\section{Introduction and statement of results}\label{Sect1}

It is a classical Abelian theorem of Frobenius \cite{Frobenius} that for an arithmetic function $f:\mathbb N \to \mathbb C$ (for which we will set $f(0):=0$ in this study), if the limit of the average value
$L:=\lim_{N\to \infty}\  \frac{1}{N}\sum_{1\leq k \leq N}f(k)$ exists then for $q\in\mathbb C, |q|<1$, when $q\to 1$ {radially}
we have

\begin{equation}\label{Frob}
\lim_{q\to 1}\  (1-q)\sum_{n\geq 1}f(n)q^n =L.
\end{equation}
Equation \eqref{Frob} is the statement of {\it Abel's convergence theorem} if the sequence $\{f(n)\}$ converges; if $\{f(n)\}$ diverges, one may assign limiting value $L$ to the sequence so long as the average value converges.
In fact, the limit \eqref{Frob} holds if $q\to 1$ through any path in a so-called {\it Stolz sector} of the unit disk, viz. a region with vertex at $z=1$ such that $\frac{|1-q|}{1-|q|}\leq M$ for some $M> 0$. See \cite{Divergent, Tauberian} for more about Abelian theorems and the complementary class of Tauberian theorems.

Note that if $f(n)$ is the indicator function for a subset $S\subseteq \mathbb N$ with arithmetic density $d_S$, then \eqref{Frob} gives the limiting value $L=d_S$ as $q\to 1$. In \cite{Paper1, Paper2}, inspired by work of Alladi \cite{A}, the author and my collaborators
exploit this idea to prove partition-theoretic and $q$-series formulas for $d_S$ with $q\to 1$, as well as at other roots of unity $\zeta$.
In this paper, intended as a complement to \cite{Paper2},
we employ similar techniques to prove partition and $q$-series analogues of \eqref{Frob}.

Let $\mathcal P$ denote the integer partitions. For $\lambda \in \mathcal P$, let $|\lambda|$ denote the {size} of $\lambda$ (sum of parts), $\ell(\lambda)$ denote the {length} (number of parts), and let $\operatorname{sm}(\lambda), \operatorname{lg}(\lambda)$ denote the smallest part and largest part of $\lambda$, respectively, noting $|\emptyset|=\ell(\emptyset)= \operatorname{sm}(\emptyset)= \operatorname{lg}(\emptyset):=0$ for $\lambda=\emptyset$ the empty partition.
For $z,q\in \mathbb C, |q|<1$, let $(z;q)_n:=\prod_{0\leq k <n}(1-zq^k)$ denote the {$q$-Pochhammer symbol}, with $(z;q)_{\infty}:=\lim_{n\to \infty} (z;q)_{n}$. (For further reading on partition theory, see \cite{And}.)

Throughout,
we take $q\to 1$ in a Stolz sector of the unit disk, $L:=\lim_{N\to \infty}\frac{1}{N}\sum_{1\leq k \leq N}f(k)$.
We postpone proofs for Section \ref{Sect2}.

\begin{theorem}\label{thm1}
For $f(n)$ an arithmetic function, we have
\begin{flalign*}
 \lim_{q\to 1}\  (q;q)_{\infty}\sum_{\lambda \in \mathcal P} f\left(\operatorname{lg}(\lambda)\right)q^{|\lambda|}\
=\  L,
\end{flalign*}
where the sum is taken over all partitions $\lambda$. \end{theorem}

Theorem \ref{thm1} is a partition analogue of Frobenius' formula \eqref{Frob}; if $f(n)\to L$ as $n\to \infty$, then the theorem yields a partition analogue of Abel's convergence theorem. Applying partition generating function methods gives further formulas to compute the limit $L$.
We will require the {partition-theoretic M\"{o}bius function} defined in \cite{Schneider_arithmetic}:
\begin{equation}\label{mudef}
\mu_{\mathcal{P}}(\lambda) := \begin{cases} 0 & \rm{if} \ \lambda \ \rm{has \ any\  part\  repeated}, \\
(-1)^{\ell(\lambda)} & \rm{otherwise}. \end{cases}
\end{equation}

\begin{corollary}\label{cor1}
For $f(n)$ an arithmetic function, we have
\begin{flalign*}
 -\lim_{q\to 1}\   \sum_{\lambda \in \mathcal P}\mu_{\mathcal P}(\lambda)f\left(\operatorname{sm}(\lambda)\right)q^{|\lambda|}\   =\  L.
\end{flalign*}
\end{corollary}

\begin{corollary}\label{cor1.5}
For $f(n)$ an arithmetic function, we have
\begin{flalign*}
\lim_{q\to 1}\ (q;q)_{\infty}\sum_{n\geq 1}\frac{f(n)q^n}{(q;q)_{n}}\  =\  -\lim_{q\to 1}\  \sum_{n\geq 1}\sum_{k\geq 1}\frac{(-1)^k f(n) q^{nk+\frac{k(k-1)}{2}}}{(q;q)_{k-1}}\  =\  L.
\end{flalign*}
\end{corollary}

\begin{remark}
Setting $f(n)$ equal to the indicator function for $S\subseteq \mathbb N$ in Theorem \ref{thm1} and Corollaries \ref{cor1} and \ref{cor1.5} re-proves the main arithmetic density results of \cite{Paper1, Paper2} with $\zeta=1$.
\end{remark}

\begin{remark}
That the left-hand limit in Corollary \ref{cor1.5} is equal to $L$ confirms the conjecture the author made just below Example E.1.1 in \cite{SchneiderPhD} with $\zeta=1$.
\end{remark}

\begin{example}
Let $f(n)=\frac{\varphi(n)}{n}$ in Corollary \ref{cor1.5} with $\varphi(n)$ the Euler phi function; it is a well-known fact that the average value of $f(n)$ approaches $L=6/\pi^2$ as $n\to \infty$.  Then we have
$$\lim_{q\to 1}\ (q;q)_{\infty}\sum_{n\geq 1}\frac{\varphi(n)q^n}{n\cdot (q;q)_{n}} \  =\  \frac{6}{\pi^2}.
$$
\end{example}

\begin{remark}
This re-proves Example E.1.1 of \cite{SchneiderPhD} with $\zeta=1$.
\end{remark}

Somewhat surprisingly, if one replaces ``$\operatorname{lg}$'' with ``$\operatorname{sm}$'' in Theorem \ref{thm1}, the limit still holds.

\begin{theorem}\label{thm2}
For $f(n)$ an arithmetic function, we have
\begin{flalign*}
 \lim_{q\to 1}\  (q;q)_{\infty}\sum_{\lambda \in \mathcal P} f\left(\operatorname{sm}(\lambda)\right)q^{|\lambda|}\
=\  L.
\end{flalign*}
 \end{theorem}

