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+{"text":"\\section{Introduction}\n\nThe phase diagram of  Quantum Chromodynamics (QCD) is investigated\nin large-scale lattice gauge theory simulations\n\\cite{Karsch:2006sm} and heavy-ion collision experiments at\nCERN-SPS and RHIC Brookhaven \\cite{Muller:2006ee}, where the\napproximately baryon-free region at finite temperatures is\naccessible and consensus about the critical temperature for the\noccurence of a strongly correlated quark-gluon plasma phase (sQGP)\nis developing. The region of low temperatures and high baryon\ndensities, however, which is interesting for the astrophysics of\ncompact stars, is not accessible to lattice QCD studies yet and\nheavy-ion collision experiments such as the CBM experiment at FAIR\nDarmstadt are still in preparation \\cite{Senger:2006}. The most\nstringent of the presently available constraints on the EoS of\nsuperdense hadronic matter from compact stars and heavy-ion\ncollisions have recently been discussed in\nRef.~\\cite{Klahn:2006ir} and may form the basis for future\nsystematic investigations of the compatibility of dense quark\nmatter models with those phenomenological constraints. Therefore,\nthe question arises for appropriate models describing the\nnonperturbative properties of strongly interacting matter such as\ndynamical chiral symmetry breaking and hadronic bound state\nformation in the vacuum and at finite temperatures and densities.\n\nThe Nambu--Jona-Lasinio (NJL) model has proven very useful for\nproviding results to this question within a simple, but\nmicroscopic formulation, mostly on the mean-field level, see\n\\cite{Buballa:2003qv}. The state of the art phase diagrams of\nneutral quark matter for compact star applications have recently\nbeen obtained in\n\\cite{Ruster:2005jc,Blaschke:2005uj,Aguilera:2004ag,Abuki:2005ms}\nwhere also references to other approaches can be found. One of the\nshortcomings of the NJL model is the absence of confinement, the\nother is its nonrenormalizability. It is customary to speak of the\nNJL model in its form with a cut-off regularization, where\nphysical observables can be defined and calculated. The cutoff in\nmomentum space, however, defines a range of the interaction and\nmakes the NJL model nonlocal. It has been suggested that the\ncutoff-regularized NJL model can be considered as a limiting case\nof a more general formulation of nonlocal chiral quark models\nusing separable interactions \\cite{Schmidt:1994di}. In this form\none can even make contact with the Dyson-Schwinger equation\napproach to QCD by defining a separable representation of the\neffective gluon propagator \\cite{Blaschke:2000gd}, or to the\ninstanton liquid model, see \\cite{GomezDumm:2005hy}.\n\nWe have made extensive use of the parametrization given in\n\\cite{Schmidt:1994di} for studies of quark matter phases in\ncompact stars \\cite{Blaschke:2003yn,Grigorian:2003vi} where the\nrole of the smoothness of the momentum dependence for the\nquark-hadron phase transition and compact star structure has been\nexplored. These investigations have been also used in simulations\nof hybrid star cooling \\cite{Grigorian:2004jq,Popov:2005xa}, which\ncan be selective for the choice of the quation of state (EoS) of\nquark matter by comparing to observational data feor surface\ntemperature and age of compact stars. As a result of these\nstudies, color superconducting phases with small gaps of the order\nof 10 keV - 1 MeV appear to be favorable for the cooling\nphenomenology. A prominent candidate, the color-spin-locking (CSL)\nphase, has been investigated more in detail within the NJL model\nwith satisfactory results  \\cite{Aguilera:2005tg}. However, its\ngeneralization to formfactors with a smooth momentum dependence\nrevealed a severe sensitivity resulting in variations of the CSL\ngaps over four orders of magnitude \\cite{Aguilera:2005uf}.\n\nUnfortunately, with the NJL parametrization given in\n\\cite{Schmidt:1994di} it was not possible to reproduce results\nwith NJL parametrizations given in \\cite{Buballa:2003qv} and used,\ne.g., in  Refs.\n\\cite{Ruster:2005jc,Blaschke:2005uj,Aguilera:2005tg}. Therefore,\nin the present work a new parametrization of the model presented\nin Ref. \\cite{Schmidt:1994di} is performed with a special emphasis\non reproducing NJL parametrizations given in \\cite{Buballa:2003qv}\nin the limiting case of a sharp cutoff formfactor. We also take\ninto account the strangeness degree of freedom and consider\nLorentzian-type formfactor models where the form of the momentum\ndependence for the quark-quark interaction can be varied\nparametrically thus being most suitable for a quantitative\nanalysis the phase diagram and high-density EoS under the above\nmentioned constraints from compact star and heavy-ion collision\nphenomenology.\n\n\n\\section{Basic formulation}\n\nWe consider a nonlocal chiral quark model with separable\nquark-antiquark interaction in the color singlet\nscalar\/pseudoscalar isovector channel \\cite{Gocke:2001ri} where\nthe formfactors are given in the  instantaneous approximation, in\nthe same way as it was suggested in \\cite{Schmidt:1994di}.\n\nThe Lagrangian density of the quark model is given by\n($i,j=u,d,s$)\n\\begin{equation}\n{\\cal L}= \\bar{q}_{i}(i \\gamma_\\mu \\partial^\\mu - m_{i,0})q_{i } +\nG_{S}\\sum_{a=0}^{8}\\left[ (\\bar{q_i}\\tilde{g}(x)~\\lambda\n_{ij}^{a}q_j)^{2}+(\\bar{q_i}(i~\\tilde{g}(x)\\gamma _{5})\\lambda\n_{ij}^{a}q_j)^{2}\\right],\n\\end{equation}\nwhere indices occuring twice are to be summed over and the\nformfactor $\\tilde{g}(x)$ for the  nonlocal current-current\ncoupling has been introduced. Here $m_0=m_{u,0}=m_{d,0}$ and\n$m_{s,0}$ are the current quark masses of the light and strange\nflavors, respectively, $\\lambda _{ij}^a$ are the Gell-Mann\nmatrices of the $SU(3)$ flavor group and $\\gamma _{\\mu}$, $\\gamma\n_5$ are Dirac matrices.\n\nThe nonlocality of the current-current interaction in the\nquark-antiquark ($q \\bar{q}$) channel is implemented in the\nseparable approximation via the same formfactor functions for all\ncolors and flavors. In our calculations we use the Gaussian (G),\nLorentzian (L), Woods-Saxon (WS) and cutoff (NJL) formfactors in\nmomentum space defined as (see Ref. \\cite{Schmidt:1994di})\n\\begin{eqnarray}\ng_{{\\rm G}}(p) &=&\\exp (-p^{2}\/\\Lambda _{{\\rm G}}^{2})~,\n  \\nonumber \\\\\ng_{{\\rm L}}(p) &=&[1+(p\/\\Lambda _{{\\rm L}})^{2\\alpha }]^{-1},\n\\nonumber \\\\\ng_{{\\rm WS}}(p)&=& [1+\\exp(- \\alpha)]\/\\{1+\\exp[\\alpha~\n(p^2\/\\Lambda_{{\\rm WS}}^2-1]\\},\\nonumber \\\\\n g_{{\\rm NJL}}(p) &=&\\theta (1-p\/\\Lambda\n_{{\\rm NJL}})~.\\nonumber\n\\end{eqnarray}\nThe formfactors can be introduced in a manifestly covariant way\n(see \\cite{Gocke:2001ri}), but besides technical complications at\nfinite $T$ and $\\mu$, where Matsubara summations have to be\nperformed numerically, it is not a priori obvious that such a\nformulation shall be superior to an instantaneous approximation\n(3D) which could be justified as a separable representation of a\nCoulomb-gauge potential model \\cite{Blaschke:1994px}.\n\nTypically, three-flavor NJL type models use a 't Hooft determinant\ninteraction that induces a U$_{A}$(1) symmetry breaking in the\npseudoscalar isoscalar meson sector, which can be adjusted such\nthat the $\\eta $-$\\eta^{\\prime }$ mass difference is described. In\nthe present approach this term is neglected using the motivation\ngiven in \\cite{Blaschke:2005uj}, so that the flavor sectors\ndecouple in the mean-field approximation.\n\nThe dynamical quark mass functions are then given by\n$M_i(p)=m_{i,0}+\\phi _i~g(p)$, where the chiral gaps fulfill the\ngap equations\n\\begin{eqnarray}\n\\phi_{i} &=& 4G_{S}\\frac{N_{c}}{\\pi ^{2}}\\int dp\np^{2}g(p)\\frac{M_{i}(p)}{E_{i}(p)}~, \\label{disprel}\n\\end{eqnarray}\ncorresponding to minima of the thermodynamic potential with\nrespect to variations of the order parameters $\\phi_{i}$, the\nquark dispersion relations are $E_{i}(p)=\\sqrt{p^{2}+M_{i}^2(p)}$.\n\nThe basic set of equations should be chosen to fix the parameters\nincluded in the model, which are the current masses, coupling\nconstant and cutoff parameter ($m_{0},m_{s,0},G_{S}$ and\n$\\Lambda$).\n\nIn order to do that we use the properties of bound states of\nquarks in the vacuum given by the pion decay constant $f_{\\pi\n}=92.4$~MeV, the masses of the pion $M_{\\pi }=135$ MeV and the\nkaon $M_{K}=494$ MeV and either the constituent quark mass\n$M(p=0)=m_0+\\phi_u$ or the chiral condensate of light quarks,\ndefined as\n\\begin{equation}\n\\langle u \\bar{u}\\rangle _0 =-\\frac{N_{c}}{\\pi ^{2}}\\int dp\np^{2}\\frac{M_{u}(p)-m_{0}}{E_{u}(p)}~, \\label{condensate}\n\\end{equation}\nwith a phenomenological value from QCD sum rules \\cite{Dosch} of\n$190$ MeV $\\le -\\langle u \\bar{u}\\rangle^{1\/3}_0\\le 260$ MeV. The\nchiral condensate generally is not properly defined in the case of\nnonlocal interactions. The subtraction of the $m_{0}$ term has\nbeen included to make the integral convergent.\n\nThe pion ~decay ~constant can be expressed in the form\n\\begin{equation}\n\\label{fpi} f_{\\pi }=\\frac{3~g_{\\pi q\\bar{q}}}{2\\pi ^{2}} \\int dp\np^{2}g(p)\\frac{M_{u}(p)}{E_{u}(p)(E_{u}(p)^{2}-M_{\\pi }^{2}\/4)},\n\\end{equation}%\nwhere the pion wave function renormalization factor $g_{\\pi\nq\\bar{q}}$  is\n\\[\ng_{\\pi q\\bar{q}}^{-2}= \\frac{3}{2\\pi ^{2}}\\int dp p^{2}g^{2}(p)\n\\frac{E_{u}(p)}{(E_{u}(p)^{2}-M_{\\pi }^{2}\/4)^{2}}.\n\\]\nThe masses of pion and kaon are obtained from a direct\ngeneralization of the well-known NJL model\n\\cite{Rehberg:1995kh,Costa:2005cz} by introducing formfactors with\nthe momentum space integration and replacing constituent quark\nmasses by the momentum dependent mass functions $M(p)$\n\\begin{eqnarray}\n\\label{Mpi} M_{\\pi } &=&\\left[ \\left(\n\\frac{1}{2G_{S}}-2I_{u}^{(1)}\\right) \/I_{uu}^{(2)}\n\\right] ^{1\/2}, \\\\\nM_{K} &=&\\left[ \\left(\n\\frac{1}{2G_{S}}-(I_{u}^{(1)}+I_{s}^{(1)})\\right)\n\/I_{us}^{(2)}\\right] ^{1\/2},\n\\end{eqnarray}\nIn these mass formulae, the following abbreviations for integrals\nhave been used\n\\begin{eqnarray}\nI_{i}^{(1)} &=&\\frac{3}{\\pi ^{2}}\\int dp\np^{2}g^{2}(p)\\frac{1}{E_{i}(p)},\n\\nonumber \\\\\nI_{uu}^{(2)} &=&\\frac{3}{2\\pi ^{2}} \\int dp\np^{2}\\frac{g^{2}(p)}{E_{u}(p)(E_{u}(p)^{2}- M_{\\pi }^{2}\/4)},\n\\nonumber \\\\\nI_{us}^{(2)} &=&\\frac{3}{\\pi ^{2}}\\int dp p^{2} g^{2}(p)\n\\frac{E_{u}(p)+E_{s}(p)}{E_{u}(p)E_{s}(p)[(E_{u}(p)+E_{s}(p))^{2}-M_{K}^{2}]}.\n\\end{eqnarray}\nWe can use these notations to give an estimate of the validity of\nlow-energy theorems for this nonlocal generalization of the NJL\nmodel. To this end we rewrite Eq. (\\ref{fpi}) for $f_\\pi$ and\n$g_{\\pi q\\bar{q}}$ as\n\n\\begin{eqnarray}\n\\label{fpi-2} f_{\\pi }&=&g_{\\pi q\\bar{q}}\\left(\\phi_u~\nI_{uu}^{(2)}\n+ m_0 \\langle g^{-1}(p) \\rangle^{(2)}\\right)\\\\\ng_{\\pi q\\bar{q}}^{-2}&\\approx& I_{uu}^{(2)} + M_{\\pi }^{2}\/4 \\cdot\n\\langle E^{-2}_{u}(p) \\rangle^{(2)}~, \\label{gpi-2}\n\\end{eqnarray}\nwhere the mean values of a distribution $F(p)$ are defined using\nthe integral $I^{(2)}_{uu}$ as an operator: $\\langle\nF(p)\\rangle^{(2)}=I^{(2)}_{uu}[F(p)]$. To leading order in an\nexpansion at the chiral limit ($m_0\\to 0$, $M_\\pi\\to 0$) one\nobtains the Goldberger-Treiman relation\n\\begin{equation}\n\\label{GT} f_{\\pi }g_{\\pi q\\bar{q}}=\\phi_u~.\n\\end{equation}\nRewriting the gap equation (\\ref{disprel}) for the light flavor as\n\\begin{equation}\n\\label{gap} \\phi_u[1-4G_S~I_{u}^{(1)}]=- m_0 \\langle u\\bar{u}\n\\rangle_0 \\cdot 4 G_S\n\\end{equation}\nand the pion mass formula (\\ref{Mpi}) as\n\\begin{equation}\n\\label{Mpi2} M_{\\pi }^2 =\\frac{1}{2G_{S}}\\left(1 -4\nG_S~I_{u}^{(1)}\\right) \/I_{uu}^{(2)}~,\n\\end{equation}\nwe obtain by combining (\\ref{Mpi2}) with (\\ref{gap}), (\\ref{GT})\nand (\\ref{fpi-2}) in leading order the Gell-Mann--Oakes--Renner\nrelation (GMOR)\n\\begin{equation}\n\\label{GMOR} M_{\\pi }^{2}f_{\\pi }^{2}=- 2 m_{0}\\langle u\\bar{u}\n\\rangle_0~.\n\\end{equation}\nAs an indicator of the validity of this low-energy theorem we will\nshow the GMOR value for the light current quark mass\n\\begin{equation}\nm_{0}^{GMOR}=-\\frac{M_{\\pi }^{2}f_{\\pi }^{2}}{2\\langle u\n\\bar{u}\\rangle_0} \\label{m0}\n\\end{equation}\ntogether with the result of the parametrization of  $m_{0}$.\n\nSince we have no 't Hooft term, there is no mixing of flavor\nsectors, and one can consider the light quark sector independent\nof the strange one. The equation for the kaon mass fixes the\nstrange quark's current mass $m_{s,0}$, whereby a self-consistent\nsolution of the strange quark gap equation is implied.\n\n\n\\section{Results}\n\nIn the present parametrization scheme the gap equation plays a\nspecial role. Although the gap is not an observable quantity, we\nwill use it as an phenomenological input instead of the\ncondensate, which in some cases does not fulfill the\nphenomenological constraints. Moreover, for each formfactor model\nthere is some minimal value of $G_{S}\\Lambda ^{2}$ for which the\ncondensate has a minimum: for the Gaussian model it is 7.376, for\nthe Lorentzian model with $\\alpha =2$ it is 3.795, and for $\\alpha\n=10$ it is 2.825. For the NJL model this minimal value is 2.588.\nThe corresponding values of the condensate are given in Table\n\\ref{LorsysM}. These values for the finite current masses are\nshifted to the left as it is shown in Fig. \\ref{patalogy} and for\nthem the parameter sets are fixed (see Table \\ref{LorsysM}). When\nthe condensate is chosen there are two possible values of\n$G_{S}\\Lambda ^{2}$ (the lower and higher branches) for which one\ncan fix the parameters of the model.  We show that the constraint\non the condensate from QCD sum rules \\protect\\cite{Dosch} with an\nupper limit at $260$ MeV can be fulfilled only for values of\n$\\alpha$ exceeding 3-5 for both Lorentzian and Woods- Saxon\nformfactor models  . For the particular choice of $ - <u \\bar\nu>^{1\/3} = 280$ MeV and 260 MeV we fixed the parameters for both\nbranches of solutions (see Tables \\ref{LL260}\n-\\ref{NJL}).\n\nIn order to obtain the parameter sets we choose three values for\nthe non-observable value of the constituent quark mass\n$M(p=0)=330$ MeV, 335 MeV, 367.5 MeV, 380 MeV, and 400 MeV. The\nvalues are taken such that the mass $3M(p=0)$ is larger than the\nmass of the nucleon as a bound state of three quarks. The results\nof the parametrizations are given in the Tables\n\\ref{L330}-\\ref{L400}.\n\nIn Fig. \\ref{patalogy} the dependence of the chiral condensate is\nshown as a function of $G_S\\Lambda^2$ for different formfactors in\nthe chiral limit band for an appropriate choice of the current\nmass $\\sim 0.01~\\Lambda$. It is shown that the minimal possible\nvalue of the condensate varies from one formfactor model to\nanother and only in the NJL model the appropriate values of\ncondensate in the range of QCD sum rule values $230\\pm 10$ MeV can\nbe reached. The Figs. \\ref{ffcomp} and \\ref{ffpcomp} show the gap\nfunction and the diagonal elements of the separable interaction\nfor different formfactors in order to demonstrate the systematics\nof the changes related to the degree of the softening given by the\nparameter $\\alpha$ in Lorentzian functions.\n\n\\section{Conclusions}\n\nWe have presented parametrizations of nonlocal chiral quark models\nwith instantaneous, separable interactions defined by momentum\ndependent formfactors which interpolate between the soft Gaussian\ntype and the hard cut off (NJL) in tabulated form. The\nintroduction of a Lorentzian and\/or Woods-Saxon-type function with\nan additional parameter allowed a systematic investigation of the\nNJL model limit, where existing parametrizations could be\nrecovered.\n\nWe have shown that the instantaneous nonlocal models have an\nessential problem for the softest formfactors, where it is\nimpossible to obtain acceptable values for the chiral condensate.\nHowever, for the astrophysical applications this problem could be\nconsidered as of minor importance relative to the insights which a\nsystematic variation of the interaction model offers for the\nbetter understanding of mechanisms governing the quark matter EoS\non a microscopic level.\n\nWe show numerically that the Goldberger-Treiman relation and GMOR\nas low-energy theorems hold also for the nonlocal chiral quark\nmodel.\n\nThe present approach to nonlocal chiral quark models can be\napplied subsequently for systematic studies of constraints on the\nEoS of superdense matter coming from the phenomenology of heavy\nion collisions and compact stars.\n\n\\subsection*{Acknowledgement}\n\nI thank David Blaschke, Norberto Scoccola and Yuri Kalinovsky for\nthe initiation of this work and their constructive discussions. I\nam grateful to A. Dorokhov, O. Teryaev and V. Yudichev for their\ninterest in this work and support during my visit at the JINR\nDubna. The research was supported in part by\nDFG under grant No. 436 ARM 17\/4\/05 and by\nthe DAAD partnership program between the Universities of Rostock\nand\nYerevan.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMekler's construction \\cite{mekler1981stability} provides a general method to interpret any structure in a finite relational language in a pure $2$-nilpotent group of finite exponent (the resulting group is typically not finitely generated). This is not a bi-interpretation, however it tends to preserve various model-theoretic tameness properties. First Mekler proved that for any cardinal $\\kappa$ the constructed group is $\\kappa$-stable if and only if the initial structure was \\cite{mekler1981stability}. Afterwards,  it was shown by Baudisch and Pentzel that simplicity of the theory is preserved, and by Baudisch that, assuming stability, CM-triviality is also preserved \\cite{baudisch2002mekler}. See \\cite[Section A.3]{hodges1993model} for a detailed exposition of Mekler's construction.\n\n\n\n\nThe aim of this paper is to investigate further preservation of various generalized stability-theoretic properties from Shelah's classification program \\cite{shelah1990classification}. We concentrate on the classes of $k$-dependent and $\\operatorname{NTP}_2$ theories.\n\n \nThe classes of $k$-dependent theories (see Definition \\ref{def: k-dependence}), for each $k \\in \\mathbb{N}$,  were defined by Shelah in \\cite{shelah2014strongly}, and give a generalization of the class of NIP theories (which corresponds to the case $k=1$). See \\cite{shelah2007definable, hempel2016n, chernikov2014n} for some further results about $k$-dependent groups and fields and connections to combinatorics. In Theorem \\ref{thm: k-dependence is preserved} we show that Mekler's construction preserves $k$-dependence.\nOur initial motivation was to obtain algebraic examples that witness the strictness of the $k$-dependence hierarchy. For $k \\geq 2$, we will say that a theory is \\emph{strictly $k$-dependent} if it is $k$-dependent, but not $(k-1)$-dependent. The usual combinatorial example of a strictly $k$-dependent theory is given by the random $k$-hypergraph. The first example of a strictly $2$-dependent group was given in \\cite{hempel2016n} (it was also considered in \\cite[Example 4.1.14]{wagner2002simple}):\n\n\\begin{expl}\\label{ex: extraspecial}\nLet $G$ be $\\oplus_{\\omega}\\mathbb F_p$, where $\\mathbb F_p$ is the finite field with $p$ elements. Consider the structure\n$\\mathcal{G} = (G, \\mathbb F_p, 0, +,\\cdot)$, where $0$ is the neutral element, $+$ is addition in $G$, and $\\cdot$ is the bilinear form $(a_i)_i \\cdot (b_i)_i = \\sum_i a_i b_i$ from $G$ to $\\mathbb F_p$. This structure is not NIP, but is $2$-dependent. In the case $p=2$, $\\mathcal{G}$ is interpretable in an extra-special $p$-group $\\mathcal{H} = (H,\\cdot, 1)$, and conversely $\\mathcal{H}$ is interpretable in $\\mathcal{G}$ (see \\cite[Proposition 3.11]{macpherson2008one} and the discussion around it, or the appendix in \\cite{millietdefinable}). Hence $\\mathcal{H}$ provides an example of a strictly $2$-dependent pure group.\n\n \\end{expl} \n\nIn Corollary \\ref{cor: strictly k-dep groups} we use Mekler's construction to show that for every $k$, there is a strictly $k$-dependent pure group.\n\nThe class of $\\operatorname{NTP}_2$ theories was defined in \\cite{shelah1980simple} (see Definition \\ref{def: NTP2}). It gives a common generalization of simple and NIP theories (along with containing many new important examples), and more recently it was studied in e.g. \\cite{chernikov2012forking, chernikov2014theories, yaacov2014independence, chernikov2015groups}. In Theorem \\ref{thm: NTP2 is preserved} we show that Mekler's construction preserves $\\operatorname{NTP}_2$.\n\nThe paper is organized as follows. In Section \\ref{sec: Mekler} we review Mekler's construction and record some auxiliary lemmas, including the key lemma about type-definability of partial transversals and related objects (Proposition \\ref{prop: type def}). In Section \\ref{sec: NIP} we prove that NIP is preserved. In Section \\ref{sec: kDep} we discuss indiscernible witnesses for $k$-dependence and give a proof that Mekler's construction preserves $k$-dependence. As an application, for each $k\\geq 2$ we construct a strictly $k$-dependent pure group\nand discuss some related open problems.\nFinally, in Section \\ref{sec: NTP2} we prove that Mekler's construction preserves $\\operatorname{NTP}_2$.\n\n\\subsection*{Acknowledgements}\nWe would like to thank the anonymous referee for a very detailed report and many useful suggestions on improving the paper. We also thank JinHoo Ahn for pointing out some typos in the preliminary version.\n\nBoth authors were partially supported by the NSF Research Grant DMS-1600796, by the NSF CAREER grant DMS-1651321 and by an Alfred P. Sloan Fellowship.\n\n\\section{Preliminaries on Mekler's construction}\\label{sec: Mekler}\n\nWe review Mekler's construction from \\cite{mekler1981stability}, following the exposition and notation in \\cite[Section \nA.3]{hodges1993model} (to which we refer the reader for further details).\n\n\\begin{defn}\nA graph (binary, symmetric relation without self-loops) is called \\emph{nice} if it satisfies the following two properties:\n\\begin{enumerate}\n\\item there are at least two vertices, and for any two distinct vertices $a$ and $b$ there is some vertex $c$ different from $a$ and $b$ such that $c$ is joined to $a$ but not to $b$;\n\\item there are no triangles or squares in the graph.\n\\end{enumerate}\n\\end{defn}\n\n\n\n\nFor any graph $C$ and an odd prime $p$, we define a $2$-nilpotent group of exponent $p$ denoted  by $G(C)$ which is generated freely in the variety of $2$-nilpotent groups of exponent $p$ by the vertices of $C$ by imposing that two generators commute if and only if they are connected by an edge in $C$.\n\nNow, let $C$ be a nice graph and consider the group $G(C)$. Let $G$ be any model of $\\operatorname{Th}(G(C))$.  We consider the following $\\emptyset$-definable equivalence relations on the elements of $G$. \n\n\\begin{defn}\nLet $g$ and $h$ be elements of $G$, then\n\n\\begin{itemize}\n\\item $g \\sim h$, if $C_G(g) = C_G(h)$.\n\\item $g \\approx h$ if there is some  natural number $r$  and $c$ in $Z(G)$ such that $g = h^r \\cdot c$.\n\\item $g \\equiv_Z h$ if $g\\cdot Z(G)= h\\cdot  Z(G)$.\n\\end{itemize}\n\n\n\\end{defn}\n\nNote that  $g \\equiv_Z h$ implies $g \\approx h$, which implies $g \\sim h$.\n\n\\begin{defn}\nLet $g$ be an element of $G$ and let $q$ be a natural number. We say that $g$ is \\emph{of type $q$} if there are $q$ different $\\approx$-equivalence classes in the $\\sim$-class $[g]_{\\sim}$ of $g$. Moreover, we say that $g$ is \\emph{isolated} if all non central $h \\in G$ which commute with $g$ are $\\approx$-equivalent to $g$.\n\\end{defn}\n\n\nAll non-central elements of $G$ can be partitioned into four different $\\emptyset$-definable classes (see \\cite[Lemma A.3.6 - A.3.10]{hodges1993model} for the details): \n\\begin{enumerate}\n\n\\item elements of type $1$ which are not isolated, also referred to as \\emph{elements of type $1^{\\nu}$} (in $G(C)$ this class includes the elements given by the vertices of $C$), \n\\item elements of type $1$ which are isolated, also referred to as \\emph{elements of type  $1^{\\iota}$},\n\\item elements of type $p$, and\n\\item elements of type $p-1$.\n\\end{enumerate}\nThe elements of the latter two types are always non-isolated (it is easy to see from the definition that only an element of type $1$ can be isolated).\n\n\nBy \\cite[Lemma A.3.8, (a) $\\Leftrightarrow$ (b)]{hodges1993model}, for every element $g \\in G$ of type $p$, the non-central elements of $G$ which commute with $g$ are precisely the elements $\\sim$-equivalent to $g$, and an element $b$ of type $1^{\\nu}$ together with the elements $\\sim$-equivalent to $b$.\n\n\\begin{defn}\\label{def: handle}\nFor every element $g \\in G$ of type $p$, we call an element $b$ of type $1^{\\nu}$ which commutes with $g$ a \\emph{handle of $g$}. \n\\end{defn}\n\n\n\\begin{fact}\\label{fact: handle}\n\tBy the above, we obtain immediately that a handle is definable from $g$ up to $\\sim$-equivalence.\n\t\\end{fact}\n\nNote here, that the center of $G$ as well as the quotient $G\/Z(G)$ are elementary abelian $p$-groups. Hence they can be viewed as $\\mathbb F_p$-vector spaces. From now on, \\emph{independence} over some supergroup of $Z(G)$ will refer to linear independence in terms of the corresponding $\\mathbb F_p$-vector space.\n\n\n\n\n\\begin{defn}\\label{def: transversal}\nLet $G$ be a model of $\\operatorname{Th}(G(C))$. We define the following:\n\\begin{itemize}\n\\item A \\emph{$1^{\\nu}$-transversal} of $G$ is a set $X^{\\nu}$ consisting of one representative for each $\\sim$-class of elements of type $1^{\\nu}$ in $G$.\n\\item An element is \\emph{proper} if it is not a product of any elements of type $1^{\\nu}$ in $G$.\n\\item A \\emph{$p$-transversal} of $G$ is a set $X^p$ of pairwise $\\sim$-inequivalent proper elements of type $p$ in $G$ which is maximal with the property that if $Y$ is a finite subset of $X^p$ and all elements of $Y$ have the same handle, then $Y$ is independent modulo the subgroup generated by all elements of type $1^{\\nu}$ in $G$ and $Z(G)$.\n\\item A \\emph{$1^{\\iota}$-transversal} of $G$ is a set $X^{\\iota}$  of representatives of $\\sim$-classes of proper elements of type $1^{\\iota}$ in $G$ which is maximal independent modulo the subgroup generated by all elements of types $1^{\\nu}$ and $p$ in $G$, together with $Z(G)$.\n\\item A set $X \\subseteq G$ is a \\emph{transversal of $G$} if $X= X^{\\nu} \\sqcup X^p \\sqcup X^{\\iota}$, where $X^{\\nu}, X^p$ and $X^{\\iota}$ are some transversals of the corresponding types.\n\\item A subset $Y$ of a transversal is called a \\emph{partial transversal} if it is closed under handles (i.\\ e.\\ for any element $a$ of type $p$ in $Y$ there is an element of type $1^{\\nu}$ in $Y$ which is a handle of $a$).\n\\item For a given (partial) transversal $X$, we denote by $X^{\\nu}$, $ X^p$, and $X^{\\iota}$ the elements in $X$ of the corresponding types.\n\\end{itemize}\n\\end{defn} \n\n\n\\begin{lemma}\\label{lem: transversal type def}\n\tLet $G \\models \\operatorname{Th}(G(C))$. Given a small tuple of variables $\\bar{x} = \\bar x^{\\nu \\frown} \\bar{x} ^{p \\frown} \\bar{x}^{\\iota}$, there is a partial type $\\Phi(\\bar{x})$ such that for any tuples $\\bar{a}^\\nu, \\bar{a}^p$ and $\\bar{a}^{\\iota}$ in $G$,  we have that $G \\models \\Phi(\\bar{a}^\\nu, \\bar{a}^p, \\bar{a}^{\\iota})$ if and only if every element in  $\\bar{a}^\\nu, \\bar{a}^p$ and $\\bar{a}^{\\iota}$ is of type $1^{\\nu}, p$ and $1^{\\iota}$, respectively, and $\\bar{a} = \\bar{a} ^{\\nu \\frown} \\bar{a} ^{p \\frown} \\bar{a}^{\\iota}$ can be extended to a transversal of $G$.\n\t\\end{lemma}\n\\begin{proof}\nBy inspecting Definition \\ref{def: transversal}. For example, let's describe the partial type $\\Phi^p(\\bar{x}^p)$ expressing that $\\bar{x}^p = (x_i^p : i < \\kappa)$ can be extended to a $p$-transversal (the conditions on $\\bar{x}^\\nu$ and $\\bar{x}^\\iota$ are expressed similarly). For $q \\in \\mathbb{N}$, let $\\phi_q(x)$ be the formula defining the set of all elements of type $q$ in $G$, let $\\phi_\\iota(x)$ define the set of isolated elements, and let $\\phi_{\\textrm{h}}(x_1, \\ldots, x_q)$ express that $x_1, \\ldots, x_q$ have the same handle.\nThe set of proper elements is defined by \n$$\\Phi_{\\textrm{prop}}(x) := \\{ \\forall y_1 \\ldots \\forall y_{n-1} (\\bigwedge_{i<n} \\phi_1(y_i) \\land \\neg \\phi_\\iota(y_i)) \\rightarrow  (x \\neq \\prod_{i<n} y_i ) : n \\in \\mathbb{N} \\}.$$\nThe independence requirement in the definition of a $p$-transversal can be expressed as a partial type \n$\\Phi_{\\textrm{ind}}(\\bar{x}^p)$ containing the formula\n\n\n$$ \\phi_{\\textrm{h}}(x^p_{i_0}, \\ldots, x^p_{i_{n-1}}) \\rightarrow $$\n$$\\left( \\forall y_0 \\ldots \\forall y_{m-1} \\forall z \\bigwedge_{i<m}(\\phi_1(y_i) \\land \\neg \\phi_\\iota(y_i) \\land z \\in Z(G)) \\rightarrow z \\cdot \\prod_{i<m} y_i^{r_i} \\neq \\prod_{j<n} (x^p_{i_j})^{q_j}  \\right)$$ \n\nfor each $m, n \\in \\mathbb{N}$, $i_0 < \\ldots < i_{n-1} < \\kappa$ and $q_0, \\ldots, q_{n-1}, r_0, \\ldots, r_{m-1} \\in \\mathbb{N}, 0 < q_j <p$.\n\n Then we can take $\\Phi^p(\\bar{x}^p) :=  \\{ \\phi_p(x_i^p) : i < \\kappa\\} \\cup \\{ \\neg( x_i^p \\sim x_j^p) : i<j<\\kappa \\} \\cup \\bigcup_{i < \\kappa} \\Phi_{\\textrm{prop}}(x_i^p) \\cup \\Phi_{\\textrm{ind}}(\\bar{x}^p)$.\n\n\\end{proof}\n\nThe following can be easily deduced from \\cite[Corollary A.3.11]{hodges1993model}.\n\n\\begin{fact}\nLet $C$ be a nice graph. \nThere is an interpretation $\\Gamma$ such that for any model $G$ of $\\operatorname{Th}(G(C))$, we have that $\\Gamma (G)$ is a model of $\\operatorname{Th}(C)$. More specifically, the graph $\\Gamma(G) = (V,R)$ is given by the set of vertices $V=\\{ g \\in G : g \\textrm{ is of type }1^\\nu, g \\notin Z(G) \\} \/ \\approx$ and the (well-defined) edge relation $([g]_{\\approx}, [h]_{\\approx}) \\in R$ $\\iff$ $g,h$ commute.\n\\end{fact}\n\n\nThe full set of transversal gives another important graph, a so called cover of a nice graph, which we define below.\n\n\\begin{defn}\\label{def: cover} \\begin{enumerate}\n\\item\n\tLet $C$ be an infinite nice graph. A graph $C^+$ containing $C$ as a subgraph is called a \\emph{cover} of $C$ if for every vertex $b \\in C^+ \\setminus C$, one of the following two statements holds:\n\t\\begin{itemize}\n\t\t\\item there is a unique vertex $a$ in $C^+$ that is joined to $b$, and  moreover $a$ is in $C$ and it has infinitely many adjacent vertices in $C$;\n\t\t \n\t\t \\item $b$ is joined to no vertex in $C^+$.\n\t\t\\end{itemize}\n\t\n\\item\tA cover $C^+$ of $C$ is a \\emph{$\\lambda$-cover} if \n\t\\begin{itemize}\n\t\t\\item for every vertex $a$ in $C$ the number of vertices in $C^+\\setminus C$ joined to $a$ is $\\lambda$ if $a$ is joined to infinitely many vertices in $C$, and zero otherwise;\n\t\t\\item the number of new vertices in $C^+ \\setminus C$ which are not joined to any other vertex in $C^+$ is $\\lambda$.\n\t\t\\end{itemize}\n\t\\end{enumerate}\n\n\\end{defn}\nObserve that a proper cover of a nice graph is never a nice graph.\n\n\\begin{fact}\\cite[Corollary A.3.14(a) + proof of (c)]{hodges1993model}. \\label{rem: X as a cover}\nGiven a $1^\\nu$-transversal $X^\\nu$ of $G$, we identify  the elements of $X^\\nu$ with the set of vertices of $\\Gamma(G)$ by mapping $x \\in X^\\nu$ to its class $[x]_{\\approx}$. Then a transversal $X$ can be viewed as a cover of the nice graph given by the elements of type $1^\\nu$ in $X$, with the edge relation given by commutation. (See  Remark \\ref{rem: cover explanation} for an explanation.)\n\\end{fact}\n\n\n\\begin{fact}\\cite[Theorem A.3.14, Corollary A.3.15]{hodges1993model}\\label{fac: decomposition}\nLet $G$ be a model of $\\operatorname{Th}(G(C))$ and let $X$ be a transversal of $G$. \n\\begin{enumerate}\n\\item There is a subgroup of $Z(G)$ which we denote by $K_X$ such that $G = \\langle X \\rangle \\times K_X$.\n\\item The group $K_X$ is an elementary abelian $p$-group, in particular $\\operatorname{Th}(K_X)$ is stable and eliminates quantifiers.\n\\item If $G$ is saturated and uncountable, then both the graph $\\Gamma(G)$ and the group $K_X$ are also saturated (as $|K_X| = |G|$ and $\\operatorname{Th}(K_X)$ is uncountably categorical).\n\\item If $G$ is a saturated model of $\\operatorname{Th}(G(C))$, then every automorphism of $\\Gamma(G)$ can be lifted to an automorphism of $G$.\n\\item $\\langle X \\rangle \\cong G(X)$ via an isomorphism which is the identity on the elements in $X$ (where $X$ is viewed as a graph as in Remark \\ref{rem: X as a cover}, note that it doesn't have to be nice).\n\\item If $G$ is saturated then $X$ (where $X$ is viewed as a graph as in Remark \\ref{rem: X as a cover}) is a $|G|$-cover of $\\Gamma(G)$.\n\n\\end{enumerate}\nAlternatively, one could work with a special model instead of a saturated one to avoid any set-theoretic issues. \n\\end{fact}\n\n\\begin{remark}\\label{rem: unbounded commutator products}\nThe reason given for (3) in \\cite{hodges1993model} appears to assume that $K_X$ is definable, which is not the case, so we provide a proof.\n\n\nAssume that $C$ is an infinite nice graph. \nGiven $m,n \\in \\omega$, let $\\psi_{m,n}$ be the sentence\n$$ \\forall g_1, \\ldots, g_{n} \\in Z \\exists x \\in Z \\left( \\forall z_1, \\ldots, z_{2m} \\bigwedge \\left(g^{\\alpha_1}_1 \\cdot \\ldots \\cdot g_{n}^{\\alpha_{n}} \\cdot x^\\alpha \\neq \\prod_{j=1}^{m}[z_{2j-1}, z_{2j}]  \\right) \\right),$$\nwhere the conjunction is over all $(\\alpha_1, \\ldots, \\alpha_{n-1}, \\alpha) \\in \n\\{ 0, \\ldots, p-1 \\}^{n+1}$ with $\\alpha \\neq 0$.\n\nWe claim that $G(C) \\models \\psi_{m,n}$ for all $m,n \\in \\omega$. Indeed, by \\cite[Lemma A.3.2(b)+(d)]{hodges1993model}, a basis for $G(C)'=Z(G(C))$ is given by the set of all elements of the form $[a,b]$ with $a,b$ elements in $C$ not connected by an edge. Since $C$ is nice, given any vertices $v \\neq w$ there is at most one vertex to which both $v$ and $w$ are connected. As $C$ is infinite, we can find  pairwise distinct elements $a$ and $\\{ b_i : i < \\omega \\}$ in $C$ such that $a$ is not connected by an edge to any of the $b_i$'s. Indeed, take any pairwise distinct elements $a', a'', (c_i : i < \\omega)$ in $C$. For each $i$, at most one of $a', a''$ is connected to $c_i$, hence for one of $a,a'$ there are infinitely many $i<\\omega$ such that it is not connected to $c_i$.\nHence $\\{[a,b_i] : i < \\omega \\}$ is a linearly independent set and $Z(G(C))$ has infinite dimension. Now, given arbitrary $m,n$ and $g_i \\in Z(G(C))$, assume that each $g_i$ is a linear combination of at most $t$ vectors from the basis. If we take $x$ to be a product of more than $nt + m$ non-trivial commutators, the equality above cannot hold (as $\\alpha \\neq 0$, and for any $l \\in \\omega$ the sum of $l$ vectors in a basis cannot be written as a linear combination of less than $l$ vectors in the same basis).\n\nNow let $G$ be a saturated model of $\\operatorname{Th}(G(C))$, $\\lambda := |G|$. In view of Remark \\ref{rem: invariance of tranversal}, it is enough to find a sequence $(h_i : i <\\lambda)$ of elements in $Z(G)$ which are linearly independent in the elementary abelian $p$-group $Z(G)$ over $G'$. This can be done by transfinite induction using saturation of $G(C)$ and the fact that given any finitely many elements $g_1, \\ldots, g_n \\in Z(G)$, we can find an element $x \\in Z(G)$ independent from them over $G'$ since $G \\models \\psi_{m,n}$ for all $m \\in \\omega$ (see Definition \\ref{def: indep over derived subgp}).\n \n\\end{remark}\n\n\n\\begin{remark}\\label{rem: cover explanation}\nWe provide some details concerning Fact \\ref{fac: decomposition}(6).\nThe fact that this is a cover in any model of $\\operatorname{Th}(G(C))$ doesn't seem to follow explicitly from the statement of Axiom 10, as stated in the proof of \\cite[Corollary A.3.14(c)]{hodges1993model}. However, it follows by an argument similar to the proof of Remark \\ref{rem: unbounded commutator products}. This time, working in the vector space $ G\/Z(G)$ and using \\cite[Lemma A.3.9]{hodges1993model}, the proof of Axiom 10, and \\cite[Lemma A.3.2(c)]{hodges1993model} in $G(C)$, all the relevant sets in Definition \\ref{def: cover} are infinite, so by saturation can be demonstrated to be of size $|G|$ in a saturated model $G$.  \n\\end{remark}\n\n\nThe following lemma is a refinement of Fact \\ref{fac: decomposition}(4) and \\cite[Lemma 4.12]{baudisch2002mekler}. \n\n\n\\begin{lemma} \\label{Lem_GlueAut}\n\tLet $G$ be a saturated model of $\\operatorname{Th}(G(C))$, $X$ be a transversal, and $K_X \\leq Z(G)$ be such that  $G = \\langle X \\rangle \\times K_X$. Let  $Y$ and $Z$ be two small subsets  of $X$ and let $\\bar h_1, \\bar h_2$ be two tuples in $K_X$. Suppose that \n\t%\n\t\\begin{itemize}\n\t\t\\item there is a bijection $f$ between  $Y$ and $Z$ which respects the $1^{\\nu}$-, $p$-, and $1^{\\iota}$-parts, the handles, and $\\operatorname{tp}_{\\Gamma} (Y^{\\nu}) = \\operatorname{tp}_{\\Gamma} (f(Y^{\\nu}))$,\n\t\t\\item $\\operatorname{tp}_{K_X}(\\bar h_1) = \\operatorname{tp}_{K_X}(\\bar h_2)$.\n\t\t\\end{itemize}\nThen there is an automorphism of $G$ coinciding with $f$ on $Y$ and sending  $\\bar h_1$ to  $\\bar h_2$.\n\\end{lemma}\n\n\\begin{proof}\nBy Remark \\ref{rem: X as a cover}, we identify $\\Gamma(G)$ with $X^\\nu$. By saturation of $\\Gamma(G)$, $f \\restriction Y_\\nu$ extends to an automorphism $\\sigma$ of the graph $X^\\nu$. As $X$ is a $|G|$-cover of $X^\\nu$ by saturation of $G$, and $f$ respects the $1^{\\nu}$-, $p$-, and $1^{\\iota}$-parts and the handles, $\\sigma$ extends to an automorphism $\\tau$ of the graph $X$ agreeing with $f$ using a straightforward back-and-forth argument. By Fact \\ref{fac: decomposition}(5), we have that $\\langle X \\rangle \\cong G(X)$ and $\\tau$ lifts to an automorphism of the group $G(X)$, hence to an automorphism $\\tilde{\\tau}$ of $\\langle X \\rangle$ extending $f$ by construction. As $K_X$ is saturated by Fact \\ref{fac: decomposition}(3), there is an automorphism $\\rho$ of $K_X$ which maps $\\bar h_1$ to $\\bar h_2$. We can now take the cartesian product of $\\tilde{\\tau} \\times \\rho$ to obtain an automorphism of $G$ which extends $f$ and maps $\\bar h_1$ to $\\bar h_2$.  \n\\end{proof}\n\n\n\n\n\nNext, we observe that in Fact \\ref{fac: decomposition} the choice of a transversal and an elementary abelian subgroup of the center in the decomposition of $G$ can be made entirely independently of each other.\n\n\\begin{lemma}\\label{lem: invariance of tranversal}\nLet $G$ be any model of $\\operatorname{Th}(G(C))$ and let $X$ be a transversal of $G$. Then we have $G' = \\langle X\\rangle '$.\n\\end{lemma}\n\n\n\\proof Let $K$ be a subgroup of $Z(G)$ as in Fact \\ref{fac: decomposition}, such that $G = \\langle X\\rangle \\times K$. It is enough to show that for all $g, g' \\in G$, we have that $[g,g']$ is in $\\langle X\\rangle '$. We choose\n$x, y \\in \\langle X \\rangle $ and $h,k \\in K$ such that $g =  x \\cdot  h$ and $g' = y \\cdot  k$. Then, using that $G$ is $2$-nilpotent, we have $$[g,g'] = [  x \\cdot h, y \\cdot k] = [x, y],$$ which is in $\\langle X\\rangle '$.\n\\qed\n\n\\begin{remark}\\label{rem: invariance of tranversal}\nLet $X$ and $Y$ be two transversals, and let $K_X$ be such that $G = \\langle X \\rangle \\times K_X$. Then the proof of \\cite[Theorem A.3.14(d)]{hodges1993model} and Lemma \\ref{lem: invariance of tranversal} imply that $G = \\langle Y \\rangle \\times K_X$. Furthermore, that proof also implies that for any set $B$ of elements in $Z(G)$ which are linearly independent in  the elementary abelian $p$-group $Z(G)$ over $G'$ (seen as an $\\mathbb F_p$-vector space), there is a subgroup $K$ of $G$ containing $B$ such that for any transversal $X$ of $G$, we have that $G=\\langle X\\rangle \\times K$.\n\t\\end{remark}\n\n\n\\begin{defn}\\label{def: indep over derived subgp}\n\n\nFor any small cardinal $\\kappa$, let $\\Psi((y_i)_{ i \\in \\kappa})$ be the partial type  consisting of the formulas ``$y_i \\in Z(G)$'' for all $i < \\kappa$, and the formulas   \n$$ \\forall z_0, \\dots, z_{2m-1}\n\\left(\n\\bigwedge_{(\\alpha_0, \\dots , \\alpha_{n-1}) \\in \\{0, \\ldots, p-1 \\}^{ n}\\setminus ( 0, \\dots, 0)} \n\\left(\\, y_{i_0}^{\\alpha_0} \\cdot \\ldots \\cdot y_{i_{n-1}}^{\\alpha_{n-1}}\n \\neq \n \\prod_{j=0}^{m-1} [z_{2j}, z_{2j+1}] \\ \n\n \\right)\\right)$$\n \nfor all $m,n \\in \\mathbb{N}$ and $i_0, \\ldots, i_n \\in \\kappa$. \n\nAn easy inspection yields that for any tuple $\\bar{b}$, $\\bar b \\models \\Psi(\\bar y)$ if and only if $\\bar{b}$ is a tuple of central elements linearly independent in  the elementary abelian $p$-group $Z(G)$ over $G'$ (seen as an $\\mathbb F_p$-vector space).\n\\end{defn}\n\nLet now $\\Phi(\\bar x)$, with $\\bar x = \\bar{x}^{\\nu \\frown} \\bar{x}^{p \\frown} \\bar{x}^{\\iota}$, be the partial type given by Lemma \\ref{lem: transversal type def}.\n Consider the partial type $$\\pi(\\bar x ,\\bar{y}) = \\Phi(\\bar x) \\cup  \\Psi(\\bar y).$$\n \n\n\n\\begin{prop} \\label{prop: type def}\nLet $G$ be a model of $\\operatorname{Th}(G(C))$. Then $G \\models \\pi(\\bar a ,\\bar{b})$ if and only if we can extend $\\bar a$ to a transversal $X$ of $G$ and find a subset $H \\subseteq Z(G)$ containing $\\bar{b}$ which is linearly independent over $G'$, so that $G = \\langle X \\rangle \\times \\langle H \\rangle$.\n\\end{prop}\n\\begin{proof}\n\n\nCombining this with Remark \\ref{rem: invariance of tranversal}, there is a subgroup $K$ of $G$ containing $\\bar b$ such that for any transversal $X$ of $G$, we have that $G=\\langle X\\rangle \\times K$.\nCombining this with Lemma \\ref{lem: transversal type def}, we can conclude.\n\n\n\\end{proof}\n\n\n\n\n\\if 0\n\n\\section{Indiscernible sequences in Mekler's construction}\\label{sec: Indiscernibles decomposition}\nWe will need the following classical result of Morley and Shelah on the existence of indiscernible subsequences in stable theories.\n\\begin{fact}\nLet $T$ be a stable theory, and \n\\end{fact}\n\nWe remark that no analog of this result holds for NIP theories without any set-theoretic assumptions (see ***).\n\n\n\n\\begin{prop}\nLet $C$ be an infinite nice graph. Then $\\operatorname{Th}(C)$    is  NIP if and only if $\\operatorname{Th}(G(C))$ is NIP.\n\\end{prop}\n\n\\proof\nAs the graph $C$ is interpretable in any model of  $\\operatorname{Th}(G(C))$, the implication from left to right is obviously true. \n\nTo proof the reverse implication assume that the contrary, namely that there is a nice graph $C$ such that  $\\operatorname{Th}(C)$ is NIP but  $\\operatorname{Th}(G(C))$ has IP. By Fact \\ref{factNIP}, we can find  an $\\emptyset$-indiscernible sequence  $(a_i)_{i \\in \\kappa}$ of elements  in a saturated model $G$ of  $\\operatorname{Th}(G(C))$ that witnesses IP.\n\n\nNow, fix a transversal $X= X_{\\nu} \\cup X_p \\cup X_{\\iota}$.\nAs every element is a product of finitely many elements in the transversal multiplied by an element of $H$, we may assume, modulo passing to a subsequence of the same length,  that there is  is natural numbers $n$,\nelements in the transversal $ (x_{ji})_{i <\\kappa,\\, j\\leq n}$ and elements $(h_i)_{i < \\kappa}$ in $H$ such that $ a_i = \\Pi_{j=0}^{n-1} x_{ji}\\cdot  h_i$. \n\nThe first step is to pass to a subsequence and choose a priori different transversal in way that each $a_i$ can be written as a $n$-product of elements in the transversal (i.\\ e.\\ we can find a subsequence and a transversal for which all $h_i$'s are $1$).\n\nAs there are only three different types of elements, modulo passing to a subsequence of the same length, we may assume that for a fixed $0\\leq j\\leq n-1$, the elements $x_{ji}$ for $i<\\kappa$ is of the same type, i.\\ e.\\ all are of type $1^\\nu$, $p$, or $1^\\iota$. Furthermore, for each coordinate, we may assume that the elements are all equal or all distinct and if for some $t$ and $s$ the sequence $(x_{ji})_{i< \\kappa}$ constant and $(x_{ts})_{s< \\kappa}$ is non constant, we have that $x_{is} \\not= x_{j0}$.\n\nNow, without loss of generality suppose that $ x_{0i}$'s are all distinct. Let \n$$\\beta_\\alpha = \\min\\{\\beta : \\exists 0< k <n,\\ x_{\\beta, \\, k}= x_{\\alpha, \\, 0}\\}. $$\n Consider the function $f: \\kappa \\rightarrow \\kappa,$ given by\n$$ f(\\alpha)=\\begin{cases}  \\beta_\\alpha \\text{ if  } \\beta_\\alpha < \\alpha   \\\\\n\n0 \\text{ otherwise} \n\\end{cases}\n$$\nBy Fodor's Lemma and the easily verifiable  fact that for some $\\gamma$ different than $0$, the set $\\{\\alpha < \\kappa: f(\\alpha)= \\gamma\\}$ has size at most $n-1$, there is a stationary subset $S$ of $\\kappa$ such that $f: S \\rightarrow \\kappa$ is constantly zero. As $S$ still has order type $\\kappa$, we may assume that our sequence is the form $ (x_{0i}, \\dots, x_{n-1,i})_{i <\\kappa}$ such that for all $j<i<\\kappa$ and $0<k<n$ we have that $x_{0i} \\neq x_{kj}$. \nNow, we want to replace our set of transversal recursively on $i<\\kappa$ in a way such that each $(a_i =\\Pi_{j=0}^{n-1} y_{ji} )$ for some $y_{ji} $ in the transversal. Suppose you have succeeded for all ordinals less than $\\alpha$. Now, let $m$ the size of the set $\\{k: x_{\\alpha, k} = x_{\\alpha, 0}\\}$ (i.\\ e.\\ the number of appearances of $x_{0\\alpha}$ in the product which gives $a_\\alpha$). Now replace $x_{\\alpha,0}$ by $x_{0 \\alpha}\\cdot h_i^q$ with $q<p$ such that $qn\\equiv 1\\ (\\text{mod } p)$. Then,\n$$ a_\\alpha = \\Pi_{j=0}^{n-1} x_{j \\alpha } .$$\nMoreover, it is easy to check that this set remains a transversal. By the choice of the sequence $x_{ji}$'s, the element $x_{0 \\alpha}$ does not appear in the product which gives $a_i$ for $i<\\alpha$. Thus, for this new transversal such an $a_i$ remains equal to the product $\\Pi_{j=0}^{n-1} x_{ji}$. Now, for each $\\gamma>\\alpha$ in which $x_{0 \\alpha}$  makes an appearance in the product of $a_\\gamma$, we can replace this element by $x_{0 \\alpha}\\cdot h_i^q$  and change $h_{\\gamma}$ accordingly such that $ a_\\gamma = \\Pi_{j=0}^{n-1} x_{j \\gamma }\\cdot  h_{\\gamma}$. This finishes our construction.\n\n Next we replace the sequences of $n$-tuples $\\bar x_i$ by an indiscernible one. To do so, we first add the handles of each of the elements of type $p$ to the end of our sequence to insure that the tuples $\\bar x_i$ are closed under handles. Note that these tuples will still have the same length. Now we can find an indiscernible sequence $(\\bar y_i)_{i<\\kappa}$ such that\n$$ \\operatorname{EM}( a_i, \\bar x_i: i \\in \\kappa) \\subset \\operatorname{tp}(\\bar a_i,\\bar y_i: i \\in \\kappa). $$\nNote that $ a_i $ remains to be equal to $  \\Pi_{j=0}^{n-1} y_{ji}$ and that $\\bigcup_{i\\in \\kappa} \\bar y_i$ can be completed to a transversal $Y$ of $G$. \n\n\nAs the original sequence witnesses IP, we can find a $m$-tuple  $\\bar g= (g_0, \\dots, g_{m-1})$ of elements in $G$ such that $\\operatorname{tp}(a_i \/ \\bar g)$ alternates. Choose elements $(z_{ij})_{i<m, j<n_m}$ in the transversal and elements $(k_i)_{i<m}$ in $H$ such that $g_i = \\Pi_{j=1}^{n_i} z_{ij}\\cdot k_i$. Again, we may prolong the tuples $\\bar z_i = (z_{i0}, \\dots, z_{i,n_m-1})$ to include all handles of elements of type $p$. After possibly cutting an initial segment of our sequence, we may assume that if $(y_{ij})_{j< \\kappa}$ is non constant, then non of these elements coincide with one of the $z_{ij}$'s.\n \nNow let $Y_i= Y_{\\nu}^{(i)} \\cup Y_{p}^{(i)} \\cup Y_{\\iota}^{(i)}$ where $Y_{\\nu}^{(i)}$, $Y_{p}^{(i)}$, and $Y_{\\iota}^{(i)}$ are the elements of type $1\\nu$, $p$, and $1^\\iota$ respectively in the tuple $\\bar y_i$. Moreover, let $Z= Z_{\\nu} \\cup Z_p \\cup Z_{\\iota}$ be elements of type $1\\nu$, $p$, and $1^\\iota$ respectively in the set $\\{z_{ij}: i< m, j<n_m\\}$. We have that $Y_{\\nu}^{(i)}$ are an indiscernible sequence in $\\operatorname{Th}(C)$.  