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+{"text":"\\section{Introduction}\n\\vspace{-2mm}\nMusic listeners typically rely on a combination of listening contexts to find music including elements of mood, theme, time of day, location and activity. This scenario can be handled by defining a dictionary of contextual terms and directly associating them with music as a class label \\cite{yan2015improving, ibrahim2020audio}. However, such a music tagging approach (i.e., multi-label classification) is severely limited in considering contextual expression complexities that listeners can use from a natural language perspective. For example, a listener may use `club' to search for electronic dance music, and unless a model is trained with this specific word, it is not possible to consider the word as a query string. This issue has been addressed by representing tag words with embedding vectors and associating them with music in several different settings such as zero-shot learning \\cite{choi2019zero}, query-by-blending \\cite{watanabe2019query} and multi-task music representation learning \\cite{schindler2019multi}. The aforementioned approaches were based on system training utilizing word embedding with either general text (e.g., Wikipedia or Gigaword) or music-specific corpus (e.g., tags, lyrics, artist IDs, track IDs). What is noteworthy here is that the general text training approach is limited in reflecting \"musical\" dimensions, whereas music-specific corpus limits incorporation of listening contexts which are not directly related to music while simultaneously suffering from small vocabulary size. \nIn this work, we investigate various word embedding spaces trained with  combinations of general and music-specific text data to bridge the gap between listening contexts and music. \n\n\n\\vspace{-3mm}\n\\section{Datasets and Method} \n\\vspace{-2mm}\nWe conducted our research using the latest \\textit{Wikipedia} dump\\footnote{\\url{https:\/\/dumps.wikimedia.org\/enwiki\/20200601\/}} for general text data and a hybrid music corpus for music-specific text data. The music corpus is composed of \\textit{Amazon} album review, \\textit{AllMusic} tags\\footnote{\\url{https:\/\/www.allmusic.com}}, and artist\/track IDs. The \\textit{Amazon} album review data contain consumer opinions about the music \\cite{He2016}, which was obtained from the MuMu dataset \\footnote{\\url{https:\/\/www.upf.edu\/web\/mtg\/mumu}} \\cite{oramas2017multi}. The \\textit{Allmusic} dataset includes music tags (genre, style) and context tags (mood and theme) \\cite{schindler2019multi}. The artist\/track IDs were obtained from the MSD dataset \\cite{bertin2011million}. The IDs are also regarded as a unique word associated with the corresponding music \\cite{watanabe2019query}. \nWe used Word2Vec based on Continuous Bag of Words (CBOW) to learn word embedding \\cite{mikolov2013efficient}. For music corpus, we clustered review texts, tags, and artist\/track IDs for each music track using MSD track\\_id \\footnote{\\url{http:\/\/millionsongdataset.com\/}} and MusicBrainz id \\footnote{\\url{https:\/\/musicbrainz.org\/}} and also took the context window within the cluster. Additionally, we shuffled words within clusters to address data augmentation. Although this method broke review sentence order, it improved capture of word co-occurrences in the hybrid set with greater spread.\n\n\n\n\n\\begin{table*}[!t]\n\\begin{center}\n\\vspace{-4mm}\n\\caption{Compare ranking evaluation metric between 7 embedding spaces.}\n\\label{tab:1}\n\\begin{tiny}\n\\begin{sc}\n\\resizebox{0.9\\textwidth}{!}{\n\\begin{tabular}{ccccccccc}\n\\toprule\n\\multirow{2}{*}[-0.7ex]{Corpus} & \\multirow{2}{*}[-0.7ex]{Size} & \\multirow{2}{*}[-0.7ex]{\\begin{tabular}[c]{@{}c@{}}Unique \\\\ Word\\end{tabular}} & \\multirow{2}{*}[-0.7ex]{\\begin{tabular}[c]{@{}c@{}}Unique \\\\ Track\\end{tabular}} & \\multirow{2}{*}[-0.7ex]{\\begin{tabular}[c]{@{}c@{}}Unique \\\\ Artist\\end{tabular}} & \\multicolumn{2}{c}{AllMusic (Seen)} & \\multicolumn{2}{c}{LastFm (Unseen)} \\\\ \\cmidrule(lr){6-7} \\cmidrule(lr){8-9}\n &  &  &  &  & spearmanr & nDCG@30 & spearmanr & nDCG@30 \\\\ \\midrule\n{[}AllMusic Tags   + Amazon Music Reviews{]} (Augmented) + Wikipedia & 1.98B & 11,622,471 & 521,778 & 28,330 & 0.194 & 0.327 & 0.312 & 0.591 \\\\ \\cmidrule{1-9}\nAllMusic Tags + Amazon Music Reviews +   Wikipedia & 1.8B & 11,622,471 & 521,778 & 28,330 & 0.187 & 0.233 & 0.226 & 0.548 \\\\ \\cmidrule{1-9}\nAllMusic Tags   + Wikipedia & 1.76B & 11,163,229 & 507,435 & 25,203 & 0.157 & 0.215 & 0.183 & 0.526 \\\\ \\midrule\n{[}AllMusic Tags   + Amazon Music Reviews{]} (Augmented) & 0.27B & 664,163 & 521,778 & 28,330 & \\textbf{0.267} & \\textbf{0.339} & \\textbf{0.407} & \\textbf{0.626} \\\\ \\cmidrule{1-9}\nAllMusic Tags + Amazon Music Reviews & 45.3m & 664,163 & 521,778 & 28,330 & 0.187 & 0.232 & 0.358 & 0.600 \\\\ \\cmidrule{1-9}\nAllMusic Tags & 7.1m & 1,401 & 507,435 & 25,203 & 0.252 & 0.242 &  &  \\\\ \\midrule\nWikipedia & 1.75B & 11,163,055 & 0 & 0 & 0.098 & 0.167 & 0.162 & 0.551\n\\\\\\bottomrule\n\\end{tabular}\n}\n\\end{sc}\n\\end{tiny}\n\\end{center}\n\\vspace{-3mm}\n\\end{table*}\n\n\n\\begin{figure*}[!t]\n\\begin{center}\n\\centerline{\\includegraphics[width=0.95\\textwidth]{imgs\/re_2col_padding.png}}\n\\vspace{-2mm}\n\\caption{Tag visualization of different type of embedding using t-SNE. The music corpus includes \\textit{AllMusic} Tags, \\textit{Amazon} music reviews, MSD Artist\/Track IDs. \n}\n\\label{figure:fig1}\n\\end{center}\n\\vspace{-8mm}\n\\end{figure*}\n\n\n\\vspace{-3mm}\n\\section{Experiments}\n\\vspace{-2mm}\n\nWe trained the word embedding model with vector size 100, window size 15, and five iterations. To test word embedding, we used test tags from two different datasets and thus, different characteristics. \\textit{Allmusic} was used to enable a balanced distribution of music terms and context terms consisting of 1,401 genre\/style music tags and mood\/theme context tags. This dataset was also used for training (seen) as part of the music corpus. The \\textit{last.fm} dataset with genre, mood, and eras tags was also used. We selected the top 100 tags with maximal occurrence frequency. The latter dataset which focuses on music terms was not included in the training (unseen) phase. To measure word-to-word similarity performance of the proposed word embedding system, we employed a co-occurrence of tags scheme for ground truth creation. We then measured spearman's rank correlation and normalized discounted cumulative gain at \\textit{k} (nDCG@k) between ground-truth co-occurrence and word-to-word similarity of word embeddings. For the nDCG evaluation, we use the top \\textit{k} retrieved words ($k=30$).\n\n\n\n\n\n\n\n\n\\vspace{-3mm}\n\\section{Results and Discussion}\n\\vspace{-2mm}\nTable \\ref{tab:1} shows performance results, size of the training corpus, unique words, unique tracks, and unique artists of each method. The results show that the two word embeddings including music corpus significantly outperform the model trained with \\textit{Wikipedia} only. This is expected as the test sets were based on music tag datasets. Between music corpus only and music corpus with Wikipedia, the result depended on how many music terms and context terms are balanced in the test sets. When music terms are concentrated (\\textit{last.fm} tags), word embedding trained with music corpus only outperformed that with both music corpus and Wikipedia. However, in the balanced case (\\textit{AllMusic} tags), word embedding trained with both music corpus and Wikipedia resulted in improved performances. Table \\ref{tab:1} also shows that the augmented music corpus achieved notable high performance results. This suggests that the proposed data augmentation is beneficial when the order of words is not important. \nThe t-SNE plot in Figure \\ref{figure:fig1} provides a more intuitive visualization of our research results. Here, we used two music genre terms `electronic' and `house' and three listening context terms `club', `club\\_dance', and `partying' as relevant words. In \\textit{Wikipedia}, the gaps between terms are significant with only `house' and `club' in close proximity. In the music corpus, the two genre terms and `club' and `club\\_dance' are tightly clustered while `partying' is significantly beyond the cluster centroid. In the music corpus with \\textit{Wikipedia}, while the context term `partying' is still outside of the cluster containing all of the other terms, it is substantially closer than the music corpus example. This indicates that using both general and music-specific data has the potential of capturing a more balanced correlation between music and listening context (for examples of music retrieval tasks using context words, please refer to \\footnote{\\url{https:\/\/dohppak.github.io\/MusicWordVec}}).\n\n\n\n\n\n\\vspace{-3mm}\n\\section{Future Work}\n\\vspace{-2mm}\nOur current plan is to expand on findings as reported in this paper and build a set of user-annotated word-to-word similarity pairs to directly measure the relationship between general words, music contexts, and music tracks. We also plan to additionally use the musical word embedding system from trained word embedding as a prototype vector for each music track in the context of audio-based music regression \\cite{van2013deep}, classification, and metric learning \\cite{choi2019zero} settings. This will allow us to construct a more nuanced audio embedding system as conventional music classification is in the order hundreds of labels as class prototypes, while the proposed approach allows for half a million of track prototypes that are strongly reflective of millions of music context terms. \n\n\n\n\n\n\\nocite{langley00}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nProperties of Laplacian eigenvalues on bounded domains subject to various boundary conditions are a classical topic in spectral theory due to the fact that these eigenvalues can be interpreted, e.g., as frequencies of vibrating membranes; see Lord Rayleigh's famous book {\\em The Theory of Sound}~\\cite{Rayleigh}. A prominent line of research in this context is related to inequalities between Dirichlet and Neumann eigenvalues; its history dates back at least to the 1950s. On a bounded, sufficiently regular, connected domain $\\Omega \\subset \\R^d$ denote by \n\\begin{align*}\n 0 < \\lambda_1 < \\lambda_2 \\leq \\lambda_3 \\leq \\cdots\n\\end{align*}\nthe eigenvalues of the (negative) Laplacian subject to a Dirichlet boundary condition on the boundary $\\partial \\Omega$ and by\n\\begin{align*}\n 0 = \\mu_1 < \\mu_2 \\leq \\mu_3 \\leq \\cdots\n\\end{align*}\nthe eigenvalues corresponding to a Neumann condition. A variational argument easily implies $\\mu_k \\leq \\lambda_k$ for all $k \\in \\N$, but in fact several non-trivial improvements of this inequality were found in the course of time. In 1952 P\\'olya~\\cite{P52} proved $\\mu_2 <\\lambda_1$ in the two-dimensional case, see also Szeg\\H{o}'s contribution~\\cite{S54}. Shortly after, in 1955 Payne~\\cite{P55} showed $\\mu_{k + 2} < \\lambda_k$ for all $k \\in \\N$ for convex, two-dimensional $\\Omega$ with $C^2$-boundary. This result was extended and generalized three decades later by Levine and Weinberger~\\cite{LW86}, who (amongst other estimates) obtained the inequality $\\mu_{k + d} \\leq \\lambda_k$ for all $k \\in \\N$, for arbitrary convex domains. For not necessarily convex bounded $C^1$-domains the inequality $\\mu_{k + 1} \\leq \\lambda_k$ for all $k \\in \\N$ was established by Friedlander~\\cite{F91} in 1991, which is valid for all $d$. In 2004 Filonov~\\cite{F05} showed that even $\\mu_{k + 1} < \\lambda_k$ for all $k \\in \\N$ holds in every space dimension $d \\geq 2$ and for every bounded Lipschitz (and even more general) domain.\n\nThe present paper focuses on Laplacian eigenvalues for the mixed case of a Dirichlet boundary condition on a nonempty part $\\Gamma = \\Gamma_{\\rm D}$ of $\\partial \\Omega$ and a Neumann condition on the complement $\\Gamma_{\\rm N}$ of $\\Gamma_{\\rm D}$ in $\\partial \\Omega$. These boundary conditions are ``in between'' the Neumann and Dirichlet problems in the sense that the corresponding eigenvalues\n\\begin{align*}\n 0 < \\lambda_1^\\Gamma < \\lambda_2^\\Gamma \\leq \\lambda_3^\\Gamma \\leq \\cdots\n\\end{align*}\nsatisfy \n\\begin{align}\\label{eq:trivial}\n \\mu_k \\leq \\lambda_k^\\Gamma \\leq \\lambda_k \\quad \\text{for~all}~k \\in \\N;\n\\end{align}\nthis is a trivial consequence of variational principles. Our aim here is to investigate the position of the eigenvalues of the mixed problem in comparison with the Neumann and Dirichlet eigenvalues in more detail. In general this position will depend on the size of the Dirichlet and Neumann parts $\\Gamma_{\\rm D}$ and $\\Gamma_{\\rm N}$, respectively, as well as on their geometries. In  this paper we study the case of Lipschitz domains which are polygonal or polyhedral or have some polyhedra-like properties.\n\nIn our first main result, Theorem~\\ref{thm:NeumannMixed}, we provide an improvement of the first inequality in~\\eqref{eq:trivial} comparing Neumann and mixed Laplacian eigenvalues. Here we assume that $\\Omega$ is a bounded Lipschitz domain and that the ``Neumann part'' $\\Gamma_{\\rm N}$ of the boundary is small enough in the sense that there exists a nontrivial vector being tangential to almost all points in $\\Gamma_{\\rm N}$. Under these conditions we obtain the inequality\n\\begin{align}\\label{eq:INeumann}\n \\mu_{k + 1} \\leq \\lambda_k^\\Gamma \\quad \\text{for~all}~k \\in \\N.\n\\end{align}\nThis result applies to several configurations. For instance, the assumptions of the theorem are satisfied if $\\Gamma_{\\rm N}$ is a part of $\\partial \\Omega$ having zero curvature into at least one direction or if, in three or more space dimensions, $\\Gamma_{\\rm N}$ consists of two flat parts of the boundary, e.g.\\ two faces of a polyhedron; cf.\\ the corollaries and examples in Section~\\ref{sec:Neumann}.\n\nOur second main result deals with the comparison of mixed and Dirichlet eigenvalues, aiming at an improvement of the second inequality in~\\eqref{eq:trivial}. Due to the methods of proof used in this part of the paper we restrict ourselves to the case that $\\Omega$ is a polygonal (for $d = 2$) or polyhedral (for $d \\geq 3$), convex domain. Letting $l$ be the number of linearly independent vectors which are tangential to almost all points of $\\Gamma_{\\rm D}$, in Theorem~\\ref{thm:DirichletMixed} we show the inequality\n\\begin{align}\\label{eq:DIdentity}\n \\lambda_{k + l}^\\Gamma \\leq \\lambda_k \\quad \\text{for~all}~k \\in \\N.\n\\end{align}\nIf, for instance, $\\Gamma_{\\rm D}$ is one face of the polyhedral domain $\\Omega \\subset \\R^d$ then~\\eqref{eq:DIdentity} implies\n\\begin{align*}\n \\lambda_{k + d - 1}^\\Gamma \\leq \\lambda_k \\quad \\text{for~all}~k \\in \\N.\n\\end{align*}\nThus the comparison of mixed and Dirichlet eigenvalues exhibits some dimension dependence similar to the comparison of Neumann and Dirichlet eigenvalues found in~\\cite{LW86}. On the other hand, if $\\Gamma_{\\rm D}$ consists of at most $d - 1$ pairwise non-parallel faces then~\\eqref{eq:DIdentity} yields\n\\begin{align*}\n \\lambda_{k + 1}^\\Gamma \\leq \\lambda_k \\quad \\text{for~all}~k \\in \\N.\n\\end{align*}\nFor further consequences of Theorem~\\ref{thm:DirichletMixed} we refer the reader to the corollaries in Section~\\ref{sec:Dirichlet}.\n\nWe point out that in general none of the inequalities~\\eqref{eq:INeumann} and~\\eqref{eq:DIdentity} is strict. This can be seen from simple examples of a square or a cube and proper choices of the Dirichlet and Neumann parts of the boundary, see Example~\\ref{ex:square} and Example~\\ref{ex:cube} below. However, under additional assumptions on the choice of $\\Gamma_{\\rm D}$ and $\\Gamma_{\\rm N}$ strict inequality can be obtained, see Corollary~\\ref{cor:strict} and Corollary~\\ref{cor:DirichletMixedOneFace}.\n\nThe proofs of our main results are based on variational principles and proper choices of test functions. For the proof of~\\eqref{eq:INeumann} we choose an exponential function suitable to the joint tangent vector of $\\Gamma_{\\rm N}$; cf.~\\cite{F05} for the use of an exponential test function in the comparison of Neumann and Dirichlet eigenvalues. For the proof of~\\eqref{eq:DIdentity} we employ appropriate linear combinations of derivatives of Dirichlet eigenfunctions as test functions; this is motivated by~\\cite{LW86}. However, our calculations differ essentially from those made in~\\cite{LW86} as the mentioned work makes use of differential geometric tools and curvature properties of the boundary while the proof of our Theorem~\\ref{thm:DirichletMixed} relies on an integral identity for polyhedral domains (which fails for general, curved domains). For $d = 2$ this identity is contained in Grisvard's classical book~\\cite{G85}; in the appendix of the present work we provide a proof of it for arbitrary dimensions.\n\nFinally, let us mention that eigenvalue inequalities for Laplacians and more general elliptic operators where studied recently in~\\cite{AM12, BRS16,FL10,GM09,K10, LR15, R14}. Especially inequalities for Laplacian eigenvalues of particular polygonal domains like triangles and rhombi have attracted interest recently due to applications to the hot spots conjecture and other problems, see, e.g.,~\\cite{S15,S16}. For further literature on mixed elliptic boundary value problems (sometimes also called Zaremba problems) we refer the reader to~\\cite{A11,B94,G11,P16,S02,S68}. For elliptic boundary value problems on polygonal and polyhedral domains see the monographs~\\cite{D88, G85,MR10}.\n\n\n\n\n\n\n\\section{Preliminaries: Laplacian eigenvalue problems with mixed boundary conditions}\n\\label{sec:prelim}\n\nLet us first fix some notation and recall some basic facts. Throughout the whole paper $\\Omega \\subset \\R^d$, $d \\geq 2$, is a bounded, connected Lipschitz domain. Recall that by Rademacher's theorem for almost all $x' \\in \\partial \\Omega$ there exists a well-defined outer unit normal vector $\\nu (x')$. Consequently, the $(d - 1)$-dimensional tangential hyperplane\n\\begin{align}\\label{eq:tangential}\n T_{x'} = \\bigg\\{\\tau = (\\tau_1, \\dots, \\tau_d)^\\top \\in \\R^d : \\sum_{j = 1}^d \\tau_j \\nu_j (x') = 0 \\bigg\\}\n\\end{align}\ncan be defined for almost all $x' \\in \\partial \\Omega$. We denote by $H^k (\\Omega)$ the Sobolev spaces of orders $k \\geq 1$ on $\\Omega$ and by $H^s (\\partial \\Omega)$ the Sobolev spaces of orders $s \\in [- 1\/2, 1\/2]$ on $\\partial \\Omega$; in particular, for $s \\in [0, 1\/2]$ the space $H^{- s} (\\partial \\Omega)$ is the dual of $H^{s} (\\partial \\Omega)$. For $u \\in H^1 (\\Omega)$ we denote by $u |_{\\partial \\Omega} \\in H^{1\/2} (\\partial \\Omega)$ the trace of $u$. Moreover, if $u \\in H^1 (\\Omega)$ with $\\Delta u \\in L^2 (\\Omega)$ distributionally then the normal derivative $\\partial_\\nu u |_{\\partial \\Omega} \\in H^{- 1\/2} (\\partial \\Omega)$ of $u$ on $\\partial \\Omega$ can be defined via Green's identity\n\\begin{align}\\label{eq:Green}\n \\int_\\Omega \\nabla u \\cdot \\overline{\\nabla v} \\dd x = - \\int_\\Omega \\Delta u \\overline{v} \\dd x + (\\partial_\\nu u |_{\\partial \\Omega}, v |_{\\partial \\Omega})_{\\partial \\Omega}, \\quad v \\in H^1 (\\Omega),\n\\end{align}\nwhere $(\\cdot, \\cdot)_{\\partial \\Omega}$ is the (sesquilinear) duality between $H^{- 1\/2} (\\partial \\Omega)$ and $H^{1\/2} (\\partial \\Omega)$; cf., e.g.,~\\cite[Chapter~4]{McL}. If $u$ is sufficiently regular up to the boundary, for instance $u \\in H^2 (\\Omega)$, then $\\partial_\\nu u |_{\\partial \\Omega} = \\nu \\cdot \\nabla u |_{\\partial \\Omega}$ almost everywhere on $\\partial \\Omega$; in this case the duality in~\\eqref{eq:Green} turns into the boundary integral of $\\partial_\\nu u |_{\\partial \\Omega} \\overline{v |_{\\partial \\Omega}}$ with respect to the standard surface measure on $\\partial \\Omega$. In the following, for a relatively open subset $\\omega$ of $\\partial \\Omega$ we write $\\partial_\\nu u |_{\\omega} = 0$ if\n\\begin{align}\\label{eq:partNeumann}\n (\\partial_\\nu u |_{\\partial \\Omega}, v |_{\\partial \\Omega} )_{\\partial \\Omega} = 0 \\quad \\text{for~all}~v \\in \n H^1 (\\Omega)~\\text{such~that}~v |_{\\partial \\Omega \\setminus \\overline \\omega} = 0.\n\\end{align}\nNote that for $u$ being sufficiently regular in a neighborhood of $\\omega$ the condition~\\eqref{eq:partNeumann} simply means $(\\nu \\cdot \\nabla u) |_\\omega = 0$.\n\nIn order to write down the mixed Dirichlet--Neumann eigenvalue problem, we make the following assumptions.\n\n\\begin{hypothesis}\\label{hyp}\nWe assume that $\\Gamma = \\Gamma_{\\rm D}$ and $\\Gamma_{\\rm N}$ are two relatively open, non\\-empty subsets of $\\partial \\Omega$ such that $\\Gamma_{\\rm D} \\cap \\Gamma_{\\rm N} = \\emptyset$ and $\\partial \\Omega \\setminus (\\Gamma_{\\rm D} \\cup \\Gamma_{\\rm N})$ has measure zero.\n\\end{hypothesis}\n\nUnder the assumption of Hypothesis~\\ref{hyp} we define\n\\begin{align*}\n H_{0, \\Gamma}^1 (\\Omega) = \\left\\{u \\in H^1 (\\Omega) : u |_{\\Gamma} = 0 \\right\\},\n\\end{align*}\nthe space of functions in $H^1 (\\Omega)$ whose trace vanishes on $\\Gamma$. The negative Laplacian subject to a Dirichlet boundary condition on $\\Gamma = \\Gamma_{\\rm D}$ and a Neumann boundary condition on $\\Gamma_{\\rm N}$ is given by\n\\begin{align*}\n - \\Delta_\\Gamma u = - \\Delta u, \\quad \\dom (- \\Delta_\\Gamma) = \\left\\{ u \\in H_{0, \\Gamma}^1 (\\Omega) : \\Delta u \\in L^2 (\\Omega), \\partial_\\nu u |_{\\Gamma_{\\rm N}} = 0 \\right\\}.\n\\end{align*}\nThe operator $- \\Delta_\\Gamma$ is selfadjoint in $L^2 (\\Omega)$ and has a purely discrete spectrum. In fact, $- \\Delta_\\Gamma$ corresponds to the closed, nonnegative, symmetric sesquilinear form $\\{u, v\\} \\mapsto \\int_\\Omega \\nabla u \\cdot \\overline{\\nabla v} \\dd x$ with domain $H_{0, \\Gamma}^1 (\\Omega)$; cf.~\\cite{BS87,Kato,S12} for more details on semi-bounded selfadjoint operators and corresponding quadratic forms. Therefore the eigenvalues of the mixed Laplacian $- \\Delta_\\Gamma$, ordered nondecreasingly and counted with multiplicities, are given by the min-max principle\n\\begin{align}\\label{eq:minmax}\n \\lambda_k^\\Gamma = \\min_{\\substack{L \\subset H_{0, \\Gamma}^1 (\\Omega) \\\\ \\dim L = k}} \\,\\, \\max_{u \\in L \\setminus \\{0\\}} \\frac{\\int_\\Omega |\\nabla u|^2 \\dd x}{\\int_\\Omega |u|^2 \\dd x}, \\quad k \\in \\N.\n\\end{align}\nAs is well-known, the eigenvalues of the selfadjoint Laplacian with a Neumann boundary condition on the whole boundary $\\partial \\Omega$ are given by\n\\begin{align}\\label{eq:minMaxNeumann}\n \\mu_k = \\min_{\\substack{L \\subset H^1 (\\Omega) \\\\ \\dim L = k}} \\,\\, \\max_{u \\in L \\setminus \\{0\\}} \\frac{\\int_\\Omega |\\nabla u|^2 \\dd x}{\\int_\\Omega |u|^2 \\dd x}, \\quad k \\in \\N.\n\\end{align}\n\n\nIn the following we provide a first, simple observation on the behavior of the eigenvalues of the mixed problem when the Dirichlet part of the boundary is increased. The next, preparatory lemma is a simple consequence of a unique continuation principle; it can be proven similar to~\\cite[Lemma~3.1]{R14}.\n\n\\begin{lem}\\label{lem:UCP}\nLet $\\Omega$ be a bounded, connected Lipschitz domain, let $\\lambda \\in \\R$ and let $u \\in H^1 (\\Omega)$ be such that $- \\Delta u = \\lambda u$. If $\\omega \\subset \\partial \\Omega$ is a relatively open, nonempty set such that $u |_\\omega = 0$ and $\\partial_\\nu u |_\\omega = 0$ then $u = 0$ identically on $\\Omega$.