diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzcuqy" "b/data_all_eng_slimpj/shuffled/split2/finalzzcuqy"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzcuqy"
@@ -0,0 +1,5 @@
+{"text":"\\section{Statement of results}\n\n\\noindent\nLet $X$ be a smooth, projective variety over a characteristic $0$\nfield $k$, and denote by $\\Kbm{0,r}{X,\\beta}$ the Kontsevich moduli\nspace of genus $0$, $r$-pointed stable maps to $X$ of class $\\beta$.\nBehrend and Fantechi defined a \\emph{perfect obstruction theory} for\n$\\Kbm{0,r}{X,\\beta}$, i.e., a complex $E^\\bullet$ perfect of amplitude\n$[-1,0]$ together with a map to the cotangent complex $\\phi:E^\\bullet\n\\rightarrow L^\\bullet_{\\Kbm{0,r}{X,\\beta}}$ such that $h^0(\\phi)$ is\nan isomorphism and $h^{-1}(\\phi)$ is surjective.  In many cases $\\phi$\nis a quasi-isomorphism, and then the dualizing sheaf on\n$\\Kbm{0,r}{X,\\beta}$ is the determinant $\\text{det}(E^\\bullet)$.  For\nthis reason $\\text{det}(E^\\bullet)$ is called the \\emph{virtual\ncanonical bundle}.  This paper gives a formula,\nProposition~\\ref{prop-vc}, for the virtual canonical bundle in terms\nof tautological divisor classes on $X$, modulo torsion.\n\n\\medskip\\noindent\nGrothendieck-Riemann-Roch applies in a straightforward manner, but the\nresulting formula is not optimal: it is difficult to determine whether\nthe virtual canonical bundle is NEF, ample, etc.  The main work in\nSection~\\ref{sec-div} proves divisor class relations yielding a\nsuccinct formula for the virtual canonical bundle.  The proof reduces\nto local computations for the universal family over the Artin stack of\nall prestable curves of genus $0$, cf. Section~\\ref{sec-local}.\nBecause of this, most results are stated for Artin stacks.  This leads\nto one \\emph{ad hoc} consruction: since there is as yet no theory of\ncycle class groups for Artin stacks admitting Chern classes for all\nperfect complexes of bounded amplitude, a Riemann-Roch theorem for all\nperfect morphisms relatively representable by proper algebraic spaces,\nand arbitrary pullbacks for all cycles coming from Chern classes, a\nstand-in $Q_\\pi$ is used, cf. Section~\\ref{sec-Qpi} (also by avoiding\nRiemann-Roch, this allows some relations to be proved ``integrally''\nrather than ``modulo torsion'').  Also, although the relative Picard\nfunctor of the universal family of genus $0$ curves is well-known, a\ndescription is included in Sections ~\\ref{sec-dec} and ~\\ref{sec-cpct}\nfor completeness.\n\n\\medskip\\noindent  \nIn the special case $X={\\mathbb P}^n_k$, Pandharipande proved most of\nthese divisor class relations, ~\\cite{QDiv}, and the formula for the\nvirtual canonical bundle, ~\\cite{Pand97}, modulo numerical\nequivalence.  This was certainly our inspiration, but our proofs are\ncompletely different, yield a more general virtual canonical bundle\nformula, and hold modulo torsion (and sometimes ``integrally'') rather\nthan modulo numerical equivalence.\n\n\\section{Decorated prestable curves} \\label{sec-dec}\n\\marpar{sec-dec}\n \n\\noindent\nThere exists an Artin stack parametrizing prestable curves whose dual\ngraph is a given modular graph.  This section describes a variant\nArtin stack obtained by ``decorating'' the modular graph.  Although\nthe variation is simple, it arises often enough to warrant a few\nwords.  This variant is used in the next section to describe the\nclosure of the identity section in the relative Picard functor for the\nuniversal family of prestable curves of compact type.\nThe reference for this section is ~\\cite{BM}.\n\n\\medskip\\noindent\nA \\emph{modular graph} is a (not necessarily connected) graph $\\sigma$\n-- edges are undirected and \\emph{tails} or half-edges are allowed --\ntogether with a \\emph{genus function} $g:\\text{Vertex}(\\sigma)\n\\rightarrow \\mathbb{Z}_{\\geq 0}$.  There are 2 collections of morphisms\nbetween modular graphs: \\emph{contractions} are surjective on\nvertices, roughly contracting subgraphs of the domain to vertices of\nthe target, and \\emph{graph inclusions} are inclusions of\nsubgraphs. (In ~\\cite{BM}, the \\emph{combinatorial morphisms} are\nobtained from graph inclusions by adjoining formal inverses of certain\n``stabilizing'' contractions; stability is not an issue here, so graph\ninclusions are more appropriate).  Also, for every diagram,\n$$\n\\begin{CD}\n& & \\sigma_3 \\\\ \n& & @VV a V \\\\\n\\sigma_2 @> \\phi >> \\sigma_1\n\\end{CD}\n$$\nof a contraction $\\phi$ and a graph inclusion $a$, there is a\n\\emph{pullback diagram},\n$$\n\\begin{CD}\n\\sigma_4 @> a^*\\phi >> \\sigma_3 \\\\\n@V \\phi^* a VV @VV a V \\\\\n\\sigma_2 @> \\phi >> \\sigma_1\n\\end{CD}\n$$\nof a contraction $a^*\\phi$ and a graph inclusion $\\phi^*a$ such that\nthe maps on vertices, $a\\circ a^*\\phi, \\phi \\circ\n\\phi^*a:\\text{Vertex}(\\sigma_4) \\rightarrow \\text{Vertex}(\\sigma_1)$\nare equal.  The diagram is unique up to a unique isomorphism (both as\na contraction and a graph inclusion) of $\\sigma_4$.  The category of\nmodular graphs is denoted $\\mathfrak{G}$.\n\n\\medskip\\noindent\nTo each prestable curve there is an associated modular graph, and to\neach modular graph $\\sigma$ there is an Artin stack $\\mathfrak{M}(\\sigma)$\nparametrizing prestable curves along with a contraction of the\nassociated modular graph to $\\sigma$.  This defines a lax 2-functor\nfrom $\\mathfrak{G}$ to the 2-category of Artin stacks, covariant for\ncontractions, contravariant for graph inclusions, and such that for\nevery pullback diagram there is a 2-equivalence $\\mathfrak{M}(a)\\circ\n\\mathfrak{M}(\\phi) \\Rightarrow \\mathfrak{M}(a^*\\phi) \\circ \\mathfrak{M}(\\phi^*a)$.\n\n\\begin{defn} \\label{defn-decg}\n\\marpar{defn-decg}\nA \\emph{category of decorated modular graphs} is a category $\\mathfrak{H}$\nwith 2 sets of morphisms -- $\\mathfrak{H}$-contractions and $\\mathfrak{H}$-graph\ninclusions -- together with a functor $p:\\mathfrak{H} \\rightarrow \\mathfrak{G}$\ncompatible with both contractions and graph inclusions satisfying the\nfollowing axioms,\n\\begin{enumerate}\n\\item[(i)]\nfor every $\\mathfrak{H}$-contraction $\\phi:\\tau_2 \\rightarrow \\tau_1$ and\n$\\mathfrak{H}$-graph inclusion $a:\\tau_3 \\rightarrow \\tau_1$, there exists\nan object $\\tau_4$, an $\\mathfrak{H}$-contraction $a^*\\phi:\\tau_4\n\\rightarrow \\tau_3$ and an $\\mathfrak{H}$-graph inclusion $\\phi^*a:\\tau_4\n\\rightarrow \\tau_2$ mapping under $p$ to a pullback diagram, moreover\nthis is unique up to unique isomorphism of $\\tau_4$, and\n\\item[(ii)]\nfor every object $\\tau$ in $\\mathfrak{H}$ and every contraction $\\phi:p(\\tau)\n\\rightarrow \\sigma$ in $\\mathfrak{G}$, there is a $\\mathfrak{H}$-contraction $\\psi$\nsuch that $p(\\psi)=\\phi$, and $\\psi$ is unique up to unique isomormphism.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{constr} \\label{constr-semigp}\n\\marpar{constr-semigp}\nLet $A$ be an Abelian semigroup.  Define $\\mathfrak{G}_A$ to be the category\nwhose objects are pairs $(\\sigma,\\alpha)$ of a modular graph $\\sigma$\ntogether with a function $\\alpha:\\text{Vertex}(\\sigma) \\rightarrow A$,\nwhere $\\mathfrak{G}_A$-contractions, $\\phi:(\\sigma_1,\\alpha_1) \\rightarrow\n(\\sigma_2,\\alpha_2)$, are contractions $\\phi:\\sigma_1 \\rightarrow\n\\sigma_2$ such that $\\alpha_2(v) = \\sum_{w\\in \\phi^{-1}(v)}\n\\alpha_1(w)$ for every $v\\in \\text{Vertex}(\\sigma_2)$, and where\n$\\mathfrak{G}_A$-graph inclusions, $a:(\\sigma_1,\\alpha_1) \\rightarrow\n(\\sigma_2,\\alpha_2)$, are graph inclusions $a:\\sigma_1 \\rightarrow\n\\sigma_2$ such that $\\alpha_2(a(v)) = \\alpha_1(v)$ for every vertex\n$v\\in \\text{Vertex}(\\sigma_1)$.  Define $p:\\mathfrak{G}_A \\rightarrow\n\\mathfrak{G}$ to be the obvious forgetful functor.  This is a category of\ndecorated modular graphs; the only one used in the rest of this paper.\n\\end{constr}\n\n\\medskip\\noindent\nThe aim of this section is to construct for every object $\\tau$ of\n$\\mathfrak{H}$ an Artin stack $\\mathfrak{M}_{\\mathfrak{H}}(\\tau)$ parametrizing\nprestable curves along with a lifting of the associated modular graph\nto an object of $\\mathfrak{H}$ contracting to $\\tau$.  The association $\\tau\n\\mapsto \\mathfrak{M}_{\\mathfrak{H}}(\\tau)$ should define a lax 2-functor from\n$\\mathfrak{H}$ to the 2-category of Artin stacks, covariant for\n$\\mathfrak{H}$-contractions, contravariant for $\\mathfrak{H}$-graph inclusions,\nand such that for every pullback diagram there is an associated\n2-equivalence.