# Bifurcation

And so ethereally walked the shadowy figure alongside the river fork.  Perhaps yet again shall his new companion meet its kin; perhaps too shall his kin beget a new companion.  He had hitherto accompanied the river, and it him.  Its origin is now but a subtle memory, an ephemeral effervescence.  It beget to him a reflection, a projection of his own intricacy.  Yet it solemnly made for a banal companion.  The bifurcation incited a novel awareness, yet a dismal prospect, for his new companion was much like the previous: it beget nothing more than a reflection.

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# Strange Military Law

Major Nidal Hasan, the accused 2009 Fort Hood shooter, attempted to plead guilty to the shootings, but was denied this plea by the judge.  The government is seeking the death penalty in this case.  As [1] states, “under military law, Hasan is not allowed to plead guilty because the premeditated murder charges carry death as the maximum sentence and the government is pursuing the death penalty in Hasan’s case”.  [1] states that under military law, he could plead guilty to lesser charges that do not carry the death penalty (which presumably would concurrently prevent him from being charged with the more serious crimes).  The telegram reporter is quoted as saying “‘Judge Gross said he would enter not guilty pleas in behalf of Hasan, if necessary'”.

[2] clarified some of my confusion.  It simply said “military law does not allow for guilty pleas in death penalty trials”.  It’s fine that the prosecution seeks the death penalty, but it’s absurd that the defendant doesn’t have a guilty plea option under military law, or that a judge could enter a “not guilty” plea against the defendant’s will.  It’s a technicality that really needs to be fixed.

Also, “On Wednesday, for the fifth time, the judge started the hearing with a contempt charge against Hasan and fined him \$1,000, for showing up unshaven”.  This seems pretty ridiculous too, although [2] says the beard is “in violation of Army regulations”.

# Lessons in Venn Diagrams

The Romney campaign released the following diagram several weeks back.  Alright those familiar with set theory, what is the problem with this diagram?  Also, if you figure it out, the Romney campaign may want to hire you.

Some other interesting ones:

# Fluid Mechanics in Seven Dimensions?

Cayley-Dickson Algebras:
The Cayley-Dickson algebras $\mathbb{A}_n$ are special real algebras satisfying

$\dim_\mathbb{R}\mathbb{A}_n=2^n$

for $n\geq 0.$  If we let $\{e_0,...,e_{2^n-1}\}$ be the basis of $\mathbb{A}_n,$ then the algebra is generated by the relations

$e_i^2=e_1\cdots e_{2^n-1}=-1$

where $i\geq 1.$  Hence $\mathbb{A}_0=\mathbb{R}$ and $\mathbb{A}_1=\mathbb{C}.$  $\mathbb{A}_2$ is called the quaternions and is often denoted $\mathbb{H}.$  $\mathbb{A}_3$ and $\mathbb{A}_4$ are called the octonions and sedenions respectively and are respectively also denoted $\mathbb{O}$ and $\mathbb{S}.$

The Cayley-Dickson algebras can also be constructed inductively using a recursive unary operation (called conjugation).  It proceeds as follows, let $\mathbb{A}_0=\mathbb{R}.$  We define $\mathbb{A}_1$ constructively.  Let $\mathbb{A}_1=\mathbb{R}\oplus\mathbb{R}$ and define the following operations:

$\begin{array}{rcl}(a,b)+(c,d)&=&(a+b,c+d)\\(a,b)(c,d)&=&(ac-bd,ad+bc)\\(a,b)^*&=&(a,-b).\end{array}$

One can verify this structure is isomorphic to $\mathbb{C}.$  Next we let $\mathbb{A}_2=\mathbb{A}_1\oplus\mathbb{A}_1=\mathbb{C}\oplus\mathbb{C}$ and define its structure

$\begin{array}{rcl}(a,b)(c,d)&=&(ac-d^*b,da+bc^*)\\(a,b)^*&=&(a^*,-b)\end{array}$

and addition remaining the same and the inner conjugation being complex conjugation.  How does is this retain isomorphism to our original definition of $\mathbb{A}_2?$  Note the dimension is retained by the fact that

$\mathbb{A}_2=\mathbb{C}\oplus\mathbb{C}=\mathbb{R}\oplus\mathbb{R}\oplus\mathbb{R}\oplus\mathbb{R}.$

Hence using this as our inclusion, we show the original requirements that

$i^2=j^2=k^2=ijk=-1.$

We have

$\begin{array}{rcl}i^2&=&(0,1,0,0)^2\\&=&\left((0,1),(0,0)\right)^2\\&=&\left((0,1),(0,0)\right)\left((0,1),(0,0)\right)\\&=&\left((0,1)(0,1)-(0,0)^*(0,0),(0,0)(0,1)+(0,0)(0,1)^*\right)\\&=&\left((-1,0)-(0,0),(0,0)+(0,0)(0,-1)\right)\\&=&\left((-1,0),(0,0)\right)\\&=&\left(-1,0,0,0\right).\end{array}$

