# Categorification of the Müller-Wichards System Performance Estimation Model: Model Symmetries, Invariants, and Closed Forms

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## Abstract

**:**

## 1. Introduction

## 2. Definitions, Terminology, and Preliminary Lemmas

#### 2.1. Categories and Morphisms

- For every pair of objects $X,Y\in Ob{j}_{\mathit{C}}$ there is a (possibly empty) set $Mo{r}_{\mathit{C}}\left(X,Y\right)$ of morphisms from $X$ to $Y$;
- For any $X,Y,Z\in Ob{j}_{\mathit{C}}$ there is a composition “$\circ $” of morphisms$$\circ :Mo{r}_{\mathit{C}}\left(X,Y\right)\times Mo{r}_{\mathit{C}}\left(Y,Z\right)\to Mo{r}_{\mathit{C}}\left(X,Z\right)$$
- For every $X\in Ob{j}_{\mathit{C}}$ there is an identity morphism ${1}_{X}\in Mo{r}_{\mathit{C}}\left(X,X\right)$ such that for $f\in Mo{r}_{\mathit{C}}\left(X,Y\right)$ and $g\in Mo{r}_{\mathit{C}}\left(Y,X\right)$, $f\circ {1}_{X}=f$ and ${1}_{X}\circ g=g$;
- When defined, composition of morphisms is associative, i.e., $\left(f\circ g\right)\circ h=f\circ \left(g\circ h\right)$.

- The category $\mathit{S}\mathit{e}\mathit{t}$ where $Ob{j}_{\mathit{S}\mathit{e}\mathit{t}}$ is the collection of all sets, the morphisms are the ordinary mappings between sets, and $\circ $ is the usual composition of maps.
- The category $\mathit{G}\mathit{r}\mathit{p}$ where $Ob{j}_{\mathit{G}\mathit{r}\mathit{p}}$ is the collection of all groups, the morphisms are the ordinary group homomorphisms, and $\circ $ is the usual composition of homomorphisms.

**Grp**satisfy items 3 and 4 above.

#### 2.2. Functors

- $F{1}_{X}={1}_{FX}$ for every $X\in Ob{j}_{\mathit{C}}$; and
- When $f\circ g=h$ is defined in $\mathit{C}$, then $Ff\circ Fg=Fh$ is defined in $\mathit{D}$ and $Ff\circ Fg=F\left(f\circ g\right)=Fh$.

- The identity functor ${1}_{\mathit{C}}:\mathit{C}\to \mathit{C}$ which makes the assignments ${1}_{\mathit{C}}X=X$ for every $X\in Ob{j}_{\mathit{C}}$ and ${1}_{\mathit{C}}f=f$ for every $f\in Mo{r}_{\mathit{C}}\left(X,Y\right)$.
- The forgetful functor $U:\mathit{G}\mathit{r}\mathit{p}\to \mathit{S}\mathit{e}\mathit{t}$ which assigns to every group $G\in Ob{j}_{\mathit{G}\mathit{r}\mathit{p}}$ its underlying set $UG\in Ob{j}_{\mathit{S}\mathit{e}\mathit{t}}$ and to each homomorphism $f\in Mo{r}_{\mathit{G}\mathit{r}\mathit{p}}\left(G,H\right)$ the set map $Uf\in Mo{r}_{\mathit{S}\mathit{e}\mathit{t}}\left(UG,UH\right)$ (i.e., $U$ forgets group structure going from $\mathit{G}\mathit{r}\mathit{p}$ to $\mathit{S}\mathit{e}\mathit{t}$).

#### 2.3. Preliminary Lemmas

**Lemma**

**1 [9].**

**Lemma**

**2 [9].**

**Lemma**

**3 [24]**.

**Lemma**

**5 [24]**.

## 3. The Müller-Wichards Performance Categories

- If ${g}_{1}=\left({r}_{1},{w}_{1},{u}_{1}\right)\in \wp $ and ${g}_{2}=\left({r}_{2},{w}_{2},{u}_{2}\right)\in {\wp}^{\prime}$, then ${g}_{1}{\otimes}^{q}{g}_{2}=\left(r,w,u\right)$, where$$u={u}_{1}\u02c5{u}_{2},$$$$w=Re\left[{u}_{1}{w}_{1}+{u}_{2}{w}_{2}+\neg u\left({w}_{1}+{w}_{2}\right)\right],$$$$r=\{\begin{array}{c}\frac{\left|w\right|}{{\left[{\left(\frac{\left|{w}_{1}\right|}{{r}_{1}}\right)}^{q}+{\left(\frac{\left|{w}_{2}\right|}{{r}_{2}}\right)}^{q}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}},q\infty \\ \frac{\left|w\right|}{\underset{}{\mathrm{max}}\left[\frac{\left|{w}_{1}\right|}{{r}_{1}},\frac{\left|{w}_{2}\right|}{{r}_{2}}\right]},q=\infty \end{array}$$
- If ${g}_{1},{g}_{2}\in V$, then ${g}_{1}{\otimes}^{q}{g}_{2}=\left(r,i,u\right),$ where $u$ and $r$ are given by Equations $\left(8\right)$ and $\left(10\right)$, respectively.

