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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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 $u, m, t \in \mathbb N_{0}.$
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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

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 ...
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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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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 ...
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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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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 ...
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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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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 $x(4)=2\times 3\times 5\times 7=210$.
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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{...
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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 ...
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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 ...
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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 ...
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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 $\...
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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\}$...
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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 ...
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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 ...
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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 ...
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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 ...
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$1$ 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:...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...