Similar computations can be done for $j$ and $k$ and for showing that

$(0,1,0,0)(0,0,1,0)(0,0,0,1)=(-1,0,0,0).$

It turns out $\mathbb{A}_3$ and beyond are not associative, so in our initial requirement that

$e_1\cdots e_{2^n-1}=-1,$

we will clarify that we mean

$\left(\cdots((e_1e_2)e_3)\cdots e_{2^n-2}\right)e_{2^n-1}=-1.$

We continue inductively be setting $\mathbb{A}_{n+1}=\mathbb{A}_n\oplus\mathbb{A}_n$ and defining product and conjugation as before (and componentwise addition).  One can establish the following table of properties

$\begin{array}{|c|c|c|c|c|}\hline\mathbb{R}&\mathbb{C}&\mathbb{H}&\mathbb{O}&\mathbb{S}\\\hline\mbox{division algebra}&\mbox{division algebra}&\mbox{division algebra}&\mbox{division algebra}&\phantom{normed}\\\mbox{associative}&\mbox{associative}&\mbox{associative}&&\\\mbox{commutative}&\mbox{commutative}&&&\\\mbox{trivial conjugation}&&&&\\\end{array}$

where associativity and commutativity refer to the multiplication (the addition is always both).  All of the above properties (save the division algebra structure) can at this point (albeit with some tediousness) be shown.  If $a=(a_0,...,a_{2^n-1})\in\mathbb{A}_n,$ we can define its real part as

$\mbox{Re}\,a=a_0.$

Note for $\mathbb{R}$ we have

$a^*a=aa=a^2.$

For $a\in\mathbb{R}$ we can define $\|a\|=\sqrt{\mbox{Re}\,a^*a}=\sqrt{a^2}=|a|.$  For $\mathbb{C}$ we have

$(a,b)^*(a,b)=(a,-b)(a,b)=(a^2+b^2,ab-ba)=(a^2+b^2,0).$

Similarly we can define $\|(a,b)\|=\sqrt{\mbox{Re}\,(a,b)^*(a,b)}=\sqrt{a^2+b^2}.$  For $\mathbb{H}$ we have

$(a,b)^*(a,b)=(a^*,-b)(a,b)=(a^*a+b^*b,a^*b-a^*b)=(\|a\|^2+\|b\|^2,0).$

Hence $\|(a,b,c,d)\|=\sqrt{a^2+b^2+c^2+d^2}.$  And similarly for $\mathbb{O}$ we have

$(a,b)^*(a,b)=(a^*,-b)(a,b)=(\|a\|^2+\|b\|^2,0),$

so $\|(a_0,...,a_7)\|=\sqrt{a_0^2+\cdots+a_7^2}.$  Inductively, one can see that the norm on $\mathbb{A}_n$ coincides with the euclidean norm in $\mathbb{R}^{2^n}.$  We can also note that the multiplicative inverse of an element $x\in\mathbb{A}_n$ is

$x^{-1}=\frac{x^*}{\|x\|^2}.$

Although it turns out that in the sedenions and above ($\mathbb{A}_i$ for $i\geq 4$), there are zero divisors (for example, $(e_3+e_{10})(e_6-e_{15})=0$), so the algebra fails to be a division algebra.

Theorem 1.  (Hurwitz’ Theorem)  If $A$ is a real normed division algebra with identity, then $A\in\{\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}\}.$

Corollary 2.  (Frobenius’ Theorem)  If $A$ is an associative real division algebra, then $A\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}.$

Generalized Cross Products:
The cross product of two elements $x,y\in\mathbb{R}^3$ can be written

$x\times y=\begin{vmatrix}e_1&e_2&e_3\\x_1&x_2&x_3\\y_1&y_2&y_3\end{vmatrix}=(x_2y_3-x_3y_2,x_3y_1-x_1y_3,x_1y_2-x_2y_1).$

There is a natural way to define a cross product of $n-1$ elements in $\mathbb{R}^n$ by

$\mathsf{X}(v_1,...,v_{n-1})=\begin{vmatrix}e_1&\cdots&e_n\\v_{1_1}&\cdots&v_{1_n}\\\vdots&\ddots&\vdots\\v_{{n-1}_1}&\cdots&v_{{n-1}_n}\end{vmatrix}.$

However, our ultimate objective is to define vorticity in other dimensions.  So we will want a binary cross product that we can apply to our differential operator and a vector field: $\nabla\times f.$  We want $\times$ to satisfy some conditions

$\begin{array}{rcl}(w+x)\times(y+z)&=&(w+x)\times y+(w+x)\times z=w\times y+x\times y+w\times z+x\times z\\ 0&=&x\cdot(x\times y)=(x\times y)\cdot y\\\|x\times y\|^2&=&\|x\|^2\|y\|^2-(x\cdot y)^2.\end{array}$

