Mathematics

Most Popular

more tags
1500 questions
89
votes
3 answers

Why did mathematicians introduce the concept of uniform continuity?

I have solved many problems regarding uniform continuity, but still I can't understand the following: Is there any practical application of this concept, or it is just a theoretical concept? Is there any wide application of this concept in any…
ramanujan
  • 1,624
89
votes
5 answers

Product of two Gaussian PDFs is a Gaussian PDF, but Product of two Gaussian Variables is not Gaussian

The Product of Two Gaussian Random Variables is not Gaussian distributed: Is the product of two Gaussian random variables also a Gaussian? Also Wolfram Mathworld So this is saying $X \sim N(\mu_1, \sigma_1^2)$ , $Y \sim N(\mu_2, \sigma_2^2)$ then…
Frames Catherine White
  • 1,720
89
votes
4 answers

A strange integral: $\int_{-\infty}^{+\infty} {dx \over 1 + \left(x + \tan x\right)^2} = \pi.$

While browsing on Integral and Series , I found a strange integral posted by @ Sangchul Lee . His post doesn't have a response for more than a month, so I decide to post it here. I hope he doesn't mind because the integral looks very interesting to…
Venus
  • 10,214
88
votes
6 answers

Proof of $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}$

I've just started to learn about the tensor product and I want to show: $$(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}.$$ Can you tell me if my proof is right: $\mathbb{Z}/m\mathbb{Z}$…
Rudy the Reindeer
  • 48,632
88
votes
2 answers

Thurston's 37th way of thinking about the derivative

In Thurston's superb essay On proof and progress in mathematics , he makes this observation: Of course there is always another subtlety to be gleaned, but I would like to at least think that I have absorbed the main intuition behind each element of…
Zev Chonoles
  • 133,510
88
votes
5 answers

What's the intuition with partitions of unity?

I've been studying Spivak's Calculus on Manifolds and I'm really not getting what's behind partitions of unity. Spivak introduces the topic with the following theorem: Let $A\subset \Bbb R^n$ and let $\mathcal{O}$ be an open cover of $A$ . Then…
Gold
  • 28,441
88
votes
1 answer

Does $X\times S^1\cong Y\times S^1$ imply that $X\times\mathbb R\cong Y\times\mathbb R$?

This question came up in a recent video series of lectures by Mike Freedman available through Max Planck Institut's website. He proves the "difficult" converse direction, that $X\times \mathbb R\cong Y\times \mathbb R$ implies $X\times S^1\cong…
Cheerful Parsnip
  • 28,517
88
votes
5 answers

Calculating the volume of a restaurant take-away box that is circular on the bottom and square on the top

Having a bit of a problem calculating the volume of a take-away box: I originally wanted to use integration to measure it by rotating around the x-axiz, but realised that when folded the top becomes a square, and the whole thing becomes rather…
Nemui
  • 867
  • 6
  • 7
88
votes
9 answers

How to find the inverse modulo $m$?

For example: $$7x \equiv 1 \pmod{31} $$ In this example, the modular inverse of $7$ with respect to $31$ is $9$ . How can we find out that $9$ ? What are the steps that I need to do? Update If I have a general modulo equation: $$5x + 1 \equiv 2…
roxrook
  • 12,457
88
votes
21 answers

List of interesting integrals for early calculus students

I am teaching Calc 1 right now and I want to give my students more interesting examples of integrals. By interesting, I mean ones that are challenging, not as straightforward (though not extremely challenging like Putnam problems or anything). For…
GeoffDS
  • 11,430
88
votes
11 answers

Expected number of unpecked chicks - NYT article

In this article , the winner of the math competition answered this question correctly: In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately to its left or right. What is the expected number of…
AAC
  • 1,085
88
votes
7 answers

Understanding Borel sets

I'm studying Probability theory, but I can't fully understand what are Borel sets. In my understanding, an example would be if we have a line segment [0, 1], then a Borel set on this interval is a set of all intervals in [0, 1]. Am I wrong? I just…
Tikhon Belousko
  • 1,261
  • 2
  • 11
  • 11
88
votes
6 answers

Alice and Bob play the determinant game

Alice and Bob play the following game with an $n \times n$ matrix, where $n$ is odd. Alice fills in one of the entries of the matrix with a real number, then Bob, then Alice and so forth until the entire matrix is filled. At the end, the determinant…
pad
  • 3,085
88
votes
9 answers

In Linear Algebra, what is a vector?

I understand that a vector space is a collection of vectors that can be added and scalar multiplied and satisfies the 8 axioms, however, I do not know what a vector is. I know in physics a vector is a geometric object that has a magnitude and a…
Paul Lee
  • 1,035
88
votes
1 answer

Direct proof that $\pi$ is not constructible

Is there a direct proof that $\pi$ is not constructible, that is, that squaring the circle cannot be done by rule and compass? Of course, $\pi$ is not constructible because it is transcendental and so is not a root of any polynomial with rational…
lhf
  • 223,222
1 2 3
…
99 100