Top new questions this week:
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I have found an interesting family of infinite products. The most interesting one of them being:
$\sqrt{2}-1=\dfrac{1\cdot7\cdot9\cdot15\cdot17\cdot23\cdots}{3\cdot5\cdot11\cdot13\cdot19\cdot21\cdots}$...
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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 \...
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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 ...
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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 \...
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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 ...
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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 ...
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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$ ...
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Greatest hits from previous weeks:
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$\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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
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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 ...
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Can you answer these questions?
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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 ...
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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 ...
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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 ...
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