Theorem \ref{thm2} is a second partition analogue of \eqref{Frob}. As with Theorem \ref{thm1}, generating function methods yield further formulas to compute $L$.

\begin{corollary}\label{cor2}
For $f(n)$ an arithmetic function, we have
\begin{flalign*}
-\lim_{q\to 1}\   \sum_{\lambda \in \mathcal P}\mu_{\mathcal P}(\lambda)f\left(\operatorname{lg}(\lambda)\right)q^{|\lambda|} \  =\  L.
\end{flalign*}
\end{corollary}

\begin{corollary}\label{cor2.5}
For $f(n)$ an arithmetic function, we have
\begin{flalign*}
\lim_{q\to 1}\ \sum_{n\geq 1}f(n)q^n(q;q)_{n-1}\  = \  \lim_{q\to 1}\ \sum_{n\geq 1}\sum_{k\geq 1}\frac{f(n)q^{nk}}{(q;q)_{k-1}}\  =\  L.
\end{flalign*}
\end{corollary}

\begin{remark}
Setting $f(n)$ equal to the indicator function for $S\subseteq \mathbb N$, then that the second limit in Corollary \ref{cor2} is equal to $L=d_{S}$ re-proves Theorem 3.6 of \cite{Paper2}.
\end{remark}

Now we give an application of these ideas, to an operator from statistical physics that has sparked much recent work in the theories of partitions and modular forms. For $a(\lambda)$ a function defined on partitions, and $|q|<1$, recall the {\it $q$-bracket of Bloch and Okounkov} introduced in \cite{B-O},
\begin{equation} \left<a\right>_q:=\frac{\sum_{\lambda \in \mathcal P}a(\lambda)q^{|\lambda|}}{\sum_{\lambda \in \mathcal P}q^{|\lambda|}}\  =\  (q;q)_{\infty}\sum_{\lambda \in \mathcal P}a(\lambda)q^{|\lambda|}.
\end{equation}
The $q$-bracket can be interpreted as the expected value (i.e., average value) of $a(\lambda)$ over all partitions, given certain hypotheses from statistical physics;
it is connected to modular, quasi-modular and $p$-adic modular forms \cite{B-O,BOW, Padic, Zagier}.

Let us also recall the statistic ${A}_n(q), n\geq1,$ defined in \cite{Paper2}, which serves to distinguish between different classes of limiting phenomena in that paper:
\begin{equation}\label{A_ndef}
{A}_n(q):=a\left((n)\right)+\sum_{\operatorname{sm}(\lambda){\geq}^* n}\left[a(\lambda\cdot (n))-a(\lambda)\right] q^{|\lambda|},
\end{equation}
with $(n)\in \mathcal P$ the partition of $n$ with one part, $\lambda\cdot(n)\in \mathcal P$ the partition formed by adjoining part $n$ to partition $\lambda$, and with ${\geq}^*$ denoting $>$ if $a(\lambda)=0$ at partitions with any part repeated, and denoting $\geq$ otherwise. For $n=0$ we will set $A_0(q):=a(\emptyset)$, which is consistent with \eqref{A_ndef}.

Interestingly, the statistic $A_n(q)$ is intertwined with the $q$-bracket $\left<a\right>_q$.

\begin{theorem}\label{cor3}
For $a(\lambda)$ a function defined on partitions, $|q|<1$, we have that
$$\left<a\right>_q\  =\  (q;q)_{\infty}\sum_{n\geq 0}\frac{A_n(q)q^n}{(q;q)_n}.$$
If $A_k:=\lim_{q\to 1}A_k(q)$ exists for every $k\geq 1$, let
${A}:=\lim_{N\to \infty}\frac{1}{N}\sum_{1\leq k \leq N}{A}_k$. Then
$$\lim_{q\to 1}\  \left<a\right>_q\  =\ A.$$
\end{theorem}

We discuss in passing another, conjectural application. Euler's continued fraction formula \cite{Euler} converts any $q$-hypergeometric series with multiplicative coefficients into a $q$-continued fraction (see \cite{SchneiderPhD}); if $f(n)$ is completely multiplicative, then the $q$-series identities in Corollaries \ref{cor1.5} and \ref{cor2.5} translate to $q$-continued fractions that converge to $L$ as $q\to 1$ in a Stolz sector. Perhaps then applying convergence acceleration and continued fraction techniques like those summarized in \cite{Vanderpoorten} to $q$-series and $q$-continued fractions, could yield approaches to prove irrationality of constants $L$ such as $\zeta(m), m\geq 2$.\footnote{The author
discussed this idea with J. Lagarias and L. Rolen after my talk at the Vanderbilt University Number Theory Seminar,  Oct. 27, 2020.}

\begin{remark}
One anticipates similar limiting formulas hold as $q$ approaches other roots of unity.
\end{remark}

\section{Proofs of our results}\label{Sect2}

The proofs in this section begin with a rewriting of \eqref{Frob}, multiplied by an additional factor of $q$ (which does not affect the asymptotic below), as $q\to1$ in a Stolz sector of the unit disk:
\begin{equation}\label{qasymp}
\sum_{n\geq 1}f(n)q^n \  \sim\  L\cdot \frac{q}{1-q},
\end{equation}
with $L:=\lim_{N\to \infty}\  \frac{1}{N}\sum_{1\leq k \leq N}f(k)$ as above. This asymptotic equality can be used as a building block for producing further $q$-series formulas to compute the limit $L$. Theorem \ref{thm1} and its corollaries result from a generalization of the proof of Theorem 3.5 in \cite{Paper2}.

\begin{proof}[Proof of Theorem \ref{thm1}]
Take $q\mapsto q^k$ in \eqref{qasymp}.  Multiply through by $(-1)^{k} q^{\frac{k(k-1)}{2}}(q;q)_{k-1}^{-1}$ and sum both sides over $k\geq 1$, swapping order of summation on the left-hand side to give
\begin{equation}\label{qleft}
\sum_{n \geq 1} \sum_{k\geq 1} \frac{(-1)^{k} f(n) q^{nk+\frac{k(k-1)}{2}}}{(q;q)_{k-1}}.
\end{equation}
For each $k\geq 1$, the factor $(q;q)_{k-1}^{-1}$ generates partitions with largest part strictly $<k$. The factor $q^{nk}$ adjoins a largest part $k$ with multiplicity $n$ to each partition, for every $n\geq 1$. The $q^{k(k-1)/2}=q^{1+2+3+...+(k-1)}$ factor guarantees at least one part of each size $<k$. Thus \eqref{qleft} is the generating function for partitions $\gamma$ with every natural number $<\operatorname{lg}(\gamma)$ appearing as a part, weighted by $(-1)^{\operatorname{lg}(\gamma)}f\left(m_{\operatorname{lg}}(\gamma) \right)=(-1)^k f(n)$, where $m_{\operatorname{lg}}(\gamma)=n$ denotes the multiplicity of the largest part of each partition $\gamma$.

