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The Unipodal Algebra is the algebra generated by taking all the complex scalar
multiples of the unity 1 and a unit vector $u$, and then closing off the set under
multiplication. The resulting algebra is commutative, with $u^2=1$. What follows
is a collection of papers that utilize the unipodal algebra to perform integrals and
to prove certain identities about with some trigonometric/hyperbolic functions.

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On this page we have an assortment of recreational math problems from Olympiad
math contests, SATs, university entrance exams, and the like. The point of this page
is to show how some of these problems can be attacked from the vantage of using the
unipodal algebra. The probems I choose to try often require some degree of cleverness
from the student just to solve them with traditional methods. The use of the unipodal
algebra allows us to fashion proofs that sometimes require less cleverness and at
other times, more cleverness. But that's were the fun is!!

Recreational Math Olympiad Problems using the Unipodal Algebra.

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The Unipodal Quadratic Theorem.

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In this paper we prove the advantage of using the unipodal algebra when performing
a convolution integral that resulted from the process of a Laplace transform.

The Laplace Transform and Convolution Theorem.

In this paper we prove the special relativistic Addition of Velocities formula and
Lorentz transformation by use of the unipodal algebra. $X' = X e^{-u\theta}$ where $\theta$ is
the Rapidity.

The relativistic Addition of Velocities formula.

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I invented a problem to solve that I suspected the unipodal algebra could be used
favorably to solve it. If you are not familair with the unipodal algebra, this website
has a lot of material on it.

Statement of the problem:

Solve for the value of the expression (where $x$ is real) \begin{equation} \left(x + \frac{1}{x}\right)^{n} - \left(x - \frac{1}{x}\right)^{n}\quad (\mbox{$n$ a positive integer})\,,\label{eq:expression} \end{equation} given the two relations \begin{equation} \left(x + \frac{1}{x}\right)^{n} + \left(x - \frac{1}{x}\right)^{n} = x\,,\label{eq:Relation1} \end{equation} and \begin{equation} x^2 - \frac{1}{x^2} = B\,,\label{eq:Relation2} \end{equation} where $B$ is some nonzero real number.

A Unipodal Experiment An interesting experimental use of the unipodal
algebra applied to a certain Olympiad-like problem.

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Here is the unipodal algebra is used to solve for the Normal Modes of a Loaded
String with two masses and two different spring constants. Examples with and without
using the Laplace transform.

The Normal Modes of a Loaded String..

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Here is the unipodal integral $\int \cos x \cosh x\, dx = \half (\cos x \sinh x + \sin x \cosh x)$.

The integral: $\int \cos x\cosh x\, dx$.

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Here is the unipodal integral $\int (a \cosh x - \sinh x) \cosh bx dx $.

The integral: $\int (a \cosh x - \sinh x) \cosh bx\, dx$.

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As broadening the algebra from the reals to the complex can help; broadening from the complex to the unipodal can also help.

The differential equation $$\left(\frac{dy}{dx}\right)^2 - 1 = x^2$$ is to be solve with help from the unipodal algebra.

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Here is the unipodal hyperbolic identity $\sinh^{-1} y = \log\, [ y +\sqrt{y^2 + 1}]$.

The identity: $\sinh^{-1} y = \log\, [ y +\sqrt{y^2 + 1}]$.

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Here is the Main Integral Theorem $ \int e^{xu}\,dx = ue^{xu}$.

The Main Hyperbolic Integral Theorem: $\int e^{xu}\,dx = ue^{xu}$.

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Here is the unipodal hyperbolic identity $\int \sinh mx \sinh nx\,dx = \frac{\sinh(m + n)x}{2(m + n)} - \frac{\sinh(m - n)x}{2(m - n)}$.

The integral: $\int \sinh mx \sinh nx\, dx = \frac{\sinh(m + n)x}{2(m + n)} - \frac{\sinh(m - n)x}{2(m - n)}$.

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Here is the unipodal derivation of the logarithm and the square root of a vector $u$.

The logarithm and the square root of a vector $u$.

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