This is a brief return to the topic of Irrational Sunflowers. The sunflower associated with a real number 
Here is the sunflower of 
Seven rays emanate from the center because 
a = np.pi n = 5000 x = np.mod(a*np.arange(n, 2*n), 1) np.sum(np.diff(np.sort(x)) > 1/n)
This code computes the polar angles of sunflower points indexed 
Here is the same sunflower with the points alternatively colored red and blue.
The colors blur into purple when the rational approximation pattern is strong. But they are clearly seen in the transitional period from 22/7 approximation to 355/113.
- How many points would we need to see the next rational approximation after 355/113?
- What will that approximation be? Yes, 22/7 and 355/113 and among the convergent of the continued fraction of
. But so is 333/106 which I do not see in the sunflower. Are some convergents better than others?
Finally, the code I used to plot sunflowers.
import numpy as np
import matplotlib.pyplot as plt
a = np.pi
k = np.arange(10000)
r = np.sqrt(k)
t = a*2*np.pi*k
plt.axes().set_aspect('equal')
plt.plot(r*np.cos(t), r*np.sin(t), '.')
plt.show()
be the vertex set of a graph. Write 
if 
are adjacent vertices. Given a function 
, define 
.
of all functions 
. It is symmetric: 
and positive semidefinite: 
. Since equality is attained for constant 
, 0 is always an eigenvalue of 
.
of functions with zero mean: 
. Indeed, 
where 
.
is equivalent to 
. Indeed, since 
, it follows that 
for every 
. So, the identity 
holds on 
denote the diameter of the graph. Then the number of distinct Laplacian eigenvalues is at least 
and 
elsewhere. Also let 
for 
. Note that 
for all 
. One can prove by induction that 
when the distance from 
to 
is greater than 
, and 
when the distance from 
when 
and 
. This shows that 
is not a linear combination of 
. Since 
, it follows that 
is not a linear combination of 
. Hence the minimal polynomial of ![{[2-\sqrt{2}, 2, 2+\sqrt{2}]}](https://s0.wp.com/latex.php?latex=%7B%5B2-%5Csqrt%7B2%7D%2C+2%2C+2%2B%5Csqrt%7B2%7D%5D%7D&bg=ffffff&fg=000000&s=0 "{[2-\sqrt{2}, 2, 2+\sqrt{2}]}")
.![{[4, 4, 4]}](https://s0.wp.com/latex.php?latex=%7B%5B4%2C+4%2C+4%5D%7D&bg=ffffff&fg=000000&s=0 "{[4, 4, 4]}")
. This pattern continues for other complete graphs: since the complement is the empty graph (
.
.
but they are obviously non-isomorphic (consider the vertex degrees):
and are non-isomorphic.
be the remainder of the division of 
by 
if and only if 
. It also looks like 
, provided by Max Alexeyev on the OEIS page for 
for some 
. Let 
be the smallest prime divisor of 
. This means that the order of 
is a divisor of 
elements, and the only divisor of 
. Thus, the order of 
, which is absurd.
is 4700063497 deserves to be mentioned here. But I would like to consider the most frequent values of 
integers with 
. The frequency order remained stable at the top. These are the most common 11 values, with their frequency (in %) listed for the aforementioned six ranges.
with 
is even, we are computing 
and that is likely to be a power of 
. This boosts the frequency of the even (generally, composite) powers of 2 in the sequence.
ensures the suitable sign of the discriminant. It describes a parabola with vertex (1, 1) and focus (2, 2), contained in the first quadrant and tangent to the axes at (4, 0) and (0, 4). Within the orange region there are no real roots.
and want to find the sum of all distances 
over 
. Why? Maybe because over five years ago, the gradient flow of this quantity was used for "clustering by collision" (
, then absolute values are taken, and then they are all added up. Double-counted, hence the factor of 0.5.
; the cost of computing them is linear in time and memory. For each fixed 
with 
is simply 
, or, equivalently, 
. So, all we have to do is add these up. Still one line of code (after sorting), but a much faster one:
, the average distance of a sample taken from a distribution converges to the expected value of |X-Y| where X, Y are two independent variables with that distribution. Let’s express this in terms of the probability density function 
. By symmetry, we can integrate over 
and double the result:
, and the boundary terms are zero.
, this simplifies nicely:
is simply the integral of 
. I don’t often see CDF squared, like here. Some examples: for the uniform distribution on [0,1] we get
, it is
, the sum (or average) of distances to all other points. To do this, we don’t sum over 
. The latter sum is simply 
where 
is the total sum. So, all we need is
compute the correlation coefficient of 
with 
. That the lag does not exceed 
ensures the entire sequence participates in the computation, so we are not making a conclusion about its periodicity after comparing a handful of terms at the beginning and the end. In other words, we are not going to detect periodicity if the period is more than half of the observed time period.
as above. After 5 million trials with a sequence of length 100, the distribution of R was as follows:
is two orders of magnitude less, etc. Also, |R| is unlikely to be very close to zero unless the data is structured in some strange way. Some kind of correlation ought to be present in the white noise.
. The number of trials was not very large (10000) so there is some fluctuation, but the pattern is clear.