Under conjugation,
this set of partitions $\gamma$ maps to partitions $\lambda$ into distinct parts weighted by $\mu_{\mathcal P}(\lambda)f\left(\operatorname{sm}(\lambda)\right)=(-1)^{\ell(\lambda)}f\left(\operatorname{sm}(\lambda)\right)=(-1)^k f(n)$, which is nonzero since $\lambda$ has no repeated part.
Thus, noting $f(0):=0$, and multiplying through by a factor of $-1$ to produce the desired end result, we have
\begin{flalign}\label{nexttolast}
-\sum_{n\geq 1}\sum_{k\geq 1}\frac{(-1)^k f(n) q^{nk+\frac{k(k-1)}{2}}}{(q;q)_{k-1}}\  &=\  -\sum_{\lambda \in \mathcal P} \mu_{\mathcal P}(\lambda)f\left(\operatorname{sm}(\lambda)\right)q^{|\lambda|}=\  \sum_{n\geq 1} f(n)q^n(q^{n+1};q)_{\infty}\  \\ \nonumber &= \  (q;q)_{\infty}\sum_{n\geq 1}\frac{f(n)q^n}{(q;q)_n}\  =\  (q;q)_{\infty}\sum_{\lambda \in \mathcal P} f\left(\operatorname{lg}(\lambda)\right)q^{|\lambda|},
\end{flalign}
using standard partition generating function methods. Manipulating the right-hand side of \eqref{qasymp} accordingly gives
\begin{equation}\label{finalstep}
-\sum_{n\geq 1}\sum_{k\geq 1}\frac{(-1)^k f(n)  q^{nk+\frac{k(k-1)}{2}}}{(q;q)_{k-1}}\  \sim \   -L\cdot \sum_{k\geq 1}\frac{(-1)^k q^{\frac{k(k+1)}{2}}}{(q;q)_{k}}\  =\  -L\cdot \left((q;q)_{\infty}-1 \right),\end{equation}
using an identity of Euler in the final step, which approaches $L$ as $q\to 1$. Comparing the right-hand sides of \eqref{nexttolast} and \eqref{finalstep} in the limit, completes the proof. \end{proof}

\begin{proof}[Proof of Corollaries \ref{cor1} and \ref{cor1.5}]
These two corollaries record alternative expressions for the limit $L$ derived during the proof of Theorem \ref{thm1} above.
\end{proof}

The proof of Theorem \ref{thm2} generalizes the proof of Theorem 3.6 in \cite{Paper2}.

\begin{proof}[Proof of Theorem \ref{thm2}]
Take $q\mapsto q^k$ in \eqref{qasymp}. Multiply both sides by $(q;q)_{k-1}^{-1}$, sum over $k\geq 1$, then swap order of summation and make the change of indices $k\mapsto k+1$ on the left, to give
\begin{flalign}\label{asymp3}
\sum_{n\geq 1}\sum_{k\geq 1}\frac{f(n)q^{nk}}{(q;q)_{k-1}}=\sum_{n\geq 1}f(n)q^n\sum_{k\geq 0}\frac{q^{nk}}{(q;q)_{k}}&\  \sim\  L \cdot \sum_{k\geq 1}\frac{q^k}{(q;q)_k}.
\end{flalign}
By the $q$-binomial theorem, the inner sum over $k\geq 0$ in the second double series is equal to $(q^n;q)_{\infty}^{-1}$, and the summation on the right is $(q;q)_{\infty}^{-1}-1$. Multiplying both sides of \eqref{asymp3} by $(q;q)_{\infty}$ gives, from standard generating function arguments,
as $q\to 1$:
 \begin{flalign}\label{asymp4}
(q;q)_{\infty}\sum_{\lambda \in \mathcal P} f\left(\operatorname{sm}(\lambda)\right)q^{|\lambda|}=(q;q)_{\infty}\sum_{n\geq 1} \frac{f(n)q^n}{(q^{n};q)_{\infty}}\ &=\  \sum_{n\geq 1}f(n)q^n(q;q)_{n-1}\\ \nonumber &=\  -\sum_{\lambda\in \mathcal P}\mu_{\mathcal P}(\lambda) f\left(\operatorname{lg}(\lambda)\right)q^{|\lambda|}\  \sim \  L.\end{flalign}
That the left side is asymptotically equal to $L$, is equivalent to the statement of the theorem.
\end{proof}

\begin{proof}[Proof of Corollaries \ref{cor2} and \ref{cor2.5}]
These corollaries record alternative expressions for the limit $L$ derived during the proof of Theorem \ref{thm2} above. \end{proof}

\begin{proof}[Proof of Theorem \ref{cor3}]
The first identity of the theorem is the special case $f(n)=1$ of Lemma 4.3 in \cite{Paper2}, after adding $a(\emptyset)$ to both sides of the lemma, in light of \cite{Paper2}, eq. (21). Then under the hypotheses in the statement of the theorem, we have
\begin{equation}\left<a\right>_q\  =\  (q;q)_{\infty}\sum_{n\geq 0}\frac{A_n(q)q^n}{(q;q)_n}\  \sim\  (q;q)_{\infty}\sum_{n\geq 1}\frac{A_nq^n}{(q;q)_n}\  \sim\  A
\end{equation}
as $q\to 1$, noting $a(\emptyset)\cdot (q;q)_{\infty}\to 0$ so the $n=0$ term vanishes in the first asymptotic, and the final asymptotic is the $f(n)=A_n, L=A$ case of Corollary \ref{cor1.5}.
\end{proof}

The asymptotic equality \eqref{qasymp} can be involved in $q$-series manipulations in diverse ways, much like the formulas involving the $q$-density statistic in \cite{Paper2}, which themselves could be generalized to give new limit formulas as in this paper. For example, if one takes $q\mapsto q^k$ in \eqref{qasymp} and sums over all $k\geq 1$, swapping order of summation on the left in the absolutely convergent double series, then geometric series and classical Lambert series identities yield as $q\to 1$ that
\begin{equation}
\lim_{q\to 1}\  \frac{\sum_{n\geq 1}\frac{f(n)q^n}{1-q^n}}{\sum_{n\geq 1}\frac{q^n}{1-q^n}} \  =\  L \end{equation}
with limit $L$ as defined previously, a generalization of Example 2 in \cite{Paper2}.

Indeed, using techniques from Lambert series, $q$-hypergeometric series, modular forms, et al. (not to mention repeated application of L'Hospital's rule), the creative analyst might produce any number of further ``$q$-Abelian'' theorems.