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

**Theorem**

**5.**

**Theorem**

**6.**

**Theorem**

**7.**

## 4. The Category of Systems and the Müller-Wichards Performance Functors

**Theorem**

**8**.

**Theorem**

**9**.

**Theorem**

**10**.

**Theorem**

**11.**

## 5. The CMW Fundamental Symmetry Group and Associated Invariants

**Theorem**

**12.**

**Corollary**

**1.**

**Theorem**

**13.**

**Theorem**

**14.**

## 6. System Factorization and Product Performance Models

**Theorem**

**15.**

**Theorem**

**16.**

**Corollary**

**2.**

**Corollary**

**3.**

## 7. Closed Form Models

**Theorem**

**17.**

**Theorem**

**18.**

**Theorem**

**19.**

## 8. Statistical Properties of $r$ for Closed Form Models

#### 8.1. Fixed Rates and Independent Random Weights

#### 8.2. Random Rates and Random Weights

## 9. Special Case Closed Form Models

**SCCFM**

**1.**

**SCCFM**

**2.**

**SCCFM**

**3.**

**SCCFM**

**4.**

## 10. Applications of Special Case Closed Form Models

#### 10.1. The Effect of $q$ Upon the ${t}_{q}^{X}-{n}_{X}$ Dependence, $X\in \left\{C,D,V\right\}$

#### 10.2. A Stationary Action Principle for SCCFM4 System Performance

#### 10.3. Invariance of the Equation of Motion

## 11. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Proof of Theorem**

**2.**

**Proof of Theorem**

**3.**

**Proof of Theorem**

**4.**

**Proof of Theorem**

**5.**

**Proof of Theorem**

**6.**

**Proof of Theorem**

**7.**

**Proof of Theorem**

**8.**

**Proof of Theorem**

**9.**

**Proof of Theorem**

**10.**

**Proof of Theorem**

**11.**

**Proof of Theorem**

**12.**

**Proof of Theorem**

**13.**

**Proof of Theorem**

**14.**

**Proof of Theorem**

**15.**

**Proof of Theorem**

**16.**

**Proof of Corollary**

**2.**

**Proof of Corollary**

**3.**

**Proof of Theorem**

**17.**

**Proof of Theorem**

**18.**

**Proof of Theorem**

**19.**

**Proof of**

**SCCFM1.**

**Proof of**

**SCCFM3.**

**Proof of**

**SCCFM4.**

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**Figure 3.**Probability density functions for $r$ for various $q$ values when ${r}_{I}=1,n=10,$ and $\lambda =10,20,40$ (

**top**to

**bottom**).

**Figure 4.**Probability density functions for $r$ when ${r}_{I}=1,q\in \left\{1.5,2,2.5,3,4,\infty \right\}$, and $n\in \left\{2,4,8,10,16\right\}.$

**Figure 5.**Probability density functions for $r$ when $n=10,\lambda =10,\gamma =2\left(top\right),$ $\gamma =10$ (

**bottom**) and $q\in \left\{1.5,2,2.5,3,4,\infty \right\}$ (left to right).

**Figure 6.**Probability density functions for $r$ when $n=10,\lambda =10,\gamma \in \left\{1,2,3\right\},$ and $q\in \left\{1,1.5,2,2.5,3,4,\infty \right\}$. Each $r$ value for $\gamma \in \left\{2,3\right\}$ has been divided by the associated value of $\gamma $.

**Figure 7.**Probability density functions for $r$ when $n=10,\lambda =10\mathrm{and}\text{}q\in \left\{1.5,2,2.5,3,4\right\}$.

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Parks, A.D.; Marchette, D.J.
Categorification of the Müller-Wichards System Performance Estimation Model: Model Symmetries, Invariants, and Closed Forms. *Systems* **2019**, *7*, 6.
https://doi.org/10.3390/systems7010006

**AMA Style**

Parks AD, Marchette DJ.
Categorification of the Müller-Wichards System Performance Estimation Model: Model Symmetries, Invariants, and Closed Forms. *Systems*. 2019; 7(1):6.
https://doi.org/10.3390/systems7010006

**Chicago/Turabian Style**

Parks, Allen D., and David J. Marchette.
2019. "Categorification of the Müller-Wichards System Performance Estimation Model: Model Symmetries, Invariants, and Closed Forms" *Systems* 7, no. 1: 6.
https://doi.org/10.3390/systems7010006