Thus, as $\\operatorname{Th}(C)$ is NIP, we have that $\\operatorname{tp}(Y_\\nu^i)$ stabilizes over $Z_\\nu$, in particular, we may assume that $Y_\\nu^i$ is indiscernible over $Z_\\nu$.\n \n \nand a finite subset $Z =Z_{\\nu} \\cup Z_p \\cup Z_{\\iota}$ of $Y$ such that $g= s(Z_{\\nu} , Z_p, Z_{\\iota}, h)$. Assume again that $Z$ is closed under handles.  As $\\operatorname{Th}(C)$ is NIP  and $Y^i$ are indiscernible, we have that $\\operatorname{tp}(Y_\\nu^i)$ stabilizes over $Z_\\nu$, in particular, we may assume that $Y_\\nu^{(i)}$ is indiscernible over $Z_\\nu$. Moreover, by indiscernibility  of $Y^{(i)}$ we have that for all $i, j$ in $\\kappa$,\n$$Y_p^{(i)} \\cap Z_p = Y_p^{(j)} \\cap Z_p \\ \\ \\ \\  \\mbox{ and } \\ \\ \\ \\ Y_\\iota^{(i)} \\cap Z_\\iota = Y_\\iota^{(j)} \\cap Z_\\iota.$$\nPutting all together, we obtain that for any two sequences $i_0 < i_1 < \\dots < i_l,$ and $j_0 < j_1 < \\dots < j_l$ in $\\kappa$ we can find a bijection $\\sigma$ between  $(Y^{i_0},  \\dots, Y^{i_l})$ and $(Y^{j_0},  \\dots, Y^{j_l})$ such that \n\\begin{itemize}\n\\item $\\operatorname{tp}_\\Gamma (Y_\\nu^{i_0},  \\dots, Y^{i_l}, Z_\\nu) = \\operatorname{tp}_\\Gamma (\\sigma(Y^{i_0},  \\dots, \\sigma(Y^{i_l}), Z_\\nu) $.\n\\item the map $\\sigma$ fixes $Z$.\n\\item  the map $\\sigma$ respects the $1^{\\nu}$-, $p$-, and $1^{\\iota}$-parts and the handles;\n\n\\end{itemize}\nThus by Fact \\ref{Fact_Bau} we have that $$\\operatorname{tp} (Y^{i_0},  \\dots, Y^{i_l}, Z_\\nu) = \\operatorname{tp} (\\sigma(Y^{i_0}),  \\dots, \\sigma(Y^{i_l}), Z)  = \\operatorname{tp} (Y^{j_0},  \\dots, Y^{j_l}, Z) .$$\nor in other words there is an automorphism $f$ fixing $Z$ mapping $(Y^{i_0},  \\dots, Y^{i_l})$ to  $(Y^{j_0},  \\dots, Y^{j_l})$. Furthermore, we may assume that this automorphism fixes $H$.\nThis implies that the sequence $a_i =   \\Pi_{j=0}^{n-1} y_{ji}$ is indiscernible over $\\bar g$ contradicting the initial assumption.\n\n\\qed\n\n\nLet $G$ be a saturated model of $\\operatorname{Th}(G(C))$.\n\n\\begin{defn}\n\tLet $g$ be an element of $G$. For a given transversal $X$ of $G$ and the corresponding subgroup $H$ of $G$ such that $G= \\langle X \\rangle \\times H$ and $(x_1, \\dots, x_n)$ in $X$ and $h$ in $H$ such that $g=\\Pi_{i=1}^n x_i \\cdot h$ we call $x_i$ the \\emph{$i$th coordinate} of $g$.\n\t\\end{defn}\n\t\n\t\\begin{lemma}\\label{Lem_IndStep}\n\t\n\t\tLet \n\t\t%\n\t\t$$ (\\bar a_{\\alpha}= (a_{\\alpha}^0, \\dots, a_{\\alpha}^{d}), b_\\alpha, (x_{\\alpha}^{0})_0, \\dots, (x_{\\alpha}^0)_{m_0},\t\\dots  (x_{\\alpha}^{d})_0, \\dots, (x_{\\alpha}^{d})_{m_d}): \\alpha <2^\\kappa) $$ \n\t\t%\n\t\tbe an indiscernible sequence in $G$  such that the collection \n\t\t%\n\t\t$$X_0 = \\{  (x_{\\alpha}^i)_j ; \\alpha< 2^\\kappa, i < d, j <m_i\\}$$\n\t\t%\n\t\t is a part of a transversal and  for $i\\leq d$ we have that\n\t\t%\n\t\t$$a_\\alpha^i = \\Pi_{j=0}^{n-1} (x_{\\alpha}^i)_j. $$\n\t\t%\n\t\t\n\t\n\t\tSuppose there is a transversal $Y$ extending $X_0$ and an a natural number $m$ such that\n\t\n\tfor the sequence\t$b_{\\alpha}  = \\Pi_{j=0}^{m} (y_{\\alpha})_j \\cdot h_{\\alpha}$\n\t\tat least one of the coordinates is pairwise different, say the $r$ one, and moreover for each $\\alpha$ this coordinate differs from all the coordinates of the elements $a_{\\alpha}^1, \\dots, a_{\\alpha}^{d}$, i.\\ e.\\  $(y_{\\alpha})_r$ is different from any $(x_{\\alpha}^{i})_r$ for $0\\leq i <d$ and $1\\leq j \\leq m_i$.\n\t\n\t\t\n\tThen there is a subsequence $( b_\\alpha)_{\\alpha \\in I}$ of $ ( b_{\\alpha})_{\\alpha <2^\\kappa}$ of length at least $\\kappa$, a transversal $X$ extending $X_0$ and $(x_{\\alpha})_0, \\dots, (x_{\\alpha})_{m}$ in $X$ such that \n\n\t$$b_\\alpha = \\Pi_{j=0}^{m} (x_{\\alpha})_j,$$\n\t\n\tand \n\t\t$$ (\\bar a_{\\alpha},  b_\\alpha,[(x_{\\alpha}^{0})_0, \\dots, (x_{\\alpha}^0)_{m_0}],\\dots  \t [(x_{\\alpha}^{d})_0,  \\dots, (x_{\\alpha}^{d})_{m_d}], [(x_{\\alpha})_0, \\dots, (x_{\\alpha})_{m} )]: \\alpha \\in I) $$ \n\t\t%\n\t\tis still an indiscernible sequence in $G$.\n\n\t\t\\end{lemma}\n\\proof\nAs there are only 3 different types of elements in the transversal, up to passing to a subsequence and using the pigeonhole principle we may assume that for each $\\alpha \\in 2^\\kappa $ the $i$th coordinate of each $b_\\alpha$ is of the same type.\n\nSecondly using Erdos Rado, there is a subsequence  index by $I$ of length at least $\\kappa$ such that  for each pair of coordinates, we may assume that uniformly equal in the following sense: Given $i\\leq d$, $k\\leq m_i$, and $s\\neq t  \\leq m$ then\n\\begin{eqnarray*}\n\\mbox{either} & (x_{\\alpha}^{i})_k = (y_\\beta)_s \\mbox { for all  }  \\alpha \\neq \\beta \\mbox{ in } I\\\\\n\\mbox{or} & (x_{\\alpha}^{i})_k \\neq (y_\\beta)_s \\mbox { for all  }  \\alpha \\neq \\beta \\mbox{ in } I\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n\\mbox{either} & (x_{\\alpha})_t= (y_\\beta)_s \\mbox { for all  }  \\alpha \\neq \\beta \\mbox{ in } I\\\\\n\\mbox{or} & (x_{\\alpha})_t \\neq (y_\\beta)_s \\mbox { for all  }  \\alpha \\neq \\beta \\mbox{ in } I\n\\end{eqnarray*}\nand for all $\\alpha \\in I$\n\\begin{eqnarray*}\n\\mbox{either} &  (y_{\\alpha})_t= (y_\\alpha)_s \\\\\n\\mbox{or} & (y_{\\alpha})_t \\neq (y_\\alpha)_s \n\\end{eqnarray*}\nThis implies moreover, that if for a given $\\alpha \\in I$ and $j\\leq m$, the coordinate $(y_\\alpha)_t$ belongs to \n$$X_0 \\setminus \\{ (x_{\\alpha}^{i})_k: 0\\leq i \\leq d, 0\\leq k \\leq m_i \\},$$\nthen this coordinate has to be constant.\nNow, consider the $r$ coordinate of $b_\\alpha$ which is by assumption pairwise different and does not coincide with any of the $(x_{\\alpha}^{i})_k$. By the above, we have that for all $\\alpha \\in I$, the element  $(y_\\alpha)_r$ does not belong to $X_0$. Moreover there is a natural number $\\ell$ such that $\\ell$ many of the coordinates of $b_\\alpha$ coincides with $(y_\\alpha)_r$. \n\nNext, replace $ (y_{\\alpha})_r$ by $ (y_{\\alpha})_r \\cdot (h_{\\alpha})^q$  with $q<p$ such that $q\\ell \\equiv 1\\ (\\text{mod } p)$ in the transversal $Y$. By Lemma \\ref{Lem_RemTrans} this remains a transversal. Then,\n$$ b_\\alpha = \\Pi_{j=0}^{m} (y_\\alpha)_j  $$\nin the transversal $X$. Furthermore, as we haven't replaced any of the elements in $X_0$, we have still that  each $a_{\\alpha}^{i}$ is equal to $\\Pi_{j=1}^{m_1} (x_{\\alpha}^{i})_j$. Now we can find elements $(\\bar x_\\alpha)_{\\alpha \\in I}$ such that\n\\begin{eqnarray*}\n\t & \\operatorname{EM}( (\\bar a_{\\alpha},  b_\\alpha,[(x_{\\alpha}^{0})_0, \\dots, (x_{\\alpha}^0)_{m_0}],\\dots  \t [(x_{\\alpha}^{d})_0,  \\dots, (x_{\\alpha}^{d})_{m_d}], [(y_{\\alpha})_0, \\dots, (y_{\\alpha})_{m} ]): \\alpha \\in I)  \\\\\n\t \\subset& \\operatorname{tp} (\\bar a_{\\alpha},  b_\\alpha,[(x_{\\alpha}^{0})_0, \\dots, (x_{\\alpha}^0)_{m_0}],\\dots  \t [(x_{\\alpha}^{d})_0,  \\dots, (x_{\\alpha}^{d})_{m_d}], [(x_{\\alpha})_0, \\dots, (x_{\\alpha})_{m} )]: \\alpha \\in I)  . \n\t \\end{eqnarray*}\nand \n$$(\\bar a_{\\alpha},  b_\\alpha,[(x_{\\alpha}^{0})_0, \\dots, (x_{\\alpha}^0)_{m_0}],\\dots  \t [(x_{\\alpha}^{d})_0,  \\dots, (x_{\\alpha}^{d})_{m_d}], [(x_{\\alpha})_0, \\dots, (x_{\\alpha})_{m} )]$$\nis moreover indiscernible. Note that $ X_1 = X_0 \\cup \\{(x_{\\alpha})_j: \\alpha \\in \\kappa, 0\\leq j\\leq m\\}$ can be completed to a transversal $X$ of $G$ and that \n$$b_\\alpha = \\Pi_{j=0}^{m} (x_{\\alpha})_j.$$\nThis completes the proof.\n\\qed \n\n\\begin{lemma}\\label{Lem_a->X}\nLet $(\\bar a_\\alpha: \\alpha \\in \\lambda)$ is an indiscernible sequence and let $\\ell$ be $|\\bar a_\\alpha|$. Suppose that  $\\lambda \\geq \\operatorname{exp}_\\ell(\\kappa)$. \nThen there is \n\\begin{itemize}\n\t\\item a subset $I$ of $\\lambda$ of size at least $\\kappa$;\n\t\\item a transversal $X =X^\\nu\\cup X^p\\cup X^\\iota $;\n\t\\item  a finite tuple of elements $\\bar c$ in $X$;\n\t\\item an indiscernible sequence $(X^\\nu_\\alpha, X^p_\\alpha, X^\\iota_\\alpha, \\bar h_\\alpha: \\alpha \\in I)$ where $ X^\\nu_\\alpha\\subset  X^\\nu$, $ X^p_\\alpha\\subset  X^p$, $ X^\\iota_\\alpha\\subset  X^\\iota$, $\\bar h_\\alpha$ is a tuple in $H_X$, and $X^\\nu_\\alpha$ contains all handles of elements in $X^p_\\alpha$;\n\t\n\n\\item terms $t_0(\\bar z), \\dots, t_{\\ell}(\\bar z)$;\n\\end{itemize}\n such that\n$$ a_\\alpha^i= t_i (X^\\nu_\\alpha, X^p_\\alpha, X^\\iota_\\alpha, \\bar h_\\alpha, \\bar c)$$\n\\end{lemma}\n\\proof\nUp to passing to a subsequence and permuting the tuple, using inductively Lemma \\ref{Lem_IndStep} we may assume that there is a transversal $X$ and $t\\leq d$ such that for every $k\\leq t$, there is a natural number $m_k$ and elements $\\{(x_{\\alpha}^{k})_{j})\\}_{\\alpha\\in \\kappa, 1\\leq j\\leq m_k}$ in $X$ such that \n\n\\begin{itemize}\n\\item $a_\\alpha^k = \\Pi_{j=0}^{m_k}(x_{\\alpha}^{k})_{j}$;\n\\item $(\\bar a_\\alpha, [(x_{\\alpha}^{0})_0, \\dots, (x_{\\alpha}^{0})_{m_0}],\\dots,  [(x_{\\alpha}^{t})_0, \\dots, (x_{\\alpha}^{t})_{m_t}])$ is indiscernible;\n\\item for $k> t$, we have that for all $\\alpha$\n$a_{\\alpha}^{k}  = \\Pi_{j=1}^{m_k} (x_{\\alpha}^{k})_j \\cdot h_{\\alpha}^{k}$\n with $ (x_{\\alpha}^{k})_j \\in X$ and  $h_{\\alpha}^{k} \\in H_X$, and for each coordinate $j$ which is pairwise different, this coordinate is equal to one of the coordinates of $a_\\alpha^s$ for some $s \\leq t$.\n\\end{itemize}\n\n\nNow, for each coordinate sequence of the sequence $a_\\alpha^k$ with $k>t$  that is constant, we add this element to the tuple $\\bar c$. \n\nAs the theory of $H_X$ is stable, there is $I$ of size $\\kappa$ such that for every $t\\leq k\\leq d$, the sequence $(h_\\alpha^k)_{\\alpha \\in I}$ is indiscernible in the sense of $H_X$. Now, we may add the sequence  $(\\bar h_\\alpha = (h_\\alpha^{t+1}, \\dots, h_\\alpha^{d}))$ to the above sequence obtain \n$$(\\bar a_\\alpha, [(x_{\\alpha}^{0})_0, \\dots, (x_{\\alpha}^{0})_{m_0}],\\dots,  [(x_{\\alpha}^{t})_0, \\dots, (x_{\\alpha}^{t})_{m_t}], \\bar h_\\alpha)$$\n As we can clue any automorphism of $H_X$ with any automorphism on $\\langle X \\rangle$ to obtain an automorphism of $G$, this sequence remains indiscernible.\n \nNow, let $X^\\nu_\\alpha$ be the elements of type $1^\\nu$, let $X^p$ be the elements of type $p$, and let $ X^\\iota$ be the elements of type $1^\\iota$ in the set $\\{ x_{\\alpha}^{k})_i: 0\\leq i \\leq d, 0\\leq k \\leq m_i\\}$. \nAs the handle of an element $x$ of type $p$ is definable over $x$, we may add these to the set $X^\\nu_\\alpha$ and assume our set is closed under handles. Since for each $i\\leq d$, the element $a_\\alpha^i$ is a product of $m_i$ many elements and possibly $h_\\alpha^i$, we can fix a term $t_i$ such that \n$$a_\\alpha^i= t_i (X^\\nu_\\alpha, X^p_\\alpha, X^\\iota_\\alpha, \\bar h_\\alpha, \\bar c)$$\n and we can conclude.\n\\qed \n\n\n\n\\proof[Proof with tuples 1. try]\nSuppose that $\\phi(x; \\bar y)$ has IP, let $\\ell$ be $|\\bar y|$ and $\\lambda \\geq \\operatorname{exp}_\\ell(\\kappa)$. Choose an indiscernible sequence $(\\bar a_\\alpha = a_{\\alpha}^1, \\dots, a_{\\alpha}^d)_{\\alpha \\in \\lambda}$ witnessing it. By Lemma \\ref{Lem_a->X} we can find \n\\begin{itemize}\n\t\\item a subset $I$ of $\\lambda$ of size at least $\\kappa$;\n\t\\item a transversal $X =X^\\nu\\cup X^p\\cup X^\\iota $;\n\t\\item  a finite tuple of elements $\\bar c$ in $X$;\n\t\\item an indiscernible sequence $(X^\\nu_\\alpha, X^p_\\alpha, X^\\iota_\\alpha, \\bar h_\\alpha: \\alpha \\in I)$ where $ X^\\nu_\\alpha\\subset  X^\\nu$, $ X^p_\\alpha\\subset  X^p$, $ X^\\iota_\\alpha\\subset  X^\\iota$, $\\bar h_\\alpha$ is a tuple in $H_X$, and $X^\\nu_\\alpha$ contains all handles of elements in $X^p_\\alpha$;\n\\item terms $t_0(\\bar z), \\dots, t_{\\ell}(\\bar z)$;\n\\end{itemize}\n such that\n$$ a_\\alpha^i= t_i (X^\\nu_\\alpha, X^p_\\alpha, X^\\iota_\\alpha, \\bar h_\\alpha, \\bar c)$$\nNow let $\\psi(x, \\bar z, \\bar y)$ be the formula such that\n$$ \\psi(x, \\bar c, X^\\nu_\\alpha, X^p_\\alpha, X^\\iota_\\alpha, \\bar h_\\alpha) \\leftrightarrow \\phi(x; t_0 (X^\\nu_\\alpha, X^p_\\alpha, X^\\iota_\\alpha, \\bar h_\\alpha, \\bar c), \\dots , t_\\ell (X^\\nu_\\alpha, X^p_\\alpha, X^\\iota_\\alpha, \\bar h_\\alpha, \\bar c) ).$$\n\nAs the sequence witnesses IP of the formula $\\phi(x; \\bar y)$, we can choose $g$ in $G$ such that\n$$\\models \\phi(g; t_0 (X^\\nu_\\alpha, X^p_\\alpha, X^\\iota_\\alpha, \\bar h_\\alpha, \\bar c), \\dots , t_\\ell (X^\\nu_\\alpha, X^p_\\alpha, X^\\iota_\\alpha, \\bar h_\\alpha, \\bar c) ) \\mbox{ if and only if $i$ is odd }$$\nThus \n$$\\models \\psi(g, \\bar c; X^\\nu_\\alpha, X^p_\\alpha, X^\\iota_\\alpha, \\bar h_\\alpha ) \\mbox{ if and only if $i$ is odd }.$$\n\nChoose elements $(z_i)_{i<m}$ in the transversal $X$ and an element $k$ in $H$ such that $g = \\Pi_{j=1}^{m} z_i \\cdot k$. Again, we may prolong the tuples $\\bar z$ to include all handles of elements of type $p$. After possibly cutting an initial segment of our sequence, we may assume that if $((x_\\alpha^k)_j)_{\\alpha < \\kappa}$ is non constant, then non of these elements coincide with one of the $z_i$'s.\n \nNow, let $Z= Z^{\\nu} \\cup Z^p \\cup Z^{\\iota}$ be elements of type $1^\\nu$, $p$, and $1^\\iota$ respectively in the set $\\{z_i: i< m\\} \\cup \\bar c$. We have that $(X^{\\nu}_{\\alpha})_{\\alpha < \\kappa}$ is an indiscernible sequence in $\\operatorname{Th}(C)$ (remark that this is true because $C$ is interpretable in $G$ using $X^\\nu$).  \nThus, as $\\operatorname{Th}(C)$ is NIP, we have that $\\operatorname{tp}(X^{\\nu}_{\\alpha})$ stabilizes over $Z^\\nu$, in particular, we may assume that $X^{\\nu}_{\\alpha}$ is indiscernible over $Z^\\nu$. Moreover, by indiscernibility  of $ X^p_\\alpha, X^\\iota_\\alpha$ we have that for all $\\alpha, \\beta$ in $\\kappa$,\n$$X^p_\\alpha \\cap Z^p = X^p_\\beta \\cap Z^p \\ \\ \\ \\  \\mbox{ and } \\ \\ \\ \\ X^\\iota_\\alpha \\cap Z^\\iota = X^\\iota_\\beta \\cap Z^\\iota.$$\nPutting all together, we obtain that for any two sequences $\\alpha_0 < \\alpha_1 < \\dots < \\alpha_l,$ and $\\beta_0 < \\beta_1 < \\dots < \\beta_l$ in $\\kappa$ we can find a bijection $\\sigma$ between  $(X_{\\alpha_0},  \\dots, X_{\\alpha_l})$ and $(X_{\\beta_0},  \\dots, X_{\\beta_l})$ such that \n\\begin{itemize}\n\\item $\\operatorname{tp}_\\Gamma (X_{\\alpha_0},  \\dots, X_{\\alpha_l}, Z_\\nu) = \\operatorname{tp}_\\Gamma (\\sigma(X_{\\beta_0}),  \\dots, \\sigma(X_{\\beta_l}), Z_\\nu) $.\n\\item the map $\\sigma$ fixes $Z$.\n\\item  the map $\\sigma$ respects the $1^{\\nu}$-, $p$-, and $1^{\\iota}$-parts and the handles;\n\n\\end{itemize}\nThus by Fact \\ref{Fact_Bau} we have that $$\\operatorname{tp} (X_{\\alpha_0},  \\dots, X_{\\alpha_l}, Z^\\nu) = \\operatorname{tp} (\\sigma(X_{\\alpha_0}),  \\dots, \\sigma(X_{\\alpha_l}), Z \\cap X^\\nu)  = \\operatorname{tp} (X_{\\beta_0},  \\dots, X_{\\beta_l}, Z) .$$\nor in other words there is an automorphism $\\rho$ fixing $Z$ mapping $(X_{\\alpha_0},  \\dots, X_{\\alpha_l})$ to  $(X_{\\beta_0},  \\dots, X_{\\beta_l})$. \n\nFurthermore, by indiscernibility of $(h_\\alpha)$, there is also an automorphism $\\tau $ of $H$ mapping $(h_{\\alpha_0}, \\dots, h_{\\alpha_l})$ to $(h_{\\beta_0}, \\dots, h_{\\beta_l})$. Now the automorphism $\\rho \\times \\tau$ gives us the desired automorphism of $G$.\nThis \n\n\\qed\n\n\\fi\n\n\\section{Preservation of NIP}\\label{sec: NIP}\nWe begin with the simplest case demonstrating that NIP is preserved. Recall the following basic characterization of NIP.\n\n\\begin{fact} (see e.g. \\cite{adler2008introduction}) \\label{fac: char of NIP}\nLet $T$ be a complete first-order theory and let $\\mathbb{M} \\models T$ be a monster model. Let $\\kappa$ be the regular cardinal $ |T|^+$. Then the following are equivalent.\n\\begin{enumerate}\n\\item $T$ is NIP.\n\\item For every indiscernible sequence $I = (\\bar a_i : i\\in \\kappa)$ of finite tuples and a finite tuple $\\bar b$ in $\\mathbb{M}$, there is some $\\alpha < \\kappa$ such that $\\operatorname{tp}(\\bar b \\bar a_i) = \\operatorname{tp}(\\bar b \\bar a_j)$ for all $i,j > \\alpha$.\n\\end{enumerate}\n\\end{fact}\n\nAs in Section \\ref{sec: Mekler},  let $C$ be a nice graph and let $G(C)$ be the $2$-nilpotent group of exponent $p$ which is freely generated (in the variety of 2-nilpotent groups) by the vertices of $C$ by imposing that two generators commute if and only if they are connected by an edge in $C$.\n\n\\begin{theorem} \\label{thm: NIP}\n$\\operatorname{Th}(C)$ is NIP if and only if $\\operatorname{Th}(G(C))$ is NIP.\n\\end{theorem}\n\\begin{proof}\nIf $\\operatorname{Th}(G(C))$ is NIP, then $\\operatorname{Th}(C)$ is also NIP as $C$ is interpretable in $G(C)$.\n\nNow, we want to prove the converse. Let $G \\models \\operatorname{Th}(G(C))$ be a saturated model, and assume that $\\operatorname{Th}(G(C))$ has IP but $\\operatorname{Th}(C)$ is NIP. Fix $\\kappa$ to be $ (\\aleph_0)^+$. Then there is some formula $\\phi(\\bar{x},\\bar{y}) \\in L_G$, and a sequence $I = (\\bar{a}_i : i \\in \\kappa)$ in $G$ shattered by $\\phi(\\bar{x},\\bar{y})$, i.e. such that for every $S \\subseteq \\kappa$, there is some $\\bar{b}_S$ in $G$ satisfying $G \\models \\phi(\\bar{b}_S, \\bar{a}_i)$ if and only if $ i \\in S$.\n\nLet $X$ be a transversal for $G$ and $H \\subseteq Z(G)$ a set of elements linearly independent over $G'$ and such that $G = \\langle X \\rangle \\times \\langle H \\rangle$. Then for each $i \\in \\kappa$ we have, slightly abusing notation, $\\bar{a}_i = t_i (\\bar{x}_i , \\bar{h}_i)$ for some $L_G$-term $t_i$ and some finite tuples $\\bar{x}_i= \\bar{x}_i^{\\nu \\frown} \\bar{x}_i^{p \\frown} \\bar{x}_i^{\\iota}$ from $X$  where $\\bar{x}_i^\\nu, \\bar{x}_i^p, \\bar{x}_i^\\iota$ list all of the elements  of type $1^\\nu, p, 1^\\iota$ in $\\bar{x}_i$, respectively, and $\\bar{h}_i$ from $H$. After adding some elements of type $1^\\nu$ to the beginning of the tuple and changing the term $t_i$ accordingly, we may assume that for each $i\\in \\kappa$ and $j< |\\bar x_i^p|$, the handle of the j-$th$ element of  $\\bar{x}_i^p$ is the $j$-th element of  $\\bar{x}_i^\\nu$ (there might be some repetitions of elements of type $1^\\nu$ as different elements of type $p$ might have the same handle). As $\\kappa > |L_G| + \\aleph_0$, passing to a cofinal subsequence and reordering the tuples if necessary, we may assume that:\n\\begin{enumerate}\n\\item $t_i = t \\in L_G$ and $|\\bar{x}_i|$ and $ |\\bar{h}_i|$ are constant for all $i \\in \\kappa$,\n\\item\n$|\\bar{x}_i^\\nu|, |\\bar{x}_i^p|, |\\bar{x}_i^\\iota|$ are constant for all $i \\in \\kappa$.\n\\end{enumerate}\n\nConsider the $L_G$-formula $\\phi'(\\bar{x},\\bar{y}') = \\phi(\\bar{x}; t(\\bar{y}_1 , \\bar{y}_2))$ with $\\bar{y}' := \\bar{y}_1^{\\frown} \\bar{y}_2$ and $|\\bar y_1|= |\\bar x_i|$ and $|\\bar y_2|= |\\bar h_i|$. Let $\\bar{a}'_i := \\bar{x}_i^{\\frown}\\bar{h}_i$. Then the sequence $I' := (\\bar{a}'_i : i \\in \\kappa)$ is  shattered by $\\phi'(\\bar{x},\\bar{y}')$. Note however that $I'$ is generally not indiscernible. \n\nTo fix this, let $J = ((\\bar{x}'_i)^\\frown \\bar{h}'_i  : i \\in \\kappa)$ be an $L_G$-indiscernible sequence of tuples in $G$ with the same EM-type as $I'$. Then we have:\n\n\\begin{enumerate}\n\\item $J$ is still shattered by $\\phi'(\\bar{x}, \\bar{y}')$,\n\\item for each $i\\in \\kappa$ and $j< |x_i^p|$, we have that the handle of the $j$-th element of  $(\\bar{x}'_i)^p$ is the $j$-th element of  $(\\bar{x}'_i)^\\nu$ (since being a handle is a definable condition, see Definition \\ref{fact: handle}, and the corresponding property was true on all elements in $I'$).\n\\item The set of all elements of $G$ appearing in the sequence $(\\bar{x}'_i : i \\in \\kappa)$ still can be extended to some transversal $X'$ of $G$.\n\\item The set of all elements of $G$ appearing in the sequence $(\\bar{h}'_i : i \\in \\kappa)$ can be extended to some set $H' \\subseteq Z(G)$ linearly independent over $G'$ and such that $G = \\langle X' \\rangle \\times \\langle H' \\rangle $.\n\\end{enumerate}\n\nThe last two conditions hold as the sets of all elements appearing in the sequences $(\\bar{x}_i : i\\in \\kappa)$ and $(\\bar{h}_i : i \\in \\kappa)$ satisfied the respective conditions, these conditions are type-definable by Proposition \\ref{prop: type def} and $J$ has the same EM-type as $I'$.\n\n\n\nNow let $\\bar{b} \\in G$ be such that both sets $\\{i \\in \\kappa: G \\models \\phi'(\\bar{b}, \\bar{a}'_i)\\}$ and $ \\{ i \\in \\kappa :  G \\models \\neg \\phi'(\\bar{b}, \\bar{a}'_i) \\}$ are cofinal in $\\kappa$. Then $\\bar{b} = s (\\bar{z}, \\bar{k})$ for some term $s \\in L_G$ and some finite tuples $\\bar{z}$ in $X'$ and $\\bar{k}$ in $H'$. Write $\\bar{z} = \\bar{z}^{\\nu \\frown} \\bar{z}^{p \\frown} \\bar{z}^{\\iota}$, with $\\bar{z}^{\\nu}, \\bar{z}^p, \\bar{z}^{\\iota}$ listing the elements of the corresponding types in $\\bar{z}$. In the same way as extending $\\bar x_i$, we may add elements to the tuple $\\bar{z}$ and assume that the handle of the $j$-th element of  $\\bar{z}^p$ is the $j$-th element of  $\\bar{z}^\\nu$. \n\nConsider all of the elements in $\\bar z^\\nu$ and $((\\bar{x}'_i)^\\nu: i \\in \\kappa )$ as elements in $\\Gamma(G)$ --- a saturated model of $\\operatorname{Th}(C)$, and note that as $\\Gamma(G)$ is interpretable in $G$ we have that the sequence $((\\bar{x}'_i)^\\nu : i \\in \\kappa)$ is also indiscernible in $\\Gamma(G)$. As $\\operatorname{Th}(\\Gamma(G))$ is NIP, by Fact  \\ref{fac: char of NIP} there is some $\\alpha < \\kappa$ such that $\\operatorname{tp}_{\\Gamma}(\\bar{z}^\\nu, (\\bar{x}'_i)^\\nu) = \\operatorname{tp}_{\\Gamma}(\\bar{z}^\\nu, (\\bar{x}'_j)^\\nu)$ for all $i,j > \\alpha$.\nMoreover, using indiscernibility of the sequence $(\\bar{x}'_i)$ and possibly throwing away finitely many elements from the sequence, we have that \n$$(\\bar{x}'_i)^p \\cap \\bar{z}^p = (\\bar{x}'_j)^p \\cap \\bar{z}^p, (\\bar{x}'_i)^\\iota \\cap \\bar{z}^\\iota = (\\bar{x}'_j)^\\iota \\cap \\bar{z}^\\iota\\ \\mbox{(as tuples)}$$ \nand $\\bar{x}'_i \\cap \\bar{x}'_j$ is constant, for all $i,j \\in \\kappa$. Thus, for any $i,j > \\alpha$, the bijection $\\sigma_{i,j}$ sending $\\bar{x}'_i \\bar{z}$ to  $\\bar{x}'_j \\bar{z}$ and preserving the order of the elements satisfies:\n\\begin{enumerate}\n\\item $\\operatorname{tp}_{\\Gamma}((\\bar{x}'_i)^\\nu \\bar{z}^\\nu) = \\operatorname{tp}_{\\Gamma}(\\sigma_{i,j}((\\bar{x}'_i)^\\nu \\bar{z}^\\nu))$,\n\\item the map $\\sigma_{i,j}$ fixes $\\bar{z}$,\n\\item the map $\\sigma_{i,j}$ respects the $1^{\\nu}$-, $p$- and $1^\\iota$-parts and the handles (since the handle of the $j$-th element of  $(\\bar{x}'_i)^p$ is the $j$-th element of  $(\\bar{x}'_i)^\\nu$).\n\\end{enumerate}\n\n\nNow consider $\\bar{k}$ and $(\\bar{h}'_i: i \\in \\kappa)$ as tuples of elements in $\\langle H' \\rangle$, which is a model of the stable theory $\\operatorname{Th}(\\langle H' \\rangle)$. Moreover, as $(\\bar{h}'_i : i \\in \\kappa)$ is $L_G$-indiscernible and $\\operatorname{Th}(\\langle H' \\rangle)$ eliminates quantifiers, $(\\bar{h}'_i : i \\in \\kappa)$ is also indiscernible in the sense of $\\operatorname{Th}(\\langle H' \\rangle)$. Hence, by stability, there is some $\\beta \\in \\kappa$ such that $\\operatorname{tp}_{\\langle H' \\rangle}(\\bar{k}\\bar{h}'_i) = \\operatorname{tp}_{\\langle H' \\rangle}(\\bar{k} \\bar{h}'_j)$ for all $i,j > \\beta$.\n\nNow,  Lemma \\ref{Lem_GlueAut} gives us an automorphism of $G$ sending $\\bar{x}'_i \\bar{h}'_i \\bar{z} \\bar k$ to $\\bar{x}'_j \\bar{h}'_j \\bar{z} \\bar k$, so $\\operatorname{tp}_G(\\bar{x}'_i \\bar{h}'_i\/ \\bar{z} \\bar k) = \\operatorname{tp}_G(\\bar{x}'_j \\bar{h}'_j\/ \\bar{z} \\bar k)$ for all $i, j > \\operatorname{max}\\{\\alpha, \\beta\\}$. This contradicts the choice of $\\bar{b} = s (\\bar{z}, \\bar{k})$.\n\\end{proof}\n\n\n\n\n\n\\subsection*{An alternative argument for NIP}\nAn alternative proof can be provided relying on the previous work of Mekler and set-theoretic absoluteness.\n\nRecall that the \\emph{stability spectrum} of a complete theory $T$ is defined as the function\n$$ f_T(\\kappa) := \\sup \\{ |S_1(M)| : M \\models T, |M| = \\kappa \\}$$\nfor all infinite cardinals $\\kappa$. Furthermore, for every infinite cardinal $\\kappa$, let $$\\operatorname{ded} \\kappa := \\sup \\{ \\lambda : \\textrm{exists a linear order of size }\\leq \\kappa \\textrm{ with } \\lambda \\textrm{-many cuts} \\}.$$\nSee \\cite{chernikov2016number} and \\cite[Section 6]{chernikov2016non} for a general discussion of the function $\\operatorname{ded} \\kappa$ and its connection to NIP. We will only need the following two facts.\n\\begin{fact}[Shelah \\cite{shelah1990classification}]\\label{fac: NIP by counting types} Let $T$ be a theory in a countable language.\n\\begin{enumerate}\n\\item It $T$ is NIP, then $f_T(\\kappa) \\leq (\\operatorname{ded} \\kappa)^{\\aleph_0}$ for all infinite cardinals $\\kappa$.\n\\item If $T$ has IP, then $f_T(\\kappa) = 2^\\kappa$ for all infinite cardinals $\\kappa$.\n\\end{enumerate}\n\\end{fact}\n\nIt is possible that in a model of ZFC, $\\operatorname{ded} \\kappa = 2^\\kappa$ for all infinite cardinals $\\kappa$ (e.g. in a model of the Generalized Continuum Hypothesis). However, there are models of ZFC in which these two functions are different.\n\n\n\\begin{fact}[Mitchell \\cite{mitchell1972aronszajn}]\nFor every cardinal $\\kappa$ of uncountable cofinality, there is a cardinal preserving Cohen extension such that $(\\operatorname{ded} \\kappa)^{\\aleph_0} < 2^\\kappa$.\n\\end{fact}\n\nIn the original paper of Mekler \\cite{mekler1981stability} it is demonstrated that if $C$ is a nice graph and $\\operatorname{Th}(C)$ is stable, then $\\operatorname{Th}(G(C))$ is stable. More precisely, the following result is established (in ZFC).\n\\begin{fact} \\label{fac: stability is preserved}\nLet $C$ be a nice graph. Then $f_{\\operatorname{Th}(G(C))} (\\kappa) \\leq f_{\\operatorname{Th}(C)}(\\kappa) + \\kappa$ for all infinite cardinals $\\kappa$.\n\\end{fact}\n\nFinally, note that the property ``$T$ is NIP'' is a finitary formula-by-formula statement, hence set-theoretically absolute. Thus in order to prove Theorem \\ref{thm: NIP}, it is enough to prove it in \\emph{some} model of ZFC. Working in Mitchell's model for some $\\kappa$ of uncountable cofinality (hence $(\\operatorname{ded} \\kappa)^{\\aleph_0} + \\kappa < 2^\\kappa$), it follows immediately by combining Facts \\ref{fac: NIP by counting types} and \\ref{fac: stability is preserved}.\n\n\n\\section{Preservation of $k$-dependence}\\label{sec: kDep}\n\nWe are following the notation from \\cite{chernikov2014n}, and begin by recalling some of the facts there.\n\n\n\\begin{defn}\\rm \\label{def: k-dependence}\nA formula $\\varphi\\left(x;y_{0},\\ldots,y_{k-1}\\right)$ has the \\emph{$k$-independence property} (with respect to a theory $T$), if in some model there is a sequence $\\left(a_{0,i},\\ldots,a_{k-1,i}\\right)_{i\\in\\omega}$\nsuch that for every $s\\subseteq\\omega^{k}$ there is $b_{s}$ such\nthat \n\\[\n\\models\\phi\\left(b_{s};a_{0,i_{0}},\\ldots,a_{k-1,i_{k-1}}\\right)\\Leftrightarrow\\left(i_{0},\\ldots,i_{k-1}\\right)\\in s\\mbox{.}\n\\]\nHere $x,y_0, \\ldots, y_{k-1}$ are tuples of variables.\nOtherwise we say that $\\varphi\\left(x,y_{0},\\ldots,y_{k-1}\\right)$ is \\emph{$k$-dependent}.\nA theory is \\emph{$k$-dependent} if it implies that every formula is\n$k$-dependent.\n\n\\end{defn}\n\nTo characterize $k$-dependence in a formula-free way, we have to work with a more complicated form of indiscernibility.\n\n\\begin{defn}\nFix a language $L_{\\operatorname{opg}}^k=\\{R(x_0,\\ldots ,x_{k-1}),<, P_0(x),\\ldots , P_{k-1}(x)\\}$. \nAn \\emph{ordered $k$-partite hypergraph} is an $L^{k}_{\\operatorname{opg}}$-structure $ \\mathcal{A} = \\left(A; <, R, P_0, \\ldots , P_{k-1} \\right)$ such that:\n\\begin{enumerate}\n\\item\n$A$ is the (pairwise disjoint) union $P^{\\mathcal{A}}_0 \\sqcup\\ldots \\sqcup P^{\\mathcal{A}}_{k-1}$,\n\\item $R^{\\mathcal{A}}$ is a symmetric relation so that if $(a_0,\\ldots ,a_{k-1})\\in R^{\\mathcal{A}}$ then $P_i\\cap \\{a_0, \\ldots, a_{k-1}\\}$ is a singleton for every $i<k$,\n\\item\n$<^{\\mathcal{A}}$ is a linear ordering on $A$ with $P_0(A)<\\ldots <P_{k-1}(A)$.\n\\end{enumerate}\n\n\\end{defn}\n\n\\begin{fact} \\label{fac: Gnp} \\cite[Fact 4.4 + Remark 4.5]{chernikov2014n}\nLet $\\mathcal{K}$ be the class of all finite ordered $k$-partite hypergraphs. Then $\\mathcal{K}$ is a Fra\\\"{i}ss\\'e class, and its limit is called the \\emph{ordered $k$-partite random hypergraph}, which we will denote by $G_{k,p}$.  An ordered $k$-partite hypergraph $\\mathcal{A}$ is a model of $\\operatorname{Th}(G_{k,p})$ if and only if:\n\\begin{itemize}\n\\item\n$(P_i(A), <)$ is a model of DLO for each $i<k$,\n\\item\nfor every $j<k$, any finite disjoint sets \n$A_0,A_1\\subset {\\prod_{i<k, i\\neq j}P_i(A)}$\n and\n$b_0<b_1\\in P_j(A)$, there is $b_0<b<b_1$ such that: $R(b,\\bar{a})$ holds for every $\\bar{a} \\in A_0$ and $\\neg R(b, \\bar{a})$ holds for every $\\bar{a} \\in A_1$.\n\\end{itemize}\n\\end{fact}\n\\noindent We denote by $O_{k,p}$ the reduct of $G_{k,p}$ to the language $L^{k}_{\\operatorname{op}} = \\{<, P_0(x),\\ldots , P_{k-1}(x)\\}$.\n\n\n\\begin{defn}\\rm\nLet $T$ be a theory in the language $L$, and let $\\operatorname{\\mathbb{M}}$ be a monster\nmodel of $T$.\n\\begin{enumerate}\n\\item Let $I$ be a structure in the language $L_0$.\nWe say that $\\bar{a}=\\left(a_{i}\\right)_{i\\in I}$, \nwith $a_{i}$ a tuple in $\\operatorname{\\mathbb{M}}$, is \\emph{$I$-indiscernible} over a set of parameters $C \\subseteq \\operatorname{\\mathbb{M}}$ if for\nall $n\\in\\omega$ and all $i_{0},\\ldots,i_{n}$ and $j_{0},\\ldots,j_{n}$\nfrom $I$ we have:\n$$\n\\operatorname{qftp}_{L_0}\\left(i_{0}, \\ldots, i_{n}\\right)=\\operatorname{qftp}_{L_0}\\left(j_{0},\\ldots,j_{n}\\right)\n\\Rightarrow $$\n$$\\operatorname{tp}_{L}\\left(a_{i_{0}}, \\ldots, a_{i_{n}} \/ C\\right)=\\operatorname{tp}_{L}\\left(a_{j_{0}}, \\ldots, a_{j_{n}} \/ C \\right)\n.$$\n\\item For $L_0$-structures $I$ and $J$, we say that $\\left(b_{i}\\right)_{i\\in J}$\nis \\emph{based on} $\\left(a_{i}\\right)_{i\\in I}$ over a set of parameters $C \\subseteq \\operatorname{\\mathbb{M}}$ if for any finite\nset $\\Delta$ of $L(C)$-formulas,  and for any finite tuple $\\left(j_{0},\\ldots,j_{n}\\right)$\nfrom $J$ there is a tuple $\\left(i_{0},\\ldots,i_{n}\\right)$ from\n$I$ such that:\n\n\\begin{itemize}\n\\item $\\operatorname{qftp}_{L_0}\\left(j_{0},\\ldots,j_{n}\\right)=\\operatorname{qftp}_{L_0}\\left(i_{0},\\ldots,i_{n}\\right)$ and\n\\item $\\operatorname{tp}_{\\Delta}\\left(b_{j_{0}},\\ldots,b_{j_{n}}\\right)=\\operatorname{tp}_{\\Delta}\\left(a_{i_{0}},\\ldots,a_{i_{n}}\\right)$.\n\\end{itemize}\n\\end{enumerate}\n\\end{defn}\n\nThe following fact gives a method for finding $G_{k,p}$-indiscernibles using structural Ramsey theory.\n\n\\begin{fact}\\cite[Corollary 4.8]{chernikov2014n}\\label{fac: random hypergraph indiscernibles exist}\nLet $C \\subseteq \\operatorname{\\mathbb{M}}$ be a small set of parameters.\n\\begin{enumerate}\n\\item For any $\\bar{a}=\\left(a_{g}\\right)_{g\\in O_{k,p}}$, there is some $\\left(b_{g}\\right)_{g\\in O_{k,p}}$ which is $O_{k,p}$-indiscernible over $C$ and is based on $\\bar{a}$ over $C$.\n\\item For any $\\bar{a}=\\left(a_{g}\\right)_{g\\in G_{k,p}}$, there is some $\\left(b_{g}\\right)_{g\\in G_{k,p}}$ which is $G_{k,p}$-indiscernible over $C$ and is based on $\\bar{a}$ over $C$.\n\\end{enumerate}\n\n\n\\end{fact}\n\n\n\n\n\n\\begin{fact} \\cite[Proposition 6.3]{chernikov2014n} \\label{fac: char of NIP_k by preserving indisc}\nLet $T$ be a complete theory and let $\\mathbb{M} \\models T$ be a monster model. For any $k \\in \\mathbb{N}$, the following are\nequivalent:\n\\begin{enumerate}\n\\item $T$ is $k$-dependent.\n\\item For any $\\left(a_{g}\\right)_{g\\in G_{k,p}}$ and $b$ with $a_g, b$ finite tuples in $\\mathbb{M}$,\nif $\\left(a_{g}\\right)_{g\\in G_{n,p}}$ is $G_{n,p}$-indiscernible over\n$b$ and $O_{k,p}$-indiscernible (over $\\emptyset$), then it is $O_{k,p}$-indiscernible over $b$.\n\\end{enumerate}\n\n\\end{fact}\n\nWe are ready to prove the main theorem of the section.\n\\begin{theorem}\\label{thm: k-dependence is preserved}\nFor any $k \\in \\mathbb{N}$ and a nice graph $C$, $\\operatorname{Th}(C)$ is $k$-dependent if and only if $\\operatorname{Th}(G(C))$ is $k$-dependent.\n\\end{theorem}\n\n\\begin{proof}\nLet $G \\models \\operatorname{Th}(G(C))$ be a saturated model, let $X$ be a transversal, and  let $H$ be a set in $Z(G)$ which is linearly independent over $G'$  such that $G = \\langle X \\rangle \\times \\langle H \\rangle$. Moreover, fix $\\kappa$ to be $\\aleph_0^+$.\n\nAs in the NIP case, if $\\operatorname{Th}(G(C))$ is $k$-dependent, then $\\operatorname{Th}(C)$ is also $k$-dependent as $C$ is interpretable in $G(C)$.\n\n\nNow suppose that $\\operatorname{Th}(C)$ is $k$-dependent but $\\operatorname{Th}(G(C))$ has the $k$-independence property witnessed by the formula  $\\varphi\\left(x;y_{0},\\ldots,y_{k-1}\\right) \\in L_G$.\nBy compactness we can find a sequence $\\left(a_{0,\\alpha},\\ldots,a_{k-1,\\alpha}\\right)_{\\alpha \\in \\kappa}$ such that for any $s\\subseteq \\kappa^{k}$ there is some $b_{s}$ such\nthat \n\\[\n\\models\\phi\\left(b_{s};a_{0,\\alpha_{0}},\\ldots,a_{k-1,\\alpha_{k-1}}\\right)\\Leftrightarrow\\left(\\alpha_{0},\\ldots,\\alpha_{k-1}\\right)\\in s\\mbox{.}\n\\]\nBy the choice of $X$ and $H$, for each $i < k$ and $\\alpha \\in \\kappa$, there is some term $t_{i, \\alpha} \\in L_G$ and some finite tuples $\\bar{x}_{i, \\alpha}$ from $X$ and $\\bar{h}_{i, \\alpha}$ from $H$ such that $a_{i, \\alpha} = t_{i,\\alpha}(\\bar{x}_{i, \\alpha}, \\bar{h}_{i,\\alpha})$. \nAs $\\kappa > |L_G| + \\aleph_0$, passing to a subsequence of length $\\kappa$ for each $i<k$ we may assume that $t_{i, \\alpha} = t_i$ and $\\bar{x}_{i, \\alpha} = \\bar{x}_{i, \\alpha}^{\\nu \\frown} \\bar{x}_{i, \\alpha}^{p \\frown} \\bar{x}_{i, \\alpha}^{\\iota}$ with $\\bar{x}_{i, \\alpha}^{\\nu}, \\bar{x}_{i, \\alpha}^{p}, \\bar{x}_{i, \\alpha}^{\\iota}$ listing all elements of the corresponding type in $\\bar{x}_{i, \\alpha}$ and $|\\bar{x}_{i, \\alpha}^{\\nu}|, |\\bar{x}_{i, \\alpha}^{p}|, |\\bar{x}_{i, \\alpha}^{\\iota}|$ constant for all $i<j$ and $\\alpha \\in \\kappa$. \nMoreover, in the same way as in the NIP case, we add the handles of the elements in the tuple $\\bar x_{i,\\alpha}^p$ to the beginning of $\\bar{x}_{i, \\alpha}^{\\nu}$. Taking $\\psi(x; y'_0, \\ldots, y'_{k-1}) := \\phi(x; t_0(y'_0), \\ldots, t_{k-1}(y'_{k-1}))$, we see that the sequence $(\\bar{x}^{\\frown}_{0,\\alpha} \\bar{h}_{0,\\alpha}, \\ldots,  \\bar{x}^{\\frown}_{k-1,\\alpha} \\bar{h}_{k-1,\\alpha}: \\alpha \\in \\kappa)$ is shattered by $\\psi$, i.\\ e.\\ for each $A \\subset \\kappa^k $ there is some $\\bar b$ such that $G \\models \\psi(\\bar{b}; \\bar{x}_{0, i_0}^\\frown \\bar{h}_{0, i_0}, \\ldots, \\bar{x}_{k-1, i_{k-1}}^\\frown \\bar{h}_{k-1, i_{k-1}})$ if and only if $  (i_0, \\dots, i_{k-1}) \\in A$. \nWe define an $L_{\\operatorname{op}}$-structure on $\\kappa$ by interpreting each of the $P_i, i < k$ as some countable disjoint subsets of $\\kappa$, and choosing any ordering isomorphic to $(\\mathbb{Q}, <)$ on each of the $P_i$'s. \nWe pass to the corresponding subsequences of $(\\bar x_{i,\\alpha}^{\\frown} \\bar{h}_{i, \\alpha}: \\alpha \\in \\kappa)$, namely for each $i \\in k$, we consider the sequence given by $(\\bar x_{i,\\alpha}^{\\frown} \\bar{h}_{i, \\alpha}: \\alpha \\in P_i)$.\nTaking these $k$ different sequences together we obtain the sequence $(\\bar{x}_g^\\frown \\bar{h}_g : g \\in O_{k,p})$ indexed by $O_{k,p}$. This sequence is shattered in the following sense: for each $A \\subset P_0 \\times \\dots \\times P_{k-1} $ there is some $\\bar b \\in G$ such that $G \\models \\psi(\\bar{b}; \\bar{x}_{g_0}^\\frown \\bar{h}_{g_0}, \\ldots, {\\bar{x}_{g_{k-1}}}^{\\frown} \\bar{h}_{g_{k-1}})$ if and only if $  (g_0, \\dots, g_{k-1}) \\in A$. \n\n\n\nBy Fact \\ref{fac: random hypergraph indiscernibles exist}(1), let $(\\bar{y}_{g}^{\\frown} \\bar{m}_g : g \\in O_{k,p})$ be an $O_{k,p}$-indiscernible in $G$ based on $(\\bar{x}_g^\\frown \\bar{h}_g : g \\in O_{k,p})$. Observe that, using Proposition \\ref{prop: type def} as in the proof of Theorem \\ref{thm: NIP}, we still have:\n\\begin{enumerate}\n\\item $(\\bar{y}_{g}^{\\frown} \\bar{m}_g : g \\in O_{k,p})$ is shattered by $\\psi$,\n\\item the handle for each $j$th element in the tuple $\\bar y_{g}^p$ is the $j$th element of the tuple $\\bar y_{g}^\\nu$,\n\\item the set of all elements of $G$ appearing in $(\\bar{y}_g : g \\in O_{k,p})$ is a partial transversal, hence can be extended to a transversal $Y$ of $G$,\n\\item the set of all elements of $G$ appearing in $(\\bar{m}_g : g \\in O_{k,p})$ is still a set of elements in $Z(G)$ linearly independent over $G'$, hence can be extended to a linearly independent set $M$ such that $G = \\langle Y \\rangle \\times \\langle M \\rangle$.\n\\end{enumerate}\nWe can expand $O_{k,p}$ to $G_{k,p}$ (see Fact \\ref{fac: Gnp}). As $(\\bar{y}_{g}^{\\frown} \\bar{m}_g : g \\in O_{k,p})$ is shattered by $\\psi$, we can find an element $b \\in G$ such that $G \\models \\psi(b; \\bar{y}_{g_0}^{\\frown}  \\bar{m}_{g_0}, \\ldots, \\bar{y}_{g_{k-1}}^{\\frown} \\bar{m}_{g_{k-1}}) \\iff G_{k,p} \\models R(g_0, \\ldots, g_{k-1})$, for all $g_{i}\\in P_{i}$. We can write $b = s(\\bar{z}, \\bar{\\ell})$ for some term $s \\in L_G$ and some finite tuples $\\bar{z} = \\bar{z}^{\\nu \\frown} \\bar{z}^{p \\frown} \\bar{z}^\\iota$ in $Y$ and $\\bar{\\ell}$ in $M$. As usual, extending $\\bar{z}^\\nu$ if necessary, we may assume that $\\bar{z}$ is closed under handles. Taking $\\theta(x'; y'_0, \\ldots, y'_{k-1}) := \\psi(s(x'); y'_0, \\ldots, y'_{k-1})$, we still have that $$G \\models \\theta(\\bar{z}^{\\frown}\\bar{\\ell}; \\bar{y}_{g_0}^{\\frown}  \\bar{m}_{g_0}, \\ldots, \\bar{y}_{g_{k-1}}^{\\frown} \\bar{m}_{g_{k-1}}) \\iff G_{k,p} \\models R(g_0, \\ldots, g_{k-1})$$ for all $g_{i}\\in P_{i}$.\n\nBy Fact \\ref{fac: random hypergraph indiscernibles exist}(2), we can find $(\\bar{z}_g^{\\frown} \\bar{\\ell}_g : g \\in G_{k,p})$ which is $G_{k,p}$-indiscernible over $\\bar{z}^{\\frown}\\bar{\\ell}$ and is based on $(\\bar{y}_{g}^{\\frown} \\bar{m}_g : g \\in G_{k,p})$ over $\\bar{z}^{\\frown}\\bar{\\ell}$. Then   we have:\n\\begin{enumerate}\n\\item $G \\models \\theta(\\bar{z}^{\\frown}\\bar{\\ell}; \\bar{z}_{g_0}^{\\frown}  \\bar{\\ell}_{g_0}, \\ldots, \\bar{z}_{g_{k-1}}^{\\frown} \\bar{\\ell}_{g_{k-1}}) \\iff G_{k,p} \\models R(g_0, \\ldots, g_{k-1})$, for all $g_{i}\\in P_{i}$;\n\\item for $\\bar{z}_g = \\bar{z}_g^{\\nu \\frown} \\bar{z}_g^{p \\frown} \\bar{z}_g^{\\iota}$ we have that:\n\\begin{itemize}\n\t\\item all of these tuples are of fixed length and list elements of the corresponding type,\n\t\\item  the handle of the $j$-th element of  $\\bar{z}_g^p$ is the $j$-th element of  $\\bar{z}_g^\\nu$;\n\t\\end{itemize}\n\\item the set of all elements of $G$ appearing in $\\bar{z}$ and $(\\bar{z}_g : g \\in G_{k,p})$ is a partial transversal, hence can be extended to some transversal $Z$ of $G$;\n\\item the set of all elements of $G$ appearing in $\\bar{\\ell}$ and $(\\bar{\\ell}_g : g \\in G_{k,p})$ is still a set of elements in $Z(G)$ linearly independent over $G'$, hence can be extended to a linearly independent set $L$ such that $G = \\langle Z \\rangle \\times \\langle L \\rangle$;\n\\item $(\\bar{z}_g^{\\frown} \\bar{\\ell}_g : g \\in G_{k,p})$ is $O_{k,p}$-indiscernible over $\\emptyset$ (follows since $(\\bar{z}_g^{\\frown} \\bar{\\ell}_g : g \\in G_{k,p})$ is based on $(\\bar{y}_{g}^{\\frown} \\bar{m}_g : g \\in G_{k,p})$, which was $O_{k,p}$-indiscernible, as in the proof of \\cite[Lemma 6.2]{chernikov2014n}).\n\\end{enumerate}\n\n\nConsider now all of the elements in $\\bar{z}^\\nu$ and $(\\bar{z}_{g}^{\\nu} : g \\in G_{k,p})$ as elements in $\\Gamma(G)$, a saturated model of $\\operatorname{Th}(C)$, and note that as $\\Gamma(G)$ is interpretable in $G$, we have that the sequence $(\\bar{z}^{\\nu}_g : g \\in G_{k,p})$ is also $G_{k,p}$-indiscernible over $\\bar{z}^{\\nu}$ and is $O_{k,p}$-indiscernible over $\\emptyset$, both in $\\Gamma(G)$. As $\\operatorname{Th}(C)$ is $k$-dependent, it follows by Fact \\ref{fac: char of NIP_k by preserving indisc} that $(\\bar{z}_{g}^{\\nu} : g \\in G_{k,p})$ is $O_{k,p}$-indiscernible over $\\bar{z}^\\nu$ in $\\Gamma(G)$. \nHence for any finite tuples $g_0, \\ldots, g_n, q_0, \\ldots, q_n \\in G_{k,p}$ such that $\\operatorname{tp}_{L_{\\operatorname{op}}^k}(\\bar{g})  = \\operatorname{tp}_{L_{\\operatorname{op}}^k}(\\bar{q})$, we have that  $\\operatorname{tp}_{\\Gamma}(\\bar{z}^\\nu_{g_0}, \\ldots, \\bar{z}^\\nu_{g_n}\/ \\bar{z}^\\nu )$ is equal to $\\operatorname{tp}_{\\Gamma}(\\bar{z}^\\nu_{q_0}, \\ldots, \\bar{z}^\\nu_{q_n}\/ \\bar{z}^\\nu)$. \nNow, using  $O_{k,p}$-indiscernibility and that $\\bar z$ is finite, for each $i < k$ there is some finite $\\lambda_i \\subseteq  P_i $ such that for all $g \\neq q \\in P_i$ with $g,q > \\lambda_i$ (i.e. $g > h$ for every element $h \\in \\lambda_i$, and the same for $q$) we have\n$$ \\bar{z}_{g}^p  \\cap \\bar{z}^p = \\bar{z}_{q}^p \\cap \\bar{z}^p, \\bar{z}_{g}^\\iota \\cap \\bar{z}^\\iota = \\bar{z}_{q}^\\iota\\cap \\bar{z}^\\iota\\ \\mbox{(as tuples)}\n$$\nand $\\bar{z}_{g} \\cap \\bar{z}_{q}$ is constant. Thus, for any $g_0, \\dots ,g_{k-1}, q_0, \\dots, q_{k-1}$  such that  $g_i, q_i > \\lambda_i$ and $g_i, q_i \\in P_i$, we get that mapping $\\bar{z}_{g_0}, \\dots,\\bar{z}_{g_{k-1}},  \\bar{z}$ to  $\\bar{z}_{q_0}, \\dots,\\bar{z}_{q_{k-1}},   \\bar{z}$ preserving the positions of the elements in the tuples defines a bijection $\\sigma_{\\bar g,\\bar q}$\nsuch that:\n\\begin{enumerate}\n\\item $\\operatorname{tp}_{\\Gamma}(\\bar{z}_{g_0}^\\nu , \\dots,\\bar{z}_{g_{k-1}}^\\nu ,\\bar{z}^\\nu) = \\operatorname{tp}_{\\Gamma}(\\sigma_{\\bar g,\\bar q}(\\bar{z}_{g_0}^\\nu , \\dots,\\bar{z}_{g_{k-1}}^\\nu, \\bar{z}^\\nu)),$\n\\item the map $\\sigma_{\\bar g,\\bar q}$ fixes $\\bar{z}$,\n\\item the map $\\sigma_{\\bar g,\\bar q}$ respects the $1^{\\nu}$-, $p$- and $1^\\iota$-parts and the handles.\n\\end{enumerate}\n\n\n\n\n\nNext we consider all the elements in $\\bar{\\ell}$ and $(\\bar{\\ell}_g : g \\in G_{k,p})$ as elements in $\\langle L \\rangle$, a saturated model of the stable theory $\\operatorname{Th}(\\langle L \\rangle)$. By quantifier elimination, we still have that $(\\bar{\\ell}_g : g \\in G_{k,p})$ is both $O_{k,p}$-indiscernible and $G_{k,p}$-indiscernible over $\\bar{\\ell}$ in $\\langle L \\rangle$. As $\\langle L \\rangle$ is stable, so in particular $k$-dependent, by Fact \\ref{fac: char of NIP_k by preserving indisc}, $(\\bar{\\ell}_g : g \\in G_{k,p})$ is  $O_{k,p}$-indiscernible over $\\bar{\\ell}$.\n\nNow let $\\bar{g}, \\bar{q} \\in G_{k,p}$ be such that $g_i, q_i > \\lambda_i$ and $g_i,q_i \\in P_i$ for all $i < k$, and such that $G_{k,p} \\models R(g_0, \\ldots, g_{k-1}) \\land \\neg R(q_0, \\ldots, q_{k-1})$ holds. Then by the choice of $\\bar{z}^{\\frown} \\bar{\\ell}$ we have that $G \\models \\theta(\\bar{z}^{\\frown}\\bar{\\ell}; \\bar{z}_{g_0}^{\\frown} \\bar{\\ell}_{g_0}, \\ldots, \\bar{z}_{g_{k-1}}^{\\frown} \\bar{\\ell}_{g_{k-1}}) \\land \\neg \\theta(\\bar{z}^{\\frown}\\bar{\\ell}; \\bar{z}_{q_0}^{\\frown}  \\bar{\\ell}_{q_0}, \\ldots, \\bar{z}_{q_{k-1}}^{\\frown} \\bar{\\ell}_{q_{k-1}})$. On the other hand, combining the last two paragraphs and using Lemma \\ref{Lem_GlueAut}, \nwe find an automorphism of $G$ sending $(\\bar{z}_{g_0}^\\frown  \\bar{\\ell}_{g_0}, \\ldots, \\bar{z}_{g_{k-1}}^\\frown \\bar{\\ell}_{g_{k-1}})$ to $(\\bar{z}_{q_0}^\\frown  \\bar{\\ell}_{q_0}, \\ldots, \\bar{z}_{q_{k-1}}^\\frown  \\bar{\\ell}_{q_{k-1}})$ and fixing $\\bar{z}^{\\frown}\\bar{\\ell}$ --- a contradiction.\n\n\n\\end{proof}\n\n\n\\begin{cor} \\label{cor: strictly k-dep groups}\n\nFor every $k \\geq 2$, there is a strictly $k$-dependent  pure group $G$. Moreover, we can find such a $G$ with a simple theory.\n\\end{cor}\n\\begin{proof}\nFor each $k \\geq 2$, let $A_k$ be the random $k$-hypergraph. It is well-known that $\\operatorname{Th}(A_k)$ is simple. Moreover, $A_{k}$ is clearly not $(k-1)$-dependent, as witnessed by the edge relation, and it is easy to verify that $A_{k}$ is $k$-dependent (as it eliminates quantifiers and all relation symbols are at most $k$-ary, see e.g. \\cite[Proposition 6.5]{chernikov2014n}).\n\nNow $A_k$, as well as any other structure in a finite relational language, is bi-interpretable with some nice graph $C_k$ by \\cite[Theorem 5.5.1 + Exercise 5.5.9]{hodges1993model}, so $C_k$ also has all of the aforementioned properties. Then Mekler's construction produces a group $G(C_k)$ with all of the desired properties, by Theorem \\ref{thm: k-dependence is preserved} and  preservation of simplicity from \\cite{baudisch2002mekler}.\n\\end{proof}\n\nThis corollary gives first examples of strictly $k$-dependent groups, however many other questions about the existence of strictly $k$-dependent algebraic structures remain.\n\n\\begin{problem}\n\\begin{enumerate}\n\\item Are there pseudofinite strictly $k$-dependent groups, for $k>2$? \n\n\\item Are there $\\aleph_0$-categorical strictly $k$-dependent groups, for $k>2$? \n\n\\end{enumerate}\nWe note that the strictly $2$-dependent group in Example \\ref{ex: extraspecial} is  both pseudofinite and $\\aleph_0$-categorical (see \\cite[Proposition 3.11]{macpherson2008one} and the discussion around it). However, Mekler's construction does not preserve $\\aleph_0$-categoricity in general (this is mentioned in \\cite[Introduction]{baudisch2002mekler}), e.g. because the proof in Remark \\ref{rem: unbounded commutator products} shows that if $C$ is an infinite nice graph, then in $G(C)$ there are infinitely many pairwise inequivalent formulas $\\phi_n(x)$ expressing that $x$ is a product of at most $n$ commutators.\n\\end{problem}\n\n\n\\begin{problem}\nAre there strictly $k$-dependent fields, for any $k \\geq 2$? We conjecture that there aren't any with a simple theory. It is proved in \\cite{hempel2016n} that any $k$-dependent PAC field is separably closed, and there are no known examples of fields with a simple theory which are not PAC.\n\\end{problem}\n\n\n\\section{Preservation of $\\operatorname{NTP}_2$} \\label{sec: NTP2}\nWe recall the definition of $\\operatorname{NTP}_2$ (and refer to \\cite{chernikov2014theories} for further details).\n\\begin{defn} \\label{def: NTP2}\n\\begin{enumerate}\n\\item A formula $\\phi(x,y)$, with $x,y$ tuples of variables, has $\\operatorname{TP}_2$ if there is an array $( a_{i,j} : i,j \\in \\omega)$ of tuples in $\\operatorname{\\mathbb{M}} \\models T$ and some $k \\in \\omega$ such that:\n\\begin{enumerate}\n\\item for all $i \\in \\omega$, the set $\\{ \\phi(x, a_{i,j}) : j \\in \\omega \\}$ is $k$-inconsistent.\n\\item for all $f : \\omega \\to \\omega$, the set $\\{ \\phi(x, a_{i, f(i)}) : i \\in \\omega \\}$ is consistent.\n\\end{enumerate}\n\n\\item A theory $T$ is $\\operatorname{NTP}_2$ if no formula has $\\operatorname{TP}_2$ relatively to it.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{remark} \\label{rem : 2-incons}\\cite[Lemma  3.2]{chernikov2014theories}\nIf $T$ is not $\\operatorname{NTP}_2$, one can find a formula as in Definition \\ref{def: NTP2}(1) with $k = 2$.\n\\end{remark}\n\nWe will use the following formula-free characterization of $\\operatorname{NTP}_2$ from \\cite[Section 1]{chernikov2014theories}.\n\n\\begin{fact} \\label{fac: char of NTP2} Let $T$ be a theory and $\\operatorname{\\mathbb{M}} \\models T$ a monster model. Let $\\kappa := |T|^+$. The following are equivalent:\n\\begin{enumerate}\n\\item $T$ is $\\operatorname{NTP}_2$.\n\\item For any array $(a_{i,j} : i \\in \\kappa, j \\in \\omega)$ of finite tuples with \\emph{mutually indiscernible rows} (i.e. for each $i \\in \\kappa$, the sequence $\\bar{a}_i := (a_{i,j} : j \\in \\omega)$ is indiscernible over $\\{ a_{i',j} : i' \\in \\kappa \\setminus \\{i \\}, j \\in \\omega \\}$) and a finite tuple $b$, there is some $\\alpha \\in \\kappa $ satisfying the following:\nfor any $i > \\alpha$ there is some  $b'$ such that $\\operatorname{tp}(b\/ a_{i,0}) = \\operatorname{tp}(b' \/ a_{i,0})$ and $\\bar{a}_i$ is indiscernible over $b'$.