Questions tagged [experimental-mathematics]
The utilization of advanced computing technology in mathematical research: new mathematical results discovered partly or entirely with the aid of computer-based tools.
177
questions
0
votes
0answers
6 views
integral transform integration over finite interval
please anyone help me to find the value of this integral $\int_{0}^{2 \pi} \frac{ (\omega)^{u-m}}{(\omega - \theta)^{t}} d\omega$ = ? where $u, m, t \in \mathbb N_{0}.$
0
votes
0answers
13 views
Is probability perspective dependent?
Suppose there be a person 'A' doing coin unbiased coin flips and he gets 4 heads in a row, then the chance of the next being a tail is 97%, which is very very likely.
But if person 'B' unaware of 'A'...
12
votes
2answers
197 views
Does the fraction of distinct substrings in prefixes of the Thue–Morse sequence of length $2^n$ tend to $73/96$?
Recall that the Thue–Morse sequence$^{[1]}$$\!^{[2]}$$\!^{[3]}$ is an infinite binary sequence that begins with $\,t_0 = 0,$ and whose each prefix $p_n$ of length $2^n$ is immediately followed by its ...
2
votes
0answers
107 views
On a conjecture about the arithmetic function that counts the number of twin primes
In this post we denote (for a fixed positive integer or real number $x$) as $$\pi_2(x)=\#\{\text{ primes }p\leq x\,:\,p+2\text{ is also a prime}\}$$
the arithmetic function that counts the number of ...
0
votes
0answers
54 views
On conjectures about the arithmetic function that counts the number of Sophie Germain primes
In this post we denote (for a fixed positive integer or real number $x$) as $$\operatorname{Germain}(x)=\#\{\text{ primes }p\leq x\,|\,2p+1\text{ is also prime}\}$$
the arithmetic function that counts ...
0
votes
0answers
28 views
Sign of a function that involves the arithmetic function that counts the number of Ramanujan primes less than a given integer
In this post we denote the arithmetic function that counts the number of Ramanujan $R$ primes less than $x$ as
$$\pi_{_{R}}(x)=\#\{\text{Ramanujan primes }R\text{ such that }R\leq x\}$$
thus is the ...
1
vote
3answers
46 views
Probability for heads from an experiment
You are given a (perhaps biased) coin with probability for heads $p$ for some unknown $p$. Construct a random experiment such that with probability at least $0.99$ you can determine the value of $p$ ...
2
votes
2answers
116 views
Point picking in a 1x1 square: probability of line segments connecting 2 random interior points to catacorner vertices intersecting in the square.
Like many of the best problems I found this one on twitter.
"In a square with side length 1, two random points in the square are connected by segments to two opposite vertices. How likely is it that ...
2
votes
0answers
48 views
Understanding of the definition “statistical experiment” (Blackwell, 1951)
I'm interesting for the understanding of the definition "statistical experiment".
Formally statistical experiment, statistical model or just experiment was defined as a triple
$$\mathscr{P}=(\Omega,\...
1
vote
1answer
797 views
When to use Pooled Variance?
In the following examples, why is pooled variance used in the first video but not in the other? When do you use pooled variance?
https://classroom.udacity.com/courses/ud257/lessons/4018018619/...
1
vote
1answer
38 views
Is division purely an arithmetic operator? Is it not the combination of arithmetic operation and relational operation?
An algorithm for computing the ratio of two positive numbers is the following: determine the number of times that the divisor can be subtracted from the numerator, until the accumulated effect of the ...
0
votes
2answers
64 views
Multiplication with negative multiplier [duplicate]
Multiplication is often expressed as repeated addition.
Such as
$$5\cdot 3=5+5+5$$
$$-5\cdot 3=(-5)+(-5)+(-5)$$
Above in both the cases multiplier is positive.In case of multiplier is negative how ...
2
votes
0answers
116 views
How many times should I perform an experiment to get a high level of accuracy?
I have an experiment with 4 factors (independent variables). The first and second variable have 8 levels, the third 5 and the fourth only 3 levels. Theres one dependent variable. How many times should ...