\section*{Acknowledgments}
The author is indebted to George Andrews, Matthew R. Just, Ken Ono, Paul Pollack, A. V. Sills and Ian Wagner for discussions that strongly informed this study. %


\section*{Abstract}
We show that the Backus (1962) equivalent-medium average, which is an average over a spatial variable, and the Gazis et al. (1963) effective-medium average, which is an average over a symmetry group, do not commute, in general.
They commute in special cases, which we exemplify.
\section{Introduction}
Hookean solids are defined by their mechanical property relating linearly the stress tensor,~$\sigma$\,, and the strain tensor,~$\varepsilon$\,,
\begin{equation*}
\sigma_{ij}=\sum_{k=1}^3\sum_{\ell=1}^3c_{ijk\ell}\varepsilon_{k\ell}\,,\qquad i,j=1,2,3
\,.
\end{equation*}
The elasticity tensor,~$c$\,, belongs to one of eight material-symmetry classes shown in Figure~\ref{fig:orderrelation}.
\begin{figure}
\begin{center}
\includegraphics[scale=0.7]{FigPartialOrder.pdf}
\end{center}
\caption{\small{Order relation of material-symmetry classes of elasticity tensors:
Arrows indicate subgroups in this partial ordering. For instance, monoclinic is a subgroup of all nontrivial symmetries, in particular, of both orthotropic and trigonal, but orthotropic is not a subgroup of trigonal or {\it vice-versa}.}}
\label{fig:orderrelation}
\end{figure}

The Backus (1962) moving average allows us to quantify the response of a wave propagating through a series of parallel layers whose thicknesses are much smaller than the wavelength.
Each layer is a Hookean solid exhibiting a given material symmetry with given elasticity parameters.
The average is a Hookean solid whose elasticity parameters---and, hence, its material symmetry---allow us to model a long-wavelength response.
This material symmetry of the resulting medium, to which we refer as {\sl equivalent}, is a consequence of symmetries exhibited by the averaged layers.

The long-wave-equivalent medium to a stack of isotropic or transversely isotropic layers
with thicknesses much less than the signal wavelength was
shown by Backus (1962) to be a homogeneous or nearly homogeneous transversely
isotropic medium, where a {\it nearly\/} homogeneous medium is a consequence of a {\it moving\/} average.
Backus (1962) formulation is reviewed by Slawinski (2016) and Bos et al.\ (2016),
where formulations for generally anisotropic, monoclinic, and orthotropic thin layers
are also derived.   Bos et al.\ (2016) examine the underlying assumptions and
approximations behind the Backus (1962) formulation, which is derived by
expressing rapidly varying stresses and strains in terms of products of
algebraic combinations of rapidly varying elasticity parameters with
slowly varying stresses and strains.
The only mathematical approximation in the formulation
is that the average of a product of a rapidly varying function and a slowly
varying function is approximately equal to the product of the averages of
the two functions.

According to Backus (1962), the average of $f(x_3)$ of ``width''~$\ell'$  is
\begin{equation}
\label{eq:BackusOne}
\overline f(x_3):=\int\limits_{-\infty}^\infty w(\zeta-x_3)f(\zeta)\,{\rm d}\zeta
\,,
\end{equation}
where $w(x_3)$ is the weight function with the following properties:
\begin{equation*}
w(x_3)\geqslant0\,,
\quad w(\pm\infty)=0\,,
\quad
\int\limits_{-\infty}^\infty w(x_3)\,{\rm d}x_3=1\,,
\quad
\int\limits_{-\infty}^\infty x_3w(x_3)\,{\rm d}x_3=0\,,
\quad
\int\limits_{-\infty}^\infty x_3^2w(x_3)\,{\rm d}x_3=(\ell')^2\,.
\end{equation*}
These properties define $w(x_3)$ as a probability-density function with mean~$0$ and standard deviation~$\ell'$\,, explaining the use of the term ``width'' for $\ell'$\,.

Gazis et al.~(1963) average allows us to obtain the closest symmetric counterpart---in the Frobenius sense---of a chosen material symmetry to a generally anisotropic Hookean solid.
The average is a Hookean solid, to which we refer as {\sl effective}, whose elasticity parameters correspond to the symmetry chosen {\it a priori}.

Gazis average is a projection given by
\begin{equation}
\widetilde c^{\,\,\rm sym}:=\intop_{G^{\rm sym}}(g\circ c)\,\mathrm{d}\mu(g)
\,,
\label{eq:proj}
\end{equation}
where the integration is over the symmetry group, $G^{\rm sym}$\,, whose elements are $g$\,, with respect to the invariant measure, $\mu$\,, normalized so that $\mu(G^{\rm sym})=1$\,; $\widetilde c^{\,\,\rm sym}$ is the orthogonal projection of $c$\,, in the sense of the Frobenius norm,  on the linear space containing all tensors of that symmetry, which are $ c^{\,\,\rm sym}$\,.
Integral~(\ref{eq:proj}) reduces to a finite sum for the classes whose symmetry groups are finite, which are all classes except isotropy and transverse isotropy.

The Gazis et al.\  (1963) approach is reviewed and extended by Danek et al.\ (2013, 2015)
in the context of random errors.
Therein, elasticity tensors are not constrained to the same---or even different but known---orientation of the coordinate system.

