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Top new questions this week:

Interesting infinite product $\sqrt{2}-1=\dfrac{1\cdot7\cdot9\cdot15\cdot17\cdot23\cdots}{3\cdot5\cdot11\cdot13\cdot19\cdot21\cdots}$

I have found an interesting family of infinite products. The most interesting one of them being: $\sqrt{2}-1=\dfrac{1\cdot7\cdot9\cdot15\cdot17\cdot23\cdots}{3\cdot5\cdot11\cdot13\cdot19\cdot21\cdots}$...

sequences-and-series infinite-product  
user avatar asked by Lithium Score of 21
user avatar answered by Martin R Score of 4

A daunting double integral with a simple closed form

In the following, I'll present a curious double integral that despite its daunting look has a very nice closed form, $$\int _0^{\pi/2}\int _0^{\pi/2}\cot (x) \csc ^2(y) \log (\cos (y)) \log \left(1-2 \...

real-analysis calculus integration sequences-and-series definite-integrals  
user avatar asked by user97357329 Score of 14
user avatar answered by Zacky Score of 3

Strapping down a cylinder

I want to strap down a big heavy cylinder on a flatbed truck. The strap is attached to the truck bed as shown in the picture and also behind. Will the strap slip off as in the next picture? PS. This ...

user avatar asked by cdupont Score of 14
user avatar answered by Intelligenti pauca Score of 15

How does the common definition of a function satisfy the precise definition of a function?

Before my reading on Linear Algebra by Hoffman/Kunze, I was under the impression that a function was "a rule (or mathematical object) that maps/assigns each $x \in X$ (domain) to an element $y \...

functions definition  
user avatar asked by IDK Score of 12
user avatar answered by Ethan Bolker Score of 5

Is a function $\mathbb{R}^n \to \mathbb{R}$ which has a closed and connected graph necessarily continuous?

It has been proved here and here that a function $\mathbb{R} \to \mathbb{R}$ which has a closed and connected graph is continuous. This fact is also proved in a nice article by Burgess. I don't know ...

user avatar asked by Geoffrey Sangston Score of 11

Applications of Category Theory in Abstract Algebra

Almost every text on Category theory uses categories such as Ab, Grp, and so on as examples to work with but can category theoretic methods actually help us understand the structures better? In ...

abstract-algebra category-theory  
user avatar asked by Acharyachakit Score of 11
user avatar answered by Qiaochu Yuan Score of 11

$f(xy) = f(x+y)$ : A variation of an old problem

I have seen the following problem in various places, such as Toppr and Quora. Suppose that the function $f:\mathbb{N}\to\mathbb{N}$ satisfies $f(x+y)=f(xy)$ for all $x,y\in\mathbb{N}$. Show that $f$ ...

contest-math recreational-mathematics functional-equations  
user avatar asked by Timothy Chow Score of 10
user avatar answered by Mastrem Score of 2

Greatest hits from previous weeks:

Does $\pi$ contain all possible number combinations?

$\pi$ Pi Pi is an infinite, nonrepeating $($sic$)$ decimal - meaning that every possible number combination exists somewhere in pi. Converted into ASCII text, somewhere in that infinite string of ...

number-theory irrational-numbers pi  
user avatar asked by Chani Score of 774

How many even numbers of four digits can be formed with the digits 0,1,2,3,4,5 and 6 no digit being used more?

My attempt to solve this problem is: First digit cannot be zero, so the number of choices only $6 (1,2,3,4,5,6)$ The last digit can be pick from $0,2,4,6$, so the number of choices only 4 Second ...

user avatar asked by akusaja Score of 4
user avatar answered by akusaja Score of 3

Find the coordinates of a point on a circle

I have a circle like so $r$, with angle $\theta$ to the $y$-axis"> Given a rotation θ and a radius r, how do I find the coordinate (x,y)? Keep in mind, this rotation could be anywhere between 0 and ...

geometry trigonometry circles rotations  
user avatar asked by CoderTheTyler Score of 41
user avatar answered by André Nicolas Score of 48

Probability of winning a prize in a raffle

My work is having it's annual Christmas raffle today. 1600 tickets have been sold, and there are 40 prizes to win. I have bought ten tickets. What are the odds I will win a prize? While an initial ...

probability combinatorics binomial-coefficients recreational-mathematics problem-solving  
user avatar asked by Clarkey Score of 11
user avatar answered by Srivatsan Score of 22

What does the dot product of two vectors represent?

I know how to calculate the dot product of two vectors alright. However, it is not clear to me what, exactly, does the dot product represent. The product of two numbers, $2$ and $3$, we say that it ...

geometry vectors  
user avatar asked by Saturn Score of 137
user avatar answered by King Squirrel Score of 93

Probability of getting exactly 2 heads in 3 coins tossed with order not important?

I have been thinking of this problem for the post 3-4 hours, I have come up with this problem it is not a home work exercise Let's say I have 3 coins and I toss them, Here order is not important ...

user avatar asked by Max Score of 7
user avatar answered by Sammy Black Score of 10

If a club has 24 members, In how many ways can 4 officers be chosen from the members of the club?

I understand the concept of combinations and permutations. However, I am not getting how to apply the formulas. I believe understanding exactly how to do this would help.A club has 24 members. a. In ...

permutations combinations  
user avatar asked by XxTIBZxX Score of 4
user avatar answered by DeepSea Score of 7

Can you answer these questions?

Notation related question about a two variable functor fixing one of its variables.

The following question is taken from "Arrows, Structures and Functors the categorical imperative" by Arbib and Manes $\color{Green}{Background:}$ $\textbf{(1)}$ $\textbf{Definition:}$ A ...

category-theory notation functors  
user avatar asked by Seth Score of 1

Show that $a(\gcd (n,k))$ is generated from roots of generating function of $\sum _{h=0}^{\infty } \left(\sum _{k=1}^n x^{h n+k} a(\gcd (n,k))\right)$

Let $a(n)$ be the Dirichlet inverse of the Euler totient function: $$a(n) = \sum\limits_{d|n} d \cdot \mu(d) \tag{1}$$ And let the matrix $T$ be: $$T(n,k)=a(\gcd(n,k)) \tag{2}$$ Compute the ordinary ...

number-theory polynomials roots generating-functions gcd-and-lcm  
user avatar asked by Mats Granvik Score of 1

Bound the sum of the prime factors for a special case

For an integer $n = \prod_{i=1}^k p_i$ where all $p_i$'s are distinct primes and $k \ge 2$, can we give an upper bound of the sum of these prime factors, namely $\sum_{i=1}^k p_i$ ? (or some ...

elementary-number-theory prime-numbers combinatorial-number-theory  
user avatar asked by hx1a Score of 1
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