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 the fact that you're self-studying is what your question is _about_.

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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
98 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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93 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
169 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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31 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 $\mathbf{X}(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\...
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1answer
103 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 ...
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1answer
61 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
43 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
40 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 f(x)dx|\leq\dfrac{...
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2answers
66 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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1answer
34 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
49 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
51 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
426 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
709 views

Most efficient way to learn mathematics [closed]

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 ...
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0answers
134 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 proof of the 2-D hairy ball theorem. When trying to generalize this proof to higher dimensions I get the term $$\nabla_{[i}(x^i \nabla_{j]...
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2answers
74 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 $a_1,\ldots,a_i\...
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0answers
66 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 ...
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1answer
185 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'} @>>{...
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1answer
29 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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40 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 $r$, $p(r)$....
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0answers
29 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 ...
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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 ...
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1answer
34 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
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1answer
78 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 ...
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1answer
56 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 $F_{y'y'}F_{...
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0answers
63 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 ...
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0answers
58 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
99 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 $x$ ...
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1answer
28 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 f}{\...
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1answer
77 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
21 views

Showing there is a cartesian coordinate system in Euclidean geometry.

I'm pretty sure I should just show there is a bijection between the points in Euclidean Geometry and elements of $\mathbb{R^2}$. How do I do this?
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1answer
24 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 $X_1-\Sigma_{12} \Sigma_{22}^{-1}X_2$ ...
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0answers
81 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 ...
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1answer
57 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 ...
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4answers
682 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 ...
0
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1answer
270 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
21 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
60 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
69 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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1answer
55 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
74 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
41 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 \...
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1answer
72 views

Prove $\frac{d}{dx}{\rm arctanh}(\ln \cosh x) = \frac{\tanh x}{1-(\ln \cosh x)^2}$

In the book "Lehrbuch der Analysis Teil I" of Heuser page 303, there was a task: Prove $$\frac{d}{dx}{\rm arctanh}(\ln \cosh x) = \frac{\tanh x}{1-(\ln \cosh x)^2}.$$ When I tried, I ended up with $$\...
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1answer
73 views

How was the explicit closed form for this implicit function derived?

The problem comes from reading this [0] paper but I think I can express it in a self contained question. Consider the implicit function $H(z)$ defined by the relation: $$F_z(z+H(z))-F_z(z-H(z))=0....
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1answer
34 views

Where can I find simple integration problems (and other computational exercises) involving special functions?

Working lots of computational exercises in my pre-calculus and calculus classes has given me a great deal of intuition in dealing with elementary functions. Thanks to these years of practice, I can ...
2
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1answer
125 views

Is this function Riemann integrable in $[0,1]$?

The function is $f(x) = 1$ for $ 0 \le x \lt 1 $ and $f(x) = 2$ for $x = 1$ I calculate the upper sum $$U(P,f) = \sum_{i=1}^n M_i \Delta x_i = \sum_{i=1}^{n-1} 1\,\Delta x_i + 2 \,\Delta x_n = 1(x_{...
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2answers
556 views

Weak topologies and weak convergence - Looking for feedbacks

I am currently trying to get exactly what the weak and the weak* topologies are, in particular in connection to the concept of weak convergence in measure, however I am not completely sure on what I ...
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2answers
319 views

Convexity of mutual information $I(X;Y)$ in conditional $p(y \mid x)$

I'm trying to understand the proof that $I(X;Y)$ is convex in conditional distribution $p(y \mid x)$ - from Elements of Information Theory by Cover & Thomas, theorem 2.7.4. In the proof we fix $p(...
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2answers
454 views

How to brush up on calculus?

It's been years since I took calculus, and while I have a good understanding of the theorems of single variable calculus from my real analysis courses, computationally I am a bit slow. It takes me ...