a bijection between the points in EG and elements of R^2. How do I do this? note: EG=Euclidean Geometry
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1answer
22 views

Independence of two multivariate normals.

Suppose we have two multivariate normals $X_1 \sim N(u_1, \Sigma_{11}\Sigma_{22}$) and $X_2 \sim N(u_2, \Sigma_{21} \Sigma_{22})$ . Why are $X_2 $ and $X_1-\Sigma_{12} \Sigma_{22}^{-1}X_2$ ...
0
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0answers
57 views

What are some interesting, atypical mathematical topics that a student who has taken an introductory calculus sequence can learn about?

I understand that usually the next step after $3$ semesters of calculus and $1$ semester of ordinary differential equations (plus one semester of linear algebra, for some) is something like an ...
0
votes
1answer
49 views

Conditional expectation of $X$ given $Z$, where $Z = 1$ if $X > Y$ and $-1$, otherwise

Let $X\sim\operatorname{Exp}(1)$ and $Y\sim\operatorname{Exp}(2)$ be independent random variables. Define $Z$ by $$ Z = \begin{cases} 1,& X>Y\\ -1,& X\leqslant Y. \end{cases} $$ I want to ...
0
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0answers
22 views

To sketch a “typical” plot of a specific time series model

Let X have a distribution with mean $\mu$ and variance $\sigma^2$, and let $Y_t = X$ for all t. Sketch a “typical” time plot of $Y_t$. My thoughts: This process $Y_t$ is stationary with mean $\mu$, ...
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0answers
15 views

On Conditional distribution of the multivariate normal.

Following the answer to this question. Where we are talking about a multivariate normal than has mean and covariance matrix that can be decomposed as: $\boldsymbol\mu = \begin{bmatrix} ...
4
votes
3answers
184 views

Outline for high school combinatorics class?

I am a high school student and I have taken all the math classes that my school provides (through calculus AB). I have been looking at a possible independent study for next year and I have landed on ...
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votes
1answer
70 views

sum of two dependent random variables

Let $X$ be a cotinuous random variable uniformly distributed over $[-10,10]$. Let $Y$ be a random variable with pdf $f_Y(y) = \frac{1}{40}\ln \frac{20}{|y|}, -20 \leq y \leq 20$. $X$ and $Y$ ARE NOT ...
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0answers
14 views

transformation and functions of random variables

Let $X,Y$ be independent random variables. I already have the distribution of $XY$ over a certain subinterval of $\mathbb{R}$, by convolution. My question is, is it possible to get the distribution of ...
2
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1answer
42 views

The inverse of the sum of two matrices in *Applied statistical decision theory *.

I am following Applied statistical decision theory [by] Raiffa, Howard. Which can be consulted online here. A theorem at the page linked states that if two matrices $A,B$ are non-singular and of ...
3
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2answers
58 views

Logic behind a proof in Topological Vector Spaces

I found the following result at the beginning of some notes on topological vector spaces (TVS). This is a quite fundamental result, that apparently is considered the corresponding version of the ...
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0answers
28 views

A math proof within a question about homogeneous Poisson process

We know that a homogeneous Poisson process is a process with a constant intensity $\lambda$. That is, for any time interval $[t, t+\Delta t]$, $P\left \{ k \;\text{events in}\; [t, t+\Delta t] \right ...
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0answers
35 views

sum/product combination of random variables

Let $X$ and $Y$ be independent random variables. If I am asked about the distribution of random variable $XY+Y$, is it ok if I compute $XY$ first and then add the result to $Y$ (via convolution, or ...
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0answers
32 views

Proof about a homogeneous Poisson process

We know that a homogeneous Poisson process is a process with a constant intensity $\lambda$. That is, for any time interval $[t, t+\Delta t]$, $P\left \{ k \;\text{events in}\; [t, t+\Delta t] \right ...