\n\\end{enumerate}\n\n\n\\end{fact}\n\n\nThe following can be proved using finitary Ramsey theorem and compactness, see \\cite[Section 1]{chernikov2014theories} for the details.\n\n\\begin{fact}\\label{fac: extracting mut ind}\n\nLet $(a_{\\alpha,i} : \\alpha,i \\in \\kappa)$ be an array of tuples from $\\operatorname{\\mathbb{M}} \\models T$.\nThen there is an array $(b_{\\alpha,i} : \\alpha, i \\in \\kappa)$ with mutually indiscernible rows \\emph{based on $(a_{\\alpha,i} : \\alpha,i \\in \\kappa)$}, i.e. such that for every finite set of formulas $\\Delta$, any $\\alpha_0, \\ldots, \\alpha_{n-1} \\in \\kappa$ and any strictly increasing finite tuples $\\bar{j}_0, \\ldots, \\bar{j}_{n-1}$ from $\\kappa$, there are some strictly increasing tuples $\\bar{i}_0, \\ldots, \\bar{i}_{n-1}$ from $\\kappa$ such that \n$$\\models \\Delta( (b_{\\alpha_0,i} : i \\in \\bar{j}_0), \\ldots, (b_{\\alpha_{n-1}, i} : i \\in \\bar{j}_{n-1}) ) \\iff$$\n$$\\models \\Delta( (a_{\\alpha_0,i} : i \\in \\bar{i}_0), \\ldots, (a_{\\alpha_{n-1}, i} : i \\in \\bar{i}_{n-1}) ).$$\n\\end{fact}\n\n\\begin{remark} \\label{rem: extraction preserves witness TP2}\nIf $\\phi(x,y)$ and $(a_{\\alpha,i} : \\alpha,i \\in \\kappa)$ satisfy the condition in Definition \\ref{def: NTP2}(1) and $(b_{\\alpha,i} : \\alpha, i \\in \\kappa)$ is based on it, then $\\phi(x,y)$ and $(b_{\\alpha,i} : \\alpha, i \\in \\kappa)$ also satisfy the condition in Definition \\ref{def: NTP2}(1).\n\\end{remark}\n\n\n\n\\begin{theorem}\\label{thm: NTP2 is preserved}\nFor any nice graph $C$, we have that $\\operatorname{Th}(G(C))$ is $\\operatorname{NTP}_2$ if and only if $\\operatorname{Th}(C)$ is $\\operatorname{NTP}_2$.\n\\end{theorem}\n\\begin{proof}\n\t\n\tAs before, let $G \\models \\operatorname{Th}(G(C))$ be a monster model, let $X$ be a transversal, and  let $H$ be a set in $Z(G)$ which is linearly independent over $G'$ such that $G = \\langle X \\rangle \\times \\langle H \\rangle$. Moreover, fix $\\kappa$ to be $\\aleph_0^+$. If $\\operatorname{Th}(G(C))$ is $\\operatorname{NTP}_2$ then $\\operatorname{Th}(C)$ is also $\\operatorname{NTP}_2$ as $C$ is interpretable in $G(C)$.\n\n\n\tNow suppose that $\\operatorname{Th}(C)$ is $\\operatorname{NTP}_2$, but $\\operatorname{Th}(G(C))$ has $\\operatorname{TP}_2$. By compactness and Remark \\ref{rem : 2-incons} we can find some formula $\\phi(x,y)$ and an array $(\\bar a_{i,j} : i,j \\in \\kappa)$ of tuples in $G$ witnessing $\\operatorname{TP}_2$ as in Definition \\ref{def: NTP2}(1) with $k=2$.\n\tThen for all $i,j \\in \\kappa$ we have $\\bar a_{i,j} = t_{i,j}(\\bar{x}_{i,j}, \\bar{h}_{i,j})$ for some terms $t_{i,j} \\in L_G$ and some finite tuples $\\bar{x}_{i,j}$ from $X$ and $\\bar{h}_{i,j}$ from $H$.\n\nAs $\\kappa > |L_G| + \\aleph_0$, passing to a subsequence of each row, and then to a subsequence of the rows, we may assume that $t_{i,j} = t \\in L_G$ and $\\bar{x}_{i,j} = \\bar{x}_{i,j}^{\\nu \\frown} \\bar{x}_{i,j}^{p \\frown} \\bar{x}_{i,j}^{\\iota}$ with $|\\bar{x}_{i,j}^\\nu|, |\\bar{x}_{i,j}^p|, |\\bar{x}_{i,j}^\\iota|, |\\bar{h}_{i,j}|$ constant for all $i,j \\in \\kappa$. Again as in the NIP case, we add the handles of the elements in the tuple $\\bar x_{i,\\alpha}^p$ to the beginning of $\\bar{x}_{i, \\alpha}^{\\nu}$ for all $i,j \\in \\kappa$. Taking $\\psi(x,y') := \\phi(x, t(y') )$ with $|y'| = |\\bar{x}_{i,j}^\\frown \\bar{h}_{i,j}|$ and $\\bar{b}_{i,j} :=  \\bar{x}_{i,j}^\\frown \\bar{h}_{i,j}$, we have that $\\psi(x,y') \\in L_G$ and the array $(\\bar{b}_{i,j} : i,j \\in \\kappa)$ still satisfy the condition in Definition \\ref{def: NTP2}(1) with $k=2$. \n\nBy Fact \\ref{fac: extracting mut ind}, let $(\\bar{c}_{i,j} : i,j \\in \\kappa)$ with $\\bar{c}_{i,j} = \\bar{y}_{i,j}^\\frown \\bar{m}_{i,j}$ be an array with mutually indiscernible rows based on $(\\bar{b}_{i,j} : i,j \\in \\kappa)$. Then, arguing as in the proofs of Theorems \\ref{thm: NIP} and  \\ref{thm: k-dependence is preserved} using type-definability of the relevant properties from Proposition \\ref{prop: type def} and Remark \\ref{rem: extraction preserves witness TP2}, we have:\n\\begin{enumerate}\n\\item $\\psi(x,y')$ and the array $(\\bar{c}_{i,j} : i,j \\in \\kappa)$ satisfy the condition in Definition \\ref{def: NTP2}(1) with $k=2$;\n\\item For $\\bar{y}_{i,j}= \\bar{y}_{i,j}^{\\nu \\frown} \\bar{y}_{i,j}^{p \\frown} \\bar{y}_{i,j}^{\\iota}$ we have that:\n\\begin{itemize}\n\t\\item all of these tuples are of fixed length and list elements of the corresponding type,\n\t\\item  the handle of the $n$-th element of  $\\bar{y}_{i,j}^p$ is the $n$-th element of  $\\bar{y}_{i,j}^\\nu$;\n\t\\end{itemize}\n\\item the set of all elements of $G$ appearing in $(\\bar{y}_{i,j} : i,j \\in \\kappa)$ is a partial transversal of $G$ and can be extended to a transversal $Y$ of $G$;\n\\item the set of all elements of $G$ appearing in $(\\bar{m}_{i,j} : i,j \\in \\kappa)$ is a set of elements in $Z(G)$ linearly independent over $G'$, hence can be extended to a set of generators $M$ such that $G = \\langle Y \\rangle \\times \\langle M \\rangle$.\n\\end{enumerate}\n\nLet now $\\bar{b}$ be a tuple in $G$ such that $G \\models \\{ \\psi(\\bar b, \\bar{c}_{i,0}) : i \\in \\kappa\\}$. We have that $\\bar b =\ns(\\bar{y}, \\bar{m})$ for some term $s \\in L_G$ and some finite tuples $\\bar{y}$ in $Y$ and $\\bar{m}$ in $M$. Let\n$\\bar{y} = \\bar{y}^{\\nu \\frown} \\bar{y}^{p\\frown} \\bar{y}^{\\iota}$, each listing the elements of the corresponding type. In\nthe same way as for each of the $\\bar y_{i,j}$'s, we add the handles of the elements in the tuple $\\bar y^p$ to the\nbeginning of $\\bar{y}^{\\nu}$ so that the handle of the $n$-th element of $\\bar{y}^p$ is the $n$-th element of\n$\\bar{y}^\\nu$.\nTaking $\\theta(x',y') := \\psi(s(x'), y')$, we still have that $\\bar{y}^\\frown \\bar{m} \\models \\{ \\theta(x', \\bar{c}_{i,0}) : i \\in \\kappa  \\}$ and the set of formulas $\\{ \\theta(x', \\bar{c}_{i,j}) : j \\in \\kappa \\}$ is $2$-inconsistent for each $i \\in \\kappa$. Moreover, after possibly throwing away finitely many rows, we may assume that \nthe rows are mutually indiscernible over $\\bar{y}^\\frown \\bar{m} \\cap \\bigcup \\{\\bar{c}_{i,0} : i \\in \\kappa \\} $ (if an element of $\\bar{y}^\\frown \\bar{m}$ appears in $\\bar{c}_{i,0}$, then the rows of the array $(\\bar{c}_{i',j} : i' \\in \\kappa, i'\\neq i, j \\in \\kappa)$ are mutually indiscernible over it). This implies that if $z \\in \\bar{y} \\cap \\bar{y}_{i,0}$ for some $i$ and $z$ is the $n$-th element in the tuple $\\bar{y}_{i,0} $, then it is the $n$-th element in any tuple $\\bar{y}_{j,0} $ with $j \\in \\kappa$.\n\n\nConsider all of the elements in $\\bar{y}^{\\nu}$ and $(\\bar{y}_{i,j}^{\\nu} : i,j \\in \\kappa)$ as elements in $\\Gamma(G)$, a saturated model of $\\operatorname{Th}(C)$, and note that as $\\Gamma(G)$ is interpretable in $G$ we have that the array $(\\bar{y}_{i,j}^\\nu : i,j \\in \\kappa)$ has mutually indiscernible rows in $\\Gamma(G)$. As $\\operatorname{Th}(\\Gamma(G))$ is $\\operatorname{NTP}_2$, it follows by Fact \\ref{fac: char of NTP2}  that there is some $\\alpha \\in \\kappa$ such that for each $i> \\alpha$ there is some tuple $\\bar{y}^{\\prime \\nu}$ such that $\\operatorname{tp}_{\\Gamma}(\\bar{y}^{\\nu}\/ \\bar{y}^{\\nu}_{i,0}) = \\operatorname{tp}_{\\Gamma}(\\bar{y}^{\\prime \\nu} \/ \\bar{y}^{\\nu}_{i,0})$ and the sequence $(\\bar{y}_{i,j}^{\\nu} : j \\in \\kappa)$ is $L_{\\Gamma}$-indiscernible over $\\bar{y}^{\\prime \\nu}$, i.\\ e.\\  $\\operatorname{tp}_{\\Gamma}(\\bar{y}^{\\nu}, \\bar{y}^{\\nu}_{i,0}) = \\operatorname{tp}_{\\Gamma}(\\bar{y}^{\\prime \\nu}, \\bar{y}^{\\nu}_{i,0})$.\nLet   $\\sigma_0$ be the bijection which maps   $ \\bar{y}^{\\nu \\frown} \\bar{y}_{i,0}$ to $\\bar{y}^{\\prime \\nu \\frown} \\bar{y}_{i,0}$.\nNow we want to extend this bijection $\\sigma_0$ to a bijection $\\sigma$ defined on each element of  $\\bar y^{\\frown} \\bar{y}_{i,0}$ in a type and handle preserving way. To do so, we have to choose an image for each element in $\\bar y^{p^\\frown} \\bar y^\\iota$. Let $z$ be the $n$-th element of $ \\bar y^p$ and let $u$ be the $n$-th element of $\\bar y^\\nu$ (i.\\ e.\\ the handle of $z$).\n\\begin{itemize}\n\t\\item If $z\\not \\in  \\bar y_{i,0}^p$, then choose $\\sigma(z)$ to be any element in $Y^p$ which has handle $\\sigma_0(u)$ and is not contained in $ \\bar y_{i,0}^p$ (as $Y$ is a $|G|$-cover and $\\operatorname{tp}_\\Gamma(u) = \\operatorname{tp}_\\Gamma(\\sigma_0(u))$, using Fact \\ref{fac: decomposition}(4) there must be infinitely many elements in $Y^p$ for which $\\sigma_0(u)$ is a handle, so we can choose one of them which is not contained in the finite tuple $\\bar{y}^p_{i,0}$).\n\t\\item If $z \\in  \\bar y_{i,0}^p$, then we have that $\\sigma_0$ fixes $z$ as well as the handle $u$ of $z$. In this case let $\\sigma(z)$ be equal to $z$.\n\t\\end{itemize}\nFinally, we define $\\sigma$ on each element of $\\bar y^\\iota$ as the identity map. Let $\\bar y'= \\bar y^{\\prime \\nu \\frown} \\sigma(\\bar{y}^p)^{\\frown} \\bar{y}^\\iota $. Then we have that  for all $y \\in \\bar{y}^{\\frown} \\bar{y}_{i,0}$:\n\\begin{enumerate}\n\t\\item $\\sigma$ is well defined;\n\\item $\\sigma$ fixes all elements in $\\bar{y}_{i,0}$;\n\\item $\\sigma$ respects types and handles by construction;\n\\item $\\operatorname{tp}_{\\Gamma}(\\bar{y}^{\\nu}, \\bar{y}^{\\nu}_{i,0}) = \\operatorname{tp}_{\\Gamma}(\\sigma(\\bar{y}^{\\nu}, \\bar{y}^{\\nu}_{i,0}))$ as $\\sigma(y) = \\sigma_0(y)$ for all $y \\in \\bar{y}^{\\nu \\frown} \\bar{y}^\\nu_{i,0}$.\n\\end{enumerate}\n\nNow consider $\\bar{m}$ and $(\\bar{m}_{i,j} : i,j \\in \\kappa)$ as tuples of elements in $\\langle M \\rangle$, which is a model of the stable theory $\\operatorname{Th}(\\langle M\\rangle)$. Moreover, as $(\\bar{m}_{i,j} : i,j \\in \\kappa)$ has $L_G$-mutually indiscernible rows and $\\operatorname{Th}(\\langle M \\rangle)$ eliminates quantifiers, $(\\bar{m}_{i,j} : i,j \\in \\kappa)$ has mutually indiscernible rows in the sense of $\\operatorname{Th}(\\langle M \\rangle)$. Hence, by Fact \\ref{fac: char of NTP2} again, there is some $\\beta \\in \\kappa$ such that for each $i > \\beta$ there is some $\\tau \\in \\operatorname{Aut}(\\langle M \\rangle)$ fixing $\\bar{m}_{i,0}$ and such that $(\\bar{m}_{i,j} : j \\in \\kappa)$ is indiscernible over $\\bar{m}' := \\tau( \\bar{m})$. \n\nFix some $i > \\max\\{\\alpha, \\beta\\}$ and let $\\bar y'$ and  $\\bar m'$ be chosen as above. Then by Lemma \\ref{Lem_GlueAut} we find an automorphism of $G$ which maps  $\\bar{y} \\bar{m}^{\\frown} \\bar{y}_{i,0} \\bar{m}_{i,0}$ to $\\bar{y}' (\\bar{m}')^{\\frown} \\bar{y}_{i,0} \\bar{m}_{i,0}$, hence \n$$\\operatorname{tp}_G(\\bar{y}' \\bar{m}' \/ \\bar{y}_{i,0} \\bar{m}_{i,0}) = \\operatorname{tp}_G(\\bar{y} \\bar{m} \/ \\bar{y}_{i,0} \\bar{m}_{i,0}).$$ \nIn particular, $G \\models \\theta(\\bar{y}' \\bar{m}' , \\bar{y}_{i,0} \\bar{m}_{i,0})$. We will show that\n$$\\operatorname{tp}_G( \\bar{y}_{i,0} \\bar{m}_{i,0}\/ \\bar{y}' \\bar{m}') = \\operatorname{tp}_G(\\bar{y}_{i,1} \\bar{m}_{i,1}\/ \\bar{y}' \\bar{m}'),$$\n which would then contradict the assumption that $\\{ \\theta(x', \\bar{y}_{i,j} \\bar{m}_{i,j}) : j \\in \\kappa \\}$ is $2$-inconsistent.\n\nWe show that sending    $ \\bar y' \\bar{y}_{i,0}$ to $ \\bar y'\\bar{y}_{i,1}$ is a well-defined bijection $f_0$. The only thing to check is that if the $n$-th element $z$ of $\\bar y_{i,0}$ is an element of $\\bar y'$, then the $n$-th element of $\\bar y_{i,1}$ is equal to $z$. This is true as by construction  we have that the sequence $(\\bar y_{i,j}: j\\in \\kappa)$ is indiscernible over $\\bar y'\\cap  \\bar y_{i,0}$ (as $\\bar y'\\cap  \\bar y_{i,0} = \\bar y \\cap  \\bar y_{i,0}$ by construction, and $(\\bar y_{i,j}: j\\in \\kappa)$ is indiscernible over $\\bar y \\cap  \\bar y_{i,0}$ by assumption). Moreover, we have the following properties for $f_0$:\n\\begin{enumerate}\n\\item $f_0$ fixes all elements in $\\bar{y}'$ (by construction);\n\\item $f_0$ respects types and handles (by construction);\n\\item $\\operatorname{tp}_{\\Gamma}(\\bar{y}^{\\prime \\nu}, \\bar{y}^{\\nu}_{i,0}) = \\operatorname{tp}_{\\Gamma}(f_0(\\bar{y}^{\\prime \\nu}, \\bar{y}^{\\nu}_{i,0}))$ (since by the choice of $\\bar{y}^{\\prime \\nu}$ above, we have that $(\\bar{y}^\\nu_{i,j}: j\\in \\kappa)$ is indiscernible over $\\bar{y}^{\\prime \\nu}$ in $\\Gamma(G)$).\n\\end{enumerate}\n\n\n\n\nSimilarly, by the choice of $\\bar{m}'$ above, the sequence $(\\bar{m}_{i,j} : j \\in \\kappa)$ is indiscernible over $\\bar{m}'$, so $\\operatorname{tp}_{\\langle M \\rangle}(\\bar{m}_{i,0}, \\bar {m}') = \\operatorname{tp}_{\\langle M \\rangle}(\\bar{m}_{i,1}, \\bar {m}') $\n\nAgain, Lemma \\ref{Lem_GlueAut} gives us an automorphism of $G$ sending $\\bar{y}_{i,0} \\bar{m}_{i,0}$ to $\\bar{y}_{i,1} \\bar{m}_{i,1}$ and fixing $\\bar{y}' \\bar{m}'$, as wanted.\n\n\n\n\n\\end{proof}\n\n\\begin{remark}\nA slight modification of the same proof shows that $\\operatorname{Th}(G(C))$ is \\emph{strong} if and only if $\\operatorname{Th}(C)$ is strong (see \\cite[Sections 2 and 3]{chernikov2014theories} for the relevant definitions).\n\\end{remark} \n\nHowever, since in the proof we have to throw away a finite, but unknown number of rows, this leaves the following problem. \n\\begin{problem}\nAssume that $\\operatorname{Th}(C)$ is of finite burden. Does $\\operatorname{Th}(G(C))$ also have to be of finite burden?\n\\end{problem}\n\nFinally, it would be interesting to investigate what other properties from generalized stability are preserved by Mekler's construction. For example:\n\n\\begin{conj}\nIf $\\operatorname{Th}(C)$ is NSOP$_1$ then $\\operatorname{Th}(G(C))$ is also NSOP$_1$.\\end{conj}\n\nWe expect that this could be verified using the methods of this paper and the criterion from \\cite{chernikov2016model} and \\cite{kaplan2017kim}.\n\n\n\\if 0\n\n\\subsection{A criterion for $2$-dependence}\n\nWe want to provide a formula-free characterization of $n$-dependence\nwhich doesn't include any assumption of indiscernibility of the witnessing\nsequence over the additional parameters (as it is the case in my paper\nwith Kota and Daniel). We can do it for $2$-dependence under some\nset-theoretic assumption.\n\\begin{lemma}\n\\label{lem: basic no shattering}Let $\\phi\\left(x;y_{1},y_{2}\\right)$\nbe $2$-dependent. Then there is some $n\\in\\mathbb{N}$ such that\nfor any $c\\in\\mathbb{M}_{x}$ and $I\\subseteq\\mathbb{M}_{y_{1}},J\\subseteq\\mathbb{M}_{y_{2}}$\nendless mutually indiscernible sequences, for any $A\\subseteq I$\nof size $>n$ there is some $b_{A}\\in J$ such that $A$ cannot be\nshattered by the family $\\left\\{ \\phi\\left(c,y_{1},b\\right):b\\in J,b>b_{A}\\right\\} $.\\end{lemma}\n\\begin{proof}\nAssume that $I,J$ are endless mutually indiscernible sequences and\n$c$ is such that the conclusion is not satisfied for any $n\\in\\omega$.\nLet $D\\subseteq I\\times J$ be any finite set. Let $a_{1}<\\ldots<a_{m}$\nand $b_{1}<\\ldots<b_{n}$ list the projections of $D$ on $I$ and\n$J$, respectively. By assumption, there is some $A\\subseteq I$ of\nsize $m$ such that for any $b'\\in B$, $A$ is shattered by the family\n$\\left\\{ \\phi\\left(c,y_{1},b\\right):b\\in J,b>b'\\right\\} $. List $A$\nas $a_{1}'<\\ldots<a_{m}'$. Then we can choose some $b_{1}'<\\ldots<b_{n}'\\in J$\nsuch that $\\models\\phi\\left(c,a'_{i},b_{j}'\\right)\\iff\\left(a_{i},b_{j}\\right)\\in D$.\nAs $I,J$ are mutually indiscernible, taking an automorphism of $\\mathbb{M}$\nsending $a_{i}'$ to $a_{i}$ and $b_{j}'$ to $b_{j}$, for all $1\\leq i\\leq m,1\\leq j\\leq n$,\n$c$ is sent to some $c_{D}$ such that $\\models\\phi\\left(c_{D},a_{i},b_{j}\\right)\\iff\\left(a_{i},b_{j}\\right)\\in D$.\nThis implies that $\\phi\\left(x;y_{1},y_{2}\\right)$ is not $2$-dependent,\na contradiction. Hence the conclusion holds for $c,I,J$ for some\n$n$.\n\nBy compactness it is not hard to conclude that $n$ can be chosen\ndepending only on $\\phi$ (and not on $I,J,c$).\n\\end{proof}\nWe will need the following lemma (originally from Shelah, with simplifications\nby Adler and Casanovas, see e.g. ***)\n\\begin{fact}\n\\label{fact: ded lemma}If $\\kappa$ is an infinite cardinal, $\\mathcal{F}\\subseteq2^{\\kappa}$\nand $\\left|\\mathcal{F}\\right|>\\operatorname{ded}\\kappa$, then for each $n\\in\\omega$\nthere is some $S\\subseteq\\kappa$ such that $\\left|S\\right|=n$ and\n$\\mathcal{F}\\restriction S=2^{S}$.\n\\end{fact}\nGiven sets $A\\subseteq\\mathbb{M}_{x}$, $B\\subseteq\\mathbb{M}_{y}$\nand a formula $\\phi\\left(x,y\\right)\\in\\mathcal{L}$, we denote by\n$S_{\\phi,B}\\left(A\\right)$ the set of all $\\phi$-types over $A$\nrealized in $B$, and by $S_{B}\\left(A\\right)$ the set of all complete\ntypes over $A$ realized in $B$.\n\\begin{prop}\n\\label{prop: few types on a tail}Let $T$ be $2$-dependent, let\n$\\kappa\\geq\\left|T\\right|$ be an infinite cardinal, and let $\\lambda>\\kappa$\nbe a regular cardinal. Then for any mutually indiscernible sequences\n$I=\\left(a_{i}:i\\in\\kappa\\right),J=\\left(b_{j}:j\\in\\lambda\\right)$\nof finite tuples and a finite tuple $c$, there is some $\\beta\\in\\lambda$\nsuch that $\\left|S_{J_{>\\beta}}\\left(Ic\\right)\\right|\\leq\\left(\\operatorname{ded}\\kappa\\right)^{\\left|T\\right|}$.\\end{prop}\n\\begin{proof}\nLet $I,J$ and $c$ be given. We will show that for each $\\phi\\left(x,y_{1},y_{2}\\right)\\in\\mathcal{L}$\nthere is some $\\beta_{\\phi}\\in\\lambda$ such that $\\left|S_{\\phi,J_{>\\beta_{\\phi}}}\\left(Ic\\right)\\right|\\leq\\operatorname{ded}\\kappa$.\nThis is enough, as then we can take any $\\beta\\in\\lambda$ with $\\beta>\\beta_{\\phi}$\nfor all $\\phi\\in\\mathcal{L}$ (possible as $\\lambda=\\operatorname{cf}\\left(\\lambda\\right)>\\left|T\\right|$),\nand $\\left|S_{J_{>\\beta}}\\left(Ic\\right)\\right|\\leq\\left|\\prod_{\\phi\\in\\mathcal{L}}S_{\\phi,J_{>\\beta_{\\phi}}}\\left(Ic\\right)\\right|\\leq\\left(\\operatorname{ded}\\kappa\\right)^{\\left|T\\right|}$.\n\nSo let $\\phi\\in\\mathcal{L}$ be fixed, and assume that for any $\\beta\\in\\lambda$,\n$\\left|S_{\\phi,J_{>\\beta_{\\phi}}}\\left(Ic\\right)\\right|>\\operatorname{ded}\\kappa$.\nThen by Fact \\ref{fact: ded lemma}, considering $\\mathcal{F}=\\left\\{ f_{p}:p\\in S_{\\phi,J_{>\\beta_{\\phi}}}\\left(Ic\\right)\\right\\} $\n(where $f_{p}\\in2^{\\kappa}$ is given by $f_{p}\\left(\\alpha\\right)=1\\iff\\phi\\left(c,a_{\\alpha},y_{2}\\right)\\in p$,\nfor all $\\alpha\\in\\kappa$), for any $n\\in\\omega$ there is \\emph{some}\n$S\\subseteq I$, $\\left|S\\right|=n$, such that $S$ is shattered\nby the family $\\left\\{ \\phi\\left(c,y_{1},b_{j}\\right):j\\in\\lambda,j>\\beta\\right\\} $.\nUsing regularity of $\\lambda$, we can choose by transfinite induction\na strictly increasing sequence $\\left(\\beta_{\\alpha}:\\alpha\\in\\lambda\\right)$\nwith $\\beta_{\\alpha}\\in\\lambda$ such that for each $\\alpha\\in\\lambda$\nthere is some $S_{\\alpha}\\subseteq I,\\left|S_{\\alpha}\\right|=n$ shattered\nby the family $\\left\\{ \\phi\\left(c,y_{1},b_{j}\\right):j\\in\\lambda,\\beta_{\\alpha}<j<\\beta_{\\alpha+1}\\right\\} $.\nAs $\\lambda>\\kappa=\\kappa^{n}$ is regular, passing to a subsequence\nwe may assume that there is some $S\\subseteq I,\\left|S\\right|=n$\nsuch that $S_{\\alpha}=S$ for all $\\alpha\\in\\lambda$, i.e. this set\n$S$ can be shattered arbitrarily far into the sequence. Now by Lemma\n\\ref{lem: basic no shattering}, this contradicts $2$-dependence\nof $\\phi$ if we take $n$ large enough.\\end{proof}\n\\begin{remark}\nWe can also give a finitary counterpart, with polynomial bound in\nplace of $\\operatorname{ded}$.\n\\end{remark}\n\n\\begin{lemma}\n\\label{lem: extension exists}For any cardinal $\\kappa$ and any regular\ncardinal $\\lambda\\geq2^{\\kappa}$ there is a bipartite graph $\\mathcal{G}_{\\kappa,\\lambda}=\\left(\\kappa,\\lambda,E\\right)$\nsatisfying the following: for any sets $A,A'\\subseteq\\kappa$ with\n$A\\cap A'=\\emptyset$ and $b\\in\\lambda$ there is some $b^{*}\\in\\lambda$,\n$b^{*}>b$ satisfying $\\bigwedge_{a\\in A}E\\left(a,b^{*}\\right)\\land\\bigwedge_{a'\\in A'}E\\left(a',b^{*}\\right)$.\\end{lemma}\n\\begin{proof}\nLet $\\lambda\\geq2^{\\kappa}$ be any regular cardinal. Let \n\\[\nD:=\\left\\{ \\left(A,A',b\\right):A,A'\\subseteq\\kappa,\\,A\\cap A'=\\emptyset,\\,b\\in\\lambda\\right\\} \\mbox{.}\n\\]\nThen $\\left|D\\right|\\leq\\lambda$ by assumption, let's enumerate it\nas $\\left(\\left(A_{\\alpha},A'_{\\alpha},b_{\\alpha}\\right):\\alpha<\\lambda\\right)$.\nWe define $E_{\\alpha}\\subseteq\\kappa\\times\\lambda$ by transfinite\ninduction on $\\alpha<\\lambda$. On step $\\alpha$, we choose some\n$c_{\\alpha}\\in\\lambda$ such that $c_{\\alpha}>\\left\\{ b_{\\beta},c_{\\beta}:\\beta<\\alpha\\right\\} $\n--- possible by regularity of $\\lambda$, and we take $E_{\\alpha}:=\\left\\{ \\left(a,c_{\\alpha}\\right):a\\in A_{\\alpha}\\right\\} $.\nLet $E:=\\bigsqcup_{\\alpha<\\lambda}E_{\\alpha}$ --- it satisfies the\nrequirement by construction.\\end{proof}\n\\begin{defn}\nWe say that a theory $T$ is \\emph{globally $2$-dependent} if there\nare cardinals $\\kappa\\leq\\lambda$ as above such that the following\nholds. Given any mutually indiscernible sequences $I=\\left(a_{i}:i\\in\\kappa\\right),J=\\left(b_{j}:j\\in\\lambda\\right)$\nof finite tuples and a finite tuple $c$, if $\\mathcal{G}_{\\kappa,\\lambda}$\nis as above the there are some $i\\in\\kappa$ and $j,j'\\in\\lambda$\nsuch that $ca_{i}b_{j}\\equiv ca_{i}b_{j'}$ but $E\\left(i,j\\right)\\land\\neg E\\left(i,j'\\right)$\nholds.\n\\end{defn}\nSo the idea is that $T$ is globally $2$-dependent if on mutually\nindiscernible sequences, we cannot distinguish the edges from the\nnon-edges of a random graph not only by any single formula formula,\nbut also by complete types.\n\\begin{remark}\nIf $T$ is not $2$-dependent, then it is not globally $2$-dependent.\\end{remark}\n\\begin{proof}\nLet $\\phi\\left(x,y_{1},y_{2}\\right)$ be a formula witnessing failure\nof $2$-dependence. Then for any $\\kappa,\\lambda$ we can find some\nmutually indiscernible sequences $I,J$ such that the family $\\left\\{ \\phi\\left(c,y_{1},y_{2}\\right):c\\in\\mathbb{M}\\right\\} $\nshatters $I\\times J$. In particular, we can find $c$ such that $\\mathbb{M}\\models\\phi\\left(c,a_{i},b_{j}\\right)\\iff\\mathcal{G}_{\\kappa,\\lambda}\\models E\\left(a_{i},b_{j}\\right)$,\ncontradicting global $2$-dependence.\\end{proof}\n\\begin{prop}\nLet $T$ be a countable $2$-dependent theory and assume that there\nis some cardinal $\\kappa$ such that $\\operatorname{ded}\\kappa<2^{\\kappa}$. Then\n$T$ is globally $2$-dependent.\\end{prop}\n\\begin{proof}\nFix such a $\\kappa$, and let $\\lambda$ be any regular cardinal $\\geq2^{\\kappa}$.\nLet $\\mathcal{G}_{\\kappa,\\lambda}$ be as given by Lemma \\ref{lem: extension exists}.\nLet $I,J$ and $c$as above be given. By Proposition \\ref{prop: few types on a tail},\nthere is some $\\beta\\in\\lambda$ such that $\\left|S_{J_{>\\beta}}\\left(I\\right)\\right|\\leq\\left(\\operatorname{ded}\\kappa\\right)^{\\aleph_{0}}$.\nOn the other hand, by definition of $\\mathcal{G}_{\\kappa,\\lambda}$,\nwe still have $\\left|S_{E,\\left\\{ \\alpha\\in\\lambda:\\alpha>\\beta\\right\\} }\\left(\\kappa\\right)\\right|=2^{\\kappa}>\\left(\\operatorname{ded}\\kappa\\right)^{\\aleph_{0}}$\nby assumption. Then we can find some $j,j'\\in\\lambda$ such that $\\operatorname{tp}_{E}\\left(j\/\\kappa\\right)\\neq\\operatorname{tp}_{E}\\left(j'\/\\kappa\\right)$\nbut $\\operatorname{tp}\\left(b_{j}\/Ic\\right)=\\operatorname{tp}\\left(b_{j'}\/Ic\\right)$. But then\nthere is some $i\\in\\kappa$ such that $E\\left(i,j\\right)\\leftrightarrow\\neg E\\left(i,j'\\right)$\nand still $b_{j}a_{i}c\\equiv b_{j'}a_{i}c$, as wanted.\n\\end{proof}\nBy a theorem of Mitchell, for any $\\kappa$ with $\\operatorname{cf}\\left(\\kappa\\right)>\\aleph_{0}$\nit is consistent that $\\operatorname{ded}\\kappa<2^{\\kappa}$. Hence this criterion\ncan always be used to determine $2$-dependence, in some model of\nZFC (and then sometimes set-theoretic absoluteness can be applied).\n\\begin{problem}\nIs it true that $n$-dependent implies globally $n$-dependent (defined\nanalogously), in ZFC, or at least consistently for $n>2$?\\end{problem}\n\\begin{remark}\nLet $T$ be $n$-dependent and $\\omega$-categorical. Then $T$ is\nglobally $n$-dependent (since every type in finitely many variables\nis equivalent to a formula, hence $n$-dependent and can't define\nthe random $n$-hypergraph on mutually indiscernible sequences).\n\\end{remark}\n\n\\fi\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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#1}}\n\\newcommand{\\dao}[1]{{\\color{docol} #1}}\n\\newcommand{\\commentxl}[1]{{\\color{ascol} \\footnotesize \\it [XL: #1]}}\n\\newcommand{\\discdo}[1]{{\\color{docol} DO: #1}}\n\\newcommand{\\discco}[1]{{\\color{cocol} CO: #1}}\n\\newcommand{\\discas}[1]{{\\color{ascol} AS: #1}}\n\\newcommand{\\todo}[1]{\\marginpar{\\bfseries !}{\\color{docol} \\small[TODO: #1]}}\n\n\n\\def\\Xint#1{\\mathchoice\n{\\XXint\\displaystyle\\textstyle{#1}}%\n{\\XXint\\textstyle\\scriptstyle{#1}}%\n{\\XXint\\scriptstyle\\scriptscriptstyle{#1}}%\n{\\XXint\\scriptscriptstyle\\scriptscriptstyle{#1}}%\n\\!\\int}\n\\def\\XXint#1#2#3{{\\setbox0=\\hbox{$#1{#2#3}{\\int}$ }\n\\vcenter{\\hbox{$#2#3$ }}\\kern-.6\\wd0}}\n\\def\\Xint-{\\Xint-}\n\\def\\Xint={\\Xint=}\n\\def\\Xint-{\\Xint-}\n\\title{Force-Based Atomistic\/Continuum Blending for Multilattices}\\thanks{DO was supported by the NSF PIRE Grant OISE-0967140 and NSF RTG program DMS-1344962.  XL was supported by the Simons Collaboration Grant with Award ID: 426935.  CO was supported by ERC Starting Grant 335120. }\n\\author{Derek Olson, Xingjie Li, Christoph Ortner, Brian Van Koten}\n\n\\begin{document}\n\n\\begin{abstract}\n   We formulate the blended force-based quasicontinuum (BQCF) method for\n   multilattices and develop rigorous error estimates in terms of the\n   approximation parameters: atomistic region, blending region and continuum\n   finite element mesh.  Balancing the approximation parameters yields a\n   convergent atomistic\/continuum multiscale method for multilattices with point\n   defects, including a rigorous convergence rate in terms of the computational\n   cost. The analysis is illustrated with numerical results for a Stone--Wales\n   defect in graphene.\n\\end{abstract}\n\n\\maketitle\n\n\n\n\n\\section{Introduction}\n\nA full twenty years has passed since the original proposal of the quasicontinuum method~\\cite{ortiz1996quasicontinuum} captivated the materials science community with the potential to model material phenomena spanning vastly different length scales.  The quasicontinuum (QC) method was among the first of the so-called atomistic-to-continuum (AtC) coupling algorithms which sought to bridge the gap between length scales from the nano to macroscale.  A remarkable number of these AtC methods have been proposed since (see e.g.~\\cite{tadmor2011,miller2009,acta.atc} for a thorough discussion of many of these), and recently a mathematical framework has begun to emerge to analyze and compare several of these methods for defects in crystalline materials comprised of a Bravais lattice.  Indeed, all three of the blended force-based quasicontinuum method (BQCF), blended energy-based quasicontinuum (BQCE), and blended ghost force correction (BGFC) methods have recently been analyzed in the context of a single defect in a two or three dimensional Bravais lattice~\\cite{blended2014,OrtnerZhang2014bgfc} as has the optimization-based AtC approach of~\\cite{olson2015}.  Analyses in two and three dimensional Bravais lattices also exist for the AtC method of~\\cite{lu2013}, but this has not yet been extended to allow for defects.  Meanwhile, the methods~\\cite{MakridakisMitsoudisRosakis2012,shapeev_2011,shapeev2012} have been shown to be consistent (or free of ghost forces) for pair potential interactions only.\n\n\nIn the present work, we resolve the long-standing challenge to develop a rigorous numerical analysis for AtC methods in the context of \\textit{multilattices}, which allows for more than one atom to be present in the unit cell of the crystal. This description includes important materials such as hcp metals, diamond structures, and recently discovered 2D materials such as graphene and hexagonal boron nitride.\n\nConcretely, we generalise the formulation and analysis of the blended force-based quasicontinuum (BQCF) method. Our main result is that, for a point defect in a homogeneous host crystal, the BQCF method for multilattices exhibits the same rate of convergence as in the Bravais lattice case. This is in sharp contrast with the blended energy-based quasicontinuum method for which a reduced convergence rate is expected in the multilattice setting \\cite{OrtnerZhang2014bgfc}.\n\nThe present work represents the first analysis that has been undertaken that remains valid for an AtC method which permits defects in a two or three dimensional multilattice. Even analyses of AtC methods for defect-free multilattices remain extremely sparse:  the homogenized QC method~\\cite{AbdulleLinShapeevII,AbdulleLinShapeev2012}, for example, only allows for dead load external forces while the cascading Cauchy--Born method was rigorously analyzed only in one-dimensional multilattices for phase transforming materials~\\cite{dobson2007multilattice}.\n\nAs its name entails, the BQCF method is a force-based AtC method where a hybrid force operator is constructed instead of a hybrid energy functional~\\cite{dobson_esaim,Shenoy:1999a,shilkrot2002coupled,lu2013,Bochev_08_MMS,Bochev_08_OUP}.  The primary advantage of force-based methods is that the forces can easily be defined in a way to avoid spurious interface effects (ghost forces); that is, the defect-free perfect crystal is a bona fide equilibrium configuration of the AtC force operator.  The cost of defining the BQCF method and other force-based methods to be free of ghost forces is that these force fields are no longer conservative, which creates significant challenges in their numerical analysis \\cite{dobson2010sharp, lu2014}.  The blended force-based methods, originally studied in~\\cite{li2012positive,Bochev_08_MMS,Bochev_08_OUP, lu2013}, seek to overcome this problem by a smooth blending between atomistic and continuum forces over a region called the blending, overlap, or handshake region.  Similar force-based blending methods have also been applied to coupling peridynamics with classical elasticity~\\cite{seleson2013}.\n\nAn alternative to the force-based paradigm is the energy-based paradigm where a global, hybrid energy is defined which is some combination of atomistic and continuum energies.  This encompasses the original quasicontinuum method and many other offshoots and ancestors~\\cite{ortiz1996quasicontinuum,xiao2004,abraham1998,E2006,shimokawa,datta2004,eidel2009,Bauman_08_CM}.  The peril of these methods is the aforementioned ghost forces, and it remains open to construct a general, ghost-force free, energy-based AtC method for Bravais lattices in two or three dimensions.  As such we do not concern ourselves with an energy-based AtC method for multilattices; however, see~\\cite{OrtnerZhang2014bgfc,shapeevMulti} for promising directions.\n\n\n\\subsection{Outline}\nWe begin in Section~\\ref{model} by formulating an atomistic model for a multilattice material describing a single point defect embedded in an infinite homogeneous crystal.  This is a canonical extension of the framework adopted for Bravais lattices in~\\cite{olson2015,blended2014,bqcf13,Ehrlacher2013,OrtnerZhang2014bgfc}.\n\nIn Section~\\ref{bqcf} we then formulate the BQCF method for this model and state our main results: (1) existence of a solution to the multilattice BQCF method and (2) a sharp error estimate. We also convert this error estimate to an estimate in terms of the computational complexity of the BQCF method in Section~\\ref{num} which in particular allows us to balance approximation parameters to obtain a formulation optimised for the error \/ cost ratio. We present a numerical verification of these rates by testing the method on a Stone--Wales defect in graphene.  The complexity estimates obtained for the BQCF method for point defects in multilattices match those estimates in~\\cite{blended2014} for Bravais lattices.\n\nFinally, Section~\\ref{analysis} covers the technical details needed to prove our main result, Theorem~\\ref{main_thm}.  These technical details can be seen as generalizations of the results of Bravais lattices, and the primary new component is having to account for shifts between atoms in the same unit cell.\n\n\\subsection{Notation}\nWe introduce new notation throughout the paper required to carry out the analysis.  For the convenience of the reader, we have listed many of these in Appendix~\\ref{sec:appnotation}.  Here, we briefly establish several basic conventions we make throughout.  We use $d$ and $n$ to denote the dimensions of the domain and range respectively, calligraphic fonts (e.g. $\\mathcal{L}, \\mathcal{M}$) to denote lattices, sans-serif fonts (e.g. ${\\sf F}, {\\sf G}$) for $n \\times d$ matrices, the lower case Greek letters $\\alpha, \\beta, \\gamma, \\delta, \\iota, \\chi$ are used as subscripts denoting atomic species, and the lower case Greek letters $\\rho, \\tau, \\sigma$ denote vectors (bond directions) between lattice sites.\n\nThe symbol $| \\cdot |$ is used to denote the $\\ell^2$ norm of a single vector in $\\mathbb{R}^m$, while $\\| \\cdot\\|$ is used to denote either an $\\ell^p$ or $L^p$ norm over a specified set.  We use $\\cdot$ for the dot product between two vectors, $\\otimes$ as the tensor product, and $:$ as the inner product on tensors.\n\nDerivatives of functions $f: \\mathbb{R}^d \\to \\mathbb{R}^n$ are denoted by $\\nabla f : \\mathbb{R}^d \\to \\mathbb{R}^{d \\times n}$ and higher order derivatives by $\\nabla^j f$.  Given $F:X \\to Y$ where $X$ and $Y$ are Banach spaces, we denote Fr\\'echet or Gateaux derivatives by $\\delta^j F$, $j$ indicating the order. We will most commonly interpret these derivatives as (multi-)linear forms and use them when $Y = \\mathbb{R}$, in which case we will then write the Gateaux derivatives as\n\\begin{align*}\n&\\<\\delta F(x), y\\> , \\quad x,y \\in X\\\\\n&\\<\\delta^2 F(x)z,y\\>, \\quad x,y,z \\in X \\quad \\mbox{and so on for higher order derivatives.}\n\\end{align*}\nWe reserve $D$ for specific finite difference operators (defined in \\eqref{finite_diff1} and \\eqref{finite_diff2}), and use $B_R$ to denote the ball of radius $R$ about the origin.\n\nWe use the modified Vinogradov notation $x \\lesssim y$ throughout the manuscript to mean there exists a positive constant $C$ such that $x \\leq Cy$. Where appropriate, we clarify what the constant $C$ is allowed to depend on; in particular if there is any dependence on approximation parameters then it will always be made explicit.\n\n\n\n\\section{Atomistic Model}\\label{model}\n\n\\subsection{Defect-free Multilattice}\nWe consider an infinite Bravais lattice, or simply a {\\em lattice}, {$\\mathcal{L}$, defined by\n\\[\n\\mathcal{L} := {\\sf F}\\mathbb{Z}^d, \\quad \\text{ for some } {\\sf F} \\in \\mathbb{R}^{d \\times d}, \\quad \\det({\\sf F}) = 1, \\quad \\mbox{and $d \\in \\{2,3\\},$}\n\\]\nwhere the requirement $\\det({\\sf F}) = 1$ is purely a notational convenience.  From a physical standpoint by taking symmetry into account, it can be shown that there are only 14 unique physical lattices in 3D and five in 2D (see e.g.~\\cite{tadmor2011}); however, we consider the lattice to merely be a mathematical framework. A multilattice is then obtained by associating a basis of $S$ atoms to each lattice site, and this is also referred to as a crystal when the Bravais lattice is interpreted as one of the unique physical lattices.}\n\nFor each site $\\xi \\in \\mathcal{L}$, these\n$S$ atoms are located inside the unit cell of $\\xi$ at positions $\\xi +\np_\\alpha^{\\rm ref}$ for $p_\\alpha^{\\rm ref} \\in \\mathbb{R}^d$ and $\\alpha = 0, \\ldots, S-1$.  The\nmultilattice is then defined by\n\\[\n\\mathcal{M} := \\bigcup_{\\alpha = 0}^{S-1}\\mathcal{L} + p_\\alpha^{\\rm ref}.\n\\]\nWe call each $\\mathcal{L} + p_\\alpha^{\\rm ref}$ a sublattice; {{here the addition ``+'' means a translation of the lattice $\\mathcal{L}$ by the\nvector $p_{\\alpha}^{\\rm ref}$.}} Without loss of generality, we further assume $p_0^{\\rm ref} = 0$ (one atom is always located at a lattice site).  Furthermore, we make the distinction between a lattice site, which we use to refer to a site in the Bravais lattice, $\\mathcal{L}$, and an atom which is an element in the multilattice $\\mathcal{M}$.\n\nTwo simple examples of multilattices are shown in Figure~\\ref{fig:multilattice}\nincluding the 2D hexagonal lattice (e.g., graphene) for which\n\\begin{equation}\\label{graph_param}\n\\mathcal{L} = a_0\\begin{pmatrix} \\sqrt{3} &\\sqrt{3}\/2 \\\\ 0 &3\/2\\end{pmatrix}\\mathbb{Z}^2, \\quad S = 2, \\quad p_0 = \\begin{pmatrix} 0 \\\\ 0\\end{pmatrix}, \\quad p_1 = a_0\\begin{pmatrix} \\sqrt{3}\/2 \\\\ 1\/2\\end{pmatrix}, \\quad a_0 = \\frac{\\sqrt{2}}{3^{3\/4}}.\n\\end{equation}\n(The $a_0 = \\frac{\\sqrt{2}}{3^{3\/4}}$ prefactor is due to the normalisation $\\det({\\sf F}) = 1$.)\n\n\\begin{figure}\n\\subfigure[2D graphene: the dashed circles indicate the interaction\n   neighbourhoods of the highlighted atoms.]{\n   \\includegraphics[width=0.42\\textwidth]{rep_graphene}}\n\\qquad\n\\subfigure[3D rock salt: the interior cube represents a possible\n      choice of unit cell.]{\n   \\includegraphics[width=0.42\\textwidth]{fcc_rock_salt1}}\n\\caption{Examples of multilattice structures.} \\label{fig:multilattice}\n\\end{figure}\n\nFor each species of atom, we define the deformation field $y_\\alpha(\\xi)$ as the deformation of the atom of species $\\alpha$ at site $\\xi$.  We note that $y_\\alpha:\\mathcal{L} \\to \\mathbb{R}^n$ where the range dimension $n \\in \\{2,3\\}$ may be different than the domain dimension $d$ to allow, e.g., for out of plane displacements in $2D$. {However, we remark that our later assumptions on stability of the multilattice (Assumption~\\ref{assumption2}) will place a restriction on the out of plane behavior; for example bending, or rippling, cannot currently be incorporated into the analysis.  We further discuss the issues involved in this in our concluding discussion, Section~\\ref{discussion}.}  In the case of these out of plane displacements, we will use $\\xi \\in \\mathbb{R}^2$ as both a vector in $\\mathbb{R}^2$ and as the vector $\\begin{pmatrix} \\xi \\\\ 0 \\end{pmatrix} \\in \\mathbb{R}^3$.  (We remark that though we will not consider dislocations, we could also consider $n = 1$ for an anti-plane screw dislocation model by fixing a second coordinate to be constant in this framework.)\n\nThe set of all sublattice deformations is denoted by $\\bm{y}:= (y_{\\alpha})_{\\alpha=0}^{S-1}$ and displacements by $\\bm{u}:= (u_{\\alpha})_{\\alpha=0}^{s-1}$. Equivalently we can describe the kinematics of a multilattice by a pair $(Y, \\bm{p})$ where $Y : \\mathcal{L} \\to \\mathbb{R}^n$ is a deformation field and $p_0, \\dots, p_{S-1} : \\mathcal{L} \\to \\mathbb{R}^n$ are shift fields. The two descriptions are related by\n\\begin{align*}\nY(\\xi) =~ y_0(\\xi), \\quad p_\\alpha(\\xi) = y_\\alpha(\\xi) - y_0(\\xi); \\qquad \\mbox{and} \\qquad y_\\alpha(\\xi) =~ Y(\\xi) + p_\\alpha(\\xi),\n\\end{align*}\nand analogous expressions hold for displacements as well.\n\nWe now turn to a description of the energy. We will make the fundamental modeling\nassumption that the total potential energy of the system can be written as a sum\nof \\textit{site potentials}---that is,\n\\begin{equation}\\label{atDefEnergy}\n   \\hat{\\mathcal{E}}^{\\rm a}_{\\rm hom}(\\bm{y}) := \\sum_{\\xi \\in \\mathcal{L}} \\hat{V}(D\\bm{y}(\\xi)),\n\\end{equation}\nwhere the various new symbols introduced are specified in the following. \\dao{We also note that this assumption is not restrictive as almost any reasonable classical potential such as an $n$-body potential, pair functional, or bond-order potential may be written in this form.} \\dao{The main restriction is that long-range Coulomb interaction is excluded.}\n\nWe use $D\\bm{y}(\\xi)$ to denote the collection of finite differences (relative atom positions)\nneeded to compute the energy at site $\\xi$. More precisely, we specify a\n{\\em finite} set of triples\n\\begin{equation*}\n   \\mathcal{R} \\subset \\mathcal{L} \\times \\{0, 1, \\ldots, S-1\\} \\times \\{0, 1, \\ldots, S-1\\}\\setminus \\bigcup_{\\alpha = 0}^{S-1}\\{ (0\\alpha\\alpha)\\},\n\\end{equation*}\nand use\n\\begin{equation}\\label{finite_diff1}\n   D_{(\\rho \\alpha\\beta)}\\bm{y}(\\xi) := y_\\beta(\\xi + \\rho) - y_\\alpha(\\xi)\n\\end{equation}\nto denote the relative positions of species $\\beta$ at site $\\xi+\\rho$ and\nspecies $\\alpha$ at site $\\xi.$ The collection of finite differences, or\nfinite difference {\\dao{stencils}}, $D\\bm{y}$, is then defined by\n\\begin{equation}\\label{finite_diff2}\nD\\bm{y}(\\xi) := \\left(D_{(\\rho \\alpha\\beta)}\\bm{y}(\\xi)\\right)_{(\\rho \\alpha\\beta) \\in \\mathcal{R}}.\n\\end{equation}\nIn terms of $(Y,\\bm{p})$, this this notation becomes\n\\[\nD_{(\\rho \\alpha\\beta)}(Y,\\bm{p}) := Y(\\xi + \\rho) - Y(\\xi) + p_\\beta(\\xi+\\rho) - p_\\alpha(\\xi)\n\\quad \\text{and} \\quad\n D(Y,\\bm{p}) := \\big( D_{(\\rho \\alpha\\beta)} (Y,\\bm{p}))_{(\\rho \\alpha\\beta) \\in \\mathcal{R}}.\n\\]\nFor future reference we remark that we can write\n\\begin{equation*}\n   D_{(\\rho \\alpha\\beta)} \\bm{y} = D_\\rho y_\\beta(\\xi) + p_\\beta(\\xi)-p_\\alpha(\\xi),\n  \n  \n\\end{equation*}\nwhere $D_\\rho f(\\xi) := f(\\xi + \\rho) - f(\\xi)$.  Moreover, we define the set of lattice vectors in $\\mathcal{R}$ as\n\\[\n\\mathcal{R}_1 := \\{ \\rho \\in \\mathcal{L} : \\exists (\\rho \\alpha\\beta) \\in \\mathcal{R}\\}.\n\\]\n\nThe site potential is then a function $\\hat{V} : (\\mathbb{R}^n)^\\mathcal{R} \\to \\mathbb{R} \\cup \\{+\\infty \\}$, where $+\\infty$ allows for singularities in the potential (though we will later assume certain smoothness of the potential for convenience of the analysis).\n\nSince the homogeneous reference configuration, $\\bm{y}^{\\rm ref}$, defined by\n\\begin{equation}\\label{ref_config}\n\\bm{y}^{\\rm ref}_\\alpha(\\xi) := \\xi + p^{\\rm ref}_\\alpha,\n\\end{equation}\nfor constant $p_\\alpha^{\\rm ref} \\in \\mathbb{R}^n$\nyields infinite energy, {\\dao{(due to an infinite sum over constant values of the site potential in the reference configuration),}} we thus will consider an energy difference functional defined on displacements from the reference state instead of \\eqref{atDefEnergy}. For a displacement\n$\\bm{u} \\equiv (U, \\bm{p})$ from the reference state $\\bm{y}^{\\rm ref}$ let\n\\begin{equation*}\n   V(D\\bm{u}(\\xi)) = \\hat{V}(D(\\bm{y}^{\\rm ref} + \\bm{u})(\\xi)),\n\\end{equation*}\nand then the associated energy difference functional is defined by\n\\begin{equation}\\label{atDispEnergy}\n\\mathcal{E}^{\\rm a}_{\\rm hom}(\\bm{u}) := \\sum_{\\xi \\in \\mathcal{L}} V(D\\bm{u}(\\xi)) - V(0).\n\\end{equation}\nwhere $V(0)$ is a constant which will not affect minimization or force computations, so for simplicity, we \\dao{assume without loss of generality that $V(0)=0$}.  In Theorem~\\ref{well_defined} \\dao{below}, we recall a result \\dao{of~\\cite{olsonOrtner2016}} that characterizes for which displacements\\dao{,} $\\bm{u}$\\dao{,}\n$\\mathcal{E}^{\\rm a}_{\\rm hom}(\\bm{u})$ is well-defined.\n\nA convenient notation for derivatives of $V$ is the following: if  $(\\rho \\alpha\\beta), (\\tau\\gamma\\delta) \\in \\mathcal{R}$ and $\\bm{g} = (\\bm{g}_{(\\rho \\alpha\\beta)})_{(\\rho \\alpha\\beta) \\in \\mathcal{R}} \\in (\\mathbb{R}^n)^{\\mathcal{R}}$, we set\n\\begin{align*}\n[V_{,(\\rho \\alpha\\beta)}(\\bm{g})]_{i} :=~& \\frac{\\partial V(\\bm{g})}{\\partial \\bm{g}_{(\\rho \\alpha\\beta)}^i }, \\quad i = 1,\\ldots, n, \\\\\nV_{,(\\rho \\alpha\\beta)}(\\bm{g}) :=~& \\frac{\\partial V(\\bm{g})}{\\partial \\bm{g}_{(\\rho \\alpha\\beta)}}, \\\\\n[V_{,(\\rho \\alpha\\beta)(\\tau\\gamma\\delta)}(\\bm{g})]_{ij} :=~& \\frac{\\partial^2 V(\\bm{g})}{\\partial \\bm{g}_{(\\tau\\gamma\\delta)}^j \\partial \\bm{g}_{(\\rho \\alpha\\beta)}^i},  \\quad i,j = 1,\\ldots, n, \\\\\nV_{,(\\rho \\alpha\\beta)(\\tau\\gamma\\delta)}(\\bm{g}) :=~& \\frac{\\partial^2 V(\\bm{g})}{\\partial \\bm{g}_{(\\tau\\gamma\\delta)} \\partial \\bm{g}_{(\\rho \\alpha\\beta)}},\n\\end{align*}\nand {\\dao{note that this}} can be extended to derivatives of arbitrary order.  Furthermore, we adopt the convention that if $(\\rho \\alpha\\beta) \\notin \\mathcal{R}$, then $V_{,(\\rho \\alpha\\beta)} = 0$.\n\n\nThe following standing assumptions on the interaction range and site potentials are made.\n\\begin{assumption}\\label{assumption1}\n\\hspace{2em}\n\\begin{enumerate}\n\\item[(V.1)] The interaction range, $\\mathcal{R}$, satisfies\n\\begin{align*}\n    & \\mbox{\\dao{For each $\\alpha \\in \\{0,\\ldots,S-1\\}$, the set of vectors $\\rho$ such that $(\\rho\\alpha\\alpha) \\in \\mathcal{R}$ spans $\\mathbb{R}^d$,}} \\\\\n   & \\mbox{and } (0\\alpha\\beta) \\in \\mathcal{R} \\quad \\text{for all $\\alpha \\neq \\beta \\in \\{0,\\ldots,S-1\\}$ }.\n\\end{align*}\n\\item[(V.2)] $V$ is four times continuously differentiable with uniformly bounded derivatives and satisfies $V(0) = 0$ (for simplicity of notation).  Since $V:(\\mathbb{R}^n)^{\\mathcal{R}} \\to \\mathbb{R}$, the statement that $V$ has uniformly bounded derivatives means there exists $M$ such that for any multi-index $\\gamma$ with $|\\gamma| \\leq 4$, $|\\partial_\\gamma V| \\leq M$.\n\\end{enumerate}\n\\end{assumption}\n\nWe remark that (V.1) may always be met by enlarging the interaction range, \\dao{$\\mathcal{R}$}. \\co{On the other hand, (V.2) is made for simplicity of the analysis; it can be weakened to admit interatomic potentials with typical singularities under collisions of atoms, but this would introduce several additional technicalities in our analysis.}\n\nNext, we specify the function space over which $\\mathcal{E}_{\\rm hom}^{\\rm a}(\\bm{u})$ is defined, which can be achieved in several equivalent ways. A convenient route is by first defining a continuous, piecewise linear interpolant of an atomistic displacement.  Let $\\mathcal{T}_{\\rm a}$ be a simplicial decomposition of $\\mathcal{L}$ obtained as in~\\cite{blended2014}: first let $\\hat{T} := {\\rm conv} \\{0, e_1, e_2\\}$ \\dao{(where ${\\rm conv}$ represents the convex hull of a set)} be the unit triangle in $2D$ and $\\hat{T}_1, \\ldots, \\hat{T}_6$ six congruent tetrahedra in $3D$ that subdivide the unit cube \\dao{(see Figure 1 in~\\cite{blended2014} for an illustration in $3D$)} and then define\n\\begin{equation*}\\label{eq:Ta}\n\\mathcal{T}_{\\rm a} =\n\\begin{cases}\n\\{\\xi + {\\sf F} \\hat{T}, \\xi -{\\sf F}\\hat{T}: \\xi \\in \\mathcal{L} \\}, \\quad \\mbox{if $d = 2$,} \\\\\n\\{\\xi + \\dao{{\\sf F}}\\hat{T}_i: \\xi \\in \\mathcal{L}, i = 1, \\ldots, 6 \\}, \\quad \\mbox{if $d = 3$}.\n\\end{cases}\n\\end{equation*}\nWe will often refer to this as the atomistic triangulation or \\textit{fully refined} triangulation.  As noted before, we may always enlarge the interaction range, $\\mathcal{R}$, so we may assume without loss of generality that\n \\begin{equation*} \\label{assumption:mesh}\n\\dao{     \\text{if } {\\rm conv}\\{ \\xi, \\xi+\\rho \\} \\text{ is an edge of $\\mathcal{T}_{\\rm a}$,\n     then there exist $\\alpha, \\beta$ such that } (\\rho \\alpha\\beta) \\in \\mathcal{R}.