8
votes
4answers
348 views
The closed-form solution of the family $\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(pn+m)}$?
(The results below extend this post.) Given the Clausen function $\operatorname{Cl}_n\left(z\right)$. And,
$$\begin{aligned}
\operatorname{Cl}_2\left(\frac\pi2\right) &= \text{Catalan's constant}\...
2
votes
0answers
69 views
A conjectured continued fraction involving non-polynomial patterns
After having been devoting some time for many years to experimental mathematics, I am thinking to publish the details of some of my most fruitful computing workflows for discovering identities. I ...
77
votes
1answer
1k views
Conjectured formula for the Fabius function
The Fabius function is the unique function ${\bf F}:\mathbb R\to[-1, 1]$ satisfying the following conditions:
a functional–integral equation$\require{action}
\require{enclose}{^{\texttip{\dagger}{a ...
0
votes
1answer
60 views
Mathematics of war references
I have an upcoming talk in on data science. Now a ways such topics have been hijacked by talks on the use of machine learning. Deviating from the trend, I want to focus of core advancement of ...
1
vote
0answers
41 views
An infinite family of BBP type formulas for Gieseking's constant $\kappa = \rm{Cl}_2\left(\frac\pi3\right)$?
(This is related to a recent post.)
In Mathworld's Figure Eight knot, it mentioned several formulas for the hyperbolic volume $V = 2\kappa$ where $\kappa$ is Gieseking's constant $\kappa \approx 1....
6
votes
0answers
156 views
On the closed-form of the triple integral $\int_0^\infty\int_0^\infty\int_0^\infty\frac1{xyz\left(x+y+z+1/x+1/y+1/z\right)^2}\rm{dx\,dy\,dz}$
While doing research for my recent post on the Clausen function $\rm{Cl}_m(x)$, I came across in p. 19 of this paper (by one of the Borwein brothers) the remarkable integral,
$$I_3 =\frac4{3!}\int_0^\...
6
votes
0answers
130 views
Is there an identity between the Clausen function $\rm{Cl}_8\left(\frac\pi3\right)$ and $\sum_{n=1}^\infty \frac{1}{n^9\,\binom {2n}n}$?
Given the log sine integral,
$$\rm{Ls}_m\Big(\frac{\pi}3\Big) = \int_0^{\pi/3}\Big(\ln\big(2\sin\tfrac{\theta}{2}\big)\Big)^{m-1}\,d\theta$$
we have in this post,
$$\begin{aligned}
\frac\pi{2!}\,\...
12
votes
5answers
407 views
Prove that $\int_0^\infty\left(\arctan \frac1x\right)^2 \mathrm d x = \pi\ln 2$
Prove $$\int_0^\infty\left(\arctan \frac1x\right)^2 \mathrm d x = \pi\ln 2$$
Out of boredom, I decided to play with some integrals and Inverse Symbolic Calculator and accidentally found this to my ...
12
votes
1answer
181 views
An asymptotic series for $\small\prod_{k=n}^\infty\operatorname{sinc}\left({2^{-k}}\pi\right),\,n\to\infty$
Using empirical methods, I conjectured that$^{[1]}$$\!^{[2]}$
$$\small\prod_{k=n}^\infty\operatorname{sinc}\left({2^{-k}}\pi\right)=1-\frac{2\pi^2}9\,4^{-n}+\frac{38 \,\pi ^4}{2025}\,4^{-2n}-\frac{...
9
votes
1answer
156 views
Semiperimeter of isosceles Heronian triangles.
A Heronian triangle is a triangle with integer sides and area, named after Heron's formula which states that the area of a triangle with sides $a$, $b$, and $c$ is $$
A = \sqrt{s(s-a)(s-b)(s-c)}
$$ ...
1
vote
1answer
16 views
Understanding “Comparison” method in statistics
I am struggling a lot with statistics so I decided to try David Freedman's Statistics book.