Concluding this introduction, let us emphasize that the fundamental distinction between the two averages is their domain of operation.
The Gazis et al.\ (1963) average is an average over symmetry groups at a point and the Backus (1962) average is a spatial average over a distance.
Both averages can be used, separately or together, in quantitative seismology.
Hence, an examination of their commutativity might provide us with an insight into their physical meaning and into allowable mathematical operations.
\section{Generally anisotropic layers and monoclinic medium}
Let us consider a stack of generally anisotropic layers to obtain a monoclinic medium.
To examine the commutativity between the Backus and Gazis averages, let us study the following diagram,
\begin{equation}
\label{eq:CD2}
\begin{CD}
\rm{aniso}@>\rm{B}>>\rm{aniso}\\
@V\mathrm{G}VV                         @VV\rm{G}V\\
\rm{mono}@>>\rm{B}>\rm{mono}
\end{CD}
\end{equation}
and Proposition~\ref{thm:One}, below,
\begin{prop}
\label{thm:One}
In general, the Backus and Gazis averages do not commute.
\end{prop}
\begin{proof}
To prove this proposition and in view of Diagram~\ref{eq:CD2}, let us begin with the following corollary.
\begin{corollary}
For the generally anisotropic and monoclinic symmetries, the Backus and Gazis averages do not commute.
\end{corollary}
\noindent To understand this corollary, we invoke the following lemma, whose proof is in \ref{AppOne1}.
\begin{lemma}
\label{lem:Mono}
For the effective monoclinic symmetry, the result of the Gazis average is tantamount to replacing each $c_{ijk\ell}$\,, in a generally anisotropic tensor, by its corresponding $c_{ijk\ell}$ of the monoclinic tensor, expressed in the natural coordinate system, including replacements of the anisotropic-tensor components by the zeros of the corresponding monoclinic components.
\end{lemma}
\noindent Let us first examine the counterclockwise path of Diagram~\ref{eq:CD2}.
Lemma~\ref{lem:Mono} entails a corollary.
\begin{corollary}
\label{col:Mono}
For the effective monoclinic symmetry, given a generally anisotropic tensor,~$C$\,,
\begin{equation}
\label{eq:GazisMono}
\widetilde{C}^{\,\rm mono}=C^{\,\rm mono}
\,;
\end{equation}
where $\widetilde{C}^{\,\rm mono}$ is the Gazis average of~$C$\,, and $C^{\,\rm mono}$ is a monoclinic tensor whose nonzero entries are the same as for~$C$\,.
\end{corollary}
\noindent According to Corollary~\ref{col:Mono}, the effective monoclinic
tensor is obtained simply by setting  to zero---in the generally anisotropic tensor---the
components that are zero for the monoclinic tensor.
Then, the second counterclockwise branch of Diagram~\ref{eq:CD2} is performed as follows.
Applying the Backus average, we obtain (Bos et al., 2015)
\begin{equation*}
\langle c_{3333}\rangle=\overline{\left(\frac{1}{c_{3333}}\right)}^{\,\,-1}\,,
\qquad
\langle c_{2323}\rangle=\frac{\overline{\left(\frac{c_{2323}}{D}\right)}}{2D_2}\,,
\end{equation*}
\begin{equation*}
\langle c_{1313}\rangle=\frac{\overline{\left(\frac{c_{1313}}{D}\right)}}{2D_2}\,,
\qquad
\langle c_{2313}\rangle=\frac{\overline{\left(\frac{c_{2313}}{D}\right)}}{2D_2}\,,
\end{equation*}
where $D\equiv 2(c_{2323}c_{1313}-c_{2313}^2)$ and
$D_2\equiv (\overline{c_{1313}/D})(\overline{c_{2323}/D})-(\overline{c_{2313}/D})^2$\,.
We also obtain
\begin{equation*}
\langle c_{1133}\rangle=
\overline{\left(\frac{1}{c_{3333}}\right)}^{\,\,-1}
\overline{\left(\frac{c_{1133}}{c_{3333}}\right)}\,,
\quad
\langle c_{2233}\rangle=
\overline{\left(\frac{1}{c_{3333}}\right)}^{\,\,-1}
\overline{\left(\frac{c_{2233}}{c_{3333}}\right)}\,,
\quad
\langle c_{3312}\rangle=
\overline{\left(\frac{1}{c_{3333}}\right)}^{\,\,-1}
\overline{\left(\frac{c_{3312}}{c_{3333}}\right)}\,,
\end{equation*}
\begin{equation*}
\langle c_{1111}\rangle=
\overline{c_{1111}}-\overline{\left(\frac{c_{1133}^2}{c_{3333}}\right)}+
\overline{\left(\frac{1}{c_{3333}}\right)}^{\,\,-1}
\overline{\left(\frac{c_{1133}}{c_{3333}}\right)}^{\,2}\,,
\end{equation*}
\begin{equation*}
\langle c_{1122}\rangle=
\overline{c_{1122}}-\overline{\left(\frac{c_{1133}\,c_{2233}}{c_{3333}}\right)}+
\overline{\left(\frac{1}{c_{3333}}\right)}^{\,\,-1}
\overline{\left(\frac{c_{1133}}{c_{3333}}\right)}\,\,
\overline{\left(\frac{c_{2233}}{c_{3333}}\right)}\,,
\end{equation*}
\begin{equation*}
\langle c_{2222}\rangle=
\overline{c_{2222}}-\overline{\left(\frac{c_{2233}^2}{c_{3333}}\right)}+
\overline{\left(\frac{1}{c_{3333}}\right)}^{\,\,-1}
\overline{\left(\frac{c_{2233}}{c_{3333}}\right)}^{\,2}\,,
\end{equation*}
\begin{equation*}
\langle c_{1212}\rangle=
\overline{c_{1212}}-\overline{\left(\frac{c_{3312}^2}{c_{3333}}\right)}+
\overline{\left(\frac{1}{c_{3333}}\right)}^{\,\,-1}
\overline{\left(\frac{c_{3312}}{c_{3333}}\right)}^{\,2}\,,
\end{equation*}
\begin{equation*}
\langle c_{1112}\rangle=
\overline{c_{1112}}-\overline{\left(\frac{c_{3312}\,c_{1133}}{c_{3333}}\right)}+
\overline{\left(\frac{1}{c_{3333}}\right)}^{\,\,-1}
\overline{\left(\frac{c_{1133}}{c_{3333}}\right)}\,\,
\overline{\left(\frac{c_{3312}}{c_{3333}}\right)}
\end{equation*}
and
\begin{equation*}
\langle c_{2212}\rangle=
\overline{c_{2212}}-\overline{\left(\frac{c_{3312}\,c_{2233}}{c_{3333}}\right)}+
\overline{\left(\frac{1}{c_{3333}}\right)}^{\,\,-1}
\overline{\left(\frac{c_{2233}}{c_{3333}}\right)}\,\,
\overline{\left(\frac{c_{3312}}{c_{3333}}\right)}\,,
\end{equation*}
where angle brackets denote the equivalent-medium elasticity parameters.
The other equivalent-medium elasticity parameters are zero.