\n\n\\begin{defn} \\label{defn-sat}\n\\marpar{defn-sat}\nLet $\\mathfrak{H}$ be a category of decorated modular graphs, considered as\na usual category whose morphisms are $\\mathfrak{H}$-contractions.  A\nsubcategory $\\mathfrak{H}'$ is \\emph{saturated} if $\\mathfrak{H}'$ contains every\n$\\mathfrak{H}$-contraction whose domain is in $\\mathfrak{H}'$.  A subcategory\n$\\mathfrak{H}'$ is \\emph{$p$-embedding} if the functor of categories with\ncontractions as morphisms, $p:\\mathfrak{H}' \\rightarrow \\mathfrak{G}$, is an\nequivalence to a (necessarily full) subcategory of $\\mathfrak{G}$.\n\n\\noindent\nLet $\\tau$ be an object of $\\mathfrak{H}$ and denote by $\\mathfrak{H}_\\tau$ the\ncategory whose objects are contractions $\\phi:\\tau_\\phi \\rightarrow\n\\tau$ and whose morphisms are commutative diagrams of contractions.  A\nsubcategory $\\mathfrak{H}'$ of $\\mathfrak{H}_\\tau$ is \\emph{saturated} if\n$\\mathfrak{H}'$ contains every morphism in $\\mathfrak{H}_\\tau$ whose domain is in\n$\\mathfrak{H}'$.  A subcategory $\\mathfrak{H}'$ of $\\mathfrak{H}_\\tau$ is\n\\emph{$p$-embedding} if the functor $p:\\mathfrak{H}' \\rightarrow\n\\mathfrak{G}_{p(\\tau)}$ is an equivalence to a (necessarily full)\nsubcategory of $\\mathfrak{G}_{p(\\tau)}$.  Denote by\n$\\text{Sat}(\\mathfrak{H}_\\tau)$ the set of saturated, $p$-embedding\nsubcategories of $\\mathfrak{H}_\\tau$ directed by reverse inclusion of\nsubcategories.\n\\end{defn}\n\n\\medskip\\noindent\nLet $\\tau$ be an object of $\\mathfrak{H}$ and let $\\mathfrak{H}'$ be a saturated,\n$p$-embedding subcategory of $\\mathfrak{H}_\\tau$.  Define\n$U_{\\mathfrak{H}'}(\\tau)$ to be the open substack of $\\mathfrak{M}(p(\\tau))$ whose\ncomplement is the union of the images of all 1-morphisms\n$\\mathfrak{M}(\\phi):\\mathfrak{M}(\\sigma) \\rightarrow \\mathfrak{M}(p(\\tau))$ such that\n$\\phi$ is not in the image of $p:\\mathfrak{H}' \\rightarrow\n\\mathfrak{G}_{p(\\tau)}$.  It is straightforward that $u_{\\mathfrak{H}'}(\\tau)$ is\nopen: the intersection with every quasi-compact open substack of\n$\\mathfrak{M}(p(\\tau))$ is open, and $U_{\\mathfrak{H}'}(\\tau)$ is the union of\nthese open sets.\n\n\\medskip\\noindent\nLet $\\mathfrak{H}' \\subset \\mathfrak{H}''$ be saturated, $p$-embedding\nsubcategories of $\\mathfrak{H}_\\tau$.  Then $U_{\\mathfrak{H}'}(\\tau) \\subset\nU_{\\mathfrak{H}''}(\\tau)$ as subsets of $\\mathfrak{M}(p(\\tau))$.  Therefore\n$\\mathfrak{H}' \\mapsto U_{\\mathfrak{H}'}(\\tau)$ is a directed system of open\nimmersions of Artin stacks indexed by $\\text{Sat}(\\mathfrak{H}_\\tau)$.\nBecause this is a directed system of open immersions, the direct limit\nis an Artin stack.\n\n\\begin{notat} \\label{notat-MHtau}\n\\marpar{notat-MHtau}\nDenote by $\\mathfrak{M}_{\\mathfrak{H}}(\\tau)$ the direct limit of the directed\nsystem $\\mathfrak{H}' \\mapsto U_{\\mathfrak{H}'}(\\tau)$.  Denote by\n$\\mathfrak{M}_p(\\tau): \\mathfrak{M}_{\\mathfrak{H}}(\\tau) \\rightarrow \\mathfrak{M}(p(\\tau))$\nthe natural 1-morphism.  If $\\mathfrak{H}=\\mathfrak{G}_A$, also denote\n$\\mathfrak{M}_{\\mathfrak{H}}(\\tau)$ by $\\mathfrak{M}_A(\\tau)$.\n\\end{notat}\n\n\\medskip\\noindent\nThe ``points'' of $\\mathfrak{M}_{\\mathfrak{H}}(\\tau)$ have a simple description.\n\n\\begin{defn} \\label{defn-strict}\n\\marpar{defn-strict}\nFor every modular graph $\\sigma$, define\n$\\mathfrak{M}^{\\text{strict}}(\\sigma)$ to be the open substack of\n$\\mathfrak{M}(\\sigma)$ that is the complement of the images of all\n$\\mathfrak{M}(\\phi)$ where $\\phi:\\sigma' \\rightarrow \\sigma$ is a\nnon-invertible contraction.\n\\end{defn}\n\n\\begin{lem} \\label{lem-pts}\n\\marpar{lem-pts}\nLet $\\tau$ be an object of $\\mathfrak{H}$ and let $\\phi:\\sigma \\rightarrow\np(\\tau)$ be a contraction.  The 2-fibered product\n$\\mathfrak{M}^{\\text{strict}}(\\sigma) \\times_{\\mathfrak{M}(p(\\tau))}\n\\mathfrak{M}_{\\mathfrak{H}}(\\tau)$ is equivalent to a disjoint union of copies of\n$\\mathfrak{M}^{\\text{strict}}(\\sigma)$ indexed by equivalence classes of\ncontractions $\\psi$ in $\\mathfrak{H}_\\tau$ such that $p(\\psi)=\\phi$.\n\\end{lem}\n\n\\begin{proof}\nLet $(\\eta,\\zeta,\\theta)$ be an object of the 2-fibered product, i.e.,\na triple of an object of $\\mathfrak{M}^{\\text{strict}}(\\sigma)$, an object\nof $\\mathfrak{M}_{\\mathfrak{H}}(\\tau)$ and an equivalence\n$\\theta:\\mathfrak{M}(\\phi)(\\eta) \\rightarrow \\mathfrak{M}_{p}(\\tau)(\\zeta)$.\nThere is a saturated, $p$-embedding subcategory $\\mathfrak{H}'$ such that\n$\\mathfrak{M}_p(\\tau)(\\zeta)$ is in $U_{\\mathfrak{H}'}(\\tau)$.  Because this is in\nthe image of $\\mathfrak{M}(\\phi)$, there is a contraction $\\psi:\\tau_\\psi\n\\rightarrow \\tau$ in $\\mathfrak{H}'$ such that $\\phi=p(\\psi)$.  Because\n$\\mathfrak{H}'$ is $p$-embedding, $\\psi$ is unique up to unique isomorphism.\nBy the nature of the direct limit, $\\psi$ is independent of the choice\nof $\\mathfrak{H}'$.\n\n\\medskip\\noindent\nConversely, given an object $\\eta$ of $\\mathfrak{M}^{\\text{strict}}(\\sigma)$\nand a contraction $\\psi:\\tau_\\psi \\rightarrow \\tau$ in $\\mathfrak{H}_\\tau$\nsuch that $p(\\psi) = \\phi$, define $\\mathfrak{H}'$ to be the subcategory of\n$\\mathfrak{H}_\\tau$ consisting of all contractions through which $\\psi$\nfactors.  By Definition~\\ref{defn-decg}(ii), this is a saturated,\n$p$-embedding subcategory.  And $\\mathfrak{M}(\\phi)(\\eta)$ is in\n$U_{\\mathfrak{H}}(\\tau)$.  The image in the direct limit is an object\n$\\zeta$, and there is a canonical isomorphism $\\theta:\n\\mathfrak{M}(\\phi)(\\eta) \\rightarrow \\mathfrak{M}_p(\\tau)(\\zeta)$.  Thus\n$(\\eta,\\zeta,\\theta)$ is an object of the 2-fibered product.\n\n\\medskip\\noindent\nIt is left to the reader to verify these operations give an\nequivalence of stacks.\n\\end{proof}\n\n\\medskip\\noindent\n\\textbf{Note:}  \nThe functorialities are only sketched.  Given an $\\mathfrak{H}$-contraction\n$\\phi:\\tau_1 \\rightarrow \\tau_2$, Definition~\\ref{defn-decg}(ii) gives\na map of directed sets $\\text{Sat}(\\phi):\\text{Sat}(\\mathfrak{H}_{\\tau_1})\n\\rightarrow \\text{Sat}(\\mathfrak{H}_{\\tau_1})$, and composition with\n$\\mathfrak{M}(p(\\phi))$ gives a compatible family of 1-morphisms of directed\nsystems.  This defines the 1-morphism $\\mathfrak{M}_{\\mathfrak{H}}(\\phi)$.  Given\nan $\\mathfrak{H}$-graph inclusion $a:\\tau_1 \\rightarrow \\tau_2$, existence\nof pullback diagrams, Definition~\\ref{defn-decg}(i), gives a map of\ndirected sets $\\text{Sat}(a):\\text{Sat}(\\mathfrak{H}_{\\tau_2}) \\rightarrow\n\\text{Sat}(\\mathfrak{H}_{\\tau_1})$, and composition with $\\mathfrak{M}(p(a))$\ngives a compatible family of 1-morphisms of directed systems.  This\ndefines the 1-morphism $\\mathfrak{M}_{\\mathfrak{H}}(a)$.  The rest of the\ncompatibilities are straightforward.\n\n\\section{The universal relative Picard for curves of compact type}\n\\label{sec-cpct}\n\\marpar{sec-cpct}\n\n\\noindent\nThe results in this section are well-known, and easily follow from\n~\\cite{Raynaud} and ~\\cite{Ner}.  It is useful in the rest of the\npaper to gather the results here.\n\n\\begin{notat} \\label{notat-H}\n\\marpar{notat-H}\nDenote by $\\mathfrak{H} \\subset \\mathfrak{G}_\\mathbb{Z}$ the full subcategory of objects\n$(\\sigma,\\alpha)$ such that $\\sigma$ is a forest of trees, i.e., the\ngraph has no cycles.  For each triple of integers $g,n \\geq 0$ and\n$e$, denote by $\\tau_{g,n}(e)$ the object of $\\mathfrak{H}$ consisting of a\ntree $\\sigma_{g,n}$ with a single vertex of genus $g$ and $n$ flags,\nsuch that $\\alpha(v)=n$.\n\\end{notat}\n\n\\medskip\\noindent\nDenote by $\\pi:\\mathcal{C} \\rightarrow \\mathfrak{M}_{\\mathfrak{H}}(\\tau_{g,0}(0))$ the\npullback from $\\mathfrak{M}(\\sigma_{g,0})$ of the universal curve.  For each\n4-tuple $A=((g',g''),(e',e''))$ of integers $g',g''\\geq 0$,\n$g'+g''=g$, and integers $e',e''$, $e'+e''=0$, denote by $\\tau_A$ the\ntree with vertices $v',v''$ such that $g(v')=g', \\alpha(v')=e'$ and\n$g(v'')=g'', \\alpha(v'')=e''$.  Denote by $\\phi:\\tau_A \\rightarrow\n\\tau_{g,0}(0)$ the canonical contraction.  The 2-fibered product\n$\\mathfrak{M}_{\\mathfrak{H}}(\\tau_A) \\times_{\\mathfrak{M}_{\\mathfrak{H}}(\\tau_{g,0}(0))}\n\\mathcal{C}$ has 2 irreducible components $\\mathcal{C}', \\mathcal{C}''$ corresponding\nto the vertices $v', v''$.  