1.5.7 Smoothing the Minimal Element Basis

Con­tin­u­ous func­tions define there are no gaps be­tween points on the func­tion and that it is dif­fer­en­tiable every­where. This pro­vides the min­i­mum re­quire­ments for a func­tion to be smooth, and the smooth­ness of a func­tion is a prop­er­ty re­lat­ed to the number of con­tin­u­ous de­riv­a­tives it has over a domain. If a func­tion has con­tin­u­ous de­riv­a­tives for all orders in its domain, this func­tion is re­ferred to as a C-in­fin­i­ty func­tion $C^{\infty}.$We also note that the min­i­mal el­e­ment basis is the lim­iter in terms of the angle co­ef­fi­cients that can be chosen. Mind­ful of this, we return to

and state the fol­low­ing the­o­rem.

The­o­rem (1.5.108) – For any so­lu­tions of $\mathrm{K}_{\mathcal{me}}$, the wind­ing number $k$ of any re­duced ra­tio­nal number chosen for $\frac{\mathcal{n}}{\mathcal{d}}$, re­sults in a wind­ing number equal to $\mathcal{n}$. For non-re­duced ra­tio­nals, and $\frac{\mathcal{n}}{\mathcal{d}}$ in ra­di­ans, the wind­ing number $k$ will be equal to the re­duced number $\mathcal{n}$ for that ra­tio­nal.

Proof – Sim­i­lar the pre­vi­ous sec­tion, for $\mathrm{K}_{\mathcal{me}}$, the number of el­e­ments in the min­i­mal el­e­ment basis, $\mathcal{n}$, the nu­mer­a­tor of the ra­tio­nal ex­pres­sion $\in \mathbb{Z}$, $\mathcal{d}$, the de­nom­i­na­tor of the ra­tio­nal ex­pres­sion $\in \mathbb{Z}$, and $k$, the wind­ing number, we first solve the fol­low­ing for $k$

We can solve the above to show that $\mathrm{K}_{\mathcal{me}} = \frac{k\mathcal{d}}{\mathcal{n}}$, but we can also show that $\mathrm{K}_{\mathcal{me}}= \mathcal{d},$ based on the fact that the min­i­mal el­e­ment will be $2i\pi\left( \frac{1}{\mathcal{d}} \right)$ for all re­duced ra­tio­nals, and for $\mathrm{K}_{\mathcal{me}} = \frac{k\mathcal{d}}{\mathcal{n}},$ this im­plies that $k = \mathcal{n}$, and then arrive at $\mathrm{K}_{\mathcal{me}}= \mathcal{d}$, stated ex­plic­it­ly, as

To prove that $\mathrm{K}_{\mathcal{me}}= \mathcal{d}$, we can use the finite $\mathcal{d} \times \mathcal{d}$ ma­tri­ces of the cyclic groups of the form $Z_{\mathcal{d}}$, for $\mathcal{d}$, the in­te­ger value of the ra­tio­nal de­nom­i­na­tor we have been work­ing with. The el­e­ments of the group will have the fol­low­ing matrix tem­plate, where the column number of the first row serves as the nu­mer­a­tor value $\mathcal{n} + 1$, and all other matrix en­tries, apart from the en­tries with value $e^{2i\pi\left( \frac{\mathcal{n}}{\mathcal{d}} \right)}$, will be set to zero

The entry, $a_{1,1} = e^{2i\pi\left( \frac{0}{\mathcal{d}} \right)}$, which con­tin­ues down the main di­ag­o­nal is the iden­ti­ty el­e­ment $\epsilon.$ The min­i­mal el­e­ment is the first entry to the right of $a_{1,1}$, for $a_{1,2} = e^{2i\pi\left( \frac{1}{\mathcal{d}} \right)}$, and con­tin­ues along its di­ag­o­nal path to form the min­i­mal el­e­ment matrix. These ma­tri­ces are modulo $\mathcal{d}$, which rep­re­sents a ro­ta­tion of $2i\pi$ ra­di­ans on the unit circle. In the cyclic groups, the prim­i­tive root is the same thing we have been re­fer­ring to as the min­i­mal el­e­ment, and as such, every el­e­ment in the cyclic group can be thought of as a power of this prim­i­tive root. So, an­oth­er way to con­sid­er every entry in the matrix tem­plate is for the set of en­tries that is sim­i­lar to our orig­i­nal de­f­i­n­i­tion, but with the $\mathcal{n}$ vari­able set to 1. This acts on the min­i­mal el­e­ment, or prim­i­tive root, by