\n\\end{lem}\n\nThe previous lemma can be used to derive the following strict monotonicity principle, which will be used in the following sections.\n\n\\begin{prop}\\label{prop:monotonicity}\nAssume that $\\Gamma \\subset \\Gamma' \\subset \\partial \\Omega$ are nonempty, relatively open sets such that $\\Gamma' \\setminus \\Gamma$ has a nontrivial interior. Then\n\\begin{align*}\n \\lambda_k^\\Gamma < \\lambda_k^{\\Gamma'}\n\\end{align*}\nholds for all $k \\in \\N$.\n\\end{prop}\n\n\\begin{proof}\nLet $k \\in \\N$ and $\\lambda = \\lambda_k^{\\Gamma'}$. By the min-max principle~\\eqref{eq:minmax} there exists a subspace $L \\subset H_{0, \\Gamma'}^1 (\\Omega)$ with $\\dim L = k$ such that\n\\begin{align*}\n \\int_\\Omega |\\nabla u|^2 \\dd x \\leq \\lambda \\int_\\Omega |u|^2 \\dd x, \\quad u \\in L.\n\\end{align*}\nHence for all $u \\in L$ and all $v \\in \\ker (- \\Delta_\\Gamma - \\lambda)$ we have $u + v \\in H_{0, \\Gamma}^1 (\\Omega)$ and\n\\begin{align}\\label{eq:usualCalc}\n\\begin{split}\n \\int_\\Omega |\\nabla (u + v)|^2 \\dd x & = \\int_\\Omega |\\nabla u|^2 \\dd x + 2 \\Real \\int_\\Omega \\nabla v \\cdot \\overline{\\nabla u} \\dd x + \\int_\\Omega |\\nabla v|^2 \\dd x \\\\\n & \\leq \\lambda \\int_\\Omega |u|^2 \\dd x + 2 \\lambda \\Real \\int_\\Omega v \\overline{u} \\dd x + \\lambda \\int_\\Omega |v|^2 \\dd x \\\\\n & = \\lambda \\int_\\Omega |u + v|^2 \\dd x,\n\\end{split}\n\\end{align}\nwhere we have used Green's identity~\\eqref{eq:Green} as well as $u |_\\Gamma = 0$ and $\\partial_\\nu v |_{\\partial \\Omega \\setminus \\overline \\Gamma} = 0$. Moreover, $L \\cap \\ker (- \\Delta_\\Gamma - \\lambda) = \\{0\\}$, which follows from Lemma~\\ref{lem:UCP} when choosing $\\omega$ to be the interior of $\\Gamma' \\setminus \\Gamma$. Thus\n\\begin{align*}\n \\dim \\big( L + \\ker (- \\Delta_\\Gamma - \\lambda) \\big) = k + \\dim \\ker (- \\Delta_\\Gamma - \\lambda)\n\\end{align*}\nand with~\\eqref{eq:usualCalc} and the min-max principle it follows\n\\begin{align}\\label{eq:dasIsses}\n \\lambda_k^\\Gamma \\leq \\lambda_{k + \\dim \\ker (- \\Delta_\\Gamma - \\lambda)}^\\Gamma \\leq \\lambda.\n\\end{align}\nSince $\\lambda_k^\\Gamma = \\lambda$ together with~\\eqref{eq:dasIsses} would imply $\\lambda_k^\\Gamma = \\lambda_{k + \\dim \\ker (- \\Delta_\\Gamma - \\lambda)}^\\Gamma = \\lambda$, i.e., $\\lambda$ is an eigenvalue of $-\\Delta_\\Gamma$ of multiplicity $\\dim \\ker (- \\Delta_\\Gamma - \\lambda) + 1$ or larger, a contradiction, it follows $\\lambda_k^\\Gamma < \\lambda = \\lambda_k^{\\Gamma'}$.\n\\end{proof}\n\nPolygonal and (multidimensional) polyhedral domains play an important role in the following sections. In order to avoid ambiguities we give the following definition.\n\n\\begin{dfn}\\label{def:poly}\nLet $\\Omega \\subset \\R^d$, $d \\geq 2$, be a bounded, connected Lipschitz domain. \n\\begin{enumerate}\n \\item If $d = 2$ we say that $\\Omega$ is a polyhedral (or polygonal) domain if $\\partial \\Omega$ is the union of finitely many line segments.\n \\item Recursively, if $d \\geq 3$ we say that $\\Omega$ is a polyhedral domain if for each ($d - 1$)-dimensional affine hyperplane $H \\subset \\R^d$ the intersection $H \\cap \\Omega$ is either a polyhedral domain in $\\R^{d - 1}$ (where we identify $H$ with $\\R^{d - 1}$) or empty.\n\\end{enumerate}\n\\end{dfn}\n\nNote that in the case $d = 3$ a bounded Lipschitz domain is polyhedral if and only if its boundary is the union of finitely many polygonal faces.\n\n\n\\section{Inequalities for Neumann and mixed eigenvalues}\\label{sec:Neumann}\n\nIn this section we compare Neumann and mixed Laplacian eigenvalues for polyhedral and more general domains in any space dimension $d \\geq 2$. We assume that $\\Omega \\subset \\R^d$, $d \\geq 2$, is a bounded, connected Lipschitz domain and that Hypothesis~\\ref{hyp} is satisfied. As before we denote by $0 = \\mu_1 < \\mu_2 \\leq \\mu_2 \\leq \\dots$ the Neumann Laplacian eigenvalues and by $\\lambda_1^\\Gamma < \\lambda_2^\\Gamma \\leq \\lambda_3^\\Gamma \\leq \\dots$ the eigenvalues of $- \\Delta_\\Gamma$. \n\nFor the following theorem recall that the tangential hyperplane $T_{x'}$ exists for almost all $x' \\in \\partial \\Omega$; cf.~\\eqref{eq:tangential}. We define $\\hat \\Gamma_{\\rm N}$ to be the set of all $x' \\in \\Gamma_{\\rm N}$ such that $T_{x'}$ exists. We define the linear subspace\n\\begin{align*}\n \\cS(\\Gamma_{\\rm N}) := \\bigcap_{x' \\in \\hat \\Gamma_{\\rm N}} T_{x'}\n\\end{align*}\nof $\\R^d$ consisting of all vectors being tangential to all $x' \\in \\Gamma_{\\rm N}$ apart from a set of measure zero. With this notation the main result of this section looks as follows.\n\n\\begin{thm}\\label{thm:NeumannMixed}\nLet $\\Omega \\subset \\R^d$, $d \\geq 2$, be a bounded, connected Lipschitz domain and let Hypothesis~\\ref{hyp} be satisfied. If $\\dim \\cS(\\Gamma_{\\rm N}) \\geq 1$ then \n\\begin{align}\\label{eq:Jawoll}\n \\mu_{k+1} \\leq \\lambda_k^\\Gamma\n\\end{align}\nholds for all $k\\in\\dN$.\n\\end{thm}\n\n\\begin{proof}\nLet $k \\in \\N$ and define $\\lambda = \\lambda_k^\\Gamma > 0$. By the min-max principle~\\eqref{eq:minmax} there exists a subspace $L$ of $H_{0, \\Gamma}^1 (\\Omega)$ such that $\\dim L = k$ and\n\\begin{align}\\label{eq:EigenspaceMinMax}\n \\int_\\Omega |\\nabla u|^2 \\dd x \\leq \\lambda \\int_\\Omega |u|^2 \\dd x\n\\end{align}\nholds for all $u \\in L$. Due to the assumption $\\dim \\cS (\\Gamma_{\\rm N}) \\geq 1$ there exists a vector $\\omega_0 \\in \\cS (\\Gamma_{\\rm N})$ such that $|\\omega_0| = \\sqrt{\\lambda}$ holds. Letting $v (x) = e^{i \\omega_0 \\cdot x}$, $x \\in \\Omega$, we have $v \\in H^2 (\\Omega)$, $\\nabla v = i \\omega_0 v$, and $- \\Delta v = \\lambda v$. With the help of~\\eqref{eq:EigenspaceMinMax}, for each $u \\in L$ and each $c \\in \\dC$ we obtain\n\\begin{align}\\label{eq:anfang}\n\\begin{split}\n \\int_\\Omega |\\nabla (u + c v)|^2 \\dd x & = \\int_\\Omega |\\nabla u|^2 \\dd x + 2 \\Real \\int_\\Omega c \\nabla v \\cdot \\overline{\\nabla u} \\dd x + \\int_\\Omega |c \\nabla v|^2 \\dd x \\\\\n & \\leq \\lambda \\int_\\Omega |u|^2 \\dd x + 2 \\Real \\int_\\Omega c \\nabla v \\cdot \\overline{\\nabla u} \\dd x + \\lambda \\int_\\Omega |c v|^2 \\dd x.\n\\end{split}\n\\end{align}\nMoreover, Green's identity~\\eqref{eq:Green} together with $u |_{\\Gamma_{\\rm D}} = 0$ and $\\omega_0 \\cdot \\nu |_{\\Gamma_{\\rm N}} = 0$ yields\n\\begin{align}\\label{eq:ende}\n\\begin{split}\n \\int_\\Omega \\nabla v \\cdot \\overline{\\nabla u} \\dd x & = - \\int_\\Omega \\Delta v \\overline{u} \\dd x + (\\partial_\\nu v |_{\\partial \\Omega}, u |_{\\partial \\Omega})_{\\partial \\Omega} \\\\\n & = \\lambda \\int_\\Omega v \\overline{u} \\dd x + \\int_{\\partial \\Omega} i v \\overline{u} \\omega_0 \\cdot \\nu \\dd \\sigma \\\\\n & = \\lambda \\int_\\Omega v \\overline{u} \\dd x,\n\\end{split}\n\\end{align}\nwhere $\\sigma$ is the standard surface measure on $\\partial \\Omega$. Combining~\\eqref{eq:anfang} and~\\eqref{eq:ende} we arrive at\n\\begin{align}\\label{eq:letzte}\n \\int_\\Omega |\\nabla (u + c v)|^2 \\dd x & \\leq \\lambda \\int_\\Omega |u + c v|^2 \\dd x \n\\end{align}\nfor all $u \\in L$ and all $c \\in \\C$. Moreover, the function $v$ does not belong to $L$ as all functions in $L$ vanish on $\\Gamma$. Hence $\\dim (L + \\spann \\{v\\}) = k + 1$ and~\\eqref{eq:letzte} together with the min-max principle~\\eqref{eq:minMaxNeumann} implies the assertion of the theorem.\n\\end{proof}\n\nThe following corollaries are direct consequences of Theorem~\\ref{thm:NeumannMixed}. They illustrate the application of Theorem~\\ref{thm:NeumannMixed} to domains with partially flat boundaries. \n\n\\begin{cor}\\label{cor:eins}\nLet $\\Omega$ be a bounded, connected Lipschitz domain in $\\R^2$ and assume that $\\Gamma_{\\rm N}$ is contained in the union of parallel line segments. Then~\\eqref{eq:Jawoll} holds for all $k\\in\\dN$.\n\\end{cor}\n\n\\begin{cor}\\label{cor:zwei}\nLet $\\Omega$ be a bounded, connected Lipschitz domain in $\\R^3$ and assume that $\\Sigma \\subset \\partial \\Omega$ is the union of two plane parts and all plane parts of the boundary which are parallel to one of these two. If $\\Gamma_{\\rm N} \\subset \\Sigma$ then~\\eqref{eq:Jawoll} holds for all $k\\in\\dN$.\n\\end{cor}\n\nThe domains $\\Omega_1$ and $\\Omega_2$ in Figure~\\ref{fig:Neumann} are examples to which the previous corollaries apply.\n\n\\begin{figure}[h]\n\\begin{tikzpicture}\n\\pgfsetlinewidth{0.8pt}\n\\color{gray}\n\\pgfputat{\\pgfxy(2.3,-0.4)}{\\pgfbox[center,base]{$\\Gamma_{\\rm N}$}}\n\\color{gray}\n\\pgfxyline(-0.7,-0.8)(-1.5,0)\n\\pgfxyline(1.6,0)(2.6,-1)\n\\color{black}\n\\pgfxyline(-1.5,0)(1.6,0)\n\\pgfxyline(2.6,-1)(1.3,-1.3)\n\\draw (1.3,-1.3) arc [radius=2, start angle=45, end angle= 107];\n\\pgfputat{\\pgfxy(1.2,-0.6)}{\\pgfbox[center,base]{$\\Omega_1$}}\n\\end{tikzpicture} \n\\qquad \\qquad\n\\begin{tikzpicture}\n\\pgfsetlinewidth{0.8pt}\n\\fill[color=lightgray] (-0.5,-1) -- (0.5,-1) -- (0.5, -0.5) -- (-0.5,-0.5) -- (-0.5,-1);\n\\fill[color=lightgray] (-1,0) -- (1,0) -- (1, 0.5) -- (-1,0.5) -- (-1,0);\n\\fill[color=lightgray] (-0.5,-1) -- (-1,0) -- (-1,0.5) -- (-0.5,-0.5) -- (-0.5,-1);\n\\draw[densely dashed] (-1,0) -- (1,0);\n\\draw (1,0) -- (0.5,-1) -- (-0.5, -1) -- (-1,0);\n\\draw (-1,0.5) -- (1,0.5) -- (0.5,-0.5) -- (-0.5, -0.5) -- (-1,0.5);\n\\draw (-1,0) -- (-1,0.5);\n\\draw (1,0) -- (1,0.5);\n\\draw (-0.5,-1) -- (-0.5,-0.5);\n\\draw (0.5,-1) -- (0.5,-0.5);\n\\pgfputat{\\pgfxy(0.2,-0.35)}{\\pgfbox[center,base]{$\\Omega_2$}}\n\\color{gray}\n\\pgfputat{\\pgfxy(-1.3,-0.4)}{\\pgfbox[center,base]{$\\Gamma_{\\rm N}$}}\n\\end{tikzpicture}\n\\qquad \\qquad\n\\begin{tikzpicture}\n\\pgfsetlinewidth{0.8pt}\n\\fill[color=black] (-0.6,0) circle (0.6cm and 0.15cm);\n\\fill[color=black] (-0.6,1.2) circle (0.6cm and 0.15cm);\n\\fill[left color=lightgray!50!black,right color=lightgray!50!black,middle color=lightgray!50,shading=axis,opacity=0.25] (0,0) -- (0,1.2) arc (360:180:0.6cm and 0.15cm) -- (-1.2,0) arc (180:360:0.6cm and 0.15cm);\n\\draw (-1.2,1.2) -- (-1.2,0) arc (180:360:0.6cm and 0.15cm) -- (0,1.2) ++ (-0.6,0) circle (0.6cm and 0.15cm);\n\\draw[densely dashed] (-1.2,0) arc (180:0:0.6cm and 0.15cm);\n\\pgfputat{\\pgfxy(-0.5,0.4)}{\\pgfbox[center,base]{$\\Omega_3$}}\n\\pgfputat{\\pgfxy(0.7,0.5)}{\\pgfbox[center,base]{$\\Gamma_{\\rm D}$}}\n\\pgfsetlinewidth{0.3pt}\n\\pgfxyline(0.4,0.5)(0.1,0.1)\n\\pgfxyline(0.4,0.8)(0.1,1.1)\n\\pgfsetlinewidth{0.8pt}\n\\color{gray}\n\\pgfputat{\\pgfxy(-1.6,0.5)}{\\pgfbox[center,base]{$\\Gamma_{\\rm N}$}}\n\\pgfsetlinewidth{0.8pt}\n\\color{gray}\n\\pgfxyline(-1.2,1.2)(-1.2,0)\n\\pgfxyline(0,1.2)(0,0)\n\\end{tikzpicture}\n\\caption{Three configurations for which the inequality~\\eqref{eq:Jawoll} holds; cf.\\ Corollary~\\ref{cor:eins}--\\ref{cor:zwei} and Example~\\ref{ex:cylinder}. \n}\n\\label{fig:Neumann}\n\\end{figure}\n\nThe next example shows that Theorem~\\ref{thm:NeumannMixed} can also be applied to non-polyhedral three-dimensional domains.\n\n\\begin{example}\\label{ex:cylinder}\nLet $\\Omega \\subset \\R^3$ be a cylinder with possibly deformed top and bottom faces. Moreover, assume that $\\Gamma_{\\rm N}$ is contained in the shell of $\\Omega$. Then $\\dim \\cS (\\Gamma_{\\rm N}) = 1$ and Theorem~\\ref{thm:NeumannMixed} implies~\\eqref{eq:Jawoll} for all $k\\in\\dN$. For the simplest case of a non-deformed cylinder see the domain $\\Omega_3$ in Figure~\\ref{fig:Neumann}.\n\\end{example}\n\nTheorem~\\ref{thm:NeumannMixed} asserts that the inequality $\\mu_{k + 1} \\leq \\lambda_k^\\Gamma$ holds for all $k$ if $\\Gamma_{\\rm N}$ is not too large in a certain sense. The following example shows that the eigenvalue inequality is violated if $\\Gamma_{\\rm N}$ is too large.\n\n\\begin{example}\nConsider the domain $\\Omega := [0,\\pi]^2 \\subset \\dR^2$ and set $\\Gamma_{\\rm D} := (0,\\pi) \\times \\{0\\}$, i.e., we impose a Dirichlet boundary condition on one side of the square $\\Omega$ and Neumann boundary conditions on the rest of the boundary; in this case $\\dim \\cS (\\Gamma_{\\rm N}) = 0$. The Laplacian eigenfunctions and eigenvalues corresponding to the mixed and the pure Neumann problem on $\\Omega$ can be calculated explicitly using separation of variables. For the mixed problem the eigenvalues are given by the numbers $(n - 1)^2 + (m - 1\/2)^2$ with $n, m \\in \\N$, while the eigenvalues of the pure Neumann problem are $(n - 1)^2 + (m - 1)^2$ with $n, m \\in \\N$. In particular,\n\\begin{align*}\n \\mu_2 = 1 > 1\/4 = \\lambda_1^\\Gamma,\n\\end{align*}\nso that the inequality~\\eqref{eq:Jawoll} fails already for $k = 1$.\n\\end{example}\n\nThe following example shows that in general no strict inequality holds in the situation of Theorem~\\ref{thm:NeumannMixed}.\n\n\\begin{example}\\label{ex:square}\nLet again $\\Omega = [0, \\pi]^2 \\subset \\R^2$ and let $\\Gamma_{\\rm D} = (0, \\pi) \\times \\{0, \\pi\\}$ consist of two parallel faces. Then the Neumann eigenvalues are $(n - 1)^2 + (m - 1)^2$ and the mixed eigenvalues are $(n - 1)^2 + m^2$, $n, m \\in \\N$, yielding\n\\begin{align*}\n \\mu_2 = 1 = \\lambda_1^\\Gamma.\n\\end{align*}\nHowever, $\\dim \\cS (\\Gamma_{\\rm N}) = 1$, i.e., the assumptions of Theorem~\\ref{thm:NeumannMixed} are satisfied.\n\\end{example}\n\nTheorem~\\ref{thm:NeumannMixed} can be combined with Proposition~\\ref{prop:monotonicity} implying the following result. Roughly speaking, it states that the inequality~\\eqref{eq:Jawoll} is strict if $\\Gamma_{\\rm N}$ can be enlarged nontrivially such that the condition on the dimension of the joint tangential space is not violated.\n\n\\begin{cor}\\label{cor:strict}\nLet $\\Omega \\subset \\R^d$, $d \\geq 2$, be a bounded, connected Lipschitz domain and let Hypothesis~\\ref{hyp} be satisfied. Moreover, let $\\Gamma' = \\Gamma_{\\rm D}'$ and $\\Gamma_{\\rm N}'$ be relatively open, nonempty subsets of $\\partial \\Omega$ such that $\\Gamma_{\\rm D}' \\cap \\Gamma_{\\rm N}' = \\emptyset$, $\\partial \\Omega \\setminus (\\Gamma_{\\rm D}' \\cup \\Gamma_{\\rm N}')$ has measure zero, and $\\Gamma' \\subset \\Gamma$. If $\\Gamma \\setminus \\Gamma'$ has a nonempty interior and $\\dim \\cS (\\Gamma_{\\rm N}') \\geq 1$ then\n\\begin{align}\\label{eq:nochmalStrict}\n \\mu_{k+1} < \\lambda_k^\\Gamma\n\\end{align}\nholds for all $k\\in\\dN$.\n\\end{cor}\n\nWe provide an exemplary application of Corollary~\\ref{cor:strict} in the next example.\n\n\\begin{example}\nLet $\\Omega \\subset \\R^3$ be a polyhedral domain whose boundary contains two parallel faces $\\Gamma_1, \\Gamma_2$. If we choose $\\Gamma_{\\rm N} = \\Gamma_1$ and $\\Gamma = \\Gamma_{\\rm D}$ contains all faces of $\\Omega$ except $\\Gamma_1$ then the assumptions of Corollary~\\ref{cor:strict} are satisfied with $\\Gamma_{\\rm N}' = \\Gamma_1 \\cup \\Gamma_2$. Hence~\\eqref{eq:nochmalStrict} is satisfied.\n\\end{example}\n\n\n\\section{Inequalities for Dirichlet and mixed eigenvalues on polygonal and polyhedral domains}\\label{sec:Dirichlet}\n\nIn this section we provide inequalities which compare the eigenvalues $\\lambda_1^\\Gamma < \\lambda_2^\\Gamma \\leq \\lambda_3^\\Gamma \\leq \\dots$ of the operator $- \\Delta_\\Gamma$ subject to mixed boundary conditions with the eigenvalues $\\lambda_1 < \\lambda_2 \\leq \\lambda_3 \\leq\\dots$ of the Dirichlet Laplacian. Throughout this section we make an additional restriction on the class of domains. We assume that $\\Omega \\subset \\R^d$, $d \\geq 2$, is a polyhedral, convex, bounded domain; cf.\\ Definition~\\ref{def:poly}. Moreover, we assume that $\\Gamma_{\\rm D}$ and $\\Gamma_{\\rm N}$ are chosen according to Hypothesis~\\ref{hyp}. For the main result of this section let $\\hat \\Gamma_{\\rm D}$ denote the set of points $x' \\in \\Gamma_{\\rm D}$ such that the tangential hyperplane $T_{x'}$ exists, see~\\eqref{eq:tangential}, and define the linear subspace\n\\begin{align*}\n \\cS (\\Gamma_{\\rm D}) = \\bigcap_{x' \\in \\hat \\Gamma_{\\rm D}} T_{x'}\n\\end{align*}\nof $\\R^d$ consisting of all vectors being tangential to almost all points of $\\Gamma_{\\rm D}$. Note that $\\dim \\cS (\\Gamma_{\\rm D}) \\in \\{0, \\dots, d - 1\\}$. The main result of this section reads as follows; its proof relies heavily on Lemma~\\ref{lem:det} in the appendix.\n\n\\begin{thm}\\label{thm:DirichletMixed}\nLet Hypothesis~\\ref{hyp} be satisfied and assume, in addition, that $\\Omega$ is polyhedral and convex. Then\n\\begin{align}\\label{eq:mixedDirichlet}\n \\lambda_{k + \\dim \\cS (\\Gamma_{\\rm D})}^\\Gamma \\leq \\lambda_k\n\\end{align}\nholds for all $k \\in \\N$.\n\\end{thm}\n\n\\begin{proof}\nLet $k \\in \\N$ and let $u_j$ be real-valued Dirichlet Laplacian eigenfunctions corresponding to the eigenvalues $\\lambda_j$, $j = 1, \\dots, k$, being pairwise orthogonal in $L^2 (\\Omega)$. For $a_1, \\dots, a_k, b_1, \\dots, b_d \\in \\C$ define\n\\begin{align}\\label{eq:PhiPsi}\n \\Phi = \\sum_{j = 1}^k a_j u_j \\in H^2 (\\Omega) \\cap H_0^1 (\\Omega) \\quad \\text{and} \\quad \\Psi = \\sum_{j = 1}^d b_j \\partial_j u_k \\in H^1 (\\Omega).\n\\end{align}\nNote that by Green's identity\n\\begin{align*}\n \\int_\\Omega \\nabla u_j \\cdot \\nabla u_l \\dd x & = - \\int_\\Omega \\Delta u_j u_l \\dd x = \\lambda_j \\int_\\Omega u_j u_l \\dd x = 0, \\quad j, l \\in \\{1, \\dots, k\\},~j \\neq l.\n\\end{align*}\nNote further that $- \\Delta \\Psi = \\lambda_k \\Psi$ holds in the distributional sense. With these observations and $\\Phi |_{\\partial \\Omega} = 0$ we get\n\\begin{align}\\label{eq:erste}\n\\begin{split}\n \\int_\\Omega |\\nabla (\\Phi + \\Psi)|^2 \\dd x &  = \\sum_{j = 1}^k \\int_\\Omega |a_j \\nabla u_j|^2 \\dd x + 2 \\Real \\int_\\Omega \\nabla \\Psi \\cdot \\overline{\\nabla \\Phi} \\dd x + \\int_\\Omega |\\nabla \\Psi|^2 \\dd x \\\\\n & = \\sum_{j = 1}^k \\lambda_j \\int_\\Omega |a_j u_j|^2 \\dd x + 2 \\lambda_k \\Real \\int_\\Omega \\Psi \\overline{\\Phi} \\dd x + \\int_\\Omega |\\nabla \\Psi|^2 \\dd x.\n\\end{split}\n\\end{align}\nMoreover, for the last integral with the help of Lemma~\\ref{lem:det} we obtain\n\\begin{align}\\label{eq:istJaToll}\n\\begin{split}\n \\int_\\Omega |\\nabla \\Psi|^2 \\dd x & = \\sum_{m = 1}^d \\int_\\Omega \\bigg| \\sum_{j = 1}^d b_j \\partial_{m j} u_k \\bigg|^2 \\dd x \\\\\n & = \\sum_{m = 1}^d \\int_\\Omega \\bigg( \\sum_{j = 1}^d |b_j \\partial_{m j} u_k |^2 + 2 \\Real \\bigg( \\sum_{j = 1}^d \\sum_{l < j} b_l \\overline{b_j} (\\partial_{m l} u_k) (\\partial_{m j} u_k) \\bigg) \\bigg) \\dd x \\\\\n & = \\sum_{m = 1}^d \\int_\\Omega \\bigg( \\sum_{j = 1}^d |b_j|^2 (\\partial_{j j} u_k) (\\partial_{m m} u_k) \\\\\n & \\qquad \\qquad + 2 \\Real \\bigg( \\sum_{j = 1}^d \\sum_{l < j} b_l \\overline{b_j} (\\partial_{l j} u_k) (\\partial_{m m} u_k) \\bigg) \\bigg) \\dd x \\\\\n & = \\sum_{m = 1}^d \\int_\\Omega \\sum_{l, j = 1}^d b_l \\overline{b_j} (\\partial_{l j} u_k) (\\partial_{m m} u_k) \\dd x.\n\\end{split}\n\\end{align}\nOn the other hand, defining the $d \\times d$-matrix\n\\begin{align*}\n B = \\big(b_l \\overline{b_j} \\big)_{l, j = 1}^d\n\\end{align*}\nand using integration by parts we get\n\\begin{align}\\label{eq:super}\n\\begin{split}\n \\lambda_k \\int_\\Omega |\\Psi|^2 \\dd x & = \\lambda_k \\int_\\Omega \\sum_{l, j = 1}^d b_l \\overline{b_j} (\\partial_{l} u_k) (\\partial_j u_k) \\dd x \\\\\n & = \\lambda_k \\int_\\Omega \\nabla u_k \\cdot B \\nabla u_k \\dd x \\\\\n & = \\int_\\Omega (\\Delta u_k) \\diver (B \\nabla u_k) \\dd x \\\\\n & = \\sum_{m = 1}^d \\int_\\Omega (\\partial_{m m} u_k) \\sum_{l, j = 1}^d b_l \\overline{b_j} \\partial_{l j} u_k \\dd x.\n\\end{split}\n\\end{align}\nCombining~\\eqref{eq:istJaToll} and~\\eqref{eq:super} and plugging the result into~\\eqref{eq:erste} yields\n\\begin{align}\\label{eq:dritteNeu}\n\\begin{split}\n \\int_\\Omega |\\nabla (\\Phi + \\Psi)|^2 \\dd x & = \\sum_{j = 1}^k \\lambda_j \\int_\\Omega |a_j u_j|^2 \\dd x + 2 \\lambda_k \\Real \\int_\\Omega \\Psi \\overline \\Phi \\dd x + \\lambda_k \\int_\\Omega |\\Psi|^2 \\dd x \\\\\n & \\leq \\lambda_k \\int_\\Omega |\\Phi + \\Psi|^2 \\dd x.\n\\end{split}\n\\end{align}\n\nIn order to apply the min-max principle~\\eqref{eq:minmax}, our aim is to estimate the dimension of the linear space consisting of functions of the form $\\Phi + \\Psi$ as in~\\eqref{eq:PhiPsi} which additionally belong to $H_{0, \\Gamma}^1 (\\Omega)$. For this note first that \n\\begin{align}\\label{eq:siehMalAn}\n \\dim \\spann \\left\\{u_1, \\dots, u_k, \\partial_1 u_k, \\dots, \\partial_d u_k \\right\\} = k + \\dim \\spann \\{\\partial_1 u_k, \\dots, \\partial_d u_k\\}.\n\\end{align}\nIn fact, by assumption we have $\\dim \\spann \\{u_1, \\dots, u_k\\} = k$. Moreover, let\n\\begin{align*}\n w \\in \\spann \\left\\{u_1, \\dots, u_k \\right\\} \\cap \\spann \\left\\{ \\partial_1 u_k, \\dots, \\partial_d u_k \\right\\}.\n\\end{align*}\nThen $w \\in H_0^1 (\\Omega)$ and $w = \\sum_{j = 1}^d b_j \\partial_j u_k$ for certain $b_1, \\dots, b_d \\in \\C$. For a contradiction assume first that the vector $(\\Real b_1, \\dots, \\Real b_d)^\\top$ is nontrivial. Let $\\Lambda$ be a face of $\\partial\\Omega$ such that the vector $(\\Real b_1, \\dots, \\Real b_d)^\\top$ is not tangential to $\\Lambda$ and let $\\tau^1, \\dots, \\tau^{d - 1}$ be linearly independent tangential vectors of $\\Lambda$. Then the system $\\{\\tau^1, \\dots, \\tau^{d - 1}, (\\Real b_1, \\dots, \\Real b_d)^\\top\\}$ is linearly independent, and due to $u_k |_\\Lambda = 0$ we have\n\\begin{align}\\label{eq:bisschenNull}\n \\tau^j \\cdot \\nabla u_k |_\\Lambda = 0, \\quad j = 1, \\dots, d - 1.\n\\end{align}\nMoreover, \n\\begin{align}\\label{eq:nochmehrNull}\n (\\Real b_1, \\dots, \\Real b_d)^\\top \\cdot \\nabla u_k |_\\Lambda = (\\Real w) |_\\Lambda = 0.\n\\end{align}\nFrom~\\eqref{eq:bisschenNull} and~\\eqref{eq:nochmehrNull} it follows \n\\begin{align*}\n \\partial_\\nu u_k |_\\Lambda = \\nu \\cdot \\nabla u_k |_\\Lambda = 0\n\\end{align*}\nas the constant outer unit normal $\\nu$ on $\\Lambda$ can be written as a linear combination of $\\tau^1, \\dots, \\tau^{d - 1}$ and $(\\Real b_1, \\dots, \\Real b_d)^\\top$. Together with $u_k |_\\Lambda = 0$, by Lemma~\\ref{lem:UCP} this implies $u_k = 0$, a contradiction; thus $\\Real b_1 = \\dots = \\Real b_d = 0$. Analogously we obtain $\\Imag b_1 = \\dots = \\Imag b_d = 0$ and thus $w = 0$. From this we conclude~\\eqref{eq:siehMalAn}. \n\nLet us now derive from~\\eqref{eq:dritteNeu} and~\\eqref{eq:siehMalAn} the assertion of the theorem. In fact, the linear space $\\cS (\\Gamma_{\\rm D})$ is tangential to all of $\\Gamma_{\\rm D}$ and $u_k$ vanishes on $\\Gamma_{\\rm D}$. Thus \n\\begin{align*}\n \\sum_{j = 1}^d b_j \\partial_j u_k |_{\\Gamma_{\\rm D}} = (b_1, \\dots, b_d)^\\top \\cdot \\nabla u_k |_{\\Gamma_{\\rm D}} = 0\n\\end{align*}\nholds for all $(b_1, \\dots, b_d)^\\top \\in \\cS (\\Gamma_{\\rm D})$, that is, \n\\begin{align}\\label{eq:schonFast}\n \\sum_{j = 1}^d b_j \\partial_j u_k \\in H_{0, \\Gamma}^1 (\\Omega) \\quad \\text{for all}~(b_1, \\dots, b_d)^\\top \\in \\cS (\\Gamma_{\\rm D}).\n\\end{align}\nNext, note that $\\partial_1 u_k, \\dots, \\partial_d u_k$ are linearly independent. For this let $b_1, \\dots, b_d \\in \\C$ be such that \n\\begin{align*}\n \\sum_{j = 1}^d b_j \\partial_j u_k = 0\n\\end{align*}\nin $\\Omega$ and assume for contradiction that we are off the case $b_1 = \\dots = b_d = 0$. Then without loss of generality the vector $(\\Real b_1, \\dots, \\Real b_d)^\\top$ is nontrivial and the derivative of $u_k$ in the direction of this vector vanishes on all of $\\Omega$. From this and $u_k |_{\\partial \\Omega} = 0$ it follows $u_k = 0$ on $\\Omega$, a contradiction. In particular, linearly independent vectors $(b_1, \\dots, b_d)^\\top \\in \\cS (\\Gamma_{\\rm D})$ lead to linearly independent functions $\\sum_{j = 1}^d b_j \\partial_j u_k \\in H_{0, \\Gamma}^1 (\\Omega)$, see~\\eqref{eq:schonFast}. Hence\n\\begin{align*}\n \\dim \\left( \\spann \\{\\partial_1 u_k, \\dots, \\partial_d u_k\\} \\cap H_{0, \\Gamma}^1 (\\Omega) \\right) \\geq \\dim \\cS (\\Gamma_{\\rm D}).