Following the clockwise path of Diagram~\ref{eq:CD2},
the upper branch is derived in matrix form in Bos et al.\ (2015).
Then, from Bos et al. (2015) the result of the right-hand branch is
derived by setting entries in the generally anisotropic tensor
that are zero for the monoclinic tensor to zero.
The nonzero entries, which are too complicated to display explicitly, are---in general---not the same as the result of the counterclockwise path.
Hence, for generally anisotropic and monoclinic symmetries, the Backus and Gazis averages do not commute.
\end{proof}
\section{Higher symmetries}
\subsection{Monoclinic layers and orthotropic medium}
\label{sec:mono}
Proposition~\ref{thm:One} remains valid for layers exhibiting higher material symmetries, and simpler expressions of the corresponding elasticity tensors allow us to examine special cases that result in commutativity.
Let us consider the following corollary of Proposition~\ref{thm:One}.
\begin{corollary}
\label{thm:Two}
For the monoclinic and orthotropic symmetries, the Backus and Gazis averages do not commute.
\end{corollary}
\noindent To study this corollary, let us consider the following diagram,
\begin{equation}
\label{eq:CD}
\begin{CD}
\rm{mono}@>\rm{B}>>\rm{mono}\\
@V\mathrm{G}VV                         @VV\rm{G}V\\
\rm{ortho}@>>\rm{B}>\rm{ortho}
\end{CD}
\end{equation}
and the lemma, whose proof is in \ref{AppOne2}.
\begin{lemma}
\label{lem:Ortho}
For the effective orthotropic symmetry, the result of the Gazis average is tantamount to replacing each $c_{ijk\ell}$\,, in a generally anisotropic---or monoclinic---tensor, by its corresponding $c_{ijk\ell}$ of the orthotropic tensor, expressed in the natural coordinate system, including the replacements by the corresponding zeros.
\end{lemma}
\noindent Lemma~\ref{lem:Ortho} entails a corollary.
\begin{corollary}
\label{col:Ortho}
For the effective orthotropic symmetry, given a generally anisotropic---or monoclinic---tensor,~$C$\,,
\begin{equation}
\label{eq:GazisOrtho}
\widetilde{C}^{\,\rm ortho}=C^{\,\rm ortho}
\,.
\end{equation}
where $\widetilde{C}^{\,\rm ortho}$ is the Gazis average of~$C$\,, and $C^{\,\rm ortho}$ is an orthotropic tensor whose nonzero entries are the same as for~$C$\,.
\end{corollary}
\noindent Let us consider a monoclinic tensor and proceed counterclockwise along the first branch of Diagram~\ref{eq:CD}.
Using the fact that the monoclinic symmetry is a special case of general anisotropy, we invoke Corollary~\ref{col:Ortho} to conclude that $\widetilde{C}^{\,\rm ortho}=C^{\,\rm ortho}$\,,
which is equivalent to setting $c_{1112}$\,, $c_{2212}$\,, $c_{3312}$ and $c_{2313}$ to zero
in the monoclinic tensor.
We perform the upper branch of Diagram~\ref{eq:CD}, which is the averaging of a stack of monoclinic layers to get a monoclinic equivalent medium, as in the case of the lower branch of Diagram~\ref{eq:CD2}.
Thus, following the clockwise path, we obtain
\begin{equation*}
c_{1212}^\circlearrowright=
\overline{c_{1212}}-\overline{\left(\frac{c_{3312}^2}{c_{3333}}\right)}+
\overline{\left(\frac{1}{c_{3333}}\right)}^{\,\,-1}
\overline{\left(\frac{c_{3312}}{c_{3333}}\right)}^{\,2}\,,
\end{equation*}
\begin{equation*}
c_{1313}^\circlearrowright=\overline{\left(\frac{c_{1313}}{D}\right)}/(2D_2)\,,\qquad
c_{2323}^\circlearrowright=\overline{\left(\frac{c_{2323}}{D}\right)}/(2D_2)
\end{equation*}
Following the counterclockwise path, we obtain
\begin{equation*}
c_{1212}^\circlearrowleft=\overline{c_{1212}}\,,\quad
c_{1313}^\circlearrowleft=\overline{\left(\frac{1}{c_{1313}}\right)}^{\,\,-1}\,,\quad
c_{2323}^\circlearrowleft=\overline{\left(\frac{1}{c_{2323}}\right)}^{\,\,-1}\,.
\end{equation*}
The other entries are the same for both paths.

In conclusion, the results of the clockwise and  counterclockwise paths are the same if $c_{2313}=c_{3312}=0$\,, which is a special case of monoclinic symmetry.
Thus, the Backus average and Gazis average commute for that case, but not in general.
\subsection{Orthotropic layers and tetragonal medium}
\label{sec:ortho}
In a manner analogous to Diagram~\ref{eq:CD}, but proceeding from the the upper-left-hand corner orthotropic tensor to lower-right-hand corner tetragonal tensor by the counterclockwise path,
\begin{equation}
\label{eq:CD3}
\begin{CD}
\rm{ortho}@>\rm{B}>>\rm{ortho}\\
@V\mathrm{G}VV                         @VV\rm{G}V\\
\rm{tetra}@>>\rm{B}>\rm{tetra}
\end{CD}
\end{equation}
we obtain
\begin{equation*}
c_{1111}^\circlearrowleft=\overline{\frac{c_{1111}+c_{2222}}{2}-
\frac{\left(\frac{c_{1111}+c_{2222}}{2}\right)^2}{c_{3333}}}+
\overline{\left(\frac{c_{1111}+c_{2222}}{2c_{3333}}\right)}^2
\overline{\left(\frac{1}{c_{3333}}\right)}^{\,\,-1}
\,.
\end{equation*}
Following the clockwise path, we obtain
\begin{equation*}
c_{1111}^\circlearrowright=\overline{\frac{c_{1111}+c_{2222}}{2}-
\frac{c_{1133}^2+c_{2233}^2}{2c_{3333}}}+
\frac{1}{2}\left[\overline{\left(\frac{c_{1133}}{c_{3333}}\right)}^2+
\overline{\left(\frac{c_{2233}}{c_{3333}}\right)}^2\right]
\overline{\left(\frac{1}{c_{3333}}\right)}^{\,\,-1}\,.
\end{equation*}
These results are not equal to one another, unless $c_{1133}=c_{2233}$\,, which is a special case of orthotropic symmetry.
Also $c_{2323}$ must equal $c_{1313}$ for $c_{2323}^\circlearrowright=c_{2323}^\circlearrowleft$.
The other entries are the same for both paths.
Thus, the Backus average and Gazis average do commute for $c_{1133}=c_{2233}$ and $c_{2323}=c_{1313}$\,, which is a special case of orthotropic symmetry, but not in general.

Let us also consider the case of monoclinic layers and a tetragonal medium to examine the process of combining the Gazis averages, which is tantamount to combining Diagrams~(\ref{eq:CD}) and~(\ref{eq:CD3}),
\begin{equation}
\begin{CD}
\label{eq:CD4}
\rm{mono}@>\rm{B}>>\rm{mono}\\
@V\mathrm{G}VV                         @VV\rm{G}V\\
\rm{ortho}@>>\rm{B}>\rm{ortho}\\
@V\mathrm{G}VV                         @VV\rm{G}V\\
\rm{tetra}@>>\rm{B}>\rm{tetra}
\end{CD}
\end{equation}
In accordance with Proposition~\ref{thm:One}, there is---in general---no commutativity.
However, the outcomes are the same as for the corresponding steps in Sections~\ref{sec:mono} and  \ref{sec:ortho}.
In general, for the Gazis average, proceeding directly, $\rm{aniso}\xrightarrow{\rm{G}}\rm{iso}$\,, is tantamount to proceeding along arrows in Figure~\ref{fig:orderrelation}, $\rm{aniso}\xrightarrow{\rm{G}}\cdots\xrightarrow{\rm{G}}\rm{iso}$\,.
No such combining of the Backus averages is possible, since, for each step, layers become a homogeneous medium.
\subsection{Transversely isotropic layers}
Lack of commutativity can also be exemplified by the case of transversely isotropic layers.
Following the clockwise path of Diagram~\ref{eq:CD}, the Backus average results in a transversely isotropic medium, whose Gazis average---in accordance with Figure~\ref{fig:orderrelation}---is isotropic.
Following the counterclockwise path, Gazis average results in an isotropic medium, whose Backus average, however, is transverse isotropy.
Thus, not only the elasticity parameters, but even the resulting material-symmetry classes differ.