There is a unique effective Cartier\ndivisor $\\mathcal{D} \\subset \\mathcal{C}$ such that for every\n$A=((g',g''),(e',e''))$,\n$$\n\\mathfrak{M}^{\\text{strict}}_{\\mathfrak{H}}(\\tau_A)\n\\times_{\\mathfrak{M}_{\\mathfrak{H}}(\\tau_{g,0}(0))} \\mathcal{D}\n$$ \nis empty if $e'=e''=0$ and is $e'\\mathcal{C}''$ if\n$e'>0$.  \n\n\\medskip\\noindent\nLet $U\\subset \\mathfrak{M}(\\sigma_{g,0})$ denote the open substack that is\nthe image of $\\mathfrak{M}_p(\\tau_{g,0}(0))$, i.e., $U$ is the Artin stack\nof $n$-pointed, genus $g$ curves of \\emph{compact type}.  The\n1-morphism $\\pi:\\mathcal{C}_U \\rightarrow U$ is cohomologically flat, so by\n~\\cite[Thm. 7.3]{Artin} the relative Picard functor of the universal\ncurve over $U$ is a 1-morphism $\\text{pr}:\\text{Pic}_{\\mathcal{C}_U\/U}\n\\rightarrow U$ relatively representable by \\emph{non-separated}\nalgebraic spaces.  The closure $E_{\\mathcal{C}_U\/U}$ of the identity\nsection gives a closed substack of $\\text{Pic}_{\\mathcal{C}_U\/U}$ which is\nrelatively representable over $U$ by non-separated group algebraic\nspaces, ~\\cite[Prop. 5.2]{Raynaud}.  The quotient $Q_{\\mathcal{C}_U\/U}$ of\n$\\text{Pic}_{\\mathcal{C}_U\/U}$ by $E_{\\mathcal{C}_U\/U}$ is a stack that is\nrelatively representable over $U$ by a countable disjoint union of\nsmooth, proper group algebraic spaces, ~\\cite[Thm. 4.1.1]{Raynaud}\n(properness requires a bit more, see ~\\cite[Ex. 8, p. 246]{Ner}).  The\nnext lemma describes $E_{\\mathcal{C}_U\/U}$.\n\n\\medskip\\noindent\nThe invertible sheaf $\\mathcal O_{\\mathcal{C}}(\\mathcal{D})$ defines a\n1-morphism $f:\\mathfrak{M}_{\\mathfrak{H}}(\\tau_{g,0}(0)) \\rightarrow\n\\text{Pic}_{\\mathcal{C}_U\/U}$, and there is a natural 2-equivalence of\n$\\text{pr}\\circ f$ with $\\mathfrak{M}_{\\mathfrak{H}}(\\tau_{g,0}(0))$.\n\n\\begin{lem} \\label{lem-subgp} \n\\marpar{lem-subgp}\nThe 1-morphism $f$ defines an equivalence to $E_{\\mathcal{C}_U\/U}$, the\nclosure of the \nidentity section of $\\text{Pic}_{\\mathcal{C}_U\/U}$.  \nDenoting by\n$Q^0_{\\mathcal{C}_U\/U}$ the identity component of the quotient and by\n$\\text{Pic}^0_{\\mathcal{C}_U\/U}$ the preimage, there are 1-morphisms\n$$\n\\text{Pic}^0_{\\mathcal{C}_U\/U} \\rightleftarrows\n\\mathfrak{M}_{\\mathfrak{H}}(\\tau_{g,0}) \\times\nQ^0_{\\mathcal{C}_U\/U}\n$$ \ngiving an equivalence of stacks\nover $U$, and splitting the extension of group algebraic spaces over\n$U$.\n\\end{lem}  \n\n\\begin{proof}\nIt is easy to see $f$ is an equivalence to its image which is a\nsubgroup of $E_{\\mathcal{C}_U\/U}$.  To prove the image of $f$ is all of\n$E_{\\mathcal{C}_U\/U}$, by the valuative criterion of closedness it suffices\nto check equality of pullbacks for every map of a DVR to\n$\\mathfrak{M}_{g,0}$ sending the generic point to\n$\\mathfrak{M}_{g,0}^{\\text{strict}}$. By ~\\cite[Prop. 6.1.3]{Raynaud}, the\nsections of $E_{\\mathcal{C}_U\/U}$ over are a DVR are just the quotient of\nthe free Abelian group on the irreducible components of the closed\nfiber by the subgroup generated by the entire fiber.  By\nLemma~\\ref{lem-pts} the same is true for the pullback of\n$\\mathfrak{M}_{\\mathfrak{H}}(\\tau_{g,0}(0))$, and it is clear the map between them\nis an isomorphism.\n\n\\medskip\\noindent\nThe splitting of $\\text{Pic}^0_{\\mathcal{C}_U\/U} \\rightarrow\nQ^0_{\\mathcal{C}_U\/U}$ is given by the subfunctor of\n$\\text{Pic}^0_{\\mathcal{C}_U\/U}$ of invertible sheaves whose degree on\nevery irreducible component of every fiber is $0$, denoted by $P^0$ in\n\\cite{Raynaud}.\n\\end{proof}\n\n\\medskip\\noindent\nIn the special case that $g=0$, more is true.  First of all,\n$\\mathfrak{M}_{\\mathbb{Z}}(\\tau) = \\mathfrak{M}_{\\mathfrak{H}}(\\tau)$ for every $\\tau$ of genus\n$0$.  Secondly,  $U=\\mathfrak{M}_{0,n}$.  \n\n\\begin{cor}[Raynaud, Prop. 9.3.1, ~\\cite{Raynaud}] \\label{cor-subgp}\n\\marpar{cor-subgp}\nFor $g=0$, $\\text{Pic}^0_{\\mathcal{C}\/\\mathfrak{M}_{0,0}}$ is equivalent\nto $\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(0))$.\n\\end{cor}\n\n\\medskip\\noindent\nMoreover, the union $\\cup_{e\\in \\mathbb{Z}} \\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(e))$ is a\ngroup algebraic space over $\\mathfrak{M}_{0,0}$ containing\n$\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(0))$ as a subgroup algebraic space over\n$\\mathfrak{M}_{0,0}$.  Essentially, given a contraction\n$\\phi:\\sigma\\rightarrow \\sigma_{0,0}$ and given liftings\n$\\psi_i:(\\sigma,\\alpha_i) \\rightarrow \\tau_{0,0}(e_i)$ for $i=1,2$,\naddition is determined by $\\psi_1+\\psi_2=\\psi:(\\sigma,\n\\alpha_1+\\alpha_2) \\rightarrow \\tau_{0,0}(e_1+e_2)$.  The \\emph{total\ndegree map} gives an isomorphism of $\\text{Pic}\/\\text{Pic}^0$ with\n$\\mathbb{Z}$.  The following result is easy.\n\n\\begin{lem} \\label{lem-g0}\n\\marpar{lem-g0}\nFor $g=0$, for every $e$ there is an equivalence of stacks over\n$\\mathfrak{M}_{0,0}$, $\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(e)) \\rightleftarrows\n\\text{Pic}^e_{\\mathcal{C}\/\\mathfrak{M}_{0,0}}$, such that the equivalence\n$\\cup_{e\\in \\mathbb{Z}} \\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(e)) \\rightleftarrows\n\\text{Pic}_{\\mathcal{C}\/\\mathfrak{M}_{0,0}}$ is an equivalence of group algebraic\nspaces over $\\mathfrak{M}_{0,0}$ and is compatible with the equivalence in\nCorollary~\\ref{cor-subgp}.\n\\end{lem}\n\n\\subsection{Notation for boundary divisor classes}\n\\label{subsec-bound}\n\\marpar{subsec-bound}\n\n\\noindent\nLet $r\\geq 0$ be an integer, and let $e_1,\\dots,e_r$ be integers.\nDenote by $\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(e_1,\\dots,e_r))$ the 2-fibered\nproduct,\n$$\n\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(e_1)) \\times_{\\mathfrak{M}_{0,0}}\n\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(e_2)) \\times_{\\mathfrak{M}_{0,0}} \\dots\n\\times_{\\mathfrak{M}_{0,0}} \\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(e_r)).\n$$\nFor each $i=1,\\dots,r$, let $(e'_i,e''_i)$ be a pair of integers such\nthat $e'_i + e''_i = e_i$.  Let $\\sigma$ be the modular graph with two\nvertices $v',v''$ with $g(v')=g(v'')=0$, one edge connecting $v',\nv''$, and no tails.  Let $\\phi:\\sigma \\rightarrow \\sigma_{0,0}$ be the\ncanonical contraction.  Denote by\n$\\zeta:\\mathfrak{M}^{\\text{strict}}(\\sigma) \\rightarrow\n\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{g,0}(e_1,\\dots,e_r))$ the 1-morphism whose\nprojection to the $i^{\\text{th}}$ factor is determined via\nLemma~\\ref{lem-pts} by the lifting $\\psi_i:(\\sigma,\\alpha_i)\n\\rightarrow \\tau_{g,0}(e_i)$ of $\\phi$ such that $\\alpha_i(v') = e'_i,\n\\alpha_i(v'')=e_i''$.  Define $\\Delta_{(e'_1,e''_1,\\dots,e'_r,e''_r)}$\nto be the effective Cartier divisor on\n$\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(e_1,\\dots,e_r))$ that is the closure of the\nimage of $\\zeta$.\n\n\\medskip\\noindent\nLet $\\pi:C\\rightarrow M$ be a flat 1-morphism relatively represented\nby proper algebraic spaces whose geometric fibers are connected,\nat-worst-nodal curves of arithmetic genus $0$.  Let $D_1,\\dots,D_r$ be\nCartier divisor classes on $C$ of relative degrees $e_1,\\dots,e_r$.\nLet $f(e'_1,e''_1,\\dots,e'_r,e''_r)$ be a function on $\\mathbb{Z}^{2r}$ with\nvalues in $\\mathbb{Z}$, resp. $\\mathbb{Q}$, etc.  Denote by $\\xi: M \\rightarrow\n\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(e_1,\\dots,e_r))$ the 1-morphism whose\nprojection to the $i^\\text{th}$ factor,\n$\\text{Pic}^{e_i}_{\\mathfrak{C}\/\\mathfrak{M}_{0,0}}$ is determined by\n$\\mathcal O_C(D_i)$.\n\n\\begin{notat} \\label{notat-boundary}\n\\marpar{notat-boundary}\nDenote by,\n$$\n\\sum_{(\\beta',\\beta'')} f(\\langle D_1,\\beta' \\rangle, \\langle\nD_1, \\beta'' \\rangle, \\dots, \\langle D_r, \\beta' \\rangle, \\langle D_r,\n\\beta'' \\rangle) \\Delta_{\\beta',\\beta''}\n$$\nthe Cartier divisor class, resp. $\\mathbb{Q}$-Cartier divisor class, etc., \nthat is the pullback by $\\xi$ of the Cartier divisor class, etc.,\n$$\n\\sum_{(e'_1,e''_1,\\dots,e'_r,e''_r)} f(e'_1,e''_1,\\dots,e'_r,e''_r)\n\\Delta_{(e'_1,e''_1,\\dots,e'_r,e''_r)},\n$$\nthe summation over all sequences $(e'_1,e''_1,\\dots,e'_r,e''_r)$ with\n$e'_i + e''_i = e_i$.  \nIf $f(e'_1,e''_1,\\dots,e'_r,e''_r) = f(e''_1,e'_1,\\dots,e''_r,e'_r)$,\ndenote by,\n$$\n{\\sum_{(\\beta',\\beta'')}}^\\prime f(\\langle D_1,\\beta' \\rangle, \\langle\nD_1, \\beta'' \\rangle, \\dots, \\langle D_r, \\beta' \\rangle, \\langle D_r,\n\\beta'' \\rangle) \\Delta_{\\beta',\\beta''}\n$$\nthe pullback by $\\xi$ of,\n$$\n{\\sum_{(e'_1,e''_1,\\dots,e'_r,e''_r)}}^\\prime\nf(e'_1,e''_1,\\dots,e'_r,e''_r)\n\\Delta_{(e'_1,e''_1,\\dots,e'_r,e''_r)},\n$$\nwhere the summation is over equivalence classes of sequences\n$(e'_1,e''_1,\\dots,e'_r,e''_r)$ such that $e'_i+e''_i=e_i$ \nunder the equivalence relation\n$(e'_1,e''_1,\\dots,e'_r,e''_r) \\sim (e''_1,e'_1,\\dots,e''_r,e'_r)$.  \n\\end{notat}\n\n\\begin{ex} \\label{ex-boundary}\n\\marpar{ex-boundary}\nLet $n\\geq 0$ be an integer and let $(A,B)$ be a partition of\n$\\{1,\\dots,n\\}$.  For the universal family over $\\mathfrak{M}_{0,n}$, denote\nby $s_1,\\dots,s_n$ the universal sections.  Then,\n$$\n\\sum_{\\beta',\\beta''} \\prod_{i\\in A} \\langle s_i,\\beta' \\rangle\n\\cdot \\prod_{j\\in B} \\langle s_j, \\beta'' \\rangle\n\\Delta_{\\beta',\\beta''} \n$$\nis the Cartier divisor class of the boundary divisor $\\Delta_{(A,B)}$.  \n\\end{ex}\n\n\n\\section{The functor $Q_\\pi$} \\label{sec-Qpi}\n\\marpar{sec-Qpi}\n\n\\noindent\nLet $M$ be an Artin stack, and let $\\pi:C \\rightarrow M$ be a flat\n1-morphism, relatively representable by proper algebraic spaces whose\ngeometric fibers are connected, at-worst-nodal curves of arithmetic\ngenus $0$.  There exists an invertible dualizing sheaf $\\omega_\\pi$,\nand the relative trace map, $\\text{Tr}_\\pi: R\\pi_* \\omega_\\pi[1]\n\\rightarrow \\mathcal O_M$ is a quasi-isomorphism.  In particular,\n$\\text{Ext}^1_{\\mathcal O_C}(\\omega_\\pi,\\mathcal O_C)$ is canonically isomorphic to\n$H^0(M,\\mathcal O_M)$.  Therefore $1\\in H^0(M,\\mathcal O_M)$ determines an extension\nclass, i.e., a short exact sequence,\n$$\n\\begin{CD}\n0 @>>> \\omega_\\pi @>>> E_\\pi @>>> \\mathcal O_C @>>> 0.\n\\end{CD}\n$$\nThe morphism $\\pi$ is perfect, so for every complex $F^\\bullet$ perfect\nof bounded amplitude on $C$, $R\\pi_* F^\\bullet$ is a perfect complex of\nbounded amplitude on $M$.  By ~\\cite{detdiv}, the\ndeterminant of a perfect complex of bounded amplitude is defined.\n\n\\begin{defn} \\label{defn-E}\n\\marpar{defn-E}\nFor every complex $F^\\bullet$ perfect of bounded amplitude on $C$, \ndefine $Q_\\pi(F^\\bullet) = \\text{det}(R\\pi_* E_\\pi\\otimes F^\\bullet)$.\n\\end{defn}\n\n\\medskip\\noindent\nThere is another interpretation of $Q_\\pi(F^\\bullet)$.\n\n\\begin{lem} \\label{lem-interp}\n\\marpar{lem-interp}\nFor every complex $F^\\bullet$ perfect of bounded amplitude on $C$, \n$$\nQ_\\pi(F^\\bullet) \\cong\n\\text{det}(R\\pi_*(F^\\bullet)) \\otimes \n\\text{det}(R\\pi_*((F^\\bullet)^\\vee))^\\vee.\n$$\n\\end{lem}\n\n\\begin{proof}\nBy the short exact sequence for $E_\\pi$, $Q_\\pi(F^\\bullet) \\cong\n\\text{det}(R\\pi_*(F^\\bullet)) \\otimes \\text{det}(R\\pi_*(\\omega_\\pi\n\\otimes F^\\bullet))$.  The lemma follows by duality.\n\\end{proof}\n\n\\medskip\\noindent\nIt is straightforward to compute $F^\\bullet$ whenever there exist\ncycle class groups for $C$ and $M$ such that Chern classes are defined\nfor all perfect complexes of bounded amplitude and such that\nGrothendieck-Riemann-Roch holds for $\\pi$.\n\n\\begin{lem} \\label{lem-GRR}\n\\marpar{lem-GRR}\nIf there exist cycle class groups for $C$ and $M$ such that Chern\nclasses exist for all perfect complexes of bounded amplitude and such\nthat Grothendieck-Riemann-Roch holds for $\\pi$, then modulo $2$-power\ntorsion, the first Chern class of $Q_\\pi(F^\\bullet)$ is\n$\\pi_*(C_1(F^\\bullet)^2 - 2C_2(F^\\bullet))$.\n\\end{lem}\n\n\\begin{proof}\nDenote the Todd class of $\\pi$ by $\\tau = 1 + \\tau_1 + \\tau_2 +\n\\dots$.  Of course $\\tau_1 = -C_1(\\omega_\\pi)$.  By GRR,\n$\\text{ch}(R\\pi_* \\mathcal O_C) = \\pi_*(\\tau)$.  The canonical map $\\mathcal O_M\n\\rightarrow R\\pi_*\\mathcal O_C$ is a quasi-isomorphism.  Therefore\n$\\pi_*(\\tau_2)=0$, modulo $2$-power torsion.  By additivity of the\nChern character, $\\text{ch}(E_\\pi) = 2 + C_1(\\omega_\\pi) +\n\\frac{1}{2}C_1(\\omega_\\pi)^2 + \\dots$.  Therefore,\n$$\n\\text{ch}(E_\\pi)\\cdot \\tau = 2 + 2\\tau_2 + \\dots\n$$\nSo for any complex $F^\\bullet$ perfect of bounded amplitude,\n$$\n\\begin{array}{c}\n\\text{ch}(E_\\pi\\otimes F^\\bullet)\\cdot \\tau = \\text{ch}(F^\\bullet) \\cdot\n\\text{ch}(E_\\pi) \\cdot \\tau = \\\\\n(\\text{rk}(F^\\bullet) + C_1(F^\\bullet) +\n\\frac{1}{2}(C_1(F^\\bullet)^2 - 2C_2(F^\\bullet)) +\\dots)(2 + 2\\tau_2+\n\\dots). \n\\end{array}\n$$\nApplying $\\pi_*$ gives,\n$$\n2\\pi_*(C_1(F^\\bullet)) + \\pi_*(C_1(F^\\bullet)^2 - 2C_2(F^\\bullet)) + \\dots\n$$\nTherefore the first Chern class of $\\text{det}(R\\pi_*(E_\\pi\\otimes\nF^\\bullet))$ is $\\pi_*(C_1(F^\\bullet)^2 - 2C_2(F^\\bullet))$, modulo\n$2$-power torsion.\n\\end{proof}\n\n\\begin{rmk} \\label{rmk-Q}\nThe point is this.  In every reasonable case, $Q_\\pi$ is just\n$\\pi_*(C_1^2-2C_2)$.  Moreover $Q_\\pi$ is compatible with base-change\nby arbitrary 1-morphisms.  This allows to reduce certain computations\nto the Artin stack of all genus $0$ curves.  As far as we are aware,\nno one has written a definition of cycle class groups for all locally\nfinitely presented Artin stacks that has Chern classes for all perfect\ncomplexes of bounded amplitude, has pushforward maps and\nGrothendieck-Riemann-Roch for perfect 1-morphisms representable by\nproper algebraic spaces, and has pullback maps by arbitrary\n1-morphisms for cycles coming from Chern classes.  Doubtless such a\ntheory exists; whatever it is, $Q_\\pi = \\pi_*(C_1^2-2C_2)$.\n\\end{rmk}\n\n\\medskip\\noindent\nLet the following diagram be 2-Cartesian,\n$$\n\\begin{CD}\nC' @> \\zeta_C >> C \\\\\n@V \\pi' VV @VV \\pi V \\\\\nM' @> \\zeta_M >> M\n\\end{CD}\n$$\ntogether with a 2-equivalence $\\theta:\\pi\\circ \\zeta_C \\Rightarrow\n\\zeta_M \\circ \\pi'$.  \n\n\\begin{lem} \\label{lem-pullback}\n\\marpar{lem-pullback}\nFor every complex $F^\\bullet$ perfect of bounded amplitude on $C$, \n$\\zeta_M^* Q_\\pi(F^\\bullet)$ is isomorphic to $Q_{\\pi'}(\\zeta_C^*\nF^\\bullet)$.\n\\end{lem}\n\n\\begin{proof}\nOf course $\\zeta_C^* E_\\pi = E_{\\pi'}$.  And $\\zeta_M^* R\\pi_*$ is\ncanonically equivalent to $R(\\pi')_* \\zeta_C^*$ for perfect\ncomplexes of bounded amplitude.  Therefore $\\zeta_M^*\nQ_\\pi(F^\\bullet)$ equals $\\text{det}(\\zeta_M^* R\\pi_*(E_\\pi\\otimes\nF^\\bullet))$ equals $\\text{det}(R(\\pi')_* \\zeta_C^*(E_\\pi \\otimes\nF^\\bullet))$ equals $\\text{det}(R(\\pi')_* E_{\\pi'}\\otimes \\zeta_C^*\nF^\\bullet)$ equals $Q_{\\pi'}(\\zeta_C^* F^\\bullet)$.\n\\end{proof}\n\n\\begin{lem} \\label{lem-inv}\n\\marpar{lem-inv}\nLet $L$ be an invertible sheaf on $C$ of relative degree $e$ over\n$M$.  For every invertible sheaf $L'$ on $M$, $Q_\\pi(L\\otimes \\pi^*L')\n\\cong Q_\\pi(L)\\otimes (L')^{2e}$.  In particular, if $e=0$,\n$Q_\\pi(L\\otimes \\pi^* L') \\cong Q_\\pi(L)$.\n\\end{lem}\n\n\\begin{proof}\nTo compute the rank of $R\\pi_*(E_\\pi\\otimes F^\\bullet)$ over any\nconnected component of $M$, it suffices to base-change to the spectrum\nof a field mapping to that component.  Then, by\nGrothendieck-Riemann-Roch, the rank is $2\\text{deg}(C_1(F^\\bullet))$.\nIn particular, $R\\pi_*(E_\\pi \\otimes L)$ has rank $2e$.  \n\n\\medskip\\noindent\nBy the projection formula, $R\\pi_*(E_\\pi\\otimes L\\otimes \\pi^* L')\n\\cong R\\pi_*(E_\\pi \\otimes L)\\otimes L'$.  Of course\n$\\text{det}(R\\pi_*(E_\\pi \\otimes L)\\otimes L') = Q_\\pi(L)\\otimes\n(L')^\\text{rank}$.  This follows from the uniqueness of $\\text{det}$:\nfor any invertible sheaf $L'$ the association $F^\\bullet \\mapsto\n\\text{det}(F^\\bullet \\otimes L')\\otimes\n(L')^{-\\text{rank}(F^\\bullet)}$ also satisfies the axioms for a\ndeterminant function and is hence canonically isomorphic to\n$\\text{det}(F^\\bullet)$.  Therefore $Q_\\pi(L\\otimes \\pi^*L') =\nQ_\\pi(L)\\otimes (L')^{2e}$.\n\\end{proof}\n\n\\section{Local computations} \\label{sec-local}\n\\marpar{sec-local}\n\n\\noindent\nThis section contains 2 computations: $Q_\\pi(\\omega_\\pi)$ and\n$Q_\\pi(L)$ for every invertible sheaf on $C$ of relative degree $0$.\nBecause of Lemma~\\ref{lem-pullback} the first computation reduces to\nthe universal case over $\\mathfrak{M}_{0,0}$.  