\n\\end{align*}\nFrom this and~\\eqref{eq:siehMalAn} we conclude\n\\begin{align*}\n \\dim \\left( \\spann \\{u_1, \\dots, u_k, \\partial_1 u_k, \\dots, \\partial_d u_k \\} \\cap H_{0, \\Gamma}^1 (\\Omega) \\right) \\geq k + \\dim \\cS (\\Gamma_{\\rm D}).\n\\end{align*}\nHence~\\eqref{eq:dritteNeu} together with the definition of $\\Phi$ and $\\Psi$ in~\\eqref{eq:PhiPsi} yields\n\\begin{align*}\n \\int_\\Omega |\\nabla u|^2 \\dd x \\leq \\lambda_k \\int_\\Omega |u|^2 \\dd x\n\\end{align*}\nfor all $u$ in a subspace of $H_{0, \\Gamma}^1 (\\Omega)$ of dimension $k + \\dim \\cS (\\Gamma_{\\rm D})$ or larger. This leads to the assertion of the theorem.\n\\end{proof}\n\nWe collect several immediate consequences of Theorem~\\ref{thm:DirichletMixed}; cf.\\ Figure~\\ref{fig:Dirichlet}. First we consider the case of polyhedral domains and a Dirichlet boundary condition on only one face of $\\Omega$. The second assertion of the following corollary makes use of Proposition~\\ref{prop:monotonicity} additionally.\n\n\\begin{cor}\\label{cor:DirichletMixedOneFace}\nLet $\\Omega \\subset \\dR^d$, $d \\geq 2$, be a polyhedral, convex, bounded domain and let $\\Gamma = \\Gamma_{\\rm D} \\subset \\Sigma$, \nwhere $\\Sigma \\subset \\partial \\Omega$ is either one face of $\\partial \\Omega$ or the union of two parallel faces. Then\n\\begin{align*}\n \\lambda_{k + d - 1}^\\Gamma  \\leq \\lambda_k\n\\end{align*}\nholds for all $k \\in \\N$. If, in addition, $\\Sigma \\setminus \\Gamma_{\\rm D}$ has a nonempty interior then\n\\begin{align*}\n \\lambda_{k + d - 1}^\\Gamma < \\lambda_k\n\\end{align*}\nholds for all $k \\in \\N$.\n\\end{cor}\n\nTheorem~\\ref{thm:DirichletMixed} has also nontrivial implications if $\\Gamma_{\\rm D}$ is larger than only one face (or a pair of parallel faces). This is illustrated in the three-dimensional case in the following corollary.\n\n\\begin{cor}\\label{cor:DirichletMixed3D}\nLet $\\Omega \\subset \\R^3$ be a polyhedral, convex, bounded domain. If $\\Sigma_j$ is a part of $\\partial \\Omega$ consisting of parallel faces, $j = 1, 2$, and $\\Gamma = \\Gamma_{\\rm D} \\subset \\Sigma_1 \\cup \\Sigma_2$ then\n \\begin{align*}\n \\lambda_{k + 1}^\\Gamma \\leq \\lambda_k\n\\end{align*}\nholds for all $k \\in \\N$.\n\\end{cor}\n\n\\begin{figure}[h]\n\\begin{tikzpicture}\n\\pgfsetlinewidth{0.8pt}\n\\fill[color=black] (-0.5,-1) -- (-1,0) -- (-1,0.5) -- (-0.5,-0.5) -- (-0.5,-1);\n\\draw[densely dashed] (-1,0) -- (1,0);\n\\draw (1,0) -- (0.5,-1) -- (-0.5, -1) -- (-1,0);\n\\draw (-1,0.5) -- (1,0.5) -- (0.5,-0.5) -- (-0.5, -0.5) -- (-1,0.5);\n\\draw (-1,0) -- (-1,0.5);\n\\draw (1,0) -- (1,0.5);\n\\draw (-0.5,-1) -- (-0.5,-0.5);\n\\draw (0.5,-1) -- (0.5,-0.5);\n\\pgfputat{\\pgfxy(0.2,-0.35)}{\\pgfbox[center,base]{$\\Omega_1$}}\n\\pgfputat{\\pgfxy(-1.3,-0.4)}{\\pgfbox[center,base]{$\\Gamma_{\\rm D}$}}\n\\end{tikzpicture}\n\\qquad \\qquad\n\\begin{tikzpicture}\n\\pgfsetlinewidth{0.8pt}\n\\fill[color=black] (-0.5,-1) -- (-1,0) -- (-1,0.5) -- (-0.5,-0.5) -- (-0.5,-1);\n\\fill[color=black] (0.5,-1) -- (1,0) -- (1,0.5) -- (0.5,-0.5) -- (0.5,-1);\n\\draw[densely dashed] (-1,0) -- (1,0);\n\\draw (1,0) -- (0.5,-1) -- (-0.5, -1) -- (-1,0);\n\\draw (-1,0.5) -- (1,0.5) -- (0.5,-0.5) -- (-0.5, -0.5) -- (-1,0.5);\n\\draw (-1,0) -- (-1,0.5);\n\\draw (1,0) -- (1,0.5);\n\\draw (-0.5,-1) -- (-0.5,-0.5);\n\\draw (0.5,-1) -- (0.5,-0.5);\n\\pgfputat{\\pgfxy(0.2,-0.35)}{\\pgfbox[center,base]{$\\Omega_2$}}\n\\pgfputat{\\pgfxy(-1.3,-0.4)}{\\pgfbox[center,base]{$\\Gamma_{\\rm D}$}}\n\\end{tikzpicture}\n\\caption{For the example of $\\Omega_1$ the inequality $\\lambda_{k + 2}^\\Gamma \\leq \\lambda_k$ holds for all $k \\in \\N$, see Corollary~\\ref{cor:DirichletMixedOneFace}. For $\\Omega_2$ one has $\\lambda_{k + 1}^\\Gamma \\leq \\lambda_k$ for all $k \\in \\N$, see Corollary~\\ref{cor:DirichletMixed3D}.}\n\\label{fig:Dirichlet}\n\\end{figure}\n\nThe following example demonstrates that, in general, the number $\\dim \\cS (\\Gamma_{\\rm D})$ in the eigenvalue inequality~\\eqref{eq:mixedDirichlet} cannot be increased and the inequality~\\eqref{eq:mixedDirichlet} is not strict. \n\n\\begin{example}\\label{ex:cube}\nWe consider the cube $\\Omega := [0,\\pi]^3 \\subset \\dR^3$ and suppose that $\\Gamma_{\\rm N} := [0,\\pi]^2 \\times \\{0,\\pi\\}$, i.e., a Neumann boundary condition is imposed on two opposite faces of $\\Omega$ and Dirichlet boundary conditions prevail on the rest of the boundary. In this case obviously $\\dim \\cS (\\Gamma_{\\rm D}) = 1$ and Theorem~\\ref{thm:DirichletMixed} yields $\\lambda_{k + 1}^\\Gamma \\leq \\lambda_k$ for all $k \\in \\N$. Indeed an inequality of the form $\\lambda_{k + 2}^\\Gamma \\leq \\lambda_k$ does not hold for all $k \\in \\N$. In fact, the eigenvalues of the mixed problem can be calculated via separation of variables. They are given by the numbers $(n-1)^2 + m^2 + l^2$ with $n, m, l \\in \\dN$. On the other hand the Dirichlet Laplacian eigenvalues can be calculated analogously and have the form $n^2 + m^2 + l^2$ with $n, m, l \\in \\N$.  Thus\n\\begin{align*}\n \\lambda_3^\\Gamma = 5 > 3 = \\lambda_1.\n\\end{align*}\nMoreover, in this example the eigenvalue inequality $\\lambda_{k + 1}^\\Gamma \\leq \\lambda_k$ for all $k \\in \\N$ obtained from Theorem~\\ref{thm:DirichletMixed} is not strict since we see\n\\begin{align*}\n \\lambda_2^\\Gamma = 3 = \\lambda_1.\n\\end{align*}\n\\end{example}\n\n\\begin{remark}\nThe reasoning in the proof of Theorem~\\ref{thm:DirichletMixed} can be used directly to derive the inequality\n\\begin{align}\\label{eq:LW}\n \\mu_{k + d} \\leq \\lambda_k \\quad \\text{for~all}~k \\in \\N\n\\end{align}\non any polyhedral, convex domain. Levine and Weinberger~\\cite{LW86} proved this for every smooth, convex domain and extended their result to arbitrary convex domains by an approximation step. For the polyhedral case the method of the present paper is more direct.\n\\end{remark}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAttentional sequence-to-sequence models have become the new standard for machine translation over the last two years, and with the unprecedented improvements in translation accuracy comes a new set of technical challenges. One of the biggest challenges is the high training and decoding costs of these neural machine translation (NMT) system, which is often at least an order of magnitude higher than a phrase-based system trained on the same data. For instance, phrasal MT systems were able achieve single-threaded decoding speeds of 100-500 words\/sec on decade-old CPUs \\cite{quirk2007}, while \\citet{jean2015} reported single-threaded decoding speeds of 8-10 words\/sec on a shallow NMT system. \\citet{googlenmt} was able to reach CPU decoding speeds of 100 words\/sec for a deep model, but used 44 CPU cores to do so. There has been recent work in speeding up decoding by reducing the search space \\cite{kim2016}, but little in computational improvements.\n\nIn this work, we consider a production scenario which requires low-latency, high-throughput NMT decoding. We focus on CPU-based decoders, since GPU\/FPGA\/ASIC-based decoders require specialized hardware deployment and logistical constraints such as batch processing. Efficient CPU decoders can also be used for on-device mobile translation. We focus on single-threaded decoding and single-sentence processing, since multiple threads can be used to reduce latency but not total throughput.\n\nWe approach this problem from two angles: In Section~\\ref{sec:speedups}, we describe a number of techniques for improving the speed of the decoder, and obtain a 4.4x speedup over a highly efficient baseline. These speedups do not affect decoding results, so they can be applied universally. In Section~\\ref{sec:model_improvements}, we describe a simple but powerful network architecture which uses a single RNN (GRU\/LSTM) layer at the bottom with a large number of fully-connected (FC) layers on top, and obtains improvements similar to a deep RNN model at a fraction of the training and decoding cost.\n\n\\section{Data Set}\nThe data set we evaluate on in this work is WMT English-French NewsTest2014, which has 380M words of parallel training data and a 3003 sentence test set. The NewsTest2013 set is used for validation. In order to compare our architecture to past work, we train a word-based system without any data augmentation techniques. The network architecture is very similar to \\citet{bahdanau2014}, and specific details of layer size\/depth are provided in subsequent sections. We use an 80k source\/target vocab and perform standard unk-replacement \\cite{jean2015} on out-of-vocabulary words. Training is performed using an in-house toolkit.\n\n\\section{Baseline Decoder}\nOur baseline decoder is a standard beam search decoder \\cite{sutskever2014} with several straightforward performance optimizations:\n\\begin{itemize}[noitemsep]\n\\item It is written in pure C++, with no heap allocation done during the core search.\n\\item A candidate list is used to reduce the output softmax from 80k to \\texttildelow$500$. We run word alignment \\cite{brown1993} on the training and keep the top 20 context-free translations for each source word in the test sentence.\n\\item The Intel MKL library is used for matrix multiplication, as it is the fastest floating point matrix multiplication library for CPUs.\n\\item Early stopping is performed when the top partial hypothesis has a log-score of $\\delta = 3.0$ worse than the best completed hypothesis.\n\\item Batching of matrix multiplication is applied when possible. Since each sentence is decoded separately, we can only batch over the hypotheses in the beam as well as the input vectors on the source side.\n\\end{itemize}\n\n\\section{Decoder Speed Improvements}\n\\label{sec:speedups}\n\nThis section describes a number of speedups that can be made to a CPU-based attentional sequence-to-sequence beam decoder. Crucially, none of these speedups affect the actual mathematical computation of the decoder, so they can be applied to any network architecture with a guarantee that they will not affect the results.\\footnote{Some speedups apply quantization which leads to small random perturbations, but these change the BLEU score by less than 0.02.}\n\nThe model used here is similar to the original implementation of \\citet{bahdanau2014}. The exact target GRU equation is:\n{\\vskip -0.18in}\n{\\fontsize{10.0}{10.0}\\selectfont\n\\begin{eqnarray*}\nd_{ij} & = & {\\rm tanh}(W_a{h_{i-1}} + V_a{x_i}){\\cdot}{\\rm tanh}(U_as_j) \\\\\n\\alpha_{ij} & = & \\frac{e^{d_{ij}}}{\\sum_{j'}e^{d_{ij'}}} \\\\\nc_{i} &=& \\sum_{j} \\alpha_{ij}s_j \\\\\nu_i & = & \\sigma(W_u{h_{i-1}} + V_u{x_i} + U_u{c_i} + b_u) \\\\\nr_i & = & \\sigma(W_r{h_{i-1}} + V_r{x_i} + U_r{c_i} + b_r) \\\\\n\\hat{h}_i & = & \\sigma(r_i{\\odot}(W_h{h_{i-1}}) + V_h{x_i} + U_h{c_i} + b_h) \\\\\nh_i & = & u_ih_{i-1} + (1 - u_i)\\hat{h}_i\n\\end{eqnarray*}\n}\nWhere $W_*$, $V_*$, $U_*$, $b_*$ are learned parameters, $s_j$ is the hidden vector of the $j^{\\rm th}$ source word, $h_{i-1}$ is the previous target recurrent vector, $x_i$ is the target input (e.g., embedding of previous word).\n\nWe also denote the various hyperparameters: $b$ for the beam size, $r$ for the recurrent hidden size, $e$ is the embedding size, $|S|$ for the source sentence length, and $|T|$ for the target sentence length, $|E|$ is the vocab size.\n\n\\subsection{16-Bit Matrix Multiplication}\nAlthough CPU-based matrix multiplication libraries are highly optimized, they typically only operate on 32\/64-bit floats, even though DNNs can almost always operate on much lower precision without degredation of accuracy \\cite{han2015}. However, low-precision math (1-bit to 7-bit) is difficult to implement efficiently on the CPU, and even 8-bit math has limited support in terms of vectorized (SIMD) instruction sets. Here, we use 16-bit fixed-point integer math, since it has first-class SIMD support and requires minimal changes to training. Training is still performed with 32-bit floats, but we clip the weights to the range [-1.0, 1.0] the {\\tt relu} activation to [0.0, 10.0] to ensure that all values fit into 16-bits with high precision. A reference implementation of 16-bit multiplication in C++\/SSE2 is provided in the supplementary material, with a thorough description of low-level details.\\footnote{Included as ancillary file in Arxiv submission, on right side of submission page.}\n\nA comparison between our 16-bit integer implementation and Intel MKL's 32-bit floating point multiplication is given in Figure~\\ref{fig:matrix_mult}. We can see that 16-bit multiplication is 2x-3x faster than 32-bit multiplication for batch sizes between 2 and 8, which is the typical range of the beam size $b$. We are able to achieve greater than a 2x speedup in certain cases because we pre-process the weight matrix offline to have optimal memory layout, which is a capability BLAS libraries do not have.\n\n\\begin{figure}[thb]\n\\begin{center}\n\\includegraphics[width=225px]{images\/matrix_mult_no_cache.png}\n{\\vskip -0.1in}\n\\caption{{\\small Single-threaded matrix multiplication using our 16-bit fixed-point vs. Intel MKL's 32-bit float, averaged over 10,000 multiplications. Both use the AVX2 instruction set.\n}}\n{\\vskip -0.1in}\n\\label{fig:matrix_mult}\n\\end{center}\n\\end{figure}\n\n\\label{sec:matrix_mult}\n\n\\subsection{Pre-Compute Embeddings}\nIn the first hidden layer on the source and target sides, $x_i$ corresponds to word embeddings. Since this is a closed set of values that are fixed after training, the vectors $V{x_i}$ can be pre-computed \\cite{devlin2014} for each word in the vocabulary and stored in a lookup table. This can only be applied to the first hidden layer.\n\nPre-computation does increase the memory cost of the model, since we must store $r \\times 3$ floats per word instead of $e$. However, if we only compute the $k$ most frequently words (e.g., $k = 8,000$), this reduces the pre-computation memory by 90\\% but still results in 95\\%+ token coverage due to the Zipfian distribution of language.\n\n\\subsection{Pre-Compute Attention}\nThe attention context computation in the GRU can be re-factored as follows: \n{\n\\setlength{\\belowdisplayskip}{2pt} \\setlength{\\belowdisplayshortskip}{2pt}\n\\setlength{\\abovedisplayskip}{2pt} \\setlength{\\abovedisplayshortskip}{2pt}\n\n\\[U{c_i} = U(\\sum_j \\alpha_{ij}s_j) = \\sum_j \\alpha_{ij}(Us_j)\\]\n}\nCrucially, the hidden vector representation $s_j$ is only dependent on the source sentence, while $a_{ij}$ is dependent on the target hypothesis. Therefore, the original computation $U{c_i}$ requires total $|T| \\times b$ multiplications per sentence, but the re-factored version $Us_j$ only requires total $|S|$ multiplications. The expectation over $\\alpha$ must still be computed at each target timestep, but this is much less expensive than the multiplication by $U$. \n\n\\subsection{SSE \\& Lookup Tables}\nFor the element-wise vector functions use in the GRU, we can use vectorized instructions (SSE\/AVX) for the {\\tt add} and {\\tt multiply} functions, and lookup tables for {\\tt sigmoid} and {\\tt tanh}. Reference implementations in C++ are provided in the supplementary material.\n\n\\subsection{Merge Recurrent States}\nIn the GRU equation, for the first target hidden layer, $x_i$ represents the previously generated word, and $h_{i-1}$ encodes the hypothesis up to {\\it two} words before the current word. Therefore, if two partial hypotheses in the beam only differ by the last emitted word, their $h_{i-1}$ vectors will be identical. Thus, we can perform matrix multiplication $Wh_{i-1}$ only on the {\\it unique} $h_{i-1}$ vectors in the beam at each target timestep. For a beam size of $b = 6$, we measured that the ratio of unique $h_{i-1}$ compared to total $h_{i-1}$ is approximately 70\\%, averaged over several language pairs. This can only be applied to the first target hidden layer.\n\n\\begin{table}[thb]\n\\begin{center}\n\\begin{tabular}{lcc}\n\\toprule\n& \\textbf{Words\/Sec.} & \\textbf{Speedup} \\\\\n\\textbf{Type}    & {\\fontsize{8.0}{8.0}\\selectfont  \\textbf{(Single-Threaded)}} & \\textbf{Factor} \\\\\n\\midrule\nBaseline & 95    & 1.00x \\\\\n+ 16-Bit Mult.   & 248 & 2.59x \\\\\n+ Pre-Comp. Emb. & 311 & 3.25x \\\\\n+ Pre-Comp. Att. & 342 & 3.57x \\\\\n+ SSE \\& Lookup  & 386 & 4.06x \\\\\n+ Merge Rec.     & 418 & 4.37x \\\\\n\\bottomrule\n\\end{tabular}\n\\label{table:seq_to_seq_results}\n\\end{center}\n{\\vskip -0.1in}\n\\caption{{\\small Decoding speeds on an Intel E5-2660 CPU, processing each sentence independently.}}\n\\label{table:speedup_results}\n{\\vskip -0.1in}\n\\end{table}\n\n\\subsection{Speedup Results}\nCumulative results from each of the preceding speedups are presented in Table~\\ref{table:speedup_results}, measured on WMT English-French NewsTest2014. The NMT architecture evaluated here uses 3-layer 512-dimensional bidirectional GRU for the source, and a 1-layer 1024-dimensional attentional GRU for the target. Each sentence is decoded independently with a beam of 6. Since these speedups are all mathematical identities excluding quantization noise, all outputs achieve 36.2 BLEU and are 99.9\\%+ identical.\n\nThe largest improvement is from 16-bit matrix multiplication, but all speedups contribute a significant amount. Overall, we are able to achieve a 4.4x speedup over a fast baseline decoder. Although the absolute speed is impressive, the model only uses one target layer and is several BLEU behind the SOTA, so the next goal is to maximize model accuracy while still achieving speeds greater than some target, such as 100 words\/sec.\n\n\\begin{table*}[thb]\n{\\fontsize{10.5}{10.1}\\selectfont\n\\begin{center}\n\\begin{tabular}{lcc}\n\\toprule\n& & \\\\ \\noalign{\\vskip -0.1in}\n& & \\textbf{Words\/Sec} \\\\\n\\textbf{System} & \\textbf{BLEU} & {\\fontsize{8.0}{8.0}\\selectfont \\textbf{(Single-Threaded)}} \\\\\n\\midrule \n& & \\\\ \\noalign{\\vskip -0.1in}\nBasic Phrase-Based MT \\citep{schwenk2014} & 33.1 & - \\\\\nSOTA Phrase-Based MT \\citep{durrani2014} & 37.0 & - \\\\\n6-Layer Non-Attentional Seq-to-Seq LSTM \\citep{luong2014} & 33.1 & - \\\\\nRNN Search, 1-Layer Att. GRU, w\/ Large Vocab \\citep{jean2015} & 34.6 & $\\dagger$ \\\\\nGoogle NMT, 8-Layer Att. LSTM, Word-Based \\citep{googlenmt} & 37.9 & $\\flat$ \\\\\nGoogle NMT, 8-Layer Att. LSTM, WPM-32k \\citep{googlenmt} & 39.0$^\\ddagger$ & $\\flat$ \\\\\nBaidu Deep Attention, 8-Layer Att. LSTM \\cite{zhou2016} & 39.2 & - \\\\\n\\midrule\n& & \\\\ \\noalign{\\vskip -0.1in}\n(S1) Trg: 1024-AttGRU & 36.2 & 418 \\\\\n(S2) Trg: 1024-AttGRU + 1024-GRU & 36.8 & 242 \\\\\n(S3) Trg: 1024-AttGRU + 3-Layer 768-FC-Relu + 1024-FC-Tanh & 37.1 & 271 \\\\\n(S4) Trg: 1024-AttGRU + 7-Layer 768-FC-Relu + 1024-FC-Tanh & 37.4 & 229 \\\\\n(S5) Trg: 1024-AttGRU + 7-Layer 768-FC-Relu + 1024-GRU & 37.6 & 157 \\\\\n(S6) Trg: 1024-AttGRU + 15-Layer 768-FC-Relu + 1024-FC-Tanh & 37.3 & 163 \\\\\n(S7) Src: 8-Layer LSTM, Trg: 1024-AttLSTM + 7-Layer 1024-LSTM$^\\S$ & 37.8 & 28 \\\\\n\\midrule\n& & \\\\ \\noalign{\\vskip -0.1in}\n\\textbf{(E1) Ensemble of 2x Model (S4)} & \\textbf{38.3} & \\textbf{102} \\\\\n(E2) Ensemble of 3x Model (S4) & 38.5 & 65 \\\\\n\\bottomrule\n\\end{tabular}\n\\label{table:seq_to_seq_results}\n\\end{center}\n\\caption{{\\small Results on WMT English-French NewsTest2014. Models (S1)-(S6) use a 3-layer 512-dim bidirectional GRU for the source side. The CPU is an Intel Haswell E5-2660. $\\dagger$ Reported as {\\texttildelow}8 words\/sec on one CPU core. $\\flat$ Reported as {\\texttildelow}100 words\/sec, parallelized across 44 CPU cores. $\\ddagger$ Uses word-piece tokenization, all others are word-based. $\\S$ Reproduction of Google NMT, Word-Based.}}\n\\label{table:model_results}\n}\n\\end{table*}\n\n\\section{Model Improvements}\n\n\\label{sec:model_improvements}\nIn NMT, like in many other deep learning tasks, accuracy can be greatly improved by adding more hidden layers, but training and decoding time increase significantly \\cite{luong2014, zhou2016, googlenmt}. Several past works have noted that convolutional neural networks (CNNs) are significantly less expensive than RNNs, and replaced the source and\/or target side with a CNN-based architecture \\cite{gehring2016, kalchbrenner2016}. However, these works have found it is difficult to replace the target side of the model with CNN layers while maintaining high accuracy. The use of a recurrent target is especially important to track attentional coverage and ensure fluency.\n\nHere, we propose a mixed model which uses an RNN layer at the bottom to both capture full-sentence context and perform attention, followed by a series of fully-connected (FC) layers applied on top at each timestep. The FC layers can be interpreted as a CNN without overlapping stride. Since each FC layer consists of a single matrix multiplication, it is $1\/6^{\\rm th}$ the cost of a GRU (or $1\/8^{\\rm th}$ an LSTM). Additionally, several of the speedups from Section~\\ref{sec:speedups} can only be applied to the first layer, so there is strong incentive to only use a single target RNN.\n\nTo avoid vanishing gradients, we use ResNet-style skip connections \\cite{he2016}. These allow very deep models to be trained from scratch and do not require any additional matrix multiplications, unlike highway networks \\cite{srivastava2015}. With 5 intermediate FC layers, target timestep $i$ is computed as:\n{\\vskip -0.18in}\n{\\fontsize{10.0}{10.0}\\selectfont\n\\begin{eqnarray*}\nh^B_{i} &=& {\\rm AttGRU}(h^B_{i-1}, x_i, S) \\\\\nh^1_{i} &=& {\\rm relu}(W^1h^B_i) \\\\\nh^2_{i} &=& {\\rm relu}(W^2h^1_i) \\\\\nh^3_{i} &=& {\\rm relu}(W^3h^2_i + h^1_i) \\\\\nh^4_{i} &=& {\\rm relu}(W^4h^3_i) \\\\\nh^5_{i} &=& {\\rm relu}(W^5h^4_i + h^3_i) \\\\\nh^T_{i} &=& {\\rm tanh}(W^Th^5_i)\\ {\\rm {\\bf or}}\\ {\\rm GRU}(h^T_{i-1}, h^5_{i}) \\\\\ny_i &=& {\\rm softmax}(Vh^T_{i})\n\\end{eqnarray*}\n{\\vskip -0.09in}\n}\n\nWe follow \\citet{he2016} and only use skip connections on every other FC layer, but do not use batch normalization. The same pattern can be used for more FC layers, and the FC layers can be a different size than the bottom or top hidden layers. The top hidden layer can be an RNN or an FC layer. It is important to use {\\tt relu} activations (opposed to {\\tt tanh}) for ResNet-style skip connections. The GRUs still use {\\tt tanh}.\n\n\\subsection{Model Results}\nResults using the mixed RNN+FC architecture are shown in Table~\\ref{table:model_results}, using all speedups. We have found that the benefit of using RNN+FC layers on the source is minimal, so we only perform ablation on the target. For the source, we use a 3-layer 512-dim bidi GRU in all models (S1)-(S6).\n\nModel (S1) and (S2) are one and two layer baselines. Model (S4), which uses 7 intermediate FC layers, has similar decoding cost to (S2) while doubling the improvement over (S1) to 1.2 BLEU. We see minimal benefit from using a GRU on the top layer (S5) or using more FC layers (S6). In (E1) and (E2) we present 2 and 3 model ensembles of (S4), trained from scratch with different random seeds. We can see that the 2-model ensemble improves results by 0.9 BLEU, but the 3-model ensemble has little additional improvment. Although not presented here, we have found these improvement from decoder speedups and RNN+FC to be consistent across many language pairs.