}\n \\end{equation*}\n\nGiven a {\\helen{discrete set of displacement values}} $u: \\mathcal{L} \\to \\mathbb{R}^n$, we then denote the continuous, piecewise linear interpolant of $u$ with respect to $\\mathcal{T}_{\\rm a}$ by $Iu \\equiv \\bar{u}$. We will use both notations, \\dao{$Iu$ and $\\bar{u}$}, depending on which is notationally more convenient. Subsequently, we define the function space\n\\begin{equation*}\\label{disSpace}\n\\begin{split}\n\\mathcal{U} :=~& \\left\\{ \\bm{u} = (u_\\alpha)_{\\alpha = 0}^{S-1} : u_\\alpha:\\mathcal{L} \\to \\mathbb{R}^n, \\|\\bm{u}\\|_{\\rm a} < \\infty \\right\\}, \\, \\mbox{where} \\\\\n\\|\\bm{u}\\|_{\\rm a}^2 :=~& \\sum_{\\alpha = 0}^{S-1}\\|\\nabla Iu_\\alpha\\|_{L^2(\\mathbb{R}^d)}^2 + \\sum_{\\alpha \\neq \\beta}\\| Iu_\\alpha - Iu_\\beta\\|_{L^2(\\mathbb{R}^d)}^2.\n\\end{split}\n\\end{equation*}\n\n\nClearly, $\\|\\cdot\\|_{\\rm a}$ is not a norm on $\\mathcal{U}$ since $\\|\\bm{u}\\|_{\\rm a} = 0$ only implies that each $u_\\alpha(\\xi)$  is a constant independent of $\\alpha$.  However, $\\|\\cdot\\|_{\\rm a}$ is a semi-norm on $\\mathcal{U}$ and hence a true norm on the quotient space\n\\begin{equation*}\n   \\bm{\\mathcal{U}} := \\mathcal{U}\/\\mathbb{R}^n\n      := \\big\\{  \\{(u_\\alpha + C)_{\\alpha = 0}^{S-1} : C \\in \\mathbb{R}^n \\}\n               \\,:\\, \\bm{u} \\in \\mathcal{U} \\big\\}.\n\\end{equation*}\nSince the atomistic energy is invariant with respect to addition by constants, it is exactly this quotient space which we utilize as our function space.  {\\helen{We also note that $\\bm{u}$ and $(U,\\bm{p})$ are two equivalent descriptions for the displacements}, and an equivalent norm on this space which will be convenient in terms of the $(U,\\bm{p})$ description is}\n\\begin{equation*\n\\|(U,\\bm{p})\\|_{\\rm a} := \\|\\nabla IU\\|_{L^2(\\mathbb{R}^d)}^2 + \\sum_{\\alpha = 1}^{S-1} \\| Ip_\\alpha\\|_{L^2(\\mathbb{R}^d)}^2.\n\\end{equation*}\n\nA dense subspace of $\\mathcal{U}$ that we will use as a test function space is $\\bm{\\mathcal{U}}_0$ where\n\\begin{equation*}\\label{testSpace}\n\\begin{split}\n\\mathcal{U}_0 :=~& \\left\\{\\bm{u} {\\helen{\\in \\mathcal{U}}} : {\\rm supp}(\\nabla Iu_0), \\,  \\mbox{and} \\, {\\rm supp}(Iu_\\alpha - Iu_0) \\, \\mbox{are compact}\\right\\}, \\\\\n\\bm{\\mathcal{U}}_0 :=~& \\mathcal{U}_0\/\\mathbb{R}^n.\n\\end{split}\n\\end{equation*}\nAs proven in~\\cite{olsonOrtner2016}, this test space is dense in $\\bm{\\mathcal{U}}$.\n\\begin{lemma}\\cite[Lemma A.1]{olsonOrtner2016}\\label{lem:dense}\nThe quotient space $\\bm{\\mathcal{U}}_0$ is dense in $\\bm{\\mathcal{U}}$.\n\\end{lemma}\n\n\n\n\\subsection{Point Defect}\nWe\nnow introduce a framework to embed a point defect in a homogeneous\nmultilattice. This problem has been heavily used in analyzing and comparing\ndifferent AtC methods for simple lattices\nin~\\cite{olson2015,blended2014,acta.atc,OrtnerZhang2014bgfc} as it allows for a\nrange of non-trivial benchmark problems and serves as a first step in analyzing\nmore complicated scenarios such as interacting defects~\\cite{hudson2015}.  {\\helen{Point\ndefects can be}} thought of as zero-dimensional defects representing a\nchange to a single site in the lattice. Common examples include vacancies,\ninterstitials, substitutions, and in graphene, the Stone--Wales defect which we\nuse for our numerical verification.\n\nOur first task is to define an analog of $\\mathcal{E}^{\\rm a}_{\\rm hom}$ for point\ndefects, which is well-defined on the function space $\\bm{\\mathcal{U}}$.\nWe accomplish this through a site-dependent site potential, $V_\\xi$, which must take\ninto account the defective structure of the lattice near the defect core, \\dao{which we assume to be at or near the\norigin}. We then write the atomistic potential energy as\n\\begin{equation}\\label{defEnergy}\n   \\mathcal{E}^{\\rm a}(\\bm{u}) := \\sum_{\\xi \\in \\mathcal{L}}\n               V_\\xi(D\\bm{u}(\\xi)).\n\\end{equation}\n\nAs in Assumption~\\ref{assumption1}, we require certain smoothness of the site-dependent site potential in addition to homogeneity outside of a defect core.\n\\begin{assumption}\\label{assumptionSite}\n\\quad\n\\begin{enumerate}\n\\item[(V.3)] There exists $R_{\\rm def} > 0$ such that $V_\\xi \\equiv V$ for all \\dao{$|\\xi| \\geq R_{\\rm def}$}.\n\\item[(V.4)] Each $V_\\xi$ is four times continuously differentiable with uniformly bounded derivatives.\n\\end{enumerate}\n\\end{assumption}\n\nWe now recall from \\cite[Theorem 2.2]{olsonOrtner2016} that $\\mathcal{E}^{\\rm a}$\nand $\\mathcal{E}^{\\rm a}_{\\rm hom}$ are well-defined on $\\bm{\\mathcal{U}}$; \\dao{the main idea of the proof is that both are defined on displacements having compact support, and by density of $\\bm{\\mathcal{U}}_0$ in $\\bm{\\mathcal{U}}$, they may be uniquely extended by continuity to all of $\\bm{\\mathcal{U}}$}.\n\\begin{theorem}\\cite[Lemma 3.3]{olsonOrtner2016}\\label{well_defined}\nAssume the reference configuration $\\bm{y}^{\\rm ref}$ with $y^{\\rm ref}_\\alpha(\\xi) = \\xi + p^{\\rm ref}_\\alpha$ is an equilibrium configuration of the defect free energy meaning that\n\\begin{equation}\\label{ostrich1}\n\\sum_{\\xi \\in \\mathcal{L}} \\sum_{(\\rho \\alpha\\beta) \\in \\mathcal{R}} \\hat{V}_{,(\\rho \\alpha\\beta)}(D\\bm{y}^{\\rm ref}(\\xi)) \\cdot D\\bm{v}(\\xi) = 0, \\quad \\forall \\, \\bm{v} \\in \\bm{\\mathcal{U}}_0.\n\\end{equation}\nThen $\\mathcal{E}^{\\rm a}_{\\rm hom}(\\bm{u})$ and $\\mathcal{E}^{\\rm a}(\\bm{u})$ may be uniquely extended to continuous functions on $\\bm{\\mathcal{U}}$ which are ${\\rm C}^3$ (three times continuously differentiable) on $\\bm{\\mathcal{U}}$.\n\\end{theorem}\n\n\\begin{remark}\nThe condition~\\eqref{ostrich1} that the reference configuration be an equilibrium is equivalent to requiring the shifts are equilibrated within each cell.  See~\\cite[Lemma 9]{olsonOrtner2016} for details. Such reference configurations are thus straightforward to generate numerically.\n\\end{remark}\n\n\\medskip\n\nSince we will eventually be working with a finite domain on which there is no difference between the original functionals and their extensions, we make no distinction between an energy and its continuous extension.\n\nWe are now able to pose the defect equilibration problem which we wish to\napproximate with the BQCF method, {\\helen{that is, to find $\\bm{u}^\\infty \\in\\bm{\\mathcal{U}} $ such that}}\n\\begin{equation}\\label{def_problem}\n\\bm{u}^\\infty \\in \\arg\\min_{\\bm{u} \\in \\bm{\\mathcal{U}}} \\mathcal{E}^{\\rm a}(\\bm{u}),\n\\end{equation}\nwhere {\\helen{$\\arg\\min$}} represents the set of local minima of a functional.\n\nWhile Assumptions~\\ref{assumption1} and~\\ref{assumptionSite} can be readily\nweakened in various ways, the next assumption concerning existence and stability\nof a defect configuration minimizing $\\mathcal{E}^{\\rm a}$ is essential for our\nanalysis:\n\\begin{assumption}\\label{assumption2} (Strong Stability)\n   There exists a solution, $\\bm{u}^\\infty$, to~\\eqref{def_problem} and a constant $\\gamma_{\\rm a} > 0$\n   such that\n\\[\n\\<\\delta^2\\mathcal{E}^{\\rm a}(\\bm{u}^\\infty)\\bm{v},\\bm{v}\\> \\geq \\gamma_{\\rm a} \\|\\bm{v}\\|_{\\rm a}^2\n\\qquad \\forall \\bm{v} \\in \\bm{\\mathcal{U}}_0.\n\\]\n\\end{assumption}\n\n\\medskip\n\nProving Assumption~\\ref{assumption2} turns out to be notoriously difficult;\nindeed the only result of this kind we are aware of is for a special case of\na screw dislocation in a simple lattice~\\cite[Remark 3.2]{hudson2015} under anti-plane\ndeformation. Nevertheless, we expect it to hold for {\\em virtually all}\nrealistic defects and realistic interatomic potentials. We also mention\nthat it can be numerically checked \\textit{a posteriori} once the defect\nconfiguration has been computed.\n\n\n\nA useful consequence of Assumption~\\ref{assumption2} is the following\nregularity result, which is proven in \\cite{olsonOrtner2016} and\nextends the analogous simple lattice result~\\cite{Ehrlacher2013}. \\co{These decay rates will be an essential component for converting the BQCF\nerror estimates in terms of solution regularity that are presented in Section~\\ref{bqcf}\ninto complexity estimates that are numerically verified in Section~\\ref{num}.}\n\n\\begin{theorem}\\cite[Theorem 2.5]{olsonOrtner2016}\\label{decay_thm}\nFor $\\bm{\\rho} = \\rho_1 \\dots \\rho_k$, the defect solution $(U^\\infty, \\bm{p}^\\infty)$ satisfies\n\\begin{equation}\\label{decay_est}\n\\begin{split}\n|D_{\\bm{\\rho}} U^\\infty(\\xi)| \\lesssim~& (1 + |\\xi|)^{1-d-k}, \\quad \\mbox{for $1 \\leq k \\leq 3$}, \\\\\n|D_{\\bm{\\rho}} p_\\alpha^\\infty(\\xi)| \\lesssim~& (1+|\\xi|)^{-d-k}, \\quad \\mbox{for $0 \\leq k \\leq 2$, and all $\\alpha = 0, \\ldots, S-1$.}\n\\end{split}\n\\end{equation}\nThe implied constant is allowed to depend on the interaction range through the maximum of $|\\rho|$ for $\\rho \\in \\mathcal{R}_1$, the site potential, and $\\gamma_{\\rm a}$.\n\\end{theorem}\n\n\\medskip\n\nThese decay rates will be an essential component for converting the BQCF\nerror estimates in terms of solution regularity that are presented in Section~\\ref{bqcf}\ninto complexity estimates that are numerically verified in Section~\\ref{num}.\n\nSince we will compare discrete atomistic configurations\n with continuous finite element functions, it will be useful to reformulate\nTheorem~\\ref{decay_thm} in terms of gradients of smooth interpolants, which\nwe define in the next lemma (see~\\cite{blended2014} for further details and the proof).\n\n\n\\begin{lemma}\\label{smoothInterpolant}\nLet $u:\\mathcal{L} \\to \\mathbb{R}^n$, then there exists a unique function $\\tilde{I}u:\\mathbb{R}^d \\to \\mathbb{R}^n$ with  $\\tilde{I}u \\in {\\rm C}^{2,1}(\\mathbb{R}^d)$ such that\n\\begin{enumerate}\n\\item $\\tilde{I}u$ is multiquintic in $\\xi + {\\sf F}(0,1)^d$ for each $\\xi \\in \\mathcal{L}$.\n\\item Given any multiindex $\\gamma$ with $|\\gamma| \\leq 2$, the interpolant\n   satisfies $\\partial_\\gamma \\tilde{I}u(\\xi) = D^{{\\rm nn}}_\\gamma u(\\xi)$\n   where $D^{{\\rm nn}}_\\gamma$ are nearest-neighbor finite difference operators,\n\\begin{align*}\nD^{{\\rm nn},0}_i u(\\xi) :=~& u(\\xi), \\\\\nD^{{\\rm nn},1}_i u(\\xi) :=~& \\frac{1}{2}(u(\\xi + {\\sf F} e_i) - u(\\xi - {\\sf F} e_i)) \\quad (e_i \\mbox{ is the $i$th standard basis vector}), \\\\\nD^{{\\rm nn},2}_i u(\\xi) :=~& u(\\xi + {\\sf F} e_i) -2u(\\xi) + u(\\xi - {\\sf F} e_i), \\\\\nD^{{\\rm nn}}_\\gamma u(\\xi) :=~& D^{{\\rm nn},|\\gamma_1|}_{1}\\cdots D^{{\\rm nn},|\\gamma_d|}_{d}u(\\xi).\n\\end{align*}\n\\end{enumerate}\n\\end{lemma}\n\n\\medskip\n\nWe will apply $\\tilde{I}$ to both displacements and shifts using the notation\n\\[\n\\tilde{I}(U, \\bm{p}) = (\\tilde{I}U, \\tilde{I}\\bm{p}) = (\\tilde{U}, \\tilde{\\bm{p}}).\n\\]\nThen, combining Theorem~\\ref{decay_thm} and Lemma \\eqref{smoothInterpolant}\nyields the following result.\n\n\\begin{theorem}\\label{decay_thm1}\nThe defect solution $(U^\\infty, \\bm{p}^\\infty)$ satisfies\n\\begin{equation}\\label{decay_est_cont}\n\\begin{split}\n|\\nabla^j \\tilde{I}U^\\infty(x)| \\lesssim~& (1+|x|)^{1-d-j}, \\quad \\mbox{for $j = 1,2$}, \\\\\n|\\nabla^j \\tilde{p}_\\alpha^\\infty(x)| \\lesssim~& (1+|x|)^{-d-j}, \\quad \\mbox{for $j = 0,1,2$, and all $\\alpha = 0, \\ldots, S-1$,}\n\\end{split}\n\\end{equation}\nwhere the implied constant is again allowed to depend on the interaction range, the site potential, and $\\gamma_{\\rm a}$.\n\\end{theorem}\n\n\n\\section{BQCF Method Formulation and Main Results}\\label{bqcf}\n\nAny AtC approximation of the defect problem~\\eqref{def_problem}\nmust include the following ingredients: the atomistic and continuum\ndomains, a coarsened finite element mesh in the continuum region, a\nspecification of the continuum model, and finally and most importantly\na mechanism for coupling the atomistic and continuum components.\n\nWe define the atomistic and continuum domains for the multilattice BQCF method\nby making similar choices as in the BQCF method for Bravais\nlattices~\\cite{blended2014}. We first give an intuitive description of the\ndomains involved, but will (re-)define them again below after introducing the\n\\textit{blending function}. Choose a computational domain $\\Omega \\subset \\mathbb{R}^d$\nto be a (large) polygonal domain containing the origin (the defect). Fix a\n``defect core'' region $\\Omega_{\\rm core}$ such that, if $V_\\xi \\not\\equiv V$,\nthen $\\xi \\in \\Omega_{\\rm core}$.  Then take $\\Omega_{\\rm a}$, the atomistic domain,\nto be a polygonal domain with $\\Omega_{\\rm core} \\subset \\Omega_{\\rm a} \\subset\n\\Omega$, and set $\\Omega_{\\rm c}$, the continuum domain to be $\\Omega_{\\rm c} =\n\\Omega\\setminus\\Omega_{\\rm core}$. In blending methods, the atomistic and\ncontinuum domains overlap in a blending region $\\Omega_{\\rm b} = \\Omega_{\\rm c} \\cap\n\\Omega_{\\rm a}$ over which the atomistic and continuum forces will be blended.\n\nNext, we define a finite element mesh $\\mathcal{T}_h$ over $\\Omega$ with\nnodes $\\mathcal{N}_h$. For now we only require that the finite element mesh\nis fully refined over $\\Omega_{\\rm a}$, that is,\nif $T \\cap \\Omega_{\\rm a} \\neq \\emptyset$, then $T \\in \\mathcal{T}_h$ if and only if $T \\in \\mathcal{T}_{\\rm a}$, but we will state further assumptions in\nSection~\\ref{sec:approx_params}.\n\n\nThe continuum model we adopt is the Cauchy--Born model~\\cite{cauchy, born1954, ortiz1996quasicontinuum}, a nonlinear\nhyperelastic model, which is amenable to AtC couplings due to the definition of the\nstrain energy density function in terms of the atomistic potential $V$,\n\\[\nW_{\\rm CB}({\\sf G}, \\bm{p}) := V\\Big(({\\sf G} \\rho + p_\\beta - p_\\alpha)_{(\\rho \\alpha\\beta) \\in \\mathcal{R}}\\Big) \\quad \\mbox{for ${\\sf G} \\in \\mathbb{R}^{n \\times d}$ and $\\bm{p} \\in (\\mathbb{R}^n)^{S}$},\n\\]\nwithout resorting to any constitutive laws. {\\helen{We note that $G$ here is the deformation gradient of lattice sites in a unit cell while $\\bm{p}$\nare the displacements of shift vectors; in contrast with typical continuum treatments of multilattices, we maintain the shift vectors as degrees of freedom in the Cauchy--Born model and do not minimize them out.}}\n\nFor $W^{1,\\infty}$ displacement fields, $U$, and $L^\\infty$ shift fields, $\\bm{p}$,\nthis leads to a Cauchy--Born energy functional, formally (for now) defined by\n\\begin{equation*}\\label{cb_energy}\n\\mathcal{E}^{\\rm c}(U, \\bm{p}) := \\int_{\\mathbb{R}^d} W_{\\rm CB}(\\nabla U(x), \\bm{p}(x))\\, dx = \\int_{\\mathbb{R}^d} V\\big(\\nabla(U,\\bm{p})\\big)\\, dx\n\\end{equation*}\nwhere\n\\begin{equation*}\\label{cont_grad}\n\\nabla(U,\\bm{p}) := \\big(\\nabla_{(\\rho \\alpha\\beta)}(U,\\bm{p})\\big)_{(\\rho \\alpha\\beta) \\in \\mathcal{R}} := \\big(\\nabla_\\rho U + p_\\beta - p_\\alpha\\big)_{(\\rho \\alpha\\beta) \\in \\mathcal{R}}\n\\end{equation*}\nis a continuum variant of the atomistic \\dao{finite difference stencil\n\\begin{equation*}\nD(U,\\bm{p})(x) = \\big(D_{(\\rho \\alpha\\beta)}(U,\\bm{p})(x)\\big)_{(\\rho \\alpha\\beta) \\in \\mathcal{R}} := \\big(D_\\rho U(x) + p_\\beta(x+\\rho) - p_\\alpha(x)\\big)_{(\\rho \\alpha\\beta) \\in \\mathcal{R}}.\n\\end{equation*}}\n\nThe admissible finite element space we consider will be $\\mathcal{P}_1$ finite elements for both the displacements and the shifts subject to homogeneous boundary conditions.  However, we will again consider equivalence classes of finite element functions by taking a quotient space.  Thus, we define\n\\begin{equation*}\\label{fin_spaces}\n\\begin{split}\n\\mathcal{U}_h :=~& \\left\\{u \\in {\\rm C}^0(\\Omega) : u|_{T} \\in \\mathcal{P}_1(T), \\quad \\forall \\, T \\in \\mathcal{T}_h\\right\\}, \\\\\n\\bm{\\mathcal{U}}_h :=~& \\mathcal{U}_h \/ \\mathbb{R}^n, \\\\\n\\mathcal{U}_{h,0} :=~& \\left\\{u \\in {\\rm C}^0(\\mathbb{R}^d): u|_{T} \\in \\mathcal{P}_1(T), \\quad \\forall \\, T \\in \\mathcal{T}_h, u = 0 \\mbox{ on } \\mathbb{R}^d\\setminus\\Omega \\right\\}, \\\\\n\\bm{\\mathcal{U}}_{h,0} :=~& \\mathcal{U}_{h,0}\/ \\mathbb{R}^n, \\\\\n\\bm{\\mathcal{P}}_{h,0}\n:=~& \\dao{\\left\\{\\bm{p}=(p_0,\\dots,p_{S-1}): p_0=0, \\text{ and }p_1,\\dots,p_{S-1} \\in \\big(\\mathcal{U}_{h,0}\\big)^{S-1}\\right\\}}.\n\\end{split}\n\\end{equation*}\nThese spaces are endowed with the norm\n\\begin{equation*}\\label{eq:ml}\n\\|(U, \\bm{p})\\|_{\\rm ml}^2 := \\|\\nabla U \\|_{L^2(\\mathbb{R}^d)}^2 + \\sum_{\\alpha = 0}^{S-1}\\|p_\\alpha\\|^2_{L^2(\\mathbb{R}^d)} = \\|\\nabla U \\|_{L^2(\\mathbb{R}^d)}^2 + \\|\\bm{p}\\|^2_{L^2(\\mathbb{R}^d)},\n\\end{equation*}\nwhere $\\|\\bm{p}\\|^2_{L^2(\\mathbb{R}^d)} = \\sum_{\\alpha = 0}^{S-1}\\|p_\\alpha\\|^2_{L^2(\\mathbb{R}^d)}$ is used for brevity.  Along with the finite element space, we also introduce the standard piecewise linear finite element interpolant, $I_h$, defined as usual through $I_h u(\\nu) = u(\\nu)$ for $\\nu \\in \\mathcal{N}_h$.\n\nThe BQCF method is defined by blending forces on each degree of freedom, \\dao{$(\\nu,\\alpha) \\in \\mathcal{N}_h \\times \\{0,\\ldots,S-1\\}$}, where the forces are defined by a weighted average of atomistic and continuum forces:\n\\begin{equation}\\label{f_bqcf}\n\\mathcal{F}^{{\\rm bqcf}}_{\\nu,\\alpha}(U,\\bm{p}) :=  (1-\\varphi(\\nu))\\frac{ \\partial \\mathcal{E}^{\\rm a}(U,\\bm{p})}{\\partial u_\\alpha(\\nu)} + \\varphi(\\nu)\\frac{ \\partial \\mathcal{E}^{\\rm c}(U,\\bm{p})}{\\partial u_\\alpha(\\nu)},\n\\end{equation}\nwhere \\dao{the \\textit{blending function}, $\\varphi$, satisfies} $\\varphi \\in \\rm{C}^{2,1}(\\mathbb{R}^d)$ with $\\varphi = 0$ in $\\Omega_{\\rm core}$\nand $\\varphi = 1$ in $\\mathbb{R}^d \\setminus \\Omega_{\\rm a}$.  The BQCF method then seeks to solve $\\mathcal{F}^{{\\rm bqcf}}_{\\nu,\\alpha}(U,\\bm{p}) = 0$ for all $\\nu \\notin \\partial \\Omega$. Equivalently, we can write the force balance equations in weak form using the variational operator\n{\\helen{\n\\begin{align}\n\\<&\\mathcal{F}^{{\\rm bqcf}}(U,\\bm{p}), (W,\\bm{r})\\> \\nonumber\\\\\n&\\; := \\sum_{\\nu}\\sum_{\\alpha}\\mathcal{F}^{{\\rm bqcf}}_{\\nu,\\alpha}(U,\\bm{p})\\cdot \\left(W+r_{\\alpha}\\right)(\\nu)\\nonumber\\\\\n&\\; = \\sum_{\\nu}\\sum_{\\alpha} (1-\\varphi(\\nu))\\frac{ \\partial \\mathcal{E}^{\\rm a}(U,\\bm{p})}{\\partial u_\\alpha(\\nu)}\\cdot\\left(W+r_{\\alpha}\\right)(\\nu) + \\varphi(\\nu)\\frac{ \\partial \\mathcal{E}^{\\rm c}(U,\\bm{p})}{\\partial u_\\alpha(\\nu)}\\cdot\\left(W+r_{\\alpha}\\right)(\\nu)\\nonumber\\\\\n&\\; = \\<\\delta\\mathcal{E}^{\\rm a}(U,\\bm{p}),((1-\\varphi)W,(1-\\varphi)\\bm{r})\\>\\nonumber\\\\\n&\\qquad\\qquad+  \\<\\delta\\mathcal{E}^{\\rm c}(U,\\bm{p}),(I_h(\\varphi W),I_h(\\varphi \\bm{r}))\\>,\\label{v_bqcf}\n\\end{align}\n}}\nwhere the last equal sign comes from direct calculation.\nThe BQCF approximation to the defect optimization problem~\\eqref{def_problem}\nis then to {\\it find $(U, \\bm{p}) \\in \\bm{\\mathcal{U}}_{h,0} \\times \\bm{\\mathcal{P}}_{h,0}$\nsuch that }\n\\begin{equation}\\label{bqcf_approx}\n   \\<\\mathcal{F}^{{\\rm bqcf}}(U,\\bm{p}), (W,\\bm{r})\\>  = 0,\n    \\quad \\forall (W,\\bm{r}) \\in \\bm{\\mathcal{U}}_{h,0} \\times \\bm{\\mathcal{P}}_{h,0}.\n\\end{equation}\nThe variational formulation is preferred for the\nanalysis while the force-based formulation (from which the name BQCF is derived) is\npreferred for implementation.  The pointwise formulation~\\eqref{f_bqcf} was essentially how the original BQCF method was proposed for Bravais lattices~\\cite{badia2007force}, and this was analyzed in a finite-difference framework without defects for Bravais lattices in~\\cite{lu2013, li2012positive}.  The variational formulation~\\eqref{v_bqcf} was introduced in~\\cite{blended2014} for Bravais lattices, and its subsequent analysis led to one of the first complete analyses of an AtC method capable of modeling defects.\n\n\n\n\\subsection{Assumptions on the Approximation Parameters}\n\\label{sec:approx_params}\nWe now summarise the precise technical requirements on the approximation\nparameters, $\\varphi, \\Omega, \\Omega_{\\rm a}, \\Omega_{\\rm b}, \\Omega_{\\rm c}, \\mathcal{T}_h$,\nwhich will be analogous to those in \\cite{blended2014}.\n\nWe begin by summarising basic assumptions on the blending function:\n\\begin{enumerate}\n\\item $\\varphi \\in \\rm{C}^{2,1}$ and $0 \\leq \\varphi \\leq 1$\n\n\\item If $V_\\xi \\not\\equiv V$, then $\\varphi(\\xi) = 0$.  This means that\n   $\\varphi$ vanishes near any defect, hence the pure atomistic force is\n   employed in those regions.\n\n\\item There exists $K > 0$ such that $\\varphi(x) = 1$ if $|x| \\geq K$.\n      That is, $\\varphi$ is identically one far away from the defect.\n\\end{enumerate}\n\nAs the second step we specify the computational domain $\\Omega$ and its\ncorresponding partition $\\mathcal{T}_h$. {\\helen{First, we shall require that ${\\rm supp}(1-\\varphi)\\subset\\Omega $ always holds.}} To state the required properties\nfor $\\mathcal{T}_h$, we first precisely specify the sub-domains in terms of\n$\\varphi$ and $\\Omega$. Let\n\\begin{equation*}\n   r_{\\rm cut} := \\max \\{ |\\rho| :  (\\rho \\alpha\\beta) \\in \\mathcal{R} \\}\n\\end{equation*}\nbe an interaction cut-off radius, {let $r_{\\rm cell}$ be the radius of the smallest ball circumscribing the unit cell of $\\mathcal{L}$, and define $r_{\\rm buff} := \\max\\{ r_{\\rm cut}, r_{\\rm cell}\\}$}. Then we set\n\\begin{align*}\n\\Omega_{\\rm a} :=~& {\\rm supp}(1-\\varphi) + B_{4r_{\\rm buff}}, \\quad \\Omega_{\\rm b} :=~ {\\rm supp}(\\nabla \\varphi) + B_{4r_{\\rm buff}}, \\\\\n\\Omega_{\\rm c} :=~& {\\rm supp}(\\varphi) \\cap \\Omega + B_{4r_{\\rm buff}}, \\quad \\Omega_{\\rm core} :=~ \\Omega\\setminus\\Omega_{\\rm c}.\n\\end{align*}\nThe size and shape regularity of the various subdomains are parameterized\nin terms of inner and outer radii: for ${\\rm t} \\in \\{ {\\rm a}, {\\rm c}, {\\rm b}, {\\rm core}\\}$,\nwe set\n\\[\n   r_{\\rm t} := \\sup_{r}\\{r > 0: {\\helen{ B_{r} }} \\subset\n                        \\Omega_{\\rm t} \\cup \\Omega_{\\rm core} \\},\n   \\quad R_{\\rm t} := \\inf_{R} \\{R > 0: \\Omega_{\\rm t} \\subset {\\helen{B_{R}}}\\},\n\\]\n{\\helen{where we recall the notation $B_R$ to denote the ball of radius $R$ about the origin. }}  The corresponding outer and inner radii for the complete domain $\\Omega$\nare, respectively, denoted by $R_{\\rm o}$ and $r_{\\rm i}$:\n\\dao{\n\\[\n   r_{\\rm i} := \\sup_{r}\\{r > 0: {\\helen{ B_{r} }} \\subset\n                        \\Omega \\},\n   \\quad R_{\\rm o} := \\inf_{R} \\{R > 0: \\Omega \\subset {\\helen{B_{R}}}\\}.\n\\]\n}\nFinally, we define an overlapping exterior domain,\n\\begin{align*}\n   \\Omega_{\\rm ext} := \\mathbb{R}^d \\setminus {\\helen{ B_{r_{\\rm i}\/2}}},\n\\end{align*}\nwhich will be used to quantify the far-field error made by truncating\nto a finite computational domain.\n\n\\begin{figure}\n   \\centering\n   {\\includegraphics[width=0.45\\textwidth]{radiiMod}}\n   \\caption{A diagram showing a selected number of domains and their inner and outer radii.}\\label{fig:domains}\n\\end{figure}\n\nFor the sake of completeness, we now restate a crucial condition on the finite\nelement mesh:\n\\begin{enumerate}[resume]\n\\item  The finite element mesh is fully refined over $\\Omega_{\\rm a}$, that is,\n  if $T \\cap \\Omega_{\\rm a} \\neq \\emptyset$, then $T \\in \\mathcal{T}_h$ if and only if $T \\in \\mathcal{T}_{\\rm a}$.\n\\end{enumerate}\n\nTo conclude this discussion we note that only the blending function $\\varphi$\nand the finite element mesh $\\mathcal{T}_h$ are free approximation parameters,\nwhile the subdomains and corresponding radii are derived (in particular,\n$\\Omega = \\bigcup \\mathcal{T}_h$). In our analysis we will require bounds\non the ``shape regularity'' of $\\varphi$, $\\mathcal{T}_h$, and the domains defined above:\n\n\\begin{assumption} \\label{assumption-shapereg}\n   In addition to (1)--(4) there exist constants $C_{\\mathcal{T}_h}, C_\\varphi > 0$, which shall be\n   fixed throughout, such that\n   \\begin{align*}\n      \\|\\nabla^j \\varphi\\|_{L^\\infty} \\leq C_\\varphi R_{\\rm a}^{-j} \\qquad\n      \\text{for $j = 1,2,3$, \\quad and} \\qquad\n      \\max_{T \\in \\mathcal{T}_h} \\frac{\\sigma_T}{\\rho_T} \\leq C_{\\mathcal{T}_h},\n   \\end{align*}\n   where $\\sigma_T$ denotes the radius of the smallest ball\ncircumscribing $T$ and $\\rho_T$ the radius of the largest ball contained in $T$.  Defining the mesh size function\n\\[\nh(x) := \\max_{\\substack{T \\in \\mathcal{T}_h: \\\\ x \\in T}} \\sigma_T,\n\\]\nthere exists $s \\geq 1$ such that the mesh satisfies the growth condition\n\\[\n|h(x)| \\leq C_{\\mathcal{T}_h}\\Big(\\frac{|x|}{R_{{\\rm a}}}\\Big)^s, \\quad   |x| \\geq R_{\\rm a}.\n\\]\n\nMoreover, there exists  $C_{\\rm o} > 0$ and a positive integer $\\lambda$ such that\n\\begin{equation}\\label{def_constant_Co}\n  R_{\\rm o} \\leq C_{\\rm o} R_{\\rm core}^{\\lambda} \\quad \\mbox{\\dao{and} } \\quad \\frac{1}{4} R_{{\\rm a}} \\leq R_{\\rm core} \\leq \\frac{3}{4} R_{\\rm a}.\n\\end{equation}\n\\end{assumption}\n\nWhile $C_\\varphi$ will feature heavily in our analysis, the parameter\n$C_{\\mathcal{T}_h}$ will only enter implicitly in the form of constants in\ninterpolation error estimates.  The condition $\\frac{1}{4} R_{{\\rm a}} \\leq R_{\\rm core} \\leq \\frac{3}{4} R_{\\rm a}$ greatly simplifies the analysis.  It is likely this could be weakened by an extremely refined analysis as can be done in one dimension~\\cite{li2012positive}, but the asymptotic estimates obtained would be unchanged with the exception of an improved prefactor so we do not pursue this.  \\dao{Moreover, though one can generate blending functions which satisfy these assumptions using splines, we point out that in practical implementations one can relax the regularity requirements on the blending functions, and this has provided no loss in performance in simulations carried out for  lattices in~\\cite{bqcf13}.}\n\n\n\\subsection{Main Result}\n\\label{sec:main-result-subsec}\nOur main result concerns the existence of a solution to~\\eqref{bqcf_approx} and an estimate on the error committed.\n\\begin{theorem}\\label{main_thm}\n   Suppose that Assumptions~\\ref{assumption1},~\\ref{assumptionSite}, and~\\ref{assumption2} are valid.\n   Then there exists $R_{\\rm core}^*$ such that, for any approximation parameters\n   satisfying Assumption~\\ref{assumption-shapereg} as well as\n   $R_{\\rm core} \\geq R_{\\rm core}^*$,\n   %\n   there exists a solution $(U^{{\\rm bqcf}}, \\bm{p}^{{\\rm bqcf}}) \\in\n   \\bm{\\mathcal{U}}_{h,0} \\times \\bm{\\mathcal{P}}_{h,0}$ to the BQCF\n   equations~\\eqref{bqcf_approx} that satisfies\n\\begin{equation}\\label{main_estimate}\n\\begin{split}\n\\|\\nabla IU^\\infty -& \\nabla U^{{\\rm bqcf}} \\|_{L^2(\\mathbb{R}^d)} +\n \\| I \\bm{p}^\\infty - \\bm{p}^{{\\rm bqcf}}\\|_{L^2(\\mathbb{R}^d)}\n \\lesssim~ \\gamma_{\\rm tr} \\Big(\n   \\|h \\nabla^2 \\tilde{I}U^{\\infty}\\|_{L^2(\\Omega_{\\rm c})} \\\\\n   &\\qquad + \\|h \\nabla \\tilde{I}\\bm{p}^\\infty \\|_{L^2(\\Omega_{\\rm c})} +\n   \\|\\nabla\\tilde{I} {U}^{\\infty}\\|_{L^2(\\Omega_{\\rm ext})}\n   + \\|\\tilde{I} \\bm{p}^\\infty\\|_{L^2(\\Omega_{\\rm ext})}\\Big),\n\\end{split}\n\\end{equation}\nwhere\n\\[\n\\gamma_{\\rm tr} := \\begin{cases} &\\sqrt{1 + \\log(R_{\\rm o}\/R_{{\\rm a}})}, \\quad \\mbox{if $d = 2$,} \\\\\n                                 &1, \\quad \\mbox{if $d = 3$.} \\end{cases}\n\\]\nThe implied constant, as well as $R_{\\rm core}^*$, may depend on $C_\\varphi$ and $C_{\\mathcal{T}_h}$,\nthe interatomic potentials $V, V_\\xi$, the maximum of $|\\rho|$ for $\\rho \\in \\mathcal{R}_1$, and the stability constant, $\\gamma_{\\rm a}$.\n\\end{theorem}\n\n\\begin{remark}\n   The quantity $\\gamma_{\\rm tr}$ arises from a trace inequality that is needed when estimating interpolants on the\n   atomistic mesh in terms of interpolants on the continuum mesh, c.f.~[Lemma\n   4.6]\\cite{blended2014}.\n\\end{remark}\n\n\\medskip\n\n\nSection~\\ref{analysis} is devoted to proving Theorem~\\ref{main_thm}, but before\nwe embark on this, we first demonstrate how the error estimate can be combined\nwith the regularity estimates of Theorem~\\ref{decay_thm1} to yield an optimised\nBQCF scheme with balanced approximation parameters. This is followed by a\nnumerical test on a Stone--Wales defect in graphene, validating our theoretical\nconvergence rates.\n\n\\subsection{Optimal parameter choices}\nOnce we restrict ourselves to a Cauchy--Born energy with $\\mathcal{P}_1$ discretisation as the continuum model, the free parameters in the design of the BQCF method are the domain, $\\Omega$;\nblending function, $\\varphi$; and finite element mesh, $\\mathcal{T}_h$ in the\nsense that once these are set according to Section~\\ref{sec:approx_params},\nthen the BQCF method~\\eqref{bqcf_approx} is fully formulated.  Ideally, these\nparameters should be chosen in an optimal way so as to obtain the most efficient\nmethod.\n\n\nThe choice of blending function is, in the case of the BQCF method,\narbitrary as long as Assumption~\\ref{assumption-shapereg} is satisfied.\nThere are many choices to make for the blending\nfunction which meet these requirements, see e.g.~\\cite{bqce12}.\n\nThe finite element mesh and hence the choice of $\\Omega$ may, however, be optimized.\nThe key to choosing the finite element mesh and size of $\\Omega$ lies in applying the\ndecay results of Theorem~\\ref{decay_thm1} to our error estimate~\\eqref{main_estimate},~\\cite{acta.atc,bqcf13,bqce12}.  \\dao{In obtaining our optimized parameters, we do not provide rigorous proofs but instead use heuristic assumptions to arrive at approximate choices which can then be rigorously analyzed numerically.  To start, we}\n assume that the mesh size function $h(x)$ is radial, i.e.,\n$h(x) \\equiv h(|x|)$. Then, ignoring logarithmic factors in $\\gamma_{\\rm tr}$ \\dao{and employing the estimate $|1+r|^{-1} \\lesssim r^{-1}$ for $r \\geq 1$}, the error\nestimate~\\eqref{main_estimate} can be further estimated by\n\\begin{equation*}\n\\begin{split}\n&\\|\\nabla IU^\\infty - \\nabla U^{{\\rm bqcf}} \\|_{L^2(\\mathbb{R}^d)}^2 +\n\\| I\\bm{p}^\\infty - \\bm{p}^{{\\rm bqcf}}\\|_{L^2(\\mathbb{R}^d)}^2\n\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\lesssim \\int_{r_{\\rm core}}^{R_{{\\rm c}}}|h(r)|^2r^{-3-d} \\, dr  + \\int_{1\/2r_{\\rm i}}^{\\infty}r^{-1-d} \\, dr\n\\end{split}\n\\end{equation*}\n\\dao{Next, we note that from the definitions of $\\Omega_{\\rm c}, \\Omega$, and $r_{\\rm i}$, we have $r_{\\rm i} = R_{\\rm c} + 4r_{\\rm buff}$ so that we may make the replacement $r_{\\rm i} \\approx R_{\\rm c}$.} Denoting the number of degrees of freedom by ${\\rm DoF}$ (nodes in the continuum finite element mesh times the number of species in the multilattice), we can then carry out an optimization problem consisting of minimizing this error estimate subject to a fixed number of degrees of freedom, ${\\rm DoF}$.   This problem is exactly the same as for the Bravais lattice and is\n\\begin{align*}\n\\min_{h \\in L^2, R_{{\\rm c}} > 0}  \\int_{r_{\\rm core}}^{R_{{\\rm c}}}|h(r)|^2r^{-3-d} \\, dr  + \\int_{1\/2\\dao{R_{\\rm c}}}^{\\infty}r^{-1-d} \\, dr.\n\\end{align*}\n\\dao{This problem is solved in~\\cite{olsonThesis} where it is found that there are approximate minimisers of the form $h(r) =\n\\big(r \/ R_{\\rm a} \\big)^{\\frac{1+d}{1+d\/2}}$.  A simplified approximate solution can be obtained by first minimizing $\\int_{r_{\\rm core}}^{R_{{\\rm c}}}|h(r)|^2r^{-3-d} \\, dr$ with respect to $h$ where the same expression for $h$ will result, but instead of also minimizing with respect to $R_{\\rm c}$, one can simply note that the error then becomes\n\\begin{equation}\\label{error_dof}\n\\int_{r_{\\rm core}}^{R_{{\\rm c}}}|h(r)|^2r^{-3-d} \\, dr  + \\int_{1\/2R_{\\rm c}}^{\\infty}r^{-1-d} \\, dr \\lesssim r_{\\rm core}^{-d-2} + R_{{\\rm c}}^{-d} \\lesssim~ R_{\\rm a}^{-d-2} + R_{{\\rm c}}^{-d}.\n\\end{equation}\nIn order to balance the sources of error, one should take  $R_{{\\rm c}} = R_{\\rm a}^{\\dao{2\/d}+1}$.  Finally, by simply writing the number of degrees of freedom as the sum of those in the atomistic and continuum regions, it is possible to derive the result that $\\#{\\rm DoF} \\approx  R_{\\rm a}^d$; further details can be found in~\\cite{olsonThesis,Dev2013,acta.atc,blended2014}.}\n\nAfter making the estimation $\\gamma_{\\rm tr} \\leq (\\log {\\rm DoF})^{1\/2}$~\\cite{blended2014} for $d = 2$,  the main error estimate, ~\\eqref{main_estimate}, currently written in terms of solution regularity, may now be replaced \\dao{by an estimate of~\\eqref{error_dof} in terms of computational cost since $\\#{\\rm DoF} \\approx  R_{\\rm a}^d$:}\n\\begin{equation}\\label{main_esty2}\n\\begin{split}\n&\\|\\nabla IU^\\infty - \\nabla U^{{\\rm bqcf}} \\|_{L^2(\\mathbb{R}^d)}^2 +\n   \\| I \\bm{p}^\\infty - \\bm{p}^{{\\rm bqcf}} \\|_{L^2(\\mathbb{R}^d)}^2 \\\\\n\t&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\lesssim~ \\begin{cases} ({\\rm DoF})^{-1-2\/d} \\log{\\rm DoF}, &\\quad d = 2 , \\\\\n\t({\\rm DoF})^{-1-2\/d}, &\\quad d = 3,\n\t\\end{cases}\n\\end{split}\n\\end{equation}\nwhich exactly matches the rate for the Bravais lattice case~\\cite{blended2014}.\nThis is due to the fact that the limiting factor in both error estimates is\nthe $\\mathcal{P}_1$ finite element approximation.\n\n\\begin{remark}\n   \\dao{In the Bravais lattice analysis~\\cite{blended2014}, the expression of $R_{{\\rm c}}$ in terms of $R_{{\\rm a}}$\n\t is incorrect which has led to an error in the expression for the error estimate in terms of the degrees of freedom. In that paper, a different mesh scaling is also used, but should the same mesh scaling be used, the error estimates in terms of the degrees of freedom would be identical up to a constant prefactor.}\n\\end{remark}\n\n\\subsection{Numerical tests} \\label{num}\nIn addition to providing a means to estimating the computational cost of the\nBQCF  method, the estimate~\\eqref{main_esty2} is also convenient to verify\n numerically. We have carried this out for\na Stone--Wales defect in graphene using both the BQCF method and a fully\natomistic method.\n\nFor the latter we simply minimize the full atomistic\nenergy over displacements that are non-zero only on the computational domain\n$\\Omega$ (clamped boundary conditions).  Using the methods\ndiscussed in Section~\\ref{analysis}, it is not difficult to show that the\nsolution, $(U^{\\rm Dir},\\bm{p}^{\\rm Dir})$, to this atomistic Galerkin method\nexists and satisfies the error estimate\n\\begin{equation}\\label{minor_esty1}\n   \\|\\nabla IU^\\infty - \\nabla U^{\\rm Dir} \\|_{L^2(\\mathbb{R}^d)}\n      + \\| I\\bm{p}^\\infty - \\bm{p}^{\\rm Dir}\\|_{L^2(\\mathbb{R}^d)}\n      \\lesssim~ ({\\rm DoF})^{-1\/2}.\n\\end{equation}\n\nWe now set the model up for the Stone--Wales defect in graphene, recalling first\n the multilattice parameter values given\nin Section~\\ref{model}.  We choose a Stillinger-Weber~\\cite{stillinger1985}\ntype interatomic potential with a pair potential and bond angle potential component.\nThe interaction range we consider is\n\\begin{align*}\n\\mathcal{R} = \\big\\{&(\\rho_1 00), (\\rho_2 00), (-\\rho_1 00),(-\\rho_2 00),(\\rho_1-\\rho_2 00),(\\rho_2-\\rho_1 00),\\\\\n&(001),(010),(-\\rho_2 01),(\\rho_2 10),(-\\rho_1 01),(\\rho_1 10),\\\\\n& (\\rho_1 11), (\\rho_2 11), (-\\rho_1 11),(-\\rho_2 11),(\\rho_1-\\rho_2 11),(\\rho_2-\\rho_1 11)\\big\\},\n\\end{align*}\nwhich is  depicted in Figure~\\ref{fig:multilattice}.  In this notation, the site potential is given by\n\\begin{align*}\n&\\hat{V}(D\\bm{y}(\\xi)) = \\sum_{(\\rho \\alpha\\beta) \\in \\mathcal{R}} \\frac{1}{2}\\phi(D_{(\\rho \\alpha\\beta)}\\bm{y}(\\xi)) + \\vartheta(D_{(-\\rho_1 01)}\\bm{y}(\\xi), D_{(-\\rho_1-\\rho_2 01)}\\bm{y}(\\xi))  \\\\\n&\\qquad+ \\vartheta(D_{(-\\rho_1 01)}\\bm{y}(\\xi), D_{(-\\rho_2 01)}\\bm{y}(\\xi)) + \\vartheta(D_{(-\\rho_1-\\rho_2 01)}\\bm{y}(\\xi), D_{(-\\rho_2 01)}\\bm{y}(\\xi)) \\\\\n&\\qquad+ \\vartheta(D_{(\\rho_1 10)}\\bm{y}(\\xi), D_{(\\rho_1+\\rho_2 10)}\\bm{y}(\\xi)) + \\vartheta(D_{(\\rho_1 10)}\\bm{y}(\\xi), D_{(\\rho_2 10)}\\bm{y}(\\xi)) \\\\\n&\\qquad+ \\vartheta(D_{(\\rho_1+\\rho_2 10)}\\bm{y}(\\xi), D_{(\\rho_2 10)}\\bm{y}(\\xi)),\n\\end{align*}\nwhere $\\phi(r) = r^{-12} - 2r^{-6}$ is a pair potential term and\n\\[\n\\vartheta(r_1, r_2) = \\Big(\\frac{r_1\\cdot r_2}{|r_1| \\,  |r_2|}+1\/2\\Big)^2\n\\]\nis a three-body term that penalizes angles that differ from $\\frac{2\\pi}{3}$.\n\n\n\nThe Stone--Wales defect shown in Figure~\\ref{fig:stone} is obtained by rotating\nthe bond between the two carbon atoms at the origin site by ninety degrees about\nthe midpoint of this bond. One way of incorporating this defect into our\nframework is to define a reference configuration $(Y_0, p_1)$ where $Y_0(\\xi) =\n{\\sf F} \\xi$ for all $\\xi \\neq 0$ with ${\\sf F}$ and $p_1$ given by the graphene\nparameters in~\\eqref{graph_param}. At the origin, we set $Y_0(0) = \\mbox{Rot}(0) $\nand $p_1(0) = \\mbox{Rot}(p_1)$, where $\\mbox{Rot}$ represents the ninety degree\nrotation about the midpoint of the segment ${\\rm conv}\\{0, p_1\\}$.  Then we set\n$V_\\xi(D(U,p)(\\xi)) = \\hat{V}(D(Y_0 + U,p_1 + p)(\\xi))$.\n\n\\begin{figure}\n   \\centering\n\\subfigure[A perfect graphene sheet. ]\n{\\includegraphics[width=0.35\\textwidth]{swleft}}\n\\subfigure[An unrelaxed Stone--Wales defect. ]\n{\\includegraphics[width=0.35\\textwidth]{swright}}\n\\caption{Examples of a perfect graphene sheet and a Stone--Wales defect.\nThe dotted lines in the right display indicate bonds that are broken during the rotation of the highlighted atoms.}\\label{fig:stone}\n\\end{figure}\n\nWe choose hexagonal domains for $\\Omega_{\\rm core}, \\Omega_{\\rm a}, \\Omega$, etc.,\nand use a blending function which approximately minimizes the $L^2$ norm of\n$\\nabla^2 \\varphi$ on $\\Omega_{{\\rm b}}$~\\cite{bqce12}.  We select the inner width,\n$r_{\\rm core}$, of the hexagon $\\Omega_{\\rm core}$ to be from the range ${{R_{{\\rm a}}=}}\\left\\{8,\n12,16,20,24\\right\\}$ with $\\kappa = 1\/2$, and then the remaining domains are\nchosen as scaled hexagons satisfying the requirements of Section~\\ref{bqcf} and\nTheorem~\\ref{main_thm} \\dao{(see Figure 10 in~\\cite{bqcf13} for a representative illustration of this domain decomposition for a Bravais lattice)}.  Finally, our finite element mesh is graded\nradially with approximate mesh size $h(r) = \\big(\\frac{r}{R_{\\rm a}}\\big)^{3\/2}$ as described earlier in this\nsection with $d = 2$.  The BQCF equations were solved by a preconditioned nonlinear conjugate\ngradient algorithm with line-search based on force-orthogonality only\n{\\helen{(in BQCF there is no energy functional for which descent can be imposed).}}\n\nIn Figure~\\ref{fig:error} we show the error in the displacement gradients and the single graphene\nshift vector for the computed BQCF solution versus the number of degrees of\nfreedom. Both match our theoretical predictions\nfrom~\\eqref{main_esty2} and indeed demonstrate that the error\nestimates are sharp (up to logarithms). We also show the error committed by the\natomistic Galerkin method (which is estimated in~\\eqref{minor_esty1}),\nto demonstrate the practical gain achieved by the BQCF method.\n\n\n\\begin{figure}\n   \\centering\n\\subfigure[Error in displacement field for Stone--Wales defect.]\n   {\\includegraphics[width=0.45\\textwidth]{NNNSW}}\n\\subfigure[Error in shift field for Stone--Wales defect. ]\n{\\includegraphics[width=0.45\\textwidth]{NNSWSHIFT}}\n\\caption{BQCF error plotted against degrees of freedom. {\\helen{We have also plotted the ``purely atomistic'' error, denoted by ATM, which is the solution obtained by truncating the infinite dimensional atomistic problem to a finite domain using homogeneous Dirichlet boundary conditions.}}\n}\\label{fig:error}\n\\end{figure}\n\n\\FloatBarrier\n\n\n\\section{Proofs} \\label{analysis}\n\nThe remainder of this paper is devoted to proving our main result, Theorem~\\ref{main_thm}.  As in \\cite{blended2014}, the abstract\nframework for the proof is provided by the\ninverse function theorem~\\cite{acta.atc,ortnerInverse,hubbard2009}, {\\helen{which we recall for reference and which is used to establish well-posedness of the nonlinear BQCF variational equation in Theorem ~\\ref{main_thm}.}}\n\n\\begin{theorem}[Inverse Function Theorem \\cite{ortnerInverse,hubbard2009}]\\label{inverseFunctionTheorem}\nLet $X$ and $Y$ be Banach spaces with $f:X \\to Y$, $f \\in {\\rm{C}}^1(U)$ with $U \\subset X$ an open set containing $x_0$.\nSuppose that $\\eta > 0, \\sigma > 0$, and $L > 0$ exist such that $\\|f(x_0)\\|_Y < \\eta$,  $\\delta f(x_0)$ is invertible with $\\|\\delta f(x_0)^{-1}\\|_{\\mathcal{L}(Y,X)} < \\sigma$, $B_{2\\eta\\sigma}(x_0) \\subset U$, $\\delta f$ is Lipschitz continuous on $B_{2\\eta\\sigma}(x_0)$ with Lipschitz constant $L$, and $2L\\eta\\sigma^2 < 1$. Then there exists a ${\\rm C}^1$ inverse function $g:B_{\\eta}(y_0) \\to B_{2\\eta\\sigma}(x_0)$ and thus an element $\\bar{x} \\in X$ such that $f(\\bar{x}) = 0$ and\n\\begin{align*}\n\\|x_0 - \\bar{x}\\|_{X} <~& 2\\eta\\sigma.\n\\end{align*}\n\\end{theorem}\n\n\n\\medskip\n\nThe nonlinear operator we consider is the variational BQCF operator $\\mathcal{F}^{\\rm BQCF}(U, \\bm{p})$, and the point about which we linearize is $x_0 = (U_h, \\bm{p}_h) := \\Pi_h(U^\\infty, \\bm{p}^\\infty)$ where $\\Pi_h$ is a projection operator defined in the following section.  In Section~\\ref{cons} we prove a consistency estimate on the residual $\\mathcal{F}^{\\rm BQCF}(U_h, \\bm{p}_h)$:\n\\begin{equation}\\label{cons_est}\n\\begin{split}\n\\sup_{\\|(W,\\bm{r})\\|_{\\rm ml} = 1} \\big|\\< \\mathcal{F}^{\\rm BQCF}(U_h, \\bm{p}_h), (W,\\bm{r}) \\>\\big| &\\lesssim \\|h \\nabla^2 \\tilde{U}^\\infty\\|_{L^2(\\Omega_{\\rm c})} + \\|h\\nabla \\tilde{\\bm{p}}^\\infty\\|_{L^2(\\Omega_{\\rm c})}  \\\\\n&\\qquad\\qquad + \\|\\nabla \\tilde{U}^\\infty\\|_{L^2(\\Omega_{\\rm ext})}\n  + \\|\\tilde{\\bm{p}}^\\infty\\|_{L^2(\\Omega_{\\rm ext})}.\n\\end{split}\n\\end{equation}\nThe invertibility condition on the derivative of $\\mathcal{F}^{{\\rm bqcf}}$ is proven as a coercivity condition in Section~\\ref{stab} where we show that\n\\begin{equation}\\label{stab_est}\n\\<\\delta \\mathcal{F}^{\\rm BQCF}(U_h, \\bm{p}_h){\\helen{(W,\\bm{r})}},(W,\\bm{r}) \\> \\gtrsim \\|(W,\\bm{r})\\|_{\\rm ml}^2, \\quad \\forall (W,\\bm{r}) \\in \\bm{\\mathcal{U}}_{h,0} \\times \\bm{\\mathcal{P}}_{h,0},\n\\end{equation}\nprovided that the atomistic region is sufficiently large.\n\nAfter we prove these two estimates, in Section \\ref{sec:proof_main_result} we\ncombine them with a Lipschitz estimate on $\\delta\\mathcal{F}^{\\rm bqcf}$ and\napply the inverse function theorem to prove Theorem~\\ref{main_thm}.\n\n\n\nThroughout this analysis, we continue to use the modified Vinogradov notation $x \\lesssim y$, where the implied constants are allowed to depend on the shape regularity constants $C_{\\mathcal{T}_h}, C_{\\rm o}$ ({\\helen{which are defined in Assumption~\\ref{assumption-shapereg} and \\eqref{def_constant_Co} }}), the interatomic potentials (and their interaction range), and the stability constant $\\gamma_{\\rm a}$.\n\n\n\\subsection{Cauchy--Born Modeling Error}\\label{tech}\nIn preparation for the consistency analysis in Section~\\ref{cons} we first establish several auxiliary results about the Cauchy--Born model.\n\nA central technical tool in the analysis of AtC coupling methods is the ability to compare discrete atomistic displacements which are the natural atomistic kinematic variables ({\\helen{recall that the atomistic displacements are equivalent to atomistic site displacements plus atomistic shift vectors}}), and continuous displacement and shift fields which capture the continuum kinematics.  We have already introduced several interpolants which serve this task: a micro-interpolant, $I$; a finite element interpolant, $I_h$; and a smooth interpolant, $\\tilde{I}$.  We will also introduce a quasi-interpolant in this section which will allow us to define an analytically convenient atomistic version of stress~\\cite{theil2012}.\n\nWe use $\\bar{\\zeta}(x)$ to denote the nodal basis function associated with the origin for the atomistic finite element mesh $\\mathcal{T}_{\\rm a}$ and $\\bar{\\zeta}_\\xi(x) := \\bar{\\zeta}(x-\\xi)$ to denote the nodal basis function at site $\\xi$.   We may then write the micro-interpolant $Iu = \\bar{u}$ as\n\\begin{equation*}\\label{def_baru}\n\\bar{u}(x) = \\sum_{\\xi \\in \\mathcal{L}} u(\\xi) \\bar{\\zeta}(x-\\xi).\n\\end{equation*}\nThe quasi-interpolant of $u$ is then defined by a convolution with $\\bar{\\zeta}$\n\\begin{equation}\\label{def_quasi_interp}\nu^*(x) := (\\bar{\\zeta} * \\bar{u})(x).\n\\end{equation}\n\nIt will later be important that this convolution operation is invertible and stable. This is a consequence of~\\cite[Lemma 5]{atInterpolant}, which we state here for reference.\n\n\\begin{lemma}\\cite[Lemma 5]{atInterpolant}\\label{iso_lemma}\nFor a given atomistic displacement, $u$, there exists a unique atomistic displacement $\\acute{u}$ with the property that $\\bar{\\zeta} * \\bar{\\acute{u}}(\\xi) = u(\\xi)$ for all $\\xi \\in \\mathcal{L}$.\n\\end{lemma}\n\n\\medskip\n\nOne of the primary uses of the $u^*$ interpolant will be the development of an \\textit{atomistic stress function} which can be compared to the continuum stress in the Cauchy--Born model~\\cite{theil2012}.  The first variation of the continuum model may be written in terms of a stress tensor,\n\\begin{equation}\\label{cont_tensor_eq1}\n\\begin{split}\n&\\langle \\delta \\mathcal{E}^{{\\rm c}} (U, \\bm{q}), (W, \\bm{r})\\rangle\n=~ \\int_{\\mathbb{R}^d} \\sum_{(\\rho \\alpha\\beta)} V_{,(\\rho \\alpha\\beta)}((\\nabla_{\\tau} U+q_{\\delta}-q_{\\gamma})_{(\\tau\\gamma\\delta) \\in \\mathbb{R}})\n\\cdot \\left(\\nabla_{\\rho} W+r_{\\beta}-r_{\\alpha}\\right) \\\\\n&~=~ \\int_{\\mathbb{R}^d} \\sum_{(\\rho \\alpha\\beta)} V_{,(\\rho \\alpha\\beta)}(\\nabla(U, \\bm{q})) \\otimes \\rho : \\nabla W  + \\int_{\\mathbb{R}^d} \\sum_{(\\rho \\alpha\\beta)} V_{,(\\rho \\alpha\\beta)}(\\nabla(U, \\bm{q}))_{(\\tau\\gamma\\delta) \\in \\mathbb{R}})\n\\cdot \\left(r_{\\beta}-r_{\\alpha}\\right)\n\\\\\n&~=~ \\int_{\\mathbb{R}^d} \\sum_{\\beta}[{\\rm{S}^{\\rm c}_{\\rm d}}(U,\\bm{q})(x)]_{\\beta} : \\nabla W\n      + \\int_{\\mathbb{R}^d} \\sum_{\\alpha,\\beta} [{\\rm{S}^{\\rm c}_{\\rm s}}(U,\\bm{q})(x)]_{\\alpha\\beta}\n               (r_\\beta - r_\\alpha),\n\\end{split}\n\\end{equation}\nwhere we defined\n\\begin{equation}\\label{cont_stress_tensor}\n\\begin{split}\n[{\\rm{S}^{\\rm c}_{\\rm d}}(U,\\bm{q})(x)]_{\\beta} :=~& \\sum_{\\substack{\\alpha, \\rho: \\\\ (\\rho \\alpha\\beta)\\in\\mathcal{R}}} V_{,(\\rho \\alpha\\beta)}(\\nabla(U, \\bm{q})(x))\\otimes \\rho, \\\\\n[{\\rm{S}^{\\rm c}_{\\rm s}}(U,\\bm{q})(x)]_{\\alpha\\beta} :=~& \\sum_{\\rho \\in \\mathcal{R}_1} V_{,(\\rho \\alpha\\beta)}(\\nabla(U, \\bm{q})(x)).\n\\end{split}\n\\end{equation}\n\nTo compare the atomistic and continuum models, we now construct an analogous atomistic stress tensor. Its definition will make it clear why we introduced the seemingly unnecessary sum over $\\beta$ in the first group in \\eqref{cont_tensor_eq1}. \\dao{The basic idea is to extend the construction of~\\cite{theil2012}: the argument $\\nabla(U, \\bm{q})(x)$ in~\\eqref{cont_stress_tensor} will be replaced by a local averaging of first order finite difference approximations $D(U, \\bm{q})(\\xi)$ for $\\xi$ near $x$.}\n\n\\begin{lemma}\n   For $(U,\\bm{q}) \\in \\bm{\\mathcal{U}}$, define the atomistic stress tensors\n   \\begin{align}\n      \\label{eq:defn_Sa}\n    \\begin{split}\n   [\\,{\\rm S}^{\\rm a}_{\\rm d}(U,\\bm{q})(x)]_\\beta :=~& \\sum_{\\substack{\\alpha, \\rho: \\\\ (\\rho \\alpha\\beta)\\in\\mathcal{R}}} \\sum_{\\xi \\in \\mathcal{L}} \\big( V_{,(\\rho \\alpha\\beta)}\\big(D(U,\\bm{q})(\\xi)\\big) \\otimes \\rho \\big) \\omega_{\\rho}(\\xi-x),\\\\\n   [\\,{\\rm S}^{\\rm a}_{\\rm s}(U,\\bm{q})(x)]_{\\alpha\\beta} :=~& \\sum_{\\rho \\in \\mathcal{R}_1} \\sum_{\\xi\\in\\mathcal{L}} V_{,(\\rho \\alpha\\beta)} \\big(D(U,\\bm{q})(\\xi)\\big) \\omega_{0}(\\xi-x).\n   \\end{split} \\\\\n   \\label{omega_rho}\n   \\text{where} \\qquad \\omega_{\\rho}(x) &:= \\int_{0}^1 \\bar{\\zeta}(x+t\\rho)dt.