and since the prim­i­tive root ro­tates coun­ter­clock­wise by one el­e­ment at a time on unit circle, this trans­lates as one column shift to the right on the tem­plate matrix. We can ex­press any value of $\left( e^{2i\pi\left( \frac{\mathcal{n}}{\mathcal{d}} \right)} \right)^{\mathrm{K}_{\mathcal{me}}}$, as its min­i­mal el­e­ment, or prim­i­tive root, rep­re­sen­ta­tion $\left( e^{2i\pi\left( \frac{1}{\mathcal{d}} \right)} \right)^{\mathcal{n}\mathrm{K}_{\mathcal{me}}}$, for $\mathcal{n}$, the power to which we raise the min­i­mal el­e­ment, which is also equal to the nu­mer­a­tor in our orig­i­nal form. In this form, it is clear to see that every number in the set of el­e­ments

is a power of the min­i­mal el­e­ment. Since $\mathrm{K}_{\mathcal{me}}$ is the non-zero ex­po­nen­tial power that the min­i­mal el­e­ment is raised to in order to equal $\epsilon$, for $\mathrm{K}_{\mathcal{me}}= \mathcal{d}$, we have our first iden­ti­ty $\epsilon$, for a non-zero ex­po­nen­tial. As we have stated in the de­f­i­n­i­tion, $\mathrm{K}_{\mathcal{me}}$ is the number of unique el­e­ments in the set of the min­i­mal el­e­ment basis, after pass­ing $\mathrm{K}_{\mathcal{me}}= \mathcal{d}$, for it­er­a­tive powers $\left( e^{2i\pi\left( \frac{1}{\mathcal{d}} \right)} \right)^{\mathrm{K}_{\mathcal{me}}}$, the cyclic nature of the el­e­ment re­peats itself. There­fore, the total number of unique el­e­ments in the set of the min­i­mal el­e­ment basis equals $\mathrm{K}_{\mathcal{me}}$, which is equal to $\mathcal{d}$. We can now con­sid­er the fol­low­ing equa­tion with subtle changes, and since we are taking powers of the min­i­mal el­e­ment, we know that the wind­ing number will in­crease by one after every factor $\mathrm{K}_{\mathcal{me}}= \mathcal{d}$. Now we can solve again for $\mathrm{K}_{\mathcal{me}}$, having proved that it is equal to $\mathcal{d}$ for the fol­low­ing,

there­fore, $k$ must be equal to the value $\mathcal{n}$.

To clar­i­fy the matrix tem­plate in­tro­duced in the pre­vi­ous the­o­rem, we present an ex­am­ple of a $6 \times 6$ matrix. We start first by the zero-wind iden­ti­ty per­mu­ta­tion $\epsilon_{\mathcal{wind}(0)},$

and the dis­tance trav­elled from $\epsilon_{\mathcal{wind} = 0}$ to the min­i­mal dis­tance point on the unit circle is $e^{2i\pi\left( \frac{1}{6} \right)} = e^{i\pi\left( \frac{1}{3} \right)}$, which is the min­i­mal el­e­ment per­mu­ta­tion $\mathcal{m}\mathcal{e}_{\mathcal{wind}(0)}$