\n\nAll together, we were able to achieve a BLEU score of 38.3 while decoding at 100 words\/sec on a single CPU core. As a point of comparison, \\citet{googlenmt} achieves similar BLEU scores on this test set (37.9 to 38.9) and reports a CPU decoding speed of {\\texttildelow}100 words\/sec (0.2226 sents\/sec), but parallelizes this decoding across 44 CPU cores. System (S7), which is our re-implementation of \\citet{googlenmt}, decodes at 28 words\/sec on one CPU core, using all of the speedups described in Section~\\ref{sec:speedups}. \\citet{zhou2016} has a similar computational cost to (S7), but we were not able to replicate those results in terms of accuracy.\n\nAlthough we are comparing an ensemble to a single model, we can see ensemble (E1) is over 3x faster to decode than the single model (S7). Additionally, we have found that model (S4) is roughly 3x faster to train than (S7) using the same GPU resources, so (E1) is also 1.5x faster to train than a single model (S7).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nA real matrix $A$ is said to be \\emph{eventually nonnegative} (\\emph{positive}) if there exists a nonnegative integer $p$ such that $A^k$ is entrywise nonnegative (positive) for all $k \\geq p$. If $p$ is the smallest such integer, then $p$ is called the \\emph{power index of $A$} and is denoted by $p(A)$.\n\nEventual nonnegativity  has been the subject of study in several papers (see, e.g., \\cite{cnm2002, cnm2004,f1978, jt2004, mpt2014, n2006, nt2009, zm2003, zt1999}) and it is well-known that the notions of eventual positivity and nonnegativity are associated with properties of the eigenspace corresponding to the spectral radius.\n\nA real matrix $A$ is said to possess the \\emph{Perron-Frobenius property} if its spectral radius is a positive eigenvalue corresponding to an entrywise nonnegative eigenvector. The \\emph{strong Perron-Frobenius property} further requires that the spectral radius is simple; that it dominates in modulus every other eigenvalue of $A$; and that it has an entrywise positive eigenvector.\n\nSeveral challenges regarding the theory and applications of eventually nonnegative matrices remain unresolved. For example, eventual positivity of $A$ is equivalent to $A$ and $A^\\top$ possessing the strong Perron-Frobenius property, however, the Perron-Frobenius property for $A$ and $A^\\top$ is a necessary but not sufficient condition for eventual nonnegativity of $A$.\n\nA \\emph{matrix pth-root} or \\emph{matrix root} of a matrix $A$ is any matrix that satisfies the equation $X^p - A = 0$. An eventually nonnegative (positive) matrix with power index $p=p(A)$ is, by definition, a $p$th-root of the nonnegative (positive) matrix $A^p$. As a consequence, in order to gain more insight into the powers of an eventually nonnegative (positive) matrix, it is natural to examine the roots of matrices that possess the (strong) Perron-Frobenius property. In \\cite{mpt2014}, the matrix-roots of eventually positive matrices were classified. In this research, we classify the matrix roots of irreducible imprimitive nonnegative matrices; in particular, the main results in Section 3 provide necessary and sufficient conditions for the existence of an eventually nonnegative matrix $p$th-root of such a matrix. In addition, our proofs demonstrate how to construct these roots given a Jordan canonical form of such a matrix. \n\n\\section{Background}\n\nDenote by $\\ii$ the imaginary unit, i.e., $\\ii := \\sqrt{-1}$. When convenient, an indexed set of the form $\\{ x_i, x_{i+1}, \\dots, x_{i+j} \\}$ is abbreviated to $\\left\\{ x_k \\right\\}_{k=i}^{i+j}$. \n\nFor $h$ a positive integer greater than one, let  \n\\begin{align}\nR(h) &:= \\{ 0, 1, \\dots, h - 1 \\}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\nonumber \t\t\\\\\n\\omega =\\omega_h &:= \\exp{\\left( 2 \\pi \\ii\/h \\right)} \\in \\bb{C}, \t\t\t\t\t\t\t\t\t\\nonumber \t\t\\\\\n\\Omega_h &:= \\set{\\omega^k}{k}{0}{h-1} = \\set{\\omega_h^k}{k}{0}{h-1} \\subseteq \\bb{C}, \t\t\t\t\\label{bigomegah} \n\\end{align}\nand\n\\begin{align}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\\nu_h := \n\\left( 1, \\omega, \\dots, \\omega^{h-1} \\right) \\in \\bb{C}^h.\t\t\t\t\t\t\t\t\t\t\\label{omegastar}\n\\end{align}\nWith $\\omega$ as defined, it is well-known that for $\\alpha$, $\\beta \\in \\bb{Z}$,\n\\begin{align}\n\\alpha \\equiv {\\beta} \\bmod{h} &\\Longrightarrow \\omega^\\alpha = \\omega^\\beta  \t\t\t\t\t\t\\label{omalphbeta} \n\\end{align}\n\nDenote by $\\mat{n}{\\bb{C}}$ ($\\mat{n}{\\bb{R}}$) the algebra of complex (respectively, real) $n \\times n$ matrices. Given $A \\in \\mat{n}{\\bb{C}}$, the \\textit{spectrum} of $A$ is denoted by $\\sig{A}$, the \\emph{spectral radius} of $A$ is denoted by $\\sr{A}$, and the \\emph{peripheral spectrum}, denoted by $\\peri{A}$, is the multi-set given by \n\\begin{align*}\n\\peri{A} = \\{ \\lambda \\in \\sig{A} : |\\lambda| = \\sr{A} \\}.\n\\end{align*} \n\n The \\emph{direct sum} of the matrices $A_1, \\dots, A_k$, where $A_i \\in \\mat{n_i}{\\bb{C}}$, denoted by $A_1 \\oplus \\dots \\oplus A_k$, $\\bigoplus_{i=1}^k A_i$, or $\\diag{A_1,\\dots,A_k}$, is the $n \\times n$ matrix \n \\[ \n \\left[\n \\begin{array}{ccc}\n A_1 &  & \\multirow{2}{*}{\\Large 0} \\\\\n \\multirow{2}{*}{\\Large 0} & \\ddots &  \\\\\n  &  & A_k\n \\end{array}\n \\right],\n \\]\nwhere $n = \\sum_{i=1}^k n_i$.\n\nFor $\\lambda \\in \\bb{C}$, $\\jordan{n}{\\lambda}$ denotes the $n \\times n$ \\emph{Jordan block} with eigenvalue $\\lambda$. For $A \\in \\mat{n}{\\bb{C}}$, denote by $J = \\inv{Z} A Z = \\bigoplus_{i=1}^t \\jordan{n_i}{\\lambda_i} = \\bigoplus_{i=1}^t J_{n_i}$, where $\\sum n_i = n$, a Jordan canonical form of $A$. Denote by $\\lambda_1,\\dots,\\lambda_s$ the \\textit{distinct} eigenvalues of $A$, and, for $i=1,\\dots,s$, let $m_i$ denote the \\textit{index} of $\\lambda_i$, i.e., the size of the largest Jordan block associated with $\\lambda_i$.\n \nFor $z = r \\exp{\\left( \\ii \\theta \\right)} \\in \\bb{C}$, where $r >0$, and an integer $p >1$, let \n\\begin{align*} \nz^{1\/p} := r^{1\/p} \\exp{( \\ii \\theta\/p )}, \n\\end{align*}\nand, for $j \\in R(p)$, let\n\\begin{align}  \nf_j (z) := z^{1\/p} \\exp{(\\ii2 \\pi j\/ p)} = r^{1\/p} \\exp{\\left( \\ii \\left( \\theta + 2 \\pi j \\right) \/ p \\right)}, \t\t\t\t\\label{rtf}\n\\end{align}\ni.e., $f_j$ denotes the $(j+1)$st-branch of the $p$th-root function.\n\n\\subsection{Combinatorial Structure}\n\nFor notation and definitions concerning the \\emph{combinatorial stucture of a matrix}, i.e., the location of the zero-nonzero entries of a matrix, we follow \\cite{br1991} and \\cite{h2009}; for further results concerning \\emph{combinatorial matrix theory}, see \\cite{br1991} and references therein.\n\nA \\emph{directed graph} (or simply \\emph{digraph}) $\\Gamma = (V,E)$ consists of a finite, nonempty set $V$ of \\emph{vertices}, together with a set $E \\subseteq V \\times V$ of \\emph{arcs}. For $A \\in \\mat{n}{\\bb{C}}$, the \\emph{directed graph} (or simply \\emph{digraph}) of $A$, denoted by $\\Gamma = \\dg{A}$, has vertex set $V = \\{ 1, \\dots, n \\}$ and arc set $E = \\{ (i, j) \\in V \\times V : a_{ij} \\neq 0\\}$. If $R$, $C \\subseteq \\{1,\\dots, n\\}$, then $A[R|C]$ denotes the submatrix of $A$ whose rows and columns are indexed by $R$ and $C$, respectively. \n\nA digraph $\\Gamma$ is \\textit{strongly connected} or \\textit{strong} if for any two distinct vertices $u$ and $v$ of $\\Gamma$, there is a walk in $\\Gamma$ from $u$ to $v$ (following \\cite{br1991}, we consider every vertex of $V$ as strongly connected to itself). For a strongly connected digraph $\\Gamma$, the \\emph{index of imprimitivity} is the greatest common divisor of the lengths of the closed walks in $\\Gamma$. A strong digraph is \\emph{primitive} if its index of imprimitivity is one, otherwise it is \\emph{imprimitive}. \n\nFor $n \\geq 2$, a matrix $A \\in \\mat{n}{\\bb{C}}$,  is \\emph{reducible} if there exists a permutation matrix $P$ such that\n\\begin{align*}\nP^\\top A P =\n\\begin{bmatrix}\nA_{11} & A_{12} \\\\\n0 & A_{22}\n\\end{bmatrix},\n\\end{align*}\nwhere $A_{11}$ and $A_{22}$ are nonempty square matrices and $0$ is a rectangular zero block. If $A$ is not reducible, then A is called \\emph{irreducible}. The connection between reducibility and the digraph of $A$ is as follows: $A$ is irreducible if and only if $\\dg{A}$ is strongly connected\\footnote{Following \\cite{br1991}, vertices are strongly connected to themselves so we take this result to hold for all $n \\in \\bb{N}$.} (see, e.g., \\cite[Theorem 3.2.1]{br1991} or \\cite[Theorem 6.2.24]{hj1990}). \n\nFor $h \\geq 2$, a digraph $\\Gamma = ( V, E )$ is \\emph{cyclically $h$-partite} if there exists an ordered partition $\\Pi = (\\pi_1,\\dots, \\pi_h)$ of $V$ into $h$ nonempty subsets such that for each arc $(i, j) \\in E$, there exists $\\ell \\in \\{ 1, \\dots, h \\}$ such that $i \\in \\pi_\\ell$ and $j \\in \\pi_{\\ell+1}$ (where, for convenience, we take $V_{h + 1} := V_1$). For $h \\geq 2$, a strong digraph $\\Gamma$ is cyclically $h$-partite if and only if $h$ divides the index of imprimitivity (see, e.g., \\cite[p. 70]{br1991}). A matrix $A \\in \\mat{n}{\\bb{C}}$ is called \\emph{h-cyclic} if $\\dg{A}$ is cyclically $h$-partite and if $\\dg{A}$ is cyclically $h$-partite with ordered partition $\\Pi$, then $A$ is said to be \\emph{h-cyclic with partition} $\\Pi$ or that $\\Pi$ \\emph{describes} the $h$-cyclic structure of A. The ordered partition $\\Pi = (\\pi_1,\\dots, \\pi_h)$ is \\emph{consecutive} if $\\pi_1 = \\{1,\\dots, i_1\\}$, $\\pi_2 = \\{i_1 + 1,\\dots, i_2\\},\\dots, \\pi_h = \\{ i_{h - 1} + 1, \\dots, n \\}$. If $A$ is $h$-cyclic with consecutive ordered partition $\\Pi$, then $A$ has the block form\n\\begin{align}\n\\begin{bmatrix} \n0 \t& A_{12} \t& 0 \t\t& \\cdots \t& 0 \t\t\\\\\n0 \t& 0 \t\t& A_{23} \t& \\ddots \t& \\vdots\t\\\\\n\\vdots & \\vdots \t& \\ddots \t& \\ddots \t& 0\t\t\\\\\n0 \t& 0\t\t& \\cdots\t& 0 \t\t& A_{(h-1)h}\t\\\\\nA_{h1} & 0 & 0 &\\cdots & 0\n\\end{bmatrix}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\label{cyclic_form}\n\\end{align}\nwhere $A_{i,i+1} = A[\\pi_i|\\pi_{i+1}]$ (\\cite[p. 71]{br1991}). For any $h$-cyclic matrix $A$, there exists a permutation matrix $P$ such that $P^\\top AP$ is $h$-cyclic with consecutive ordered partition. The \\emph{cyclic index} or \\emph{index of cyclicity} of $A$ is the largest $h$ for which $A$ is $h$-cyclic.\n\nAn irreducible nonnegative matrix $A$ is \\emph{primitive} if $\\dg{A}$ is primitive, and the \\emph{index of imprimitivity} of $A$ is the index of imprimitivity of $\\dg{A}$.  If $A$ is irreducible and imprimitive with index of imprimitivity $h \\geq 2$, then $h$ is the cyclic index of $A$, $\\dg{A}$ is cyclically $h$-partite with ordered partition $\\Pi = (\\pi_1,\\dots, \\pi_h)$, and the sets $\\pi_i$ are uniquely determined (up to cyclic permutation of the $\\pi_i$) (see, for example, \\cite[p. 70]{br1991}). Furthermore, $\\dg{A^h}$ is the disjoint union of $h$ primitive digraphs on the sets of vertices $\\pi_i$, $i = 1,\\dots, h$ (see, e.g., \\cite[\\S 3.4]{br1991}).\n\nFollowing \\cite{h2009}, given an ordered partition $\\Pi = \\left( \\pi_1,\\dots,\\pi_h \\right)$ of $\\{1, \\dots, n\\}$ into $h$ nonnempty subsets, the \\emph{cyclic characteristic matrix}, denoted by  $\\chi_\\Pi$, is the $n \\times n$ matrix whose $(i,j)$-entry is 1 if there exists $\\ell \\in \\{1,\\dots, h\\}$ such that $i \\in \\pi_\\ell$ and $j \\in \\pi_{\\ell + 1}$, and 0 otherwise. For an ordered partition $\\Pi = \\left( \\pi_1,\\dots,\\pi_h \\right)$ of $\\{1, \\dots, n\\}$ into $h$ nonnempty subsets, note that \n\\begin{enumerate}[label=(\\arabic*)]\n\\item $\\chi_\\Pi$ is $h$-cyclic and $\\dg{\\chi_\\Pi}$ contains every arc $(i,j)$ for $i \\in \\pi_\\ell$ and $j \\in \\pi_{\\ell+1}$; and\n\\item $A \\in \\mat{n}{\\bb{C}}$ is $h$-cyclic with ordered partition $\\Pi$ if and only if $\\dg{A} \\subseteq \\dg{\\chi_\\Pi}$.\n\\end{enumerate}\n\nWe recall the Perron-Frobenius Theorem for irreducible, imprimitive matrices.\n\n\\begin{thm}[see, e.g., \\cite{bp1994} or \\cite{hj1990}] \\label{pftirr}\nLet $A \\in \\mat{n}{\\bb{R}}$, $n \\geq 2$, and suppose that $A$ is irreducible, nonnegative, imprimitive and suppose that $h>1$ is the cyclic index of $A$. Then\n\\begin{enumerate}[label=(\\alph*)]\n\\item $\\rho >0$;\n\\item $\\rho \\in \\sig{A}$;\n\\item there exists a positive vector $x$ such that $Ax = \\rho x$; \n\\item $\\rho$ is an algebraically (and hence geometrically) simple eigenvalue of $A$; and\n\\item $\\peri{A} = \\left\\{ \\rho \\exp{( \\ii 2 \\pi k\/ h )} : k \\in R(h) \\right\\}$.\n\\item $\\omega^k \\sig{A}=\\sig{A}$ for $k \\in R(h)$.\n\\end{enumerate}\n\\end{thm}\n\n\\subsection{Matrix Functions}\n\nFor background material concerning matrix functions, we follow \\cite{h2008}; for more results concerning matrix functions, see, e.g., \\cite{h2008}, \\cite[Chapter 6]{hj1994}, or \\cite[Chapter 9]{lt1985}. Herein it is assumed that the complex matrix $A$ has $s \\leq t$ distinct eigenvalues.\n\n\\begin{mydef} \nLet $f : \\bb{C} \\longrightarrow \\bb{C}$ be a function and let $f^{(k)}$ denote the $k$th derivative of $f$. The function $f$ is said to be \\textit{defined on the spectrum of $A$} if the values\n\\begin{align*}\n\\begin{array}{c c c}\nf^{(k)}(\\lambda_i), & k=0,\\dots,m_i-1, & i=1,\\dots,s,\n\\end{array}\n\\end{align*}\ncalled \\textit{the values of the function $f$ on the spectrum of $A$}, exist.  \n\\end{mydef}\n\n\\begin{mydef}[Matrix function via Jordan canonical form] \nIf $f$ is defined on the spectrum of $A \\in \\mat{n}{\\bb{C}}$, then\n\\begin{align*} \nf(A) := Z f(J) \\inv{Z} = Z \\left( \\bigoplus_{i=1}^t f(J_{n_i}) \\right) \\inv{Z}, \n\\end{align*}\nwhere\n\\begin{align}\nf(J_{n_i}) := \n\\begin{bmatrix} \nf(\\lambda_i) & f'(\\lambda_i) & \\dots   & \\frac{f^{(n_i-1)}(\\lambda_i)}{(n_i - 1)!} \t\\\\\n\t          & f(\\lambda_i)  & \\ddots & \\vdots \t\t\t\t\t\t\\\\\n\t          &                       & \\ddots & f'(\\lambda_i)\t\t\t\t\t\t\\\\\n\t          &  \t  \t   &             & f(\\lambda_i)\n\\end{bmatrix}. \t\t\t\t\t\t\t\t\t\t\t\t\t\t\\label{fjb}\n\\end{align}\n\\end{mydef}\n\n\\begin{thm}[see, e.g., {{\\cite[\\S 1.9]{h2008}} or \\cite[Theorem 2.2.5]{pp_d2013}}] \\label{jordanformtheorem}\nLet $f$ be defined on the spectrum of a nonsingular matrix $A \\in \\mat{n}{\\bb{C}}$ and suppose that $f'(\\lambda_i) \\neq 0$ for $i=1,\\dots,t$. If $J = \\bigoplus_{i=1}^t J_{n_i} (\\lambda_i) = \\inv{Z} A Z$ is a Jordan canonical form of $A$, then \n\\begin{align*}\nJ_f := \\bigoplus_{i=1}^t J_{n_i} (f (\\lambda_i)) \n\\end{align*}\nis a Jordan canonical form of $f(A)$.\n\\end{thm}\n\n\\begin{thm}[{\\cite[Theorems 2.1 and 2.2]{s2003}}] \\label{thm_class_rts} \nIf $A \\in \\mat{n}{\\bb{C}}$ is nonsingular, then $A$ has precisely $p^s$ $p$th-roots that are expressible as polynomials in $A$, given by \n\\begin{align}\nX_j = Z \\left( \\bigoplus_{i=1}^t f_{j_i} (J_{n_i}) \\right) \\inv{Z}, \t\t\t\t\t\t\t\\label{prim_roots}\n\\end{align}\nwhere $j = \\begin{pmatrix} j_1, \\dots, j_t \\end{pmatrix}$, $j_i \\in \\{0,1,\\dots,p-1\\}$, and $j_i = j_k$ whenever $\\lambda_i = \\lambda_k$. \n\nIf $s < t$, then $A$ has additional $p$th-roots that form parameterized families \n\\begin{align}\nX_j (U) = Z U \\left( \\bigoplus_{i=1}^t f_{j_i} (J_{n_i}) \\right) \\inv{U} \\inv{Z}, \t\t\t\t\t\\label{nnprim_roots}\n\\end{align}\nwhere $U$ is an arbitrary nonsingular matrix that commutes with $J$ and, for each $j$, there exist $i$ and $k$, depending on $j$, such that $\\lambda_i = \\lambda_k$, while $j_i \\neq j_k$.\n\\end{thm}\n\nIn the theory of matrix functions, the roots given by \\eqref{prim_roots} are called the {\\it primary roots} of $A$, and the roots given by \\eqref{nnprim_roots}, which exist only if $A$ is {\\it derogatory} (i.e., some eigenvalue appears in more than one Jordan block), are called the {\\it nonprimary roots} \\cite[Chapter 1]{h2008}.\n\n\\subsection{Other results}\n\n\\begin{thm}[{\\cite[Theorem 13]{p2013}}]\\label{omegahfracpwr} \nLet $f_k$ be defined as in \\eqref{rtf}. If\n\\begin{align*}\n\\mathcal{B} := \n\\left\\{ j = \\left( j_0, j_1, \\dots, j_{h-1} \\right) : j_k \\in R(p), \\forall k \\in R(h) \\right\\}, \n\\end{align*}\nand \n\\begin{align*}\n(\\Omega_h)_j^{1\/p} := \\set{f_{j_k} (\\omega^k)}{k}{0}{h-1}, \n~ j \\in \\mathcal{B}, \n\\end{align*}\nthen there exists a unique $j \\in \\mathcal{B}$ such that $(\\Omega_h)_j^{1\/p} = \\Omega_h$ if and only if $\\gcd{(h,p)}=1$.\n\\end{thm}\n\n\\begin{rem} \\label{remarkrootomega} \nFor every $k \\in R(h)$ and $j = (j_0,j_1,\\dots,j_{h-1}) \\in \\mathcal{B}$, it is easy to verify that $f_{j_k} ( \\omega^k ) = \\omega^{(k + h j_k)\/p}$,\nwhence it follows that\n\\begin{align*}\n(\\Omega_h)_j^{1\/p} \n= \\set{\\omega^{(k + h j_k)\/p}}{k}{0}{h-1}.\n\\end{align*}\n\nIn \\cite{p2013} it is shown that if $j$ is the unique $h$-tuple as specified in \\hyp{omegahfracpwr}{Theorem}, then the set  \n\\begin{align*} \n\\mathcal{E}_j := \\set{\\frac{k + h j_k}{p}}{k}{0}{h-1} \n\\end{align*}\nis integral and a \\emph{complete residue system modulo h}, i.e., the map $\\phi: \\mathcal{E}_j \\longrightarrow R(h)$, where \n\\[ (k + hj_k)\/p \\longrightarrow ((k + hj_k)\/p) \\bmod{h}, \\] \nis bijective. Moreover, because $0 \\leq (k + h j_k)\/p \\leq h-1$ (the lower-bound is trivial and the upper bound follows because for any $k$,\n\\begin{align*}\n\\frac{k + h j_k}{p} \\leq \\frac{(h-1) + h(p-1)}{p} = \\frac{hp-1}{p} = h - \\frac{1}{p} < h)\n\\end{align*}\nit follows that \n\\begin{align} \n\\mathcal{E}_j = R(h).\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\label{fracequalsRh}\n\\end{align}\n\nIf $( \\nu_h )_j^{1\/p} := ( f_{j_0}(1), f_{j_1}(\\omega), \\dots, f_{j_{h-1}}(\\omega^{h-1}))$, then, following \\eqref{fracequalsRh}, $\\left( \\nu_h \\right)_j^{1\/p}$ corresponds to a permutation of $\\Omega_h$. \n\nBecause $\\gcd{(h,p)}=1$, it follows that $((k + h k j_1)\/p) \\equiv ((k + h j_k)\/p) \\bmod{h}$, which, along with \\eqref{omalphbeta}, yields \n\\begin{align*}\n(\\omega^{(1 + h j_1)\/p})^k = \\omega^{(k + h k j_1)\/p} = \\omega^{(k + h j_k)\/p},~k \\in \\{ 2, \\dots, h-1 \\}.\t\n\\end{align*}\nHence\n\\begin{align*}\n\\left( \\nu_h \\right)_j^{1\/p} \n&= ( 1, \\omega^{(1 + h j_1)\/p}, \\dots, \\omega^{((h-1) + hj_{h-1})\/p})\t\t\t\\\\\n&= ( 1, ( \\omega^{(1 + h j_1)\/p})^1, \\dots, ( \\omega^{(1 + h j_1)\/p})^{h-1}).\n\\end{align*}\nNote that\n\\begin{align*}\n\\left( \\nu_h \\right)^q\n:= ( 1, \\omega^q, \\dots, \\omega^{q(h-1)} )\n= ( 1, ( \\omega^q )^1, \\dots, ( \\omega^q )^{h-1} ) \t\n\\end{align*}\nis a permutation of the elements in $\\Omega_h$ if and only if $\\gcd{(h,q)} = 1$ (\\cite[Corollary 7]{p2013}); following \\eqref{omalphbeta}, for every $h$ there are $\\varphi(h)$ such permutations, where $\\varphi$ denotes {Euler's \\emph{totient} function}. Thus, if $\\gcd(h,p) = 1$, then there exists $q \\in \\bb{N}$, $\\gcd{(h,q)} = 1$, such that\n\\begin{align}\n\\left( \\nu_h \\right)_j^{1\/p} = \\left( \\nu_h \\right)^q.\t\t\t\t\t\t\t\t\\label{fracequalspwr}\n\\end{align} \n\\end{rem}\n\nNext, we state results concerning the structure of the Jordan chains of $h$-cyclic matrices. It is assumed that $A \\in \\mat{n}{\\bb{C}}$ is nonsingular, $h$-cyclic with ordered-partition $\\Pi$, and has the form \\eqref{cyclic_form}.\n\n\\begin{cor} [{\\cite[Corollary 3.2]{mp2014}}] \\label{jblockcor}\nIf $\\jordan{r}{\\lambda}$ is a Jordan block of $J$, then $\\jordan{r}{\\lambda \\omega^k}$ is a Jordan block of $J$ for $k \\in R(h)$.\n\\end{cor}\n\n\\begin{rem} \nWith $\\nu_h$ as defined in \\eqref{omegastar} and\n\\begin{align}\nJ \\left( \\lambda \\nu_h, r \\right) := \n\\begin{bmatrix}\n\\jordan{r}{\\lambda} \t\t\t\t\\\\\n& \\jordan{r}{\\lambda \\omega} \t\t\t\\\\\n& & \\ddots \t\t\t\t\t\t\\\\\n& & & \\jordan{r}{\\lambda \\omega^{h-1}}\n\\end{bmatrix} \\in \\mat{rh}{\\bb{C}},\t\t\t\t\t\t\t\t\t\t\\label{cyclicjordanblocks}\n\\end{align}\nit follows that a Jordan form of a nonsingular $h$-cylic matrix $A$ has the form\n\\begin{align*}\n\\inv{Z} A Z = J = \\bigoplus_{i=1}^{t'} J\\left( \\lambda_i \\nu_h, r_i \\right),~t'|t.\n\\end{align*}\n\\end{rem}\n\n\\begin{lem}[{\\cite[Theorem 3.6]{mp2014}}] \\label{cycliccommutator}\nFor $i=1,\\dots,t'$, if \n\\begin{align}\nA_{\\lambda_i} :=\nZ\n\\diag{0, \\dots, 0, \\overbrace{J\\left( \\lambda_i \\nu_h, r_i \\right)}^i,0,\\dots,0}\n\\inv{Z} \\in \\mat{n}{\\bb{C}},\t\t\t\t\t\t\t\t\t\t\t\\label{cycliccommutatormatrix}\n\\end{align}\nthen $\\dg{A_{\\lambda_i}} \\subseteq \\dg{\\chi_\\Pi}$, $A_{\\lambda_i}$ commutes with $A$, and $A_{\\lambda_i} A_{\\lambda_j} = A_{\\lambda_j} A_{\\lambda_i} = 0$ for $i \\neq j$, $j=1,\\dots,t'$.\n\\end{lem}\n\n\\begin{cor} \\label{cycliccor}\nIf $x$ is a strictly nonzero right eigenvector and $y$ is a strictly nonzero left eigenvector of $A$ corresponding to $\\lambda \\in \\bb{C}$, then $A_\\lambda$ has cyclic index $h$ and $\\dg{A_\\lambda} = \\dg{\\chi_\\Pi}$.\n\\end{cor}\n\nIn particular, we examine the case when $A$ is a nonnegative, irreducible, imprimitive, nonsingular matrix with index of cyclicity $h$. Without loss of generality, it is assumed that $\\sr{A} = 1$. Following \\hyp{pftirr}{Theorem} and \\hyp{jblockcor}{Corollary}, note that the Jordan form of $A$ is\n\\begin{align*}\n\\inv{Z} A Z = \n\\begin{bmatrix}\nJ(\\nu_h,1)\t& \t\t\t\t\t\t\t\\\\\n\t\t\t& J\\left( \\lambda_2 \\nu_h,r_2 \\right) & &\t\\\\\n\t\t\t& & \\ddots \t\t\t\t\t\t\t\\\\\n\t\t\t& & & J\\left( \\lambda_t \\nu_h, r_{t'} \\right)\n\\end{bmatrix}.\t\t\t\t\t\t\t\t\t\t\t\t\t\n\\end{align*}\nConsider the matrix \n\\begin{align*}\nA_1 := Z \\begin{bmatrix} J(\\nu_h,1) & 0 \\\\ 0 & 0 \\end{bmatrix} \\inv{Z}. \n\\end{align*}\nFollowing \\hyp{cycliccor}{Corollary}, $\\dg{A_1} = \\dg{\\chi_\\Pi}$, $AA_1 = A_1 A$, and, following \\hyp{pftirr}{Theorem}, there exist positive vectors $x$ and $y$ such that $Ax = x$ and $y^\\top A = y^\\top$. If $x$ and $y$ are partitioned conformably with $A$ as\n\\begin{align*}\nx =\n\\begin{bmatrix}\nx_1 \t\\\\\nx_2 \t\\\\\n\\vdots \\\\\nx_h\n\\end{bmatrix} \\text{and }\ny^\\top = \n\\begin{bmatrix}\ny_1^\\top & y_2^\\top & \\cdots & y_h^\\top \n\\end{bmatrix}, \n\\end{align*}\nthen  \n\\begin{align*}\nA_1 = h\n\\begin{bmatrix}\n0 & x_1 y_2^\\top & \\cdots & \\cdots & 0 \t\t\\\\\n0 & 0 & x_2 y_3^\\top & \\cdots & 0 \t\t\t\\\\\n\\vdots & \\vdots & \\ddots & \\ddots  & \\vdots\t\\\\\n0 & 0 & \\cdots &  0 & x_{h-1} y_h^\\top\t\t\\\\\nx_h y_1^\\top & 0 &  \\cdots & 0 & 0\n\\end{bmatrix} \\geq 0.\n\\end{align*}\n\n\\section{Main Results}\n\nUnless otherwise noted, we assume $A \\in \\mat{n}{\\bb{R}}$ is a nonnegative, nonsingular, irreducible, imprimitive matrix with $\\sr{A} = 1$. Before we state our main results, we introduce additional concepts and notation: following Friedland \\cite{f1978}, for a multi-set $\\sigma = \\set{\\lambda_i}{i}{1}{n} \\subseteq \\bb{C}$, let $\\sr{\\sigma} := \\max_i \\{ | \\lambda_i | \\}$ and $\\bar{\\sigma} := \\set{\\bar{\\lambda}_i}{i}{1}{n} \\subseteq \\bb{C}$. If $\\sigma = \\bar{\\sigma}$, we say that $\\sigma$ is \\emph{self-conjugate}. Clearly, $\\sigma$ is self-conjugate if and only if $\\bar{\\sigma}$ is self-conjugate.\n\nThe (multi-)set $\\sigma$ is said to be a \\emph{Frobenius (multi)-set} if, for some positive integer $h \\leq n$, the following properties hold:\n\\begin{enumerate}[label = (\\roman*)]\n\\item $\\sr{\\sigma} > 0$;\n\\item $\\sigma \\cap \\{ z \\in \\bb{C}: |z| = \\sr{\\sigma }\\} = \\sr{\\sigma } \\Omega_h$; and\n\\item $\\sigma = \\omega \\sigma$, i.e., $\\sigma$ is invariant under rotation by the angle $2\\pi\/h$. \n\\end{enumerate} \nClearly, the set $\\Omega_h$ as defined in \\eqref{bigomegah} is a self-conjugate Frobenius set.