\n   \\end{align}\n   Then\n   \\begin{equation}\n      \\label{atom_tensor_eq1}\n      \\begin{split}\n         \\big\\langle \\delta \\mathcal{E}^{{\\rm a}}_{\\rm hom}(U,\\bm{q}), (W^{*}, \\bm{r}^{*}) \\big\\rangle\n   =& \\int_{\\mathbb{R}^d}  \\bigg\\{ \\sum_\\beta [\\,{\\rm S}^{\\rm a}_{\\rm d}(U,\\bm{q})]_\\beta : \\left(\\nabla \\bar{W}+\\nabla \\bar{r_\\beta} \\right) \\\\\n    &\\qquad\\qquad+ \\sum_{\\alpha,\\beta} [\\,{\\rm S}^{\\rm a}_{\\rm s}(U,\\bm{q})]_{\\alpha\\beta}\\cdot (\\bar{r}_{\\beta}-\\bar{r}_\\alpha) \\bigg\\} dx.\n\t\t\n     \n     \n      \\end{split}\n   \\end{equation}\n   {\\helen{ where $W^{*}$ and  $\\bm{r}^{*}$ are defined through  \\eqref{def_quasi_interp}. }}\n\\end{lemma}\n\\begin{proof}\nWe start by computing {\\helen{the first variation of $\\mathcal{E}^{{\\rm a}}_{\\rm hom}(U,\\bm{q})$ with the test pair $(W^{*}, \\bm{r}^{*})$:}}\n\\begin{equation}\\label{delEa_eq1}\n\\begin{split}\n&\\langle \\delta \\mathcal{E}^{{\\rm a}}_{\\rm hom}(U,\\bm{q}), (W^{*}, \\bm{r}^{*})\\rangle\\\\\n&=\\sum_{\\xi\\in\\mathcal{L}}\\sum_{(\\rho \\alpha\\beta)\\in\\mathcal{R}}V_{,(\\rho \\alpha\\beta)}\\big(D(U,\\bm{q})(\\xi)\\big)\n\\cdot\\Big(D_{\\rho}W^*(\\xi)+D_{\\rho}r_{\\beta}^*(\\xi)+r_{\\beta}^*(\\xi)-r_{\\alpha}^*(\\xi)\\Big).\n\\end{split}\n\\end{equation}\nArguing as in \\cite[Eq. (2.4)]{theil2012} we obtain\n\\begin{align}\n   \\label{convo_finite_diff}\n   D_{\\rho}W^*(\\xi)+D_{\\rho}r_{\\beta}^*(\\xi)\n   &= \\int_{\\mathbb{R}^d} \\omega_{\\rho}(\\xi-x)\\left( \\nabla_{\\rho}\\bar{W}+\\nabla_{\\rho}\\bar{r}_{\\beta}\\right)\\, dx\n   \\qquad \\text{and} \\\\\n   %\n   \\label{convo_beta_alpha}\n   r^*_{\\beta}(\\xi)-r^*_{\\alpha}(\\xi) &= \\int_{\\mathbb{R}^d}\\omega_0(\\xi-x) \\left(\\bar{r}_{\\beta}-\\bar{r}_{\\alpha}\\right)\\, dx.\n\\end{align}\n\n\n\nSubstituting \\eqref{convo_finite_diff} and \\eqref{convo_beta_alpha} into \\eqref{delEa_eq1} and recalling the definitions of the atomistic stress tensors from~\\eqref{eq:defn_Sa} yields the stated claim.\n\\end{proof}\n\nWe refer to the error between the continuum and atomistic stress functions as the \\textit{Cauchy--Born modeling error} and quantify it in the next lemma; see \\cite{theil2012} for an analogous result for Bravais lattices.\n\n\\begin{lemma}\\label{cb_error2}\n   Assume that $U \\in {\\rm C}^{2,1}(\\mathbb{R}^d; \\mathbb{R}^n)$ and $p_\\alpha \\in {\\rm C}^{1,1}(\\mathbb{R}^d, \\mathbb{R}^{n})$ for each $\\alpha$. Fix $x \\in \\mathbb{R}^d$ and set\n\t\\[\n\tr_{\\rm cut} = \\max_{\\rho \\in \\mathcal{R}_1} |\\rho|,  \\qquad \\nu_x := B_{2r_{\\rm cut}}(0).\n\t\\]\n   1. If $\\nabla U$ and $\\bm{p}$ are constant in $\\nu_x$, then\n   \\begin{equation}\\label{cb_identity1}\n   {[{\\rm S}^{\\rm a}_{\\rm d}(U, \\bm{p})(x)]_\\beta = [{\\rm S}^{\\rm c}_{\\rm d}(U, \\bm{q})(x)]_\\beta \\quad \\mbox{and} \\quad [{\\rm S}^{\\rm a}_{\\rm s}(U, \\bm{p})(x)]_{\\alpha\\beta} = [{\\rm S}^{\\rm c}_{\\rm s}(U, \\bm{q})(x)]_{\\alpha\\beta}.}\n   \\end{equation}\n\n   2. In general,\n   {\\begin{equation*}\n      \\label{eq:bound_Sa-Sc}\n\t\t\t\\begin{split}\n   \\big| [S^{\\rm a}_{\\rm d}(U,p)(x)]_\\beta - [S_{\\rm d}^{\\rm c}(U,p)(x)]_\\beta\\big|\n   \\lesssim~&\n         \\| \\nabla^2 U \\|_{L^\\infty(\\nu_x)} + \\| \\nabla \\bm{q} \\|_{L^\\infty(\\nu_x)},\\\\\n   \\big| [S^{\\rm a}_{\\rm s}(U,p)(x)]_{\\alpha\\beta} - [S_{\\rm s}^{\\rm c}(U,p)(x)]_{\\alpha\\beta}\\big|\n   \\lesssim~&\n         \\| \\nabla^2 U \\|_{L^\\infty(\\nu_x)} + \\| \\nabla \\bm{q} \\|_{L^\\infty(\\nu_x)}.\n  \\end{split}\n\t\\end{equation*}}\n   with the implied constant depending only on the interatomic potential $V$.\n\\end{lemma}\n\\begin{proof}\n   1. The identity \\eqref{cb_identity1} is an immediate consequence of the definitions \\eqref{cont_stress_tensor}, \\eqref{eq:defn_Sa} and of\n   \\begin{equation*}\\label{omega_properties_simplified}\n   \\sum_{\\xi}\\omega_\\rho(\\xi - x) = 1.\n   \\end{equation*}\n\n   2. We define an auxiliary homogeneous displacement $(U^{\\rm h}, \\bm{q}^{\\rm h})$ with $\\nabla U^{\\rm h} \\equiv \\nabla U(x)$ and $\\bm{q}^{\\rm h} \\equiv \\bm{q}(x)$. Then we have{\n   \\begin{align*}\n      [{\\rm S}^{\\rm a}_{\\rm d}(U, \\bm{q})(x)]_\\beta - [{\\rm S}^{\\rm c}_{\\rm d}(U, \\bm{q})(x)]_\\beta\n      = [{\\rm S}^{\\rm a}_{\\rm d}(U, \\bm{q})(x)]_\\beta - [{\\rm S}^{\\rm a}_{\\rm d}(U^{\\rm h}, \\bm{q}^{\\rm h})(x)]_\\beta.\n   \\end{align*}}\n   Since we assumed that $V$ is twice continuously differentiable, with globally bounded second derivatives, we\n   obtain\n   \\begin{align*}\n      \\big| [{\\rm S}^{\\rm a}_{\\rm d} &(U, \\bm{q})(x)]_\\beta - [{\\rm S}^{\\rm c}_{\\rm d}(U, \\bm{q})](x)_\\beta \\big|\n      = \\big|[{\\rm S}^{\\rm a}_{\\rm d}(U, \\bm{q})(x)]_\\beta - [{\\rm S}^{\\rm a}_{\\rm d}(U^{\\rm h}, \\bm{q}^{\\rm h})(x)]_\\beta\\big| \\\\\n      &\\dao{= \\Big|\\sum_{\\substack{\\alpha, \\rho: \\\\ (\\rho \\alpha\\beta)\\in\\mathcal{R}}} \\sum_{\\xi \\in \\mathcal{L}} \\big(\\big[ V_{,(\\rho \\alpha\\beta)}\\big(D(U,\\bm{q})(\\xi)\\big) -  V_{,(\\rho \\alpha\\beta)}\\big(D(U^{\\rm h}, \\bm{q}^{\\rm h})(\\xi)\\big)\\big]\\otimes \\rho \\big) \\omega_{\\rho}(\\xi-x)\\Big| }\\\\\n\t\t\t&\\dao{\\lesssim \\sum_{\\substack{\\alpha, \\rho: \\\\ (\\rho \\alpha\\beta)\\in\\mathcal{R}}} \\sum_{\\xi \\in \\mathcal{L}} \\big|D(U,\\bm{q})(\\xi) -  D(U^{\\rm h}, \\bm{q}^{\\rm h})(\\xi)\\big| \\omega_{\\rho}(\\xi-x)} \\\\\n      &\\lesssim \\big\\| \\nabla U - \\nabla U^{\\rm h} \\|_{L^\\infty(\\nu_x)}\n         + \\big\\| \\bm{q} - \\bm{q}^{\\rm h} \\|_{L^\\infty(\\nu_x)} + \\dao{\\| \\nabla^2 U \\|_{L^\\infty(\\nu_x)} + \\| \\nabla \\bm{q} \\|_{L^\\infty(\\nu_x)}} \\\\\n      &\\lesssim \\| \\nabla^2 U \\|_{L^\\infty(\\nu_x)} + \\| \\nabla \\bm{q} \\|_{L^\\infty(\\nu_x)},\n   \\end{align*}\n\t\\dao{where in obtaining the last two inequalities we have used a Taylor expansion of the finite differences and the fact that $\\omega_{\\rho}(\\xi-x)$ as defined in~\\eqref{omega_rho} vanishes off of $\\nu_x$.} The proof for the comparison of the ``shift'' stress tensors is nearly identical so is omitted.\n\\end{proof}\n\n\nWith this pointwise estimate, and using the fact that $\\tilde{U}$ is piecewise polynomial, it is straightforward to deduce the following Cauchy--Born modeling error estimate over $\\Omega_{\\rm c}$.\n\\begin{corollary}\\label{globel_stress}\n{For the atomistic and continuum stress tensors defined above,\n\\begin{equation*}\\label{global_est}\n\\begin{split}\n\\big\\|[{\\rm{S}}^{\\rm a}_{\\rm d}(\\tilde{U}^\\infty,\\tilde{\\bm{q}}^\\infty)]_\\beta -[{\\rm{S}}^{\\rm c}_{\\rm d}(\\tilde{U}^\\infty,\\tilde{\\bm{q}}^\\infty)]_\\beta \\big\\|_{L^2(\\Omega_{\\rm c})} \\lesssim~&  \\|\\nabla^2 \\tilde{U}^\\infty\\|_{L^2(\\Omega_{\\rm c})} + \\|\\nabla \\tilde{\\bm{q}}^\\infty\\|_{L^2(\\Omega_{\\rm c})}, \\\\\n\\big\\|[{\\rm{S}}^{\\rm a}_{\\rm s}(\\tilde{U}^\\infty,\\tilde{\\bm{q}}^\\infty)]_{\\alpha\\beta} -[{\\rm{S}}^{\\rm c}_{\\rm s}(\\tilde{U}^\\infty,\\tilde{\\bm{q}}^\\infty)]_{\\alpha\\beta} \\big\\|_{L^2(\\Omega_{\\rm c})} \\lesssim~&  \\|\\nabla^2 \\tilde{U}^\\infty\\|_{L^2(\\Omega_{\\rm c})} + \\|\\nabla \\tilde{\\bm{q}}^\\infty\\|_{L^2(\\Omega_{\\rm c})}.\n\\end{split}\n\\end{equation*}}\n\\end{corollary}\n\n\n\\begin{remark}\nThe stress estimates for a multilattice are one order lower in terms of derivatives than the corresponding Bravais lattice estimates.  A refined analysis shows that this estimate cannot be improved without an underlying point symmetry for the multilattice.  When this symmetry is present in multilattices, it is possible to define a symmetrized Cauchy--Born energy with an improved estimate~\\cite{koten2013}.\n\\end{remark}\n\n\n\n\n\\subsection{Consistency}\\label{cons}\nOur first task in completing the residual estimate~\\eqref{cons_est} is to define the projection from atomistic functions to finite element functions satisfying the Dirichlet boundary conditions so we first truncate the solution to a finite domain.  For that, let $\\eta$ be a smooth ``bump function'' with support in $B_{1}(0)$ and equal to one on $B_{3\/4}(0)$.   Let \\dao{$A_R$ be an ``annular region'' containing the support of $\\nabla (I \\eta(x\/R))$, i.e,} $A_R := B_{R+2r_{\\rm buff}}(0)\\setminus B_{3\/4R-2r_{\\rm buff}} \\supset {\\rm supp}(\\nabla (I \\eta(x\/R)))$ and define the truncation operator by\n\\begin{equation*}\\label{trunc}\nT_{R}u_\\alpha(x) = \\eta(x\/R)\\bigg(Iu_\\alpha - \\frac{1}{|A_R|}\\int\\limits_{A_R} Iu_0\\, dx\\bigg).\n\\end{equation*}\nFurther, let $S_h$ be the Scott--Zhang {\\helen{quasi-interpolation}} operator~\\cite{scott1990} onto the finite element mesh $\\mathcal{T}_h$.  We then define the projection operator by\n\\begin{align}\\label{proj_operator}\n&\\qquad\\qquad\\qquad\\quad\\Pi_{h} u_\\alpha := S_h (T_{r_{\\rm i}}u_\\alpha), \\quad \\Pi_{h} \\bm{u} :=~ \\left\\{\\Pi_{h} u_\\alpha\\right\\}_{\\alpha = 0}^{S-1}, \\\\\n&\\Pi_{h} p_\\alpha := \\Pi_{h} (u_\\alpha - u_0), \\quad \\Pi_{h} \\bm{p} :=~ \\left\\{\\Pi_{h} p_\\alpha\\right\\}_{\\alpha = 0}^{S-1}, \\qquad \\Pi_{h}(U,\\bm{p}) := (\\Pi_{h}U, \\Pi_{h}\\bm{p}).\\nonumber\n\\end{align}\n(Recall that $r_{\\rm i}$ is the radius of the largest ball inscribed in $\\Omega$.) Note that $\\nabla \\Pi_{h} u_\\alpha$ as well as\n\\[\n\\Pi_{h} u_\\alpha - \\Pi_{h} u_\\beta = S_h\\big[\\eta(x\/r_{\\rm i})\\big( I u_\\alpha - Iu_\\beta \\big)\\big]\n\\]\nhave support contained in $\\Omega$.\nWe also have the following approximation results.\n\\begin{lemma}\\label{approx_lem}\nTake $(U,\\bm{p}) = \\bm{u} \\in \\bm{\\mathcal{U}}$.  Then\n\\begin{equation*}\\label{approx_est}\n\\begin{split}\n\\|\\nabla \\bar{U} - \\nabla \\Pi_{h,R} U\\|_{L^2(\\mathbb{R}^d)} +  \\|\\bar{\\bm{p}}_\\alpha - \\Pi_{h,R} \\bm{p}_\\alpha\\|_{L^2(\\mathbb{R}^d)} \\lesssim~& \\|h \\nabla^2 \\tilde{U}^\\infty\\|_{L^2(\\Omega_{\\rm c})} + \\|h\\nabla \\tilde{\\bm{p}}^\\infty\\|_{L^2(\\Omega_{\\rm c})} \\\\\n&+ \\| \\nabla \\tilde{U}\\|_{L^2(\\Omega_{\\rm ext})} + \\|\\tilde{\\bm{p}}\\|_{L^2(\\Omega_{\\rm ext})}, \\\\\n\\|\\nabla \\tilde{U} - \\nabla \\Pi_{h,R} U\\|_{L^2(\\Omega_{\\rm c})} +  \\|\\tilde{\\bm{p}}_\\alpha - \\Pi_{h,R} \\bm{p}_\\alpha\\|_{L^2(\\Omega_{\\rm c})} \\lesssim~& \\|h \\nabla^2 \\tilde{U}^\\infty\\|_{L^2(\\Omega_{\\rm c})} + \\|h\\nabla \\tilde{\\bm{p}}^\\infty\\|_{L^2(\\Omega_{\\rm c})} \\\\\n&+ \\| \\nabla \\tilde{U}\\|_{L^2(\\Omega_{\\rm ext} \\cap \\Omega_{\\rm c})} + \\|\\tilde{\\bm{p}}\\|_{L^2(\\Omega_{\\rm ext} \\cap \\Omega_{\\rm c})}.\n\\end{split}\n\\end{equation*}\n\\end{lemma}\nThe proof is very similar to the proof of Lemma~\\ref{lem:dense} (with only additional estimates required for the finite element interpolants) and therefore omitted. See also~\\cite[Lemma 1.8]{olson2015} for similar estimates, the main difference being the usage of the Scott--Zhang interpolant which allows for $L^2$ interpolation bounds on $H^1$ functions, see~\\cite{brenner2008,scott1990}.\n\nWe can now prove the bound~\\eqref{cons_est}.\n\n\\begin{theorem}[BQCF Consistency]\\label{consistency_thm}\nDefine $(U_h, \\bm{p}_h):= \\Pi_h(U^\\infty, \\bm{p}^\\infty)$ where $(U^\\infty, \\bm{p}^\\infty)$ satisfies Assumption~\\ref{assumption2}. If Assumptions~\\ref{assumption1} and~\\ref{assumptionSite} are valid also and if the blending function, $\\varphi$, and finite element mesh, $\\mathcal{T}_h$, satisfy the requirements of Section~\\ref{bqcf}, then the BQCF consistency error is bounded by\n\\begin{equation*}\\label{cons_est1}\n\\begin{split}\n\\left|\\langle\\mathcal{F}^{{\\rm bqcf}}(U_h,\\bm{p}_h), (W,\\bm{r})\\rangle\\right| \\lesssim~ \\gamma_{\\rm tr} \\, &\\Big(\\|h\\nabla^2 \\tilde{U}\\|_{L^2(\\Omega_{\\rm c})} + \\|h\\nabla \\tilde{\\bm{p}}_\\alpha\\|_{L^2(\\Omega_{\\rm c})}  + \\|\\nabla \\tilde{U}\\|_{L^2(\\Omega_{\\rm ext})}   \\\\\n&+  \\| \\tilde{\\bm{p}}\\|_{L^2(\\Omega_{\\rm ext})}\\Big)\\cdot \\|(W,\\bm{r})\\|_{{\\rm ml}}, \\quad \\forall (W,\\bm{r}) \\, \\in \\bm{\\mathcal{U}}_{h,0} \\times \\bm{\\mathcal{P}}_{h,0},\n\\end{split}\n\\end{equation*}\nand $\\gamma_{\\rm tr}$ is a trace inequality constant (see Lemma 4.6 in \\cite{blended2014}) given by\n\\[\n\\gamma_{\\rm tr} = \\begin{cases} &\\sqrt{1 + \\log(R_{\\rm o}\/R_{{\\rm a}})}, \\quad \\mbox{if $d = 2$,} \\\\\n                                 &1, \\quad \\mbox{if $d = 3$.} \\end{cases}\n\\]\n\\end{theorem}\n\nBefore beginning the proof, we make some preliminary remarks. First, we observe that, since the Scott--Zhang interpolation operator is a projection it follows that\n\\[\n   D_{(\\rho \\alpha\\beta)}U_h(\\xi)=D_{(\\rho \\alpha\\beta)}U^\\infty(\\xi)\n   \\qquad \\text{for} \\quad\n   \\xi\\in\\mathcal{L}^{{\\rm a}},\n\\]\nwhere $\\mathcal{L}^{\\rm a} := \\mathcal{L} \\cap ({\\rm supp}(1-\\varphi) + \\mathcal{R}_1)$.\nFurthermore, since $\\delta \\mathcal{E}^{{\\rm a}}(U^\\infty, \\bm{p}^\\infty) = 0$, the residual error in the BQCF variational operator is equivalent to\n\\begin{equation}\\label{test_going}\n\\begin{split}\n\\langle & \\mathcal{F}^{{\\rm bqcf}}(U_h,\\bm{p}_h), (W,\\bm{r})\\rangle\n-\\langle \\delta \\mathcal{E}^{{\\rm a}}(U^\\infty, \\bm{p}^\\infty), (U,\\bm{q})\\rangle\\\\\n&\\quad= \\langle \\delta \\mathcal{E}^{{\\rm a}}(U^\\infty,\\bm{p}^\\infty), (1-\\varphi)(W, \\bm{r})\\rangle\n+\\langle \\delta \\mathcal{E}^{{\\rm c}}(U_h, \\bm{p}_h), \\big(I_h(\\varphi W),I_h(\\varphi\\bm{r}) \\big)\\rangle \\\\\n&\\qquad\\qquad -\\langle \\delta \\mathcal{E}^{{\\rm a}}(U^\\infty, \\bm{p}^\\infty), (U,\\bm{q})\\rangle,\n\\end{split}\n\\end{equation}\nwhere $(W, \\bm{r}) \\in \\bm{\\mathcal{U}}_{h,0} \\times \\bm{\\mathcal{P}}_{h,0}$ is an arbitrary given pair of test functions in the finite element test function space, while $(U, \\bm{q})  \\in \\bm{\\mathcal{U}} \\times \\bm{\\mathcal{P}}$ is a test pair that we are free to choose. The obvious candidate choice is $(U, \\bm{q}) = (W, \\bm{r})$ in which case we would have\n\\begin{align*}\n\\langle & \\mathcal{F}^{{\\rm bqcf}}(U_h,\\bm{p}_h), (W,\\bm{r})\\rangle\n-\\langle \\delta \\mathcal{E}^{{\\rm a}}(U^\\infty, \\bm{p}^\\infty), (U,\\bm{q})\\rangle\\\\\n&\\quad= -\\langle \\delta \\mathcal{E}^{{\\rm a}}(U^\\infty,\\bm{p}^\\infty), (\\varphi)(W, \\bm{r})\\rangle\n+\\langle \\delta \\mathcal{E}^{{\\rm c}}(U_h, \\bm{p}_h), \\big(I_h(\\varphi W),I_h(\\varphi\\bm{r}) \\big)\\rangle.\n\\end{align*}\nThe resulting residual error is concentrated only over $\\Omega_{\\rm c}$ due to $\\nabla \\varphi$ having support in $\\Omega_{\\rm c}$.  The issue in estimating this quantity is that when we convert the atomistic residual into the atomistic-stress format, the test function appears as a piecewise linear function with respect to the atomistic mesh $\\mathcal{T}_{\\rm a}$, whereas the test function is piecewise linear with respect to the graded mesh $\\mathcal{T}_h$ in the continuum portion.  For this reason, we \\dao{shall add correction terms to our previous candidate choice $(U,\\bm{q}) = (W,\\bm{r})$ via\n\\begin{equation}\\label{test_choice}\nU = W + (Z^* - \\varphi W), \\quad q_\\alpha = r_\\alpha + (z_\\alpha^* - \\varphi r_\\alpha), \\quad \\alpha = 1,\\ldots, S-1,\n\\end{equation}\nwhere $(Z,\\bm{z}) \\in \\bm{\\mathcal{U}} \\times \\bm{\\mathcal{P}}$ will be chosen to satisfy certain approximation estimates as stated in Lemma~\\ref{interpolation_lemma} below.  The reason we use $Z^*$ and $z_\\alpha^*$ instead of merely $Z$ and $z_\\alpha$ is that we shall eventually make use of the atomistic stress representation from~\\eqref{atom_tensor_eq1}.} The BQCF residual error from~\\eqref{test_going} then becomes\n\\begin{equation}\\label{residual_est_new}\n\\begin{split}\n\\langle & \\mathcal{F}^{{\\rm bqcf}}(U_h,\\bm{p}_h), (W,\\bm{r})\\rangle\n-\\langle \\delta \\mathcal{E}^{{\\rm a}}(U^\\infty, \\bm{p}^\\infty), (W + (Z^* - \\varphi W),\\bm{r} + (\\bm{z}^* - \\varphi \\bm{r}))\\rangle\\\\\n&\\quad= \\langle \\delta \\mathcal{E}^{{\\rm c}}(U_h, \\bm{p}_h), \\big(I_h(\\varphi W),I_h(\\varphi\\bm{r}) \\big)\\rangle - \\langle \\delta \\mathcal{E}^{{\\rm a}}(U^\\infty,\\bm{p}^\\infty), (Z^*, \\bm{z}^*)\\rangle\n\\end{split}\n\\end{equation}\nMoreover, since we are blending by site and using $\\mathcal{P}_1$ elements for the shifts, we may use the same form for $Z$ and $\\bm{z}$ as obtained in the simple lattice case~\\cite{blended2014} for both displacements \\textit{and} shifts.\n\n\\begin{lemma}\\label{interpolation_lemma}\nSuppose $W \\in \\bm{\\mathcal{U}}_{h,0}$ and $\\bm{r} \\in \\bm{\\mathcal{P}}_{h,0}$.  Then for $f \\in W^{1,2}_{\\rm loc}(\\mathbb{R}^d)$ and for $Z_h, Z,{z_h}_{\\alpha}, z_\\alpha$ as defined above,\n\\begin{align}\n\\int_{\\Omega_{\\rm c}} f (\\bar{Z} - Z_h) dx \\lesssim~& \\|\\nabla f\\|_{L^2(\\Omega_{\\rm c})} \\cdot \\|\\nabla Z_h\\|_{L^2(\\Omega_{\\rm c})}, \\label{weight_inter}\\\\\n\\int_{\\Omega_{\\rm c}} f\\cdot ({z_h}_{\\alpha} - \\bar{z}_\\alpha)\\, dx \\lesssim~& \\| \\nabla f\\|_{L^2(\\Omega_{\\rm c})} \\cdot \\| {z_h}_{\\alpha} \\|_{L^2(\\Omega_{\\rm c})}  \\label{za_result} \\\\\n\\| Z_h- \\bar{Z}\\|_{L^2(\\Omega_{\\rm c})} \\lesssim~& \\|\\nabla Z_h\\|_{L^2(\\Omega_{\\rm c})}, \\label{inter_diff}\\\\\n\\| {z_h}_{\\alpha} - \\bar{z}_\\alpha\\|_{L^2(\\Omega_{\\rm c})} \\lesssim~& \\| {z_h}_{\\alpha}\\|_{L^2(\\Omega_{\\rm c})}, \\label{inter_shift} \\\\\n\\|\\nabla Z_h\\|_{L^2(\\Omega_{\\rm c})} \\lesssim~& \\gamma_{\\rm tr}\\|\\nabla W \\|_{L^2(\\Omega_{\\rm c})},\\label{h1_int_norm_est} \\\\\n\\|{z_h}_{\\alpha}\\|_{L^2(\\Omega_{\\rm c})} \\lesssim~& \\|r_\\alpha\\|_{L^2(\\Omega_{\\rm c})} \\label{l2_int_norm_est}.\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nWe begin by letting $\\omega_\\xi := {\\rm supp}(\\bar{\\zeta}(x-\\xi))$ and $\\mathcal{C} := \\{\\xi \\in \\mathcal{L} : \\omega_\\xi  \\subset \\Omega_{\\rm c} \\}$.  Then we observe that $Z_h$ and $\\bar{Z}$ are constant on any patch $\\omega_\\xi$ with $\\xi \\notin \\mathcal{C}$, \\helen{and furthermore $Z_h = \\bar{Z}$.  Intuitively, this should hold because if $\\xi \\notin \\mathcal{C}$, then either $\\xi$ is near the defect core where $\\varphi = 0$ and hence $Z_h = 0$ and $\\bar{Z} = 0$; or $\\xi$ is near the exterior to the boundary of $\\Omega$ where $Z_h$ is constant. For this to rigorously hold, we need to recall the buffer, $B_{4\\rm buff}$, in the definition of $\\Omega_{\\rm c}$ which then makes proving the statement possible.  Moreover, $Z_h = \\bar{Z}$ on any patch $\\omega_\\xi$ with $\\xi \\notin \\mathcal{C}$ due to the normalization factor in the definition of $Z$.} For $f \\in W^{1,2}_{\\rm loc}(\\mathbb{R}^d)$ we then have\n   \\begin{equation}\\label{Z_result}\n\t\\begin{split}\n      &\\int_{\\Omega_{\\rm c}} f (\\bar{Z} - Z_h) dx = \\sum_{\\xi \\in \\mathcal{L}} \\int_{\\omega_\\xi \\cap \\Omega_{\\rm c}} f(x) \\big( Z(\\xi) - Z_h(x) \\big) \\bar{\\zeta}(x-\\xi) dx \\\\\n\t\t\t&= \\dao{\\sum_{\\substack{\\xi \\in \\mathcal{L}: \\\\ \\omega_\\xi \\subset \\Omega_{\\rm c}}} \\int_{\\omega_\\xi} f(x) \\big( Z(\\xi) - Z_h(x) \\big) \\bar{\\zeta}(x-\\xi) dx \\quad \\mbox{since $Z_h = Z$ is constant for $\\xi \\notin \\mathcal{C}$}}\\\\\n      &= \\sum_{\\xi \\in \\mathcal{C}} \\int_{\\omega_\\xi} \\bigg(f(x) - \\Xint-_{\\omega_\\xi} f \\bigg) \\big( Z(\\xi) - Z_h(x) \\big) \\bar{\\zeta}(x-\\xi) dx \\\\\n      &\\leq \\sum_{\\xi \\in \\mathcal{C}} \\bigg\\| f - \\Xint-_{\\omega_\\xi}\n                f  \\bigg\\|_{L^2(\\omega_\\xi)} \\| Z(\\xi) - Z_h \\|_{L^2(\\omega_\\xi)} \\\\\n      &\\lesssim \\sum_{\\xi \\in \\mathcal{C}}\\| \\nabla f \\|_{L^2(\\omega_\\xi)} \\| \\nabla Z_h \\|_{L^2(\\omega_\\xi)} \\\\\n\t\t&\\lesssim \\| \\nabla f \\|_{L^2(\\Omega_{\\rm c})} \\| \\nabla Z_h \\|_{L^2(\\Omega_{\\rm c})}.\n   \\end{split}\n\t\\end{equation}\nThis proves~\\eqref{weight_inter}.  Proving~\\eqref{za_result} is analogous:\n\\begin{equation*}\n\\int_{\\Omega_{\\rm c}} f\\cdot ({z_h}_{\\alpha} - \\bar{z}_\\alpha)\\, dx \\lesssim~ \\| \\nabla f\\|_{L^2(\\Omega_{\\rm c})} \\cdot {\\| \\nabla {z_h}_{\\alpha} \\|_{L^2(\\Omega_{\\rm c})}} {\\lesssim \\| \\nabla f\\|_{L^2(\\Omega_{\\rm c})} \\cdot \\| {z_h}_{\\alpha} \\|_{L^2(\\Omega_{\\rm c})} },\n\\end{equation*}\nwhere in obtaining the final inequality we have used that for $T \\in \\mathcal{T}_{\\rm a}$,\n\\[\n\\| \\nabla z_h \\|_{L^2(T)} \\lesssim h_T \\| z_h \\|_{L^2(T)} \\lesssim \\| z_h \\|_{L^2(T)}.\n\\]\nFor these choices, we also have the following norm estimates~\\eqref{inter_diff} and~\\eqref{inter_shift}:\n\\begin{align*}\n\\| Z_h- \\bar{Z}\\|_{L^2(\\Omega_{\\rm c})} \\lesssim~& \\|\\nabla Z_h\\|_{L^2(\\Omega_{\\rm c})}, \\\\\n\\| {z_h}_{\\alpha} - \\bar{z}_\\alpha\\|_{L^2(\\Omega_{\\rm c})} \\lesssim~& \\| {z_h}_{\\alpha}\\|_{L^2(\\Omega_{\\rm c})}.\n\\end{align*}\nTo obtain the first of these, we simply take {\\helen{$ f= \\bar{Z}-Z_h$}} in~\\eqref{weight_inter} yielding\\helen{\n\\begin{equation*}\n\\begin{split}\n&\\| Z_h- \\bar{Z}\\|_{L^2(\\Omega_{\\rm c})}^2 \\lesssim~ \\| \\nabla Z_h- \\nabla \\bar{Z}\\|_{L^2(\\Omega_{\\rm c})}\\cdot \\| \\nabla Z_h\\|_{L^2(\\Omega_{\\rm c})} \\lesssim~ \\| \\nabla Z_h\\|_{L^2(\\Omega_{\\rm c})}^2 +\\|\\nabla \\bar{Z}\\|_{L^2(\\Omega_{\\rm c})}^2 \\\\\n&~\\lesssim~ \\| \\nabla Z_h\\|_{L^2(\\Omega_{\\rm c})}^2 +\\|\\nabla Z\\|_{L^2(\\Omega_{\\rm c})}^2 \\lesssim~ \\| \\nabla Z_h\\|_{L^2(\\Omega_{\\rm c})}^2 +\\|\\nabla Z_h\\|_{L^2(\\Omega_{\\rm c})}^2,\n\\end{split}\n\\end{equation*}\nwhere we have applied Young's inequality to deduce the estimate\n\\begin{align*}\n\\|\\nabla Z\\|_{L^2(\\Omega_{\\rm c})}^2 = \\|\\nabla Z\\|_{L^2(\\mathbb{R}^d)}^2  = \\Big\\|\\frac{(\\bar{\\zeta}*\\nabla Z_h)}{\\int \\bar{\\zeta} dx}\\Big\\|^2_{L^2(\\mathbb{R}^d)} \\leq \\|\\nabla Z_h\\|_{L^2(\\mathbb{R}^d)}^2 \\|\\bar{\\zeta}\\|_{L^1(\\mathbb{R}^d)}^2 \\lesssim~ \\|\\nabla Z_h\\|_{L^2(\\Omega_{\\rm c})}^2.\n\\end{align*}\n}\nFor the second of these, we simply have\n\\begin{equation*}\n\\begin{split}\n\\| {z_h}_{\\alpha} - \\bar{z}_\\alpha\\|_{L^2(\\Omega_{\\rm c})} \\leq~& \\| {z_h}_{\\alpha}\\|_{L^2(\\Omega_{\\rm c})} +  \\|\\bar{z}_\\alpha\\|_{L^2(\\Omega_{\\rm c})} \\lesssim~ \\| {z_h}_{\\alpha}\\|_{L^2(\\Omega_{\\rm c})} +  \\|z_\\alpha\\|_{L^2(\\Omega_{\\rm c})} \\\\\n\\lesssim~& \\| {z_h}_{\\alpha}\\|_{L^2(\\Omega_{\\rm c})} + \\| {z_h}_{\\alpha}\\|_{L^2(\\Omega_{\\rm c})},\n\\end{split}\n\\end{equation*}\nwhere we have again used Young's inequality for convolutions.  Next, upon recalling the definition\n\\[\n\\gamma_{\\rm tr} = \\begin{cases} &\\sqrt{1 + \\log(R_{\\rm o}\/R_{{\\rm a}})}, \\quad \\mbox{if $d = 2$,} \\\\\n                                 &1, \\quad \\mbox{if $d = 3$,} \\end{cases}\n\\]\nwe have~\\eqref{h1_int_norm_est} and~\\eqref{l2_int_norm_est}:\n\\begin{equation*}\n\\begin{split}\n\\|\\nabla Z_h\\|_{L^2(\\Omega_{\\rm c})} \\lesssim~& \\gamma_{\\rm tr}\\|\\nabla W \\|_{L^2(\\Omega_{\\rm c})}, \\\\\n\\|{z_h}_{\\alpha}\\|_{L^2(\\Omega_{\\rm c})} \\lesssim~& \\|r_\\alpha\\|_{L^2(\\Omega_{\\rm c})}.\n\\end{split}\n\\end{equation*}\nThe first of these is a consequence of~\\cite[Lemma 7]{blended2014}.  The second is a result of $0 \\leq \\varphi \\leq 1$:\n\\begin{equation*}\n   \\|{z_h}_{\\alpha}\\|_{L^2(\\Omega_{\\rm c})} = \\|I_h(\\varphi r_\\alpha) \\|_{L^2(\\Omega_{\\rm c})} \\leq~ \\|I_h(r_\\alpha) \\|_{L^2(\\Omega_{\\rm c})} = \\|r_\\alpha\\|_{L^2(\\Omega_{\\rm c})}.\n\\end{equation*}\n\n\\end{proof}\n\nWe are now ready to prove Theorem~\\ref{consistency_thm}.\n\n\\begin{proof}[Proof of Theorem~\\ref{consistency_thm}]\n\\dao{Since $\\tilde{I}u$ interpolates $u$ at $\\xi \\in \\mathcal{L}$, we may replace discrete $U^{\\infty}$ with continuous $\\tilde{I}U=\\tilde{U}^{\\infty}$ in~\\eqref{residual_est_new} which leaves us with estimating\n\\begin{equation}\\label{residual_est}\n\\begin{split}\n\\langle & \\mathcal{F}^{{\\rm bqcf}}(U_h,\\bm{p}_h), (W,\\bm{r})\\rangle = \\langle \\mathcal{F}^{{\\rm bqcf}}(U_h,\\bm{p}_h), (W,\\bm{r})\\rangle\n-\\langle \\delta \\mathcal{E}^{{\\rm a}}(U^\\infty, \\bm{p}^\\infty), (U,\\bm{q})\\rangle\\\\\n&\\quad= \\langle \\delta \\mathcal{E}^{{\\rm c}} (U_h,\\bm{p}_h), \\big(I_h(\\varphi W), I_h(\\varphi \\bm{r})\\big)\\rangle\n-\\langle \\delta\\mathcal{E}^{{\\rm a}}(\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty), (Z^*,\\bm{z}^*)\\rangle.\n\\end{split}\n\\end{equation}\n}\nRecalling that $Z_h := I_h(\\varphi W)$, $\\bm{z}_h := I_h(\\varphi \\bm{r})$, and the atomistic and continuum stress representations of~\\eqref{eq:defn_Sa} and~\\eqref{cont_stress_tensor}, we split this into three terms {using simple algebraic manipulations as}\n\\begin{align}\n&\\langle \\delta \\mathcal{E}^{{\\rm c}} (U_h,\\bm{p}_h), \\big(I_h(\\varphi W), I_h(\\varphi \\bm{r})\\big)\\rangle\n-\\langle \\delta\\mathcal{E}^{{\\rm a}}(\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty), (Z^*,\\bm{z}^*)\\rangle\\ra\\nonumber\\\\\n&\\leq \\bigg| \\int_{\\mathbb{R}^d} \\sum_\\beta {\\big[[{\\rm S}^{\\rm c}_{\\rm d}(U_h,\\bm{p}_h)]_\\beta : \\nabla Z_h - [{\\rm S}^{\\rm a}_{\\rm d}(\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty)]_\\beta : \\nabla \\bar{Z}\\big]} \\bigg|\n  + \\bigg| \\int_{\\mathbb{R}^d} {\\sum_{\\alpha,\\beta}} [{\\rm S}^{\\rm c}_{\\rm s} (U_h,\\bm{p}_h)]_{\\alpha\\beta} \\cdot( {z_h}_{\\alpha}- {z_h}_{\\beta}) \\nonumber \\\\\n\t&\\quad - {\\sum_{\\alpha,\\beta}} [{\\rm S}^{\\rm a}_{\\rm s}(\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty)]_{\\alpha\\beta} \\cdot (\\bar{z}_\\alpha - \\bar{z}_\\beta)  \\bigg| +\\bigg| \\int_{\\mathbb{R}^d} {\\sum_\\beta [{\\rm S}^{\\rm a}_{\\rm d}(\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty)]_\\beta : \\nabla \\bar{z}_\\beta }\\bigg| \\nonumber \\\\%\\label{BQCF_consist_eq2}\\\\\n&~=: T^1_{\\rm d} + T_{\\rm s}+ T^2_{\\rm d}.\\nonumber\n\\end{align}\nNext, we analyze these terms separately.\n\n{\\it Term $T^1_{\\rm d}$: } The $T^1_{\\rm d}$ term is identical to the simple lattice case after accounting for the additional approximation of the shifts. Following the ideas\nfrom the simple lattice case~\\cite{blended2014},\n\\helen{we break down $T_{\\rm d}^{1}$ into three additional terms as in Section 6.4.1 of~\\cite{blended2014} (the difference being we do not consider a quadrature error),\n and apply the\nestimates of stress differences from Corollary~\\ref{globel_stress} and the approximating estimates from Lemma~\\ref{approx_lem} and \\eqref{h1_int_norm_est}. This produces\n\\begin{equation*}\\label{T1d_est}\n\\begin{split}\nT^1_{\\rm d} &\\lesssim~ \\bigg| \\int_{\\mathbb{R}^d} \\sum_\\beta \\big\\{[{\\rm S}^{\\rm c}_{\\rm d}(U_h,\\bm{p}_h)]_\\beta -[{\\rm S}^{\\rm c}_{\\rm d}(\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty)]_\\beta \\big\\} : \\nabla Z_h\\, dx\\bigg| \\\\\n&\\qquad +~ \\bigg| \\int_{\\mathbb{R}^d} [{\\rm S}^{\\rm c}_{\\rm d}(\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty)]_\\beta \\big\\} : (\\nabla Z_h-\\nabla \\bar{Z})\\, dx\\bigg| \\\\\n& \\qquad +~ \\bigg| \\int_{\\mathbb{R}^d} \\sum_\\beta \\big\\{[{\\rm S}^{\\rm c}_{\\rm d}(\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty)]_\\beta -[{\\rm S}^{\\rm a}_{\\rm d}(\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty)]_\\beta \\big\\} : \\nabla \\bar{Z}\\, dx\\bigg| \\\\\n&\\lesssim~ \\gamma_{\\rm tr}\\Big(\\|h \\nabla^2 \\tilde{U}^\\infty\\|_{L^2(\\Omega_{\\rm c})}  + \\|h\\nabla \\tilde{\\bm{p}}^\\infty\\|_{L^2(\\Omega_{\\rm c})} \\\\\n& \\qquad  \\qquad+ \\|\\nabla \\tilde{U}^\\infty\\|_{L^2(\\Omega_{\\rm ext})} + \\|\\tilde{\\bm{p}}\\|_{L^2(\\Omega_{\\rm ext})} \\Big)\\cdot \\|\\nabla W\\|_{L^2(\\mathbb{R}^d)}.\n\\end{split}\n\\end{equation*}\n}\n{\\it Term $T_{\\rm s}$: } For the shift term $T_{\\rm s}$, we have\n\\begin{align*}\n   T_{\\rm s} &\\lesssim  \\bigg| \\int_{\\mathbb{R}^d} {\\sum_{\\alpha,\\beta}} [{\\rm S}^{\\rm c}_{\\rm s} (U_h,\\bm{p}_h)]_{\\alpha\\beta} \\cdot( {z_h}_{\\alpha}- {z_h}_{\\beta}) - {\\sum_{\\alpha,\\beta}} [{\\rm S}^{\\rm c}_{\\rm s}(\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty)]_{\\alpha\\beta} \\cdot( {z_h}_{\\alpha}- {z_h}_{\\beta})  \\bigg| \\\\\n\t&\\quad + \\bigg| \\int_{\\mathbb{R}^d} {\\sum_{\\alpha,\\beta}} [{\\rm S}^{\\rm c}_{\\rm s} (\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty)]_{\\alpha\\beta} \\cdot( {z_h}_{\\alpha} - {z_h}_{\\beta} -(\\bar{z}_\\alpha - \\bar{z}_\\beta)) \\bigg| \\\\\n&\\quad + \\bigg| \\int_{\\mathbb{R}^d} {\\sum_{\\alpha,\\beta}} [{\\rm S}^{\\rm c}_{\\rm s} (\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty)]_{\\alpha\\beta} \\cdot(\\bar{z}_\\alpha - \\bar{z}_\\beta)  - {\\sum_{\\alpha,\\beta}} [{\\rm S}^{\\rm a}_{\\rm s}(\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty)]_{\\alpha\\beta} \\cdot (\\bar{z}_\\alpha - \\bar{z}_\\beta)  \\bigg| \\\\\n   &=: T_{\\rm s,1} + T_{\\rm s,2} + T_{\\rm s,3}.\n\\end{align*}\n\nUsing Lipschitz continuity of $\\delta V$ {(in the definition of ${\\rm S}^{\\rm c}_{\\rm s}$)} and the fact that $\\bm{z}_h$ is supported in $\\Omega_{\\rm c}$ \\dao{followed by an application of the test function estimate~\\eqref{l2_int_norm_est}}, we obtain\n\\begin{equation*}\n\\begin{split}\n   \\label{Ts1}\n   |T_{\\rm s,1}| \\lesssim~&\n   \\Big(\\| \\nabla \\Pi_h U - \\nabla \\tilde{U} \\|_{L^2(\\Omega_{\\rm c})}\n      + \\| \\Pi_h \\bm{p} - \\tilde{\\bm{p}} \\|_{L^2(\\Omega_{\\rm c})}\n      \\Big) \\| \\bm{z}_h \\|_{L^2(\\mathbb{R}^d)} \\\\\n\t\t\t\\lesssim~& \\Big(\\| \\nabla \\Pi_h U - \\nabla \\tilde{U} \\|_{L^2(\\Omega_{\\rm c})}\n      + \\| \\Pi_h \\bm{p} - \\tilde{\\bm{p}} \\|_{L^2(\\Omega_{\\rm c})}\n      \\Big) \\| \\bm{r}\\|_{L^2(\\mathbb{R}^d)}\n\\end{split}\n\\end{equation*}\nUsing the stress estimate, Corollary~\\ref{globel_stress}, \\helen{followed by the application of the test function norm estimates~\\eqref{inter_shift} and~\\eqref{l2_int_norm_est},} we get\n\\begin{align*}\n   \\label{Ts3}\n   |T_{\\rm s,3}| &\\lesssim\n      \\Big(\\| \\nabla^2 \\tilde{U} \\|_{L^2(\\Omega_{\\rm c})}\n      + \\| \\nabla \\tilde{\\bm{p}} \\|_{L^2(\\Omega_{\\rm c})} \\Big)\n      \\| \\bar{\\bm{z}} \\|_{L^2(\\mathbb{R}^d)}\\\\\n      &\\lesssim \\Big(\\| \\nabla^2 \\tilde{U} \\|_{L^2(\\Omega_{\\rm c})}\n      + \\| \\nabla \\tilde{\\bm{p}} \\|_{L^2(\\Omega_{\\rm c})} \\Big)\n      \\| \\bm{r} \\|_{L^2(\\mathbb{R}^d)}.\n\\end{align*}\nFinally, to treat $\\bm{z}_h - \\bar{\\bm{z}}$ inside $T_{\\rm s,2}$, we use~\\eqref{za_result} of Lemma~\\ref{interpolation_lemma} with\n$f= [S^{\\rm c}_s(\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty )]_{\\alpha\\beta}$  {\\helen{followed by an application of \\eqref{l2_int_norm_est}, the chain rule, and \\eqref{cont_stress_tensor}:}}\n{\\helen{\n\\begin{equation*}\\label{Ts2}\n\\begin{split}\n|T_{\\rm s,2}| &\\lesssim~ \\big\\|\\nabla \\Big( {\\rm S}^{\\rm c}_{ \\rm s }\\big(\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty\\big)\\Big)\\big\\|_{L^2(\\Omega_{\\rm c})} \\|\\bm{z}_h \\|_{L^2(\\mathbb{R}^d)}\\\\\n&\\lesssim~ \\|\\nabla {\\rm S}^{\\rm c}_{ \\rm s }\\big(\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty\\big)\n\\cdot \\nabla \\big(\\nabla\\tilde{U}^\\infty+\\tilde{\\bm{p}}^\\infty\\big)\\|_{L^2(\\Omega_{\\rm c})} \\|\\bm{r} \\|_{L^2(\\mathbb{R}^d)} .\n\\end{split}\n\\end{equation*}\n}}\nCombining our estimates for $T_{\\rm s,1}, T_{\\rm s,2}$, and $T_{\\rm s,3}$ and \\dao{appealing to Lemma~\\ref{approx_lem} to estimate $T_{\\rm s,1}$ along with the crude estimate $h \\gtrsim 1$} gives\n\\begin{align*}\\label{Ts_est}\n|T_{\\rm s}|\n\\lesssim~ \\Big(\\|h \\nabla^2 \\tilde{U}^\\infty\\|_{L^2(\\Omega_{\\rm c})} &+\\|h\\nabla \\tilde{\\bm{p}}^\\infty\\|_{L^2(\\Omega_{\\rm c})} \\\\\n&+ \\|\\nabla \\tilde{U}^\\infty\\|_{L^2(\\Omega_{\\rm ext})} + \\|\\tilde{\\bm{p}}\\|_{L^2(\\Omega_{\\rm ext})} \\Big) \\|\\bm{r} \\|_{L^2(\\mathbb{R}^d)}.\\nonumber\n\\end{align*}\n\n{\\it Term $T^2_{\\rm d}$: }\nFinally, to estimate $T^2_{\\rm d}$ we split it into\n\\begin{equation*}\\label{T2d_decomp}\n\\begin{split}\n|T^2_{\\rm d}|=~& \\bigg|\\int_{\\mathbb{R}^d}  {\\sum_\\beta[{\\rm S}^{\\rm a}_{\\rm d}(\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty)]_\\beta : \\nabla \\bar{z}_\\beta} \\bigg| \\\\\n\\lesssim~&\n \\bigg|\\int_{\\mathbb{R}^d} {\\sum_\\beta \\big({\\rm S}^{\\rm c}_{\\rm d}(\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty)]_\\beta- [{\\rm S}^{\\rm a}_{\\rm d}(\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty)]_\\beta \\big) : \\nabla \\bar{z}_\\beta} \\bigg| \\\\\n &\\quad +  \\bigg|\\int_{\\mathbb{R}^d} {\\sum_\\beta[{\\rm S}^{\\rm c}_{\\rm d}(\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty)]_\\beta : \\nabla \\bar{z}_\\beta} \\bigg| \\\\\n=:~& T^2_{\\rm d, 1} + T^2_{\\rm d,2}.\n\\end{split}\n\\end{equation*}\nTo estimate $T^2_{\\rm d, 1}$, we note that it is similar to $T^1_{\\rm d}$ in that $\\nabla_{\\rho}\\overline{z_{\\beta}}$ is zero off $\\Omega_{\\rm c}$\n \\dao{(which is due to the support of the blending function and the definition of $\\Omega_{\\rm c}$; see the proof of Lemma~\\ref{interpolation_lemma} for further explanation)}\n so we utilize the stress estimate in Corollary~\\ref{globel_stress} along with the bound\n\\dao{\n\\[\n\\|\\nabla \\bar{z}_\\beta\\| \\lesssim~ \\|\\bar{z}_\\beta\\| \\lesssim~ \\|r_\\beta\\|\n\\]\nwhich follows from\n\\begin{align*}\n\\|\\bar{z}_\\beta\\| \\lesssim~& \\| {z_h}_{\\beta}\\|_{L^2(\\Omega_{\\rm c})} \\qquad \\mbox{by~\\eqref{inter_shift}} \\\\\n\\lesssim~& \\|r_\\beta\\|_{L^2(\\Omega_{\\rm c})} \\qquad \\, \\mbox{  by~\\eqref{l2_int_norm_est}.}\n\\end{align*}\nThis produces\n\\begin{align*}\n T^2_{\\rm d, 1}\\lesssim~& \\left(\\|\\nabla^2 \\tilde{U}^\\infty\\|_{L^2(\\Omega_{\\rm c})} + \\| \\nabla \\tilde{\\bm{p}}^\\infty\\|_{L^2(\\Omega_{\\rm c})}\\right) \\| \\nabla \\bar{\\bm{z}} \\|_{L^2(\\mathbb{R}^d)} \\\\\n\\lesssim~& \\left(\\|\\nabla^2 \\tilde{U}^\\infty\\|_{L^2(\\Omega_{\\rm c})} + \\| \\nabla \\tilde{\\bm{p}}^\\infty\\|_{L^2(\\Omega_{\\rm c})}\\right) \\|  \\bm{r} \\|_{L^2(\\mathbb{R}^d)}.\n\\end{align*}}\nMeanwhile, we may integrate $T^2_{\\rm d,2}$ by parts and use the aforementioned fact that $\\|\\bar{z}_\\beta\\| \\lesssim~ \\|r_\\beta\\|$ to obtain\n\\begin{align*}\nT^2_{\\rm d,2} \\lesssim~\n{\\sum_\\beta \\left\\|{\\rm div} \\left(  [{\\rm S}^{\\rm c}_{\\rm d}(\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty )]_\\beta  \\right)\\right\\|_{L^2(\\Omega^c)} \\|\\bm{r} \\|_{L^2(\\mathbb{R}^d)}}.\n\\end{align*}\n{\\helen{Applying the chain rule to ${\\rm div} \\left(  [{\\rm S}^{\\rm c}_{\\rm d}(\\tilde{U}^\\infty,\\tilde{\\bm{p}}^\\infty )]_\\beta  \\right)$\n(just like for $T_{\\rm s, 2}$), we get }}\n\\begin{align*}\n|T^2_{\\rm d}|\\lesssim T^2_{\\rm d,1} +T^2_{\\rm d,2}&\\lesssim \\left(\\|\\nabla^2 \\tilde{U}^\\infty\\|_{L^2(\\Omega_{\\rm c})} +\\| \\nabla \\tilde{\\bm{p}}^\\infty\\|_{L^2(\\Omega_{\\rm c})}\\right) \\| \\bm{r} \\|_{L^2(\\mathbb{R}^d)}\\\\\n&\\lesssim \\left(\\|h\\nabla^2 \\tilde{U}^\\infty\\|_{L^2(\\Omega_{\\rm c})} +\\| h\\nabla \\tilde{\\bm{p}}^\\infty\\|_{L^2(\\Omega_{\\rm c})}\\right) \\| \\bm{r} \\|_{L^2(\\mathbb{R}^d)}.\n\\end{align*}\n\n\nCombining our estimates for $T^1_{\\rm d},T_{\\rm s}$,\nand $T^2_{\\rm d}$ yields the stated result.\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{Stability}\\label{stab}\nThe second key ingredient in our proof of Theorem~\\ref{main_thm} is the stability estimate~\\eqref{stab_est}; \\dao{this in turn implies a bound on the inverse of {\\helen{the linearised BQCF operator}}, which we will use in a quantitative version of the inverse function theorem to establish existence of the solution to our BQCF equations.}  Conceptually, the proof of stability is similar to that of the simple lattice case presented in~\\cite{blended2014}.\n\n\n\\begin{theorem}[Stability of BQCF]\\label{stab_theorem_full}\n Suppose that Assumptions~\\ref{assumption1},~\\ref{assumptionSite}, and~\\ref{assumption2} hold. There exists a critical size, $R_{\\rm core}^*$, of the atomistic region such that, for all shape regular meshes and blending functions meeting the requirements of Section~\\ref{bqcf} and $R_{\\rm core} \\geq R_{\\rm core}^*$,\n\\[\n\\frac{\\gamma_{{\\rm a}}}{2}\\|(W,\\bm{r})\\|_{\\rm ml}^2 \\leq~ \\<\\delta \\mathcal{F}^{{\\rm bqcf}}(\\Pi_h(U^\\infty, \\bm{p}^\\infty)) (W, \\bm{r}),(W, \\bm{r})\\>, \\quad \\forall \\, (W, \\bm{r}) \\in \\bm{\\mathcal{U}}_{h,0} \\times \\bm{\\mathcal{P}}_{h,0}.\n\\]\n\\end{theorem}\nAs an intermediate step we also prove stability of the reference state.\n\n\\begin{theorem}[Stability of BQCF at Reference State]\\label{stab_theorem}\nSuppose that Assumptions~\\ref{assumption1},~\\ref{assumptionSite}, and~\\ref{assumption2} hold.  There exists a critical size $R_{\\rm core}^*$ of the atomistic region such that, for all meshes having shape regularity constant bounded below by $C_{\\mathcal{T}_h}$ and blending functions meeting the requirements of Section~\\ref{bqcf} and $R_{\\rm core} \\geq R_{\\rm core}^*$,\n\\[\n   \\frac{3}{4}\\gamma_{\\rm a}\\|(W,\\bm{r})\\|_{\\rm ml}^2 \\leq~ \\<\\delta \\mathcal{F}^{{\\rm bqcf}}_{\\rm hom}(0) (W, \\bm{r}),(W, \\bm{r})\\>, \\quad \\forall \\, (W, \\bm{r}) \\in \\bm{\\mathcal{U}}_{h,0} \\times \\bm{\\mathcal{P}}_{h,0}.\n\\]\n\\end{theorem}\n\nBefore we present the proofs of these results in Sections~\\ref{stab_def_free} and~\\ref{stab_defect_present} we apply them to complete the proof of our main result, Theorem~\\ref{main_thm}.\n\n\n\n\\subsection{Proof of the main result}\n\\label{sec:proof_main_result}\n\n\\begin{proof}[Proof of Theorem~\\ref{main_thm}]\nWe apply the inverse function theorem, Theorem~\\ref{inverseFunctionTheorem},  to the BQCF variational operator $\\mathcal{F}^{{\\rm bqcf}}$ at the linearization point $\\Pi_h(U^\\infty, \\bm{p}^\\infty)$.  The parameters $\\eta$ and $\\sigma$ defined in Theorem~\\ref{inverseFunctionTheorem} are\n\\begin{equation*}\n\\begin{split}\n&\\eta :=~ \\gamma_{\\rm tr}\\big(\\|h\\nabla^2 \\tilde{U}\\|_{L^2(\\Omega_{\\rm c})} + \\|h\\nabla \\tilde{\\bm{p}}\\|_{L^2(\\Omega_{\\rm c})}  + \\|\\nabla \\tilde{U}\\|_{L^2(\\Omega_{\\rm ext})} \\big. \\\\\n&\\big. \\qquad\\qquad +  \\| \\tilde{\\bm{p}}\\|_{L^2(\\Omega_{\\rm ext})}\\big)\\cdot \\|(W,\\bm{r})\\|_{\\rm ml}, \\quad \\forall (W,\\bm{r}) \\, \\in \\bm{\\mathcal{U}}_{h,0} \\times \\bm{\\mathcal{P}}_{h,0},\n\\end{split}\n\\end{equation*}\nwhich is the consistency error from Theorem~\\eqref{consistency_thm}, and\n\\[\n\\sigma^{-1} := \\frac{\\gamma_{\\rm a}}{2},\n\\]\nwhich is the coercivity constant from Theorem~\\eqref{stab_theorem_full} that exists so long as $R_{\\rm core} \\geq R_{\\rm core}^*$, where $R_{\\rm core}^*$ is furnished by Theorem~\\eqref{stab_theorem_full}.  (The requirement $R_{\\rm core} \\geq R_{\\rm core}^*$ means the domain decomposition procedure meets the requirements stated in Theorem~\\ref{main_thm}.)  The Lipschitz estimate on $\\delta \\mathcal{F}^{{\\rm bqcf}}$ is a direct result of the assumptions made on the site potential in Assumption~\\ref{assumption1}.  Applying the inverse function theorem with these parameters gives existence of $(U^{{\\rm bqcf}}, \\bm{p}^{{\\rm bqcf}})$ and the stated error estimate,~\\eqref{main_estimate}, follows from the inverse function theorem and the approximation lemma, Lemma~\\ref{approx_lem}.\n\\end{proof}\n\nThe remainder of the paper is devoted to proving Theorems~\\ref{stab_theorem_full} and~\\ref{stab_theorem}.\n\n\\subsection{Stability of BQCF at defect-free reference state}\\label{stab_def_free}\n\nWe first prove Theorem \\ref{stab_theorem}, that is, coercivity of the\nhomogeneous BQCF operator,\n\\begin{align*}\n\\<\\delta \\mathcal{F}^{{\\rm bqcf}}_{\\rm hom}(0)(W,\\bm{r}),(W,\\bm{r})\\> =~& \\<\\delta^2\\mathcal{E}^{\\rm a}_{\\rm hom}(0)((1-\\varphi)W,(1-\\varphi)\\bm{r}), (W, \\bm{r})\\> \\\\\n&\\qquad +  \\<\\delta^2\\mathcal{E}^{\\rm c}(0)(I_h(\\varphi W),I_h(\\varphi \\bm{r})), (W, \\bm{r})\\>.\n\\end{align*}\nThat is, we want to show that there exists $\\gamma_{{\\rm bqcf}}$ independent of the approximation parameters such that, for sufficiently large $R_{\\rm core}$,\n\\begin{equation} \\label{eq:gamma_bqcf}\n   0 < \\gamma_{\\rm bqcf}\\|(W,\\bm{r})\\|_{\\rm ml}^2 \\leq \\<\\delta \\mathcal{F}^{{\\rm bqcf}}_{\\rm hom}(0) (W, \\bm{r}),(W, \\bm{r})\\>.\n\\end{equation}\nThe proof via contradiction is involved; hence we first outline and motivate the procedure and then give a number of technical results required to prove the theorem at the end of this section. \\dao{The main idea is that the linearized BQCF operator consists of an atomistic second variation and a continuum second variation.  Each of these can be individually shown to be coercive so intuitively, we would expect this linearized operator to be coercive for any test pair $(W,\\bm{r})$ with support concentrated near the origin (in which case the blending function is zero) and for $(W,\\bm{r})$ with support concentrated far from the origin (in which case the blending function would be one).  Thus, we expect the only possible instabilities to occur with test pairs having some support over the blending region.  Since there is no defect in the homogeneous case, any such instability should also occur for any geometric setup, i.e., we can consider the BQCF method for a sequence of growing atomistic domain sizes and should still have an unstable mode.  Thus we shall consider such a sequence and then rescale this sequence so that the atomistic region in each case is contained in a ball of fixed radius about the origin and such that these unstable modes converge (in a sense to be made precise momentarily) to some continuum limit.  We then consider evaluating the suitably rescaled linearized BQCF operator on this sequence and show using the aforementioned stability of the atomistic and continuum components \\textit{and} convergence of the test pairs $(W,\\bm{r})$ that this leads to a contradiction.  One of the main technical difficulties encountered here is that due to blending by forces, the individual atomistic\/continuum components and hence the linearized BQCF operator is not a symmetric bilinear form.  Thus we must take some care in converting the force-based formulation to a form suitable to using the existing coercivity estimates on the atomistic and continuum Hessians.}\n\nThe negation of \\eqref{eq:gamma_bqcf} is: ``for all atomistic region sizes $R_{\\rm a}$, there exists a blending function $\\varphi$ and a mesh $\\mathcal{T}_h$ compatible with the assumptions of Section~\\ref{sec:approx_params} (and in particular Assumption~\\ref{assumption-shapereg}), {\\helen{as well as a test pair, $(W, \\bm{r})$ with norm scaled to one, such that}}\n\\begin{equation} \\label{eq:negation}\n\\frac{3}{4} \\gamma_{{\\rm a}}> \\<\\delta \\mathcal{F}_{\\rm hom}^{{\\rm bqcf}}(0) (W, \\bm{r}),(W, \\bm{r})\\>.\\mbox{''}\n\\end{equation}\nThus, for contradiction, suppose that there exists a sequence $R_{{\\rm a},n} \\to \\infty$ with associated meshes $\\mathcal{T}_{h,n}$, blending functions $\\varphi_n$, finite element spaces $\\bm{\\mathcal{U}}_{h,0}^n \\times \\bm{\\mathcal{P}}_{h,0}^n$, and test pairs $(W_n, \\bm{r}_n) \\in \\bm{\\mathcal{U}}_{h,0}^n \\times \\bm{\\mathcal{P}}_{h,0}^n$ with norm one such that\n\\begin{equation}\\label{contra_seq}\n\\begin{split}\n&\\frac{3}{4}\\gamma_{\\rm a}  >\n\\sum_{\\xi \\in \\mathcal{L}} \\sum_{(\\rho\\alpha\\beta)}\\sum_{(\\tau\\gamma\\delta)} V_{,(\\rho\\alpha\\beta)(\\tau\\gamma\\delta)}:D_{(\\rho\\alpha\\beta)}((1-{\\varphi}_n) {W}_n,(1-{\\varphi}_n){\\bm{r}}_n):D_{(\\rho\\alpha\\beta)}({W}_n,{\\bm{r}}_n) \\\\\n& \\qquad \\qquad + \\int_{\\mathbb{R}^d} \\sum_{(\\rho\\alpha\\beta)}\\sum_{(\\tau\\gamma\\delta)} V_{,(\\rho\\alpha\\beta)(\\tau\\gamma\\delta)}:\\nabla_{(\\rho\\alpha\\beta)}( {I_{h}}  ( {\\varphi}_n( {W}_n,  {\\bm{r}}_n))) :\\nabla_{(\\rho\\alpha\\beta)}( {W}_n,  {\\bm{r}}_n) \\, dx,\n\\end{split}\n\\end{equation}\nwhere we have omitted the argument, $0$, in $V_{,(\\rho\\alpha\\beta)(\\tau\\gamma\\delta)}(0)$ and where $I_{h}$ is now the piecewise linear interpolant on $\\mathcal{T}_{h,n}$.\n\nWe now rescale space in \\eqref{eq:negation} and derive a continuum scaling limit,\nfrom which we will be able to obtain a contradiction.  To that end, let $\\epsilon_n = 1\/R_{{\\rm a},n}$, and define the set of scaled parameters\n\\begin{equation} \\label{eq:stab:rescaling}\n\\begin{split}\n\\hat{\\xi}_n =~& \\epsilon_n \\xi \\\\\n\\hat{x}_n =~& \\epsilon_n x \\\\\n\\hat{r}_n(\\dao{\\hat{x}_n}) =~& \\epsilon_n^{-d\/2} r_n(\\dao{\\hat{x}_n}\/\\epsilon_n) \\\\\n\\hat{W}_n(\\dao{\\hat{x}_n}) =~& \\epsilon_n^{1-d\/2} W_n(\\dao{\\hat{x}_n}\/\\epsilon_n) \\\\\n\\hat{\\varphi}_n(\\dao{\\hat{x}_n}) =~& \\varphi_n(\\dao{\\hat{x}_n}\/\\epsilon_n).\n\\end{split}\n\\end{equation}\nIn terms of these rescaled quantities, {\\helen{we define $\\hat{\\nabla}:=\\epsilon_n^{-1}\\nabla_x=\\nabla_{\\hat{x}_n}$} (when the subscript $n$ is clear we use $\\hat{\\nabla}$)} and then have\n{\\helen{\n\\begin{align*}\n\\| \\nabla_{\\hat{x}_n} \\hat{W}_n \\|^2_{L^2(\\mathbb{R}^d)} = \\| \\nabla_{x} W_n \\|^2_{L^2(\\mathbb{R}^d)},& \\,\\, \\|\\epsilon_n\\nabla_{\\hat{x}_n} \\hat{r}_n\\|^2_{L^2(\\mathbb{R}^d)} = \\|\\nabla_x r_n\\|^2_{L^2(\\mathbb{R}^d)}, \\\\\n \\| \\hat{r}_n^\\alpha \\|^2_{L^2(\\mathbb{R}^d)} =~& \\| r_n^\\alpha \\|^2_{L^2(\\mathbb{R}^d)},\n\\end{align*}\n}}\nand the rescaled BQCF operator is\n{\\helen{\n\\begin{equation}\\label{bqcfHessian}\n\\begin{split}\n&\n \\<\\delta \\mathcal{F}_{{\\rm hom},n}^{{\\rm bqcf}}(0) (\\hat{W}_n, \\hat{\\bm{r}}_n),(\\hat{W}_n, \\hat{\\bm{r}}_n)\\>  := \\\\\n&\\qquad \\epsilon^d_n \\sum_{\\hat{\\xi} \\in \\epsilon_n\\mathcal{L}} \\mathbb{C}:D_{n}((1-\\hat{\\varphi}_n)(\\hat{W}_n,\\hat{\\bm{r}}_n)):D_n(\\hat{W}_n,\\hat{\\bm{r}}_n)(\\hat{\\xi}) \\\\\n&\\qquad + \\int_{\\mathbb{R}^d} \\mathbb{C}:\\hat{\\nabla}( I_{h,n}(\\hat{\\varphi}_n(\\hat{W}_n, \\hat{\\bm{r}}_n))) :\\hat{\\nabla}(\\hat{W}_n, \\hat{\\bm{r}}_n) \\, d\\hat{x}_n,\n\\end{split}\n\\end{equation}\n}}\nwhere $I_{h,n}$ is the piecewise linear interpolant on $\\epsilon_n\\mathcal{T}_{h,n}$ and\n{\\helen{\n\\begin{align*}\nD_{n}( \\hat{W} , \\hat{\\bm{r}} ) :=~& \\big(D_{(\\rho \\alpha\\beta),n}( \\hat{W} , \\hat{\\bm{r}} )\\big)_{(\\rho \\alpha\\beta) \\in \\mathcal{R}}, \\\\\nD_{(\\rho \\alpha\\beta),n}( \\hat{W} , \\hat{\\bm{r}} ) (\\hat{\\xi}):=~& \\frac{\\hat{W}(\\hat{\\xi} + \\epsilon_n \\rho) + \\epsilon_n \\hat{r}_{n}^{\\beta}(\\hat{\\xi} + \\epsilon_n \\rho) - \\hat{W}(\\hat{\\xi}) - \\epsilon_n \\hat{r}_{n}^{\\alpha}(\\hat{\\xi})}{\\epsilon_n}.\n\\end{align*}\n}}\nThe rescaling of the shifts $\\hat{r}_n^\\alpha$ is one order lower than the rescaling of displacements,  which is due to the fact that shifts are already discrete gradients.\n\nWe also define an interpolant onto the scaled lattice $\\epsilon_n\\mathcal{L}$ by $I_n$, a projection operator from the scaled lattice to finite element spaces $\\bm{\\mathcal{U}}^{n}_{h,0} \\times \\bm{\\mathcal{P}}^n_{h,0}$ on $\\mathcal{T}_{h,n}$ by $\\Pi_{h,n} := S_{h,n} T_{r_{{\\rm i},n}}$, and the scaled finite element basis function\n\\[\n\\bar{\\zeta}_n(x) := \\epsilon_n^{-d}\\bar{\\zeta}(x\/\\epsilon_n).\n\\]\n\nSince $\\dao{\\hat{\\nabla}} \\hat{W}_n$ is bounded in $L^2$ and since each $\\hat{r}_n^\\alpha$ is also bounded (both having norm less than one), we may extract weakly convergent subsequences.  Furthermore, $\\epsilon_n \\dao{\\hat{\\nabla}} \\hat{r}_n^\\alpha$ is also bounded in $L^2$ so we may take it to be weakly convergent as well.  By replacing the original sequences with these weakly convergent subsequences (for notational convenience), we have $\\dao{\\hat{\\nabla}} \\hat{W}_n \\rightharpoonup \\dao{\\hat{\\nabla}} \\hat{W}_0$, $\\hat{r}_n^\\alpha \\rightharpoonup \\hat{r}_0^\\alpha$,\nand $\\epsilon_n \\dao{\\hat{\\nabla}} \\hat{r}_n^\\alpha \\rightharpoonup \\hat{R}^\\alpha_0$ in $L^2(\\mathbb{R}^d)$ for some functions $\\hat{W}_0, \\hat{r}_0^\\alpha$, and $\\hat{R}^\\alpha_0$ for each $\\alpha$.  However, since $\\hat{r}_n^\\alpha$ is bounded in $L^2$ and $\\epsilon_n\\hat{r}_n^\\alpha \\to 0$ in $L^2$, $\\hat{R}_0^\\alpha = 0$.\n\nNext, we choose explicit equivalence representatives for $\\hat{W}_n$; namely, we choose $\\hat{W}_n$ such that $\\int_{B_1(0)} \\hat{W}_n = 0$.   