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.
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Periodic sequences of integers generated by $a_{n+1}=\operatorname{rad}(a_{n})+\operatorname{rad}(a_{n-1})$
Let's define the radical of the positive integer $n$ as
$$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\ p\text{ prime}}}p$$
and consider the following Fibonacci-like sequence
$$a_{n+1}=\...
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Asymptotic for $\sum_{k=1}^n k^n$
Consider the OEIS sequence A031971, which is defined as:
$$a_n=\sum\limits_{k=1}^n k^n\quad\color{gray}{(1,\,5,\,36,\,354,\,4425,\,67171,\,1200304,\,.\!.\!.\!)}\tag{1}$$ I'm interested in the ...
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Are there applications for the inverse of the arc length of $ax^n$ and $a^x$? “Closed forms” found.
Based on:
How to straighten a parabola?
and
Arc length of $x^n$ found using Hypergeometric function and series. Alternate representations and solution verification needed.
Use:
$$\text{ArcLength}(...
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Extracting an asymptotic from a sequence defined by a recurrence relation
Suppose I have a sequence defined via its first term and a recurrence relation involving summation over all previous values with some coefficients. Here is the sequence I am interested in right now (...
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Minimizing a generalized partial sum of a geometric progression
Let $q$ be a rational number satisfying $1/2<q<1,$ so that $\left\{q^n\right\}_{n=0}^\infty$ is a decreasing infinite geometric progression with rational terms. Consider absolute values of ...
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Evaluating negative infinite tetration: $\lim\limits_{n\to-\infty}\,^n x=\lim_{n\to\infty}\underbrace{\log_x\left(…\log_x(0)\right)}_n=\,^{-\infty} x$
One can learn that for a power tower of height $n$:
$$y=n\big\{x^{x^{x^…}}=\,^nx\implies \log_x(y)=\boxed{\log_x(\,^nx)=\,^{n-1}x}$$
giving a recursive relation. One might see that $n<-2$ cases are ...
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I want to understand the characteristics of a particular non-linear hierarchical dynamical system
I would like to study the characteristics of the non-linear dynamical system detailed below; in particular, I would like to find its set of fixed points; and to compute (i) the maximum values of ...
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Solving $y^{(x)}(x)=ax+b$ in closed form: What function equals $ax+b$ when you take the $n$th derivative at $x=n$? (with graphs)
Based on the fun of:
Conjectured simple ODE solution: $$y^{(y(x))}(x)=f(x)\mathop \implies\limits^?(y(x))!+c_0Γ(y(x))=\int\limits_{c_1}^xf(t)(x-t)^{y(x)-1} dt $$
Imagine we had a function of which ...
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Closed form of $\frac{d}{dk}\text W_k(z)$. Derivative of W-Lambert function with respect to its branch cuts experiment.
For a change, I will ask a derivative question. Please consider the Generalized W-Lambert/Product Logarithm function $\text W_k(z)$. Let’s see what happens when we try to differentiate with respect to ...
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Any reason why these numbers are close to integers?
I've heard about Heegner numbers some time ago and have been playing today with their property which seems very intriguing to me (and, to be fair, the one I can understand most easily), namely, the ...
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Generalization of Gamma function
The Gamma function is defined by
$$\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}\mathrm{d}t\tag{1}$$
and fulfills the property
$$\Gamma(x+1)=x\Gamma(x)\tag{2}$$
I am wondering if following families of ...
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greatest common divisor for experimental numbers (closest integer)
I'm working with experimental numbers (directions in reciprocal space). That unfortunately means that the pure algorithms for GCD are not working.
The standard deviation for numbers can be as high as ...
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Z-test with Alpha Level
A medical company has performed and experiment using to treat me to groups: cancer patients taking a new drug (treatment group) and cancer patients taking a placebo (control group). They want to ...
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Is there any way to circumference of a circle with radius but without pi? [closed]
Please let me know if anybody knows how to calculate the circumference of a circle with radius but without pi?
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Method for estimating number of fish in my Aquarium
Context: I have an aquarium (50 Liters) with lots of fishes from the same specie (Guppy) and i have noticed that recently their number as grown exponentially due to their reproductive cycle. I would ...
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Polynomial regresssion with limited sample points
Is this solvable and if not, then why not?
Let's say we are given a model that looks as follows:
$y=x+ax^3+bx^5+cx^7+dx^9+\mathcal{N}(0,σ_1^2)$
Given n free choices for the input variable $x$ how can ...
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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,...
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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 ...
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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 $FGHIJ$.
...
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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 ...
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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 ...
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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'...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $p$ ...
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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 ...
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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
$$\mathscr{P}=(\Omega,\...
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When to use Pooled Variance?
In the following examples, why is pooled variance used in the first video but not in the other? When do you use pooled variance?
https://classroom.udacity.com/courses/ud257/lessons/4018018619/...
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Is division purely an arithmetic operator? Is it not the combination of arithmetic operation and relational operation?
An algorithm for computing the ratio of two positive numbers is the following: determine the number of times that the divisor can be subtracted from the numerator, until the accumulated effect of the ...
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How many times should I perform an experiment to get a high level of accuracy?
I have an experiment with 4 factors (independent variables). The first and second variable have 8 levels, the third 5 and the fourth only 3 levels. Theres one dependent variable. How many times should ...
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The closed-form solution of the family $\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(pn+m)}$?
(The results below extend this post.) Given the Clausen function $\operatorname{Cl}_n\left(z\right)$. And,
$$\begin{aligned}
\operatorname{Cl}_2\left(\frac\pi2\right) &= \text{Catalan's constant}\...
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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 ...
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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 ...
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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 ...
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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....
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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,
$$I_3 =\frac4{3!}\int_0^\...
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Is there an identity between the Clausen function $\rm{Cl}_8\left(\frac\pi3\right)$ and $\sum_{n=1}^\infty \frac{1}{n^9\,\binom {2n}n}$?
Given the log sine integral,
$$\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!}\,\...
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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 ...
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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{...
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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 $$
A = \sqrt{s(s-a)(s-b)(s-c)}
$$ ...
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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 ...
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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 ...
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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 ...
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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 $\gamma_\bigcirc(t) = (a(t),b(t)) = a(t) + ib(t)$
...
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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 ...
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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 ...
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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 ...
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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 ...
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