\n\nThe importance of Frobenius multi-sets becomes clear in view of the following result, which was introduced and stated without proof in \\cite[\\S 4, Lemma 1]{f1978} and proven rigorously in \\cite[Theorem 3.1]{zt1999}.\n\n\\begin{lem} \\label{friedlandlemma}\nLet $A$ be an eventually nonnegative matrix. If $A$ is not nilpotent, then the spectrum of $A$ is a union of self-conjugate Frobenius sets.\n\\end{lem} \n\nLet $\\lambda = r \\exp{(\\ii \\theta)} \\in \\bb{C}$, $\\Im{\\left( \\lambda \\right)} \\neq 0$, $\\lambda \\Omega_h := \\left\\{ \\lambda, \\lambda \\omega_h, \\dots, \\lambda \\omega_h^{h-1} \\right\\}$, $\\varphi = 2 \\pi k\/h$, and assume $\\gcd{(h,p)=1}$. With $f_j$ as defined in \\eqref{rtf}, a tedious but straightforward calculation shows that\n\\begin{align}\nf_i (\\lambda) f_{j_k} ( \\omega^k )= f_{j_k^{(i)}} ( \\hat{\\lambda} ), \t\t\t\t\t\t\t\\label{lambdaanalysis}\n\\end{align}\nwhere $\\hat{\\lambda} = r \\exp{( \\ii ( \\theta + \\varphi ) )}$ and $j_k^{(i)} = {\\left( i + j_k \\right)}\\bmod{p} \\in \\{ 0,1, \\dots, p-1 \\}$ (\\cite[pp.~62--63]{pp_d2013}).\n\nHence, following \\hyp{omegahfracpwr}{Theorem}, there exists a unique $j \\in \\mathcal{B}$ such that, for all $i \\in R(p)$, the set \n\\begin{align}\nf_i (\\lambda) \\left( \\Omega_h \\right)_j^{1\/p}\n:= \\left\\{ f_i (\\lambda) f_{j_0} \\left( 1 \\right), f_i (\\lambda) f_{j_1} \\left( \\omega_h \\right), \\dots, f_i (\\lambda) f_{j_{h-1}} \\left( \\omega_h^{h-1} \\right) \\right\\}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\label{lambdaanalysis2}\n\\end{align}\nis a self-conjugate Frobenius set; moreover, following \\eqref{lambdaanalysis}, there exists $j^{(i)} = \\left( j_0^{(i)}, j_1^{(i)}, \\dots, j_{h-1}^{(i)} \\right) \\in \\mathcal{B}$ such that \n\\begin{align}\nf_i (\\lambda) \\left( \\Omega_h \\right)_j^{1\/p}\n= \\left\\{ f_{j_0^{(i)}} \\left(\\lambda\\right) , f_{j_1^{(i)}} \\left( \\lambda \\omega_h \\right), \\dots, f_{j_{h-1}^{(i)}} \\left( \\lambda \\omega_h^{h-1} \\right) \\right\\}. \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\label{lambdaanalysis3}\n\\end{align}\n\nAs a consequence of \\hyp{omegahfracpwr}{Theorem} and \\hyp{friedlandlemma}{Lemma}, it should be clear that $A$ can not possess an eventually nonnegative $p$th-root if $\\gcd(h,p) > 1$; however, more can be ascertained.\n\nPartition the Jordan form of $A$ as \n\\begin{align}\n\\begin{bmatrix}\nJ_+ \t\t\t\t\\\\\n& J_- \t\t\t\t\\\\\n& & J_\\bb{C}\n\\end{bmatrix}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\label{jordformirr}\n\\end{align}\nwhere: \n\\begin{enumerate}[label=(\\arabic*)]\n\\item $J\\left( \\lambda \\nu_h, r \\right)$ is a submatrix of $J_+$ if and only if $\\sig{J\\left( \\lambda \\nu_h, r \\right)} \\cap \\bb{R}^+ \\neq \\emptyset$ and $\\sig{J\\left( \\lambda \\nu_h, r \\right)} \\cap \\bb{R}^- = \\emptyset$;\n\\item $J\\left( \\lambda \\nu_h, r \\right)$ is a submatrix of $J_-$ if and only if $\\sig{J\\left( \\lambda \\nu_h, r \\right)} \\cap \\bb{R}^- \\neq \\emptyset$; and\n\\item $J\\left( \\lambda \\nu_h, r \\right)$ is a submatrix of $J_\\bb{C}$ if and only if $\\sig{J \\left( \\lambda \\nu_h, r \\right)} \\cap \\bb{R} = \\emptyset$.\n\\item $J \\left( \\lambda \\nu_h, r \\right)$ is defined as in \\eqref{cyclicjordanblocks}, i.e.,\n\\begin{align*}\nJ \\left( \\lambda \\nu_h, r \\right) := \n\\begin{bmatrix}\n\\jordan{r}{\\lambda} \t\t\t\t\\\\\n& \\jordan{r}{\\lambda \\omega} \t\t\t\\\\\n& & \\ddots \t\t\t\t\t\t\\\\\n& & & \\jordan{r}{\\lambda \\omega^{h-1}}\n\\end{bmatrix} \\in \\mat{rh}{\\bb{C}},\n\\end{align*}\n\\end{enumerate}\nSuppose $J_+$ has $r_1$ distinct blocks of the form $J\\left( \\lambda \\nu_h, r \\right)$, $J_-$ has $r_2$ distinct blocks of the form $J\\left( \\lambda \\nu_h, r \\right)$, and $J_\\bb{C}$ has $c$ distinct blocks of the form $J\\left( \\lambda \\nu_h, r \\right)$. \n\nWe are now ready to present our main results.\n\n\\begin{thm} \\label{mainresultirr}\nLet $A \\in \\mat{n}{\\bb{R}}$, and suppose that $A \\geq 0$, nonsingular, irreducible, and imprimitive. Let $h > 1$ be the cyclic index of $A$, let $\\Pi$ describe the $h$-cyclic structure of $A$, let the Jordan form of $A$ be partitioned as in \\eqref{jordformirr}, and suppose that $\\gcd(h,p)=1$. \n\nIf $p$ is even, and\n\\begin{enumerate}[label=(\\alph*)]\n\\item $r_2 = 0$, then $A$ has $2^{r_1-1} p^c$ eventually nonnegative primary $p$th-roots; or\n\\item $r_2 >0$, then $A$ has  no eventually nonnegative primary $p$th-roots. \n\\end{enumerate}\n\nIf $p$ is odd, then $A$ has $p^c$ eventually nonnegative primary $p$th-roots.\n\\end{thm}\n\n\\begin{proof} \nFor $\\lambda \\in \\bb{C}$, $\\lambda \\neq 0$, and $j = \\left( j_0, j_1, \\dots, j_{h-1} \\right) \\in \\mathcal{B}$, let\n\\begin{align*}\nF_j \\left( J\\left( \\lambda \\nu_h, r \\right) \\right) := \n\\begin{bmatrix}\nf_{j_0} \\left( \\jordan{r}{\\lambda} \\right)\t\t\t\t\\\\\n& f_{j_1} \\left( \\jordan{r}{\\lambda \\omega} \\right) \t\t\t\\\\\n& & \\ddots \t\t\t\t\t\t\t\t\t\\\\\n& & & f_{j_{h-1}} \\left( \\jordan{r}{\\lambda \\omega^{h-1}} \\right)\n\\end{bmatrix}.\n\\end{align*}\nWe form an eventually nonnegative root $X$ by carefully selecting a root for each submatrix of every block appearing in \\eqref{jordformirr}. \n\nCase 1: \\emph{$p$ is even}. We consider the blocks appearing in \\eqref{jordformirr}:\n\\begin{enumerate}[label=(\\emph{\\roman*})]\n\\item \\label{item1} $J_+$: Following \\hyp{pftirr}{Theorem}, $J\\left( \\nu_h, 1 \\right)$ is a submatrix of $J_+$ (note that $h$ must be odd) and, following \\hyp{omegahfracpwr}{Theorem}, there exists $j = (j_0, j_1, \\dots, j_{h-1}) \\in \\mathcal{B}$ such that $\\sig{F_j \\left( J\\left( \\nu_h, 1 \\right) \\right)} = \\Omega_h$. With that specific choice of $j$, it is clear that \n\\begin{align*}\n\\sr{F_j \\left( J\\left( \\nu_h, 1 \\right) \\right)} = 1. \n\\end{align*}\n\nFor any other submatrix $J\\left( \\lambda \\nu_h, r \\right)$ of $J_+$, without loss of generality, we may assume that $\\lambda \\in \\bb{R}^+$. For every such $\\lambda$, $k$ must be chosen such that $f_k (\\lambda)$ is real (see \\cite[Corollary 2.16]{mpt2014}) and $f_k (\\lambda)$ is real if $k=0$ or $k=p\/2$; hence, there are two choices such that $f_k (\\lambda)$ is real and, following \\eqref{lambdaanalysis2} and \\eqref{lambdaanalysis3}, there are two choices such that $\\sig{F_{j^{(k)}} \\left( J\\left( \\lambda \\nu_h, r \\right) \\right)}$ is a self-conjugate Frobenius set.\n\n\\item $J_-$: If $r_2 > 0$, then $A$ does not have a real primary root so that, \\emph{a fortiori}, it can not have an eventually nonnegative primary root. \n\n\\item \\label{item3} $J_\\bb{C}$: For any submatrix $J\\left( \\lambda \\nu_h, r \\right)$ of $J_\\bb{C}$, following \\eqref{lambdaanalysis2} and \\eqref{lambdaanalysis3}, for the same $j$ chosen for $F_j \\left( J\\left( \\nu_h, 1 \\right) \\right)$ in \\hyperref[item1]{\\ref*{item1}}, there exists \\begin{align*}\nj^{(k)} = \\left( j_0^{(k)}, j_1^{(k)}, \\dots, j_{h-1}^{(k)} \\right) \\in \\mathcal{B}\n\\end{align*}\nsuch that $\\sig{F_{j^{(k)}} \\left( J\\left( \\lambda \\nu_h, r \\right) \\right)}$\nis a self-conjugate Frobenius set for all $k \\in R(p)$. Thus, there are $p^c$ ways to choose roots for blocks in $J_\\bb{C}$. \n\\end{enumerate}\nFollowing the analysis contained in \\hyperref[item1]{\\ref*{item1}}--\\hyperref[item3]{\\ref*{item3}}, note that there are $2^{r_1 - 1} p^c$ ways to form a root in this manner.\n\nCase 2: \\emph{$p$ is odd}. Following \\hyp{omegahfracpwr}{Theorem} and properties of the $p${th}-root function, if $p$ is odd, then for any submatrix $J\\left( \\lambda \\nu_h, r  \\right)$ of $J_+$ or $J_-$, there is only one choice $j \\in \\mathcal{B}$ such that $\\sig{F_j \\left( J\\left( \\lambda \\nu_h, r \\right) \\right)}$ is a self-conjugate Frobenius set. For submatrices $J\\left( \\lambda \\nu_h, r  \\right)$ of $J_\\bb{C}$, the analysis is the same as in \\hyperref[item3]{\\ref*{item3}}. In this manner, there are $p^c$ possible selections.\n\nPartition the Jordan form of $A$ as\n\\begin{align*}\n\\begin{bmatrix}\nJ(\\nu_h,1) & \\\\\n& \\hat{J}\n\\end{bmatrix}.\n\\end{align*} \nWith the above partition in mind, consider the matrix $p$th-root of $A$ given by\n\\begin{align*}\nX = \nZ\n\\begin{bmatrix}\nF_j \\left( J(\\nu_h,1) \\right) & \\\\\n & F (\\hat{J})\n\\end{bmatrix}\n\\inv{Z}\n\\end{align*}\nwhere $j$ is selected as in \\hyp{omegahfracpwr}{Theorem} and $F(\\hat{J})$ is a $p$th-root of $\\hat{J}$ containing blocks of the form $F_{j^{(k)}} \\left( J\\left( \\lambda \\nu_h, r \\right) \\right)$, chosen as in Case 1 or Case 2. \n\nThe matrix $X$ must be irreducible because every power of a reducible matrix is reducible. If $\\bar{h}$ is the cyclic index of $X$, then $1 \\leq \\bar{h} \\leq h$ (if $\\bar{h} > h$, then $X$ would have $\\bar{h}$ eigenvalues of maximum modulus and consequently so would $A$, contradicting the maximality of $h$) and $\\bar{h}$ must divide $h$. However, we claim that $\\bar{h} = h$. For contradiction, assume $\\bar{h} < h$ and consider the matrix $A_1$ given by \n\\begin{align*}\nA_1 =\nZ\n\\begin{bmatrix}\nJ(\\nu_h,1) & 0 \\\\\n0 & 0\n\\end{bmatrix}\n\\inv{Z}.\n\\end{align*}\nFollowing \\hyp{cycliccor}{Corollary}, $A_1 \\geq 0$, irreducible, $h$-cyclic, and $\\Pi$ describes the $h$-cyclic structure of $A_1$. Next, consider the matrix $X_1$ given by\n\\begin{align*}\nX_1 =\nZ\n\\begin{bmatrix}\nF_j \\left( J(\\nu_h,1) \\right) & 0 \\\\\n0 & 0\n\\end{bmatrix}\n\\inv{Z},\n\\end{align*}\nwhich is a matrix $p$th-root of $A_1$. Following \\hyp{cycliccor}{Corollary}, the cyclic index of $X_1$ is $\\bar{h}$. However, following the remarks leading up to \\eqref{fracequalspwr}, there exists $q \\in \\bb{N}$ relatively prime to $h$ such that $X_1 = A_1^q \\geq 0$ which, along with $X_1$ being a $p$th-root of $A_1$, implies $X_1 = X_1^{pq}$. Thus, $X_1$ is a nonnegative, irreducible, $h$-cyclic matrix, contradicting the maximality of $\\bar{h}$. Hence, $\\bar{h} = h$ and $X$ is $h$-cyclic. Before we continue with the proof, note that $\\dg{X} \\subseteq \\dg{X_1} = \\dg{\\chi_\\Pi^q}$.\n\nFollowing \\hyp{jordanformtheorem}{Theorem}, there exists $\\bar{Z} \\in \\mat{n}{\\bb{C}}$ such that \n\\begin{align*}\n&\\inv{\\bar{Z}} X \\bar{Z} = \\\\\n&\\begin{bmatrix}\nF_j \\left( J(\\nu_h,1) \\right)\t& \t\t\t\t\t\t\t\t\t\t\\\\\n\t\t\t& J\\left( f_{j_2}(\\lambda_2) \\left( \\nu_h \\right)_j^{1\/p},r_2 \\right) & &\t\\\\\n\t\t\t& & \\ddots \t\t\t\t\t\t\t\t\t\t\t\t\\\\\n\t\t\t& & & J\\left( f_{j_t}(\\lambda_t) \\left( \\nu_h \\right)_j^{1\/p}, r_{t'} \\right)\n\\end{bmatrix}.\n\\end{align*} \nBy construction of $\\bar{Z}$ (\\cite[Lemma 1.3.2]{pp_d2013}), note that \n\\begin{align*}\nX_1 =\nZ\n\\begin{bmatrix}\nF_j \\left( J(\\nu_h,1) \\right) & 0 \\\\\n0 & 0\n\\end{bmatrix}\n\\inv{Z}\n= \n\\bar{Z} \n\\begin{bmatrix}\nF_j \\left( J(\\nu_h,1) \\right) & 0 \\\\\n0 & 0\n\\end{bmatrix} \n\\inv{\\bar{Z}}.\n\\end{align*}\nLet \n\\begin{align*}\nX_2 :=\n\\bar{Z}\n\\begin{bmatrix}\n0\t& \t\t\t\t\t\t\t\t\t\t\\\\\n\t\t\t& J\\left( f_{j_2}(\\lambda_2) \\left( \\nu_h \\right)_j^{1\/p},r_2 \\right) & &\t\\\\\n\t\t\t& & \\ddots \t\t\t\t\t\t\t\t\t\t\t\t\\\\\n\t\t\t& & & J\\left( f_{j_t}(\\lambda_t) \\left( \\nu_h \\right)_j^{1\/p}, r_{t'} \\right)\n\\end{bmatrix}\n\\inv{\\bar{Z}}.\n\\end{align*}\nFollowing \\hyp{cycliccommutator}{Lemma}, $X_2$ is $h$-cyclic, $\\dg{X_2} \\subseteq \\dg{\\chi_\\Pi^q}$, and $X_1 X_2 = X_2 X_1 = 0$. Thus, for $k \\in \\bb{N}$, $X^k = X_1^k + X_2^k$. Because $\\sr{X_2} < 1$, $\\lim_\\rinf{k} X_2^k = 0$. The matrices $X_1$ and $X_2$ are $h$-cyclic, $\\rk{X_1^2} = \\rk{X_1}$, $\\rk{X_2^2} = \\rk{X_2}$, and $\\dg{X_2} \\subseteq \\dg{X_1} = \\dg{\\chi_\\Pi^q}$, thus, following \\cite[Theorem 2.7]{h2009}, note that $\\dg{X_2^k} \\subseteq \\dg{X_1^k} = \\dg{\\chi_\\Pi^{qk}}$. Thus, there exists $p \\in \\bb{N}$ such that  $X_1^k > X_2^k$ for all $k \\geq p$, i.e., $X$ is eventually nonnegative.\n\\end{proof}\n\n\\begin{ex}\nWe demonstrate \\hyp{mainresultirr}{Theorem} via an example: consider the matrix\n\\begin{align*}\nA =\n\\frac{1}{3}\n\\begin{bmatrix}\n0 & 0 & 2 & 1 & 0 & 0 \t\\\\\n0 & 0 & 1 & 2 & 0 & 0\t\\\\\n0 & 0 & 0 & 0 & 2 & 1\t\\\\\n0 & 0 & 0 & 0 & 1 & 2\t\\\\\n2 & 1 & 0 & 0 & 0 & 0\t\\\\\n1 & 2 & 0 & 0 & 0 & 0\n\\end{bmatrix} \\in \\mat{6}{\\bb{R}}.\n\\end{align*}\nOne can verify that $A = Z D \\inv{Z}$, where\n\\begin{align*}\nZ = \\left[\n\\begin{array}{*{6}{r}}\n1 & 1 & 1 & 1 & 1 & 1\t\t\t\t\t\t\\\\\n1 & 1 & 1 & -1 & -1 & -1\t\t\t\t\t\t\\\\\n1 & \\omega & \\omega^2 &  1 & \\omega & \\omega^2\t\t\\\\\n1 & \\omega & \\omega^2 & -1 &  -\\omega &  -\\omega^2\t\\\\\n1 & \\omega^2 & \\omega &  1 & \\omega^2 & \\omega\t\t\\\\\n1 & \\omega^2 & \\omega & -1 &  -\\omega^2 &  -\\omega\t\t\n\\end{array}\n\\right],\n\\end{align*} \nand\n\\begin{align*}\nD = \\diag{1,\\omega,\\omega^2,\\frac{1}{3},\\frac{1}{3}\\omega,\\frac{1}{3}\\omega^2}.\n\\end{align*}\nBecause $A$ has six distinct eigenvalues, following \\hyp{thm_class_rts}{Theorem}, it has $2^6 = 64$ primary square roots (and no nonprimary roots).\n\nThe matrices\n\\begin{align*} \n\\hat{D} = \\diag{1,\\omega^2,\\omega,\\frac{\\sqrt{3}}{3},\\frac{\\sqrt{3}}{3}\\omega^2,\\frac{\\sqrt{3}}{3}\\omega}\n\\end{align*}\nand\n\\begin{align*} \n\\tilde{D} = \\diag{1,\\omega^2,\\omega,-\\frac{\\sqrt{3}}{3},-\\frac{\\sqrt{3}}{3}\\omega^2,-\\frac{\\sqrt{3}}{3}\\omega}\n\\end{align*}\nare square roots of $D$, and, following \\hyp{mainresultirr}{Theorem}, the matrices\n\\begin{align*}\n\\hat{X}= Z \\hat{D} \\inv{Z} \\approx \n\\left[\n\\begin{array}{*{6}{r}}\n0 & 0 & 0 & 0 & 0.7887 & 0.2113 \\\\\n0 & 0 & 0 & 0 & 0.2113 & 0.7887 \\\\\n0.7887 & 0.2113 & 0 & 0 & 0 & 0 \\\\         \n0.2113 & 0.7887 & 0 & 0 & 0 & 0 \\\\      \n0 & 0 & 0.7887 & 0.2113 & 0 & 0\t \\\\    \n0 & 0 & 0.2113 & 0.7887 & 0 & 0  \n\\end{array}\n\\right]\n\\end{align*}\nand\n\\begin{align*}\n\\tilde{X}= Z \\tilde{D} \\inv{Z} \\approx \n\\left[\n\\begin{array}{*{6}{r}}\n0 & 0 & 0 & 0 & 0.2113 & 0.7887 \\\\\n0 & 0 & 0 & 0 & 0.7887 & 0.2113 \\\\\n0.2113 & 0.7887 & 0 & 0 & 0 & 0 \\\\         \n0.7887 & 0.2113 & 0 & 0 & 0 & 0\t\\\\      \n0 & 0 & 0.2113 & 0.7887 & 0 & 0 \\\\    \n0 & 0 & 0.7887 & 0.2113 & 0 & 0   \n\\end{array}\n\\right]\n\\end{align*}\nare the only eventually nonnegative square roots of $A$.\n\\end{ex}\n\n\\begin{rem} \\label{resultderogmatrices}\nFollowing the notation of \\hyp{mainresultirr}{Theorem}, note that all primary roots of $A$ are given by \n\\begin{align*}\nX =\nZ\n\\begin{bmatrix}\nF_{j_1} \\left( J(\\nu_h,1) \\right)\t& \t\t\t\t\t\t\t\\\\\n\t\t\t& F_{j_2} \\left( J\\left( \\lambda_2 \\nu_h, r_2 \\right) \\right) & &\t\\\\\n\t\t\t& & \\ddots \t\t\t\t\t\t\t\t\t\t\\\\\n\t\t\t& & & F_{j_t} \\left( J\\left( \\lambda_t \\nu_h, r_{t'} \\right) \\right)\n\\end{bmatrix}\n\\inv{Z}\n\\end{align*}\nwhere $j_k \\in \\mathcal{B}$ for $k = 1,\\dots, t'$, and subject to the constraint that $j_k = j_\\ell$ (because $j_k$ and $j_\\ell$ are ordered $h$-tuples, equality is meant entrywise) if $\\lambda_k = \\lambda_\\ell$.\n\nIf $A$ is \\emph{deregatory}, i.e., some eigenvalue appears in more than one Jordan block in the Jordan form of $A$, then $A$ has additional roots of the form \n\\begin{align*}\nX(U) =\nZU\n\\begin{bmatrix}\nF_{j_1} \\left( J(\\nu_h,1) \\right)\t& \t\t\t\t\t\t\t\\\\\n\t\t\t& F_{j_2} \\left( J\\left( \\lambda_2 \\nu_h, r_2 \\right) \\right) & &\t\\\\\n\t\t\t& & \\ddots \t\t\t\t\t\t\t\t\t\t\\\\\n\t\t\t& & & F_{j_t} \\left( J\\left( \\lambda_t \\nu_h, r_{t'} \\right) \\right)\n\\end{bmatrix}\n\\inv{U} \\inv{Z}\n\\end{align*}\nwhere $U$ is any matrix that commutes with $J$ and subject to the constraint that $j_k \\neq j_\\ell$, if $\\lambda_k = \\lambda_\\ell$. Select branches $j_1,j_2, \\dots, j_t$ following the proof of \\hyp{mainresultirr}{Theorem}. Following \\cite[Theorem 1, \\S 12.4]{lt1985}, note that the matrix $U$ must be of the form\n\\begin{align*}\n\\begin{bmatrix}\nu_1 \t& \t\t\t\t\t\\\\\n \t& u_2 \t& \t\t\t\t\\\\\n \t& \t& \\ddots \t\t\t\\\\\n \t& \t& \t\t& u_h \t\t\\\\\n \t& \t& \t& \t& \\hat{U} \n\\end{bmatrix}\n\\end{align*}\nand $X(U)$ is real if $U$ is selected to be real. The matrix $X(U)$ is irreducible because $X(U)^p = A$. Moreover,\n\\begin{align*}\nX_1 (U) \n&:= Z U\n\\begin{bmatrix}\nF_{j_1} \\left( J(\\nu_h,1) \\right) & 0 \\\\\n0 & 0\n\\end{bmatrix}\n\\inv{U} \\inv{Z} \n= X_1,\t\t\t\t\t\t\n\\end{align*}\nwhere $X_1$ is defined as in the proof of \\hyp{mainresultirr}{Theorem}. Thus, $X_1(U)$ is a nonnegative, irreducible, $h$-cyclic matrix that commutes with $X(U)$, so that the argument demonstrating the eventual nonnegativity of $X$ is also valid for $X(U)$.\n\\end{rem}\n\n\\begin{cor} \\label{realrootcor}\nLet $A \\in \\mat{n}{\\bb{R}}$ and suppose that $A \\geq 0$, irreducible, and imprimitive with index of cyclicity $h$. Then $A$ possesses an eventually nonnegative $p$th-root if and only if $A$ possesses a real $p$th-root and $\\gcd{(h,p)} = 1$.  \n\\end{cor}\n\n\\begin{thm} \\label{thmforgeneralirrennmatrices}\nLet $A \\in \\mat{n}{\\bb{R}}$ and suppose that $A \\geq 0$, irreducible, and imprimitive. If $\\inv{Z} A Z = J = J_0 \\oplus J_1$ is a Jordan canonical form of $A$, where $J_0$ collects all the singular Jordan blocks and $J_1$ collects the remaining Jordan blocks, and $A$ possesses a real root, then all eventually nonnegative $p$th-roots of $A$ are given by $A = Z \\left( X_0 \\oplus X_1 \\right) \\inv{Z}$, where $X_1$ is any $p$th-root of $J_1$ characterized by \\hyp{mainresultirr}{Theorem} or \\hyp{resultderogmatrices}{Remark}, and $X_0$ is a real $p$th-root of $J_0$.\n\\end{thm}\n\nAlthough the following result is known (see \\cite[Algorithm 3.1 and its proof]{h2009}), our work provides another proof.\n\n\\begin{cor} \\label{evennongen}\nIf $A \\in \\mat{n}{\\bb{R}}$ is irreducible with index of cyclicity $h$, where $h>1$, and $\\Pi$ describes the $h$-cyclic structure of $A$, then $A$ is eventually nonnegative if and only if there exists a nonsingular matrix $Z$ such that\n\\begin{align*}\n\\inv{Z} A Z =\n\\begin{bmatrix}\nJ( \\nu_h, 1) &  \\\\\n & \\hat{J}  \n\\end{bmatrix},\n\\end{align*} \nand, associated with $\\sr{A} = 1 \\in \\sig{A}$, is a positive left eigenvector $x$ and right eigenvector $y$. \n\\end{cor}\n\n\\begin{proof}\nIf $A$ is eventually nonnegative, select $p$ relatively prime to $h$ such that $A^p \\geq 0$. Then $A$ is a $p$th-root of $A^p$ and the result follows from \\hyp{thmforgeneralirrennmatrices}{Theorem}.\n\nThe converse is clear by setting \n\\[ X_1 = Z \\begin{bmatrix} J( \\nu_h, 1) & 0 \\\\ 0 & 0 \\end{bmatrix} \\inv{Z} \\] \nand \n\\[ X_2 = Z \\begin{bmatrix} 0 & 0 \\\\ 0 & \\hat{J} \\end{bmatrix} \\inv{Z}. \\] \n\\end{proof}\n\n\\begin{rem}\n\\hyp{thmforgeneralirrennmatrices}{Theorem} remains true if the assumption of nonnegativity is replaced with eventual nonnegativity.\n\\end{rem}\n\n\\begin{rem}\nFor $A \\in \\mat{n}{\\bb{C}}$ with no eigenvalues on $\\bb{R}^-$, the {\\it principal} $p$th-root, denoted by $A^{1\/p}$, is the unique $p$th-root of $A$ all of whose eigenvalues lie in the segment $\\{ z : -\\pi\/p < \\arg(z) < \\pi\/p\\}$ \\cite[Theorem 7.2]{h2008}. A nonnegative matrix $A$ is \\emph{stochastic} if $\\sum_j a_{ij} = 1$ for all $i=1,\\dots,n$. Following \\hyp{omegahfracpwr}{Theorem}, the principal $p$th-root of an imprimitve irreducible stochastic matrix is never stochastic. \n\\end{rem}\n\n\\subsection{Reducible Matrices}\n\nIdentifying the eventually nonnegative matrix roots of nonnegative reducible matrices poses many obstacles, chief of which is controlling the entries of the off-diagonal blocks in the Frobenius normal form. Moreover, the assumption that $\\gcd{(h,p)} = 1$ is not necessarily required; for instance, consider the matrix\n\\begin{align*}\nA =\n\\begin{bmatrix}\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 \\\\\n1 & 0 & 0 & 0\n\\end{bmatrix}\n\\end{align*}\nand note that the matrix\n\\begin{align*}\nB: =\n\\begin{bmatrix}\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 \\\\\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0\n\\end{bmatrix} = A^2\n\\end{align*}\nis a reducible 2-cyclic matrix; $\\sig{B} = \\Omega_2 \\cup \\Omega_2$; $B$ obviously possesses an irreducible nonnegative square root; and $\\gcd{(2,2)} = 2 > 1$.\n\n\\begin{mydef}\nA matrix $A \\in \\mat{n}{\\bb{C}}$ is said to be \\emph{completely reducible} if there exists a permutation matrix $P$ such that \n\\begin{align}\nP^\\top A P \n= \\bigoplus_{i=1}^k A_{ii}\n= \n \\left[\n \\begin{array}{ccc}\n A_{11} &  & \\multirow{2}{*}{\\large 0} \\\\\n \\multirow{2}{*}{\\large 0} & \\ddots &  \\\\\n  &  & A_{kk}\n \\end{array}\n \\right],\t\t\t\t\t\t\t\t\t\t\t\t\\label{compreducibleform}\n\\end{align}\t\nwhere $k \\geq 2$ and $A_{11}, \\dots, A_{kk}$ are square irreducible matrices.\t\t\t\t\t\t\t\t\t\n\\end{mydef} \n\n\\begin{rem}\nFollowing the definition, if $A$ is completely reducible, then, without loss of generaltiy, we may assume $A$ is in the form of the matrix on the right-hand side of \\eqref{compreducibleform}. Furthermore, it is clear that $A$ is eventually nonnegative if and only if $A_{11}, \\dots, A_{kk}$ are eventually nonnegative. \n\\end{rem}\n\nThe following is corollary to \\hyp{realrootcor}{Corollary} and \\hyp{thmforgeneralirrennmatrices}{Theorem} .\n\n\\begin{cor}\nLet $A$ be eventually nonnegative, nonsingular, and completely reducible. Let $h_i$ denote the cyclicity of $A_{ii}$ for $i=1,\\dots,k$. If each $A_{ii}$ possesses a real $p$th-root, then $A$ possesses an eventually nonnegative $p$th-root if and only if $\\gcd{(h_i,p)} = 1$ for all $i$.\n\\end{cor}\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro}Introduction}\n\n\n\nNumerous studies have shown that the key assumptions of small-scale universality, isotropy and locality of interactions are in question in various contexts of magnetohydrodynamic (MHD) turbulence \\cite{alexakisetal07,schekochihinetal08,mininni11}. Therefore, the validity of the classical phenomenology of Kolmogorov (K41) \\cite{k41a}, which provides to a good approximation the power law of the energy spectrum in hydrodynamic turbulence (besides intermittency corrections), is questionable in MHD turbulence, where several debatable phenomenological theories exist \\cite{iroshnikov64,kraichnan65,goldreichsridhar95,zhouetal04,boldyrev06,ngbhattacharjee97,galtieretal00}. \nIn summary the power law scaling exponents obtained in the various MHD turbulence phenomenologies based on weak and strong turbulence arguments both for isotropic and anisotropic fields are $-5\/3$, $-3\/2$ and $-2$. \n\nNumerical simulations to date are unable to provide a definitive answer to the scaling of the energy spectrum in MHD turbulence \\cite{mullergrappin05,mininnipouquet07}. Recently, large resolution simulations by Lee et al. \\cite{leeetal10} (using a code that enforces the symmetries of the Taylor-Green vortex to achieve higher resolution) demonstrated different scaling of the total energy spectra for different initial conditions and thereby suggested that freely decaying MHD turbulent flows are non-universal.