For this choice, we have $\\|\\hat{W}_n\\|_{L^2(B_1(0))} \\lesssim \\|\\dao{\\hat{\\nabla}} \\hat{W}_n\\|_{L^2(B_1(0))}$, and as $H^1$ is compactly embedded in $L^2$, there exists a strongly convergent subsequence, which we again denote by $\\hat{W}_n$, such that $\\hat{W}_n \\to \\hat{W}_0$ strongly in $L^2(B_1(0))$.\n\nWe also note here that $\\hat{W}_n \\rightharpoonup \\hat{W}_0$ in the space\n\\[\n\\dot{\\bm{H}}^{1} {\\helen{(\\mathbb{R}^d,\\mathbb{R}^n)}} := \\left\\{f \\in H^1_{\\rm loc}(\\mathbb{R}^d,\\mathbb{R}^n)\/\\mathbb{R}^n : \\|\\nabla f\\|_{L^2(\\mathbb{R}^d)} < \\infty \\right\\},\n\\]\nand so $\\hat{W}_0 \\in \\dot{\\bm{H}}^1(\\mathbb{R}^d, \\mathbb{R}^n)$ as well~\\cite{suli2012}.\n\nThe purpose of these subsequences is to use the pairs $(\\hat{W}_n, \\hat{\\bm{r}}_n)$ to test with $\\delta\\mathcal{F}_{{\\rm hom},n}^{{\\rm bqcf}}(0)$. However, as these test pairs only consist of weakly convergent sequences and since the inner product of two weakly convergent sequences is not necessarily convergent, we further split $\\hat{W}_n$ and $\\hat{\\bm{r}}_n$ into the sum of a strongly convergent sequence and a sequence weakly convergent to zero.\n\nThis splitting is accomplished by setting\n\\begin{equation}\\label{splitting}\n\\hat{X}_n := \\Pi_{h,n}(\\eta_{j_n}  * \\hat{W}_0), \\qquad\n\\hat{s}_n^\\alpha := \\Pi_{h,n}(\\eta_{j_n}  * \\hat{r}_0^\\alpha),\n\\end{equation}\nwhere $\\eta$ is a standard mollifier, $\\eta_j(x) = j^{-d} \\eta(x\/j)$, and $j_n \\to 0$ sufficiently slowly to ensure that  the sequences $\\hat{X}_n$ and $\\hat{s}_n^\\alpha$ are strongly convergent to, respectively, $\\hat{W}_0$ and $\\hat{r}_0^\\alpha$. We will impose several further properties on the sequence $j_n$ in Lemma~\\ref{seq_lemma} below, \nbut for the remainder of the present section, we make the following conventions for notational convenience.\n\n\\begin{remark}\\label{remark_drop_hats}\n{\\helen{\nTo simplify and lessen the notations hereafter, we drop the hat notation on the sequences $X_n, Z_n, \\bm{s}_n, \\bm{t}_n$ as well as on their derivatives, and so forth.\n}}\n\\end{remark}\n\nFurther, we define\n\\[\n   \\psi_n := 1-\\varphi_n, \\quad \\mbox{and}\n   \\quad V_{,(\\rho \\alpha\\beta)(\\tau\\gamma\\delta)} := V_{,(\\rho \\alpha\\beta)(\\tau\\gamma\\delta)}\\big(0\\big),\n\\]\nand use the notation\n\\begin{align*}\\label{stab_notation}\nV_{,(\\rho \\alpha\\beta)(\\tau\\gamma\\delta)}\\big( \\cdot \\big):v:w :=~& w^{{\\hspace{-1pt}\\top}}\\big[V_{,(\\rho \\alpha\\beta)(\\tau\\gamma\\delta)}\\big( \\cdot \\big)\\big]v \\quad \\forall v,w \\in \\mathbb{R}^n, \\nonumber \\\\\n\\mathbb{C} : D(W,\\bm{q}): D(Z,\\bm{r}) :=~& \\sum_{(\\rho \\alpha\\beta) \\in \\mathcal{R}} \\sum_{(\\tau\\gamma\\delta) \\in \\mathcal{R}} V_{,(\\rho \\alpha\\beta)(\\tau\\gamma\\delta)}:D_{(\\rho \\alpha\\beta)}(W,\\bm{q}):D_{(\\tau\\gamma\\delta)}(Z,\\bm{r}), \\\\\n\\mathbb{C} : \\nabla (W,\\bm{q}): \\nabla (Z,\\bm{r}) :=~& \\sum_{(\\rho \\alpha\\beta) \\in \\mathcal{R}} \\sum_{(\\tau\\gamma\\delta) \\in \\mathcal{R}} V_{,(\\rho \\alpha\\beta)(\\tau\\gamma\\delta)}:(\\nabla(W,\\bm{q})):(\\nabla(Z,\\bm{r})),\n\\end{align*}\nwhere the argument of $V_{,(\\rho \\alpha\\beta)(\\tau\\gamma\\delta)}(\\cdot )$ is omitted if evaluated at the reference state.\n\n\n\\begin{lemma}\\label{seq_lemma}\nThere exists $\\psi_0 \\in {\\rm C}^1$ is such that $\\psi_n \\to \\psi_0$ in ${\\rm C}^1(B_1(0))$.  Furthermore, there exists a sequence $j_n \\to 0$ such that the sequences defined by $X_n, \\bm{s}_n$ in~\\eqref{splitting} and $Z_n := W_n - X_n$ and $t_n^\\alpha := r_n^\\alpha - s_n^\\alpha$ satisfy the following convergence properties, where $\\to$ and $\\rightharpoonup$ denote respectively strong and weak $L^2(\\mathbb{R}^d)$ convergence.\n\\begin{equation}\\label{verge_result}\n\\begin{split}\n\\nabla  W_n \\rightharpoonup~& \\nabla  W_0, \\quad\nr^\\alpha_n \\rightharpoonup~ r^\\alpha_0, \\quad\n\\epsilon_n\\nabla r^\\alpha_n \\rightharpoonup 0, \\quad\n\\nabla  X_n \\to~ \\nabla  W_0, \\quad\ns_n^\\alpha \\to~  r^\\alpha_0,  \\\\\n\\epsilon_n \\nabla s^\\alpha_n \\to~& 0, \\quad\n\\nabla Z_n \\rightharpoonup~ 0, \\quad\nt_n^\\alpha \\rightharpoonup~ 0, \\quad\n\\epsilon_n \\nabla t_n^\\alpha \\rightharpoonup~ 0, \\\\\nW_n \\to~& W_0 \\, \\mbox{in $L^2(B_1(0))$,} \\quad X_n \\to~ W_0 \\, \\mbox{in $L^2(B_1(0))$}, \\quad Z_n \\to~ 0 \\, \\mbox{in $L^2(B_1(0))$}\n\\end{split}\n\\end{equation}\nMoreover, {\\helen{let $I$ denote the identity and upon defining the quantities}}\n\\begin{align*}\n &{\\rm R}^{\\rm def}_n(x) {\\helen{ :={\\rm R}^{\\rm def}_n\\left(\\psi_n\\right)(x)=~ }} \\\\\n &\\epsilon_n^d \\sum_{\\xi \\in \\epsilon_n\\mathcal{L}}\\sum_{\\substack{(\\rho \\alpha\\beta) \\\\ (\\tau\\gamma\\delta)}} V_{,(\\rho\\alpha\\beta)(\\tau\\gamma\\delta)}(0)D_{(\\tau\\gamma\\delta),n}(\\psi_n( X_n,s_n))\\otimes\\frac{\\rho}{\\epsilon_n}\\int_0^{\\epsilon_n} \\zeta_n(\\xi + t\\rho - x)\\, dt, \\\\\n&{\\rm R}^{\\rm shift}_n(x) {\\helen{ :={\\rm R}^{\\rm shift}_n\\left(\\psi_n\\right)(x)=~ }} \\\\\n&\\epsilon_n^d \\sum_{\\xi \\in \\epsilon_n\\mathcal{L}}\\sum_{\\substack{(\\rho \\alpha\\beta) \\\\ (\\tau\\gamma\\delta)}}  V_{,(\\rho\\alpha\\beta)(\\tau\\gamma\\delta)}(0)D_{(\\tau\\gamma\\delta),n}(\\psi_n{X}_n,\\psi_n{s}_n) \\bar{\\zeta}_n(\\xi- x), \\\\\n &{\\rm S}^{\\rm def}_n(x) {\\helen{ := {\\rm R}^{\\rm def}_n\\left(I\\right)(x),}} \\quad {\\rm S}^{\\rm shift}_n(x) {\\helen{ :={\\rm R}^{\\rm shift}_n\\left(I\\right)(x) }} \\\\\n \n&{\\rm S}_n^{\\rm inner}(x):=~\\\\\n&\\epsilon_n^d \\sum_{\\xi \\in \\epsilon_n\\mathcal{L}} \\sum_{\\substack{(\\rho \\alpha\\beta) \\\\ (\\tau\\gamma\\delta)}} V_{,(\\rho\\alpha\\beta)(\\tau\\gamma\\delta)}:D_{(\\rho \\alpha\\beta),n}(\\psi_n X_n, \\psi_n \\bm{s}_n):  D_{(\\tau\\gamma\\delta),n}(X_n,\\bm{s}_n),  \\\\\n\\end{align*}\nthe sequence $j_n$ may further be chosen so that\n\\begin{align}\\label{verge_res_2}\n{\\rm S}^{\\rm def}_n(x)  &\\to \\sum_{\\substack{(\\rho \\alpha\\beta) \\\\ (\\tau\\gamma\\delta)}} V_{(\\rho\\alpha\\beta)(\\tau\\gamma\\delta)}(0)\\nabla_{(\\tau\\gamma\\delta)}({W}_0,{\\bm{s}}_0),\\nonumber \\\\\n{\\rm S}^{\\rm shift}_n(x)  &\\to\\sum_{\\substack{(\\rho \\alpha\\beta) \\\\ (\\tau\\gamma\\delta)}} V_{(\\rho\\alpha\\beta)(\\tau\\gamma\\delta)}(0)\\nabla_{(\\tau\\gamma\\delta)}({W}_0,{\\bm{s}}_0),\\nonumber\\\\\n {\\helen{ {\\rm R}^{\\rm def}_n(x)}}  &\\helen{\\to  \\sum_{\\substack{(\\rho \\alpha\\beta) \\\\ (\\tau\\gamma\\delta)}} V_{,(\\rho\\alpha\\beta)(\\tau\\gamma\\delta)}(0)\\nabla_{(\\tau\\gamma\\delta)}(\\psi_0{W}_0,\\psi_0{\\bm{s}}_0), } \\\\\n{\\rm R}^{\\rm shift}_n(x) &\\to \\sum_{\\substack{(\\rho \\alpha\\beta) \\\\ (\\tau\\gamma\\delta)}} V_{,(\\rho\\alpha\\beta)(\\tau\\gamma\\delta)}(0)\\nabla_{(\\tau\\gamma\\delta)}(\\psi_0{W}_0,\\psi_0{\\bm{s}}_0), \\nonumber\\\\\n{\\rm S}_n^{\\rm inner}(x) &\\to \\int_{\\mathbb{R}^d} \\sum_{\\substack{(\\rho \\alpha\\beta) \\\\ (\\tau\\gamma\\delta)}} V_{,(\\rho \\alpha\\beta)(\\tau\\gamma\\delta)} :\\big(\\nabla_{(\\rho \\alpha\\beta)}(\\psi_0( W_0, \\bm{s}_0))\\big):\\big(\\nabla_{(\\tau\\gamma\\delta)}( W_0,\\bm{s}_0)\\big) dx,\\nonumber\n\\end{align}\nwith convergence being in $L^2(\\mathbb{R}^d)$.\n\\end{lemma}\n\n\\begin{proof}\nThe key fact in proving this result is that $j_n$ may be chosen to tend to zero sufficiently slowly such that any one of these properties holds individually, and by appropriately selecting subsequences using a diagonalization argument, they may be chosen so that all hold simultaneously.  The full proof is given in the Appendix.\n\\end{proof}\n\nWe now state a convergence result for ``cross-terms'' appearing in $\\delta \\mathcal{F}^{{\\rm bqcf}}_{{\\rm hom},n}(0)$ involving products of strongly and weakly convergent (to zero) sequences. The proof is given in the appendix.\n\n\\begin{lemma}\\label{more_lemma}\nWith $Z_n, X_n, \\bm{t}_n$, and $\\bm{s}_n$ as defined in Lemma~\\ref{seq_lemma},\n\\begin{align}\n&\\epsilon^d \\sum_{\\xi \\in \\epsilon_n\\mathcal{L}} \\mathbb{C}:D_{n}(\\psi_n {Z}_n,\\psi_n \\bm{t}_n):D_{n}({X}_n,\\bm{s}_n) \\to 0,\n\\quad \\text{and} \\label{more_1} \\\\\n&\\epsilon^d \\sum_{\\xi \\in \\epsilon_n\\mathcal{L}} \\mathbb{C}:D_{n}(\\psi_n {X}_n,\\psi_n\\bm{s}_n):D_{n}( {Z}_n, \\bm{t}_n) \\to 0 \\label{more_2}.\n\\end{align}\n\\end{lemma}\n\n\nThe next lemma manipulates the product of two weakly convergent sequences.  The idea is that we may shift the blending function \\dao{function $\\psi_n = 1-\\varphi_n$} in a way to use coercivity of the atomistic and continuum Hessians. The proof is again given in the appendix.\n\n\\begin{lemma}\\label{weak_lemma}\nLet $Z_n, X_n, \\bm{t}_n$, $\\bm{s}_n$, $\\theta_n = \\sqrt{\\psi_n}$, and $\\theta_0 = \\sqrt{\\psi_0}$ be as defined above in Lemma~\\ref{seq_lemma}.  Then\n\\begin{align*}\n&\\lim_{n\\to\\infty}\\epsilon_n^d \\sum_{\\xi \\in \\epsilon_n\\mathcal{L}} \\mathbb{C}:D_{n}(\\theta^2_nZ_n,\\theta^2_n\\bm{t}_n):D_{n}(Z_n,\\bm{t}_n) \\\\\n&=~ \\lim_{n\\to\\infty}\\epsilon_n^d \\sum_{\\xi \\in \\epsilon_n\\mathcal{L}} \\mathbb{C}:D_{n}(\\theta_nZ_n,\\theta_n\\bm{t}_n):D_{n}(\\theta_nZ_n,\\theta_n\\bm{t}_n).\n\\end{align*}\n\\end{lemma}\n\nWe are now positioned to prove Theorem~\\ref{stab_theorem}.\n\n\\begin{proof}[Proof of Theorem~\\ref{stab_theorem}, Stability of BQCF at Reference State]\nWe use the scaling~\\eqref{bqcfHessian} and substitute \\dao{(from Lemma~\\ref{seq_lemma}) the quantities} $W_n = Z_n + X_n$, $r_n^\\alpha = t_n^\\alpha + s_n^\\alpha$, $\\psi_n = 1-\\varphi_n$, and $\\theta_n = \\sqrt{1- {\\varphi}_n}$.  We divide the proof into three steps: (1) we derive an expression for the atomistic portion of $\\delta\\mathcal{F}_{{\\rm hom},n}^{{\\rm bqcf}}(0)$ in the $\\liminf$ as $n \\to \\infty$, (2) we derive an expression for the continuum component of $\\delta\\mathcal{F}_{{\\rm hom},n}^{{\\rm bqcf}}(0)$, and (3) we combine the results and use stability of the individual atomistic and continuum components to derive a contradiction.\n\n\\medskip\n\\noindent \\textit{Step 1:}  The first variation, $\\delta\\mathcal{F}_{{\\rm hom},n}^{{\\rm bqcf}}(0)$, computed in~\\eqref{bqcfHessian} is a sum of an atomistic and continuum component. The discrete, atomistic contribution is\n\\begin{align}\n& \\big\\< \\delta^2 \\mathcal{E}^{\\rm a}_{{\\rm hom}, n } (0)(1-\\varphi_n)(W_n,\\bm{r}_n),(W_n,\\bm{r}_n)\\>  \\nonumber\\\\\n&= \\epsilon_n^d \\sum_{\\xi \\in \\epsilon_n\\mathcal{L}} \\mathbb{C}:D_{n}(\\theta^2_n {W}_n,\\theta^2_n {\\bm{r}}_n):D_{n}( {W}_n, {\\bm{r}}_n) \\nonumber\\\\\n&= \\epsilon_n^d \\sum_{\\xi \\in \\epsilon_n\\mathcal{L}} \\mathbb{C}:D_{n}(\\theta^2_n {Z}_n + \\theta^2_n {X}_n,\\theta^2_n {\\bm{t}}_n + \\theta^2_n{\\bm{s}}_n):D_{n}( {Z}_n +  {X}_n, {\\bm{t}}_n +  {\\bm{s}}_n) \\nonumber\\\\\n&=~ \\epsilon_n^d \\sum_{\\xi \\in \\epsilon_n\\mathcal{L}} \\mathbb{C}:\\Big[D_{n}(\\theta^2_n {Z}_n,\\theta^2_n {\\bm{t}}_n):D_{n}( {Z}_n, {\\bm{t}}_n) +D_{n}(\\theta^2_n {Z}_n,\\theta^2_n {\\bm{t}}_n):D_{n}({X}_n,{\\bm{s}}_n)\\nonumber \\\\\n&\\quad+~ D_{n}(\\theta^2_n {X}_n,\\theta^2_n {\\bm{s}}_n):D_{n}({Z}_n,{\\bm{t}}_n) + D_{n}(\\theta^2_n {X}_n,\\theta^2_n{\\bm{s}}_n):D_{n}({X}_n,{\\bm{s}}_n)\\Big].\n\\label{eq:step1}\n\\end{align}\n\\helen{This final expression consists of four different pairings of the form $D_n(\\cdot,\\cdot): D_n(\\cdot,\\cdot)$; upon taking $\\liminf$ as $n \\to \\infty$, we use Lemma~\\ref{weak_lemma} on the first pairing, Lemma~\\ref{more_lemma} on the second and third pairings, and the final convergence property of $S_n^{\\rm inner}(x)$ from Lemma~\\ref{seq_lemma} on the fourth pairing to arrive at the following expression for the atomistic contribution}:\n\\begin{equation}\\label{soup_1}\n\\begin{split}\n    &\\liminf_{n\\to\\infty} \\big\\< \\delta^2 \\mathcal{E}^{\\rm a}_{{\\rm hom}, n} (0)(1-\\varphi)(W_n,\\bm{r}_n),(W_n,\\bm{r}_n)\\>\\\\\n    &\\;=~ \\liminf_{n\\to\\infty}\\epsilon_n^d \\sum_{\\xi \\in \\epsilon_n\\mathcal{L}} \\mathbb{C}:D_{n}(\\theta_nZ_n,\\theta_n\\bm{t}_n):D_{n}(\\theta_nZ_n,\\theta_n\\bm{t}_n)  \\\\\n\t\t&\\qquad +\\int_{\\mathbb{R}^d}\\mathbb{C}:\\nabla (\\theta_0^2 W_0, \\theta^2_0 \\bm{r}_0):\\nabla (W_0,\\bm{r}_0)\\, dx.\n\\end{split}\n\\end{equation}\n\n\\medskip\n\\noindent \\textit{Step 2:}  Meanwhile, the continuum component of $\\delta\\mathcal{F}_{{\\rm hom},n}^{{\\rm bqcf}}(0)$ from~\\eqref{bqcfHessian} is\n\\begin{equation}\\label{bqcfHessian_cont_part}\n\\begin{split}\n&\\big\\< \\delta^2 \\mathcal{E}^{\\rm c}(0)  I_{h,n}(\\varphi_n W_n,\\varphi_n \\bm{r}_n), (W_n, \\bm{r}_n) \\big\\> =~ \\int_{\\mathbb{R}^d}\\mathbb{C}:\\nabla \\big( I_{h,n}(\\varphi_n W_n), I_{h,n}(\\varphi_n\\bm{r}_n)\\big): \\nabla (W_n,\\bm{r}_n)\\, dx.\n\\end{split}\n\\end{equation}\n\nUsing standard $\\mathcal{P}_1$-nodal interpolation error estimates \\dao{and the fact that each $\\nabla \\varphi_n$ has support on $B_1$}, it is straightforward to\nprove that (c.f. Lemma~\\ref{p1_lemma})\n\\begin{equation} \\label{eq:conv-(Ih-I)(phiW)}\n   \\begin{split}\n   & \\lim_{n\\to \\infty}\\|\\nabla I_{h,n}(\\varphi_n W_n) - \\nabla (\\varphi_n W_n)\\|_{L^2(\\mathbb{R}^d)} = 0, \\\\\n   &  \\lim_{n\\to \\infty}\\|I_{h,n}(\\varphi_n r^\\alpha_n) - (\\varphi_n r^\\alpha_n)\\|_{L^2(\\mathbb{R}^d)} = 0.\n   \\end{split}\n\\end{equation}\n\nThus, taking the $\\liminf$ of~\\eqref{bqcfHessian_cont_part} and applying \\eqref{eq:conv-(Ih-I)(phiW)} we obtain\n\\begin{equation}\\label{bqcfHessian_cont_part_no_interp}\n\\liminf_{n\\to\\infty}\\big\\< \\delta^2 \\mathcal{E}^{\\rm c}(0) I_{h,n}(\\varphi W_n,\\varphi\\bm{r}_n), (W_n, \\bm{r}_n) \\big\\>\n= \\liminf_{n\\to\\infty} \\int_{\\mathbb{R}^d}\\mathbb{C}:\\nabla(\\varphi_nW_n, \\varphi_n\\bm{r}_n): \\nabla(W_n,\\bm{r}_n)\\, dx.\n\\end{equation}\nSubstituting the decomposition $(W_n, \\bm{r}_n) := (Z_n + X_n, \\bm{t}_n + \\bm{s}_n)$ into~\\eqref{bqcfHessian_cont_part_no_interp} yields\n\\begin{equation}\\label{long_limit}\n\\begin{split}\n&\\liminf_{n\\to\\infty}\\big\\< \\delta^2 \\mathcal{E}^{\\rm c}(0)  I_{h,n}(\\varphi W_n,\\varphi\\bm{r}_n), (W_n, \\bm{r}_n) \\big\\>  \\\\\n&=~\\liminf_{n\\to\\infty}\\int_{\\mathbb{R}^d}\\Big[ \\mathbb{C}: \\nabla (\\varphi_n Z_n, \\varphi_n\\bm{t}_n):\\nabla (Z_n, \\bm{t}_n) + \\mathbb{C}: \\nabla (\\varphi_n Z_n, \\varphi_n\\bm{t}_n):\\nabla (X_n, \\bm{s}_n) \\\\\n&\\qquad +  \\mathbb{C}: \\nabla (\\varphi_n X_n, \\varphi_n\\bm{s}_n):\\nabla (Z_n, \\bm{t}_n) + \\mathbb{C}: \\nabla (\\varphi_n X_n, \\varphi_n\\bm{s}_n):\\nabla (X_n, \\bm{s}_n)\\Big]\\, dx.\n\\end{split}\n\\end{equation}\n\\dao{This final expression again gives four pairings just as in step one but now of the form $\\nabla(\\cdot, \\cdot): \\nabla (\\cdot, \\cdot)$. The first pairing we momentarily leave alone, the second and third pairings both converge to zero by virtue of strong convergence of $\\nabla X_n, \\bm{s}_n$  and weak convergence of $\\nabla Z_n, \\bm{t}_n$ to $0$ from Lemma~\\ref{seq_lemma}, and the final pairing converges to $\\nabla(\\varphi_0 W_0,\\varphi_0\\bm{r}_0):\\nabla(W_0,\\bm{r}_0)$ again as a result of the strong convergence properties of $\\nabla X_n, \\bm{s}_n$ from Lemma~\\ref{seq_lemma}.  These facts simplify~\\eqref{long_limit} to}\n\\begin{equation}\\label{soup11}\n\\begin{split}\n&\\liminf_{n\\to\\infty}\\big\\< \\delta^2 \\mathcal{E}^{\\rm c}(0) I_{h,n}(\\varphi W_n,\\varphi\\bm{r}_n), (W_n, \\bm{r}_n) \\big\\> \\\\\n&=~ \\liminf_{n\\to\\infty}\\int_{\\mathbb{R}^d}\\big[\\mathbb{C}: \\nabla (\\varphi_n Z_n, \\varphi_n \\bm{t}_n): \\nabla(Z_n,\\bm{t}_n) \\\\\n&\\qquad\\qquad\\qquad\\qquad+ \\mathbb{C}:\\nabla(\\varphi_0 W_0,\\varphi_0\\bm{r}_0):\\nabla(W_0,\\bm{r}_0)\\big]\\, dx.\n\\end{split}\n\\end{equation}\nAs in the atomistic case, our goal is again to think of $\\varphi_n$ as a square, $\\varphi_n := \\sqrt{\\varphi_n}^2$ and to shift one factor of $\\sqrt{\\varphi_n}$ to each component of the duality pairing.  Using an argument very similar to that in the proof of Lemma~\\ref{weak_lemma} (which we therefore omit) we obtain\n\\begin{equation*}\\label{limit_term1_soup11}\n\\begin{split}\n&\\liminf_{n\\to\\infty}\\int_{\\mathbb{R}^d}\\mathbb{C}: \\nabla(\\varphi_n Z_n,\\varphi_n \\bm{t}_n): \\nabla(Z_n,\\bm{t}_n)\\\\\n&\\quad=~ \\liminf_{n\\to\\infty} \\int_{\\mathbb{R}^d} \\mathbb{C}: \\nabla(\\sqrt{\\varphi_n} Z_n, \\sqrt{\\varphi_n} \\bm{t}_n) : \\nabla(\\sqrt{\\varphi_n} Z_n,  \\sqrt{\\varphi_n}\\bm{t}_n).\n\\end{split}\n\\end{equation*}\n\n\nInserting the last result into \\eqref{soup11}, we obtain {\n\\begin{align}\\label{limit_bqcfH_cont_part_no_interp}\n&\\liminf_{n\\to\\infty}\\big\\< \\delta^2 \\mathcal{E}^{\\rm c}(0)  I_{h,n}(\\varphi W_n,\\varphi\\bm{r}_n), (W_n, \\bm{r}_n) \\big\\> \\\\\n&=~ \\liminf_{n\\to\\infty} \\int_{\\mathbb{R}^d} \\big[\\mathbb{C}: \\nabla(\\sqrt{\\varphi_n} Z_n, \\sqrt{\\varphi_n} \\bm{t}_n) : \\nabla(\\sqrt{\\varphi_n} Z_n,  \\sqrt{\\varphi_n}\\bm{t}_n) +\\mathbb{C}:\\nabla (\\varphi_0 W_0,\\varphi_0 \\bm{r}_0): \\nabla (W_0,\\bm{r}_0)\\big]\\, dx.\\nonumber\n\\end{align}\n}\n\\medskip\n\\noindent \\textit{Step 3:}  Upon adding the atomistic components from~\\eqref{soup_1} to {{ the continuum contributions \\eqref{limit_bqcfH_cont_part_no_interp}}} and recalling that $\\theta_0^2 = 1-\\varphi_0$, we have the following expression for $\\delta\\mathcal{F}_{{\\rm hom},n}^{{\\rm bqcf}}(0)$:\n\\begin{equation}\\label{short_limit}\n\\begin{split}\n&\\liminf_{n\\to\\infty}\\<\\delta \\mathcal{F}_{{\\rm hom},n}^{{\\rm bqcf}}(0)(W_n, \\bm{r}_n), (W_n, \\bm{r}_n)\\> \\\\\n&=~ \\liminf_{n\\to\\infty} \\int_{\\mathbb{R}^d} \\big[\\mathbb{C}: \\nabla(\\sqrt{\\varphi_n} Z_n, \\sqrt{\\varphi_n} \\bm{t}_n) : \\nabla(\\sqrt{\\varphi_n} Z_n,  \\sqrt{\\varphi_n}\\bm{t}_n) +\\mathbb{C}:\\nabla (W_0,\\bm{r}_0): \\nabla (W_0,\\bm{r}_0) \\big]\\, dx\\\\\n&\\qquad +~ \\liminf_{n\\to\\infty}\\epsilon_n^d \\sum_{\\xi \\in \\epsilon_n\\mathcal{L}} \\mathbb{C}:D_{n}(\\sqrt{1-\\varphi_n}Z_n,\\sqrt{1-\\varphi_n}\\bm{t}_n):D_{n}(\\sqrt{1-\\varphi_n}Z_n,\\sqrt{1-\\varphi_n}\\bm{t}_n)\n\\end{split}\n\\end{equation}\n\\dao{Next, using stability of the homogeneous atomistic model {\\helen{in this scaling}},\n\\[\n\\langle \\delta^2 \\mathcal{E}^{{\\rm a}}_{{\\rm hom},n}(0)(W_n, \\bm{r}_n), (W_n, \\bm{r}_n)\\rangle\n\\ge \\gamma_{\\rm a} \\|(W_n, \\bm{r}_n)\\|_{\\rm a}^2,\n\\]\n(which can easily be proven (c.f.~\\cite{olsonOrtner2016,Ehrlacher2013}) due to Assumption~\\ref{assumption2}) and the fact that atomistic stability implies Cauchy--Born Stability~\\cite[Theorem 3.6]{olsonOrtner2016}, that is,\n\\begin{equation*}\\label{CB_stability}\n\\begin{split}\n\\langle \\delta^2& \\mathcal{E}^{{\\rm c}}(0)(W_n, \\bm{r}_n), (W_n, \\bm{r}_n)\\rangle\n\\ge \\gamma_{\\rm a} \\|(W, \\bm{r})\\|_{\\rm ml}^2,\n\\end{split}\n\\end{equation*}\nwe hence have from~\\eqref{short_limit} that\n\\begin{align}\\label{red_pill}\n&\\liminf_{n\\to\\infty} \\langle \\delta  \\mathcal{F}_{{\\rm hom},n}^{{\\rm bqcf}}(0) (W_n, \\bm{r}_n),(W_n, \\bm{r}_n) \\\\\n&=~ \\liminf_{n\\to\\infty}\\big[\\langle\\delta^2 \\mathcal{E}^{\\rm c} (\\sqrt{\\varphi_n} Z_n, \\sqrt{\\varphi_n} \\bm{t}_n), (\\sqrt{\\varphi_n} Z_n, \\sqrt{\\varphi_n} \\bm{t}_n) \\rangle + \\langle\\delta^2 \\mathcal{E}^{\\rm c} (W_0,\\bm{r}_0), (W_0,\\bm{r}_0) \\rangle \\nonumber\\\\\n& \\qquad +~ \\langle \\delta^2 \\mathcal{E}^{\\rm a}_{{\\rm hom},n} (\\sqrt{1-\\varphi_n}Z_n,\\sqrt{1-\\varphi_n}\\bm{t}_n), (\\sqrt{1-\\varphi_n}Z_n,\\sqrt{1-\\varphi_n}\\bm{t}_n)\\rangle \\big]\\nonumber\\\\\n&\\geq \\liminf_{n\\to\\infty} \\gamma_{\\rm a}\\Big[\\|\\nabla (\\sqrt{\\varphi_n} Z_n)\\|^2_{L^2(\\mathbb{R}^d)} + \\|\\sqrt{\\varphi_n}\\bm{t}_n\\|^2_{L^2(\\mathbb{R}^d)} + \\|\\nabla{W}_0\\|^2_{L^2(\\mathbb{R}^d)} + \\|\\bm{r}_0\\|^2_{L^2(\\mathbb{R}^d)}  \\nonumber\\\\\n&\\qquad +  \\|\\nabla I_n(\\sqrt{1-\\varphi_n} Z_n)\\|^2_{L^2(\\mathbb{R}^d)} + \\|I_n(\\sqrt{1-\\varphi_n}\\bm{t}_n)\\|^2_{L^2(\\mathbb{R}^d)}\\Big].\\nonumber\n\\end{align}\n}\nSimilar to \\eqref{eq:conv-(Ih-I)(phiW)} (c.f. Lemma~\\ref{p1_lemma}), standard nodal interpolation error estimates imply that\n\\begin{align*}\n \\lim_{n\\to \\infty}\\|\\nabla I_n(\\sqrt{1-\\varphi_n} Z_n) - \\nabla (\\sqrt{1-\\varphi_n} Z_n)\\|_{L^2(\\mathbb{R}^d)} =& 0, \\quad \\text{and} \\\\\n   \\lim_{n\\to \\infty}\\| I_n(\\sqrt{1-\\varphi_n}\\bm{t}_n)-  (\\sqrt{1-\\varphi_n}\\bm{t}_n)\\|_{L^2(\\mathbb{R}^d)}  =& 0.\n\\end{align*}\nThus,~\\eqref{red_pill} becomes\n\\begin{equation}\\label{soup14}\n\\begin{split}\n&\\liminf_{n\\to\\infty} \\langle \\delta  \\mathcal{F}_{{\\rm hom},n}^{{\\rm bqcf}}(0) (W_n, \\bm{r}_n),(W_n, \\bm{r}_n) \\\\\n&\\quad\\geq~ \\liminf_{n\\to\\infty} \\gamma_{\\rm a}\\Big[\\|\\nabla (\\sqrt{\\varphi_n} Z_n)\\|^2_{L^2(\\mathbb{R}^d)} + \\|\\sqrt{\\varphi_n}\\bm{t}_n\\|^2_{L^2(\\mathbb{R}^d)} + \\|\\nabla{W}_0\\|^2_{L^2(\\mathbb{R}^d)} + \\|\\bm{r}_0\\|^2_{L^2(\\mathbb{R}^d)} \\\\\n&\\qquad+~ \\|\\nabla (\\sqrt{1-\\varphi_n} Z_n)\\|^2_{L^2(\\mathbb{R}^d)} + \\|\\sqrt{1-\\varphi_n}\\bm{t}_n\\|^2_{L^2(\\mathbb{R}^d)} \\Big] \\\\\n&\\quad=~ \\liminf_{n\\to\\infty}\\gamma_{\\rm a}\\Big[\\|\\nabla (\\sqrt{\\varphi_n} Z_n)\\|^2_{L^2(\\mathbb{R}^d)} + \\|\\nabla (\\sqrt{1-\\varphi_n} Z_n)\\|^2_{L^2(\\mathbb{R}^d)} + \\|\\bm{t}_n\\|^2_{L^2(\\mathbb{R}^d)} \\\\\n&\\qquad\\qquad\\qquad\\qquad  + \\|\\nabla{W}_0\\|^2_{L^2(\\mathbb{R}^d)} + \\|\\bm{r}_0\\|^2_{L^2(\\mathbb{R}^d)}\\Big].\n\\end{split}\n\\end{equation}\nNext observe\n\\begin{equation}\\label{soup15}\n\\begin{split}\n&\\|\\nabla (\\sqrt{\\varphi_n} Z_n)\\|^2_{L^2(\\mathbb{R}^d)} + \\|\\nabla (\\sqrt{1-\\varphi_n} Z_n)\\|^2_{L^2(\\mathbb{R}^d)} \\\\\n&=~ \\int \\Big[|\\nabla(\\sqrt{\\varphi_n}) \\otimes Z_n + \\sqrt{\\varphi_n}\\nabla Z_n|^2 + |\\nabla (\\sqrt{1-\\varphi_n}) \\otimes Z_n + \\sqrt{1-\\varphi_n}\\nabla Z_n|^2\\Big] \\, dx\\\\\n&=~ \\int \\Big[2\\nabla(\\sqrt{\\varphi_n}) \\otimes Z_n : \\sqrt{\\varphi_n}\\nabla Z_n + |\\nabla(\\sqrt{\\varphi_n}) \\otimes Z_n|^2 + \\varphi_n|\\nabla Z_n|^2\\Big]\\, dx  \\\\\n& +\\int \\Big[2\\nabla(\\sqrt{1-\\varphi_n}) \\otimes Z_n : \\sqrt{1-\\varphi_n}\\nabla Z_n + |\\nabla(\\sqrt{1-\\varphi_n}) \\otimes Z_n|^2 + (1-\\varphi_n)|\\nabla Z_n|^2\\big]\\, dx.\n\\end{split}\n\\end{equation}\nSince $Z_n$ converges strongly to zero in $L^2({\\rm supp}(\\nabla(\\sqrt{1-\\varphi_n})))$ by Lemma~\\ref{seq_lemma} (${\\rm supp}(\\nabla(\\sqrt{1-\\varphi_n})) \\subset B_1(0)$), it follows from~\\eqref{soup15} that\n\\begin{equation}\\label{soup16}\n\\liminf_{n\\to\\infty} \\|\\nabla (\\sqrt{\\varphi_n} Z_n)\\|^2_{L^2(\\mathbb{R}^d)} + \\|\\nabla (\\sqrt{1-\\varphi_n} Z_n)\\|^2_{L^2(\\mathbb{R}^d)} =~ \\liminf_{n\\to\\infty} \\|\\nabla Z_n\\|^2_{L^2(\\mathbb{R}^d)}.\n\\end{equation}\nSubstituting~\\eqref{soup16} into~\\eqref{soup14} produces\n\\begin{equation}\\label{soup17}\n\\begin{split}\n&\\liminf_{n\\to\\infty} \\langle \\delta  \\mathcal{F}_{{\\rm hom},n}^{{\\rm bqcf}}(0) (W_n, \\bm{r}_n),(W_n, \\bm{r}_n)\\rangle \\\\\n&\\quad\\geq~ \\liminf_{n\\to\\infty} \\gamma_{\\rm a}\\Big[\\|\\nabla Z_n\\|^2_{L^2(\\mathbb{R}^d)} + \\|\\bm{t}_n\\|^2_{L^2(\\mathbb{R}^d)} + \\|\\nabla{W}_0\\|^2_{L^2(\\mathbb{R}^d)} + \\|\\bm{r}_0\\|^2_{L^2(\\mathbb{R}^d)}\\Big] \\\\\n&\\quad\\geq~ \\liminf_{n\\to\\infty} \\gamma_{\\rm a}\\Big[\\|\\nabla Z_n\\|^2_{L^2(\\mathbb{R}^d)} + \\|\\bm{t}_n\\|^2_{L^2(\\mathbb{R}^d)} + \\|\\nabla{X}_n\\|^2_{L^2(\\mathbb{R}^d)} + \\|\\bm{s}_n\\|^2_{L^2(\\mathbb{R}^d)}\\Big] \\\\\n&\\quad=~\\liminf_{n \\to \\infty} \\gamma_{\\rm a} \\big[ \\|\\nabla {W}_n\\|^2_{L^2(\\mathbb{R}^d)} + \\|\\bm{r}_n\\|^2_{L^2(\\mathbb{R}^d)} \\big] = \\gamma_{\\rm a},\n\\end{split}\n\\end{equation}\nwhich contradicts our assumption in~\\eqref{contra_seq}.\n\\end{proof}\n\n\n\\subsection{Reference Stability Implies Defect Stability}\\label{stab_defect_present}\n\nHaving established stability of the homogeneous BQCF operator we obtain\nstability of $\\delta \\mathcal{F}^{{\\rm bqcf}}(\\Pi_{h,n}(U^\\infty,\\bm{p}^\\infty))$,\ni.e. Theorem \\ref{stab_theorem_full}, as a relatively straightforward\nconsequence. Before entering into the proof we remark that we now no longer employ the rescalings of Section~\\ref{stab_def_free}. \\dao{The basic idea of the proof is that the linearized homogeneous BQCF operator and linearized BQCF operator agree for any $(W,\\bm{r})$ which is zero in a large enough neighborhood about the origin.  Thus, to prove stability of the true linearized BQCF operator, we again consider the possibility of a sequence, $(W_n, \\bm{r}_n)$, of unstable modes whose support is contained in larger and larger balls about the origin. We will then split each $(W_n,\\bm{r}_n)$ into components concentrated near the origin (where we can use atomistic stability) and correction terms supported far from the origin where we use stability of the linearized homogeneous operator. As before, the main difficulty is converting the atomistic component of the BQCF operator to a form where we may utilize atomistic coercivity.}\n\n\\begin{proof}[Proof of Theorem~\\ref{stab_theorem_full}]\nWe prove this result by contradiction as well. Therefore suppose, as in the proof of Theorem~\\ref{stab_theorem}, that there exists $R_{{\\rm a},n} \\to \\infty$ with associated meshes $\\mathcal{T}_{h,n}$, blending functions $\\varphi_n$, {\\helen{and test pairs $(W_n, \\bm{r}_n) \\in \\bm{\\mathcal{U}}_{h,0}^n \\times \\bm{\\mathcal{P}}_{h,0}^n$ with norm scaled to one, such that}}\n\\begin{equation}\\label{contra_seq_def}\n\\begin{split}\n&\\frac{\\gamma_{\\rm a}}{2} >\n\\sum_{\\xi \\in \\mathcal{L}} \\sum_{(\\rho\\alpha\\beta)}\\sum_{(\\tau\\gamma\\delta)} V_{,(\\rho\\alpha\\beta)(\\tau\\gamma\\delta)}(D\\bm{U}_n):D_{(\\rho\\alpha\\beta)}((1-\\hat{\\varphi}_n)\\hat{W}_n,(1-\\hat{\\varphi}_n)\\hat{\\bm{r}}_n):D_{(\\rho\\alpha\\beta)}(\\hat{W}_n,\\hat{\\bm{r}}_n) \\\\\n& \\qquad \\qquad + \\int_{\\mathbb{R}^d} \\sum_{(\\rho\\alpha\\beta)}\\sum_{(\\tau\\gamma\\delta)} V_{,(\\rho\\alpha\\beta)(\\tau\\gamma\\delta)}(\\nabla \\bm{U}_n):\\nabla_{\\rho\\alpha\\beta}( I_{h,n}(\\hat{\\varphi}_n(\\hat{W}_n, \\hat{\\bm{r}}_n))) :\\nabla_{\\rho\\alpha\\beta}(\\hat{W}_n, \\hat{\\bm{r}}_n) \\, dx \\\\\n&\\quad=: \\<\\delta \\mathcal{F}^{{\\rm bqcf}}_n(\\bm{U}_n)(W_n, \\bm{r}_n),(W_n, \\bm{r}_n)\\>,\n\\end{split}\n\\end{equation}\nwhere, for notational simplicity we have defined $\\bm{U}_n := \\Pi_{h,n}(U^\\infty,\\bm{p}^\\infty)$ and redefined $\\delta \\mathcal{F}^{{\\rm bqcf}}_n$ from the previous section without a scaling by $\\epsilon$.\n\nUpon extracting a subsequence, we may assume without loss of generality that $\\nabla W_n \\rightharpoonup \\nabla W_0$ for $W_0 \\in \\dot{\\bm{H}}^1$ and $\\bm{r}_n \\rightharpoonup \\bm{r}_0 \\in L^2$. For each $R_{{\\rm a},n}$, $W_n$ and $\\bm{r}_n$ are piecewise linear with respect to the mesh $\\mathcal{T}_{\\rm a}$ on $\\Omega_{{\\rm a}, n}$.  Hence the convergence is strong on any finite collection of elements on $\\mathcal{T}_{\\rm a}$ since weak convergence implies strong convergence on finite dimensional spaces. It also follows from the full refinement of the mesh assumption on $\\Omega_{{\\rm a}, n}$ that $W_0$ and $\\bm{r}_0$ are also piecewise linear with respect to $\\mathcal{T}_{\\rm a}$.\n\nHaving established these basic facts, we will yet again split $(W_n, \\bm{r}_n)$ into the sum of a strongly convergent sequence and weakly convergent sequence as in~\\cite[Theorem 4.9]{blended2014}.  For each $n$, we take $\\eta_n(x)$ to be a smooth bump function satisfying $\\eta_n(x) = 1$ on $B_{1\/2r_{{\\rm core},n}}(0)$ and $\\eta_n(x)$ has support contained in $B_{r_{{\\rm core},n}}(0)$.  Similar to the definition of $\\Pi_h$, we then set \n\\[\n\\dao{A_{n} := B_{r_{{\\rm core},n}}\\setminus B_{(1\/2)r_{{\\rm core},n}} + B_{2r_{\\rm buff}}}\n\\]\n and\n\\begin{equation}\\label{DefectStab_test}\nX_n := I_n(\\eta_n W_0) - I_n(\\eta_n)\\Xint-_{A_n}  W_0\\, dx ,\\quad Z_n := W_n - X_n,\\quad \\bm{s}_n := I_n(\\eta_n \\bm{r}_0),\\quad \\bm{t}_n := \\bm{r}_n - \\bm{s}_n.\n\\end{equation}\nSimilar to Lemma~\\ref{approx_lem}, we have, with these definitions,\n\\begin{equation*}\\label{convo_props}\n\\nabla X_n \\to \\nabla W_0, \\quad \\mbox{and} \\quad \\nabla Z_n \\rightharpoonup 0\n   \\qquad \\text{in } L^2(\\mathbb{R}^d)\n\\end{equation*}\nand\n\\begin{align*}\\label{convo_props1}\n\\bm{s}_n \\to \\bm{r}_0, \\quad &\\mbox{and} \\quad \\bm{t}_n \\rightharpoonup 0\n\\qquad \\text{in } L^2(\\mathbb{R}^d).\n\\end{align*}\nThen we note that the norm defined by\n\\[\n\\| (U,\\bm{p}) \\|_{{\\rm a}_1}^2 := \\sum_{\\xi \\in \\mathcal{L}} |D(U,\\bm{p})(\\xi)|^2, \\quad \\mbox{where} \\quad |D(U,\\bm{p})(\\xi)|^2 := \\sum_{(\\rho \\alpha\\beta) \\in \\mathcal{R}} |D_{(\\rho \\alpha\\beta)} (U,\\bm{p})(\\xi)|^2.\n\\]\nis equivalent to the $\\|\\cdot\\|_{{\\rm a}}$ norm on $\\bm{\\mathcal{U}}$ by~\\cite[Lemma 2.1]{olsonOrtner2016}. Thus, \\dao{since we are dealing with functions which are $\\mathcal{P}^1$ with respect to $\\mathcal{T}_{\\rm a}$ on a growing atomistic region, then the continuous convergence results for $\\nabla X_n, \\nabla Z_n, \\bm{s}_n$, and $\\bm{t}_n$ imply corresponding discrete convergence results:}\n\\begin{align}\\label{convo_props2}\nD(X_n, \\bm{s}_n) \\to D(W_0, \\bm{r}_0) \\quad  &\\mbox{and} \\quad D(Z_n, \\bm{t}_n) \\rightharpoonup 0 \\qquad \\mbox{in $\\ell^2(\\mathcal{L})$}.\n\\end{align}\nWith this decomposition, \\dao{we now substitute the test pair {\\helen{$(W_n, \\bm{r_n}) = (X_n + Z_n, \\bm{s}_n + \\bm{t}_n)$ from \\eqref{DefectStab_test} into}}}\n\\begin{equation}\\label{steamroll}\n\\begin{split}\n&\\<\\delta \\mathcal{F}^{{\\rm bqcf}}_{ n}(\\bm{U}_n)(X_n + Z_n,\\bm{s}_n + \\bm{t}_n),(X_n + Z_n,\\bm{s}_n + \\bm{t}_n)\\> \\\\\n&=~ \\<\\delta  \\mathcal{F}^{{\\rm bqcf}}_{n} (\\bm{U}_n)(X_n,\\bm{s}_n),(X_n,\\bm{s}_n)\\> + \\<\\delta \\mathcal{F}^{{\\rm bqcf}}_{n} (\\bm{U}_n)(X_n,\\bm{s}_n),(Z_n,\\bm{t}_n)\\> \\\\\n&\\qquad +~  \\<\\delta  \\mathcal{F}^{{\\rm bqcf}}_{ n} (\\bm{U}_n)(Z_n,\\bm{t}_n),(X_n,\\bm{s}_n)\\> + \\<\\delta \\mathcal{F}^{{\\rm bqcf}}_{ n} (\\bm{U}_n)(Z_n,\\bm{t}_n),(Z_n,\\bm{t}_n)\\>.\n\\end{split}\n\\end{equation}\n{\\helen{\nAlso recall the definition of $\\delta\\mathcal{F}_n^{\\rm bqcf}$, which is\n\\begin{align*}\n\\langle&\\delta \\mathcal{F}^{\\rm bqcf}_n(\\bm{U}_n)(W_n, \\bm{r}_n),(W_n, \\bm{r}_n)\\rangle\\\\\n&= \\langle\\delta^2 \\mathcal{E}^{\\rm a}(\\bm{U}_n)\\big((1-\\varphi_n)(W_n, \\bm{r}_n)\\big),(W_n, \\bm{r}_n)\\rangle \\\\\n& \\qquad +\\langle\\delta^2 \\mathcal{E}^{\\rm c}(\\bm{U}_n)\\big(\\varphi_n(W_n, \\bm{r}_n)\\big),(W_n, \\bm{r}_n)\\rangle.\n\\end{align*}\n}}\nSince $D(X_n, \\bm{s}_n)$ each have support where $\\varphi_n = 0$ and $\\Pi_{h,n}(\\bm{U}_n) = (\\bm{U}_n)$ there, we can rewrite the first three terms of~\\eqref{steamroll} without the blending function as\n\\begin{equation}\\label{stew1}\n\\begin{split}\n&\\<\\delta  \\mathcal{F}^{{\\rm bqcf}}_{n} (\\bm{U}_n)(X_n + Z_n,\\bm{s}_n + \\bm{t}_n),(X_n + Z_n,\\bm{s}_n + \\bm{t}_n)\\> \\\\\n&=~ \\<\\delta^2 \\mathcal{E}^{{\\rm a}}(\\bm{U}_n)(X_n,\\bm{s}_n),(X_n,\\bm{s}_n)\\> + \\<\\delta^2 \\mathcal{E}^{{\\rm a}} (\\bm{U}_n)(X_n,\\bm{s}_n),(Z_n,\\bm{t}_n)\\> \\\\\n&\\qquad +~  \\<\\delta^2 \\mathcal{E}^{{\\rm a}} (\\bm{U}_n)(Z_n,\\bm{t}_n),(X_n,\\bm{s}_n)\\> + \\<\\delta \\mathcal{F}^{{\\rm bqcf}}_{ n} (\\bm{U}_n)(Z_n,\\bm{t}_n),(Z_n,\\bm{t}_n)\\>.\n\\end{split}\n\\end{equation}\nMoreover, $D(Z_n, \\bm{t}_n)$ has support only where $V_\\xi \\equiv V$ and so from the convergence properties~\\eqref{convo_props2}, it follows that $\\<\\delta^2 \\mathcal{E}^{{\\rm a}}(\\bm{U}_n)(X_n,\\bm{s}_n),(Z_n,\\bm{t}_n)\\>$ and $\\<\\delta^2 \\mathcal{E}^{{\\rm a}}(\\bm{U}_n)(Z_n,\\bm{t}_n),(X_n,\\bm{s}_n)\\>$ both go to zero as $n \\to \\infty$.\n\nFor the first term in \\eqref{stew1}, using the atomistic stability assumption, Assumption~\\ref{assumption2}, we obtain\n\\begin{equation}\\label{stew2}\n\\<\\delta^2 \\mathcal{E}^{{\\rm a}}  (\\bm{U}_n)(X_n,\\bm{s}_n),(X_n,\\bm{s}_n)\\> \\geq~ \\gamma_{\\rm a} \\|(X_n, \\bm{s}_n)\\|_{\\rm ml}^2.\n\\end{equation}\nThus, taking the lim inf as $n \\to \\infty$ in~\\eqref{stew1} yields\n\\begin{equation}\\label{steam_peas}\n\\begin{split}\n&\\liminf_{n \\to \\infty}\\<\\delta  \\mathcal{F}^{{\\rm bqcf}}_{n} (\\bm{U}_n)(X_n + Z_n,\\bm{s}_n + \\bm{t}_n),(X_n + Z_n,\\bm{s}_n + \\bm{t}_n)\\> \\\\\n&\\qquad \\geq \\liminf_{n \\to \\infty} \\gamma_{\\rm a} \\|(X_n, \\bm{s}_n)\\|_{\\rm ml}^2 + \\liminf_{n \\to \\infty} \\<\\delta \\mathcal{F}^{{\\rm bqcf}}_{ n} (\\bm{U}_n)(Z_n,\\bm{t}_n),(Z_n,\\bm{t}_n)\\>\n\\end{split}\n\\end{equation}\nThus, we are only left to treat $\\<\\delta \\mathcal{F}^{{\\rm bqcf}}_{ n} (\\bm{U}_n)(Z_n,\\bm{t}_n),(Z_n,\\bm{t}_n)\\>$, the\nfar-field contribution, as defined in \\eqref{DefectStab_test}. The strategy here is that far from the defect core we\nmay replace $\\delta\\mathcal{F}^{{\\rm bqcf}}_n(\\bm{U}_n)$ with $\\delta\\mathcal{F}^{{\\rm bqcf}}_{{\\rm hom},n}(0)$ and then apply Theorem \\ref{stab_theorem}. Thus, we first estimate,\n\\begin{equation}\\label{stew3}\n\\begin{split}\n&\\<\\delta \\mathcal{F}^{{\\rm bqcf}}_{n}(\\bm{U}_n)(Z_n,\\bm{t}_n),(Z_n,\\bm{t}_n)\\>  \\\\\n&=~ \\<\\delta \\mathcal{F}_{{\\rm hom}, n }^{{\\rm bqcf}}(\\bm{U}_n)(Z_n,\\bm{t}_n),(Z_n,\\bm{t}_n)\\> \\\\\n&=~ \\<\\delta \\mathcal{F}_{{\\rm hom}, n} ^{{\\rm bqcf}}(0)(Z_n,\\bm{t}_n),(Z_n,\\bm{t}_n)\\>\n   + \\big\\<\\big[ \\delta \\mathcal{F}_{{\\rm hom}, n}^{{\\rm bqcf}}(\\bm{U}_n) - \\delta \\mathcal{F}_{{\\rm hom}, n }^{{\\rm bqcf}}(0) \\big] (Z_n,\\bm{t}_n),(Z_n,\\bm{t}_n)\\big \\>\\\\\n&\\geq~ \\frac{3}{4}\\gamma_{\\rm a} \\|(Z_n, \\bm{t}_n)\\|_{\\rm ml}^2 +\n\\big\\<\\big[ \\delta \\mathcal{F}_{{\\rm hom},n}^{{\\rm bqcf}}(\\bm{U}_n) - \\delta \\mathcal{F}_{{\\rm hom},n }^{{\\rm bqcf}}(0) \\big] (Z_n,\\bm{t}_n),(Z_n,\\bm{t}_n)\\big \\>.\n\\end{split}\n\\end{equation}\nwhere we applied Theorem~\\ref{stab_theorem} in the final step. ({\\helen{Note that there is a slight}} notational discrepancy in that our $\\mathcal{F}^{\\rm bqcf}_{{\\rm hom},n}$ is indexed by $n$ here while there is no index in Theorem~\\ref{stab_theorem}.  However, we may still use this theorem since $R_{{\\rm core},n} \\to \\infty$ so we may assume $R_{{\\rm core},n} \\geq R_{\\rm core}^*$ in the statement of that theorem.)\n\nNext, we estimate the remaining group in \\eqref{stew3},\n\\begin{equation*}\\label{stew4}\n\\begin{split}\n   &\\hspace{-1cm}\\big\\<\\big[ \\delta \\mathcal{F}_{{\\rm hom}, n }^{{\\rm bqcf}}(\\bm{U}_n) - \\delta \\mathcal{F}_{{\\rm hom},n}^{{\\rm bqcf}}(0) \\big] (Z_n,\\bm{t}_n),(Z_n,\\bm{t}_n)\\big \\> \\\\\n&\\leq~ \\big|\\big\\<\\big[\\delta^2 \\mathcal{E}^{\\rm a}_{\\rm hom}(\\bm{U}_n) - \\delta^2\\mathcal{E}^{\\rm a}_{\\rm hom}(0)\\big] (1-\\varphi_n)(Z_n,\\bm{t}_n),(Z_n,\\bm{t}_n)\\> \\big| \\\\\n& \\qquad\\qquad +~ \\big|\\big\\<\\big[\\delta^2 \\mathcal{E}^{\\rm c}(\\bm{U}_n) - \\delta^2\\mathcal{E}^{\\rm c}(0)\\big]  I_{h,n}(\\varphi_n(Z_n,\\bm{t}_n)),(Z_n,\\bm{t}_n)\\big\\>\\big| \\\\\n&\\leq~  \\sum_{(\\rho \\alpha\\beta)(\\tau\\gamma\\delta)}\\|V_{,(\\rho \\alpha\\beta)(\\tau\\gamma\\delta)}(D \\bm{U}_n) - V_{,(\\rho \\alpha\\beta)(\\tau\\gamma\\delta)}(0) \\|_{\\ell^\\infty({\\rm supp}(D(Z_n,\\bm{t}_n))} \\\\\n&\\qquad \\qquad\\qquad \\cdot \\|D_{(\\rho \\alpha\\beta)}((1-\\varphi_n)(Z_n,\\bm{t}_n))\\|_{\\ell^2(\\mathbb{R}^d)}\\|D_{(\\tau\\gamma\\delta)}(Z_n,\\bm{t}_n)\\|_{\\ell^2(\\mathbb{R}^d)} \\\\\n&+~ \\sum_{(\\rho \\alpha\\beta)(\\tau\\gamma\\delta)}\\|V_{,(\\rho \\alpha\\beta)(\\tau\\gamma\\delta)}\\big(\\nabla\\bm{U}_n ) - V_{,(\\rho \\alpha\\beta)(\\tau\\gamma\\delta)}(0)\\|_{L^\\infty({\\rm supp}(\\nabla(Z_n,\\bm{t}_n))} \\\\\n&\\qquad\\qquad\\qquad \\cdot \\|\\nabla_{(\\rho \\alpha\\beta)}((1-\\varphi_n)(Z_n,\\bm{t}_n))\\|_{L^2(\\mathbb{R}^d)}\\|\\nabla_{(\\tau\\gamma\\delta)}(Z_n,\\bm{t}_n)\\|_{L^2(\\mathbb{R}^d)}.\n\\end{split}\n\\end{equation*}\n\nFrom Lemma~\\ref{approx_lem} and the decay rates from Theorem~\\ref{decay_thm1} we have\n\\begin{equation}\\label{stew5}\n\\begin{split}\n\\|V_{,(\\rho \\alpha\\beta)(\\tau\\gamma\\delta)}(D\\bm{U}_n  ) - V_{,(\\rho \\alpha\\beta)(\\tau\\gamma\\delta)}(0)\\|_{\\ell^\\infty({\\rm supp}(D(Z_n,\\bm{t}_n))} \\to~& 0,\n\\quad \\text{and} \\\\\n\\|V_{,(\\rho \\alpha\\beta)(\\tau\\gamma\\delta)}(\\nabla \\bm{U}_n) - V_{,(\\rho \\alpha\\beta)(\\tau\\gamma\\delta)}(0)\\|_{L^\\infty({\\rm supp}(\\nabla(Z_n,\\bm{t}_n))} \\to~& 0.\n\\end{split}\n\\end{equation}\nConsequently,\n\\[\n\\big\\<\\big[ \\delta \\mathcal{F}_{{\\rm hom}, n }^{{\\rm bqcf}}(\\bm{U}_n) - \\delta \\mathcal{F}_{{\\rm hom},n}^{{\\rm bqcf}}(0) \\big] (Z_n,\\bm{t}_n),(Z_n,\\bm{t}_n)\\big \\> \\to 0,\n\\]\nand from~\\eqref{stew3},\n\\begin{equation}\\label{inf_mod}\n\\liminf_{n \\to \\infty} \\<\\delta \\mathcal{F}^{{\\rm bqcf}}_{n}(\\bm{U}_n)(Z_n,\\bm{t}_n),(Z_n,\\bm{t}_n)\\> \\geq \\frac{3}{4}\\gamma_{\\rm a} \\|(Z_n, \\bm{t}_n)\\|_{\\rm ml}^2.\n\\end{equation}\n\nCombining~\\eqref{steam_peas} and~\\eqref{inf_mod}, we can therefore conclude that\n\\begin{align}\\label{stew100}\n&\\liminf_{n \\to \\infty} \\<\\delta \\mathcal{F}^{{\\rm bqcf}}_{n }(\\Pi_{h,n}(\\bm{U}_n))(X_n + Z_n,\\bm{s}_n + \\bm{t}_n),(X_n + Z_n,\\bm{s}_n + \\bm{t}_n)\\> \\nonumber\\\\\n& \\geq \\liminf_{n \\to\\infty} \\big[ \\gamma_{\\rm a} \\|(X_n, \\bm{s}_n)\\|_{\\rm ml}^2 + \\frac{3}{4}\\gamma_{\\rm a}\\|(Z_n, \\bm{t}_n)\\|_{\\rm ml}^2  \\big]  \\\\\n& {\\helen{\\geq\\liminf_{n \\to\\infty} \\frac{3}{4}\\gamma_{\\rm a}\\big[ \\|\\nabla X_n\\|_{L^2(\\mathbb{R}^d)}^2 +\\|\\bm{s}_n\\|_{L^2(\\mathbb{R}^d)}^2 + \\|\\nabla Z_n\\|_{L^2(\\mathbb{R}^d)}^2+\\|\\bm{t}_n\\|_{L^2(\\mathbb{R}^d)}^2 \\big] }}.\\nonumber\n\\end{align}\n{\\helen{\nNotice that we have\n\\[\n\\begin{split}\n\\|\\nabla W_n\\|^2_{L^2(\\mathbb{R}^d)}&=\\langle \\nabla W_n, \\nabla W_n\\rangle= \\langle \\nabla (X_n+Z_n), \\nabla (X_n+Z_n)\\rangle\\\\\n&=\\|\\nabla X_n\\|^2_{L^2(\\mathbb{R}^d)}+2\\langle \\nabla X_n, \\nabla Z_n\\rangle+ \\|\\nabla Z_n\\|^2_{L^2(\\mathbb{R}^d)},\n\\end{split}\n\\]\nso we get\n\\[\n\\|\\nabla X_n\\|^2_{L^2(\\mathbb{R}^d)}+ \\|\\nabla Z_n\\|^2_{L^2(\\mathbb{R}^d)}\n= \\|\\nabla W_n\\|^2_{L^2(\\mathbb{R}^d)}-2\\langle \\nabla X_n, \\nabla Z_n\\rangle.\n\\]\nApplying the same treatments to $\\|\\bm{r}_{n}\\|^2$, we have from \\eqref{stew100} that\n\\[\n\\begin{split}\n&\\liminf_{n \\to \\infty} \\<\\delta \\mathcal{F}^{{\\rm bqcf}}_{n }(\\Pi_{h,n}(\\bm{U}_n))(X_n + Z_n,\\bm{s}_n + \\bm{t}_n),(X_n + Z_n,\\bm{s}_n + \\bm{t}_n)\\> \\\\\n&\\geq \\liminf_{n \\to\\infty} \\frac{3}{4}\\gamma_{\\rm a}\\Big[\\|\\nabla W_n\\|^2_{L^2(\\mathbb{R}^d)} - 2(\\nabla Z_n, \\nabla X_n)_{L^2(\\mathbb{R}^d)} + \\sum_\\alpha\\|r_n^\\alpha\\|^2_{L^2(\\mathbb{R}^d)} \\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad- \\sum_\\alpha 2(s_n^\\alpha,t_n^\\alpha)_{L^2(\\mathbb{R}^d)} \\Big] \\\\\n& \\geq \\liminf_{n \\to\\infty} \\frac{3\\gamma_{\\rm a}}{4}\\Big[\\|\\nabla W_n\\|^2_{L^2(\\mathbb{R}^d)} + \\sum\\|r_n^\\alpha\\|^2_{L^2(\\mathbb{R}^d)}\\Big] = \\frac{3}{4}\\gamma_{\\rm a},\n\\end{split}\n\\]\n}}\nwhich is a contradiction to~\\eqref{contra_seq_def}.  In attaining the last equality, we have used $(\\cdot, \\cdot)_{L^2}$ to denote the $L^2$ inner product, and we have again used the fact that the inner product of {\\helen{the strongly convergent sequence $\\nabla X_n$ and weakly convergent sequence $\\nabla Z_n$ (c.f. Lemma~\\ref{seq_lemma}) converges to zero and similarly for the inner product of the strongly convergent $\\bm{s}_n$ and weakly convergent $\\bm{t}_n$.}}\n\\end{proof}\n\n\n\\section{Discussion}\\label{discussion}\n\nWe presented the first complete error analysis of an atomistic-to-continuum coupling method for multilattices capable of incorporating defects in the analysis.  Our results for the blended force-based quasicontinuum method extend the existing results for Bravais lattices~\\cite{blended2014}, with the striking conclusion that the convergence rates in the simple and multi-lattice cases coincide for the optimal mesh coarsening. Our computational results for a Stone-Wales defect in graphene confirm our theoretical predictions.\n\nWe have concerned ourselves here with the case of point defects, though we see no conceptually challenging obstacles to include dislocations in the analysis so long as there is an analogous decay result to Theorem~\\ref{decay_thm}.  However, as previously mentioned, we are still limited in our ability to model physical effects such as bending or rippling in two-dimensional materials such as graphene due to several factors.  First, our assumption concerning stability of the multilattice, Assumption~\\ref{assumption2} uses a norm, $\\| \\nabla I U\\|_{L^2} + \\|I\\bm{p}\\|_{L^2}$, which does not take any bending energy into account and so we do not guarantee our lattice is stable in this situation.  We could have of course formulated a different assumption using a discrete variant of $\\|\\nabla^2 U_3\\|$ (where $U_3$ represents the out of plane displacement), but it is a very challenging question to extend the BQCF method and its analysis to such a situation.  The next issue that must be answered is what continuum model to use since the Cauchy--Born model used herein is not adequate to model such effects.  Possible alternatives would be to use higher-order Cauchy--Born rules~\\cite{ericksen2008cauchy,yang2006generalized} which rely on higher-order strain gradients, or the so-called exponential Cauchy--Born rule~\\cite{expCauchy}. In either of these cases, to use a similar analysis to what we have presented, one would have to establish new stress estimates akin to Corollary~\\ref{globel_stress} as well ensuring that the continuum model chosen is stable provided the atomistic model is. We are also confronted with the problem of choosing a finite element space capable of approximating $H^2$ functions, which likewise challenges the analysis as well as the implementation. \n\nFinally, we remark that extensions to charged defects in ionic crystals, which represent a wide class of important multilattice crystals, represent yet another difficult challenge, largely due to the {\\helen{long-range}} nature of the interatomic forces.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n  \\label{sec:introduction}\n \n\n  Lepton flavor violation (LFV) is an important vehicle for low energy studies of physics beyond the Standard Model (BSM). Within the Standard Model (SM) with massless neutrinos, individual lepton number is conserved. Even with the addition of non-zero neutrino masses, processes that violate charged lepton number are suppressed by powers of $m_\\nu^2 \/ m_W^2$~\\cite{Raidal:2008jk}. Thus, experiments should be extremely sensitive to BSM physics that facilitate charged lepton flavor-violating (CLFV) processes.\n\n  SM phenomena are also expected to closely obey lepton-flavor universality (LFU). However, recent observations from LHCb~\\cite{Aaij:2014ora,Aaij:2015yra,Aaij:2017vbb}, BaBar~\\cite{Lees:2013uzd}, and Belle~\\cite{Huschle:2015rga,Wehle:2016yoi} show hints of LFU violation in semi-leptonic decays of $B$ mesons at the level of a few standard deviations. In response to these findings there have been many proposals introducing new physics, for example studies of $b \\rightarrow s \\mu \\mu$~\\cite{Descotes-Genon:2013wba,Altmannshofer:2013foa,Gauld:2013qja,Datta:2013kja,Buras:2013dea} and lepton-flavor non-universal interactions~\\cite{Altmannshofer:2014cfa,Sakaki:2013bfa} (for a recent review, see Ref.~\\cite{Buttazzo:2017ixm} and references therein). Although not required~\\cite{Celis:2015ara,Alonso:2015sja}, new interactions that violate LFU may also induce LFV~\\cite{Glashow:2014iga}. With the prospects of studying LFV in $B$-meson decays at LHCb and the upcoming Belle II experiment, there has been renewed theoretical attention to this type of new physics~\\cite{Bhattacharya:2014wla,Bhattacharya:2016mcc,Alok:2017jgr,Alok:2017sui,Altmannshofer:2017yso,Crivellin:2017zlb,Iguro:2017ysu,Iguro:2018qzf}.\n\nIn addition to studies of LFV in $B$ decays, some authors have proposed refined methods for direct searches at the Large Hadron Collider (LHC) to look for new TeV-scale particles that can mediate LFV \\cite{Chivukula:2017qsi}. However, it is quite possible that the new mediators are at an energy scale that is beyond the reach of the LHC. A convenient method to study effects of high-scale physics in low-energy processes involves effective field theories (EFT) \\cite{Petrov:2016azi}. If LFV happens to be at a scale $\\Lambda$ that is beyond the reach of direct searches at the LHC, studies of LFV effects at the LHC can still be done using EFT methods. The low-energy effects of BSM physics generated at a UV scale $\\Lambda$ can be characterized in terms of an effective Lagrangian $\\mathcal{L}_\\textrm{eff}$ containing terms of dimension $d \\ge 5$ suppressed by appropriate powers of the NP scale $\\Lambda$. In particular, at dimension 6 the following $SU(3)_C \\times U(1)_\\textrm{EM}$ invariant CLFV interactions are generated,\n  %\n  \\begin{align}\n    \\label{eq:4fermion_lagrangian}\n    \\mathcal{L}_\\textrm{eff}^{(6)} \\supset \\frac{1}{\\Lambda^2} \\sum_{i,j,k,l,m,n} C_{ijkl}^{mn} \\left ( \\overline{\\ell}_i \\Gamma^m \\ell_j \\right ) \\left ( \\overline{q}_k \\Gamma^n q_l \\right ) + \\textrm{h.c.} \\, ,\n  \\end{align}\n  %\n  where $i,j = 1,2,3$ label lepton generation, $k,l = 1,2,3$ label quark generation, $\\Gamma^m$ denote the Dirac structure, and $C_{ijkl}^{mn}$ are Wilson coefficients. The operators in Eq.~\\eqref{eq:4fermion_lagrangian} can be probed in a variety of ways, both at high~\\cite{Black:2002wh,Han:2010sa,Arganda:2015ija,Cai:2015poa} and low energies~\\cite{Hazard:2017udp,Hazard:2016fnc,Dreiner:2006gu,Daub:2012mu,Lindner:2016bgg,Davidson:2016edt,Crivellin:2013hpa,Crivellin:2017rmk,Celis:2014asa}.~\\footnote{For an alternative approach to studying CLFV at fixed target experiments, see e.g.~\\cite{Takeuchi:2017btl}.}\n\n The large parton luminosity for gluon-gluon interactions at high-energy $pp$ colliders, such as the LHC, implies that gluon-initiated processes might be prevalent there. However, the set of operators in Eq.~\\eqref{eq:4fermion_lagrangian} does not contain gluon fields. The lowest order effective operator that is invariant under the SM gauge group $SU(3)_C \\times SU(2)_L \\times U(1)_Y$ that couples lepton and gluon fields appears at dimension eight,\n  %\n  \\begin{align}\n    \\mathcal{L}_\\textrm{eff}^{(8)} = \\frac{g_s^2}{\\Lambda^4} \\left [ Y_{ij} \\overline{L}_L^{\\, i} H \\ell_R^{\\, j} \\, G \\cdot G + \\widetilde{Y}_{ij} \\overline{L}_L^{\\, i} H \\ell_R^{\\, j} \\, G \\cdot \\widetilde{G} \\right ] + \\textrm{h.c.} \\, ,\n  \\end{align}\n  %\n  where $L_L^{\\, i}$ represents the left-handed doublet lepton field with generation index $i$ in the gauge basis, $\\ell_R^{\\, i}$ is a right-handed lepton singlet field, and $H$ is a Higgs field. Gauge-invariant combinations of gluon fields are $G \\cdot G \\equiv G^a_{\\mu \\nu} G^{a \\, \\mu \\nu}$, and $G \\cdot \\widetilde{G} \\equiv G^a_{\\mu \\nu} \\widetilde{G}^{a \\, \\mu \\nu}$. Here $G^{a}_{\\mu \\nu}$ is a gluon field strength tensor and\n  %\n  \\begin{align}\n    \\widetilde{G}^a_{\\mu \\nu} = \\frac{1}{2} \\epsilon_{\\mu \\nu \\alpha \\beta}G^{a \\, \\alpha \\beta}\n  \\end{align}\n  %\n  is its dual. The couplings $Y_{ij} (\\widetilde{Y}_{ij})$ are in general complex.  As spontaneous symmetry breaking leads to non-diagonal lepton mass matrices, their diagonalization will result in bi-unitary transformations of $Y_{ij} (\\widetilde{Y}_{ij}) \\to y_{ij} (\\widetilde{y}_{ij})$.  Switching to a mass basis for lepton fields will then lead to LFV interactions of charged leptons $\\ell^i$,\n  %\n  \\begin{align}\n    \\label{eq:gg_lagrangian}\n    \\mathcal{L}_\\textrm{eff}^{(8)} = \\frac{v g_s^2}{\\sqrt{2} \\Lambda^4} \\left [ y_{ij} \\overline{\\ell}_L^{\\, i} \\ell_R^{\\, j} \\, G \\cdot G + \\widetilde{y}_{ij} \\overline{\\ell}_L^{\\, i} \\ell_R^{\\, j} \\, G \\cdot \\widetilde{G} \\right ] + \\textrm{h.c.} \\, ,\n  \\end{align}\n  %\n  where $v \\sim 246$~GeV is the Higgs vacuum expectation value (VEV).\n\n  For definiteness, we concentrate on the particular leptonic final state $\\mu\\tau$. In certain models of NP this final state might have the largest coupling to the new degrees of freedom, for instance due to the Cheng-Sher ansatz~\\cite{Cheng:1987rs}. Additionally, final states with muons could be preferable from the point of view of experimental detection. For instance, searches for Higgs and $Z$-boson decays to $\\mu \\tau$ are common for studies of LFV at the LHC \\cite{Arhrib:2012ax} by ATLAS \\cite{Aad:2016blu} and CMS \\cite{Khachatryan:2015kon,Sirunyan:2017xzt} collaborations.