Because of\nLemma~\\ref{lem-pullback} and Lemma~\\ref{lem-inv}, the second\ncompuation reduces to $\\mathcal O_{\\mathcal{C}}(\\mathcal{D})$ over\n$\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(0))$.  In each case the computation is\nperformed locally.\n\n\\subsection{Computation of $Q_\\pi(\\omega_\\pi)$} \\label{subsec-Qomega}\n\\marpar{subsec-Qomega}\nAssociated to $\\pi_C:C\\rightarrow M$, there is a 1-morphism $\\zeta_M:M\n\\rightarrow \\mathfrak{M}_{0,0}$, a 1-morphism $\\zeta_C: C \\rightarrow\n\\mathcal{C}$, and a 2-equivalence $\\theta:\\pi_{\\mathcal{C}} \\circ \\zeta_C\n\\Rightarrow \\zeta_M\\circ \\pi_C$ such that the following diagram is\n2-Cartesian,\n$$\n\\begin{CD}\nC @> \\zeta_C >> \\mathcal{C} \\\\\n@V \\pi_C VV @VV \\pi_{\\mathcal{C}} V \\\\\nM @> \\zeta_M >> \\mathfrak{M}_{0,0}\n\\end{CD}\n$$\nOf course $\\omega_{\\pi_C}$ is isomorphic to $\\zeta_C^*\n\\omega_{\\pi_\\mathcal{C}}$.  By Lemma~\\ref{lem-pullback},\n$Q_{\\pi_C}(\\omega_{\\pi_C}) \\cong \\zeta_M^*\nQ_{\\pi_{\\mathcal{C}}}(\\omega_{\\pi_{\\mathcal{C}}})$.  So the computation of\n$Q_{\\pi_C}(\\omega_{\\pi_C})$ is reduced to the universal family.\n\n\\medskip\\noindent\nLet the open substack $U_1\\subset \\mathfrak{M}_{0,0}$ be the complement of\nthe union of the images of $\\mathfrak{M}(\\phi):\\mathfrak{M}(\\sigma) \\rightarrow\n\\mathfrak{M}_{0,0}$ as $\\phi:\\sigma \\rightarrow \\sigma_{0,0}$ ranges over\nall contractions such that $\\#\\text{Vertex}(\\sigma) \\geq 3$.  Let $U_2\n\\subset U_1$ be the open substack\n$\\mathfrak{M}^{\\text{strict}}(\\sigma_{0,0})$.\n\n\\begin{prop} \\label{prop-Qomega}\n\\marpar{prop-Qomega}\n\\begin{enumerate}\n\\item[(i)]\nOver the open substack $U_1$, $\\omega_\\pi^\\vee$ is $\\pi$-relatively\nample.  \n\\item[(ii)]\nOver $U_1$,\n$R^1\\pi_*\\omega_\\pi^\\vee|_{U_1} = (0)$ and $\\pi_*\n\\omega_\\pi^\\vee|_{U_1}$ is locally free of rank 3.  \n\\item[(iii)]\nOver $U_2$, there is\na canonical isomorphism $i:\\text{det}(\\pi_* \\omega_\\pi^\\vee|_{U_2})\n\\rightarrow \\mathcal O_{U_2}$. \n\\item[(iv)]\nThe image of $\\text{det}(\\pi_*\\omega_\\pi^\\vee|_{U_1}) \\rightarrow\n\\text{det}(\\pi_*\\omega_\\pi^\\vee|_{U_2}) \\xrightarrow{i} \\mathcal O_{U_2}$ is\n$\\mathcal O_{U_1}(-\\Delta) \\subset \\mathcal O_{U_2}$.\n\\item[(v)]\nOver $U_1$, $Q_\\pi(\\omega_\\pi)|_{U_1} \\cong \\mathcal O_{U_1}(-\\Delta)$.\nTherefore on all of $\\mathfrak{M}_{0,0}$, $Q_\\pi(\\omega_\\pi) \\cong\n\\mathcal O_{\\mathfrak{M}_{0,0}}(-\\Delta)$.  \n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nOver $\\mathbb{Z}$, let $V = \\mathbb{Z}\\{\\mathbf{e}_0,\\mathbf{e}_1\\}$ be a free module of\nrank $2$.  Choose dual coordinates $y_0,y_1$ for $V^\\vee$.  Let\n${\\mathbb P}^1_\\mathbb{Z} = {\\mathbb P}(V)$ be the projective space with homogeneous\ncoordinates $y_0, y_1$.  Let $\\mathbb{A}^1_\\mathbb{Z}$ be the affine space with\ncoordinate $x$.  Denote by $Z\\subset \\mathbb{A}^1_\\mathbb{Z} \\times {\\mathbb P}^1_\\mathbb{Z}$ the\nclosed subscheme $\\mathbb{V}(x,y_1)$, i.e., the image of the section\n$(0,(1,0))$.  Let $\\nu:C\\rightarrow \\mathbb{A}^1_\\mathbb{Z} \\times {\\mathbb P}^1_\\mathbb{Z}$ be\nthe blowing up of $Z$.  Denote by $E\\subset C$ the exceptional\ndivisor.\n\n\\medskip\\noindent\nDefine $\\pi:C\\rightarrow \\mathbb{A}^1_\\mathbb{Z}$ to be $\\text{pr}_{\\AA^1}\\circ\n\\nu$.  This is a flat, proper morphism whose geometric fibers are\nconnected, at-worst-nodal curves of arithmetic genus $0$.  Moreover,\nno geometric fiber has more than 1 node.  Thus there is a 1-morphism\n$\\zeta:\\mathbb{A}^1_\\mathbb{Z} \\rightarrow U_1$ such that the pullback of $\\mathcal{C}$\nis equivalent to $C$.  It is straightforward that $\\zeta$ is smooth\nand is surjective on geometric points.  Thus (i) and (ii) can be\nchecked after base-change by $\\zeta$.  Also (iv) will reduce to a\ncomputation after base-change by $\\zeta$.\n\n\\medskip\\noindent\n\\textbf{(i) and (ii):} Denote by ${\\mathbb P}^2_\\mathbb{Z}$ the projective space with\ncoordinates $u_0,u_1,u_2$.  There is a rational transformation\n$f:\\mathbb{A}^1_\\mathbb{Z} \\times {\\mathbb P}^1_\\mathbb{Z} \\dashrightarrow \\mathbb{A}^1_\\mathbb{Z} \\times\n{\\mathbb P}^2_\\mathbb{Z}$ by\n$$\n\\begin{array}{ccc}\nf^*x & = & x, \\\\\nf^*u_0 & = & xy_0^2, \\\\\nf^*u_1 & = & y_0y_1, \\\\\nf^*u_2 & = & y_1^2\n\\end{array}\n$$\nBy local computation, this extends to a morphism $f:C\\rightarrow\n\\mathbb{A}^1_\\mathbb{Z} \\times {\\mathbb P}^2_\\mathbb{Z}$ that is a closed immersion and whose\nimage is $\\mathbb{V}(u_0u_2-xu_1^2)$.  By the adjunction formula,\n$\\omega_\\pi$ is the pullback of $\\mathcal O_{{\\mathbb P}^2}(-1)$.  In particular,\n$\\omega_\\pi^\\vee$ is very ample. Moreover, because\n$H^1({\\mathbb P}^2_\\mathbb{Z},\\mathcal O_{{\\mathbb P}^2}(-1)) = (0)$, also $H^1(C,\\omega_\\pi^\\vee) =\n(0)$.  By cohomology and base-change results,\n$R^1\\pi_*(\\omega_\\pi^\\vee) = (0)$ and $\\pi_*(\\omega_\\pi^\\vee)$ is\nlocally free of rank 3.  \n\n\\medskip\\noindent\n\\textbf{(iii):} The curve ${\\mathbb P}^1_\\mathbb{Z} = {\\mathbb P}(V)$ determines a morphism\n$\\eta: \\text{Spec }(\\mathbb{Z}) \\rightarrow U_2$.  This is smooth and surjective on\ngeometric points.  Moreover it gives a realization of $U_2$ as the\nclassifying stack of the group scheme $\\text{Aut}({\\mathbb P}(V)) =\n\\textbf{PGL}(V)$.  Taking the exterior power of the Euler exact\nsequence, $\\omega_{{\\mathbb P}(V)\/\\mathbb{Z}} = \\bigwedge^2(V^\\vee)\\otimes\n\\mathcal O_{{\\mathbb P}(V)}(-2)$.  Therefore $H^0({\\mathbb P}(V),\\omega_{{\\mathbb P}(V)\/\\mathbb{Z}}^\\vee)$\nequals $\\bigwedge^2(V)\\otimes \\text{Sym}^2(V^\\vee)$ as a\nrepresentation of $\\textbf{GL}(V)$.  The determinant of this\nrepresentation is the trivial character of $\\textbf{GL}(V)$.\nTherefore it is the trivial character of $\\textbf{PGL}(V)$.  This\ngives an isomorphism of $\\text{det}(\\pi_*\\omega_\\pi|_{U_2})$ with\n$\\mathcal O_{U_2}$.\n\n\\medskip\\noindent\n\\textbf{(iv):} This can be checked after pulling back by $\\zeta$.  The\npullback of $U_2$ is $\\mathbb{G}_{m,\\mathbb{Z}} \\subset \\mathbb{A}^1_\\mathbb{Z}$.  The\npullback of $i$ comes from the determinant of\n$H^0(\\mathbb{G}_{m,\\mathbb{Z}}\\times {\\mathbb P}^1_\\mathbb{Z},\\omega_\\pi^\\vee) =\n\\bigwedge^2(V) \\otimes \\text{Sym}^2(V^\\vee) \\otimes\n\\mathcal O_{\\mathbb{G}_m}$.  By the adjunction formula, $\\omega_{C\/\\mathbb{A}^1} =\n\\nu^*\\omega_{\\mathbb{A}^1\\times {\\mathbb P}^1\/\\mathbb{A}^1}(E)$.  Hence\n$\\nu_*\\omega_{C\/\\mathbb{A}^1}^\\vee = I_Z \\omega_{\\mathbb{A}^1\\times\n{\\mathbb P}^1\/\\mathbb{A}^1}$.  Therefore the canonical map,\n$$\nH^0(C,\\omega_{C\/\\mathbb{A}^1}^\\vee) \\rightarrow H^0(\\mathbb{A}^1_\\mathbb{Z} \\times\n{\\mathbb P}^1_\\mathbb{Z}, \\omega^\\vee_{\\mathbb{A}^1\\times {\\mathbb P}^1\/\\mathbb{A}^1}),\n$$ \nis given by,\n$$\n\\begin{array}{l}\n\\mathcal O_{\\mathbb{A}^1}\\{\\mathbf{f}_0,\\mathbf{f}_1,\\mathbf{f}_2\\} \\rightarrow\n\\bigwedge^2(V)\\otimes \\text{Sym}^2(V^\\vee) \\otimes \\mathcal O_{\\mathbb{A}^1}, \\\\\n\\mathbf{f}_0 \\mapsto x\\cdot (\\mathbf{e}_0\\wedge\\mathbf{e}_1)\\otimes y_0^2, \\\\\n\\mathbf{f}_1 \\mapsto  \\ \\ \\ \\     (\\mathbf{e}_0\\wedge \\mathbf{e}_1) \\otimes y_0y_1, \\\\\n\\mathbf{f}_2 \\mapsto  \\ \\ \\ \\      (\\mathbf{e}_0\\wedge \\mathbf{e}_1) \\otimes y_1^2\n\\end{array}\n$$\nIt follows that $\\text{det}(\\pi_*\\omega_\\pi^\\vee) \\rightarrow\n\\mathcal O_{\\mathbb{G}_m}$ has image $\\langle x \\rangle \\mathcal O_{\\mathbb{A}^1}$, i.e.,\n$\\eta^* \\mathcal O_{U_1}(-\\Delta)$.\n\n\\medskip\\noindent\n\\textbf{(v):} By the short exact sequence for $E_\\pi$,\n$Q_\\pi(\\omega_\\pi) = \\text{det}(R\\pi_*\\omega_\\pi)\\otimes\n\\text{deg}(R\\pi_* \\omega_\\pi^2)$.  Because the trace map is a\nquasi-isomorphism, $\\text{det}(R\\pi_* \\omega_\\pi) = \\mathcal O_{U_1}$.  By\n(ii) and duality, $$ \\text{det}(R\\pi_* \\omega_\\pi^2) \\cong\n\\text{det}(R^1\\pi_*\\omega_\\pi^2)^vee \\cong \\text{det}(\\pi_*\n\\omega_\\pi^\\vee).\n$$  \nBy (iv), this is $\\mathcal O_{U_1}(-\\Delta)$.  Therefore $Q_\\pi(\\omega_\\pi)\n\\cong \\mathcal O_{U_1}(-\\Delta)$ on $U_1$.  Because $\\mathfrak{M}_{0,0}$ is\nregular, and because the complement of $U_1$ has codimension $2$, this\nisomorphism of invertible sheaves extends to all of $\\mathfrak{M}_{0,0}$.\n\\end{proof}\n\n\\medskip\\noindent\nThe sheaf of relative differentials $\\Omega_\\pi$ is a pure coherent\nsheaf on $\\mathcal{C}$ of rank $1$, flat over $\\mathfrak{M}_{0,0}$ and is\nquasi-isomorphic to a perfect complex of amplitued $[-1,0]$.  \n\n\\begin{lem} \\label{lem-Omega}\n\\marpar{lem-Omega}\nThe perfect complex $R\\pi_*\\Omega_\\pi$ has rank $-1$ and determinant\n$\\cong \\mathcal O_{\\mathfrak{M}_{0,0}}(-\\Delta)$.  The perfect complex $R\\pi_*\nR\\textit{Hom}_{\\mathcal O_{\\mathcal{C}}}(\\Omega_\\pi,\\mathcal O_{\\mathcal{C}})$ has rank $3$\nand determinant $\\cong \\mathcal O_{\\mathfrak{M}_{0,0}}(-2\\Delta)$.\n\\end{lem}\n\n\\begin{proof}\nThere is a canonical injective sheaf homomorphism $\\Omega_\\pi\n\\rightarrow \\omega_\\pi$ and the support of the cokernel, $Z\\subset\n\\mathcal{C}$, is a closed substack that is smooth and such that\n$\\pi:Z\\rightarrow \\mathfrak{M}_{0,0}$ is unramified and is the normalization\nof $\\Delta$.  Over $U_1$, the lemma immediately follows from this and\nthe arguments in the proof of Proposition~\\ref{prop-Qomega}.  As in\nthat case, it suffices to establish the lemma over $U_1$.\n\\end{proof}\n\n\\subsection{Computation of $Q_\\pi(L)$ for invertible sheaves of degree\n  $0$} \\label{subsec-QL}\n\\marpar{subsec-QL}\n\n\\noindent\nLet $M$ be an Artin stack, let $\\pi:C\\rightarrow M$ be a flat\n1-morphism, relatively representable by proper algebraic spaces whose\ngeometric fibers are connected, at-worst-nodal curves of arithmetic\ngenus $0$.  Let $L$ be an invertible sheaf on $C$ of relative degree\n$0$ over $M$.  This determines a morphism to the relative Picard of\nthe universal curve over $\\mathfrak{M}_{0,0}$, i.e., $\\zeta_M:M \\rightarrow\n\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(0))$ such that the pullback of $\\mathcal{C}$ is\nequivalent to $C$, and such that the pullback of\n$\\mathcal O_{\\mathcal{C}}(\\mathcal{D})$ differs from $L$ by $\\pi^*L'$ for an invertible\nsheaf $L'$ on $M$.  By Lemma~\\ref{lem-pullback} and\nLemma~\\ref{lem-inv}, $Q_\\pi(L) \\cong \\zeta_M^*\nQ_\\pi(\\mathcal O_{\\mathcal{C}}(\\mathcal{D}))$.\n\n\\medskip\\noindent\nLet $\\pi:\\mathcal{C} \\rightarrow \\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(0))$ be the\nuniversal curve.\n\n\\begin{prop} \\label{prop-lcomp}\n\\marpar{prop-lcomp}\nOver $\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(0))$, $\\pi_*E_\\pi(\\mathcal{D}) = (0)$ and \n$R^1\\pi_*\nE_\\pi(\\mathcal{D})$ is a sheaf supported on $\\Delta$.  The stalk of\n$R^1\\pi_* E_\\pi(\\mathcal{D})$ at the generic point of $\\Delta_a$ is a\ntorsion sheaf of length $a^2$.  The filtration by order of vanishing\nat the generic point has associated graded pieces of length $2a-1,\n2a-3, \\dots, 3,1$.  \n\\end{prop}\n\n\\begin{proof}\nOver the open complement of $\\Delta$, the divisor $\\mathcal{D}$ is $0$.  So\nthe first part of the proposition reduces to the statement that\n$R\\pi_* E_\\pi$ \nis quasi-isomorphic to $0$.  By definition of $E_\\pi$, there is an\nexact triangle,\n$$\n\\begin{CD}\nR\\pi_* E_\\pi @>>> R\\pi_* \\mathcal O_{\\mathcal{C}} @> \\delta >> R\\pi_* \\omega_\\pi\n[1] @>>> \nR\\pi_* E_\\pi[1].\n\\end{CD}\n$$\nOf course the canonical isomorphism $R\\pi_* \\mathcal O_{\\mathcal{C}} \\cong\n\\mathcal O_{\\mathfrak{M}}$, and \n$E_\\pi$ were defined so that the composition of $\\delta$ with the trace\nmap, which is a quasi-isomorphism in this case, would be the\nidentity.  Therefore $\\delta$ is a quasi-isomorphism, so $R\\pi_*\nE_\\pi$ is quasi-isomorphic to $0$.  \n\n\\medskip\\noindent\nThe second part can be proved, and to an extent only makes sense,\nafter smooth base-change to a scheme.  Let ${\\mathbb P}^1_s$ be a copy of\n${\\mathbb P}^1$ with homogeneous coordinates $S_0,S_1$.  Let ${\\mathbb P}^1_x$ be a\ncopy of ${\\mathbb P}^1$ with homogeneous coordinates $X_0,X_1$.  Let ${\\mathbb P}^1_y$\nbe a copy of ${\\mathbb P}^1$ with homogeneous coordinates $Y_0,Y_1$.  Denote\nby $C \\subset {\\mathbb P}^1_s \\times {\\mathbb P}^1_x \\times {\\mathbb P}^1_y$ the divisor with\ndefining equation $F=S_0X_0Y_0 - S_1X_1Y_1$.  The projection\n$\\text{pr}_s:C \\rightarrow {\\mathbb P}^1_s$ is a proper, flat morphism whose\ngeometric fibers are connected, at-worst-nodal curves of arithmetic\ngenus $0$.  Denote by $L$ the invertible sheaf on $C$ that is the\nrestriction of $\\text{pr}_x^*\\mathcal O_{{\\mathbb P}^1_x}(a)\\otimes\n\\text{pr}_y^*\\mathcal O_{{\\mathbb P}^1_y}(-a)$.  This is an invertible sheaf of\nrelative degree $0$.  Therefore there is an induced $1$-morphism\n$\\zeta:{\\mathbb P}^1_s \\rightarrow \\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(0))$.  \n\n\\medskip\\noindent\nIt is straightforward that $\\zeta$ is smooth, and the image intersects\n$\\Delta_b$ iff $b=a$.  Moreover, $\\zeta^* \\Delta_a$ is the reduced\nCartier divisor $\\mathbb{V}(S_0S_1) \\subset {\\mathbb P}^1_s$.  There is an\nobvious involution $i:{\\mathbb P}^1_s\\rightarrow {\\mathbb P}^1_s$ by\n$i(S_0,S_1)=(S_1,S_0)$, and $\\zeta\\circ i$ is $2$-equivalent to\n$\\zeta$.  Therefore the length of the $R^1\\text{pr}_{s,*}\nE_{\\text{pr}_s}\\otimes L$ is $2$ times the length of the stalk of\n$R^1\\pi_* E_{\\pi}(\\mathcal{D})$ at the generic point of $\\Delta_a$; more\nprecisely, the length of the stalk at each of $(1,0),(0,1)\\in {\\mathbb P}^1_s$\nis the length of the stalk at $\\Delta_a$.  Similarly for the lengths\nof the associated graded pieces of the filtration.\n\n\\medskip\\noindent\nBecause $E_{\\text{pr}_s}$ is the extension class of the Trace mapping,\n$R^1\\text{pr}_{s,*}E_{\\text{pr}_s} \\otimes L$ is the cokernel of the\n$\\mathcal O_{{\\mathbb P}^1_s}$-homomorphisms,\n$$\n\\gamma:\\text{pr}_{s,*}(L) \\rightarrow\n\\text{Hom}_{\\mathcal O_{{\\mathbb P}^1_s}}(\\text{pr}_{s,*}(L^\\vee), \\mathcal O_{{\\mathbb P}^1_s}),\n$$\ninduced via adjointness from the multiplication map,\n$$\n\\text{pr}_{s,*}(L) \\otimes \\text{pr}_{s,*}(L^\\vee) \\rightarrow\n\\text{pr}_{s,*}(\\mathcal O_C) = \\mathcal O_{{\\mathbb P}^1_s}.\n$$\n\n\\medskip\\noindent\nOn ${\\mathbb P}^1_s\\times {\\mathbb P}^1_x \\times {\\mathbb P}^1_y$ there is a locally free\nresolution of the push-forward of $L$, resp. $L^\\vee$,\n$$\n\\begin{array}{c}\n0 \\rightarrow \\mathcal O_{{\\mathbb P}^1_s}(-1)\\boxtimes \\mathcal O_{{\\mathbb P}^1_x}(a-1)\\boxtimes\n\\mathcal O_{{\\mathbb P}^1_y}(-a-1) \\xrightarrow{F} \\mathcal O_{{\\mathbb P}^1_s}(0)\\boxtimes\n\\mathcal O_{{\\mathbb P}^1_x}(a) \\boxtimes \\mathcal O_{{\\mathbb P}^1_y}(-a) \\rightarrow L \\rightarrow\n0, \\\\\n0 \\rightarrow \\mathcal O_{{\\mathbb P}^1_s}(-1)\\boxtimes \\mathcal O_{{\\mathbb P}^1_x}(-a-1)\\boxtimes\n\\mathcal O_{{\\mathbb P}^1_y}(a-1) \\xrightarrow{F} \\mathcal O_{{\\mathbb P}^1_s}(0)\\boxtimes\n\\mathcal O_{{\\mathbb P}^1_x}(-a) \\boxtimes \\mathcal O_{{\\mathbb P}^1_y}(a) \\rightarrow L^\\vee\n\\rightarrow 0\n\\end{array}\n$$\nHence $R\\text{pr}_{s,*}L$ is the complex,\n$$\n\\mathcal O_{{\\mathbb P}^1_s}(-1)\\otimes_k H^0({\\mathbb P}^1_x,\\mathcal O_{{\\mathbb P}^1_x}(a-1)) \\otimes_k\nH^1({\\mathbb P}^1_y,\\mathcal O_{{\\mathbb P}^1_y}(-a-1)) \\xrightarrow{F} \\mathcal O_{{\\mathbb P}^1_s}\n\\otimes_k H^0({\\mathbb P}^1_x,\\mathcal O_{{\\mathbb P}^1_x}(a))\\otimes_k\nH^1({\\mathbb P}^1_y,\\mathcal O_{{\\mathbb P}^1_y}(-a)).\n$$\nSimilarly for $R\\text{pr}_{s,*}L^\\vee$.  It is possible to write out\nthis map explicitly in terms of bases for $H^0$ and $H^1$, but for the\nmain statement just observe the complex has rank $1$ and degree $-a^2$.\nSimilarly for $R\\text{pr}_{s,*}L^\\vee$.  Therefore $R^1\\pi_* E_\\pi(L)$\nis a torsion sheaf of length $2a^2$.  Because it is equivariant for\n$i$, the localization at each of $(0,1)$ and $(1,0)$ has length $a^2$.\n\n\\medskip\\noindent\nThe lengths of the associated graded pieces of the filtration by order\nof vanishing at $\\mathbb{V}(S_0S_1)$ can be computed from the\ncomplexes for $R\\text{pr}_{s,*}L$ and $R\\text{pr}_{s,*}L^\\vee$.  This\nis left to the reader.\n\\end{proof}\n\n\\begin{cor} \\label{cor-lcomp}\n\\marpar{cor-lcomp}\nIn the universal case, $Q_\\pi(\\mathcal{D}) = -\\sum_{a\\geq 0} a^2 \\Delta_a$.\nTherefore in the general case of $\\pi:C\\rightarrow M$ and an\ninvertible sheaf $L$ of relative degree $0$,\n$$\nQ_\\pi(L) = {\\sum_{\\beta',\\beta''}}^\\prime \\langle C_1(L),\\beta'\n\\rangle \\langle \nC_1(L), \\beta'' \\rangle \\Delta_{\\beta',\\beta''}.