# Questions tagged [derivation-of-formulae]

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### Derivation of an epitrochoid

I was working on an assignment and had a good idea to have a water jet model an epitrochoid to create a fountain. While the overall idea was approved by my teachers, I was told that I need to show a ...
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### What is the formula for converting an improper fraction to a mixed number [closed]

There are methods for converting improper fractions to mixed numbers, but I am interested on finding a formula to which I can input the numerator and denominator of an improper fraction and get an ...
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1 vote
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### Fourier Transform In Music

I am a senior in highschool and am currently trying to conduct an exploration on Fourier Analysis, specifically using the Discrete Fourier Transform to analyze a chord played on my piano. Basically I'...
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### How to derive A001187 formula combinatorically

I’ve learned recently that the formula for A001187 is defined recursively. The formula is: $$n2^{\binom n2}=\sum_{k=0}^{n} \binom{n}{k}kd_k2^{\binom{n-k}{2}}$$ Where $d_k$ is the number of connected ...
125 views

As you can see, here I performed a derivation of a quite simple formula, not much differing from the standard integral representation of the Polylogarithm. Seeking to make it fancier, I arrived at ...
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1 vote
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### All twin prime averages in the range $[9, 119]$ are of the form $6(5[3(z-x)]_{\pmod 7} + x)$ for some $x \in \{0,2,3\}, z \in \{0,2,3,4,5\}$.

Question. Can we come up with a general formula $f(x_5, x_7, x_{11}, \dots, x_{p_n})$ such that each twin prime average $a \in [p_n + 2, p_{n+1}^2 - 2]$ is expressible as $f(x_5, \dots, x_{p_n})$ for ...
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### Laplacian matrix and its eigenvectors and eigenvalues

I am reading paper written by Newman on finding community structure in a network. I came across with the Laplacian matrix. And there is one equation derived from some others. Before diving into that, ...
65 views

### Why is the Discrete Fourier series a sum from $0$ to $N - 1$?

I want to derive the Discrete Fourier series: $$f(x) = \sum_{k=0}^{N-1} X_k e^{i2\pi xk}$$ from the Continuous Fourier Series formula: $$f(x) = \sum_{k=-\infty}^{\infty} X_k e^{i2\pi xk}$$ ...
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### calculating limit by second moment argument

Let $0<p<1$ be fixed. Let $X_n$, $n=1,2,...$, be a sequence of random variables with non-negative integer values. Suppose their expectations satisfy $E(X_n)=\frac{n(n-1)}{2} p^{n-2}$. Question : ...
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### Mathematics behind diatonic scales [closed]

I am looking for the mathematics that can be used to calculate a diatonic scale. It is my understanding that a number of musical scales can be represented in logarithmic form using mathematical ...
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### Verifying formulas and process for surface area and volume of a spindle torus

While working on this geometry problem I reasoned that the surface area of the spindle torus is the surface area of the apple (outer surface) plus the surface area of the lemon (inner surface) while ...
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Let us take $$\mathbb{N} := \{ 1, 2, 3, \ldots \},$$ and let the function $f \colon \mathbb{N} \longrightarrow \mathbb{N} \times \mathbb{N}$ have the following values: \begin{align} & f(1) :=... • 26.6k 1 vote 2 answers 79 views ### Derive a formula to successively decrement a given value by x\% n times I have a value where it needs to be decremented by x\% successively n times For example, for\begin{align*} \text{value} &= 100 \\ x &= 2\% \\ n &= 3 \end{align*}$$we expect the ... • 109 2 votes 1 answer 100 views ### Algebraic manipulation in summations involving the floor function https://www.ivl-projecteuler.com/overview-of-problems/25-difficulty/problem-401 takes the following derivation steps: It needs to calculate:$$\sum_{k=1}^{n}{k^2\left\lfloor{\frac{n}{k}}\right\rfloor}...
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I am studying the brachistochrone problem and I have found in many books the following chain of implications for calculating the time to travel from one point $A$ to another point $B$: v = \frac{dr}{...