\n\n  It is interesting to point out that the $v\/\\Lambda^4$ suppression of the operators in Eq.~\\eqref{eq:gg_lagrangian} is not universal. Consider, for example, NP models where the effective coupling between gluons and leptons is generated after matching at one loop. This can be seen explicitly in two Higgs doublet models (2HDM) without natural flavor conservation with a heavy Higgs mediating CLFV as in Fig.~\\ref{fig:clfv_feynman}(a) or in the case of CLFV mediated by a heavy scalar or vector lepto-quark with appropriate quantum numbers as in Fig.~\\ref{fig:clfv_feynman}(b). Depending on the UV completion of the model, particles $Q$ and\/or $\\Phi^0\/Z$ could belong to the NP or SM spectra. If for both particles, $m_Q \\sim m_{\\Phi^0} \\sim \\Lambda$ in Fig.~\\ref{fig:clfv_feynman}(a) or $m_Q \\sim m_{X} \\sim \\Lambda$ in Fig.~\\ref{fig:clfv_feynman}(b), the overall scaling of the effective operators would be $\\propto (16 \\pi^2 \\Lambda^4 \/ v)^{-1}$. Yet, if $Q$ is a standard model top quark, then at low energies one should expect the scaling of the effective operators to be $\\propto (16 \\pi^2 m_t \\Lambda^2)^{-1}$. Such scaling of effective operators is standard in low-energy studies of lepton-flavor violation~\\cite{Raidal:2008jk,Celis:2014asa,Petrov:2013vka}.  Finally, the large gluon luminosity of the LHC can affect the detection probabilities, selecting effective operators with explicit gluonic degrees of freedom, even though they could be suppressed by additional powers of $1\/\\Lambda$.\n\n  It will therefore be appropriate, for the sake of a model-independent analysis, to introduce a set of dimension-full constants $\\mathcal{C}^{\\ell_1 \\ell_2}_i$ that encode all effects of relevant Wilson coefficients and scales. Once these coefficients are constrained from the LHC data, we can then use the available constraints to discuss different ultraviolet completions (and thus interpretations) of the effective theory. The Lagrangian of Eq.~\\eqref{eq:gg_lagrangian} then leads to the following interactions facilitating $\\mu \\tau$ production,\n  %\n  \\begin{align}\n    \\mathcal{L}_\\textrm{eff} =  \\sum_{i = 1}^4 \\mathcal{C}_i^{\\mu \\tau} \\mathcal{O}_i^{\\mu \\tau} + \\textrm{h.c.} \\, ,\n  \\end{align}\n  %\n  where\n  %\n  \\begin{align}\n    \\label{eq:gg_operators}\n    \\renewcommand{\\arraystretch}{1.25}\n    \\begin{array}{r l}\n      \\mathcal{O}_1^{\\mu \\tau} \\hspace*{-0.25cm} &= \\left ( \\overline{\\mu}_L \\tau_R \\right ) \\, G \\cdot G \\, , \\\\\n      \\mathcal{O}_2^{\\mu \\tau} \\hspace*{-0.25cm} &= \\left ( \\overline{\\mu}_L \\tau_R \\right ) \\, G \\cdot \\widetilde{G} \\, , \\\\\n      \\mathcal{O}_3^{\\mu \\tau} \\hspace*{-0.25cm} &= \\left ( \\overline{\\mu}_R \\tau_L \\right ) \\, G \\cdot G \\, , \\\\\n      \\mathcal{O}_4^{\\mu \\tau} \\hspace*{-0.25cm} &= \\left ( \\overline{\\mu}_R \\tau_L \\right ) \\, G \\cdot \\widetilde{G} \\, .\n    \\end{array}\n  \\end{align}\n\n  \\begin{figure}[t]\n    \\centering\n    \\begin{tabular}{m{0.025\\textwidth} m{0.35\\textwidth}}\n      (a) & \\includegraphics[width=0.325\\textwidth]{clfv_feynman_1.eps} \\\\ \\\\\n      (b) & \\includegraphics[width=0.325\\textwidth]{clfv_feynman_2.eps}\n    \\end{tabular}\n    \\caption{Example Feynman diagrams which can generate the operators in Eq.~\\eqref{eq:gg_operators}.}\n    \\label{fig:clfv_feynman}\n  \\end{figure}\n\n  The remainder of this letter proceeds as follows. In Sec.~\\ref{sec:gluonic_operators}, we place constraints on the coefficients $\\mathcal{C}_i^{\\mu \\tau}$ of gluonic operators. In Sec.~\\ref{sec:ttbar_operators} we use those constraints to put limits on lepton-flavor violating couplings of top quarks that are difficult to constrain at low energy machines. We conclude in Sec.~\\ref{sec:conclusion}.\n\n\n \n  \\section{LHC constraints on Gluonic Operators}\n  \\label{sec:gluonic_operators}\n \n\n\n \n  \\subsection{Event Selection}\n  \\label{sec:event_selection}\n \n\n  At the 13 TeV LHC, the tau decays promptly into neutrinos and either an electron, a muon, or hadrons. The cleanest signal comes from leptonic $\\tau$ decays, and since $\\mu^+ \\mu^-$ has a large SM background we will study the $\\mu e$ final state. The leading backgrounds are then $W^+ W^-$ pair production, $Z^0 \/ \\gamma^* \\rightarrow \\tau \\tau$, and $t \\overline{t}$ pair production. For this study we apply the basic cuts of Ref.~\\cite{Han:2010sa}, which are reviewed below.\n\n  For detector coverage and triggering, we require the transverse momentum $p_\\textrm{T}$ and pseudorapidity $\\eta$ to satisfy\n  %\n  \\begin{align}\n    \\label{eq:cut1}\n    p_\\textrm{T}^{\\mu,e} > 20~\\text{GeV} \\;\\;\\; \\text{ and } \\;\\;\\; | \\eta^{\\mu,e} | < 2.5 \\, ,\n  \\end{align}\n  %\n  while vetoing events with a final state jet of $p_\\textrm{T}^j > 50$~GeV and $| \\eta^j | < 2.5$. For the signal, we anticipate the $\\mu$ and $\\tau$ to be back to back in the transverse plane with $p_\\textrm{T}^\\mu = p_\\textrm{T}^\\tau$ and with the decay products of the $\\tau$ highly collimated. We therefore impose the additional requirements\n  %\n  \\begin{align}\n    \\label{eq:cut2}\n    \\renewcommand{\\arraystretch}{1.25}\n    \\begin{array}{c}\n      \\delta \\phi(p_\\textrm{T}^\\mu, p_\\textrm{T}^e) > 2.5 \\, , \\;\\;\\;\\;\\; \\delta \\phi(p_\\textrm{T}^\\textrm{miss}, p_\\textrm{T}^e) < 0.6 \\, , \\\\\n      \\Delta p_T = p_\\textrm{T}^\\mu - p_\\textrm{T}^e > 0 \\, ,\n    \\end{array}\n  \\end{align}\n  %\n  where $p_\\textrm{T}^\\textrm{miss}$ is the event's missing transverse momentum.\n\n  The signal kinematics also allows us to approximately reconstruct the $\\tau$. All of the missing energy in signal events is due to $\\tau$ decay products, which gives\n  %\n  \\begin{align}\n    \\vec{p}_\\textrm{T}^{\\; \\tau} = \\vec{p}_\\textrm{T}^{\\; e} + \\vec{p}_\\textrm{T}^\\textrm{ miss} \\, .\n  \\end{align}\n  %\n  From the expectation that the decay products of the $\\tau$ will be highly collimated such that $p_z^e \/ p_z^\\textrm{miss} \\approx p_\\textrm{T}^e \/ p_\\textrm{T}^\\textrm{miss}$. Thus, the longitudinal component of the $\\tau$ momentum should be\n  %\n  \\begin{align}\n    p_z^\\tau \\approx p_z^e \\left ( 1 + \\frac{p_\\textrm{T}^\\textrm{miss}}{p_\\textrm{T}^e} \\right ) \\, .\n  \\end{align}\n  %\n  Once the $\\tau$'s 3-momentum is reconstructed the energy is $E_\\tau^2 = \\vec{p}_\\tau^{\\; 2} + m_\\tau^2$. With the momentum of the $\\tau$ fully reconstructed for signal events, we then require the invariant mass of the $\\mu \\tau$ system to satisfy\n  %\n  \\begin{align}\n    \\label{eq:cut3}\n    M_{\\mu \\tau} > 250 \\text{ GeV} \\, ,\n  \\end{align}\n  %\n  as the missing energy present in the backgrounds does not in general come from the decay of a single $\\tau$.\n\n\n \n  \\subsection{Constraints}\n  \\label{sec:gg_constraints}\n \n\n  To estimate constraints on the operators in Eq.~\\eqref{eq:gg_operators} at the 13~TeV LHC, signal and background events were generated using {\\sc MadGraph5}~\\cite{Alwall:2014hca}. Showering and hadronization of these events, as well as decay of the $\\tau$, was then performed using {\\sc Pythia8}~\\cite{Sjostrand:2006za, Sjostrand:2007gs}, while detector effects were simulated with {\\sc Delphes}~\\cite{deFavereau:2013fsa}. The signal model file was generated using {\\sc FeynRules}~\\cite{Alloul:2013bka}. Background and signal cross sections after applying successive cuts are shown in Table~\\ref{tab:mutau_bgs}.\n  %\n  \\begin{table}[t]\n    \\begin{center}\n      \\renewcommand{\\arraystretch}{1.15}\n      \\small\n      \\begin{tabular}{| c | c | c | c | c |}\n        \\hline \\hline\n        $\\sigma$ (pb) & No cuts & $+$ Eq.~\\eqref{eq:cut1} & $+$ Eq.~\\eqref{eq:cut2} & $+$ Eq.~\\eqref{eq:cut3} \\\\\n        \\hline\n        $W W (\\mu \\tau)$ & 1.6 & 0.024 & 0.0044 & 0.0015 \\\\\n        $W W (\\mu e)$ & 1.6 & 0.35 & 0.014 & 0.0044 \\\\\n        $Z\/\\gamma^* (\\tau \\tau)$ & 2400 & 1.7 & 0.26 & 0.00083 \\\\\n        $t t (\\mu \\tau)$ & 12 & 0.043 & 0.0045 & 0.0019 \\\\\n        $t t (\\mu e)$ & 12 & 0.53 & 0.015 & 0.0081 \\\\\n        \\hline\n        $\\mathcal{O}_i^{\\mu \\tau}$ & 0.89 & 0.030 & 0.028 & 0.028 \\\\\n        \\hline \\hline\n      \\end{tabular}\n      \\caption{Background and signal cross sections at the 13~TeV LHC. The signal cross section assumes the benchmark values of $C_i^{\\mu \\tau} = 4 \\pi v \\, g_s^2 \/ \\sqrt{2} \\Lambda^4$ with $\\Lambda = 2$~TeV. Cross sections before cuts are given prior to $\\tau$ decays.}\n      \\label{tab:mutau_bgs}\n    \\end{center}\n  \\end{table}\n  %\n  Signal cross sections are calculated using the benchmark values of $\\mathcal{C}_i^{\\mu \\tau} = 4 \\pi v \\, g_s^2 \/ \\sqrt{2} \\Lambda^4$ with $\\Lambda = 2$~TeV. The running of the Wilson coefficients is assumed to be negligible. All operators are considered independently, and have the same cross section up to variations in their respective effective couplings.\n\n  At 100~fb$^{-1}$ of integrated luminosity, we estimate the $2 \\sigma$ confidence level (CL$_s$) exclusion limit and $5 \\sigma$ log-likelihood (LL) discovery significance for $C_i^{\\mu \\tau}$ to be\n  %\n  \\begin{align}\n    \\left ( \\mathcal{C}_i^{\\mu \\tau} \\right )_{2 \\sigma} &\\approx \\left ( 3300~\\textrm{GeV} \\right )^{-3} \\, ,\\\\\n    \\left ( \\mathcal{C}_i^{\\mu \\tau} \\right )_{5 \\sigma} &\\approx \\left ( 2900~\\textrm{GeV} \\right )^{-3} \\, .\n  \\end{align}\n  %\n  These estimates can be translated to general constraints on the NP scale $\\Lambda$ of Eq.~\\eqref{eq:gg_lagrangian} where $C_i^{\\mu \\tau} = 4 \\pi v \\, g_s^2 \/ \\sqrt{2} \\Lambda^4$. The values $y_{\\mu \\tau} = \\widetilde{y}_{\\mu \\tau} = 4 \\pi$ are chosen to push the perturbative limit of these operators in order to estimate the maximum sensitivity of the LHC to the various BSM scenarios discussed in Sec.~\\ref{sec:introduction}. With these assumptions we find lower bounds on $\\Lambda$ of\n  %\n  \\begin{align}\n    \\Lambda_{2 \\sigma} &\\approx 3000~\\textrm{GeV} \\, , \\\\\n    \\Lambda_{5 \\sigma} &\\approx 2800~\\textrm{GeV} \\, .\n  \\end{align}\n  %\n  If we instead anticipate $y_{\\mu \\tau} = y_{\\mu \\tau} \\sim \\mathcal{O}(1)$, we find that the scale of these operators are constrained to be $\\Lambda_{2 \\sigma} \\sim 1.6$~TeV. While we anticipate probing heavier NP scales as more data accumulates, models which generate the operators of Eq.~\\eqref{eq:gg_operators} at a single UV scale have cross sections suppressed by $\\Lambda^{-8}$ which limits the effectiveness of additional data on the ability to probe significantly higher scales at the LHC. A plot of the integrated luminosity at the 13~TeV LHC vs. $\\Lambda$ is shown in Fig.~\\ref{fig:luminosity_vs_cutoff} of Sec.~\\ref{sec:ttbar_constraints}.\n  \n  The operators of Eq.~\\eqref{eq:gg_operators} are in general also constrained by low energy experiments. For example, in Ref.~\\cite{Petrov:2013vka} the authors present an analysis of constraints from limits on LFV tau decays to a muon and one or two hadrons. The results of their analysis, converted to the normalization used in this paper, are shown in Table~\\ref{tab:tau_constraints}. The most stringent constraints, coming from $\\tau \\rightarrow \\mu \\pi^+ \\pi^-$ for $\\mathcal{O}_{1,3}^{\\mu \\tau}$ and $\\tau \\rightarrow \\mu \\eta$ for $\\mathcal{O}_{2,4}^{\\mu \\tau}$, are $\\Lambda_{1,3} \\approx 1000$~GeV and $\\Lambda_{2,4} \\approx 830$~GeV.~\\footnote{Alternative studies of similar processes offer differing estimates of the bounds from LFV tau decays (see e.g. Ref.~\\cite{Celis:2014asa}), but these estimates generally fall well below the LHC's expected sensitivity.} These bounds are several times lower than the estimated sensitivity of the LHC with 100~fb$^{-1}$ of luminosity.\n  \n  \\begin{table}[t]\n    \\centering\n    \\renewcommand{\\arraystretch}{1.15}\n    \\begin{tabular}{| c | c | c |}\n      \\hline \\hline\n      Process & $C_{1,3}^{\\mu \\tau}$ (GeV$^{-3}$) & $\\Lambda_{1,3}$ (GeV) \\\\\n      \\hline\n      $\\tau \\rightarrow \\mu \\, \\pi^+ \\pi^-$ & $780^{-3}$ & 1000 \\\\\n      $\\tau \\rightarrow \\mu \\, K^+ K^-$ & $700^{-3}$ & 950 \\\\\n      \\hline \\hline\n      Process & $C_{2,4}^{\\mu \\tau}$ (GeV$^{-3}$) & $\\Lambda_{2,4}$ (GeV) \\\\\n      \\hline\n      $\\tau \\rightarrow \\mu \\, \\eta$ & $590^{-3}$ & 830 \\\\\n      $\\tau \\rightarrow \\mu \\, \\eta^\\prime$ & $520^{-3}$ & 760 \\\\\n      \\hline\n      \\hline\n    \\end{tabular}\n    \\caption{Constraints on the coefficients of $\\mathcal{O}_i^{\\mu \\tau}$ from $\\tau$ decays, adapted from Ref.~\\cite{Petrov:2013vka}. The constraints on $\\Lambda$ are calculated using $C_i^{\\mu \\tau} = 4 \\pi v \\, g_s^2 \/ \\sqrt{2} \\Lambda^4_i$.}\n    \\label{tab:tau_constraints}\n  \\end{table}\n\n\n \n  \\section{LHC Constraints on $t \\overline{t} \\, \\mu \\tau$ Operators}\n  \\label{sec:ttbar_operators}\n \n\n  Studies of $\\mu \\tau$ production at hadron colliders mediated by four-fermion operators have been performed~\\cite{Han:2010sa}, with constraints on the NP scale obtained with the help of a single operator dominance hypothesis~\\cite{Hazard:2016fnc}. They, however, did not examine operators that include top-quark fields. As we show below, these operators can be constrained by studying $gg\\to\\mu\\tau$ processes.\n\n\n \n  \\subsection{Matching Conditions}\n  \\label{sec:matching_conditions}\n \n\n  The $SU(3)_C \\times U(1)_\\textrm{EM}$ invariant Lagrangian contributing to $\\mu \\tau$ production contains\n  %\n  \\begin{align}\n    \\label{eq:4fermion_lagrangian2}\n    \\mathcal{L}_{\\mu \\tau}^{(6)} \\supset \\frac{1}{\\Lambda^2} \\sum_{i = 1}^4 C_i^{q \\mu \\tau} \\mathcal{O}_i^{q \\mu \\tau} + \\text{h.c.} \\, ,\n  \\end{align}\n  %\n  where\n  %\n  \\begin{align}\n    \\label{eq:4fermion_operators}\n    \\renewcommand{\\arraystretch}{1.25}\n    \\begin{array}{r l}\n      \\mathcal{O}_1^{q \\mu \\tau} \\hspace*{-0.25cm} &= \\left ( \\overline{\\mu}_L \\tau_R \\right ) \\left ( \\overline{q}_L q_R \\right ) \\, , \\\\\n      \\mathcal{O}_2^{q \\mu \\tau} \\hspace*{-0.25cm} &= \\left ( \\overline{\\mu}_L \\tau_R \\right ) \\left ( \\overline{q}_R q_L \\right ) \\, , \\\\\n      \\mathcal{O}_3^{q \\mu \\tau} \\hspace*{-0.25cm} &= \\left ( \\overline{\\mu}_R \\tau_L \\right ) \\left ( \\overline{q}_L q_R \\right ) \\, , \\\\\n      \\mathcal{O}_4^{q \\mu \\tau} \\hspace*{-0.25cm} &= \\left ( \\overline{\\mu}_R \\tau_L \\right ) \\left ( \\overline{q}_R q_L \\right ) \\, .\n    \\end{array}\n  \\end{align}\n  %\n  As noted in Section~\\ref{sec:introduction}, these operators also generate the gluonic operators of Eq.~\\eqref{eq:gg_operators} via SM quark loops in the diagrams represented in Fig.~\\ref{fig:clfv_feynman}. Thus we can take advantage of the enhanced gluon luminosity at the LHC to probe these operators indirectly through gluon fusion production. A brief discussion of $SU(2)_L$ invariant operators generating those of Eq.~\\eqref{eq:4fermion_operators} is given in \\ref{sec:effective_operators}.\n\n  The coefficients of the dimension 8 operators, $C_i^{\\mu \\tau}$, are related to the Wilson coefficients of the dimension 6 operators, $C_i^{q\\mu\\tau}$ by\n  %\n  \\begin{align}\n    \\label{eq:matching1}\n    \\mathcal{C}_{1,3}^{\\mu \\tau} &= \\frac{g_s^2}{16 \\pi^2} \\frac{F_1(x)}{\\Lambda^2 m_q} \\left [ C_{1,3}^{q \\mu \\tau} + C_{2,4}^{q \\mu \\tau} \\right ]_{C_{1,3}^{q \\mu \\tau} = C_{2,4}^{q \\mu \\tau}} \\, , \\\\\n    \\label{eq:matching2}\n    \\mathcal{C}_{2,4}^{\\mu \\tau} &= \\frac{i g_s^2}{16 \\pi^2} \\frac{F_2(x)}{\\Lambda^2 m_q} \\left [ C_{1,3}^{q \\mu \\tau} - C_{2,4}^{q \\mu \\tau} \\right ]_{C_{1,3}^{q \\mu \\tau} = - C_{2,4}^{q \\mu \\tau}} \\, ,\n  \\end{align}\n  %\n  where $m_q$ is the mass of the quark running in the loop. Here $F(x)$ are functions of the parton center-of-momentum (CM) energy $\\hat{s}$, and are given by\n  %\n  \\begin{align}\n    \\label{eq:form_factor1}\n    F_1 (x) &= - \\frac{x}{2} \\left [ 4 + (4 x - 1) \\ln^2 \\left ( 1 - \\frac{1}{2x} + \\frac{\\sqrt{1 - 4 x}}{2x} \\right ) \\right ] \\, , \\\\\n    \\label{eq:form_factor2}\n    F_2 (x) &= \\frac{x}{2} \\ln^2 \\left ( 1 - \\frac{1}{2x} + \\frac{\\sqrt{1 - 4 x}}{2x} \\right ) \\, ,\n  \\end{align}\n  %\n  where $x \\equiv m_q^2 \/ \\hat{s}$. In the limit that $x \\ll 1$, the functions $F(x)$ approach $m_q^2 \/ \\hat{s}$, indicating that the contribution to $\\mu\\tau$ production from gluon fusion is dominated by the heaviest quark running in the loop. At LHC energies, provided only SM quarks contribute to this process, the top quark contribution is, therefore, expected to dominate.\n\n\n \n  \\subsection{Constraints}\n  \\label{sec:ttbar_constraints}\n \n\n  Converting the results from Section~\\ref{sec:gg_constraints}, we find, for 100~fb$^{-1}$ of integrated luminosity with the benchmark values $\\left | C_i^{q \\mu \\tau} \\right | = 4 \\pi$, chosen again to be at the perturbative limit in order to estimate the maximum potential reach of the study, the $2 \\sigma$ CL$_s$ exclusion limit and $5 \\sigma$ LL discovery significance for the $G \\cdot G$ operators $\\mathcal{O}_{1,3}^{\\mu \\tau}$ to be\n  %\n  \\begin{align}\n    \\label{eq:dim6_constraint1}\n    \\Lambda_{2 \\sigma} &\\approx 3400~\\textrm{GeV} \\, , \\\\\n    \\label{eq:dim6_constraint2}\n    \\Lambda_{5 \\sigma} &\\approx 2900~\\textrm{GeV} \\, ,\n  \\end{align}\n  %\n  while for the $G \\cdot \\widetilde{G}$ operators $\\mathcal{O}_{2,4}^{\\mu \\tau}$ they are\n  \\begin{align}\n    \\label{eq:dim6_constraint3}\n    \\Lambda_{2 \\sigma} &\\approx 4100~\\textrm{GeV} \\, , \\\\\n    \\label{eq:dim6_constraint4}\n    \\Lambda_{5 \\sigma} &\\approx 3400~\\textrm{GeV} \\, .\n  \\end{align}\n  Note that the energy scale $\\Lambda$ here is the NP scale of the dimension 6 four-fermion operators of Eq.~\\eqref{eq:4fermion_lagrangian2}.~\\footnote{The gluon interactions with $\\mu \\tau$ are clearly non-local at LHC energies. To account for this, we average the full form factors (squared) of Eqs.~\\eqref{eq:form_factor1} and \\eqref{eq:form_factor2} by reconstructing the $\\tau$ to obtain an approximate event-by-event $\\hat{s}$.} For $|C_i^{q \\mu \\tau} | \\sim \\mathcal{O}(1)$, the constraints of Eqs.~(\\ref{eq:dim6_constraint1}--\\ref{eq:dim6_constraint4}) are estimated to be only $\\Lambda_{2 \\sigma} \\sim 0.97$ (1.1)~TeV for the $G \\cdot G$ ($G \\cdot \\widetilde{G}$) operators, which may be near the scale of validity for the EFT at the LHC. However, unlike the operators discussed in Sec.~\\ref{sec:gluonic_operators}, the dimension 6 operator-induced cross-sections scale as $\\Lambda^{-4}$ and therefore stand to benefit more from the accumulation of additional data.\n\n  \\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.475\\textwidth]{LFV_Luminosity_vs_Lambda_3.eps}\n    \\caption{Luminosity goal at the 13 TeV LHC as a function of the NP scale $\\Lambda$, choosing a value of $4 \\pi$ for the dimensionless Wilson coefficients}. The solid (dashed) curves represent the $2\\sigma$ CL$_s$ exclusion limit ($5\\sigma$ LL discovery limit) on the luminosity required to rule out (discover) NP at the scale $\\Lambda$.\n    \\label{fig:luminosity_vs_cutoff}\n  \\end{figure}\n  \n  Fig.~\\ref{fig:luminosity_vs_cutoff} shows the luminosity required to set $2 \\sigma$ CL$_s$ exclusion constraints and $5 \\sigma$ LL discovery estimates as a function of the NP scale $\\Lambda$. With 100~fb$^{-1}$ of integrated luminosity, the constraints on the operators in Eq.~\\eqref{eq:gg_lagrangian} and those of Eq.~\\eqref{eq:4fermion_operators} appear to be quite similar due to the explicit inclusion of loop suppression factors via the matching conditions of Eqs.~\\eqref{eq:matching1} and \\eqref{eq:matching2}. However, the LHC becomes increasingly sensitive to interactions via SM top quark loops as integrated luminosity is increased. As data at the LHC continues to accrue, experiments will become increasingly sensitive to NP that generates the operators of Eq.~\\eqref{eq:4fermion_operators} and may be probed via gluonic processes. \n  \n  The dimension 6 operators of Eq.~\\eqref{eq:4fermion_operators} also induce at one loop couplings to the SM Higgs boson. LFV Higgs decays are stringently constrained by direct searches at the LHC, with the most recent result from the CMS collaboration at $\\sqrt{s} = 13$~TeV constraining the branching ratio to be Br$(h \\rightarrow \\mu \\tau) < 0.25\\%$~\\cite{Sirunyan:2017xzt}. In general, however, other operators may contribute to LFV Higgs couplings. Specifically, at dimension 6 the operators $(\\overline{f}_L^i f_R^j) H (H^\\dagger H)$ and $(\\overline{f}_{L,R}^i \\gamma^\\mu f_{L,R}^j) (H^\\dagger i \\overleftrightarrow{D}_\\mu H)$ induce direct LFV couplings to the Higgs boson, with potential interference between the various contributions. While a thorough study of the effects of these operators on Higgs decays is beyond the scope of this work, see e.g. Ref.~\\cite{Harnik:2012pb} for an analysis of EFT-induced LFV couplings to the SM Higgs boson.\n\n\n \n  \\section{Conclusion}\n  \\label{sec:conclusion}\n \n\n  In this letter we have examined CLFV processes initiated by gluon fusion at the $\\sqrt{s} = 13$~TeV LHC. We have demonstrated that the gluon's enhanced parton luminosity can compensate for the increased suppression from dimension 8 operators relative to the less suppressed dimension 6 quark-induced CLFV processes. This allows one to indirectly probe models of NP mediating CLFV processes that may otherwise be inaccessible at LHC energies. The LHC has already collected nearly 100~fb$^{-1}$ of data at 13~TeV. We have estimated that this data can constrain the dimensionful coefficients of gluonic CLFV-inducing operators to be $\\mathcal{C}_i^{\\mu \\tau} \\gtrsim \\left ( 3.3~\\textrm{TeV} \\right )^{-3}$.\n\n  In addition, we have presented a study of such processes occurring through SM quark loops. In models where single operator dominance is expected, we have demonstrated that it is possible to constrain the CLFV coupling of leptons to top quarks through loop-induced gluon fusion. With 100~fb$^{-1}$ of data, we have estimated that the NP scale $\\Lambda$ of $t \\overline{t} \\, \\mu \\tau$ couplings can be constrained to be $\\Lambda \\gtrsim 3.4 - 4.1$~TeV. This mechanism is especially important for models that predict an enhanced coupling to top quarks such as in certain 2HDMs with LFV. A future discovery of CLFV in the $\\mu \\tau$ final state at the LHC could be the first indication of preferential couplings to top quarks, and could be important in discriminating between the many models of CLFV.\n\n\n \n  \\section{Acknowledgements}\n  \\label{sec:acknowledgements}\n \n\n  This work has been supported in part by the U.S. Department of Energy under contract DE-SC0007983 and by the National Science Foundation under grant PHY-1460853 under the auspices of WSU Research Experience for Undergraduates program.\n\n\n  \\begin{appendix}\n\n   \n    \\section{$SU(2)_L$ Invariant Operators}\n    \\label{sec:effective_operators}\n   \n\n    The $SU(3)_C \\times U(1)_\\textrm{EM}$ invariant operators of Eq.~\\eqref{eq:4fermion_operators} arise from $SU(2)_L$ invariant forms. Specifically, $\\mathcal{O}_1^{q \\mu \\tau}$ is contained in the dimension 6 operator\n    \\begin{align}\n      \\label{eq:su2_dim6}\n      \\mathcal{O}^{LeQu} = ( \\overline{L}_L^{\\, i} e_R^{\\, j} ) \\, \\epsilon \\, ( \\overline{Q}_L^{\\, k} u_R^{\\, l} ) \\, ,\n    \\end{align}\n    where the antisymmetric tensor $\\epsilon$ contracts the suppressed $SU(2)_L$ indices, and $\\mathcal{O}_4^{q \\mu \\tau}$ is included in its Hermitian conjugate. Conversely, operators $\\mathcal{O}_{2,3}^{q \\mu \\tau}$ are first generated at dimension 8,\n    \\begin{align}\n      \\label{eq:su2_dim8}\n      \\mathcal{O}^{LHeuHQ} = ( [ \\overline{L}_L^{\\, i} H ] e_R^{\\, j} ) ( \\overline{u}_R^{\\, k} [ H^T i \\sigma_2 Q_L^{\\, l}] )\n    \\end{align}\n    and its Hermitian conjugate. Operators $\\mathcal{O}_{1,4}^{q \\mu \\tau}$ can also be generated at dimension 8 without the associated charged current interactions of Eq.~\\eqref{eq:su2_dim6}.\n\n    Because only two of the operators listed in Eq.~\\eqref{eq:4fermion_operators} appear at dimension 6 in an $SU(2)_L$ invariant form, there is in general no reason to expect the coefficients of these operators to be similar in value, as required by the matching conditions given in Eqs.~\\eqref{eq:matching1} and \\eqref{eq:matching2}. We should then generally expect a mixing of the $G \\cdot G$ and $G \\cdot \\widetilde{G}$ production mechanisms. However, Ref.~\\cite{Potter:2012yv} has demonstrated that the contributions from these operators can not be distinguished by a study of lepton pair production alone, and thus one should consider the limits presented in Section~\\ref{sec:ttbar_constraints} as estimates of the upper bounds on such processes. We postpone a more complete discussion of the four-lepton operators for a future, more detailed, analysis, where one can expect the scale of $\\mathcal{O}_{2,3}^{t \\mu \\tau}$ to be more weakly constrained than $\\mathcal{O}_{1,4}^{t \\mu \\tau}$.\n\n  \\end{appendix}\n\n\n \n  \\section*{References}\n  \\label{sec:references}\n \n\n  \\bibliographystyle{elsarticle-num}\n  ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nSensor measurements are often incomplete, noisy, and replete with outliers arising due to malfunctions or intermittent errors. Imputation of the missing entries and removal\/segregation of the outliers is a critical first step that must be carried out prior to any data analytics. Examples of applications that benefit from such a pre-processing step include estimation\/prediction of city-wide road traffic, regional air quality, electricity consumption in power distribution networks and foreground-background separation in videos. For most of these applications, the measurements can be arranged in form of a matrix, some of whose entries may be missing or contaminated with outliers. Pertinent approaches model the measurements as arising from a low-dimensional subspace whose recovery allows us to reject the noise and outliers, and impute the missing entries \\cite{candes2009exact,balzano2010online,babacan-11,giampouras2017online,candes2011robust,ding2011bayesian}.\n\\par Many real-world applications, including the aforementioned ones, involve time-varying data that arrives in a sequential manner and must be processed as such. As a result, the data matrices arising in such applications comprise of low-dimensional subspaces that evolve over time. While the classical matrix completion or robust principal component analysis (RPCA) approaches are still applicable to each snapshot of the data, the performance can generally be improved by exploiting the temporal correlations present in the measurements \\cite{liu2013tensor,balzano2010online,electrictydata,grasta,roseta}. State-of-the-art approaches for processing time-varying subspaces can mostly be classified into approaches based on tensor completion \\cite{liu2013tensor} and regularized matrix completion \\cite{NIPS2016_6160}. A common feature of these techniques is their static perspective and the resulting focus on batch processing. In contrast however, the data streaming from the sensors may be inherently dynamic, arising from subspaces that evolve over time. Theoretical guarantees for the dynamic setting have been studied in \\cite{XuDevenportNIPS}.\nDifferent from these approaches and closer to the classical time-series modeling, an online forecasting matrix completion approach was proposed in \\cite{electrictydata} where the underlying subspace was assumed to follow a linear state-space model and must be learned in an online fashion. Approaches based on matrix completion often involve a number of tuning parameters that must be correctly set in order to avoid over-fitting. However determining these parameters via cross-validation is quite challenging with time-series data, especially in the online setting \\cite{electrictydata}. Alternatively, probabilistic learning algorithms have been proposed for the static matrix completion, and are generally free of tuning parameters. Such approaches entail constructing generative models that are not only capable of modeling the data but are also simple enough to allow low-complexity updates.\n\n\n\n\n\n\n\n\n\n\n\n\\par\nThis work considers the first low-rank robust subspace filtering approach for online matrix imputation and prediction. Different from the existing matrix and tensor completion formulations, we consider low-rank matrices whose underlying subspace evolves according to a state-space model. As incomplete columns of the data matrix arrive sequentially over time, the low rank components as well as the state-space model are learned in an online fashion using the variational Bayes formalism. In particular, component distributions are chosen to allow automatic relevance determination (ARD) and unlike the matrix or tensor completion works, the algorithm parameters such as rank, noise powers, and state noise powers need not be specified or tuned. A low-complexity forward-backward algorithm is also proposed that allows the updates to be carried out efficiently. Enhancements to the proposed algorithm, capable of learning time-varying state-transition matrices, operating with a fixed lag, and robust to outliers, are also detailed.  Our approach is general and we demonstrate its efficacy on various settings.  In particular, we discuss the traffic estimation problem in detail and show that the variational Bayesian approach can be used to impute road traffic densities in an online fashion and from only a few observations. As the proposed models are generative, the resulting traffic density predictions can also be used to obtain accurate expected time-of-arrival (ETA) estimates. Additionally, the applicability of the proposed algorithm on the electricity load estimation and prediction problem is also shown. The superior performance of our algorithm vis-a-vis other state of the art subspace tracking and online matrix factorisation algorithms may be attributed to the proposed state space model as well as the flexibility in the data modeling provided by the variational Bayesian approach. In summary, the contributions of the present work are as follows:\n\\begin{enumerate}\n\t\\item We present the variational Bayesian subspace filtering (VBSF) algorithm and demonstrate its ability to perform data modeling, imputation and temporal prediction in an online setting wherein the key algorithmic parameters are automatically tuned.\n\t\\item Robust version of the VBSF algorithm is also proposed for outlier removal and data cleansing.\n\t\\item Finally, we report a comprehensive comparison of our algorithm with various relevant (offline) matrix completion as well as online subspace estimation and tracking techniques, e.g, GROUSE \\cite{balzano2010online}, Low Rank Tensor Completion (LRTC) \\cite{liu2013tensor},  GRASTA \\cite{grasta}, ROSETA \\cite{roseta}, OP-RPCA \\cite{oprca} and Online Forecasting Matrix Factorisation (OFMF) \\cite{paperarnew} over real-world traffic speed data as well as the electricity load data. \n\\end{enumerate}\n\\subsection {Related work}\nVariational Bayesian approaches for matrix completion and robust principal component analysis are well known \\cite{babacan-11,Parker-14, Parker-14-2,Wipf-16,yang2018fast,asif2016matrix,luttinen2013fast,giampouras2017online,ma2015variational}. One of the first works considered the measured matrix to be expressible as a product of low-rank matrices, associated with appropriate ARD priors \\cite{babacan-11} while faster algorithms for similar settings were proposed in \\cite{Parker-14, Parker-14-2}. More recently, other approaches towards modeling the measured matrices have also been proposed \\cite{Wipf-16}, \\cite {yang2018fast}. Moreover, variational Bayesian approaches have also been applied to road traffic estimation; see e.g. \\cite{asif2016matrix}. However, these approaches do not explicitly model the evolution of the underlying subspace. Likewise, none of the existing variational Bayesian approaches for low rank matrix completion model the evolution of the subspace \\cite{babacan-11,ma2015variational,yang2018fast}.  In contrast to these, the state-space modeling in our work is inspired from \\cite{luttinen2013fast}, where the low-complexity updates were first proposed in the context of linear dynamical models. The VBSF algorithm in the current work extends and generalizes that in \\cite{luttinen2013fast} to incorporate low-rank structure and outliers.\n\nOn a related note, temporal evolution of the additive noise is modeled in \\cite{giampouras2017online} using a forgetting factor. Different from \\cite{giampouras2017online} however, we use a state-space model to capture the evolution of the underlying subspace. An online Bayesian matrix factorization model is also proposed in \\cite{oprca} wherein the time-stamps are directly incorporated as features. In contrast, the present model is more specific and suited to a slowly time-varying system. \n\n\n\\par\nSeveral non-Bayesian algorithms have been proposed to address the online subspace estimation problem from incomplete observations\\cite{balzano2010online,grasta,oprca,roseta}. \nGROUSE \\cite{balzano2010online} is one of the early approaches that uses an update on the Grassmannian manifold to estimate the subspace. \nThe robust variant of GROUSE, namely GRASTA , handles outliers by by incorporating the $l_1$ norm cost function\\cite{grasta}. OP-RPCA\\cite{oprca} is a robust subspace estimation technique that uses alternating minimization to compute the outliers and the underlying subspace. A number of online subspace tracking algorithms, such as ROSETA \\cite{roseta}, have since been proposed. The proposed approach is compared with some of these algorithms in Sec. \\ref{results}. \n\\subsection{Applications:}\n\\subsubsection{Traffic Estimation and Prediction}\nTraffic estimation and prediction are the central components of any urban traffic congestion management system \\cite{survey_paper}.\nWith the advent of smartphones, public transportation services as well as private on-demand transportation companies are increasingly relying on the availability of real-time traffic maps for resource allocation and logistics \\cite{res_allocation}. \nSuch providers rely on probe vehicles --- GPS enabled and possibly crowd-sourced agents that upload speed measurements and corresponding location tags at sporadic times. Since traffic densities are inferred from speed measurements, they are often ridden with outliers, e.g., corresponding to random velocity changes unrelated to traffic. The traffic estimation problem entails estimating traffic densities at locations and times where no measurements are available. Finally, prediction of traffic in the near future is necessary to calculate ETA, fastest route, and other related quality of service metrics for road users. The future traffic prediction problem becomes particularly challenging in regions with diverse modes of transport, such as in India, where ETA calculations must account for the multimodal nature of traffic \\cite{mohan2013moving,goel2016access}. For instance the ETA calculations for buses should not only use traffic data meant for cars.A class of pertinent approaches have sought to visualize the traffic data as an incomplete matrix or tensor, and exploited this correlation to fill-in the missing entries \\cite{qu2009ppca,qu2008bpca,tan2016short,asif2016matrix}. Complementary to these approaches, time-series modeling focuses on learning the temporal dynamics of traffic and generate predictions in an online manner \\cite{guo2014adaptive}. While recent variants have incorporated spatial correlations as well, these techniques are generally unable to handle missing data or outliers. Finally, \\cite{paperarnew} presents the online forecasting matrix factorisation algorithm on the time series data that also handles the missing data scenario.\n\\subsubsection{Electricity Load Estimation and Prediction}\nSimilar to the traffic data, the electricity load data also exhibits the spatial and temporal structure that can be exploited to impute the missing data while simultaneously removing the noisy outliers. Due to the environmental disturbance, communication error or sensor fault, it is inevitable that load data may be lost during the collection process \\cite{zhang2018short}.\n \n\n\n\n\n\n\nThis paper is organized as follows. Sec. \\ref{vbsf} presents the online variational Bayesian subspace filtering method for traffic estimation and prediction. Sec. III presents the  online robust variational Bayesian subspace filtering method for traffic estimation and prediction in case of outliers. Results and findings for traffic prediction and electricity load prediction are discussed in Sec. \\ref{results} followed by conclusion in Sec. \\ref{conclusion}.\n\n\\section{Results}\n\\label{results}\nWe now detail the simulation results that evaluate the performance of the proposed VBSF method on variety of datasets to solve the: \n\\begin{enumerate}\n    \\item Traffic Estimation and Prediction Problem\n    \\item Electricity Load Estimation and Prediction Problem\n\\end{enumerate}\n\\subsection{Datasets}\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.7\\linewidth]{map2.PNG}\n\t\\caption{Region where traffic data is collected}\n\t\\label{gmap}\n\\end{figure}\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.7\\linewidth]{map1.png}\n\t\\caption{Map with red as missing and blue as known traffic entries}\n\t\\label{gmap2}\n\\end{figure}\n\\begin{itemize}\n\t\\item Traffic:  for traffic estimation and prediction, we use the partial road network of the city of New Delhi with an area of 200 square kms consisting of $m=519$ edges (shown in Fig. \\ref{gmap}). The road network can be modeled using a directed graph where each edge represents a road segment and nodes represent intersections. We collect the traffic data in the form of average speed of vehicles on a particular segment using the Google map APIs for nearly 3 months across 519 edges. Taking advantage of the slow varying nature of the speed in the network edges, we sample the traffic data at the rate of one sample every $t_s =15$ minutes. Note that our algorithm is agnostic of the sampling rate and would  work for higher sampling rates as well. Unlike the complete data available from the API, real-world data may have missing entries. For instance, over the smaller area shown in Fig. \\ref{gmap2}, speed measurements may be available on the blue edges but not on the red ones. Finally, we evaluate our algorithm for the twin tasks of real time traffic estimation as well as future traffic prediction. We further evaluate our algorithm for robust traffic estimation , i.e., when we the traffic data is corrupted by outliers.\n\t\\item Electricity: similar to the traffic estimation and prediction task, we evaluate the VBSF algorithm on the electricity dataset \\cite{electrictydata}, also used in \\cite{paperarnew} to evaluate the online matrix factorisation method. The data contains the hourly power consumption of 370 consumers, sampled every 15 min. The data is recorded from Jan. 1, 2012 to Jan. 1, 2015. Finally, we compare the VBSF method with various methods including the ones proposed and compared in \\cite{paperarnew}.\n\\end{itemize}\n In order to evaluate the VBSF algorithm, an incomplete data set is created by randomly sampling a fraction $p$ of the measurements. In our evaluations we consider three different cases with  75\\%, 50\\%, and 25\\% of missing data.  We select previous $h$ = 30 time intervals for traffic and, the previous $h$ = 40 time intervals for electricity dataset. We compare our algorithm with other methods that potentially solve the current traffic estimation problem in the missing data scenario. The algorithms are \n \\begin{itemize}\n\\item Low rank tensor completion (LRTC) \\cite{liu2013tensor}.\n\\item Grassmannian Rank-One Update Subspace Estimation (GROUSE) \\cite{balzano2010online}.\n\\item Historic mean, which  is simply the mean of edge speed values at a given time instance calculated using the historic data. \n\\end{itemize}\nFor the robust VBSF, we compare our algorithm with corresponding robust matrix completion frameworks. \n\\begin{itemize}\n    \\item Robust PCA via Outlier Pursuit (OP-RPCA) \\cite{oprca}.\n    \\item Robust Online Subspace Estimation and Tracking Algorithm (ROSETA) \\cite{roseta}.\n    \\item  Grassmannian Robust Adaptive Subspace Tracking Algorithm (GRASTA) \\cite{grasta}.\n\\end{itemize}\n Further, for the electricity load prediction problem, we compare our algorithm with the results of \\cite{paperarnew} and the Collaborative Kalman Filter (CKF) \\cite{paperarnew}.\n\\subsection{Traffic Estimation and Prediction Problem}\n\\subsubsection{Performance Index}\nTo measure the effectiveness of our algorithm and for the comparison with other relevant algorithms, we use mean relative error (MRE) as the performance index for the traffic data. For any time instance $\\tau$, the MRE denoted by $\\text{MRE}_\\tau$ is defined as:\n\\begin{equation}\n\\text{MRE}_\\tau= \\frac{1}{z}\\sum_{k=1}^z \\frac {\\parallel \\hat{\\mathbf{y}}_{\\tau,k}-\\mathbf{y}_{\\tau,k}\\parallel_{2}}{\\parallel \\mathbf{y}_{\\tau,k}\\parallel_2}.\n\\end{equation}\nwhere $\\mathbf{y}_{\\tau,k}$ and $\\hat{\\mathbf{y}}_{\\tau,k}$ are the ground truth and estimated data for $k^{th}$ day and $\\tau^{th}$ time instance. Since the value for the known data (sampled entries) may be modified post estimation, we compute the MRE over the whole column for a given time instance. For calculating the overall accuracy of prediction for a day, we calculate MRE averged over $z$ days. The value of $z$ is taken as 50 for weekdays and 10 for the weekends.\n\\begin{figure*}[ht]\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{figure2.pdf}\n\t\\caption{Estimation of traffic data for different percentage of missing entries}{(a) Actual Traffic data , (b) Traffic data with 25\\% entries, (c) Estimated Traffic with 25\\% known data, (d) Residual error for estimation with 25\\% data , (e) Traffic data with 50\\% entries , (f) Estimated Traffic with 50\\% known data, (g) Residual error for estimation with 50\\% data, (h) Traffic data with 75\\% entries , (i) Estimated Traffic with 75\\% known data, \\\\(j) Residual error for estimation with 75\\% data }\n\t\\label{fig:fplot}\n\\end{figure*}\n\\subsubsection{Online Real Time Traffic Estimation}\nWe now discuss simulation results for the current traffic estimation based on the current and past missing data using the VBSF algorithm. For a typical day, Fig. \\ref{fig:fplot}a shows the heatmap of the actual traffic data. The $x$-axis of each  heatmap represents time instances while the $y$-axis represents the edges. Each pixel of a heatmap indicates the speed, where higher speed is represented by a lighter colour. Figures \\ref{fig:fplot}b,  \\ref{fig:fplot}e and \\ref{fig:fplot}h are heatmaps with missing entries of varying degrees. The corresponding completed matrices using VBSF algorithm are shown in Figs. \\ref{fig:fplot}c, \\ref{fig:fplot}f, and \\ref{fig:fplot}i.   Since the proposed VBSF is an online method that completes one column at a time given the incomplete data from previous columns, the corresponding heatmaps are also generated in an online fashion. In other words, in spirit of the online methodology, window of $h+1$ incomplete columns are used to complete the last column followed by moving the window by one column.  Finally, all the completed columns form a matrix represented in these heatmaps.  Unsurprisingly, the heatmaps show that the performance of VBSF improves as the size of missing data decreases.  \\par  \nThe MRE values for real time traffic estimation using VBSF for weekends is shown in Fig. \\ref{fig:fig4}a and for weekdays in Fig. \\ref{fig:fig4}b. It is observed that the prediction error is higher during the peak traffic time (in the evening) vis-a-vis non-peak time intervals. This may be due to a greater variance in traffic during the peak time intervals. However, the difference between the MRE values for 50\\% and 25\\% missing data case is only about 0.15 in the worst case. Equivalently, the average error of estimation of speed is only around 2 km\/hr during the peak-time when the average speed is 15 km\/hr even with 75\\% missing data.  Similarly, for non-peak hours, even though the observed speed are higher (around 30-40 km\/hr), the MRE values for $p=50\\%$ and $p=25\\%$  is around 0.1, which in other words indicate an average error of 3-4 km\/hr in the estimation of speed. \\par \nThe  performance  of  the  proposed  VBSF  algorithm  is compared   with   that   of  (LRTC) \\cite{liu2013tensor}, (GROUSE) \\cite{balzano2010online}, and  the historic mean. We used a grid search based approach for rank initialization in GROUSE and choose the rank that gives the least error.  Table \\ref{tab1:table1} presents the overall results.  Further, Figs. \\ref{fig:fig5}a and \\ref{fig:fig5}b show the comparison of our algorithm for different percentage of missing traffic data. It is observed that for low missing rate of traffic data (25\\%), the LRTC (low rank tensor completion) \\cite{liu2013tensor} and VBSF obtain similar performance. But as the missing data increases,  VBSF outperforms the LRTC method.  Also, for all the cases, VBSF performs better than GROUSE.  This difference in performance can be attributed to the fact that the VBSF framework captures the temporal dependencies as well as the latent factors in the traffic matrix better than other methods. In terms of running time, VBSF is faster than LRTC and is comparable to GROUSE as shown in Table \\ref{tab1:table12}. \n\n\\begin{table}[ht!]\n\t\n\t\\begin{center}\n\t\t\n\t\t\n\t\t\\begin{tabular}{llll}\n\t\t\t\\hline \n\t\t\t&$p=0.25$ & $p=0.50$  &$p=0.75$\\\\\n\t\t\t&MRE&MRE&MRE\\\\\n\t\t\t\\hline\n\t\t\tVBSF & 0.1439 &0.11277 &0.09336\\\\\n\t\t\tGROUSE & 0.372 & 0.3446& 0.3085\\\\\n\t\t\tLRTC & 0.1921 & 0.1418&0.09578\\\\\n\t\t\tMean &0.2083&0.2083&0.2083\\\\\n\t\t\\end{tabular}\n\t\t\n\t\t\n\t\\end{center}\n\t\\caption{Performance comparison for real time traffic estimation}\n\t\\label{tab1:table1}\n\\end{table}\n\n\\begin{table}[ht!]\n\t\n\t\\begin{center}\n\t\t\\begin{tabular}{llll}\n\t\t\t\\hline \n\t\t\t&$p=0.25$ &$p=0.50$ &$p=0.75$\\\\\n\t\t\t&time($sec$)&time($sec$)&time($sec$)\\\\\n\t\t\t\\hline\n\t\t\tVBSF &  0.7001&0.8685&0.9675\\\\\n\t\t\tGROUSE &0.7935&0.85324&0.923960\\\\\n\t\t\tLRTC &2.92&4.32&6.23\\\\\n\t\t\t\n\t\t\\end{tabular}\n\t\t\n\t\t\n\t\\end{center}\n\t\\caption{Comparison of running time for different algorithms$^1$}\n\t\\label{tab1:table12}\n\\end{table}\n\\footnotetext[1]{Experiments are conducted to evaluate average running time per column on Matlab using PC: Intel i5-6200U CPU 2.4 GHz. }\n\\begin{figure*}\n\t\\centering\n\t\\begin{subfigure}[b]{1\\textwidth}\t\n\t\t\\includegraphics[width=1\\linewidth]{figure5.pdf}\t\n\t\\end{subfigure}%\n\t\n\t\\begin{subfigure}[b]{1\\textwidth}\n\t\t\\includegraphics[width=1\\linewidth]{figure62.png}\n\t\t\n\t\\end{subfigure}%\n\t\\caption{Real time Traffic Estimation  and Prediction for different missing entries}{(a) Real time traffic estimation for different missing entries (Weekend), (b) Weekday Prediction 50\\% missing entries (Weekday), (c) Weekday Prediction 50\\% missing entries, (d) Weekend Prediction 50\\% missing entries, (e) Weekday Prediction 75\\% missing entries, (f) Overall Prediction}\t\n\t\\label{fig:fig4}\t\n\\end{figure*}\n\\subsubsection{Future Traffic Prediction Problem}\nWe also test the  VBSF algorithm for speed prediction during the future time intervals assuming randomly sampled data from the current and previous time intervals. We predict traffic data up to 5 sampling intervals, that is, 15 to 75 minutes in future.  