\n$$\n\\end{cor}\n\n\\section{Some divisor class relations} \\label{sec-div}\n\\marpar{sec-div}\n\n\\noindent\nIn this section, Proposition~\\ref{prop-Qomega} and\nProposition~\\ref{prop-lcomp} are used to deduce several other divisor\nclass relations.  As usual, let $M$ be an Artin stack and let\n$\\pi:C\\rightarrow M$ be a flat 1-morphism, relatively representable by\nproper algebraic spaces whose geometric fibers are connected,\nat-worst-nodal curves of genus $0$.\n\n\\begin{hyp} \\label{hyp-GRR}\n\\marpar{hyp-GRR}\nThere are cycle class groups for $C$ and $M$ admitting Chern classes\nfor locally free sheaves, and such that Grothendieck-Riemann-Roch\nholds for $\\pi$.\n\\end{hyp}\n\n\\begin{lem} \\label{lem-rel1}\nFor every Cartier divisor class $D$ on $C$ of relative degree $\\langle\nD, \\beta \\rangle$ over $M$, modulo $2$-power torsion,\n$$\n\\pi_*(D\\cdot D) + \\langle D,\\beta \\rangle \\pi_*(D\\cdot\nC_1(\\omega_\\pi)) = {\\sum_{\\beta',\\beta''}}^\\prime \\langle D,\\beta'\n\\rangle \\langle D,\\beta'' \\rangle \\Delta_{\\beta',\\beta''}.\n$$\n\\end{lem}\n\n\\begin{proof}\nDefine $D' = 2D + \\langle D,\\beta \\rangle C_1(\\omega_\\pi)$.  This is a\nCartier divisor class of relative degree $0$.  By\nCorollary~\\ref{cor-lcomp}, \n$$\nQ_\\pi(D') = {\\sum_{\\beta',\\beta''}}^\\prime (\\langle 2D,\\beta' \\rangle\n-\\langle D,\\beta \\rangle)(\\langle 2D,\\beta'' \\rangle - \\langle D,\\beta\n\\rangle) \\Delta_{\\beta',\\beta''}.\n$$\nBy Lemma~\\ref{lem-GRR} this is,\n$$\n\\begin{array}{c}\n4\\pi_*(D\\cdot D) +4\\langle D,\\beta \\rangle \\pi_*(D\\cdot\nC_1(\\omega_\\pi) + (\\langle D,\\beta \\rangle)^2 Q_\\pi(C_1(\\omega_\\pi)) =\n\\\\\n{\\sum_{\\beta',\\beta''}}^\\prime (4\\langle D,\\beta' \\rangle \\langle\nD,\\beta'' \\rangle - (\\langle D,\\beta \\rangle)^2)\n\\Delta_{\\beta',\\beta''}.\n\\end{array}\n$$\nBy Proposition~\\ref{prop-Qomega}, $Q_\\pi(\\omega_\\pi) =\n-{\\sum_{\\beta',\\beta''}}' \\Delta_{\\beta',\\beta''}$.  Substituting this\ninto the equation, simplifying, and dividing by 4 gives the relation.\n\\end{proof}\n\n\\begin{lem} \\label{lem-rel2}\n\\marpar{lem-rel2}\nFor every pair of Cartier divisor classes on $C$, $D_1,D_2$, of\nrelative degrees $\\langle D_1,\\beta \\rangle$, resp. $\\langle D_2,\\beta\n\\rangle$, modulo $2$-power torsion,\n$$\n\\begin{array}{c}\n2\\pi_*(D_1\\cdot D_2) + \\langle D_1,\\beta \\rangle \\pi_*(D_2\\cdot\nC_1(\\omega_\\pi)) + \\langle D_2,\\beta \\rangle \\pi_*(D_1\\cdot\nC_1(\\omega_\\pi)) = \n\\\\\n\\\\\n{\\sum_{\\beta',\\beta''}}^\\prime (\\langle D_1,\\beta' \\rangle \\langle\nD_2,\\beta'' \n\\rangle + \\langle D_2,\\beta' \\rangle \\langle D_1,\\beta'' \\rangle)\n\\Delta_{\\beta',\\beta''}.\n\\end{array}\n$$\n\\end{lem}\n\n\\begin{proof}\nThis follows from Lemma~\\ref{lem-rel1} and the polarization identity\nfor quadratic forms.\n\\end{proof}\n\n\\begin{lem} \\label{lem-rel3}\n\\marpar{lem-rel3}\nFor every section of $\\pi$, $s:M\\rightarrow C$, whose image is\ncontained in the smooth locus of $\\pi$,\n$$\ns(M)\\cdot s(M) + s(M)\\cdot C_1(\\omega_\\pi).\n$$\n\\end{lem}\n\n\\begin{proof}\nThis follows by adjunction since the relative dualizing sheaf of\n$s(M)\\rightarrow M$ is trivial.\n\\end{proof}\n\n\\begin{lem} \\label{lem-rel4}\n\\marpar{lem-rel4}\nFor every section of $\\pi$, $s:M\\rightarrow C$, whose image is\ncontained in the smooth locus of $\\pi$ and for every Cartier divisor\nclass $D$ on $C$ of relative degree $\\langle D,\\beta \\rangle$ over\n$M$, modulo $2$-power torsion,\n$$\n\\begin{array}{c}\n2\\langle D,\\beta \\rangle s^*D  -\\pi_*(D\\cdot D) - \\langle D,\\beta\n\\rangle^2 \\pi_*(s(M)\\cdot s(M)) = \\\\\n\\\\\n{\\sum_{\\beta',\\beta''}}^\\prime\n(\\langle D,\\beta' \\rangle^2 \\langle s(M),\\beta'' \\rangle + \\langle\nD,\\beta'' \\rangle^2 \\langle s(M),\\beta' \\rangle )\n\\Delta_{\\beta',\\beta''}.\n\\end{array}\n$$\n\\end{lem}\n\n\\begin{proof}\nBy Lemma~\\ref{lem-rel2},\n$$\n\\begin{array}{c}\n2s^* D + \\pi_*(D\\cdot C_1(\\omega_\\pi)) + \\langle D,\\beta \\rangle\n\\pi_*(s(M)\\cdot C_1(\\omega_\\pi)) = \\\\\n\\\\\n{\\sum}^\\prime (\\langle D,\\beta' \\rangle \\langle s(M),\\beta'' \\rangle +\n\\langle D,\\beta'' \\rangle \\langle s(M),\\beta' \\rangle\n)\\Delta_{\\beta',\\beta''}.\n\\end{array}\n$$\nMultiplying both sides by $\\langle D,\\beta \\rangle$,\n$$\n\\begin{array}{c}\n2\\langle D,\\beta \\rangle s^* D + \\langle D,\\beta \\rangle \\pi_*(D \\cdot\nC_1(\\omega_\\pi)) + \\langle D, \\beta \\rangle^2 \\pi_*(s(M)\\cdot\nC_1(\\omega_\\pi)) = \\\\\n\\\\\n{\\sum}^\\prime(\\langle D,\\beta \\rangle \\langle D,\\beta' \\rangle \\langle\ns(M),\\beta'' \\rangle + \\langle D,\\beta \\rangle \\langle D,\\beta''\n\\rangle \\langle s(M),\\beta' \\rangle ) \\Delta_{\\beta',\\beta''}.\n\\end{array}\n$$\nFirst of all, by Lemma~\\ref{lem-rel4}, $\\langle D,\\beta \\rangle^2\n\\pi_*(s(M) \\cdot C_1(\\omega_\\pi)) = - \\langle D,\\beta \\rangle^2\n\\pi_*(s(M)\\cdot s(M))$.  Next, by Lemma~\\ref{lem-rel1},\n$$\n\\langle D,\\beta \\rangle \\pi_*(D\\cdot C_1(\\omega_\\pi)) = -\\pi_*(D\\cdot\nD) + {\\sum}^\\prime \\langle D,\\beta' \\rangle \\langle D,\\beta''\n\\Delta_{\\beta',\\beta''}.\n$$\nFinally,\n$$\n\\begin{array}{c}\n\\langle D,\\beta \\rangle \\langle D,\\beta' \\rangle \\langle s(M),\\beta''\n\\rangle + \\langle D,\\beta \\rangle \\langle D,\\beta'' \\rangle \\langle\ns(M),\\beta' \\rangle = \\\\ \\\\\n(\\langle D,\\beta' \\rangle + \\langle D,\\beta'' \\rangle) \\langle\nD,\\beta' \\rangle \\langle s(M),\\beta'' \\rangle + (\\langle D,\\beta'\n\\rangle + \\langle D,\\beta'' \\rangle) \\langle D,\\beta'' \\rangle \\langle\ns(M), \\beta' \\rangle = \\\\ \\\\\n\\langle D,\\beta' \\rangle^2 \\langle s(M),\\beta'' \\rangle + \\langle\nD,\\beta'' \\rangle^2 \\langle s(M),\\beta' \\rangle + \\langle D,\\beta'\n\\rangle \\langle D,\\beta'' \\rangle (\\langle s(M),\\beta' \\rangle +\n\\langle s(M),\\beta'' \\rangle) = \\\\ \\\\\n\\langle D,\\beta' \\rangle^2 \\langle s(M),\\beta'' \\rangle + \\langle\nD,\\beta'' \\rangle^2 \\langle s(M),\\beta' \\rangle + \\langle D,\\beta'\n\\rangle \\langle D,\\beta'' \\rangle.\n\\end{array}\n$$\nPlugging in these 3 identities and simplifying gives the relation.\n\\end{proof}\n\n\\medskip\\noindent\nLet $\\mathcal{C}$ be the universal curve over $\\mathfrak{M}_{0,0}$.  Let\n$\\mathcal{C}_\\text{smooth}$ denote the smooth locus of $\\pi$.  The\n2-fibered product $\\text{pr}_1:\\mathcal{C}_\\text{smooth}\n\\times_{\\mathfrak{M}_{0,0}} \\mathcal{C} \n\\rightarrow \\mathcal{C}_\\text{smooth}$ together with the diagonal\n$\\Delta:\\mathcal{C}_\\text{smooth} \\rightarrow \\mathcal{C}_\\text{smooth}\n\\times_{\\mathfrak{M}_{0,0}} \\mathcal{C}$ determine a 1-morphism\n$\\mathcal{C}_\\text{smooth} \\rightarrow \\mathfrak{M}_{1,0}$.  This extends to a\n1-morphism $\\mathcal{C} \\rightarrow \\mathfrak{M}_{1,0}$.  The pullback of the\nuniversal curve is a 1-morphism $\\pi':\\mathcal{C}' \\rightarrow \\mathcal{C}$ that\nfactors through $\\text{pr}_1:\\mathcal{C}\\times_{\\mathfrak{M}_{0,0}} \\mathcal{C}\n\\rightarrow \\mathcal{C}$.  Denote the pullback of the universal section by\n$s:\\mathcal{C} \\rightarrow \\mathcal{C}'$.  Now $\\mathcal{C}$ is regular, and the\ncomplement of $\\mathcal{C}_\\text{smooth}$ has codimension $2$.  In\nparticular, $s^*\\mathcal O_{\\mathcal{C}'}(s(\\mathcal{C}))$ can be computed on\n$\\mathcal{C}_\\text{smooth}$.  But the restriction to $\\mathcal{C}_\\text{smooth}$\nis clearly $\\omega^\\vee_\\pi$.  Therefore $s^*\\mathcal O_{\\mathcal{C}'}(s(\\mathcal{C}))\n\\cong \n\\omega_\\pi^\\vee$ on all of $\\mathcal{C}$.\n\n\\medskip\\noindent\nPulling this back by $\\zeta_C:C\\rightarrow \\mathfrak{C}$ gives a 1-morphism\n$\\pi':C'\\rightarrow C$ that factors through $\\text{pr}_1:C\\times_M\nC\\rightarrow C$.  Let $D$ be a Cartier divisor class on $C$ and\nconsider the pullback to $C'$ of $\\text{pr}_2^* D$ on $C\\times_M C$.\nThis is a Cartier divisor class $D'$ on $C'$.  Of course $s^* D' =\nD$.  Moreover, by the projection formula the pushforward to $C\\times_M\nC$ of $D'\\cdot D'$ is $\\text{pr}_2^*(D\\cdot D)$.  Therefore\n$(\\pi')_*(D'\\cdot D')$ is $(\\text{pr}_1)_*\\text{pr}_2^*(D\\cdot D)$,\ni.e., $\\pi^*\\pi_*(D\\cdot D)$.  Finally, denote by,\n$$\n\\sum_{\\beta',\\beta''} \\langle D,\\beta'' \\rangle^2\n\\widetilde{\\Delta}_{\\beta',\\beta''},\n$$\nthe divisor class on $C$,\n$$\n{\\sum_{\\beta',\\beta''}}^\\prime (\\langle D,\\beta'' \\rangle^2 \\langle\ns,\\beta' \\rangle + \\langle D,\\beta' \\rangle^2 \\langle s,\\beta''\n\\rangle)\\Delta_{\\beta',\\beta''}.