Mathematics

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1500 questions
70
votes
4 answers

How to know if a point is inside a circle?

Having a circle with the centre $(x_c, y_c)$ with the radius $r$ how to know whether a point $(x_p, y_p)$ is inside the circle?
Ivan
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70
votes
6 answers

$100$-th derivative of the function $f(x)=e^{x}\cos(x)$

I've got this task I'm not able to solve. So, I need to find the $100^\text{th}$ derivative of $$f(x)=e^{x}\cos(x)$$ where $x=\pi$ . I've tried using Leibniz's formula but it got me nowhere, induction doesn't seem to help either, so if you could…
windircurse
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70
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6 answers

Am I too young to learn more advanced math and get a teacher?

I am still 15 years old, but I am very interested in pure math. I have been teaching myself though books, from the internet and from others for the past year or so. I haven't mastered all the topics that are covered in university, just the ones that…
Argon
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70
votes
6 answers

Is it possible to formulate category theory without set theory?

I have never understood why set theory has so many detractors, or what is gained by avoiding its use. It is well known that the naive concept of a set as a collection of objects leads to logical paradoxes (when dealing with infinite sets) that can…
Matt Calhoun
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70
votes
11 answers

Why is it not true that $\int_0^{\pi} \sin(x)\; dx = 0$?

I know the following is not right, but what is the problem. So we want to calculate $$ \int_0^{\pi} \sin(x) \; dx $$ If one does a substitution $u = \sin(x)$ , then one gets $$ \int_{\sin(0) = 0}^{\sin(\pi) = 0} \text{something}\; du = 0. $$ We know…
John Doe
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70
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4 answers

Topological spaces admitting an averaging function

Let $M$ be a topological space. Define an averaging function as a continuous map $f:M \times M \to M$ which satisfies $f(a,b) = f(b,a)$ for all $a,b \in M$ and $f(a,a) = a$ for all $a \in M$. These seem like reasonable properties for a function…
Steven Gubkin
  • 10,168
70
votes
20 answers

What are some applications of elementary linear algebra outside of math?

I'm TAing linear algebra next quarter, and it strikes me that I only know one example of an application I can present to my students. I'm looking for applications of elementary linear algebra outside of mathematics that I might talk about in…
user98602
70
votes
1 answer

Semi-direct v.s. Direct products

What is the difference between a direct product and a semi-direct product in group theory? Based on what I can find, difference seems only to be the nature of the groups involved, where a direct product can involve any two groups and the semi-direct…
retro
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70
votes
8 answers

How can a set contain itself?

In Russell's famous paradox ("Does the set of all sets which do not contain themselves contain itself?") he obviously makes the assumption that a set can contain itself. I do not understand how this should be possible and therefore my answer to…
jimmyorpheus
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69
votes
4 answers

Why we consider log likelihood instead of Likelihood in Gaussian Distribution

I am reading Gaussian Distribution from a machine learning book. It states that - We shall determine values for the unknown parameters $\mu$ and $\sigma^2$ in the Gaussian by maximizing the likelihood function. In practice, it is more convenient to…
Kaidul Islam
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69
votes
4 answers

Taylor series of $\ln(1+x)$?

Compute the taylor series of $\ln(1+x)$ I've first computed derivatives (up to the $4$ th) of $\ln(1+x)$ $f^{'}(x)$ = $\frac{1}{1+x}$ $f^{''}(x) = \frac{-1}{(1+x)^2}$ $f^{'''}(x) = \frac{2}{(1+x)^3}$ $f^{''''}(x) = \frac{-6}{(1+x)^4}$ Therefore the…
Minu
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69
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2 answers

Evaluating $\int_{0}^{1}\cdots\int_{0}^{1}\bigl\{\frac{1}{x_{1}\cdots x_{n}}\bigr\}^{2}\:\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$

Here is my source of inspiration for this question. I suggest to evaluate the following new one . $$ I_{n}:= \int_0^1 \! \cdots \! \int_0^1 \left\{\frac{1}{x_1x_2 \cdots x_n}\right\}^{2} \:\mathrm{d}x_1\,\mathrm{d}\,x_2 \cdots \mathrm{d}x_n $$ where…
Olivier Oloa
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69
votes
8 answers

Why do people lose in chess?

Zermelo's Theorem , when applied to chess, states: "either white can force a win, or black can force a win, or both sides can force at least a draw [1]" I do not get this. How can it be proven? And why do people lose in chess then, if they can force…
Brika
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69
votes
3 answers

Definitions of Hessian in Riemannian Geometry

I am wondering if there is any quick way to see the following two definitions of Hessian coincide with each other without using local coordinates. $\operatorname{Hess}(f)(X,Y)= \langle \nabla_X \operatorname{grad}f,Y \ \rangle$ ; and…
user17150
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69
votes
4 answers

Open maps which are not continuous

What is an example of an open map $(0,1) \to \mathbb{R}$ which is not continuous? Is it even possible for one to exist? What about in higher dimensions? The simplest example I've been able to think of is the map $e^{1/z}$ from $\mathbb{C}$ to…
math student
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