The next per­mu­ta­tion in the cyclic group $Z_{6}$ is

and the next

The cyclic per­mu­ta­tion just prior to ar­riv­ing at the iden­ti­ty has the form $e^{2i\pi\left( \frac{\mathcal{d} - 1}{\mathcal{d}} \right)}$, and for $Z_{6}$ is

which is the per­mu­ta­tion matrix

Based on the cyclic group nature, the next it­er­a­tion brings us to the iden­ti­ty el­e­ment at the point on the unit circle, $e^{2i\pi\left( \frac{6}{6} \right)} = e^{2i\pi(1)} = e^{2i\pi},$oc­cur­ring at the cyclic el­e­ment for$\ \mathcal{k} = \mathcal{d}$, at $e^{2i\pi\left( \frac{\mathcal{k}}{\mathcal{d}} \right)}$. For this ex­am­ple, when $\mathcal{k} = 6$, the it­er­a­tion is lo­cat­ed at $e^{2i\pi\left( \frac{6}{6} \right)} = e^{2i\pi(1)} = \epsilon_{\mathcal{wind}(1)}$,

The min­i­mal el­e­ment chosen for $e^{2i\pi\left( \frac{\mathcal{n}}{\mathcal{d}} \right)}$, was $e^{2i\pi\left( \frac{1}{6} \right)},$which will always have a wind­ing number equal to $\mathcal{n} = 1$ on the com­plex unit circle. We saw the re­duced form was equal to $e^{i\pi\left( \frac{\mathcal{k}}{\frac{\mathcal{d}}{2}} \right)} = e^{i\pi\left( \frac{1}{3} \right)},$where the wind­ing number $\mathcal{k}$ is also equal to $1.$Using the non-re­duced ${e^{i\pi\left( \frac{1}{3} \right)} = e}^{2i\pi\left( \frac{1}{6} \right)}$, we can more read­i­ly see that we are work­ing with the $Z_{6}$ cyclic group. Con­tin­u­ing the ex­am­ple, we know that the next per­mu­ta­tion is going to be the min­i­mal el­e­ment per­mu­ta­tion, or more suc­cinct­ly, $e^{2i\pi\left( \epsilon_{\mathcal{wind}(1)} \right)} \times e^{2i\pi\left( \frac{1}{\mathcal{d}} \right)} = e^{2i\pi\left( 1 + \frac{1}{\mathcal{d}} \right)}= \mathcal{m}\mathcal{e}_{\mathcal{wind}(1)}$, which is the per­mu­ta­tion

There are easier and less ex­pan­sive ways of prov­ing the pre­vi­ous the­o­rem but by show­ing the con­nec­tion the min­i­mal el­e­ment has with the cyclic groups and prim­i­tive root, we can see that they are the same. By defin­ing the cyclic group tem­plate in terms of powers of the min­i­mal el­e­ment, it proves useful to de­scribe a cyclic matrix group $\mathbb{Z}_{\mathcal{d}\left( \mathcal{me} \right)}^{+}$

In $\mathbb{Z}_{\mathcal{d}\left( \mathcal{me} \right)}^{+}$, all en­tries are in terms of the min­i­mal el­e­ment, and the ex­po­nen­tial terms are re­lat­ed to the de­nom­i­na­tor value, (or the $\mathrm{K}_{\mathcal{me}}$ value) $\mathcal{d}$, the wind­ing number $k = \left\{ 0,1,2,3,\ldots \right\},$and the cyclic dis­tance it­er­a­tion $s = \left\{ 0,1,2,3,\ldots \right\}$, away from the iden­ti­ty el­e­ment $\epsilon = \left( e^{2i\pi\left( \frac{1}{\mathcal{d}} \right)} \right)^{\mathcal{d}k},$in the coun­ter­clock­wise ori­en­ta­tion. This $k$ in the cyclic dis­tance re­lates to our former value of $\mathcal{n}$ in the nu­mer­a­tor. We can also define a tem­plate for the clock­wise cyclic group $\mathbb{Z}_{\mathcal{d}\left( \mathcal{me} \right)}^{-}$

