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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Curve Fitting and Parameter Estimation Study

I have a problem with an experimental data analysis to obtain unknown parameters. The experimental data can be described by the equation: $y_k-y_{k-1}=\eta_k (x_k^{1-\alpha_k}-x_{k-1}^{1-\alpha_k})$ ...
mcakir's user avatar
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What is the periodic function that $\frac{\ln\left( \sum_{n=0}^{\lfloor x \rfloor} n! \right) - x \ln(x) + x }{\ln(x)}$ converges to?

I was analyzing the function $$\frac{\ln \left( \sum_{n=0}^{\lfloor x \rfloor} n! \right) - x \ln(x) + x}{\ln(x)} $$ I had a hunch that this object tends to a periodic function as $x \rightarrow \...
Sidharth Ghoshal's user avatar
2 votes
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Model Temperature change of a glass of hot water

I want to learn more about ordinary and partial differential equations by using real world experiments. I try to model the temperature over time of hot water in a cylindric glass. I goal is to ...
lmixa's user avatar
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Possibility of expressing ln(x) as a sum of Hermite polynomials

this might be an ill defined question, but: I have an experiment that computes the value <n/N> (where n is a random variable and N is fixed). I'm trying to compute <log(n/N)>, which due to ...
Japap_'s user avatar
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Calibration Resolution

I have a flow meter that I calibrate by doing timed catches. The best resolution I have on my scale is down to 0.01 lb. The meter I'm comparing against goes to 0.001 lb. Usually, I'll have something ...
user267587's user avatar
4 votes
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Removing four terms from the decic using a Tschirnhausen transformation?

I. Transformation In this 2021 paper, one can remove four terms from an equation of degree $n$ using a Tschirnhausen transformation of degree $n-1$ (and with radical coefficients), but only if $n\...
Tito Piezas III's user avatar
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Approximate the inverse of a given matrix using Neumann series

I know that in order to approximate the inverse of a given matrix using Neumman series, the spectral norm of a identity matrix - a given matrix has to be smaller than 1. If the norm of a identity ...
Thang Chu's user avatar
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On the elliptic curve $u(u+2186^2)(u+2188^2)=v^2$ and 7th powers?

I. Elliptic curve Given some constant integer $m$, a solution to the elliptic curve, $$E:=u\big(u+(m^7−1)^2\big)\big(u+(m^7+1)^2\big)=v^2$$ which is not a torsion point implies a polynomial identity ...
Tito Piezas III's user avatar
5 votes
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324 views

What is correlation dimension, actually?

I'm taking a course on chaotic dynamical systems, and we're talking about attractors with non-integer correlation dimensions, but I can't seem to find a satisfactory definition for this concept. ...
Frank Seidl's user avatar
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Finiteness of a series coming from expectation formula

I have found that if $$ \begin{array}{ll} &u_i=\displaystyle\sum_{j=0}^{c-1}\frac{1}{c}\binom{i-1}{c-j-1} q^{i+j-c}p^{c-j},\\ &s_i=\displaystyle\sum_{r=1}^i u_r, \end{array} $$ then $$ \...
Roberto Benito's user avatar
2 votes
1 answer
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How do I compute the cumulative probability of this experiment?

I found a pretty neat mathematical problem inside a videogame (fallout 76). Players have a specific luck value $ l $ and in the game-world there a several machines placed which can or cannot give ...
1somorph's user avatar
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Instrument resolution or Standard Error of the mean

Say I tried to measure the length of an object with an instrument that has a resolution of of 0.1mm and got the following: 12.5,12.5,12.5,12.5,12.5,12.5,12.5,12.5, 12.5,12.5 (all in mm). The standard ...
Bartosz Dlubak's user avatar
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How do I combine three or more standar deviations?

I am trying to compare some results in different experiments by using meta-analysis. Each study involved has a different number of subgroups. Since only mean and standard deviation is available for ...
Aligomez's user avatar
5 votes
1 answer
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How can I find parameter of the rational map?

I have a family of rational maps $f$ ( the Blaschke fraction ) with one complex parameter $\rho$: $$f(z) = \rho z^2 \frac{z-3}{1-3z}$$ I want to find $\rho$ such that map $f$ has a parabolic period 3 ...
Adam's user avatar
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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}=\...
Augusto Santi's user avatar
7 votes
2 answers
277 views

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 ...
Vladimir Reshetnikov's user avatar
4 votes
0 answers
175 views

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}(...
Тyma Gaidash's user avatar
14 votes
2 answers
417 views

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 (...
Vladimir Reshetnikov's user avatar
8 votes
0 answers
211 views

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 ...
Vladimir Reshetnikov's user avatar
1 vote
2 answers
144 views

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 ...
Тyma Gaidash's user avatar
2 votes
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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 ...
Sam's user avatar
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4 votes
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Solve $y^{(x)}(x)=ax+b$ using fractional calculus

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 ...
Тyma Gaidash's user avatar
5 votes
1 answer
185 views

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 ...
Тyma Gaidash's user avatar
1 vote
0 answers
132 views

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 ...
user108687's user avatar
5 votes
2 answers
269 views

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 ...
granular_bastard's user avatar
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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 ...
Kris_R's user avatar
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1 vote
1 answer
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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 ...
Kyle Reynolds's user avatar
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1 answer
212 views

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?
MATHEW PANICKER's user avatar
2 votes
3 answers
131 views

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 ...
José Braga's user avatar
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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 ...
Sadikov's user avatar
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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,...
SJA's user avatar
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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 ...
philomath's user avatar
2 votes
1 answer
106 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 $FGHIJ$. ...
Nathan Portland's user avatar
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1 answer
24 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 ...
wpkzz's user avatar
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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 ...
whoiam's user avatar
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0 answers
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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'...
Vishwa Mithra Tatta's user avatar
13 votes
2 answers
349 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 ...
Vladimir Reshetnikov's user avatar
2 votes
0 answers
157 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 ...
user759001's user avatar
2 votes
0 answers
91 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 ...
user759001's user avatar
1 vote
0 answers
40 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 ...
user759001's user avatar
1 vote
3 answers
191 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 $p$ ...
DesmondMiles's user avatar
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3 votes
2 answers
436 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 ...
futurebird's user avatar
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2 votes
0 answers
114 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,\...
Thomas J's user avatar
2 votes
1 answer
7k 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/...
CS Student's user avatar
1 vote
1 answer
63 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 ...
user42757's user avatar
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2 votes
0 answers
127 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 ...
Lewis's user avatar
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9 votes
4 answers
599 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}\...
Tito Piezas III's user avatar
2 votes
0 answers
76 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 ...
Thomas Baruchel's user avatar
84 votes
1 answer
2k 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 ...
Vladimir Reshetnikov's user avatar
1 vote
1 answer
112 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 ...
Nilotpal Sinha's user avatar

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