Help with setting up coefficient of a parametrization of an epicycloid given a predefined arc length.

I am trying to determine the coefficient q in the parametrization of a epicycloid which gives me the arc length of 4.25. The parametrization can be glimpsed in my attempt of a solution in the following Matlab code.

t = 0:0.01:2*pi;

fun = @(t,q) sqrt((c.^2).*(r.^2).sin(t).^2+(c.^2).(q.^2).*(r.^2).*sin(c.t).^2+(c.^2).(r.^2).cos(t).^2+(c.^2).(q.^2).*(r.^2).*cos(c.*t).^2+1);

fun2 = @(q) integral(@(t) fun(t,q),0,2*pi)

qsolve=fsolve(@(q) fun2(q)-4.25, 0)

The problem is that solve can not find any solution. I am very much grateful if someone can help me with this one.


Effect of remaining term length

Is going against prior art that conflicts with my patent or product with 10 years remaining or 10 days remaining the same?


If in my invention or product uses an invention that was previously patented (prior art), does it make a difference how long is remaining in term for this prior art?

Would it be harder/easier to obtain a license or fight against infringement?

Longest substrings of common length with the same parity

Given two sequences $a$ and $b$, find largest $x$ such that in $a$ there is substring $A$ and substring $B$ in $b$ meeting those conditions:

  1. length of both $A$ and $B$ is equal to $x$;
  2. sum of elements in $A$ has the same parity as sum of elements in $B$.

Lengths of $a$ and $b$ are up to $5times10^5$, so simple $O(n^2)$ solution won’t do.

$a = [0, 1, 2, 3, 4, 5]$
$b = [3, 1, 3, 6]$
Answer: 3 (one of the possible solutions is $A = [2, 3, 4], B = [3, 1, 3]$).

I’ve thought about it for hours and can’t find a solution. How to do this in linear or linearithmic time complexity?

I’m quite sure the problem can be simplified to for index $i$, storing only sum of elements up to $i$ modulo $2$. However, it doesn’t help me much.

(The problem comes from a rather-old Israeli book תכנות תחרותי: סביב אתגרים (‘Competetitive programming’) by Mordechai Ben-Ari. The book isn’t well known and I couldn’t find any solution in Hebrew, so I translated the problem into English for a better chance of getting answer.)

Any help will be appreciated.

Ubuntu Server 18.04 LTS length of support if installing Xubuntu Packages

We are using Ubuntu Server 18.04, and is supposed to have 5 years of support, but we also need a graphical GUI and we want to install Xubuntu packages.

Since Xubuntu 18.04 only has 3 years of support, what’s going to happen with this server after those 3 years?. Are we going to face security issues? All the server is “downgraded” to only 3 years of support?.

A pianist that used a wooden spacer/tool to artificially increase the length of their fingers

I’ve once heard a story about a pianist of the past who wrote a music piece that critics told him was impossible to play – the length of any person’s fingers wouldn’t be sufficient. But he used some tool to combat that and make his fingers longer and played that piece successfully. What was his name?

Is there a way to get the coordinates of the entire length of the street?

I need to end up with a set of waypoints which looks something like this when given a street name.
enter image description here

Currently I am achieving by providing the start and end waypoints of the steet.

  1. Get the start and end waypoints.
  2. Use google’s direction api to get the overview_polyline Link
  3. Based on a distance parameter get the waypoints between the ones provided by overview_polyline.

Because of how overview_polyline works my code will also work for non-straight routes / streets as well.

Question: However this is less than ideal as it requires me to know the start & end waypoints of the street. Does anyone know how I might achieve something similar where only need to provided the street name?