Questions tagged [derivation-of-formulae]
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How to find a formula for the terms of this sequence?
I saw a problem on this forum concerning the number
$$T = 1 + \frac{2 +\frac{3+ \frac {4+...}{5+...}}{4+\frac{5+...}{6+...}} }{3 + \frac{4+\frac{5+...}{6+...}}{5+\frac{6+...}{7+...}}}$$
whose rule is &...
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Does reverse homology $\ker \delta\subset \operatorname{im} \delta$ have a defining equation? I don't think it does.
Background Material. Is "reverse homology" $\ker g \subset \text{im} f$ possible?
Question. Forward (usual / historical) homology always has the defining equation $\delta^2 = 0$ for a ...
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Calculate Index of Element in Pyramid
I'm currently programming a small website that automates the calculation of Pyramid tournament placements.
Players are ranked in a pyramid like so (the numbers represent the current rank of a player):
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How was this $y\left(y’’+\frac{1-p}vy’\right)-(y+1-p)y’^2=0$ power series recurrence derived?
In György Steinbrecher’s and William Shaw’s Quantile mechanics $(47)$ to $(51)$, it can be found that:
$$\begin{aligned}y\left(y’’+\frac{1-p}vy’\right)-(y+1-p)y’^2=0,y(0)=0,y
(0)=1\iff (1-p)y’^2=\...
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Deriving the Unit Quaternion to Tait-Bryan Angles conversion.
Let me start by saying I have a working solution. But I just don't understand how to get there. I've followed the well-written paper Technical Concepts
Orientation, Rotation, Velocity and Acceleration,...
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How to prove that the feasible set of a two-asset portfolio is a hyperbola?
The question comes from ‘Mathematics for Finance: An Introduction to Financial Engineering’ by Marek Capiński (Author), Tomasz Zastawniak. The book does not give a complete proof, and I did not find a ...
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How do I derive the alveolar gas equation?
I'm taking medical physiology and of course, none of the equations are properly explained. The closest I've been able to get is this page. Even here however, steps from equations 14 to equation 15 is ...
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What is the Equation for the Batista-Costa Minimal Surface?
The Batista-Costa surface is a triply periodic minimal surface. Three photos of part of the same surface are below:
where the first two were taken form the research paper: The New Boundaries of 3D-...
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reverse construction chinese remainder theorem
How can I determine the original number $x\in[pq]$ from its remainders $x_p$ and $x_q$, when it's divided by two relatively prime numbers $p$ and $q$, given that $\gcd(p, q) = 1$? I learned about a ...
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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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What is the 1-case closed form for $\sum_{i = 1}^{x} \lfloor \frac{i - r}{d}\rfloor$?
Let all untyped variables be natural numbers.
Formula?
Given $x \geq 1$, $0 \leq r \lt d$ there are two cases to handle: $x \lt r$ and $x \geq r$.
What is the 2-case closed form for $\sum_{i = 1}^{x} \...
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What are the formulas for the circumradius, surface area and volume of each Kepler-Poinsot polyhderon based on the length of the entire edge?
Every formula I've found online is based on only a part of the total edge. If anyone knows the formulas based on each red edge below I would greatly appreciate it. A derivation of those formulas ...
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show two surface area formulas are equivalent
According to Wikipedia, for a uniform n-gonal antiprism with edge length $a$, SA = $\frac{n}{2} \left( \cot\frac{\pi}{n} + \sqrt{3} \right) a^2$ . This formula seems unnecessarily complex to me, as I ...
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Is it possible to define an implicit function for the Kth N such that N(N+1)/2 is a perfect square?
The questions asks:
Define a formula to yield the Kth N for which there exists an integer X less than or equal to N for which the sum of the integers from 0 to X (inclusive) is equal to the sum of the ...
