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.

Filter by
Sorted by
Tagged with
0
votes
0answers
13 views

Problem in a rubic cube

What guarantee that we can permute a rubic cube in such a manner that no two small square of same colour in one side are placed together(vertically and horizontally). I can make it in a systematic way,...
0
votes
0answers
30 views

Statistical significance in users research experiment

If a company sends out 100 emails to a user. These emails are opened at different hours of the day. During some of the hours, the user is more likely to open the mail. I want to understand how many ...
0
votes
0answers
16 views

Mathematical investigation of the Doppler effect experiment

I am a high school student and have attempted to model the change in observed frequency of sound experiment using a sound source, ball, net and software (phyphox) to carry out my experiment of doppler ...
2
votes
1answer
57 views

Mid point pentagon, maximization of perimeter under constraint

Let $ABCDE$ be a convex pentagon of perimeter $\mathcal{P}$. Consider $F,G,H,I,J$ the mid points of $\overline{AB}$, $\overline{BC}$, etc. Denote $\mathcal{Q}$ the perimeter of the pentagon $FGHIJ$. ...
0
votes
0answers
33 views

Doppler effect mathematical modeling

I am a high school student and I have decided to do my essay on Doppler effect by gathering my own data and analyzing rate of change in frequency etc. The problem that I’m facing at the moment is that ...
0
votes
1answer
19 views

How to know that if by rearranging a histogram it would fit a Gaussian?

Let us suppose that I have some frequencies of experimental data on a plot, a histogram. The graph of such frequency data looks completely noisy. But I have a finch that if I could reorder the ...
1
vote
0answers
27 views

How to make computers understand length, breadth and height given a 3d object?

Suppose, I am building an A.I Robot and i want that robot to bring to me that cuboidal shape object whose length is say 10 cm(in 1st case out of (40cm, 19cm, 10cm)), each time it brings the desired ...
0
votes
0answers
17 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
235 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
141 views

On a conjecture about the arithmetic function that counts the number of twin primes

I've asked the same question on MathOverflow two days ago as On a conjecture about the arithmetic function that counts the number of twin primes, I add this reference while I hope to know what about ...
2
votes
0answers
63 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 ...
1
vote
0answers
37 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
58 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
202 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
57 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,\...
2
votes
1answer
2k 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
42 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 ...
2
votes
0answers
117 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
403 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
70 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 ...
79
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
69 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
42 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
167 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^\...
8
votes
0answers
173 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
444 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
193 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{...
8
votes
1answer
201 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
17 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
351 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 ...
13
votes
0answers
227 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
290 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
247 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
157 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
483 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
65 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
79 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
78 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
42 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
178 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
157 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
42 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
134 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
48 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
113 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
366 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
40 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, ...
15
votes
1answer
335 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 ...