Also, we could---in a manner analogous to the one illustrated in Diagram~\ref{eq:CD4}\,---begin with generally anisotropic layers and obtain isotropy by the clockwise path and transverse isotropy by the counterclockwise path, which again illustrates noncommutativity.
\section{Discussion}
Herein, we assume that all tensors are expressed in the same orientation of their coordinate systems. Otherwise, the process of averaging become more complicated, as discussed---for the Gazis average---by Kochetov and Slawinski (2009a, 2009b) and as mentioned---for the Backus average---by Bos et al. (2016).

Mathematically, the noncommutativity of two distinct averages is shown by Proposition~\ref{thm:One}, and exemplified for several material symmetries.

We do not see a physical justification for special cases in which---given the same orientation of coordinate systems---these averages commute.
This behaviour might support the view that a mathematical realm, which allows for fruitful analogies with the physical world, has no causal connection with it.
\section*{Acknowledgments}
We wish to acknowledge discussions with Theodore Stanoev. This research was performed in the context of The Geomechanics Project
supported by Husky Energy. Also, this research was partially supported by the
Natural Sciences and Engineering Research Council of Canada, grant 238416-2013.
\section*{References}
\frenchspacing
\newcommand{\par\noindent\hangindent=0.4in\hangafter=1}{\par\noindent\hangindent=0.4in\hangafter=1}
\par\noindent\hangindent=0.4in\hangafter=1
Backus, G.E.,  Long-wave elastic anisotropy produced by horizontal layering,
{\it  J. Geophys. Res.\/}, {\bf 67}, 11, 4427--4440, 1962.
\setlength{\parskip}{4pt}
\par\noindent\hangindent=0.4in\hangafter=1
B\'{o}na, A., I. Bucataru and M.A. Slawinski, Space of $SO(3)$-orbits of elasticity tensors, {\it  Archives of Mechanics\/}, {\bf 60}, 2, 121--136, 2008
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Bos, L, D.R. Dalton, M.A. Slawinski and T. Stanoev,
On Backus average for generally anisotropic layers, {\it arXiv\/}, 2016.
\par\noindent\hangindent=0.4in\hangafter=1
Chapman, C. H., {\it Fundamentals of seismic wave propagation\/},  Cambridge
University Press, 2004.
\par\noindent\hangindent=0.4in\hangafter=1
Danek, T., M. Kochetov and M.A. Slawinski,  Uncertainty analysis of
effective elasticity tensors using quaternion-based global optimization and
Monte-Carlo method, {\it The Quarterly Journal of Mechanics and Applied
Mathematics\/}, {\bf 66}, 2, pp. 253--272, 2013.
\par\noindent\hangindent=0.4in\hangafter=1
Danek, T., M. Kochetov and M.A. Slawinski, Effective elasticity tensors
in the context of random errors, {\it Journal of Elasticity\/}, 2015.
\par\noindent\hangindent=0.4in\hangafter=1
Gazis, D.C., I. Tadjbakhsh and R.A. Toupin, The elastic tensor of given symmetry nearest to an anisotropic elastic tensor, {\it Acta Crystallographica\/}, {\bf 16}, 9, 917--922, 1963.
\par\noindent\hangindent=0.4in\hangafter=1
Kochetov, M. and M.A. Slawinski, On obtaining effective orthotropic
elasticity tensors, {\it The Quarterly Journal of Mechanics and Applied
Mathematics\/}, {\bf 62}, 2, pp. 149-Ð166, 2009a.
\par\noindent\hangindent=0.4in\hangafter=1
Kochetov, M. and M.A. Slawinski,  On obtaining effective transversely
isotropic elasticity tensors, {\it Journal of Elasticity\/}, {\bf 94}, 1Ð-13., 2009b.
\par\noindent\hangindent=0.4in\hangafter=1
Slawinski, M.A. {\it Wavefronts and rays in seismology: Answers to unasked questions\/},
World Scientific, 2016.
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Slawinski, M.A., {\it Waves and rays in elastic continua\/}, World Scientific, 2015.
\par\noindent\hangindent=0.4in\hangafter=1
Thomson, W., {\it Mathematical and physical papers: Elasticity, heat,
electromagnetism\/}, Cambridge University Press, 1890
\setcounter{section}{0}
\setlength{\parskip}{0pt}
\renewcommand{\thesection}{Appendix~\Alph{section}}
\section{}
\subsection{}\label{AppOne1}
Let us prove Lemma~\ref{lem:Mono}.
\begin{proof}
For discrete symmetries, we can write integral~(\ref{eq:proj}) as a sum,
\begin{equation}
\label{eq:AverageDisc}
\widetilde C^{\,\rm sym}=\frac{1}{n}\left(\tilde{A}_1^{\rm sym}\,C\,\tilde{A}_1^{\rm sym}\,{}^{^T}+\ldots+\tilde{A}_n^{\rm sym}\,C\,\tilde{A}_n^{\rm sym}\,{}^{^T}\right)
\,,
\end{equation}
where $\widetilde C^{\rm sym}$ is expressed in Kelvin's notation, in view
of  Thomson (1890, p.~110) as discussed in Chapman (2004, Section~4.4.2).