We test our algorithm for 50\\% and 75\\% of the missing entries in the traffic data. The MRE plots for traffic prediction are shown in Figs. \\ref{fig:fig4}c, \\ref{fig:fig4}d, and \\ref{fig:fig4}e.  The MRE error difference for 50\\% and 75\\% missing data is not significant. Similar to observations from the current traffic estimation simulations,  it is seen that the error increases from 5:30 to 8:00 pm. As one would expect, the prediction accuracy decreases as we predict further in future.  Interestingly, it is observed that the MRE for real-time traffic estimation with 75\\% missing entries case and for future prediction with 50\\% missing entries are comparable as can be seen in Fig. \\ref{fig:fig4}f. \\par \nThe performance of the proposed VBSF algorithm is compared with that of LRTC in Table \\ref{tab:table2}.  The VBSF performs better than the LRTC as shown in Fig. \\ref{fig:fig5}c. While predicting the speed for outlier edges (the edges which significantly deviate from their usual speed) VBSF performs better than LRTC as seen in Fig. \\ref{fig:fig5}d. \n\\begin{table}[ht!]\n\t\\begin{center}\n\t\t\n\t\t\n\t\t\\begin{tabular}{lll}\n\t\t\t\\hline\n\t\t\t&$p=0.50$ & $p=0.50$  \\\\\n\t\t\t&$15\\, mins $&$30\\,mins$\\\\\n\t\t\t\\hline\n\t\t\tVBSF & 0.15362 &0.17434 \\\\\n\t\t\tLRTC & 0.15843 & 0.1812\\\\\n\t\t\tMean & 0.2082 & 0.2073\\\\\n\t\t\\end{tabular}\n\t\t\n\t\t\n\t\\end{center}\n\t\\caption{Performance comparison for traffic prediction }\n\t\\label{tab:table2}\n\\end{table}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{compare.pdf}\n\t\\caption{Comparison between VBSF and Low rank Tensor Completion (LRTC) and Matrix Completion Algorithm (GROUSE)}{(a) Real Time Traffic Estimation for 25\\% percentage of Missing Data, (b) Real time traffic estimation for 75\\% percentage of missing data, (c) Traffic prediction for 50\\%  of missing data, (d) Traffic prediction for outliers}\n\t\\label{fig:fig5}\n\\end{figure*}\n\n\\subsubsection{Robust Traffic Estimation}\nThe GPS data that is collected using probe vehicles may be corrupted by noise and may often contain outliers which need to be removed before further processing is performed.  To mitigate the performance degradation due to outliers, we employ the robust variational Bayesian subspace filtering (RVBSF) that models the presence of outliers in the data in the sparse outlier matrix ${\\bf E}$ . To test the RVBSF algorithm, on a given day, we randomly sample a certain $p_o$ percentage of the already sampled traffic data $\\mathbf{y}_{i,\\tau}$ and replace these values with $o_{i,\\tau}$ as follows: \n\\begin{equation} \\label{outlier}\n{\\bf o}_{i,\\tau} = \\max \\left( {\\mathbf{y}}_{i,\\tau-1},{\\mathbf{y}}_{i,\\tau+1} \\right) + c \\, \\mu_t.\n\\end{equation}\nIn other words, the outlier is created by adding a large value $c\\,\\mu_t$ to the maximum of $\\mathbf{y}_{i,\\tau-1}$ and $\\mathbf{y}_{i,\\tau+1}$. Here, $\\mu_t$ is the mean of observed entries at time $t$ and c is a scaling parameter. The RVBSF algorithm is then applied to solve the real time traffic estimation problem.  The detected artificial outliers are those points residing in the matrix ${\\bf E}$.  \\par\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{outlier.pdf}\n\t\\caption{Robust Bayesian subspace filtering for traffic data }{}{(a) Comparison for VBSF and RVBSF with 5\\% outliers and $c$ = 0.75, (b) Comparison for VBSF and RVBSF with 2\\% outliers and $c$ = 0.75 (c) Comparison of VBSF and RVBSF for $c$ = 1.25, (d) Number of outliers detected for different outlier values}\n\t\\label{fig:fig6}\n\\end{figure*}\nThe accuracy of outlier detection depends on the outlier value as shown in Fig. \\ref{fig:fig6}d. The value of $c$ for simulations is chosen from the set $ [0.75, 1, 1.25, 1.5, 1.75]$.  We compare the robust VBSF (termed as RVBSF) with VBSF for two scenarios.  First,  when no outliers are added (VBSF), second,  when outliers are present in the data but only VBSF was used (VBSF\\_with\\_outliers). \nTable \\ref{tab:table3} summarises the overall performance of the RVBSF algorithm. Understandably, RVBSF improves over VBSF when outliers are present, but is still worse than the MRE of VBSF for the case when no outliers were present. For 25\\% missing entries, $p_o=5\\%$ and $c=0.75$, the plots in Fig. \\ref{fig:fig6}a illustrate the performance of the RVBSF algorithm.  Similarly for 75\\% of missing entries, $p_o=2\\%$ the results are shown in Fig. \\ref{fig:fig6}b.  When $p_o = 5\\%$ and $c= 0.75$, we observe that RVBSF detects outliers reasonably well vis-a-vis VBSF\\_with\\_outliers. Similar observation holds when outlier values increase as shown in Fig. \\ref{fig:fig6}c and Fig. \\ref{fig:fig6}d. \\par \n\n\\begin{table}[ht!]\n\t\\begin{center}\n\t\t\\begin{tabular}{llll}\n\t\t\t\\hline \n\t\t\t&$c=0.75$ & $c=0.75$ & $c=1.5$\\\\\n\t\t\t&$p_o=5$\\%& $p_o=2$\\% & $p_o=2$\\% \\\\\n\t\t\t\\hline\n\t\t\tVBSF & 0.09462 &0.09457 &0.09434\\\\\n\t\t\tVBSF\\_outlier & 0.13406 & 0.11643& 0.15318\\\\\n\t\t\tRVBSF & 0.11741 & 0.1127&0.10912\\\\\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{RVBSF: overall performance }\n\t\\label{tab:table3}\n\\end{table}\nThe performance of the proposed RVBSF algorithm is compared with that of OP-RPCA\\cite{oprca} GRASTA\\cite{grasta} and ROSETA\\cite{roseta} in Table \\ref{tab:table4}. The RVBSF algorithm performs better than the subspace estimation and tracking algorithms. The difference in performance may be due to a better modeling of the temporal structure available in the data.  \n\\begin{table}[ht!]\n\\begin{center}\n\t\\begin{tabular}{c c c c} \n\t\t\\hline\n\t\t\n\t\t&$c=0.75$ & $c=0.75$ & $c=1.5$\\\\\n\t&$p_o=5$\\%& $p_o=2$\\% & $p_o=2$\\% \\\\\n\t\t\\hline\n\t\tOP-RPCA& 0.2594 & 0.2298 & \t0.2165 \\\\ \n\t\t\\hline\n\t\tROSETA& 0.1859 & 0.1819 & \t0.1723 \\\\ \n\t\t\\hline\n\t\tGRASTA& 0.1493 & 0.1507 & \t0.1492 \\\\ \n\t\t\\hline\n\t\tRVBSF & 0.11741& 0.1127& 0.10912\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{center}\n\\caption{Performance Comparison for Robust Traffic Estimation }\n\\label{tab:table4}\n\\end{table}\nA possible limitation of the suggested robust traffic estimation framework is following. While there may be outliers present due to an erroneous speed estimation, there might be cases when the  so called outlier value may actually be a real value. The current method may not be able to distinguish between such cases. Hence, a sudden drop in speed along an edge may be treated as an outlier and its possible impact on the traffic of nearby edges be be ignored by the model. \n\\subsection{Electricity Load Prediction}\nWe now discuss the performance of the VBSF algorithm on the electricity load data set \\cite{electrictydata}. Note that the electricity load data is also a time series data with the possibility of missing entries as well as temporal correlation between successive columns.\n\\subsubsection{Performance Index}\nThe performance of the VBSF method is compared with that of \\cite{paperarnew}  using the metrics mean absolute error (MAE) and MRE, defined as: \n\\begin{equation}\n\\text{MAE}= \\frac{1}{z}\\sum_{k=1}^z \\frac {\\parallel \\hat{\\mathbf{y}}_{k}-\\mathbf{y}_{k}\\parallel_{1}}{l(\\mathbf{y}_{k})}\n\\end{equation}\n\\begin{equation}\n\\text{MRE}= \\frac{1}{z}\\sum_{k=1}^z \\frac {\\parallel \\hat{\\mathbf{y}}_{k}-\\mathbf{y}_{k}\\parallel_{2}}{\\parallel \\mathbf{y}_{k}\\parallel_{2}}\n\\end{equation}\nwhere $\\mathbf{y}_{k}$ and $\\hat{\\mathbf{y}}_{k}$ are the ground truth and estimated data for $k^{th}$ column. We run the algorithm online on dates Jan. 1, 2012 to Jan. 1, 2015 resulting into 26,304 columns. In other words, the value of $z$ is 26,304 for our simulations.\n\\subsubsection{Online Electricity Load Estimation and Prediction}\n\tWe run our algorithm for electricity data estimation and prediction. The results for real-time prediction are noted in table \\ref{tab:ele1}. It is noted as the percentage of observed data $p$ increases, the real-time prediction accuracy improves. \n\t\n\t\\begin{table}[ht!]\n\t\t\\begin{center}\n\t\t\t\\begin{tabular}{llll}\n\t\t\t\t\\hline \n\t\t\t\t&$p=0.25$\\%& $p=0.5$\\% & $p=0.75$\\% \\\\\n\t\t\t\t\\hline\n\t\t\t\t\n\t\t\t\tMRE & 0.1789 &0.101 &0.0987\\\\\n\t\t\t\tMAE(kW) & 96.95 & 66.67& 53.95\\\\\n\t\t\t\t\n\t\t\t\\end{tabular}\n\t\t\\end{center}\n\t\t\\caption{Electricity real time load prediction}\n\t\t\\label{tab:ele1}\n\t\\end{table}\n\t\n\tFurther, we predict the one-step ahead electricity load in Fig. \\ref{fig:elec}. To analyze the performance of our algorithm we compare our results with OFMF and CKF \\cite{paperarnew}. The one-step ahead prediction performance of OFMF and CKF are provided in \\cite{paperarnew}. OFMF proposes a autoregressive model based optimization to predict the one-step ahead electricity load. We compare our three cases of $p$ with the results shown in OFMF. It can be seen that our algorithm performs better than the OFMF for electricity load dataset. \n\t\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{elec_res2.pdf}\n\t\\caption{One-step ahead electricity prediction}\n\t\\label{fig:elec}\n\\end{figure}\n\n\\section{Conclusion}\n\\label{conclusion}\nThis paper considers sequentially arriving multivariate data that resides in a time-varying low-dimensional subspace. The temporal evolution of the underlying low-rank subspace is characterized via a state-space model and low-complexity variational Bayesian subspace filtering algorithms are proposed for matrix completion and outlier removal tasks. Simulation experiments quantify that the suggested model can be deployed to estimate the missing traffic data with a reasonable accuracy even with a fraction of random traffic measurements in the network. A similar result is observed on applying the VBSF algorithm on the twin tasks of imputation and prediction on the electricity data-set.  Extensive simulations on both the data sets demonstrate that the suggested model and the accompanying algorithms seem to capture the temporal evolution of the data well as compared to the current state-of-the-art matrix completion and the online subspace estimation algorithms.\n\n\n\n\\section{Variational Bayesian Subspace Filtering}\n\\label{vbsf}\nWe consider a scenario where the data with the missing entries is arriving in a sequential manner. The data can be considered  in the form of the matrix $\\mathbf{Y} \\in \\mathbb{R}^{m \\times t}$, where $t$ denotes the number of time instances over which measurements are made and $m$ denotes the number of rows of the matrix $\\mathbf{Y}$. More generally, $\\mathbf{Y}$ is an incomplete and growing matrix whose columns  arrive sequentially over time. Specifically, for each column $\\mathbf{y}_\\tau$ with $1\\leq \\tau \\leq t$, only entries from the index set $\\Omega_\\tau\\subset \\{1, \\ldots, m\\}$ are observed. The algorithms developed here will seek to achieve the following two goals:\n\\begin{itemize}\n\t\\item \\emph{imputation} which yields $\\{\\hat{y}_{i\\tau}\\}_{i\\notin\\Omega_\\tau}$ for $1\\leq \\tau \\leq t$, and\n\t\\item \\emph{prediction} which yields $\\{\\hat{\\mathbf{y}}_{t+\\tau}\\}_{\\tau = 1}^{T_p}$ where $T_p$ is the prediction horizon. \n\\end{itemize} \nThe next subsection develops a variational Bayesian algorithm for achieving the aforementioned goals. \t\n\n\\begin{figure}[ht!]\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{YAB.jpg}\n\t\\caption{Online Variational Bayesian Filtering}\n\t\\label{fig:matfac}\n\\end{figure}\n\\subsection{Hierarchical Bayesian Model}\nWe begin with detailing a generative model for the matrix $\\mathbf{Y}$. The proposed model will not only capture the rank deficient nature of $\\mathbf{Y}$ \\cite{babacan2012sparse} but also the temporal correlation between successive columns of $\\mathbf{Y}$ \\cite{asif2013low}. Recall that the standard low-rank parametrization of the full  matrix $\\mathbf{Y}$ takes the form $\\mathbf{Y} = \\mathbf{A}\\mathbf{B}$ where $\\mathbf{A} \\in \\mathbb{R}^{m \\times r}$ and  $\\mathbf{B} \\in \\mathbb{R}^{r \\times t}$. Classical non-negative matrix completion approaches seek to obtain such a factorization. In such algorithms, the choice of $r$ is critical to avoiding underfitting or overfitting. \\par\nWithin the Bayesian setting however, the measurements are modeled as arising from a distribution with unknown hyper-parameters, while various components or parameters are assigned different prior distributions. The Bayesian framework allows the use of ARD, wherein associating appropriate priors to the model parameters leads to pruning of the redundant features \\cite{babacan2012sparse}. This work uses pdfs from the exponential family that allow for tractable forms of the posterior pdf but are also flexible enough to adequately model the data. \\par \nSpecifically, the entries of $\\mathbf{Y}$ are generated as\n\\begin{align}\np(y_{i\\tau} \\mid \\mathbf{a}_{i\\boldsymbol{\\cdot}}, \\mathbf{b}_\\tau, \\beta) &= \\mathcal{N}(y_{i\\tau} \\mid \\mathbf{b}_\\tau^T\\mathbf{a}_{i\\boldsymbol{\\cdot}}, \\beta^{-1}) \n&  i \\in \\Omega_{\\tau}\n\\end{align}\nfor all $\\tau \\geq 1$, where $\\mathbf{A} \\in \\mathbb{R}^{m \\times r}$, $\\mathbf{B} \\in \\mathbb{R}^{r \\times t}$, and $\\beta \\in \\mathbb{R}_{++}$ are the (hidden) problem parameters. Unlike the deterministic setting however, the rank hyper-parameter $r$ is not critical to the imputation or prediction accuracy, but is only required to chosen according to computational considerations. The temporal evolution of the entries of $\\mathbf{Y}$ is modeled by making the columns of $\\mathbf{B}$ adhere to the following first order autoregressive model: \n\\begin{align}\\label{ss}\np(\\mathbf{b}_\\tau \\mid \\mathbf{J}, \\mathbf{b}_{\\tau-1}) &= \\mathcal{N}(\\mathbf{b}_{\\tau} \\mid \\mathbf{J}\\mathbf{b}_{\\tau-1}, \\mathbf{I}_r) & 2\\leq \\tau\\leq t\n\\end{align} \nfor $\\tau \\geq 2$, where $\\mathbf{J} \\in \\mathbb{R}^{r \\times r}$ is again a problem parameter. \nHere, $\\mathbf{J}$ captures the temporal structure of the underlying subspace, and is learned from the data itself. The scaling ambiguity present in matrix factorization allows the transition matrix $\\mathbf{J}$ to capture both slow and fast variations in $\\mathbf{b}_\\tau$ without the need to explicitly model the state noise variance. \nIt follows from \\eqref{ss} that the conditional pdf of $\\mathbf{b}_\\tau$ given $\\mathbf{J}$ is given by \n\\begin{align}\np(\\mathbf{B} \\mid \\mathbf{J}) = \\mathcal{N}(\\mathbf{b}_1; \\boldsymbol{\\mu}_1, \\boldsymbol{\\Lambda}_1 ) \\prod_{\\tau = 2}^t \\mathcal{N}(\\mathbf{b}_\\tau \\mid \\mathbf{J}\\mathbf{b}_{\\tau-1}, \\mathbf{I}_r).\n\\end{align} \nObserve that the model complexity depends on the rank $r$, which is also the number of columns in $\\mathbf{A}$ and $\\mathbf{J}$. In order to ensure the value of $r$ is learned in a data-driven fashion, the columns of $\\mathbf{A}$ and $\\mathbf{J}$ are assigned multivariate Gaussian priors with column-specific precisions, i.e., \n\\begin{align}\np(\\mathbf{A} \\mid \\boldsymbol{\\gamma}) &= \\prod_{i=1}^r \\mathcal{N}(\\mathbf{a}_i \\mid 0, \\gamma_i^{-1}\\mathbf{I}_m) \\label{paa}\\\\\np(\\mathbf{J} \\mid \\boldsymbol{\\upsilon}) &= \\prod_{i=1}^r \\mathcal{N}(\\mathbf{j}_i \\mid 0, \\upsilon_i^{-1}\\mathbf{I}_r) \\label{pja}\n\\end{align}\nwhere the precisions $\\boldsymbol{\\gamma}$ and $\\boldsymbol{\\upsilon}$ are problem parameters. It can be seen that if any of $\\gamma_i$ or $\\upsilon_i$ are large, the corresponding columns will be close to zero and consequently irrelevant. Indeed, the priors in \\eqref{paa}-\\eqref{pja} aid in automatic relevance determination since the subsequent optimization process may drive some of the precisions to infinity, yielding a low-rank factorization. \n\nFinally, the three precision variables are selected to have have non-informative Jeffrey's priors\n\\begin{align}\np(\\beta) &= \\frac{1}{\\beta}, & p(\\gamma_i) &= \\frac{1}{\\gamma_i}, & p(\\upsilon_i) &= \\frac{1}{\\upsilon_i}\n\\end{align}\nfor $1\\leq i \\leq r$. Let $\\mathbf{y}_{\\Omega}$ denote the collection of measurements $\\{y_{i\\tau}\\}_{i\\in\\Omega_\\tau, \\tau = 1}^t$. Collecting the hidden variables into $\\mathcal{H} := \\{\\mathbf{A}, \\mathbf{B}, \\mathbf{J}, \\beta, \\boldsymbol{\\gamma}, \\boldsymbol{\\upsilon}\\}$, the joint distribution of $\\{\\mathbf{y}_\\Omega, \\mathcal{H}\\}$ can be written as\n\\begin{align}\np(\\mathbf{y}_\\Omega,\\mathcal{H}) &= p(\\mathbf{y}_\\Omega | \\mathbf{A}, \\mathbf{B}, \\beta)p(\\mathbf{A} | \\boldsymbol{\\gamma})p(\\mathbf{B} | \\mathbf{J}) p(\\mathbf{J} | \\boldsymbol{\\upsilon})p(\\beta)p(\\boldsymbol{\\upsilon})p(\\boldsymbol{\\gamma}) \\nonumber \\\\\n&=\\prod_{\\tau=1}^t\\prod_{i\\in\\Omega_\\tau} \\mathcal{N}(y_{i\\tau} \\mid \\mathbf{b}_\\tau^T\\mathbf{a}_{i\\boldsymbol{\\cdot}}, \\beta^{-1}) \\nonumber\\\\\n&\\times \\prod_{i=1}^r \\left[\\mathcal{N}(\\mathbf{a}_i \\mid 0,\\gamma_i^{-1}\\mathbf{I}_m) \\mathcal{N}(\\mathbf{j}_i \\mid 0, \\upsilon_i^{-1}\\mathbf{I}_r)\\right]  \\nonumber \\\\\n&\\hspace{-1cm}\\times\\mathcal{N}(\\mathbf{b}_1; \\boldsymbol{\\mu}_1, \\boldsymbol{\\Lambda}_1 ) \\prod_{\\tau = 2}^t \\mathcal{N}(\\mathbf{b}_\\tau \\mid \\mathbf{J}\\mathbf{b}_{\\tau-1}, \\mathbf{I}_r) \\frac{1}{\\beta}\\prod_{i=1}^r \\frac{1}{\\gamma_i\\upsilon_i}\n\\end{align} \nThe full hierarchical Bayesian model adopted here is summarized in Fig. \\ref{fig:mc_algo}(a). \n\\begin{figure}[ht!]\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{block.pdf}\n\t\n\t\\caption{(a) Hierarchical Bayesian Model for Matrix Completion (b) Robust Hierarchical Bayesian Model for Matrix Completion }\n\t\\label{fig:mc_algo}\n\\end{figure}\n\n\\subsection{Variational Bayesian Inference}\nHaving specified the generative model for the data, the goal is to determine the posterior distribution $p(\\mathcal{H}|\\mathbf{y}_\\Omega)$, which would yield the corresponding point estimates and can be used for imputation and prediction tasks. However, exact full Bayesian inference is well-known to be intractable. Instead, we utilize the mean-field approximation, wherein the posterior distribution factorizes as:\n\n\\begin{align}\\label{mf}\np(\\mathcal{H} \\mid \\mathbf{y}_\\Omega) \\approx q(\\mathcal{H}) = q_{\\mathbf{A}}(\\mathbf{A})q_{\\mathbf{B}}(\\mathbf{B})q_{\\mathbf{J}}(\\mathbf{J})q_{\\boldsymbol{\\upsilon}}(\\boldsymbol{\\upsilon})q_{\\beta}(\\boldsymbol{\\beta})q_{\\boldsymbol{\\gamma}}(\\boldsymbol{\\gamma}).\n\\end{align}\n\nIn other words, the posterior is now restricted to a family of distributions that adhere to \\eqref{mf}. The factors $q_\\mathbf{A}$, $q_\\mathbf{B}$, $q_\\mathbf{J}$, $q_\\upsilon$, $q_\\beta$, and $q_\\gamma$ can be determined by minimizing the Kullback--Leibler divergence of $p(\\mathcal{H}|\\mathbf{y}_\\Omega)$ from $q(\\mathcal{H})$, usually via an alternating minimization approach \\cite{bishop2006pattern}.  \nIndeed, thanks to the choice of conjugate priors for the parameters, it can be shown that the individual factors in \\eqref{mf} take the following forms \\cite{luttinen2013fast}:\n\n\\begin{subequations}\\label{qs}\n\t\\begin{align}\n\tq_{\\mathbf{B}}(\\mathbf{B}) &= \\mathcal{N}(\\vec{\\mathbf{B}} \\mid \\boldsymbol{\\mu}^{\\mathbf{B}}, \\boldsymbol{\\Xi}^{\\mathbf{B}}) \\\\\n\tq_{\\mathbf{a}_{i\\boldsymbol{\\cdot}}} &= \\mathcal{N}({\\mathbf{a}_{i\\boldsymbol{\\cdot}}} \\mid \\boldsymbol{\\mu}_i^{\\mathbf{A}}, \\boldsymbol{\\Xi}_i^{\\mathbf{A}}) \\\\\n\tq_{\\mathbf{j}_{i\\boldsymbol{\\cdot}}} &= \\mathcal{N}({\\mathbf{j}_{i\\boldsymbol{\\cdot}}} \\mid \\boldsymbol{\\mu}_i^{\\mathbf{J}}, \\boldsymbol{\\Xi}_i^{\\mathbf{J}}) \\\\\n\tq_{\\beta}(\\beta) &= \\text{Ga}(\\beta; a^\\beta, b^\\beta) \\\\\n\tq_{\\gamma_i}(\\gamma_i) &= \\text{Ga}(\\gamma_i; a_i^\\gamma, b_i^\\gamma) \\\\\n\tq_{\\upsilon_i}(\\upsilon_i) &= \\text{Ga}(\\upsilon_i; a_i^\\upsilon, b_i^\\upsilon) \n\t\\end{align}\n\\end{subequations}\nwhere, $\\boldsymbol{\\mu}^\\mathbf{B} \\in \\mathbb{R}^{rt}$,  $\\boldsymbol{\\Xi}^{\\mathbf{B}} \\in \\mathbb{R}^{rt \\times rt}$, $\\boldsymbol{\\mu}^\\mathbf{A}_i \\in \\mathbb{R}^r$, $\\boldsymbol{\\Xi}^\\mathbf{A}_i \\in \\mathbb{R}^{r\\times r}$, $\\boldsymbol{\\mu}^\\mathbf{J}_i \\in \\mathbb{R}^r$, $\\boldsymbol{\\Xi}^\\mathbf{J}_i \\in \\mathbb{R}^{r\\times r}$, and $a^\\beta$, $b^\\beta$, $a^{\\gamma}_i$, $b^{\\gamma}_i$, $a^\\upsilon_i$, $b^\\upsilon_i \\in \\mathbb{R}_{++}$. Consequently, each iteration of alternating optimization simply involves updating the variables $\\{\\boldsymbol{\\mu}^\\mathbf{B}, \\boldsymbol{\\Xi}^\\mathbf{B}, \\{\\boldsymbol{\\mu}^\\mathbf{A}_i\\}, \\{\\boldsymbol{\\Xi}^\\mathbf{A}_i\\}, \\{\\boldsymbol{\\mu}^\\mathbf{J}_i\\}, \\{\\boldsymbol{\\Xi}^\\mathbf{J}_i\\}$, $a^\\beta, b^\\beta$, $\\{a^{\\gamma}_i\\}, \\{b^{\\gamma}_i\\}, \\{a^\\upsilon_i\\}, \\{b^\\upsilon_i\\}\\}$ in a cyclic manner. \n\nIn the present case, not all variables need to be updated explicitly and the updates may be written in a compact form. \nLet us denote $\\omega_\\tau:=\\abs{\\Omega_\\tau}$ and let $\\omega:=\\sum_\\tau \\omega_\\tau$ be the total number of observations made. Then, the updates for hyperparameters $\\{\\boldsymbol{\\upsilon},\\boldsymbol{\\gamma}\\}$ take the following form \n\\begin{subequations}\\label{upga}\n\t\\begin{align}\n\t\\hat{\\upsilon}_i &= \\frac{m}{\\sum_{k=1}^m\\left([\\boldsymbol{\\mu}^\\mathbf{J}_k]^2_i + [\\boldsymbol{\\Xi}^\\mathbf{J}_k]_{ii}\\right)} \\label{10a} \\\\\n\t\\hat{\\gamma}_i &= \\frac{m}{\\sum_{k=1}^m\\left([\\boldsymbol{\\mu}^\\mathbf{A}_k]^2_i + [\\boldsymbol{\\Sigma}^\\mathbf{A}_k]_{ii}\\right)}.\\label{10b} \n\t\\end{align}\n\\end{subequations}\nSubsequently, let $\\hat{\\boldsymbol{\\upsilon}}$ and $\\hat{\\boldsymbol{\\gamma}}$ be the vectors that collect $\\{\\hat{\\upsilon}_i\\}$ and $\\{\\hat{\\gamma}_i\\}$, respectively. Since $\\mathbf{b}_{\\tau}$ denotes the $\\tau$-th column of $\\mathbf{B}^T$, its posterior distribution may be written as $q_{\\mathbf{b}_\\tau}(\\mathbf{b}_\\tau) = \\mathcal{N}(\\mathbf{b}_\\tau \\mid \\boldsymbol{\\mu}^\\mathbf{B}_\\tau, \\boldsymbol{\\Xi}^\\mathbf{B}_\\tau)$, where $\\boldsymbol{\\mu}^\\mathbf{B}_\\tau$ and $\\boldsymbol{\\Xi}^\\mathbf{B}_\\tau$ comprise of the corresponding elements of $\\boldsymbol{\\mu}^\\mathbf{B}$ and $\\boldsymbol{\\Xi}^\\mathbf{B}$, respectively. Also define the posterior covariance matrices\n\\begin{align} \\label{sigma_up1}\n\\boldsymbol{\\Sigma}^{\\mathbf{B}}_{\\tau,\\iota} &:= \\boldsymbol{\\mu}^\\mathbf{B}_\\tau(\\boldsymbol{\\mu}^\\mathbf{B}_\\iota)^T + \\boldsymbol{\\Xi}^{\\mathbf{B}}_{\\tau,\\iota} \\\\\n\\label{sigma_up2}\n\\boldsymbol{\\Sigma}^\\mathbf{J}_i &:= \\boldsymbol{\\mu}^\\mathbf{J}_i(\\boldsymbol{\\mu}^\\mathbf{J}_i)^T + \\boldsymbol{\\Xi}^\\mathbf{J}_i \\\\\n\\label{sigma_up3}\n\\boldsymbol{\\Sigma}^\\mathbf{A}_i &:= \\boldsymbol{\\mu}^\\mathbf{A}_i(\\boldsymbol{\\mu}^\\mathbf{A}_i)^T + \\boldsymbol{\\Xi}^\\mathbf{A}_i.\n\\end{align}\nTherefore, the update for $\\hat{\\beta}$ becomes\n\\begin{align}\\label{be}\n\\hat{\\beta} = \\frac{\\omega}{\\sum_{\\tau=1}^t\\sum_{i\\in\\Omega_\\tau} \\left[y_{i\\tau}^2 - 2y_{i\\tau}(\\boldsymbol{\\mu}^\\mathbf{A}_i)^T\\boldsymbol{\\mu}^\\mathbf{B}_\\tau + \\tr{\\boldsymbol{\\Sigma}^\\mathbf{A}_{i}\\boldsymbol{\\Sigma}^{\\mathbf{B}}_{\\tau,\\tau}}\\right]}.\n\\end{align}  \n\nNext, the updates for the factors $\\mathbf{J}$ and $\\mathbf{A}$ take the following form\n\\begin{subequations}\\label{ja}\n\t\\begin{align}\n\t\\boldsymbol{\\mu}^\\mathbf{J}_i &= [\\boldsymbol{\\Xi}^\\mathbf{J}_i\\boldsymbol{\\Sigma}^{\\mathbf{B}}_{\\tau,\\tau-1}]_{\\boldsymbol{\\cdot} i} \\label{15a}\\\\\n\t\\boldsymbol{\\Xi}^\\mathbf{J}_i &= \\left(\\Diag{\\hat{\\boldsymbol{\\upsilon}}} + \\sum_{\\tau=1}^{t-1}\\boldsymbol{\\Sigma}^\\mathbf{B}_{\\tau,\\tau-1}\\right)^{-1}\\label{15b} \\\\\n\t\\boldsymbol{\\mu}^\\mathbf{A}_i &= \\hat{\\beta}\\boldsymbol{\\Xi}^\\mathbf{A}_i\\sum_{\\tau \\in\\Omega'_i} \\boldsymbol{\\mu}^\\mathbf{B}_\\tau y_{i\\tau}\\label{15c} \\\\\n\t\\boldsymbol{\\Xi}^\\mathbf{A}_i &=  \\left(\\hat{\\gamma}_i\\mathbf{I}_{r} + \\hat{\\beta}\\sum_{\\tau \\in\\Omega'_i}\\boldsymbol{\\Sigma}^\\mathbf{B}_{\\tau,\\tau} \\right)^{-1}\\label{15d}\n\t\\end{align}\n\\end{subequations}\nwhere $\\Omega'_i:=\\{\\tau \\mid i \\in \\Omega_\\tau\\}$. Observe from the updates that the rows of $\\mathbf{J}$ are independent identically distributed under the mean field approximation. The update for $\\boldsymbol{\\mu}^\\mathbf{B}$ can be written as\n\\begin{align}\n\\boldsymbol{\\mu}^\\mathbf{B} &= \\boldsymbol{\\Xi}^\\mathbf{B}\\begin{bmatrix} \\hat{\\beta}\\sum_{i\\in\\Omega_1}y_{i1}\\boldsymbol{\\mu}^\\mathbf{A}_i  + \\boldsymbol{\\Lambda}_1^{-1}\\boldsymbol{\\mu}_1\\\\\n\\hat{\\beta}\\sum_{i\\in\\Omega_2}y_{i2}\\boldsymbol{\\mu}^\\mathbf{A}_i \\\\\n\\vdots\\\\\n\\hat{\\beta}\\sum_{i\\in\\Omega_t}y_{it}\\boldsymbol{\\mu}^\\mathbf{A}_i\n\\end{bmatrix}\\label{mub}.\n\\end{align}\nFinally, $[\\boldsymbol{\\Xi}^\\mathbf{B}]^{-1}$ a block-tridiagonal matrix. Defining $\\hat{\\mathbf{J}}:=\\Ex{\\mathbf{J} \\mid \\mathbf{y}_\\Omega}$ as the matrix whose $i$-row is given by $(\\boldsymbol{\\mu}^\\mathbf{J}_{i})^T$, $\\boldsymbol{\\Sigma}^\\mathbf{A}_{(\\tau)} = \\sum_{i\\in\\Omega_\\tau'}\\boldsymbol{\\Sigma}^\\mathbf{A}_i$, and $\\boldsymbol{\\Sigma}^\\mathbf{J}:=\\sum_{i=1}^r \\boldsymbol{\\Sigma}^\\mathbf{J}_i$, the updates take the form:\n\\begin{align}\n\\left[\\boldsymbol{\\Xi}^{\\mathbf{B}}\\right]^{-1} &=  \\hat{\\beta}\\Diag{\\boldsymbol{\\Xi}^\\mathbf{A}_{(1)}, \\ldots, \\boldsymbol{\\Xi}^\\mathbf{A}_{(t)}}  + \\nonumber\\\\\n&+ \\begin{bmatrix} \\boldsymbol{\\Lambda}_1^{-1} & -\\hat{\\mathbf{J}} & \\ldots &0\\\\\n-\\hat{\\mathbf{J}} & \\mathbf{I}_r + \\boldsymbol{\\Sigma}^\\mathbf{J} & -\\hat{\\mathbf{J}}  & \\ldots \\\\\n\\vdots & \\vdots &&\\vdots \\\\\n\\ldots & 0 & -\\hat{\\mathbf{J}} &  \\mathbf{I}_r \n\\end{bmatrix}. \\label{xib}\n\\end{align}\nIt is remarked that although the $rt \\times rt$ matrix $[\\boldsymbol{\\Xi}^\\mathbf{B}]^{-1}$ is block-tridiagonal, the matrix $\\boldsymbol{\\Xi}^\\mathbf{B}$ is dense, and direct inversion would be prohibitively costly. Moreover, the classical Rauch-Tung-Striebel (RTS) smoother cannot be directly applied since evaluating the conditional expectations under $q(\\mathbf{B})$ is difficult and not amenable to the Matrix Inversion Lemma \\cite{beal2003variational}. Interestingly, observe that the updates in \\eqref{be} and \\eqref{ja} depend only on diagonal and super-diagonal blocks of $\\boldsymbol{\\Xi}^\\mathbf{B}$, namely $\\boldsymbol{\\Xi}^\\mathbf{B}_{\\tau,\\tau}$ and $\\boldsymbol{\\Xi}^\\mathbf{B}_{\\tau,\\tau-1}$, respectively. The next subsection details a low-complexity algorithm for carrying out the updates for these blocks as well as for $\\boldsymbol{\\mu}^\\mathbf{B}$.\n\\subsection{Low-complexity updates via LDL-decomposition}\nThanks to the block-tridiagonal structure of $[\\boldsymbol{\\Xi}^\\mathbf{B}]^{-1}$, it is possible to use the LDL decomposition to carry out the updates in an efficient manner. Decomposing $[\\boldsymbol{\\Xi}^\\mathbf{B}]^{-1} = \\mathbf{L}\\mathbf{D}\\mathbf{L}^T$, the key idea is that left multiplication with $\\boldsymbol{\\Xi}^\\mathbf{B}$ is equivalent to left multiplication with $\\mathbf{L}^{-T}\\mathbf{D}^{-1}\\mathbf{L}^{-1}$. Towards this end, we utilize the algorithm from \\cite{luttinen2013fast}, that comprises of two phases: the forward pass that carries out the multiplication with $\\mathbf{D}^{-1}\\mathbf{L}^{-1}$ and the backward pass that implements the multiplication with $\\mathbf{L}^{-T}$. Let us define for $2\\leq \\tau \\leq t$, \n\\begin{align}\n\\boldsymbol{\\Psi}_\\tau &:= \\hat{\\beta}\\sum_{i\\in\\Omega_\\tau}\\boldsymbol{\\Sigma}^\\mathbf{A}_{(i)}+ \\mathbf{I}_r + \\ind_{\\tau\\neq t}\\sum_{i=1}^r \\boldsymbol{\\Sigma}^\\mathbf{J}_i\t\\\\\n\\mathbf{v}_\\tau &:= \\hat{\\beta}\\sum_{i\\in\\Omega_\\tau} y_{i\\tau}\\boldsymbol{\\mu}^\\mathbf{A}_i.\\label{vbt}\n\\end{align}\nThe forward pass outputs intermediate variables $\\breve{\\boldsymbol{\\Xi}}^{\\mathbf{B}}_{\\tau,\\tau}$, $\\breve{\\boldsymbol{\\Xi}}^{\\mathbf{B}}_{\\tau,\\tau+1}$, and $\\breve{\\boldsymbol{\\mu}}_\\tau$, that are subsequently used in the backward pass. The updates take the following form:\n\\begin{enumerate}\n\t\\item Initialize $\\hat{\\boldsymbol{\\Xi}}^{\\mathbf{B}}_{1,1} = \\boldsymbol{\\Lambda}_1$ and $\\hat{\\boldsymbol{\\mu}}^{\\mathbf{B}}_1 = \\boldsymbol{\\mu}_1 + \\hat{\\beta}\\sum_{i\\in\\Omega_\\tau}y_{i\\tau}\\boldsymbol{\\Lambda}_1\\boldsymbol{\\mu}^\\mathbf{A}_i$\n\t\\item For $\\tau = 1, \\ldots, t-1$\n\t\\begin{subequations} \\label{rts}\n\t\t\\begin{align}\n\t\t\\hspace{-1cm}\\breve{\\boldsymbol{\\Xi}}^{\\mathbf{B}}_{\\tau,\\tau+1} & = -\\hat{\\boldsymbol{\\Xi}}^{\\mathbf{B}}_{\\tau,\\tau}\\hat{\\mathbf{J}} \\\\\n\t\t\\hspace{-1cm}\\breve{\\boldsymbol{\\Xi}}^{\\mathbf{B}}_{\\tau+1,\\tau+1} &= (\\boldsymbol{\\Psi}_{\\tau+1} - (\\breve{\\boldsymbol{\\Xi}}^{\\mathbf{B}}_{\\tau,\\tau+1})^T\\boldsymbol{\\Psi}^{\\mathbf{B}}_{\\tau,\\tau+1})^{-1} \\\\\n\t\t\\breve{\\boldsymbol{\\mu}}^{\\mathbf{B}}_{\\tau+1} &= \\breve{\\boldsymbol{\\Xi}}^{\\mathbf{B}}_{\\tau+1,\\tau+1}(\\mathbf{v}_{\\tau+1} - (\\breve{\\boldsymbol{\\Xi}}^{\\mathbf{B}}_{\\tau,\\tau+1})^T\\breve{\\boldsymbol{\\mu}}^{\\mathbf{B}}_{\\tau})\n\t\t\\end{align}\n\t\t\\item For $\\tau = t-1, \\ldots, 1$\n\t\t\\begin{align}\n\t\t\\hspace{-1cm}\\boldsymbol{\\Xi}^{\\mathbf{B}}_{\\tau,\\tau+1} &= - \\breve{\\boldsymbol{\\Xi}}^{\\mathbf{B}}_{\\tau,\\tau+1}\\boldsymbol{\\Xi}^{\\mathbf{B}}_{\\tau+1,\\tau+1} \\\\\n\t\t\\hspace{-1cm}\\boldsymbol{\\Xi}^{\\mathbf{B}}_{\\tau,\\tau} &= \\breve{\\boldsymbol{\\Xi}}^{\\mathbf{B}}_{\\tau,\\tau} - \\hat{\\boldsymbol{\\Xi}}^{\\mathbf{B}}_{\\tau,\\tau+1}(\\boldsymbol{\\Xi}^{\\mathbf{B}}_{\\tau,\\tau+1})^T \\\\\n\t\t\\hspace{-1cm}\\boldsymbol{\\mu}^{\\mathbf{B}}_{\\tau} &= \\breve{\\boldsymbol{\\mu}}^{\\mathbf{B}}_{\\tau} - \\breve{\\boldsymbol{\\Xi}}^{\\mathbf{B}}_{\\tau,\\tau+1}\\boldsymbol{\\mu}^{\\mathbf{B}}_{\\tau+1}\n\t\t\\end{align}\n\t\\end{subequations}\n\t\\item Output $\\{\\boldsymbol{\\Xi}^{\\mathbf{B}}_{\\tau,\\tau+1}, \\boldsymbol{\\Xi}^{\\mathbf{B}}_{\\tau,\\tau}, \\boldsymbol{\\mu}^{\\mathbf{B}}_{\\tau}\\}_{\\tau = 2}^t$\n\\end{enumerate}\nNote that while $\\boldsymbol{\\Xi}^{\\mathbf{B}}_{i,j} \\neq 0$ for $|i-j| > 1$, these blocks are neither calculated in the forward and backward passes nor required in any of the variational updates. \n\n\nFinally, the predictive distribution $p(y_{i\\tau}\\mid \\mathbf{y}_{\\Omega})$ for $\\tau \\notin \\Omega_i$ or $\\tau \\geq t+1$ is still not tractable in the present case. Instead, we simply use point estimates for estimating the missing entries. Specifically, for $\\tau \\notin \\Omega_i$, the missing entries are imputed as\n\\begin{align}\ny_{i\\tau} = (\\boldsymbol{\\mu}^\\mathbf{B}_\\tau)^T\\boldsymbol{\\mu}^\\mathbf{A}_i. \\label{pred1}\n\\end{align}\nLikewise for $\\tau \\geq t+1$, the prediction becomes\n\\begin{align}\ny_{i\\tau} = (\\hat{\\mathbf{J}}^{\\tau-t}\\boldsymbol{\\mu}^\\mathbf{B}_t)^T\\boldsymbol{\\mu}^\\mathbf{A}_i.\\label{pred2}\n\\end{align}\n\nIt can be seen that as compared to the updates in \\eqref{mub}-\\eqref{xib} that incur a complexity of $\\mathcal{O}(t^3)$, the complexity incurred due to \\eqref{rts} is only $\\mathcal{O}(t)$. Overall, the different parameters are updated cyclically until convergence for each $t = 1, 2, \\ldots$.\n\n\\subsection{EM Baysian Subspace Filtering} \nDifferent from the variational Bayesian framework used here, the EM algorithm treats $\\mathcal{H}_h:=\\{\\mathbf{A}, \\mathbf{B}, \\mathbf{J}\\}$ as hidden variables (with posterior pdf $q_h(\\mathcal{H}_h):=q_\\mathbf{B}(\\mathbf{B})q_\\mathbf{A}(\\mathbf{A})q_\\mathbf{J}(\\mathbf{J})$) and uses maximum a posteriori (MAP) estimates for the precision variables $\\mathcal{H}_p:=\\{\\boldsymbol{\\upsilon}, \\boldsymbol{\\gamma}, \\beta\\}$. Consequently, the EM algorithm for Bayesian subspace tracking starts with an initial estimate $\\mathcal{H}_p^{(0)}$ and uses the following updates at iteration $\\iota \\geq 1$,\n\\begin{itemize}\n\t\\item \\textbf{E-step:} evaluate\n\t\\begin{align}\n\tQ(\\mathcal{H}_p,\\mathcal{H}_p^{(\\iota)}):= \\mathbb{E}_{q_h(\\mathcal{H}_h)}\\left[\\log p(\\mathbf{y}_\\Omega,\\mathcal{H}_h,\\mathcal{H}_p^{(\\iota)})\\right]\n\t\\end{align}\n\t\\item \\textbf{M-step:} maximize\n\t\\begin{align}\n\t\\mathcal{H}_p^{(\\iota+1)} = \\arg\\max_{\\mathcal{H}_p} Q(\\mathcal{H}_p,\\mathcal{H}_p^{(\\iota)})\n\t\\end{align}\n\\end{itemize}\n\nInterestingly, the updates resulting from the E-step take the same form as those in \\eqref{ja} and \\eqref{rts}. On the other hand, the updates obtained from solving the M-step take the slightly different form:\n\\begin{subequations}\\label{mstep}\n\t\\begin{align}\n\t\\hat{\\upsilon}_i &= \\frac{m-2}{\\sum_{k=1}^m\\left([\\boldsymbol{\\mu}^\\mathbf{J}_k]^2_i + [\\boldsymbol{\\Xi}^\\mathbf{J}_k]_{ii}\\right)} \\\\\n\t\\hat{\\gamma}_i &= \\frac{m-2}{\\sum_{k=1}^m\\left([\\boldsymbol{\\mu}^\\mathbf{A}_k]^2_i + [\\boldsymbol{\\Sigma}^\\mathbf{A}_k]_{ii}\\right) }\\\\\n\t\\hat{\\beta} &= \\frac{\\omega-2}{\\sum_{\\tau=1}^t\\sum_{i\\in\\Omega_\\tau} \\left[y_{i\\tau}^2 - 2y_{i\\tau}(\\boldsymbol{\\mu}^\\mathbf{A}_i)^T\\boldsymbol{\\mu}^\\mathbf{B}_\\tau + \\tr{\\boldsymbol{\\Sigma}^\\mathbf{A}_{i}\\boldsymbol{\\Sigma}^{\\mathbf{B}}_\\tau}\\right]}.\n\t\\end{align}\n\\end{subequations}\nThe slight differences arise due to the difference between the mean and mode of the Gamma distribution. Specifically, for $p(x) =$ Ga$(x | a,b)$, it holds that $\\Ex{X} = a\/b$ while $\\max_x \\text{Ga}(x | a,b) = \\frac{a-1}{b}$.  \\par \n\n\\subsubsection{Remarks on the Convergence of VBSF}\nThe VB framework used in the present work is a special case of a more general mean field approximation approach. The convergence of the VB algorithm is well-known; see e.g. \\cite{tzikas2008variational}, \\cite{sato2001online}. Intuitively, the variational approximation renders the evidence lower bound convex in individual factors, and thus amenable to coordinate ascent iterations. Since the lower bound is also differentiable with respect to each factor, the coordinate ascent iterations converge to a stationary point; see \\cite{tseng2001convergence} for a more general result. However, convergence to the global optimum is not guaranteed. \n\n\\subsection{Fixed-lag tracking}\nAlgorithm \\ref{alg:ALGO} can be viewed as an offline algorithm that must be run for every $t$. In practical settings, it may be impractical to remember and process the entire history of measurements at each $t$. Moreover, given data at time $t$, estimates may only be required for entries at time  $t-\\Delta$ for some $\\Delta < h$. \nTowards this end, we consider a sliding window of measurements.\nSince $\\mathbf{A}_{t}$ and $\\mathbf{J}_{t}$ may be seen as transition matrices for the latent states and between latent state and observations, we initialize the next sliding-window with inferred approximate distributions on the transition matrices of the current window.\nFor instance, within the context of traffic density prediction, the inferred approximate distribution for a day may be used as a prior for the coming days. \n That is, the distributions for $\\mathbf{A}, \\, \\mathbf{B},$ and $\\mathbf{J}$  for a day and sliding window can be initialized with the approximate distributions obtained from the previous month's data. \n\\begin{algorithm}\n\tInitialize $  \\boldsymbol{\\gamma},\\boldsymbol{\\beta},\\mathbf{v}$, $sub=1,\\, \\Omega_\\tau, \\,\\Omega'_i,\\boldsymbol{\\Xi}^\\mathbf{A},\\boldsymbol{\\mu}^\\mathbf{A}, \\boldsymbol{\\Xi}^\\mathbf{B},\\boldsymbol{\\mu}^\\mathbf{B},\\boldsymbol{\\Xi}^\\mathbf{J}_{diag},\\boldsymbol{\\mu}^\\mathbf{J} \\boldsymbol{\\Lambda}_1,\\mu_1, $\n\t\n\t$\\hat{\\mathbf{Y}}=\\boldsymbol{\\mu}^A(\\boldsymbol{\\mu}^\\mathbf{B})^T$\n\t\n\t\\While{$Y_{conv}< 10^{-5}$}{\n\t\t$\\mathbf{Y_{old}}= \\hat{\\mathbf{Y}}$\n\t\t\n\t\t$\\boldsymbol{\\Gamma}=diag(\\boldsymbol{\\gamma})$\n\t\t\n\t\t\\uIf{$sub==1 $}\n\t\t{\t\n\t\t\t$\\text{Update using} \\, \\eqref{rts}$\n\t\t\t\n\t\t\t$sub=2$\n\t\t\t\n\t\t\t$ \\text{Update using} \\, \\eqref{10a}, \\,\\eqref{sigma_up1},\\,\\eqref{15a},\\,\\eqref{15b}\\,\\, \\forall \\,\\, 1\\leq i \\leq r$\n\t\t\t\n\t\t}\n\t\t\n\t\t\\ElseIf{$sub==2 $}  \n\t\t{\n\t\t\t$\\text{Update using} \\, \\eqref{sigma_up3},\\eqref{15c}, \\eqref{15d}, \\eqref{10b} \\, \\, \\forall \\, 1\\leq i \\leq m$\n\t\t\t\n\t\t\t$sub=1$\n\t\t}\t\t\n\t\t$\\hat{\\mathbf{Y}}=\\boldsymbol{\\mu}^A(\\boldsymbol{\\mu}^\\mathbf{B})^T$\n\t\t\n\t\t$\\text{Update using } \\eqref{be}$\n\t\t\n\t\t$Y_{conv}=\\frac{\\norm{\\mathbf{Y}-\\mathbf{Y_{old}}}_F}{\\norm{\\mathbf{Y_{old}}}_F}$\n\t}\n\t\\Return($\\hat{\\mathbf{Y}},\\boldsymbol{\\Xi}^\\mathbf{A},\\boldsymbol{\\mu}^\\mathbf{A}, \\boldsymbol{\\Xi}^\\mathbf{B},\\boldsymbol{\\mu}^\\mathbf{B},\\boldsymbol{\\Xi}^\\mathbf{J}_{diag},\\boldsymbol{\\mu}^\\mathbf{J} $)\n\t\n\t\\caption{Variational Bayesian Subspace Filtering}\n\t\\label{alg:ALGO}\n\\end{algorithm}\n\n\\section{Robust Variational Bayesian Subspace Filtering}\\label{rvbsf}\nIn this section we consider the robust version of the variational Bayesian subspace filtering problem in Sec. \\ref{vbsf}. Within this context, in addition to the missing entries in $\\mathbf{Y}$, some entries of $\\mathbf{Y}$ are also contaminated with outliers. Unlike the missing entries however, the location of these outliers is not known. These entries arise due to sensor malfunctions, communication errors, and impulse noise. The robust subspace filtering problem is more difficult as the removal of such outliers entails estimating their magnitudes as well as locations. \n\nWithin the deterministic robust PCA framework, the  matrix is modeled as taking the form $\\mathbf{Y} = \\mathbf{A}\\mathbf{B} + \\mathbf{E}$ where $\\mathbf{A} \\in \\mathbb{R}^{m \\times r}$, $\\mathbf{B} \\in \\mathbb{R}^{r \\times t}$ are low-rank matrices as before. Additionally, we also need to estimate the sparse outlier matrix $\\mathbf{E} \\in \\mathbb{R}^{m \\times t}$. As before, both $r$ and the level of sparsity in $\\mathbf{E}$ are tuning parameters that must generally be carefully selected. \n\nHere, we put forth the variational Bayesian subspace filtering algorithm that makes use of ARD priors to prune the redundant features. Consider the measurement matrix $\\mathbf{Y}$, whose entries are generated from the following pdf:\n\\begin{align}\np(y_{i\\tau} \\mid \\mathbf{a}_{i\\boldsymbol{\\cdot}}, \\mathbf{b}_\\tau, e_{i\\tau}, \\beta) &= \\mathcal{N}(y_{i\\tau} \\mid \\mathbf{b}_\\tau^T\\mathbf{a}_{i\\boldsymbol{\\cdot}} + e_{i\\tau}, \\beta^{-1})  & i \\in \\Omega_\\tau\n\\end{align}\nfor all $\\tau \\geq 1$, and apart from the matrices $\\mathbf{A}$ and $\\mathbf{B}$ defined earlier, we also have $\\{e_{i\\tau}\\}_{\\tau =1, i\\in \\Omega_\\tau}^t$ as the additional (hidden) problem parameter that captures the outliers. The generative models for $\\mathbf{A}$ and $\\mathbf{B}$ are the same as before, i.e.,\n\\begin{subequations}\\label{pbaj}\n\t\\begin{align}\n\tp(\\mathbf{B} \\mid \\mathbf{J}) &= \\mathcal{N}(\\mathbf{b}_1; \\boldsymbol{\\mu}_1, \\boldsymbol{\\Lambda}_1 ) \\prod_{\\tau = 2}^t \\mathcal{N}(\\mathbf{b}_\\tau \\mid \\mathbf{J}\\mathbf{b}_{\\tau-1}, \\mathbf{I}_r)\\label{pb}\\\\\n\tp(\\mathbf{A} \\mid \\boldsymbol{\\gamma}) &= \\prod_{i=1}^r \\mathcal{N}(\\mathbf{a}_i \\mid 0, \\gamma_i^{-1}\\mathbf{I}) \\label{pa}\\\\\n\tp(\\mathbf{J} \\mid \\boldsymbol{\\upsilon}) &= \\prod_{i=1}^r \\mathcal{N}(\\mathbf{j}_i \\mid 0, \\upsilon_i^{-1}\\mathbf{I}) \\label{pj}\n\t\\end{align} \n\\end{subequations}\nfor $\\tau \\geq 2$, and $\\boldsymbol{\\gamma}$ and $\\boldsymbol{\\upsilon}$ are problem parameters. Additionally, we also associate an ARD prior to the outliers, i.e.,\n\\begin{align}\np(e_{i\\tau}) &= \\mathcal{N}(e_{i\\tau} \\mid 0, \\alpha_{i\\tau}^{-1}) & i \\in \\Omega_\\tau\n\\end{align}\nfor $1\\leq \\tau\\leq t$, where the precision $\\alpha_{i\\tau}$ is a hidden variable, that would be driven to infinity whenever $e_{ij}$ is zero. It is remarked that the prior for $e_{i\\tau}$ is only specified for the measurements, i.e., for $i \\in \\Omega_\\tau$ and no predictions are made for the outliers. As before, we associate Jeffery's prior to the precisions $\\beta$, $\\{\\gamma_i\\}$, $\\{\\upsilon_i\\}$, and $\\{\\alpha_{i\\tau}\\}$. \n\\begin{align}\np(\\beta) &= \\frac{1}{\\beta}, & p(\\gamma_i) &= \\frac{1}{\\gamma_i}, & p(\\upsilon_i) &= \\frac{1}{\\upsilon_i}, & p(\\alpha_{i\\tau}) &= \\frac{1}{\\alpha_{i\\tau}}.\n\\end{align}\n\nLet the vectors $\\mathbf{e} \\in \\mathbb{R}^{\\omega}$ and $\\boldsymbol{\\alpha} \\in \\mathbb{R}^{\\omega}$ collect the variables $\\{e_{i\\tau}\\}$ and $\\{\\alpha_{i\\tau}\\}$, respectively. Likewise, defining all the hidden variables as $\\mathcal{H} := \\{\\mathbf{A}, \\mathbf{B}, \\mathbf{J}, \\mathbf{e}, \\beta, \\boldsymbol{\\gamma}, \\boldsymbol{\\upsilon}\\}$, the joint distribution of $\\{\\mathbf{y}_\\Omega, \\mathcal{H}\\}$ can be written as\n\\begin{align}\np(\\mathbf{y}_\\Omega,\\mathcal{H}) &\\nonumber\\\\\n&\\hspace{-1.5cm}= p(\\mathbf{y}_\\Omega | \\mathbf{A}, \\mathbf{B}, \\beta)p(\\mathbf{A} | \\boldsymbol{\\gamma})p(\\mathbf{B} | \\mathbf{J}) p(\\mathbf{J} | \\boldsymbol{\\upsilon})p(\\mathbf{e} | \\boldsymbol{\\alpha})p(\\beta)p(\\boldsymbol{\\upsilon})p(\\boldsymbol{\\gamma}) \\nonumber \\\\\n&\\hspace{-1cm}=\\prod_{\\tau=1}^t\\prod_{i\\in\\Omega_\\tau} \\mathcal{N}(y_{i\\tau} \\mid \\mathbf{b}_\\tau^T\\mathbf{a}_{i\\boldsymbol{\\cdot}}, \\beta^{-1}) \\mathcal{N}(e_{i\\tau}\\mid 0,\\alpha_{i\\tau}^{-1})\\frac{1}{\\alpha_{i\\tau}}\\nonumber\\\\\n&\\times \\prod_{i=1}^r \\left[\\mathcal{N}(\\mathbf{a}_i \\mid 0,\\gamma_i^{-1}\\mathbf{I}) \\mathcal{N}(\\mathbf{j}_i \\mid 0, \\upsilon_i^{-1}\\mathbf{I})\\right]  \\nonumber \\\\\n&\\hspace{-1cm}\\times\\mathcal{N}(\\mathbf{b}_1; \\boldsymbol{\\mu}_1, \\boldsymbol{\\Lambda}_1 ) \\prod_{\\tau = 2}^t \\mathcal{N}(\\mathbf{b}_\\tau \\mid \\mathbf{J}\\mathbf{b}_{\\tau-1}, \\mathbf{I}) \\frac{1}{\\beta}\\prod_{i=1}^r \\frac{1}{\\gamma_i\\upsilon_i}.\n\\end{align} \nThe full hierarchical Bayesian model adopted here is summarized in figure \\ref{fig:mc_algo}(b). \n\n\n\\subsection{Variational Bayesian Inference}\\label{vbi2}\nUtilizing the mean field approximation, the posterior distribution $p(\\mathcal{H} \\mid \\mathbf{y}_\\Omega)$ factorizes as\n\\begin{align}\\label{mf2}\np(\\mathcal{H} \\mid \\mathbf{y}_\\Omega) \\approx q(\\mathcal{H}) &\\nonumber\\\\\n&\\hspace{-2cm}= q_{\\mathbf{A}}(\\mathbf{A})q_{\\mathbf{B}}(\\mathbf{B})q_{\\mathbf{J}}(\\mathbf{J})q_{\\mathbf{e}}(\\mathbf{e})q_{\\boldsymbol{\\upsilon}}(\\boldsymbol{\\upsilon})q_{\\beta}(\\boldsymbol{\\beta})q_{\\boldsymbol{\\gamma}}(\\boldsymbol{\\gamma}).\n\\end{align}\nwhere the individual factors take the same forms as in \\eqref{qs}, in addition to\n\\begin{align}\nq_{\\mathbf{e}}(\\mathbf{e}) & = \\prod_{\\tau = 1}^t \\prod_{i\\in \\Omega_\\tau} \\mathcal{N}(e_{i\\tau} | \\mu_e^{i\\tau}, \\Xi_e^{i\\tau}).\n\\end{align}\nAs before, the variational inference problem can be solved by updating the variables  $\\{\\boldsymbol{\\mu}^\\mathbf{B}, \\boldsymbol{\\Xi}^\\mathbf{B}, \\{\\boldsymbol{\\mu}^\\mathbf{A}_i\\}, \\{\\boldsymbol{\\Xi}^\\mathbf{A}_i\\}, \\{\\boldsymbol{\\mu}^\\mathbf{J}_i\\}, \\{\\boldsymbol{\\Xi}^\\mathbf{J}_i\\}$, $\\{\\mu_e^{i\\tau}\\}, \\{\\Xi_e^{i\\tau}\\}, a^\\beta, b^\\beta$, $\\{a^{\\gamma}_i\\}, \\{b^{\\gamma}_i\\}, \\{a^\\upsilon_i\\}, \\{b^\\upsilon_i\\}\\}$ in a cyclic manner. However, a more compact form for the updates may be derived as follows.\n\nSpecifically, the updates for $\\{\\hat{\\upsilon}_i, \\hat{\\gamma}_i\\}$ remain the same as in \\eqref{upga}. However, the update for $\\hat{\\beta}$ takes the form:\n\\begin{align}\\label{be2}\n\\hat{\\beta} &= \\frac{\\omega}{\\sum_{\\tau=1}^t\\sum_{i\\in\\Omega_\\tau} \\nu_{i\\tau}}\\\\\n\\shortintertext{where,}\n\\nu_{i\\tau} :=& y_{i\\tau}^2 - 2(y_{i\\tau}-\\mu_e^{i\\tau})(\\boldsymbol{\\mu}^\\mathbf{A}_i)^T\\boldsymbol{\\mu}^\\mathbf{B}_\\tau - 2y_{i\\tau}\\mu_e^{i\\tau} \\nonumber\\\\\n&+ (\\mu_e^{i\\tau})^2  + \\Xi_e^{i\\tau} + \\tr{\\boldsymbol{\\Sigma}^\\mathbf{A}_{i}\\boldsymbol{\\Sigma}^{\\mathbf{B}}_{\\tau,\\tau}}.\n\\end{align}  \nFurther, the parameters $\\mu_e^{i\\tau}$ and $\\Xi_e^{i\\tau}$ are updated as\n\\begin{subequations}\\label{eup}\n\t\\begin{align}\n\t\\Xi_e^{i\\tau} &= \\frac{1}{\\hat{\\beta} + (\\mu_e^{i\\tau})^2  + \\Xi_e^{i\\tau}} \\label{36a}\\\\\n\t\\mu_e^{i\\tau} &= \\hat{\\beta} \\Xi_e^{i\\tau} (y_{i\\tau} - (\\boldsymbol{\\mu}^\\mathbf{A}_i)^T\\boldsymbol{\\mu}^\\mathbf{B}_\\tau). \\label{36b}\n\t\\end{align}\n\\end{subequations}\n\nProceeding similarly, the updates for $\\{\\boldsymbol{\\mu}^\\mathbf{J}_i\\}$, $\\{\\boldsymbol{\\Xi}^\\mathbf{J}_i\\}$, and $\\{\\boldsymbol{\\Xi}^\\mathbf{A}_i\\}$ remain the same as in \\eqref{ja}, while the updates for $\\{\\boldsymbol{\\mu}^\\mathbf{A}_i\\}$ become:\n\\begin{align}\\label{a2}\n\\boldsymbol{\\mu}^\\mathbf{A}_i &= \\hat{\\beta}\\boldsymbol{\\Xi}^\\mathbf{A}_i\\sum_{\\tau \\in\\Omega'_i} \\boldsymbol{\\mu}^\\mathbf{B}_\\tau (y_{i\\tau}-\\mu_e^{i\\tau}) .\n\\end{align}\nFinally, the updates for $\\boldsymbol{\\Xi}^{\\mathbf{B}}$ remain the same but the updates of $\\boldsymbol{\\mu}^\\mathbf{B}$ change. Specifically, the low complexity updates via LDL-decomposition remain mostly the same, except for the modified definition of $\\mathbf{v}_\\tau$ in \\eqref{vbt} which now looks like\n\\begin{align}\n\\mathbf{v}_\\tau = \\hat{\\beta} \\sum_{i \\in \\Omega_\\tau} (y_{i\\tau} - \\mu_e^{i\\tau}).\n\\end{align}\nThe full robust subspace filtering algorithm is summarized in Algorithm \\ref{alg:ALG2}. The predictions for $y_{i\\tau}$ for $i\\notin \\Omega_\\tau$ and for $\\tau \\geq t+1$ are obtained as in \\eqref{pred1} and \\eqref{pred2}, respectively. \n\\begin{algorithm}\n\t\n\tInitialize $  \\boldsymbol{\\alpha},\\boldsymbol{\\gamma},\\boldsymbol{\\beta},\\mathbf{v}$, $sub=1,\\, \\Omega_\\tau, \\,\\Omega'_i,\\boldsymbol{\\Xi}^\\mathbf{A},\\boldsymbol{\\mu}^\\mathbf{A}, \\boldsymbol{\\Xi}^\\mathbf{B},\\boldsymbol{\\mu}^\\mathbf{B},\\boldsymbol{\\Xi}^\\mathbf{J}_{diag},\\boldsymbol{\\mu}^\\mathbf{J} \\boldsymbol{\\Lambda}_1,\\mu_1, $\n\t\n\t$\\hat{\\mathbf{Y}}=\\boldsymbol{\\mu}^\\mathbf{A}(\\boldsymbol{\\mu}^\\mathbf{B})^T$\n\t\n\t\\While{$Y_{conv}< 10^{-5}$}{\n\t\t$\\mathbf{Y_{old}}= \\hat{\\mathbf{Y}}$\n\t\t\n\t\t$\\boldsymbol{\\Gamma}=diag(\\boldsymbol{\\gamma})$\n\t\t\n\t\t\\uIf{$sub==1 $}\n\t\t{\t\n\t\t\t$\\text{Update using} \\, \\eqref{rts}$\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t$sub=2$\n\t\t\t\n\t\t\t\n\t\t\t$ \\text{Update using} \\, \\eqref{10a}, \\,\\eqref{sigma_up1},\\,\\eqref{15a},\\,\\eqref{15b}\\,\\forall \\,\\, 1\\leq i \\leq r$\n\t\t\t\n\t\t\t\n\t\t}\n\t\t\n\t\t\\ElseIf{$sub==2 $} \n\t\t{\n\t\t\t\n\t\t\t$\\text{Update using} \\, \\eqref{sigma_up3},\\eqref{15c}, \\eqref{15d}, \\eqref{10b} \\, \\,\\forall \\, 1\\leq i \\leq m$\n\t\t\t$sub=3$\n\t\t\t\n\t\t\t\n\t\t}\n\t\t\n\t\t\\Else\n\t\t{\n\t\t\t$\\text{Update using} \\, \\eqref{36a},\\, \\eqref{36b} \\,\\, \\forall \\,\\, 1\\leq i \\leq m, \\,\\, \\forall \\,\\, 1\\leq \\tau \\leq t$\n\t\t\t\n\t\t\t\n\t\t\t$sub=1$\n\t\t}\n\t\t$\\hat{\\mathbf{Y}}=\\boldsymbol{\\mu}^\\mathbf{A}(\\boldsymbol{\\mu}^\\mathbf{B})^T$\n\t\t\n\t\t\n\t\t\n\t\t$\\text{Update using} \\, \\eqref{be2}$\n\t\t\n\t\t\n\t\t\n\t\t$Y_{conv}=\\frac{\\norm{\\mathbf{Y}-\\mathbf{Y_{old}}}_F}{\\norm{\\mathbf{Y_{old}}}_F}$\n\t}\n\t\\Return($\\hat{\\mathbf{Y}},\\boldsymbol{\\Xi}^\\mathbf{A},\\boldsymbol{\\mu}^\\mathbf{A}, \\boldsymbol{\\Xi}^\\mathbf{B},\\boldsymbol{\\mu}^\\mathbf{B},\\boldsymbol{\\Xi}^\\mathbf{J}_{diag},\\boldsymbol{\\mu}^\\mathbf{J} $)\n\t\\caption{Robust Variational Bayesian Subspace Filtering}\n\t\\label{alg:ALG2}\n\\end{algorithm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}