\n\nThis lack of the detailed knowledge of the energy spectrum in MHD turbulence has many implications because to predict for example heating rates in solar and space physics, the energy dissipation rate is required, which is dependent on the slope of the energy spectrum. This is also the reason why subgrid scale models, required for numerical modelling in astrophysics and geophysics, are less developed in MHD. \n\nOn the other hand, several universal small scale features have been observed in a variety of hydrodynamic turbulent flows since the seminal works by Perry, Chong and Cantwell \\cite{chongetal90,cantwell93,perrychong94} on the analysis of the velocity gradient tensor invariants. One of the most important universal results is the well known teardrop shape \\cite{tsinober02,davidson04,meneveau11} of the joint probability density function (PDF) of the invariants of the velocity gradient tensor, which describes the topology and dynamics of small scales in hydrodynamic turbulence. Other important directions of research in hydrodynamic turbulence involving the study of such invariants has been the topological classification of the coherent structures \\cite{blackburnetal96,chakrabortyetal05} and the use of the invariants in subgrid-scale modelling \\cite{bosetal02}.\n\nTherefore, due to the limited information that can be extracted just from the slopes of the energy spectra and the fact that small scale universality is one of the key assumptions in inertial range phenomenologies, we try to gain insight in this paper on the universal or\/and non-universal features of MHD turbulence by studying the structures and dynamics of small scales through joint PDFs of the invariants of the velocity gradient, magnetic field gradient and related tensors.\nThrough this analysis we also attempt to provide a classification of the structures in MHD turbulence. This investigation was carried out using Direct Numerical Simulation (DNS) data of incompressible, homogeneous, decaying MHD turbulence with no imposed symmetries and no magnetic flux either in or out of our periodic boxes.\n\nThe paper is organised as follows. The numerical method, the initial conditions and the parameters of our DNS of decaying MHD turbulent flows are provided in section \\ref{sec:dns}. In section \\ref{sec:spectra} we present the energy spectra of our flows.\nBefore presenting our results we give an outline for the classification of fluid flow topology in section \\ref{sec:defs}. Then, in sections \\ref{sec:invu} and \\ref{sec:invb} we unravel our joint PDF analysis for the invariants of gradient quantities related to the velocity and magnetic field, respectively, delineating the structure and dynamics of the examined MHD flows. At the end, in section \\ref{sec:end}, we summarise our results.\n\n\\section{\\label{sec:dns}DNS of decaying MHD turbulence}\n\\subsection{Governing equations \\& numerical method}\nWe consider the three-dimensional, incompressible MHD equations of fluid velocity $\\bm u$ and magnetic induction $\\bm b$ to be\n\\begin{align}\n \\partial_t \\bm u - \\nu \\bm\\Delta \\bm u &= (\\bm u \\times \\bm \\omega) + (\\bm j \\times \\bm b) - \\bm \\nabla P \n \\label{eq:ns} \\\\\n \\partial_t \\bm b - \\kappa \\bm\\Delta \\bm b &= \\bm \\nabla \\times (\\bm u \\times \\bm b) \n \\label{eq:induction} \\\\\n \\bm \\nabla \\cdot \\bm u &= \\bm \\nabla \\cdot \\bm b = 0\n \\label{eq:incomp}\n\\end{align}\nwith $\\nu$ the kinematic viscosity, $\\kappa$ the magnetic diffusivity,  $\\bm \\omega \\equiv \\bm \\nabla \\times \\bm u$ the vorticity, $\\bm j \\equiv \\bm \\nabla \\times \\bm b$ the current density of the magnetic field and $P = p\/\\rho + \\tfrac{1}{2}\\bm u^2$ the fluid pressure, composed by the plasma pressure $p$, the constant mass density $\\rho$ and the hydrodynamic pressure $\\tfrac{1}{2}\\bm u^2$. Note that magnetic induction has units of Alfv\\'en velocity, i.e. $\\bm b\/\\sqrt{\\rho \\mu_0}$, where $\\mu_0 = (\\kappa \\sigma)^{-1}$ is the permeability of free space with $\\sigma$ the electrical conductivity. In ideal MHD, where $\\nu = \\kappa = 0$, the total energy $E_t \\equiv \\frac{1}{2}\\avg{|\\bm u|^2 + |\\bm b|^2} = E_u + E_b$, the magnetic helicity $H_b \\equiv \\avg{\\bm u \\cdot \\bm b}$ and the cross helicity $H_c \\equiv \\avg{\\bm a \\cdot \\bm b}$ are conserved, where the angle brackets $\\avg{.}$ in this study denote spatial averages. Here, $\\bm a$ is the magnetic \npotential, which is defined as $\\bm a \\equiv - \\bm\\Delta^{-1}(\\bm \\nabla \\times \\bm b)$, since one can define $\\bm b \\equiv \\bm \\nabla \\times \\bm a$ with $\\bm \\nabla \\cdot \\bm a = 0$. \n\nOur numerical method is pseudo-spectral \\cite{gottlieborszag77}, where each component of $\\bm u$ and $\\bm b$ is represented as truncated Galerkin expansions in terms of the Fourier basis. The non-linear terms are initially computed in physical space and then transformed to spectral space using fast Fourier transforms \\cite{fftw98}. Aliasing errors are removed using the 2\/3 dealiasing rule, i.e. wavenumbers $k \\in [1,N\/3]$, where $N$ is the number of grid points in each Cartesian coordinate of our periodic box with period $2\\pi$. The non-linear terms along with the pressure term are computed in such a way that $\\bm u$ and $\\bm b$ are projected on to a divergence-free space so that Eqs. \\eqref{eq:incomp} are satisfied. The temporal integration of Eqs. \\eqref{eq:ns} and \\eqref{eq:induction} is performed using a second-order Runge-Kutta method. The code is parallelised using message passing interface (MPI) with one-dimensional domain decomposition \\cite{mpicode05}.\n\n\\subsection{Initial conditions \\& numerical parameters}\nThe initial conditions that we consider in this study are the three different cases studied in \\cite{leeetal10}. In particular, the initial velocity field is the Taylor-Green (TG) vortex \\cite{taylorgreen37} defined as \n\\begin{equation}\n \\bm u_{TG}(\\bm x) = u_0 (\\sin x \\cos y \\cos z, -\\cos x \\sin y \\cos z, 0)\n\\end{equation}\nand the initial conditions of the magnetic field are generalisation of the TG vortex symmetries. In detail, the insulating case (run ``I'' hereafter) is\n\\begin{equation}\\label{eq:icase}\n \\bm b_I(\\bm x) = b_0^I (\\cos x \\sin y \\sin z, \\sin x \\cos y \\sin z, -2 \\sin x \\sin y \\cos z)\n\\end{equation}\nwhere $\\bm j_I = \\bm \\nabla \\times \\bm b_I$ is parallel to the faces of a subvolume $[0,\\pi]^3$, which can thereby be considered as electrical insulators. Note that in this case the magnetic field $\\bm b_I = -(b_0^I\/u_0) \\bm \\nabla \\times \\bm u_{TG}$ and the magnetic as well as cross helicity are globally restricted to vanish for all times due to the TG symmetries.\nThe conducting case (run ``C'' hereafter) takes the following form\n\\begin{equation}\\label{eq:ccase}\n \\bm b_C(\\bm x) = b_0^C (\\sin 2x \\cos 2y \\cos 2z, \\cos 2x \\sin 2y \\cos 2z, -2 \\cos 2x \\cos 2y \\sin 2z)\n\\end{equation}\nwith $\\bm j_C = \\bm \\nabla \\times \\bm b_C$ perpendicular to the faces of a subvolume $[0,\\pi]^3$, which can consequently be considered as electrically conductive. In this configuration, $H_b = 0$ for all times but $H_c \\neq 0$ although negligible\n(i.e. $H_c\\ell\/E_t < 0.04$ at its maximum over time, where $\\ell$ is a typical length scale). The final case that is considered by Lee et al. \\cite{leeetal10} is an alternative (run ``A'' hereafter) to the insulating initial conditions above (see Eq. \\eqref{eq:icase}), namely\n\\begin{equation}\\label{eq:acase}\n \\bm b_A(\\bm x) = b_0^A (\\cos 2x \\sin 2y \\sin 2z, -\\sin 2x \\cos 2y \\sin 2z, 0)\n\\end{equation}\nfor which again $H_b = H_c = 0$ for all times, at least up to the peak of dissipation.\n\nThe above TG fields exhibit several intrinsic symmetries within a cubic box of size $[0,2\\pi]^3$, where periodic boundary conditions are applied. These are mirror (anti)symmetries about the planes $x=0$, $x=\\pi$, $y=0$, $y=\\pi$, $z=0$ and $z=\\pi$ as well as rotational (anti)symmetries of angle $N\\pi$ about the axes $(x,y,z)=(\\tfrac{\\pi}{2},y,\\tfrac{\\pi}{2})$ and $(x,\\tfrac{\\pi}{2},\\tfrac{\\pi}{2})$ and of angle $N\\pi\/2$ about the axis $(\\tfrac{\\pi}{2},\\tfrac{\\pi}{2},z)$ for $N \\in \\mathbb{Z}$. The above mentioned planes that possess mirror symmetries form the insulating and conducting walls of $[0,\\pi]^3$ sub-boxes, also called impermeable boxes \\cite{brachetetal83}, for the corresponding initial conditions.\n\nIt is important to mention that Lee et al. \\cite{leeetal10} imposed numerically these symmetries in order to gain substantial savings in computational resources. Unlike \\cite{leeetal10}, our computations were performed without imposing any symmetry constrains, allowing thus the turbulence to evolve freely with the view that the initial TG vortex symmetries will break at high enough Reynolds numbers. However, even for our highest resolution simulations with Taylor Reynolds number of the order of 100 the TG symmetries are not broken within the time interval of reaching the peak of dissipation. They seem to be a strong property of the MHD equations, preserved by time evolution of the solutions (see also \\cite{leeetal08}).\n\nDue to the fact that there are special global restrictions on these TG flows, we further consider a run with random initial conditions (run ``R'' hereafter) for comparison, ensuring that $H_b = H_c = 0$ and kinetic helicity $H_u \\equiv \\avg{\\bm u \\cdot \\bm \\omega} = 0$ at time $t=0$. During the time evolution magnetic and cross helicity remain zero for all times relative to the total energy. However, the kinetic helicity reaches an approximate value of $H_u\\ell\/E_t < 0.2$ at its absolute maximum over time but when dissipation is maximum $H_u\\ell\/E_t < 0.04$.\n\nWe report results based on the analysis of decaying MHD turbulence simulated with $N = 1024^3$ grid points. In order to obtain the broadest inertial range, runs I, A and C are initialised at the largest scales and\nrun R at wavenumbers $k = 1$ and 2, adding extra randomness. At time $t=0$ the fields are normalised such that the kinetic and magnetic energies are in equipartition, i.e. $E_u(t=0) = E_b(t=0) = 0.125$. Note that all flows have unit magnetic Prandtl number (i.e. $\\nu = \\kappa$). The numerical parameters of our computations are provided in Table \\ref{tbl:dnsparam}.\n\n\\begin{table}[!ht]\n  \\caption{Numerical parameters of the DNS. The values presented are taken at the peak of total dissipation. Note that $k_{max} = N\/3$, using the $2\/3$ dealiasing rule.}\n  \\label{tbl:dnsparam}\n   \\begin{ruledtabular}\n    \\begin{tabular}{*{10}{c}}\n      \\textbf{Run} & \\textbf{N} & $\\bm{\\nu}$ & $\\bm{Re_{\\lambda_t}}$ & $\\bm{L_t}$ & $\\bm{\\lambda_t}$ & $\\bm{\\eta_t}$ & $\\bm{u'}$ & $\\bm{b'}$ & $\\bm {k_{max}\\eta_t}$ \\\\\n       & & $(\\times 10^{-4})$ & & $(\\times 10^{-1})$ & $(\\times 10^{-1})$ & $(\\times 10^{-3})$ & & & \\\\\n     \\hline\n      R  & 1024 & 5.5 & 140.7 & 8.33 & 2.15 & 7.80 & 0.36 & 0.48 & 2.66 \\\\\n      I  & 1024 & 4.5 & 121.8 & 6.84 & 2.03 & 6.54 & 0.27 & 0.62 & 2.23 \\\\\n      C  & 1024 & 4.5 & 138.0 & 6.23 & 1.35 & 5.82 & 0.46 & 0.35 & 1.97 \\\\\n      A  & 1024 & 4.5 & 115.1 & 3.76 & 1.40 & 5.77 & 0.37 & 0.46 & 1.99 \\\\\n    \\end{tabular}\n  \\end{ruledtabular}\n\\end{table}\n\nThe rms velocity $u'$ is defined as\n\\begin{equation}\n  u' \\equiv \\left( \\frac{2}{3}\\int E_u(k)dk \\right)^{1\/2}.\n\\end{equation}\nand similarly for $b'$, the rms of the magnetic field.\nThe integral length scales are then defined as the total, kinetic and magnetic integral length scale, respectively\n\\begin{equation}\n L_{t,u,b} \\equiv \\frac{3\\pi}{4}\\frac{\\int k^{-1}E_{t,u,b}(k)dk}{\\int E_{t,u,b}(k)dk}\n\\end{equation}\nand likewise for the Taylor scales\n\\begin{equation}\n \\lambda_{t,u,b} \\equiv \\lrbig{5\\frac{\\int E_{t,u,b}(k)dk}{\\int k^2E_{t,u,b}(k)dk}}^{1\/2}.\n\\end{equation}\nIn Table \\ref{tbl:dnsparam}, we report the total integral and Taylor length scales as well as the Reynolds number based on $\\lambda_t$ given by $Re_{\\lambda_t} \\equiv u' \\lambda_t \/ \\nu$. Finally, the smallest length scale in our flows is defined based on K41 scaling $\\eta_t \\equiv (\\nu^3 \/ \\epsilon_t)^{1\/4}$, where $\\epsilon_t = \\nu \\avg{|\\bm \\omega|^2} + \\kappa \\avg{|\\bm j|^2}$ is the total dissipation. The time we address in this study is the moment which the dissipation reaches its maximum value and therefore the highest scale separation occurs $\\eta_t \\ll \\ell \\ll L_t$, where $\\ell$ is a typical length scale in the inertial range. Therefore, the values provided in Table \\ref{tbl:dnsparam} correspond to that moment.\n\n\\section{\\label{sec:spectra}Energy spectra}\nFigure \\ref{fig:et_spectra} presents the three-dimensional compensated total energy spectra $k^pE_t(k)$ that we obtain at the peak of dissipation for all the runs of Table \\ref{tbl:dnsparam}. The spectra are compensated with the scaling exponents $p = 2,\\; 5\/3$ and $3\/2$. The small peaks at high wavenumbers show the quality of our simulations.\n \\begin{figure}[!ht]\n  \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/R1024_Et_vs_k_expnts-eps-converted-to.pdf} \n   \\caption{Run R}\n  \\end{subfigure}\n  \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/I1024_Et_vs_k_expnts-eps-converted-to.pdf}\n   \\caption{Run I}\n  \\end{subfigure} \\\\\n  \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/C1024_Et_vs_k_expnts-eps-converted-to.pdf}\n   \\caption{Run C}\n  \\end{subfigure}\n  \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/A1024_Et_vs_k_expnts-eps-converted-to.pdf}\n   \\caption{Run A}\n  \\end{subfigure}\n  \\caption{(Color online) Three-dimensional compensated total energy spectra $k^pE_t(k)$ with scaling exponents $p = 2,\\; 5\/3,\\; 3\/2$ for a) run R, b) run I, c) run C and  d) run A of Table \\ref{tbl:dnsparam}.}\n  \\label{fig:et_spectra}\n \\end{figure}\n\nAccording to Lee et al. \\cite{leeetal10}, the energy spectrum of the magnetically dominated flow I, i.e. $E_b\/E_u > 1$ (see also $u'$ and $b'$ values in Table \\ref{tbl:dnsparam}), is close to a $k^{-2}$ power law (see Fig. \\ref{fig:et_spectra}b), which is the weak turbulence (WT) theory expectation \\cite{ngbhattacharjee97,galtieretal00}. Here, we would like to emphasize, however, that the WT scaling ($E_\\perp \\propto k_\\perp^{-2}$) is for an anisotropic energy spectrum, where perpendicular denotes the direction relative to an imposed large scale mean magnetic field $\\bm B_0$ that does not exist in this flow. In fact, it is argued that in MHD turbulence there is no prescribed cascade in the parallel direction \\cite{biskamp03}. This is based on the idea that small-scale turbulent fluctuations become anisotropic, as it is easier to shuffle strong magnetic field lines than to bend them due to the preventing action of the Lorentz force $\\bm j \\times \\bm b$.\ncaused by a large-scale field $\\bm B_0$. \n\nFurthermore, Lee et al. \\cite{leeetal10} argues that the kinetic energy dominated flow C, i.e. $E_b\/E_u < 1$ (see also Table \\ref{tbl:dnsparam}), is compatible with a $k^{-3\/2}$ slope (Fig. \\ref{fig:et_spectra}c) and the less magnetically dominated flow A is near a $k^{-5\/3}$ scaling (Fig. \\ref{fig:et_spectra}d). In addition, we report that the power law of the total energy spectrum for our also magnetically dominated run R (see Table \\ref{tbl:dnsparam}) seems to be between $k^{-5\/3}$ and $k^{-3\/2}$. The difference between these two power laws is subtle enough that any type of contamination, such as intermittency or any dissipative small-scale effects, will blur the results. However, even a $k^{-2}$ spectrum which is slightly more transparent in these high Reynolds numbers can be misinterpreted. For example, in contrast to \\cite{leeetal10}, we claim that the total energy spectrum of run A (Fig. \\ref{fig:et_spectra}d) scales like $E_t \\propto k^{-2}$ but we leave this to the readers' judgement.\n\nTherefore, the following questions are raised: How can we circumvent this ambiguity of the results? Is there a dependence of small scales on the large scale initial conditions and thereby non-universality in decaying MHD turbulence? Since limited information can be extracted just from the slopes of the spectra, we try to answer these questions by examining the topology of the small scales through the invariants of related gradient statistics, which some have shown universal characteristics for hydrodynamic turbulent flows.\n\n\\section{\\label{sec:defs}Classification of the fluid flow topology}\nAn approach that provides a well-defined and unambiguous language to describe eddying motions and flow patterns is the framework of critical point concepts from bifurcation theory \\cite{glendinning94}, which was studied extensively in the context of hydrodynamic turbulent flows by Perry, Chong, Cantwell and co-workers \\cite{chongetal90,soriaetal94,chongetal98,ooietal99}. \nHere we provide a brief outline on the background material related to the geometric invariants of second-order tensorial quantities in turbulence before going to consider various statistics of these invariants in the following sections. Extensive reviews on the subject can be found in \\cite{cantwell02,tsinober02,davidson04} and references therein. \n\nGeometric invariants remain unchanged under the full group of rotations (i.e. rotations plus reflections) \\cite{tsinober02}, being independent of the frame of reference. Any traceless second-order tensor $\\bm M$ has the following characteristic polynomial\n\\begin{equation}\n \\label{eq:cubic}\n \\det[\\bm M - \\lambda_i \\bm I] = 0 \\Rightarrow \n \\lambda_i^3 + P \\lambda_i^2 + Q \\lambda_i + R = 0\n\\end{equation}\nwhere $\\lambda_i$ are the eigenvalues of $\\bm M$ and its invariants are\n\\begin{align}\n  \\label{eq:invI}\n P &= -tr(\\bm M) = - (\\lambda_1 + \\lambda_2 + \\lambda_3) = 0 \\\\\n  \\label{eq:invII}\n Q &=  \\frac{1}{2}[P^2 - tr(\\bm M^2)] = \\lambda_1\\lambda_2 + \\lambda_2\\lambda_3 + \\lambda_3\\lambda_1 \\\\\n  \\label{eq:invIII}\n R &= -\\det(\\bm M) = - \\lambda_1\\lambda_2\\lambda_3\n\\end{align}\nThe value of the discriminant for $P = 0$ is\n\\begin{equation}\n D = \\tfrac{27}{4}R^2 + Q^3\n\\end{equation}\nand provides a general classification for the solutions of the cubic equation \\eqref{eq:cubic} dividing the (Q,R) space into the following regions\n \\begin{enumerate}\n  \\item $D > 0$: 1 real \\& 2 complex-conjugate eigenvalues\n  \\item $D = 0$: 3 real eigenvalues of which 2 are equal\n  \\item $D < 0$: 3 real distinct eigenvalues\n \\end{enumerate}\nwhich correspond to various local flow topologies. In this study, the first invariant is $P = 0$ from definition \\eqref{eq:invI} since the vector fields that we consider are solenoidal.\n\n\n\n\n\n\n\n\\section{\\label{sec:invu}Invariants of the velocity gradient, the strain rate and rotation rate tensors}\n\\subsection{Joint PDFs of the velocity gradient invariants}\nThe velocity gradient tensor $\\bm A = \\bm \\nabla \\bm u$ can be decomposed into a symmetric and skew-symmetric component,\n\\begin{equation}\n \\bm A = \\bm S + \\bm \\Omega = S_{ij} - \\tfrac{1}{2}\\epsilon_{ijk}\\omega_k\n\\end{equation}\nwhere $\\bm S = \\tfrac{1}{2}(\\bm \\nabla \\bm u + \\bm \\nabla \\bm u^T)$ and $\\bm \\Omega = \\tfrac{1}{2}(\\bm \\nabla \\bm u - \\bm \\nabla \\bm u^T)$ are the strain rate and rotation rate tensors, respectively. According to Eqs. \\eqref{eq:invII} and \\eqref{eq:invIII}, the second and third invariants of $\\bm A$ are\n\\begin{equation}\n \\label{eq:Qa}\n Q_A = \\tfrac{1}{4}[\\bm\\omega^2 - 2tr(\\bm S^2)]\n\\end{equation}\nand\n\\begin{equation}\n \\label{eq:Ra}\n R_A = -\\tfrac{1}{3}[tr(\\bm S^3) + \\tfrac{3}{4}\\omega_i\\omega_jS_{ij}],\n\\end{equation}\nrespectively. Here we are interested in the joint probability density function (PDF) of these invariants. A diagram of this joint PDF called the ($R_A,Q_A$) invariant map is presented in Fig. \\ref{fig:QaRa_map}, labelling the various topological classifications.\n\n \\begin{figure}[!ht]\n  \\includegraphics[width=0.35\\textwidth]{Plots\/QaRa_space-eps-converted-to.pdf}\n  \\caption{Diagram of the ($R_A,Q_A$) invariant map indicating the local flow topologies related to each zone.}\n  \\label{fig:QaRa_map}\n \\end{figure}\n\nIf $Q_A > 0$ then enstrophy $\\bm \\omega^2$ dominates over $tr(\\bm S^2)$ and vice versa if $Q_A < 0$. For positive values of $R_A$ the topologies are unstable, whereas for negative $R_A$ the topologies are stable. Moreover, the $D_A = 0$ line (see Fig. \\ref{fig:QaRa_map}), where $D_A = \\tfrac{27}{4} R_A ^2 + Q_A^3$ is the discriminant, divides the invariant map into two regions. One where $D_A > 0$ with one real and two complex-conjugate eigenvalues as solutions of Eq. \\eqref{eq:cubic} for the velocity gradient tensor and the other where $D_A < 0$ with three real distinct eigenvalues. Note that along the vertical $R_A = 0$ axis one of the eigenvalues is zero and therefore locally the flow topology is invariant in this direction. \n\nNow, if $Q_A$ is much greater than zero (i.e. $D_A > 0$) then $R_A \\approx -\\tfrac{1}{4}\\omega_i\\omega_jS_{ij}$. In this case, for $R_A < 0$ vortex stretching dominates over vortex compression, whereas for $R_A > 0$ vortex compression dominates (see Fig. \\ref{fig:QaRa_map}). On the other hand, if $Q_A$ is much less than zero and \n$D_A < 0$ then $R_A \\approx -\\tfrac{1}{2}tr(\\bm S^3)$. In this case, $R_A > 0$ locally is related to a sheetlike structure (or unstable node\/saddle\/saddle topologies according to the terminology of Chong et al. \\cite{chongetal90}) whereas $R_A < 0$ with a tubelike structure (or stable node\/saddle\/saddle topologies \\cite{chongetal90}). This will also become more transparent when we will deal later with the third invariant of the strain rate tensor (see section \\ref{sec:QsRs}). \n\nIn hydrodynamic turbulent flows, ranging from atmospheric boundary layers to free shear flows in wind tunnels and even simulations of compressible turbulence, there is the prominent tendency of the joint PDF of ($R_A,Q_A$) to develop an inclined teardrop shape. This shape aligns with the second and fourth quadrants, with a cusp lying along the $R_A > 0$, $D_A = 0$ branch (see for example Fig. 10.1 in \\cite{tsinober02}) and is considered to be a universal feature. Therefore, there is a preference for vortex stretching and sheetlike structures. In many visualisations of enstrophy in hydrodynamic turbulent flows the dominant structures seem to be tubelike structures but between these vortex-tubes there are sheetlike structures, where most of the dissipation is located \\cite{tanakakida93,tsinober02}.\n\nBefore analysing the results, we would like to note that the aspect ratio of the axes of all joint PDFs, that are reported in this paper, has been kept the same but the abscissa and the ordinate are different to reflect the change in magnitude of the plotted quantities in the four flows that we consider. In addition, the points near the origin correspond to low gradient values associated with the large scale motions, whereas points far away characterize the high-gradient small scales.\nAll the joint PDFs were computed at the instant of maximum dissipation.\n\nIn Fig. \\ref{fig:QaRa} we present the joint PDFs of $R_A$ versus $Q_A$ for all the decaying MHD runs of Table \\ref{tbl:dnsparam}.\n \\begin{figure}[!ht]\n  \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/R1024_v_jpdfQR-eps-converted-to.pdf}\n   \\caption{Run R}\n  \\end{subfigure}  \n  \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/I1024_v_jpdfQR-eps-converted-to.pdf}\n   \\caption{Run I}\n  \\end{subfigure} \\\\\n  \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/C1024_v_jpdfQR-eps-converted-to.pdf}\n   \\caption{Run C}\n  \\end{subfigure}\n  \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/A1024_v_jpdfQR-eps-converted-to.pdf}\n   \\caption{Run A}\n  \\end{subfigure}\n  \\caption{(Color online) Joint PDFs of the second invariant $Q_A$ and the third invariant $R_A$ of the velocity gradient tensor normalised appropriately by powers of the mean enstrophy for a) run R, b) run I, c) run C and  d) run A of Table \\ref{tbl:dnsparam}. The line $D_A = \\tfrac{27}{4} R_A ^2 + Q_A^3 = 0$ is plotted for reference.