To write the elements of the monoclinic symmetry group as $6\times 6$ matrices, we must consider orthogonal transformations in $\mathbb{R}^3$\,.
Transformation $A\in SO(3)$ of $c_{ijk\ell}$ corresponds to transformation of $C$ given by
\begin{equation}
{\footnotesize
\tilde{A}=\left[\begin{array}{cccccc}
A_{11}^{2} & A_{12}^{2} & A_{13}^{2} & \sqrt{2}A_{12}A_{13} & \sqrt{2}A_{11}A_{13} & \sqrt{2}A_{11}A_{12}\\
A_{21}^{2} & A_{22}^{2} & A_{23}^{2} & \sqrt{2}A_{22}A_{23} & \sqrt{2}A_{21}A_{23} & \sqrt{2}A_{21}A_{22}\\
A_{31}^{2} & A_{32}^{2} & A_{33}^{2} & \sqrt{2}A_{32}A_{33} & \sqrt{2}A_{31}A_{33} & \sqrt{2}A_{31}A_{32}\\
\sqrt{2}A_{21}A_{31} & \sqrt{2}A_{22}A_{32} & \sqrt{2}A_{23}A_{33} & A_{23}A_{32}+A_{22}A_{33} & A_{23}A_{31}+A_{21}A_{33} & A_{22}A_{31}+A_{21}A_{32}\\
\sqrt{2}A_{11}A_{31} & \sqrt{2}A_{12}A_{32} & \sqrt{2}A_{13}A_{33} & A_{13}A_{32}+A_{12}A_{33} & A_{13}A_{31}+A_{11}A_{33} & A_{12}A_{31}+A_{11}A_{32}\\
\sqrt{2}A_{11}A_{21} & \sqrt{2}A_{12}A_{22} & \sqrt{2}A_{13}A_{23} & A_{13}A_{22}+A_{12}A_{23} & A_{13}A_{21}+A_{11}A_{23} & A_{12}A_{21}+A_{11}A_{22}\end{array}\right]}
\,,
\label{eq:ATildeQ}
\end{equation}
which is an orthogonal matrix, $\tilde{A}\in SO(6)$ (Slawinski (2015), Section~5.2.5).\footnote{Readers interested in formulation of matrix~(\ref{eq:ATildeQ}) might refer to B\'ona et al. (2008).}

The required symmetry-group elements are
\begin{equation*}
 A_1^{\rm mono}=
\left[
\begin{array}{ccc}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1\end{array}\right]
\mapsto
\left[\begin{array}{cccccc}
1 & 0 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 1\end{array}\right]
=\tilde{A}_1^{\rm mono}
\end{equation*}
\begin{equation*}
 A_2^{\rm mono}=
\left[
\begin{array}{ccc}
-1 & 0 & 0\\
0 & -1 & 0\\
0 & 0 & 1\end{array}\right]
\mapsto
\left[\begin{array}{cccccc}
1 & 0 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & -1 & 0 & 0\\
0 & 0 & 0 & 0 & -1 & 0\\
0 & 0 & 0 & 0 & 0 & 1\end{array}\right]
=\tilde{A}_2^{\rm mono}
\,.
\end{equation*}
For the monoclinic\index{material symmetry!monoclinic} case, expression~(\ref{eq:AverageDisc}) can be stated explicitly as
\begin{equation*}
\widetilde C^{\rm mono}=
\frac{\left(\tilde{A}_1^{\rm mono}\right)\,C\,\left(\tilde{A}_1^{\rm mono}\right)^T+\left(\tilde{A}_2^{\rm mono}\right)\,C\,\left(\tilde{A}_2^{\rm mono}\right)^T}{2}
\,.
\end{equation*}
Performing matrix operations, we obtain
\begin{equation}
\widetilde C^{\rm mono}
=\left[\begin{array}{cccccc}
c_{1111} & c_{1122} & c_{1133} & 0 & 0 & \sqrt{2}c_{1112}\\
c_{1122} & c_{2222} & c_{2233} & 0 & 0 & \sqrt{2}c_{2212}\\
c_{1133} & c_{2233} & c_{3333} & 0 & 0 & \sqrt{2}c_{3312}\\
0 & 0 & 0 & 2c_{2323} & 2c_{2313} & 0\\
0 & 0 & 0 & 2c_{2313} & 2c_{1313} & 0\\
\sqrt{2}c_{1112} & \sqrt{2}c_{2212} & \sqrt{2}c_{3312} & 0 & 0 & 2c_{1212}
\end{array}\right]
\,,
\label{eq:MonoExplicitRef}
\end{equation}
which exhibits the form of the monoclinic tensor in its natural coordinate system.
In other words, $\widetilde{C}^{\rm mono}=C^{\rm mono}$\,, in accordance with Corollary~\ref{col:Mono}.
\subsection{}\label{AppOne2}
Let us prove Lemma~\ref{lem:Ortho}.

For orthotropic symmetry, $\tilde{A}_1^{\rm ortho}=\tilde{A}_1^{\rm mono}$ and
$\tilde{A}_2^{\rm ortho}=\tilde{A}_2^{\rm mono}$ and
\begin{equation*}
 A_3^{\rm ortho}=
\left[
\begin{array}{ccc}
-1 & 0 & 0\\
0 &  1 & 0\\
0 & 0 & -1\end{array}\right]
\mapsto
\left[\begin{array}{cccccc}
1 & 0 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & -1 & 0 & 0\\
0 & 0 & 0 & 0 &  1 & 0\\
0 & 0 & 0 & 0 & 0 & -1\end{array}\right]
=\tilde{A}_3^{\rm ortho}
\,,
\end{equation*}
\begin{equation*}
 A_4^{\rm ortho}=
\left[
\begin{array}{ccc}
1 & 0 & 0\\
0 &  -1 & 0\\
0 & 0 & -1\end{array}\right]
\mapsto
\left[\begin{array}{cccccc}
1 & 0 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 &  -1 & 0\\
0 & 0 & 0 & 0 & 0 & -1\end{array}\right]
=\tilde{A}_4^{\rm ortho}
\,.
\end{equation*}
For the orthotropic\index{material symmetry!orthotropic} case, expression~(\ref{eq:AverageDisc}) can be stated explicitly as
{\footnotesize
\begin{equation*}
\widetilde C^{\rm ortho}=
\frac{\left(\tilde{A}_1^{\rm ortho}\right)\,C\,\left(\tilde{A}_1^{\rm ortho}\right)^T+\left(\tilde{A}_2^{\rm ortho}\right)\,C\,\left(\tilde{A}_2^{\rm ortho}\right)^T
+\left(\tilde{A}_3^{\rm ortho}\right)\,C\,\left(\tilde{A}_3^{\rm ortho}\right)^T+\left(\tilde{A}_4^{\rm ortho}\right)\,C\,\left(\tilde{A}_4^{\rm ortho}\right)^T
}
{4}
\,.
\end{equation*}}
Performing matrix operations, we obtain
\begin{equation}
\widetilde C^{\rm ortho}
=\left[\begin{array}{cccccc}
c_{1111} & c_{1122} & c_{1133} & 0 & 0 & 0\\
c_{1122} & c_{2222} & c_{2233} & 0 & 0 & 0\\
c_{1133} & c_{2233} & c_{3333} & 0 & 0 &0\\
0 & 0 & 0 & 2c_{2323} & 0 & 0\\
0 & 0 & 0 & 0 & 2c_{1313} & 0\\
0& 0 & 0 & 0 & 0 & 2c_{1212}
\end{array}\right]
\,,
\label{eq:OrthoExplicitRef}
\end{equation}
which exhibits the form of the orthotropic tensor in its natural coordinate system.
In other words, $\widetilde{C}^{\rm ortho}=C^{\rm ortho}$\,, in accordance with Corollary~\ref{col:Ortho}.
\end{proof}
\end{document}