These conditions are respectively called bilinearity, orthogonality, and magnitude.  Hence we want $\times$ to be the product in a real normed algebra.  Moreover magnitude tells us that we want $x\times x=0.$  We also have $e_i^2=-1$ in our Cayley-Dickson algebras.  It turns out that the cross product in $\mathbb{R}^3$ can be modeled in $\mathbb{H}$ as follows.  Define $C:\mathbb{R}^3\to\mathbb{H}$ by

$C(x_1,x_2,x_3)=x_1i+x_2j+x_3k$

and $C^{-1}:\mathbb{H}\to\mathbb{R}^3$ by

$C^{-1}(a+bi+cj+dk)=(b,c,d).$

Then one can verify that for $x,y\in\mathbb{R}^3$

$x\times y=C^{-1}\left(C(x)C(y)\right).$

Note that in $\mathbb{R},$ the cross product is trivial $r\times s=0$ for all $r,s\in\mathbb{R}.$  This is obvious if we require it to satisfy orthogonality—where the dot product is just multiplication.  So the zero-product property implies the cross product must be $0.$  It is compatible with our map as well (in this case $C:\mathbb{R}\to\mathbb{C}$):

$r\times s=C^{-1}(C(r)C(s))=\mbox{Im}\,(ri)(si)=\mbox{Im}\,(-rs)=0.$

Hence we are left with one remaining cross product: the cross product in $\mathbb{R}^7$ where, hereafter, $C:\mathbb{R}^7\to\mathbb{O}$ is defined by

$C(x_1,...,x_7)=x_1e_1+\cdots+x_7e_7$

and $C^{-1}:\mathbb{O}\to\mathbb{R}^7$ is the imaginary map

$C^{-1}(a_0e_0+\cdots+a_7e_7)=(a_1,...,a_7).$

The cross product of $x,y\in\mathbb{R}^7$ is defined by

$x\times y=C^{-1}(C(x)C(y)).$

We will now define

$C(\nabla)=e_1\frac{\partial}{\partial x_1}+\cdots+e_7\frac{\partial}{\partial x_7}.$

If $f:\mathbb{R}^7\to\mathbb{R}^7$ is differentiable, then we define its curl by

$\mbox{curl}\,f=\nabla\times f=C^{-1}(C(\nabla)C(f)).$

# The -logy and -nomy Suffixes

I had mentioned before about how philosophy and mathematics used to be equivalent before math diverged by making solid, technical assumptions, and proving what followed.  Similarly is the case with regards to astrology and astronomy.  Both began by looking for correlation between the stars and X, where X was pretty much any Earthly phenomenon.  Then astronomy used counterexamples to the anthropic principle to establish more consistent theories, and astrology was properly buried.  Then for some reason in the 20th century horoscopes reincarnated astrology.1  While technically the position of the stars infinitesimally changes the net gravitational force on a human, the fact that cognitive activity is based on electric potentials that globally dominate any infinitesimal net gravitational change on the body renders their study on personality irrelevant.

This being the case, the suffix -logy on astrology seems insulting to the scientific community.  The suffix -log means “to speak”.2    Coupled with -y, it takes the meaning of “speaker” which appears to conventionally translate to “generic speaker”, or, “department/field”.  Xlogy thus becomes “study/department of X”.  In this case, X being astro–deriving from “stars”.  At the same time, -nomy derives from “law”.  How should we distinguish the two suffixes?  We could say Xlogy is the general inquiry into X, which builds upon the object Xnomy.  Xnomics could then be reserved for when Xlogy leads to multiple instances of an Xnomy.  Hence we should have the following conventional-to-literal semantic assignments:

Astrology $\mapsto\varnothing$

Astronomy (inquiry component) $\mapsto$ Astrology

Astronomy (law component) $\mapsto$ Astronomy.

What would academic departments call themselves then?  We could pick the Xlogy form.  In this case “economics” would be called “ecology”?  The prefix eco- derives from the study of environments of living organisms.  In this sense, conventional “economics” is far more abstract than it semantically pretends to be.  I would consider something along the lines of

Economics $\mapsto$ Elogy

where E derives from something like “token”.  Although trade itself may prove to be more fundamental and inherently connected to physics (think conservation of energy).  We could then have

Physics $\mapsto$ Cosmology,

which one could plausibly argue.  I’m not sure to what we could assign mathematics.  It may not even matter since math, physics, and economics may share a fundamental equivalence (all could be construed as structures of symbols=0 volume energy regions=tokens).

[1]  Campion, Nicholas, 2009. A History of Western Astrology, Vol. 2, The Medieval and Modern Worlds. London: Continuum. (via “Astrology” on Wikipedia).