In the book, first chapter, there is this explanation:
A controlled experiment to show the vaccine was ...
14
votes
1answer
306 views
On binomial sums $\sum_{n=1}^\infty \frac{1}{n^k\,\binom {2n}n}$ and log sine integrals
Seven years ago, I asked about closed-forms for the binomial sum
$$\sum_{n=1}^\infty \frac{1}{n^k\,\binom {2n}n}$$
Some alternative results have been made. Up to a certain $k$, it seems it can be ...
12
votes
0answers
214 views
The number $\pi$ in an unexpected context
[This is a follow-up question to this one: Figures and Numbers: Relating properties of geometric shapes and their Fourier series.]
Drawing shapes by some predefined Fourier series I found this square ...
13
votes
0answers
283 views
Figures and Numbers: Relating properties of geometric shapes and their Fourier series
Consider two types of parametrized curves $\gamma:[0,2\pi]\rightarrow \mathbb{R}^2$
open curves $\gamma_\sim(t) = (t,a(t) + b(t))$
closed curves $\gamma_\bigcirc(t) = (a(t),b(t)) = a(t) + ib(t)$
...
1
vote
0answers
200 views
Digits of $\pi$ in other bases?
Pi day got me interested in thinking about whether $\pi$ is a normal number - one whose digits are uniformly distributed no matter what base the number is written in. The OEIS sequence for $\pi$ in ...
0
votes
1answer
109 views
Modified Pascal's triangle
In Pascal's triangle, each number is the sum of the two numbers directly above it. I experimented with a modified Pascal's triangle. First a n-tuple of natural numbers is put to the nth row of the ...
0
votes
0answers
444 views
What is probability that: (a)a non leap year has 53 Sundays. (b)a leap year has 53 fridays.
This one really bowled me over. I don't understand from where I should begin.
I actually thought that a non leap year will have 52 weeks and 1 day. But I couldn't think how to find the probability .
I ...
2
votes
0answers
62 views
Hubs in high-dimensional point clouds (distribution of “being within nearest neighbours of how_many other points”)
The hubness problem is a phenomenon mentioned in this paper, for instance. It claims that for random data point clouds in high-dimensional spaces, there will be "many" hubs -- that is, points that are ...
2
votes
0answers
75 views
Let $x(n)=$ product of first $n$ primes.How often is $x(n)$ triangular?
$x(0)=1$, an empty product, is a triangular number. So are $x(2)=2\times 3=6$, and $x(4)=2\times 3\times 5\times 7=210$.
0
votes
0answers
18 views
Prove the following property for the given sequence.
Consider the following sequence: https://oeis.org/A079523
Let $a(n)$ denote the terms of the sequence of utterly odd numbers. Show that for each $n$ there exists a $p$ such that for each $k\in \mathbb{...
0
votes
2answers
72 views
About Utterly odd numbers.
Consider the following sequence defined here https://oeis.org/A079523 of utterly odd numbers: These are numbers such whose binary representation ends in an odd number of ones.
If $n$ is an utterly ...
2
votes
1answer
37 views
Geometry: polar equation of conics
In finding polar equations of conics we take focus as the origin but ellipse has 2 focci so which focus is taken as the origin?
Or
If if focus is not taken on origin then how to find the polar ...
5
votes
0answers
149 views
Distributions of prime numbers
When folding the number line not into a spiral (like Ulam did) but into a zig-zag pattern (like Cantor did) there are other patterns visible in the distribution of prime numbers:
[To see these ...
3
votes
1answer
132 views
Distribution of triangular, square, and pentagonal numbers
I'm doing some research in visualizing arithmetic sets (resp. properties, resp. sequences of integers). I try to create patterns (in which I hope to observe some symmetries) by injectively mapping $\...
0
votes
0answers
41 views
Lines and Points
Let $S_1$ and $S_2$ be the set of points in x-y plane (where $x>0$ and $y$ is real) defined as:
$$S_1 = \{(x,y) : | x + |x|| + |y + |y|| < 2\}$$
$$S_2 = \{(x,y) : | x - |x|| + |y - |y|| < 2\}$...