}\n  \\label{fig:QaRa}\n \\end{figure}\nThe most important outcome from the plots in Fig. \\ref{fig:QaRa} is that the shape of the joint PDF of $R_A$ with $Q_A$ is not universal in decaying MHD turbulence and small-scales seem to depend on the large-scale initial conditions. However, we should be cautious here as it is not clear whether the self-preservation of the TG vortex symmetries during the evolution restrict the dynamics in some way. \n\nOn the other hand, it is clear that there is a modest but still present trend of the ($R_A,Q_A$) map to align along the second and fourth quadrants for our simulation with random initial conditions (Fig. \\ref{fig:QaRa}a). It is noteworthy that the shape of the joint PDF is more symmetric with respect to the $R_A = 0$ axis in comparison to hydrodynamic flows (see for example \\cite{ooietal99}). Run I gives a striking joint PDF (Fig \\ref{fig:QaRa}b) with a significant percentage of its points lying in the first quadrant and with high absolute values of $Q_A$ in comparison to the rest of the runs.\nPoints of the joint PDF in the first quadrant that are far from the origin (see Fig \\ref{fig:QaRa}b) are associated with very low rates of kinetic energy dissipation. This suggests that the \nstructure is likely to be quite long-lived. Run C seems to resemble more the teardrop shape of hydrodynamic turbulence with the classic narrow cusp in $R_A > 0$, $D_A = 0$ branch (Fig. \\ref{fig:QaRa}c). Finally, Fig. \\ref{fig:QaRa}d shows the ($R_A,Q_A$) map of run A, which has a shape with features in between the random MHD and hydrodynamic turbulence. In other words, there is a modest tendency of the joint PDF to align with the second quadrant like in the random MHD run (Fig. \\ref{fig:QaRa}a) but there is a high correlation between $R_A > 0$ and $Q_A < 0$ values forming a long cusp in analogy to hydrodynamic turbulent flows.\n\n\\subsection{\\label{sec:QsRs}Joint PDFs of strain rate invariants}\nSetting $\\bm \\Omega$ to zero or essentially $\\bm \\omega$ to zero in Eqs. \\eqref{eq:Qa} and \\eqref{eq:Ra}, we can obtain the invariants of the strain rate tensor, which are\n\\begin{align}\n Q_s &= -\\tfrac{1}{2}tr(\\bm S^2) \\\\\n\\label{eq:rs}\n R_s &= -\\tfrac{1}{3}tr(\\bm S^3).\n\\end{align}\n\nThe ($R_s,Q_s$) invariant map features the geometry of the local straining of the fluid elements (see Fig. \\ref{fig:QsRs_map}).\n \\begin{figure}[!ht]\n  \\includegraphics[width=0.35\\textwidth]{Plots\/QsRs_space-eps-converted-to.pdf}\n  \\caption{Diagram of the ($R_s,Q_s$) invariant map. Each plotted curve corresponds to the following flow geometries: $\\lambda_1:\\lambda_2:\\lambda_3 = 2:-1:-1$ (axisymmetric contraction), $1:0:-1$ (two-dimensional flow), $3:1:-4$ (biaxial stretching) and $1:1:-2$ (axisymmetric stretching).}\n  \\label{fig:QsRs_map}\n \\end{figure}\nThe second invariant $Q_s$ is related to viscous dissipation $\\epsilon = 2 \\nu \\bm S^2$ through $Q_s = -\\tfrac{1}{4}\\epsilon \/ \\nu$ because the strain rate tensor is symmetric, i.e. $\\bm S^2 = S_{ij}S_{ji} = tr(\\bm S^2)$. So, locations with $Q_s$ much less than zero are highly dissipative regions. Note that $Q_s$ is negative definite. The third invariant $R_s$ has two important physical meanings. First, it is proportional to strain skewness $S_{ij}S_{jk}S_{ki}$, which appears as part of the production term in the evolution equation of $\\bm S^2$ (see \\cite{tsinober02}). Second, it can be written as a function of the eigenvalues of $S_{ij}$, viz. \n\\begin{equation}\n \\label{eq:rseig}\n R_s = -\\tfrac{1}{3}(\\lambda_1^3 + \\lambda_2^3 + \\lambda_3^3) = -\\lambda_1\\lambda_2\\lambda_3 \n\\end{equation}\nsince $tr(\\bm S) = \\lambda_1 + \\lambda_2 + \\lambda_3 = 0$ due to incompressibility, with $\\lambda_1 \\geq \\lambda_2 \\geq \\lambda_3$. Owing to the symmetry of $S_{ij}$ all eigenvalues are real and\ntherefore the ($R_s,Q_s$) invariant map is contained only in the region where $D_s = \\tfrac{27}{4} R_s ^2 + Q_s^3 \\leq 0$ (see Figs. \\ref{fig:QsRs_map} and \\ref{fig:QsRs}). So, $R_s > 0$ implies production of $\\bm S^2$ and hence of viscous dissipation with $\\lambda_1,\\lambda_2 > 0$ and $\\lambda_3 < 0$ related to sheetlike structures. On the contrary, $R_s < 0$ indicates destruction of $\\bm S^2$ with $\\lambda_1 > 0$ and $\\lambda_2,\\lambda_3 < 0$ associated with tubelike structures. Note, therefore that $\\text{sgn}(R_s) = \\text{sgn}(\\lambda_2)$. We should point out here that if we define the following ratio $a = \\lambda_2 \/\\lambda_1$ of the eigenvalues of $S_{ij}$, then each value of $a$ corresponds to a line in the ($R_s,Q_s$) plane with the following expression\n\\begin{equation}\n\\label{eq:schematic}\n R_s = (-Q_s)^{3\/2}a(1+a)(1+a+a^2)^{-3\/2}\n\\end{equation}\nwhere each line is associated with a flow topology (see caption of Fig. \\ref{fig:QsRs_map}) \\cite{cantwell02,blackburnetal96}.\n\nThe ($R_s,Q_s$) invariant map in many hydrodynamic turbulent flows away from boundaries manifests a tendency for the $R_s > 0$ region, implying a predominance of sheetlike structures related to the strain rate (see for example Fig. 8c in \\cite{ooietal99}). In particular, numerical and experimental evidences in homogeneous hydrodynamic turbulence propose the ratios of the mean eigenvalues of $S_{ij}$ to be $\\avg{\\lambda_1}$:$\\avg{\\lambda_2}$:$\\avg{\\lambda_3} =$ 3:1:-4 \\cite{ashurstetal87,tsinoberetal92} (see the corresponding line in Fig. \\ref{fig:QsRs_map}).\n\nThe joint PDFs of $R_s$ versus $Q_s$ for the four runs of Table \\ref{tbl:dnsparam} are illustrated in Fig. \\ref{fig:QsRs}.\n \\begin{figure}[!ht]\n  \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/R1024_v_jpdfQsRs-eps-converted-to.pdf}\n   \\caption{Run R}\n  \\end{subfigure}\n    \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/I1024_v_jpdfQsRs-eps-converted-to.pdf}\n   \\caption{Run I}\n  \\end{subfigure} \\\\\n    \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/C1024_v_jpdfQsRs-eps-converted-to.pdf}\n   \\caption{Run C}\n  \\end{subfigure}\n    \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/A1024_v_jpdfQsRs-eps-converted-to.pdf}\n   \\caption{Run A}\n  \\end{subfigure}\n  \\caption{(Color online) Joint PDFs of the second invariant $Q_s$ and third invariant $R_s$ of the strain rate tensor normalised appropriately by powers of the mean enstrophy for a) run R, b) run I, c) run C and  d) run A of Table \\ref{tbl:dnsparam}. The line $D_s = \\tfrac{27}{4} R_s ^2 + Q_s^3 = 0$ is plotted for reference.}\n  \\label{fig:QsRs}\n \\end{figure}\nTheir dependence on initial conditions is clearly depicted. The shape of the ($R_s,Q_s$) map for run R (Fig. \\ref{fig:QsRs}a) moves away from the $D_s = 0$ line towards the $R_s = 0$ axis expressing a more quasi two-dimensional (2D) character of the structures related to $S_{ij}$ than in hydrodynamic turbulent flows away from the boundaries. We should point out, however, that this particular shape is reminiscent to the joint PDFs of ($R_s,Q_s$) found in the buffer layer, i.e. a region very close to the wall, of wall-bounded turbulent shear flows (see for example Fig. 6f in \\cite{blackburnetal96}). The joint PDF of run I (Fig. \\ref{fig:QsRs}b) is aligned along the $R_s = 0$ with some highly dissipative small scales in contrast to the rest of the runs. The local topology in this case seems to have a strong tendency towards quasi two-dimensionality. Part of the shape of this joint PDF can be explained through two-dimensional shearing (or vortex sheet), i.e.\n\\begin{equation}\n \\label{eq:approxA}\n A_{ij} = \n  \\begin{pmatrix}\n   0 & \\partial_yu_x & 0 \\\\\n   0 & 0        & 0 \\\\\n   0 & \\partial_yu_z & 0\n  \\end{pmatrix}\n\\end{equation}\nwhich gives $Q_s = -\\tfrac{1}{4}[(\\partial_yu_x)^2+(\\partial_yu_z)^2]$ and $R_s = 0$ in analogy to the influence of the wall on the velocity gradient in wall-bounded flows \\cite{blackburnetal96}. The ($R_s,Q_s$) invariant map of run C in Fig. \\ref{fig:QsRs}c also falls away from the $D_s = 0$ branch with low correlations between $R_s$ and $Q_s$. Finally, the joint PDF of run A (Fig. \\ref{fig:QsRs}d) is almost identical in shape but less correlated with respect to Fig. \\ref{fig:QsRs}a.\n\n \nWe now try to summarise and clarify our arguments by tabulating the mean eigenvalues of the strain rate tensor and their ratios for all our runs in Table \\ref{tbl:Seig} but also by plotting the curves that can be constructed from Eq. \\eqref{eq:schematic} using the mean eigenvalues of Table \\ref{tbl:Seig} (see Fig. \\ref{fig:Seig}). \n\\begin{table}[!ht]\n \\caption{Mean eigenvalues of the strain rate tensor $S_{ij}$ and their ratios.}\n \\label{tbl:Seig}\n \\begin{ruledtabular}\n    \\begin{tabular}{*{6}{c}} \n      \\textbf{Run} & $\\bm{\\avg{\\lambda_1}}$ & $\\bm{\\avg{\\lambda_2}}$ & $\\bm{\\avg{\\lambda_3}}$ & $\\bm{\\avg{\\lambda_1}:\\avg{\\lambda_2}:\\avg{\\lambda_3}}$ \\\\\n     \\hline\n      R & 0.14 &  0.03 & -0.17 &  5 : 1 : -6 \\\\\n      I & 0.25 & -0.00 & -0.25 &  1 : 0 : -1 \\\\\n      C & 0.25 &  0.04 & -0.29 &  6 : 1 : -7 \\\\\n      A & 0.25 &  0.04 & -0.29 &  6 : 1 : -7 \\\\\n    \\end{tabular}\n \\end{ruledtabular}\n\\end{table}\n\\begin{figure}[!ht]\n  \\includegraphics[width=0.5\\textwidth]{Plots\/Qs-Rs_1024_v_mean-eigs_v2-eps-converted-to.pdf}\n  \\caption{(Color online) Plots of Eq. \\eqref{eq:schematic} using the mean eigenvalues of $S_{ij}$ from Table \\ref{tbl:Seig}. The dashed line $D_s = \\tfrac{27}{4} R_s ^2 + Q_s^3 = 0$ is plotted for reference. With HD we label the curve that corresponds to 3:1:-4, the characteristic eigenvalue ratios for homogeneous hydrodynamic turbulence.}\n  \\label{fig:Seig}\n \\end{figure}\n \nIn Fig. \\ref{fig:Seig}, we plot for reference the curve that corresponds to 3:1:-4, the characteristic eigenvalue ratios for homogeneous hydrodynamic turbulent flows that we denote as ``HD''. In that respect, all the ratios of the mean eigenvalues that we obtain are different than 3:1:-4. However, all the cases represent biaxial expansion apart from run I, which is characterised by quasi two-dimensionality with weak biaxial contraction (see Table \\ref{tbl:Seig} and Fig. \\ref{fig:Seig}). Figure \\ref{fig:Seig} makes clear that on average the flow topologies related to $S_{ij}$ of run C and A are close to run R giving weight to our argument for the similarity of their ($R_s,Q_s$) joint PDFs. The curve for run I also summarises Fig. \\ref{fig:QsRs}b by demonstrating that the quasi 2D structures associated to the strain rate tensor are weakly contracted in a average sense.\n\n\n\\subsection{Joint PDFs of the second invariants of the strain and rotation rate tensors}\nAnother important joint PDF to analyse is the one of $-Q_s$ versus the second invariant of the rotation rate tensor, $Q_\\omega$, which is in fact the only invariant for $\\bm \\Omega$. To see this, set $\\bm S$ to zero in Eqs. \\eqref{eq:Qa} and \\eqref{eq:Ra}, then\n\\begin{equation}\n Q_\\omega = -\\tfrac{1}{2}tr(\\bm \\Omega^2) = \\tfrac{1}{4}\\bm \\omega^2\n\\end{equation}\nwhich is positive definite and it is related to the second invariants of $\\bm A$ and $\\bm S$ through $Q_\\omega = Q_A - Q_s$. The ($Q_\\omega,-Q_s$) invariant map that is shown schematically in Fig. \\ref{fig:QwQs_map} identifies the relative importance of the straining and rotational part of velocity gradient tensor.\n \\begin{figure}[!ht]\n  \\includegraphics[width=0.35\\textwidth]{Plots\/QsQw_space-eps-converted-to.pdf}\n  \\caption{Diagram of the ($Q_\\omega,-Q_s$) invariant map pointing out the important regions related to strain and rotation.}\n  \\label{fig:QwQs_map}\n \\end{figure}\nA good example that describes simply the physical meanings of Fig. \\ref{fig:QwQs_map} is the Burger's vortex tube \\cite{saffman95}. As it was mentioned before $Q_s$ characterises the topology associated with viscous dissipation. So, points near the $-Q_s$ axis reflect nearly pure straining motions, i.e. regions of strong dissipation but negligible enstrophy, like outside and away from the Burger's vortex tube. On the other hand, points close to the $Q_\\omega$ axis are in nearly pure solid-body rotation, like at the centre of the Burger's vortex tube with high enstrophy but very weak dissipation. Regions with comparable strain rate and rotation map to points close to the $Q_\\omega = -Q_s$ line, which correspond to vortex sheets.\n\nGenerally, from observations in many hydrodynamic turbulent flows, regions of intense enstrophy tend to be concentrated in tubelike structures, whereas regions of high dissipation are not correlated with regions of concentrated enstrophy \\cite{tsinober02}. So, the joint PDF of $Q_\\omega$ versus $-Q_s$ is very spread for many hydrodynamic turbulent flows away from walls (see results in \\cite{jimenezetal93,blackburnetal96}).\n\n\nFigure \\ref{fig:QsQw} shows the joint PDFs of $Q_\\omega$ versus $-Q_s$, normalised with the mean enstrophy, for the four runs of Table \\ref{tbl:dnsparam}.\n \\begin{figure}[!ht]\n  \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/R1024_v_jpdfQsQw-eps-converted-to.pdf}\n   \\caption{Run R}\n  \\end{subfigure}\n  \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/I1024_v_jpdfQsQw-eps-converted-to.pdf}\n   \\caption{Run I}\n  \\end{subfigure} \\\\\n  \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/C1024_v_jpdfQsQw-eps-converted-to.pdf}\n   \\caption{Run C}\n  \\end{subfigure}  \n  \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/A1024_v_jpdfQsQw-eps-converted-to.pdf}\n   \\caption{Run A}\n  \\end{subfigure}\n  \\caption{(Color online) Joint PDFs of the second invariants of strain rate and rotation rate tensors normalised by the mean enstrophy for a) run R, b) run I, c) run C and  d) run A of Table \\ref{tbl:dnsparam}.}\n  \\label{fig:QsQw}\n \\end{figure}\nThe dependence on initial conditions is pronounced once more in these plots. The ($Q_\\omega,-Q_s$) invariant map of run R (Fig. \\ref{fig:QsQw}a) is very different to hydrodynamic turbulence away from walls. Here, the joint PDF is concentrated around the $Q_\\omega = -Q_s$ line demonstrating stronger correlation between these two variables. This result in conjunction with the outcome from Fig. \\ref{fig:Seig} confirms many visualisations of homogeneous MHD turbulent flows \\cite{biskamp03}, which illustrate large population of sheetlike rather than tubelike structures. \n\nThe shape of the joint PDF ($Q_\\omega,-Q_s$) for run I is even more extreme with a very narrow distribution along the main diagonal (Fig. \\ref{fig:QsQw}b), where regions of high dissipation are strongly correlated by high levels of enstrophy particularly for points far from the origin. The high gradients in this flow can be well approximated by Eq. \\eqref{eq:approxA} where $Q_\\omega = -Q_s = \\tfrac{1}{4}[(\\partial_yu_x)^2+(\\partial_yu_z)^2]$. According to Cantwell \\cite{cantwell02} the presence or absence of points very far from the origin, associated with quite long-lived structures, is closely related to the regularity of the initial conditions. He further mentions that such structures are much less prominent in a flow with randomised initial conditions. Here, this is transparent if one compares the run with random initial conditions (Fig. \\ref{fig:QsQw}a) with run I (Fig. \\ref{fig:QsQw}b). Moreover, it could be argued that the core of the joint PDF ($Q_\\omega,-Q_s$) of run I is similar to the joint PDF obtained in \nthe buffer layer of wall-bounded flows (see results by \\cite{blackburnetal96,chongetal98}). The ($Q_\\omega,-Q_s$) map of run A (Fig. \\ref{fig:QsQw}d) resembles Fig. \\ref{fig:QsQw}a but with weaker correlations between high dissipation and high enstrophy regions. Finally, the joint PDF of Fig. \\ref{fig:QsQw}c, which corresponds to run C, presents the weakest correlations between $Q_\\omega$ and $-Q_s$ among the four cases with a weak trend of alignment along the main diagonal.\n\n\\subsection{\\label{sec:vis} Flow structures and enstrophy dynamics}\nVarious flow field quantities were viewed interactively using a visualisations software \\cite{paraview} to get an idea of the spatial structures in our flows. In order to substantiate our approach, we present indicatively plots of iso-contours of the vorticity field in our $[0,2\\pi]^3$ periodic boxes at the moment of maximum dissipation for the four runs of Table \\ref{tbl:dnsparam} (see Fig. \\ref{fig:vis}).\n \\begin{figure}[!ht]\n  \\begin{subfigure}{0.3\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/R512_wrms3_v2.png}\n   \\caption{Run R}\n  \\end{subfigure}\n  \\begin{subfigure}{0.3\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/I512_wrms3.png}\n   \\caption{Run I}\n  \\end{subfigure} \\\\\n  \\begin{subfigure}{0.3\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/C512_wrms3_v2.png}\n   \\caption{Run C}\n  \\end{subfigure}\n  \\begin{subfigure}{0.3\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/A512_wrms3_v2.png}\n   \\caption{Run A}\n  \\end{subfigure}\n  \\caption{(Color online) Vorticity field iso-contours with $|\\omega| \\geq 3\\omega'$ for a) run R, b) run I, c) run C and d) run A of Table \\ref{tbl:dnsparam}.}\n  \\label{fig:vis}\n \\end{figure}\nFigure \\ref{fig:vis}a (run R) displays iso-contours of vorticity for $|\\omega| \\geq 3\\omega'$ where $\\omega' \\equiv (|\\bm \\omega|^2)^{1\/2}$. The predominant structures in this plot are randomly oriented sheetlike structures in support of our joint PDF analysis. In comparison to the randomly oriented structures of run R, the TG vortex symmetries become apparent in Figs. \\ref{fig:vis}b, c and d revealing their preservation in time. Remember that we did not impose any symmetries during the evolution of our runs. According to the above analysis, the peculiar run I should be prevailed by quasi two-dimensional sheetlike structures, which are shown in Fig. \\ref{fig:vis}b. These flat structures are formed on the insulating faces of the $[0,\\pi]^3$ boxes and on their mid-planes in the vertical direction, i.e. $z = \\pi\/2$. The structures of run A (Fig. \\ref{fig:vis}d) are also sheetlike but more randomly oriented in contrast to run I. In the end, run C is a more complicated TG flow.\nThis is demonstrated in Fig. \\ref{fig:vis}c for $|\\omega| \\geq 3\\omega'$.\nIt is interesting that the initial conditions of the TG velocity with the TG magnetic fields for the insulating runs I and A create less randomness in the flow fields, which are mainly dominated by quasi 2D sheetlike structures in contrast to run C.\n\nAccording to Jim\\'enez et al. \\cite{jimenezetal93}, in hydrodynamic turbulent flows away from walls, it is qualitative clear that there is no other way of production of enstrophy other than straining of weak vorticity to form stronger vortex regions. Then, strain itself is induced by vorticity and the process may become non-linear. This mechanism is called self-amplification of velocity derivatives \\cite{tsinober02,sagautcambon08}. \n\nIn order to have an initial picture of this mechanism and in particular of the formation of the vorticity fields in our MHD flows, we examine the rate of vortex stretching\n\\begin{equation}\n \\label{eq:sigma}\n \\Sigma = \\frac{\\bm \\omega \\cdot \\bm S \\cdot \\bm \\omega}{|\\bm \\omega|^2} = \\frac{R_s - R_A}{Q_\\omega}\n\\end{equation}\nwhich is essentially the part of the strain that is aligned with the local vorticity and it is the term that stretches or compresses the vortex lines in the evolution equation of the enstrophy\n\\begin{equation}\n \\label{eq:enstrophy}\n \\mathrm{d}_t (\\tfrac{1}{2} \\bm \\omega^2) = \\bm \\omega \\cdot \\bm S \\cdot \\bm \\omega \n                                   + \\nu \\bm \\omega \\cdot \\bm \\Delta \\bm \\omega \n                                   + \\bm \\omega \\cdot \\bm \\nabla \\times (\\bm j \\times \\bm b).\n\\end{equation}\nNotice that $\\Sigma$ can be written as a function of the invariants $R_A$, $R_s$ and $Q_\\omega$ (see Eq. \\eqref{eq:sigma}). \n\nFigure \\ref{fig:QwSigma} shows joint PDFs of essentially the enstrophy (i.e. $Q_\\omega$) with the rate of vortex stretching $\\Sigma$ appropriately normalised for all the flows of Table \\ref{tbl:dnsparam}.\n\\begin{figure}[!ht]\n \\begin{subfigure}{0.35\\textwidth}\n  \\includegraphics[width=\\textwidth]{Plots\/R1024_v_jpdfQwSigma-eps-converted-to.pdf}\n  \\caption{Run R}\n \\end{subfigure}\n \\begin{subfigure}{0.35\\textwidth}\n  \\includegraphics[width=\\textwidth]{Plots\/I1024_v_jpdfQwSigma-eps-converted-to.pdf}\n  \\caption{Run I}\n \\end{subfigure} \\\\\n \\begin{subfigure}{0.35\\textwidth}\n  \\includegraphics[width=\\textwidth]{Plots\/C1024_v_jpdfQwSigma-eps-converted-to.pdf}\n  \\caption{Run C}\n \\end{subfigure}\n \\begin{subfigure}{0.35\\textwidth}\n  \\includegraphics[width=\\textwidth]{Plots\/A1024_v_jpdfQwSigma-eps-converted-to.pdf}\n  \\caption{Run A}\n \\end{subfigure}\n  \\caption{(Color online) Joint PDFs of the second invariant of the rotation rate tensor $Q_\\omega$ and the vortex stretching rate $\\Sigma$ normalised appropriately by powers of the mean enstrophy for a) run R, b) run I, c) run C and  d) run A of Table \\ref{tbl:dnsparam}.}\n  \\label{fig:QwSigma}\n \\end{figure}\nVarious common features can be observed in Fig. \\ref{fig:QwSigma}. To be more specific, the highest values of enstrophy are associated with positive but low values of $\\Sigma$, i.e. stretching of vorticity, whereas high rates of stretching as well as compression correlate with regions of low $Q_\\omega$. So, there is little evidence of self-stretching by structures in the flow which have large enstrophy in analogy to hydrodynamic turbulence \\cite{jimenezetal93,ooietal99}. Another common feature in all the plots of Fig. \\ref{fig:QwSigma} is the tilt towards positive values, i.e. vorticity vectors are being more stretched than compressed. \n\n\nOn the other hand, quantitative differences are evident, such as\nthe asymmetry of the ($\\Sigma,Q_\\omega$) joint PDFs, which seems to be different for each flow. In other words, the joint PDF of run C (Fig. \\ref{fig:QwSigma}c) is shifted more towards $\\Sigma > 0$ values, akin to hydrodynamic turbulence (see results in \\cite{jimenezetal93,ooietal99}), in comparison to run A (Fig. \\ref{fig:QwSigma}d) which is closer to the joint PDF of run R (Fig. \\ref{fig:QwSigma}a). Another quantitative difference between the four flows is the very high values of enstrophy ($Q_\\omega \\simeq 6\\avg{|\\bm \\omega|^2}$) that are obtained in run I (Fig. \\ref{fig:QwSigma}b) for values of vortex stretching rate of the same order for all the flows (i.e. $\\Sigma < 0.3\\avg{|\\bm \\omega|^2}^{1\/2}$).\n\nAnother important mechanism for amplification or reduction of enstrophy that exists only in MHD turbulence is that due to the Lorentz force. This process essentially manifests from the last term of Eq. \\eqref{eq:enstrophy}, which we write here as\n\\begin{equation}\n L = \\frac{\\bm \\omega \\cdot \\bm \\nabla \\times (\\bm j \\times \\bm b)}{|\\bm \\omega|^2},\n\\end{equation}\nso that it is comparable with $\\Sigma$ (see Eq. \\eqref{eq:sigma}). In order to shed light on the dynamics of this term with respect to the enstrophy, we consider in Fig. \\ref{fig:QwL} the joint PDFs between $L$ and $Q_\\omega$, normalised appropriately, for the four runs of Table \\ref{tbl:dnsparam}.