with a de­nom­i­na­tor $\mathcal{d} \in\mathbb{Z}^{+}$, wind­ing number $k = \left\{ 0, - 1, - 2, - 3,\ldots \right\},$and a cyclic dis­tance set $\left\{ 0, - 1, - 2, - 3,\ldots, - (\mathcal{d} -1) \right\}$, away from the iden­ti­ty el­e­ment $\epsilon = \left( e^{2i\pi\left( \frac{1}{\mathcal{d}} \right)} \right)^{\mathcal{d}k},$in the clock­wise ori­en­ta­tion. The util­i­ty of these forms for the $\mathbb{Z}_{\mathcal{d}\left( \mathcal{me} \right)}^{\pm}$ groups is having a con­sis­tent one-column right­ward march of the el­e­ments for each ex­po­nent it­er­a­tion, mean­ing that for any wind­ing number $k$, and frac­tion of $2i\pi$ as it re­lates to $\frac{1}{\mathcal{d}},$the group dy­nam­ics are con­sis­tent with the prim­i­tive root el­e­ment. It should be read­i­ly ap­par­ent any el­e­ment choice will it­er­ate $\mathcal{d} =\mathrm{K}_{\mathcal{me}}$ times before re­turn­ing to the iden­ti­ty, $\epsilon = \epsilon = \left( e^{2i\pi\left( \frac{1}{\mathcal{d}} \right)} \right)^{\mathcal{d}k}.$Using our ini­tial ex­am­ple $\left( e^{2i\pi\left( \frac{5}{14} \right)} \right)^{(14)s + k}$, as ex­pect­ed, the fol­low­ing pro­gres­sion holds

The cyclic per­mu­ta­tion will have en­tries

where $k$, is the wind­ing number, $s$ is the cyclic dis­tance in the coun­ter­clock­wise ori­en­ta­tion from the iden­ti­ty $\epsilon$, and $e^{2i\pi\left( \frac{1}{\mathcal{d}} \right)}$, as the min­i­mal el­e­ment. As $d$ gets larger, the min­i­mal el­e­ment gets small­er in ar­clength and closer to the origin at $e^{2i\pi k}$. Taking the limit of the matrix en­tries in the $\mathbb{Z}_{\mathcal{d}\left( \mathcal{me} \right)}^{+}$ group

brings us to the origin for $k\in \mathbb{Z},$and we arrive at the iden­ti­ty based on the wind­ing number $k$.

As a next step, we want to ensure that the func­tion we have been work­ing with can be sim­pli­fied to a single vari­able that rep­re­sents any an­gu­lar ro­ta­tion in ra­di­ans, where the goal is to derive a func­tion $f(\theta)$, such that

Taking the limit as $\Delta(\frac{2\pi k}{\mathcal{d}}$) ap­proach­es zero shows that we can make the sub­sti­tu­tion for $\theta = \frac{2\pi k}{\mathcal{d}}$

where this rep­re­sents an in­fin­i­tes­i­mal change in $e^{i\left( \frac{2\pi k}{\mathcal{d}} \right)}$, which is the de­riv­a­tive, and it will take the same form $\frac{d}{d\theta}\left( e^{i\theta} \right) = ie^{i\theta}$, since the imag­i­nary co­ef­fi­cient $i$ on the left side de­fines the or­thog­o­nal tan­gent of $e^{i\theta}$. So, it can also be said that we have the same or­thog­o­nal­i­ty for $ie^{i\left( \frac{2\pi k}{\mathcal{d}} \right)}$, there­fore, the sub­sti­tu­tion is con­sis­tent for $\theta = \left( \frac{2\pi k}{\mathcal{d}} \right)$, and

We start­ed this sec­tion by talk­ing about smooth­ness and con­tin­u­ous func­tions. Sub­se­quent­ly, we have seen that by taking the limit as $\Delta\left( \frac{2\pi k}{\mathcal{d}} \right)$ ap­proach­es zero, we obtain an in­fin­i­tes­i­mal min­i­mal el­e­ment basis con­sis­tent with a $C^{\infty}$ func­tion since $f(\theta)$ is smooth and in­fi­nite­ly dif­fer­en­tiable.