5
votes
2answers
120 views
For any positive integer $k$, are there always primes $p$ and $q$ such that $q-p=2^k$?
Experimental evidence suggests to me that there are always primes $p$ and $q$ such that $q-p=2^k$.
Some examples include: $5-3=2$, $11-7=4$, $19-11=8$, $29-13=16$, $43-11=32$, etc.
I am now sure how ...
1
vote
0answers
44 views
Determine convergence from experimental data set
I have ran a simulation to get a set of data points represented in the image below:
Image
The $x$ values range from 5 to 400 in units of 1 (discrete x axis).
It is clear from the grapical ...
5
votes
1answer
110 views
What about sequences $\{\sum_{k=1}^n (\operatorname{rad}(k))^p\}_{n\geq 1}$ containing an infinitude of prime numbers, where $p\geq 1$ is integer?
We denote the radical of the integer $n> 1$ as
$$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\ p\text{ prime}}}p,$$
taking $\operatorname{rad}(1)=1$ that is this definition from Wikipedia.
In ...
2
votes
2answers
72 views
Why this behaviour of primes?
I calculated $$(p_k \cdot p_{k+1})\mod p_{k+2}$$ for $k=1,2,...25$ and obtained a following numbers $$1,1,2,12,7,12,1,2,16,11,40,12,24,7,13,16,48,40,72,48,40,60,15,48,12$$
We can see that there are ...
12
votes
1answer
359 views
$1$ as difference of composites with same number of prime factors and smallest examples
It is probably open can we for every $k \in \mathbb N$ find two composites $a_k$ and $b_k$ such that $a_k$ and $b_k$ have exactly $k$ prime factors and $a_k-b_k=1$.
Smallest examples found so far are:...
0
votes
0answers
39 views
Can you propose a conjectural $\text{Upper bound}(x)$ for the counting function of a sequence of primes arising from the Eratosthenes sieve?
I hope that next question is clear here, since my English is bad. In next paragraph I define a sequence of prime numbers as a sum of an even number of prime numbers minus $1$, where previous terms, ...
14
votes
1answer
285 views
Conjectures Disproven by the use of Computers?
Question: Is there a list of conjectures (famous or not so famous) that were shown to be false by employing the use of computers?
This is just curiosity more than anything. I was actually wondering ...
0
votes
2answers
125 views
Making something a control parameter or a variable when analysing a dynamical system
I am writing down a draft trying to accurately characterize some nonlinear/noninvertible discrete dynamical systems (of a former question here) and due to my lack of knowledge I am having doubts here ...
0
votes
1answer
72 views
On the solutions of an equation involving the Euler's totient function that is solved by the primes of Rassias' conjecture
Here $\varphi(n)$ denotes the Euler's totient function. I've deduced a family of prime numbers $$(x=p_1,y=p_2)$$ that solve an equation involving the Euler's totient function. These primes are the ...
7
votes
1answer
225 views
Skewes' number, and the smallest $x$ such that $\pi(x) > \operatorname{li}(x) - \tfrac1n \operatorname{li}(x^{1/2})$?
We know that Skewes' number is a yet unknown huge integer $x$ such that,
$$\pi(x) > \operatorname{li}(x)\tag1$$
where $\pi(x)$ is the prime counting function and $\operatorname{li}(x)$ is the ...
9
votes
1answer
402 views
Powers of a simple matrix and Catalan numbers
Consider $m \times m$ anti-bidiagonal matrix $M$ defined as:
$$M_{ij} = \begin{cases}
-1, & i+j=m\\
\,\,\ 1, & i+j=m+1\\
\,\,\, 0, & \text{otherwise}
\end{cases}$$
Let $S_n$ stand ...
0
votes
0answers
28 views
Method for measuring distance of a irregular curved profile
I am trying to measure a curved profile of a surface(2D) to determine the surface availability at different rate of testing. I have attached an image for a rough picture.enter image description here
...