\n \\begin{figure}[!ht]\n  \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/R1024_v_jpdfQwL_v2-eps-converted-to.pdf}\n   \\caption{Run R}\n  \\end{subfigure}\n  \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/I1024_v_jpdfQwL-eps-converted-to.pdf}\n   \\caption{Run I}\n  \\end{subfigure} \\\\\n  \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/C1024_v_jpdfQwL-eps-converted-to.pdf}\n   \\caption{Run C}\n  \\end{subfigure}\n  \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/A1024_v_jpdfQwL-eps-converted-to.pdf}\n   \\caption{Run A}\n  \\end{subfigure}\n \\caption{(Color online) Joint PDFs of the second invariant of the rotation rate tensor $Q_\\omega$ and $L$ normalised appropriately by powers of the mean enstrophy for a) run R, b) run I, c) run C and d) run A of Table \\ref{tbl:dnsparam}.}\n \\label{fig:QwL}\n\\end{figure}\n\nIt is characteristic for all the plots of Fig. \\ref{fig:QwL} that there is a preference for $L > 0$ for most of the local topology in the flow. It is also common in all the four cases that the highest values of $Q_\\omega$ are associated with regions of low but positive $L$, whereas high values of $|L|$ are related to regions of low enstrophy in a similar fashion to the self-amplification mechanism. Once more, the quantitative differences between the plots of Fig. \\ref{fig:QwL} are evident with the most notable being the joint PDF of run I (Fig. \\ref{fig:QwL}b) with the highest values of $Q_\\omega$ in terms of $L$.\n\nA comparison between the two mechanisms of amplification and reduction of enstrophy reveals the cause of high and low enstrophy regions. For the MHD flow with random initial conditions, $L$ is more correlated with regions of higher enstrophy than $\\Sigma$ but the opposite is true for the TG flows. On the other hand, the lowest enstrophy regions present correlations with higher absolute values of $L$ than $\\Sigma$ for all the runs.\n\n\\section{\\label{sec:invb} Invariants of the magnetic field gradient, the magnetic strain rate and the current density rate tensors}\nIn this section we try to classify the topology related to the magnetic field by extending the above joint PDF analysis for the invariants of magnetic field gradient tensor as well as for the invariants of its symmetric and skew-symmetric components.\n\\subsection{Joint PDFs of the magnetic field gradient invariants}\nThe magnetic field gradient $\\bm X = \\bm \\nabla \\bm b$ can be also decomposed into its symmetric and skew-symmetric component\n\\begin{equation}\n \\bm X = \\bm K + \\bm J = K_{\\alpha\\beta} - \\tfrac{1}{2}\\epsilon_{\\alpha\\beta\\gamma}j_\\gamma\n\\end{equation}\nwhere $\\bm K = \\tfrac{1}{2}(\\bm \\nabla \\bm b + \\bm \\nabla \\bm b^T)$ and $\\bm J = \\tfrac{1}{2}(\\bm \\nabla \\bm b - \\bm \\nabla \\bm b^T)$ are the magnetic strain rate and current density rate tensors, respectively. The skew-symmetric part of $\\bm X$ is related to the electric current through Ampere's law $\\bm \\nabla \\times \\bm b = \\mu_0 \\bm j$ where $\\mu_0 = (\\kappa\\sigma)^{-1}$ is the permeability of free-space and $\\sigma$ is the electrical conductivity. When the magnetic field lines are bended, current is produced providing a Lorentz force that inhibits the bending of the field lines. On the other hand, the symmetric part of $\\bm X$ characterises the force-free regions in the magnetic field, where $\\bm j = 0$ and therefore $\\bm j \\times \\bm b = 0$. An important relation one can easily derive by taking the divergence of Eq. \\eqref{eq:ns} and using the fact that our fields $\\bm u$ and $\\bm b$ are solenoidal is the following Poisson equation\n \\begin{align}\n  \\bm \\nabla^2 P &= \\bm \\nabla \\cdot [(\\bm u \\times \\bm \\omega) + (\\bm j \\times \\bm b)] \\nonumber \\\\\n            &= (\\Omega_{\\alpha\\beta}^2 - S_{\\alpha\\beta}^2) +\n               (K_{\\alpha\\beta}^2 - J_{\\alpha\\beta}^2).\n \\end{align}\nWhat is interesting in this expression is the interchange between the symmetric and skew-symmetric tensors of $\\bm \\nabla \\bm u$ and $\\bm \\nabla \\bm b$ related to $\\bm \\nabla^2 P$. It is also appealing that the viscous dissipation is related to the symmetric part of the velocity gradient, whereas the Ohmic dissipation to the skew-symmetric part of the magnetic field gradient.\n\nNow, we consider the joint PDF of the second and third invariants of $\\bm X$, which are defined according to Eqs. \\eqref{eq:invII} and \\eqref{eq:invIII} as follows\n\\begin{equation}\n \\label{eq:Qx}\n Q_X = \\tfrac{1}{4}[\\bm j^2 - 2tr(\\bm K^2)]\n\\end{equation}\n\\begin{equation}\n \\label{eq:Rx}\n R_X = -\\tfrac{1}{3}[tr(\\bm K^3) + \\tfrac{3}{4}j_\\alpha j_\\beta K_{\\alpha\\beta}].\n\\end{equation}\n\\begin{figure}[!ht]\n \\begin{subfigure}{0.35\\textwidth}\n  \\includegraphics[width=\\textwidth]{Plots\/R1024_b_jpdfQR-eps-converted-to.pdf}\n  \\caption{Run R}\n \\end{subfigure}\n \\begin{subfigure}{0.35\\textwidth}\n  \\includegraphics[width=\\textwidth]{Plots\/I1024_b_jpdfQR-eps-converted-to.pdf}\n  \\caption{Run I}\n \\end{subfigure} \\\\\n \\begin{subfigure}{0.35\\textwidth}\n  \\includegraphics[width=\\textwidth]{Plots\/C1024_b_jpdfQR-eps-converted-to.pdf}\n  \\caption{Run C}\n \\end{subfigure}\n \\begin{subfigure}{0.35\\textwidth}\n  \\includegraphics[width=\\textwidth]{Plots\/A1024_b_jpdfQR-eps-converted-to.pdf}\n  \\caption{Run A}\n \\end{subfigure}\n  \\caption{(Color online) Joint PDFs of the second invariant $Q_X$ and the third invariant $R_X$ of the magnetic field gradient tensor normalised appropriately by powers of the mean squared current density for a) run R, b) run I, c) run C and  d) run A of Table \\ref{tbl:dnsparam}. The line $D_X = \\tfrac{27}{4} R_X ^2 + Q_X^3 = 0$ is plotted for reference.}\n  \\label{fig:QxRx}\n\\end{figure}\nFor the classification of the magnetic field structures the $D_X = \\tfrac{27}{4} R_X ^2 + Q_X^3 = 0$ line was included in the plots of Fig. \\ref{fig:QxRx}. The topological classification emerging from the joint PDFs of $R_X$ and $Q_X$ can be interpreted in analogy to the invariant map of the velocity gradient (Fig. \\ref{fig:QaRa_map}). Note, however, that the individual terms of the third invariant in Eq. \\eqref{eq:Rx} do not appear in any evolution equation. Thus, $R_X$ does not have a physical meaning here but it is mathematically important for the classification of the magnetic field structures, in terms of the eigenvalues of $X_{\\alpha\\beta}$ associated with these structures.\n\nIn contrast to the invariants of the velocity gradient, the ($R_X,Q_X$) invariant map does not show a particular tendency towards any quadrant (see Fig. \\ref{fig:QxRx}). For all the runs the core shape of the joint PDF is symmetric along the $R_X = 0$ axes, meaning that there is a balance between stable and unstable structures. The small scales, on the other hand, are slightly different especially for run I (Fig. \\ref{fig:QxRx}b) and run C (Fig. \\ref{fig:QxRx}c). Moreover, the joint PDF for runs C and A (see Fig. \\ref{fig:QxRx}c and d, respectively) diminish towards the origin of the axes. In general, one could claim that this symmetric shape seems to be a general characteristic for the magnetic field gradient for all initial conditions with some small deviations, which might be due to the TG vortex symmetries.\n\n\\subsection{Joint PDFs of the magnetic strain rate invariants}\nLooking at the joint PDFs of the second and third invariants of $\\bm K$ we can study the geometry of the local magnetic straining. The invariants of the magnetic strain rate tensor can be obtained by setting $\\bm j = 0$ in Eqs. \\eqref{eq:Qx} and \\eqref{eq:Rx}, which reduce to\n\\begin{equation}\n Q_K = -\\tfrac{1}{2}tr(\\bm K^2) \n\\end{equation}\nand\n\\begin{equation}\n R_K = -\\tfrac{1}{3}tr(\\bm K^3),\n\\end{equation}\nwhere $Q_K$ is negative definite due to the symmetric nature of $\\bm K$. Note that $Q_K$ is not directly related to Ohmic dissipation in contrast to the $Q_s$ for viscous dissipation. Then, the physical interpretation of the ($R_K$,$Q_K$) invariant map is quite different from Fig. \\ref{fig:QsRs_map} but similar in terms of flow topology. So, very low values of $Q_K$ in Fig. \\ref{fig:QkRk} can be physically interpreted as regions of high magnetic-strain or regions where the Lorentz force is small. The third invariant $R_K$ can be written as the product of the eigenvalues of $K_{\\alpha\\beta}$ in analogy to $R_s$ (see Eq. \\eqref{eq:rseig}). Then, the interpretation of $R_K$ in terms of sheetlike and tubelike structures is also determined by $\\text{sgn}(R_K) = \\text{sgn}(\\lambda_2)$.\n\nThe joint PDFs between $R_K$ and $Q_K$, representing the local topology of the structures related to magnetic strain rate, appear to be symmetric along the $R_K = 0$ axis for most of the runs of Table \\ref{tbl:dnsparam} (see Fig. \\ref{fig:QkRk}).\n\\begin{figure}[!ht]\n \\begin{subfigure}{0.35\\textwidth}\n  \\includegraphics[width=\\textwidth]{Plots\/R1024_b_jpdfQkRk-eps-converted-to.pdf}\n  \\caption{Run R}\n \\end{subfigure}\n \\begin{subfigure}{0.35\\textwidth}\n  \\includegraphics[width=\\textwidth]{Plots\/I1024_b_jpdfQkRk-eps-converted-to.pdf}\n  \\caption{Run I}\n \\end{subfigure} \\\\\n \\begin{subfigure}{0.35\\textwidth}\n  \\includegraphics[width=\\textwidth]{Plots\/C1024_b_jpdfQkRk_v2-eps-converted-to.pdf}\n  \\caption{Run C}\n \\end{subfigure}\n \\begin{subfigure}{0.35\\textwidth}\n  \\includegraphics[width=\\textwidth]{Plots\/A1024_b_jpdfQkRk_v2-eps-converted-to.pdf}\n  \\caption{Run A}\n \\end{subfigure}\n  \\caption{(Color online) Joint PDFs of the second invariant $Q_K$ and the third invariant $R_K$ of the magnetic strain rate tensor normalised appropriately by powers of the mean squared current density for a) run R, b) run I, c) run C and  d) run A of Table \\ref{tbl:dnsparam}. The line $D_K = \\tfrac{27}{4} R_K ^2 + Q_K^3 = 0$ is plotted for reference.}\n  \\label{fig:QkRk}\n\\end{figure}\nIn detail, the joint PDF of run R (Fig. \\ref{fig:QkRk}a) illustrates an equipartition between tubelike and sheetlike structures associated with $\\bm K$. The shapes of the ($R_K,Q_K$) invariant map for runs I and A (see Figs. \\ref{fig:QkRk}b and \\ref{fig:QkRk}d respectively) are also symmetric and they can be well approximated by a magnetic field gradient of the form of Eq. \\eqref{eq:approxA} with $Q_K = -\\tfrac{1}{4}[(\\partial_yb_x)^2 + (\\partial_yb_z)^2]$ and $R_K = 0$. In Fig. \\ref{fig:QkRk}b there are very low values of $Q_K$ correlated with $R_K = 0$ in comparison to the rest of the flows. Therefore, this approximation for $\\bm X$ is especially valid for the small scale structures that correspond to low values of $Q_K$ in this joint PDF.\nFigure \\ref{fig:QkRk}c (run C), on the other hand, is slightly asymmetric, showing a tangible inclination of the joint PDF towards $R_K < 0$. This implies that there is a preference for the intermediate eigenvalue of $K_{\\alpha\\beta}$ to be negative and hence a tendency for tubelike structures.\n\nNow, we attempt to provide an outline of the joint PDFs of Fig. \\ref{fig:QkRk} by tabulating the mean eigenvalues of the magnetic strain rate tensor (see Table \\ref{tbl:Keig}) and by plotting the analogous expression to Eq. \\eqref{eq:schematic} for $R_K$ and $Q_K$ using the mean eigenvalues of Table \\ref{tbl:Keig} (see Fig. \\ref{fig:Keig}).\n\\begin{table}[!ht]\n \\caption{Mean eigenvalues of the magnetic strain rate tensor $K_{\\alpha\\beta}$ and their ratios.}\n \\label{tbl:Keig}\n \\begin{ruledtabular}\n    \\begin{tabular}{*{6}{c}} \n      \\textbf{Run} & $\\bm{\\avg{\\lambda_1}}$ & $\\bm{\\avg{\\lambda_2}}$ & $\\bm{\\avg{\\lambda_3}}$ & $\\bm{\\avg{\\lambda_1}:\\avg{\\lambda_2}:\\avg{\\lambda_3}}$ \\\\\n     \\hline\n      R & 0.26 &  0.00 & -0.26 & 1 : 0 : -1 \\\\\n      I & 0.44 &  0.00 & -0.44 & 1 : 0 : -1 \\\\\n      C & 0.35 & -0.02 & -0.33 & 18 : -1 : -17 \\\\\n      A & 0.44 &  0.00 & -0.44 & 1 : 0 : -1 \\\\\n    \\end{tabular}\n \\end{ruledtabular}\n\\end{table}\n\\begin{figure}[!ht]\n \\includegraphics[width=0.5\\textwidth]{Plots\/Qk-Rk_1024_b_mean-eigs_v2-eps-converted-to.pdf}\n \\caption{(Color online) Plots of Eq. \\eqref{eq:schematic} using the mean eigenvalues of $K_{\\alpha\\beta}$ from Table \\ref{tbl:Keig}. The dashed line $D_K = \\tfrac{27}{4} R_K ^2 + Q_K^3 = 0$ is plotted for reference.}\n \\label{fig:Keig}\n\\end{figure}\nThe values of the mean eigenvalue ratios tell us that on average run I and C are described by quasi two-dimensional structures in agreement with the joint PDF analysis. Moreover, the values 1:0:-1 that we obtain for run R agree with the argument that the joint PDF of Fig. \\ref{fig:QkRk}a is symmetric but also express that in an average sense the flow topology is locally invariant in one direction. The only case that deviates from two-dimensionality is run C, which is on average characterised by biaxial contraction (i.e. $\\avg{\\lambda_2} < 0$) and thereby tubelike structures.\n\n\\subsection{Joint PDFs of the second invariants of the magnetic strain and current density tensors}\nThe skew-symmetric part of the magnetic field gradient tensor, $\\bm J$ has only one invariant in analogy to the rotation rate tensor $\\bm \\Omega$. This can be obtained by letting $\\bm K$ to be zero in Eqs. \\eqref{eq:Qx} and \\eqref{eq:Rx}, then\n\\begin{equation}\n Q_j = -\\tfrac{1}{2}tr(\\bm J^2) = \\tfrac{1}{4}\\bm j^2,\n\\end{equation}\nwhich is also related to the second invariants of $\\bm X$ and $\\bm K$ through $Q_j = Q_X - Q_K$. \n\nThe ($Q_j,-Q_K$) invariant map describes the relative importance between the straining and rotational part of the magnetic field gradient in analogy to ($Q_\\omega,-Q_s$) map for the velocity gradient (see Fig. \\ref{fig:QwQs_map}). However, the important difference in this case is that the rotational part of $\\bm X$ is directly related to Ohmic dissipation and not the straining part. Hence, the points of the joint PDFs close to the $Q_j$ axis that are nearly in solid-body rotation are regions in the flow of strong Ohmic dissipation and negligible magnetic straining in contrast to the picture we get from Fig. \\ref{fig:QwQs_map}. On the other side, points adjacent to the $-Q_K$ axis express nearly pure magnetic straining motions in regions of where the current is negligible and thereby Lorentz force is suppressed.\n\nThe joint PDFs of Fig. \\ref{fig:QjQk} show that points near the axes are rare in MHD turbulent flows and are related only to the large scales of the flows, where $Q_j$ and $-Q_K$ are small in comparison to $\\avg{|\\bm j|^2}$.\n \\begin{figure}[!ht]\n  \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/R1024_b_jpdfQkQj-eps-converted-to.pdf}\n   \\caption{Run R}\n  \\end{subfigure}\n  \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/I1024_b_jpdfQkQj-eps-converted-to.pdf}\n   \\caption{Run I}\n  \\end{subfigure} \\\\\n  \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/C1024_b_jpdfQkQj-eps-converted-to.pdf}\n   \\caption{Run C}\n  \\end{subfigure}\n  \\begin{subfigure}{0.35\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/A1024_b_jpdfQkQj-eps-converted-to.pdf}\n   \\caption{Run A}\n  \\end{subfigure}\n  \\caption{(Color online) Joint PDFs of the second invariants of magnetic strain rate and current density rate tensors normalised by the mean squared current density for a) run R, b) run I, c) run C and d) run A of Table \\ref{tbl:dnsparam}.}\n  \\label{fig:QjQk}\n \\end{figure}\nMost of the points in the plots of Fig. \\ref{fig:QjQk} lie near the main diagonal, revealing that Ohmic dissipation occurs in current sheets. Here, the magnetic field gradient tensor can be well approximated by the form of Eq. \\eqref{eq:approxA}, which gives $Q_j = -Q_K = \\tfrac{1}{4}[(\\partial_yb_x)^2 + (\\partial_yb_z)^2]$. This is particularly a good approximation for runs I and A (see Figs. \\ref{fig:QjQk}b and \\ref{fig:QjQk}d respectively), where $Q_j$ and $-Q_K$ are strongly correlated for all scales. It can also be argued that this approximation is also valid for the small scales of runs R and C (see Figs. \\ref{fig:QjQk}a and \\ref{fig:QjQk}c respectively) that correspond to high values of $Q_j$ and $-Q_K$.\n\n\\subsection{Structures in the current density field}\nTo further validate our joint PDF approach, we present indicatively iso-contours of current density (Fig. \\ref{fig:visj}) in our $[0,2\\pi]^3$ periodic boxes at the moment of maximum dissipation for all the runs that we have considered (see Table \\ref{tbl:dnsparam}). All the visualisations of Fig. \\ref{fig:visj} display current density iso-contours with $|j| \\ge 6j'$ where $j' \\equiv (|\\bm j|^2)^{1\/2}$.\n \\begin{figure}[!ht]\n  \\begin{subfigure}{0.3\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/R512_jrms6.png}\n   \\caption{Run R}\n  \\end{subfigure}\n  \\begin{subfigure}{0.3\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/I512_jrms6_v2.png}\n   \\caption{Run I}\n  \\end{subfigure} \\\\\n  \\begin{subfigure}{0.3\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/C512_jrms6.png}\n   \\caption{Run C}\n  \\end{subfigure}\n  \\begin{subfigure}{0.3\\textwidth}\n   \\includegraphics[width=\\textwidth]{Plots\/A512_jrms6_v2.png}\n   \\caption{Run A}\n  \\end{subfigure}\n  \\caption{(Color online) Current density field iso-contours with $|j| \\geq 6j'$ for a) run R, b) run I, c) run C and d) run A of Table \\ref{tbl:dnsparam}.}\n  \\label{fig:visj}\n \\end{figure}\nThe field of current density for run R (Fig. \\ref{fig:visj}a) is composed by randomly oriented sheetlike structures which seem to be extremely thin, supporting the fact that the values of the mean eigenvalue ratios for the magnetic strain rate tensor are $1:0:-1$. It is clear that locally quasi two-dimensional structures are the dominant structures in Figs. \\ref{fig:visj}b and \\ref{fig:visj}d (runs I and A respectively), validating the joint PDFs of ($R_K,Q_K$). These 2D current sheets are also the structures where most of the Ohmic dissipation occurs. For run I these dominant structures are formed at the faces of the $[0,\\pi]^3$ boxes, whereas for run A these are randomly oriented. On the other hand, run C (Fig. \\ref{fig:visj}c) seem to be dominated by tubelike structures but one can also observe the coexistence of isolated thin current sheets in agreement to our analysis. Finally, the TG vortex symmetries are clearly depicted in these visualisations with each TG flow having different degree of randomness. \nThis raises again questions as to what degree these symmetries restrict the dynamics of the flows.\n\nHere, we would like to emphasize that the structures related to the magnetic field gradient have different characteristics than those related to the velocity gradient. This might well be a reason that the energy spectra that we obtain, as well as other studies, for the kinetic and magnetic energy (not shown here) seem to obey different scaling exponents. Moreover, we conjecture that these quasi two-dimensional organised structures that appear in runs I and A both in the vorticity and the current density fields are the reason to obtain a $k^{-2}$ scaling that we observe in the total energy spectra in Figs. \\ref{fig:et_spectra}b and \\ref{fig:et_spectra}d. This, however, needs to be further investigated and it will be reported elsewhere.\n\n\\section{\\label{sec:end} Conclusions}\nThe universality of the energy spectrum in MHD turbulence is in doubt by various studies. One aspect is the manifestation of different, dubious scaling exponents. In order to avoid ambiguity between scaling exponents, we explore various statistics based on the invariants of the velocity gradient and related tensors. Note that for a big family of hydrodynamic turbulent flows, the joint PDF of the invariants of the velocity gradient is generally considered to be universal. We further extend this analysis to the invariants of gradient statistics related to the magnetic field. In particular, we explore DNS data of decaying MHD turbulence with random initial conditions as well as a set of three different Taylor-Green type initial conditions without imposing any symmetry constrains in our flows during their evolution. The TG flows were chosen to be examined since recently, Lee et al. \\cite{leeetal10} reported that the scaling of the energy spectrum at the peak of dissipation depends on the initial conditions.\n\nOur study attempts to classify the structures of our MHD flows. The structures related to the strain rate tensor are predominantly sheetlike structures (i.e. $\\avg{\\lambda_2} > 0$) for all the flows apart from run I (see Fig. \\ref{fig:Seig}), which is quasi two-dimensional (i.e. $\\avg{\\lambda_2} \\simeq 0$). \nThe biaxial stretching for our MHD flows is different in comparison to hydrodynamic turbulence, namely $\\avg{\\lambda_1}:\\avg{\\lambda_2}:\\avg{\\lambda_3} = 3:1:-4$ (see Table \\ref{tbl:Seig}). Furthermore, the enstrophy dominated regions are well correlated with region of high viscous dissipation in contrast to hydrodynamic flows. We also find that viscous dissipation is an intrinsic element of vortex sheets.\n\nOn the other side, magnetic field consists of quasi two-dimensional structures, i.e. $\\avg{\\lambda_2} = 0$, for all the cases apart from run C, which is on average dominated by tubelike structures, i.e. $\\avg{\\lambda_2} < 0$ (see Table \\ref{tbl:Keig}). The correlation between magnetic strain dominated regions and regions of high Ohmic dissipation is generally stronger than the correlation between enstrophy and viscous dissipation. We also obtain that Ohmic dissipation resides in current sheets, which are thinner than the vortex sheets in the same flow. Visualisations support further our joint PDFs analysis of the invariants.\n\nOur results also demonstrate that small scales depend on the initial conditions in decaying MHD turbulence. This is dramatically illustrated through the joint PDF of $R_A$ with $Q_A$ (see Fig. \\ref{fig:QaRa}), which has a universal teardrop shape for hydrodynamic turbulence away from walls. Lack of small scale universality in decaying MHD turbulence will have important implications in modelling. The main idea of small-scale universality applied in LES (i.e. that although large scales may be dependent on boundary conditions or initial conditions, smaller scales are less flow dependent and more amenable to modelling) seems to fail for decaying MHD turbulent flows. Therefore, if MHD turbulence is non-universal, then the construction of subgrid-scale models for MHD flows might be doubtful.\n\nHowever, there is an important element regarding the TG flows that one has to address before claiming that small scale universality is absent in these flows. This element is the self-preservation of TG vortex symmetries during the evolution of the flow, which seem to be a strong property of the MHD equations. Therefore, a natural question that emerges is: what happens if we perturb the TG flows in order to break these symmetries before the peak of dissipation? Will the joint PDFs converge to a single shape or\/and the scaling of the energy spectra to a single value? What is the role of the symmetries imposed by the initial conditions in terms of the dynamics? Do we have classes of universality for these moderate Reynolds numbers or is there a universal power law in the high Reynolds number limit? We plan to address these questions in our future work.\n\n\\begin{acknowledgements}\n\nThe authors acknowledge interesting and stimulating discussions with Christos Vassilicos and Marc-Etienne Brachet. V.D. acknowledges the financial support from EU-funded Marie Curie Actions---Intra-European Fellowships (FP7-PEOPLE-2011-IEF, MHDTURB, Project No. 299973). All computations were performed using the HPC resources from\nGENCI-CINES (Grant No. 2012026421).\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}