\n$$\nThe point is this: if $\\pi$ is smooth over every generic point of $M$,\nthen the divisor class $\\widetilde{\\Delta}_{\\beta',\\beta''}$ is the\nirreducible component of $\\pi^{-1}(\\Delta_{\\beta',\\beta''})$\ncorresponding to the vertex $v'$, i.e., the irreducible component with\n``curve class'' $\\beta'$.\nPutting this all together and applying Lemma~\\ref{lem-rel4} gives the\nfollowing.\n\n\\begin{lem} \\label{lem-rel5}\n\\marpar{lem-rel5}\nFor every Cartier divisor class $D$ on $C$ of relative degree $\\langle\nD,\\beta \\rangle$ over $M$,\n$$\n\\begin{array}{c}\n2\\langle D,\\beta \\rangle D - \\pi^*\\pi_*(D\\cdot D) + \\langle D,\\beta\n\\rangle^2 C_1(\\omega_\\pi) = \\\\ \\\\\n\\sum_{\\beta',\\beta''} \\langle D,\\beta'' \\rangle^2\n\\widetilde{\\Delta}_{\\beta',\\beta''}.\n\\end{array}\n$$\nIn particular, the relative Picard group of $\\pi$ is generated by\n$C_1(\\omega_\\pi)$ and the boundary divisor classes\n$\\widetilde{\\Delta}_{\\beta',\\beta''}$.  \n\\end{lem}\n\n\\begin{rmk} \\label{rmk-rel5}\n\\marpar{rmk-rel5}\nIf $\\langle D,\\beta \\rangle \\neq 0$ then, at least up to torsion,\nLemma~\\ref{lem-rel1} follows from Lemma~\\ref{lem-rel5} by intersecting\nboth sides of the relation by $D$ and then applying $\\pi_*$.  This was\npointed out by Pandharipande, who also proved Lemma~\\ref{lem-rel4} up\nto numerical equivalence in ~\\cite[Lem. 2.2.2]{QDiv} (by a very\ndifferent method).   \n\\end{rmk}\n\n\\begin{lem} \\label{lem-rel6}\n\\marpar{lem-rel6}\nLet $s,s':M\\rightarrow C$ be sections with image in the smooth locus\nof $\\pi$ such that $s(M)$ and $s'(M)$ are disjoint.  Then,\n$$\n\\pi_*(s(M)\\cdot s(M)) + \\pi_*(s'(M)\\cdot s'(M)) = -\n\\sum_{\\beta',\\beta''} \\langle s(M),\\beta' \\rangle \\langle\ns'(M),\\beta'' \\rangle \\Delta_{\\beta',\\beta''}.\n$$\n\\end{lem}\n\n\\begin{proof}\nApply Lemma~\\ref{lem-rel2} and use $s(M)\\cdot s'(M)=0$ and\nLemma~\\ref{lem-rel3}.  \n\\end{proof}\n\n\\begin{lem} \\label{lem-rel7}\nLet $r\\geq 2$ and $s_1,\\dots,s_r:M\\rightarrow C$ be sections with image in the\nsmooth locus of $\\pi$ and which are pairwise disjoint.  Then,\n$$\n-\\sum_{i=1}^r \\pi_*(s_i(M)\\cdot s_i(M)) = (r-2)\\pi_*(s_1(M)\\cdot\ns_1(M)) + \\sum_{\\beta',\\beta''} \\langle s_1(M),\\beta' \\rangle \\langle\ns_2(M) + \\dots + s_r(M),\\beta'' \\rangle \\Delta_{\\beta',\\beta''}.\n$$\n\\end{lem}\n\n\\begin{proof}\nThis follows from Lemma~\\ref{lem-rel6} by induction.\n\\end{proof}\n\n\\begin{lem} \\label{lem-rel8}\n\\marpar{lem-rel8}\nLet $r\\geq 2$ and let $s_1,\\dots,s_r:M \\rightarrow C$ be sections with\nimage in the smooth locus of $\\pi$ and which are pairwise disjoint.\nThen,\n$$\n-\\sum_{i=1}^r \\pi_*(s_i(M)\\cdot s_i(M)) = r(r-2) \\pi_*(s_1(M)\\cdot\ns_1(M)) + \\sum_{\\beta',\\beta''} \\langle s_1(M),\\beta' \\rangle \\langle\ns_2(M) + \\dots + s_r(M),\\beta'' \\rangle^2 \\Delta_{\\beta',\\beta''}.\n$$\nCombined with Lemma~\\ref{lem-rel7} this gives,\n$$\n\\begin{array}{c}\n(r-1)(r-2)\\pi_*(s_1(M)\\cdot s_1(M)) = \\\\ \\\\\n-\\sum_{\\beta',\\beta''} \\langle\ns_1(M),\\beta' \\rangle \\langle s_2(M) + \\dots + s_r(M),\\beta'' \\rangle\n(\\langle s_2(M)+\\dots +s_r(M), \\beta'' \\rangle - 1)\n\\Delta_{\\beta',\\beta''},\n\\end{array}\n$$\nwhich in turn gives,\n$$\n\\begin{array}{c}\n-(r-1)\\sum_{i=1}^r\\pi_*(s_i(M)\\cdot s_i(M)) = \\\\ \\\\\n\\sum_{\\beta',\\beta''} \\langle s_1(M),\\beta' \\rangle \\langle s_2(M) +\n\\dots + s_r(M), \\beta'' \\rangle (r- \\langle s_2(M) +\\dots + s_r(M),\n\\beta'' \\rangle) \\Delta_{\\beta',\\beta''}.\n\\end{array}\n$$\nIn the notation of Example~\\ref{ex-boundary}, this is,\n$$\n-(r-1)(r-2)\\pi_*(s_1(M)\\cdot s_1(M)) = \\sum_{(A,B), \\ 1\\in A}\n\\#B(\\#B-1) \\Delta_{(A,B)},\n$$\nand\n$$\n-(r-1)\\sum_{i=1}^r \\pi_*(s_i(M)\\cdot s_i(M)) = \\sum_{(A,B), \\ 1\\in A}\n\\#B(r-\\#B) \\Delta_{(A,B)}.\n$$\n\\end{lem}\n\n\\begin{proof}\nDenote $D=\\sum_{i=2}^r s_i(M)$.  Apply Lemma~\\ref{lem-rel4} to get,\n$$\n\\begin{array}{c}\n2(r-1)\\cdot 0 - \\sum_{i=2}^r \\pi_*(s_i(M)\\cdot s_i(M)) -\n(r-1)^2\\pi_*(s_1(M)\\cdot s_1(M)) = \\\\ \\\\\n\\sum_{\\beta',\\beta''} \\langle s_1(M),\\beta' \\rangle \\langle\ns_2(M)+\\dots + s_r(M),\\beta'' \\rangle^2 \\Delta_{\\beta',\\beta''}.\n\\end{array}\n$$\nSimplifying,\n$$\n-\\sum_{j=1}^r \\pi_*(s_i(M)\\cdot s_i(M)) = r(r-2) \\pi_*(s_1(M)\\cdot\ns_1(M)) + \\sum \\langle s_1(M),\\beta' \\rangle \\langle s_2(M)+ \\dots\n+s_r(M),\\beta'' \\rangle^2 \\Delta_{\\beta',\\beta''}.\n$$\nSubtracting from the relation in Lemma~\\ref{lem-rel7} gives the\nrelation for $(r-1)(r-2)\\pi_*(s_1(M)\\cdot s_1(M))$.  \nMultiplying the first relation by $(r-1)$, plugging in the second\nrelation and simplifying gives the third relation.\n\\end{proof}\n\n\\begin{lem} \\label{lem-rel9}\n\\marpar{lem-rel9}\nLet $r\\geq 2$ and let $s_1,\\dots,s_r:M\\rightarrow C$ be everywhere\ndisjoint sections with image in the smooth locus.  For every $1\\leq i\n< j \\leq r$, using the notation from Example~\\ref{ex-boundary},\n$$\n\\sum_{(A,B), \\ i\\in A} \\#B(r-\\#B) \\Delta_{(A,B)} = \\sum_{(A',B'), j \\in\n  A} \\#B'(r-\\#B') \\Delta_{(A',B')}.\n$$\n\\end{lem}\n\n\\begin{proof}\nThis follows from Lemma~\\ref{lem-rel8} by permuting the roles of $1$\nwith $i$ and $j$.\n\\end{proof}\n\n\\begin{lem} \\label{lem-rel10}\n\\marpar{lem-rel10}\nLet $r \\geq 2$ and let $s_1,\\dots,s_r:M \\rightarrow C$ be everywhere\ndisjoint sections with image in the smooth locus of $\\pi$.  For every\nCartier \ndivisor class $D$ on $C$ of relative degree $\\langle D,\\beta \\rangle$,\n$$\n2(r-1)(r-2)\\langle D,\\beta \\rangle s_1^*D = (r-1)(r-2)\\pi_*(D\\cdot D)\n+ \\sum_{\\beta',\\beta''} \\langle s_1(M),\\beta' \\rangle a(D,\\beta'')\n\\Delta_{\\beta',\\beta''},\n$$\nwhere, \n$$\na(D,\\beta'') = (r-1)(r-2)\\langle D,\\beta'' \\rangle^2 -\n\\langle D,\\beta \\rangle^2 \\langle s_2(M)+\\dots +s_r(M), \\beta'' \\rangle (\n\\langle s_2(M)+\\dots + s_r(M),\\beta'' \\rangle -1).\n$$\nIn particular, if $r\\geq 3$, then modulo torsion $s_i^*D$ is in the\nspan of $\\pi_*(D\\cdot D)$ and boundary divisors for every\n$i=1,\\dots,r$.\n\\end{lem}\n\n\\begin{proof}\nThis follows from Lemma~\\ref{lem-rel4} and Lemma~\\ref{lem-rel8}.\n\\end{proof}\n\n\\begin{lem} \\label{lem-rel11}\n\\marpar{lem-rel11}\nLet $r\\geq 2$ and let $s_1,\\dots,s_r:M\\rightarrow C$ be everywhere\ndisjoint sections with image in the smooth locus of $\\pi$.  Consider\nthe sheaf $\\mathcal{E} = \\Omega_\\pi(s_1(M)+\\dots+s_r(M))$.  The perfect\ncomplex $R\\pi_*R\\textit{Hom}_{\\mathcal O_C}(\\mathcal{E},\\mathcal O_C)$ has rank $3-r$ and\nthe first Chern class of the determinant is $-2\\Delta\n-\\sum_{i=1}^r(s_i(M)\\cdot s_i(M))$.  In particular, if $r\\geq 2$, up\nto torsion,\n$$\n\\begin{array}{c}\nC_1(\\text{det}R\\pi_*\nR\\textit{Hom}_{\\mathcal O_C}(\\Omega_\\pi(s_1(M)+\\dots+s_r(M)),\\mathcal O_C)) = \\\\ \\\\\n-2\\Delta + \\frac{1}{r-1}\\sum_{(A,B),\\ 1\\in A} \\#B(r-\\#B)\n\\Delta_{(A,B)}.\n\\end{array}\n$$\n\\end{lem}\n\n\\begin{proof}\nThere is a short exact sequence,\n$$\n\\begin{CD}\n0 @>>> \\Omega_\\pi @>>> \\Omega_\\pi(s_1(M)+\\dots+s_r(M))\n@>>> \\oplus_{i=1}^r (s_i)_*\\mathcal O_M @>>> 0.\n\\end{CD}\n$$\nCombining this with Lemma~\\ref{lem-Omega}, Lemma~\\ref{lem-rel8}, and\nchasing through exact sequences gives the lemma.\n\\end{proof}\n\n\\section{The virtual canonical bundle} \\label{sec-vircan}\n\\marpar{sec-vircan}\n\n\\noindent\nLet $k$ be a field, let $X$ be a connected, smooth algebraic space\nover $k$ of dimension $n$, let\n$M$ be an Artin stack over $k$, let $\\pi:C\\rightarrow M$ be a flat\n1-morphism, \nrelatively representable by proper algebraic spaces whose geometric\nfibers are connected, at-worst-nodal curves of arithmetic genus $0$,\nlet $s_1,\\dots,s_r:M\\rightarrow C$ be pairwise disjoint sections with\nimage contained in the smooth locus of $\\pi$ (possibly $r=0$, i.e.,\nthere are no sections), and let $f:C\\rightarrow\nX$ be a 1-morphism of $k$-stacks.  In this setting, Behrend and\nFantechi introduced a perfect complex $E^\\bullet$ on $M$ of amplitude\n$[-1,1]$ and a morphism to the cotangent complex,\n$\\phi:E^\\bullet \\rightarrow L_M^\\bullet$, ~\\cite{BM}.  \nIf $\\text{char}(k)=0$ and $M$ is the Deligne-Mumford stack of stable\nmaps to $X$, Behrend and Fantechi prove $E^\\bullet$ has amplitude\n$[-1,0]$,\n$h^0(\\phi)$ is an isomorphism\nand $h^{-1}(\\phi)$ is surjective.  In\nmany interesting cases, $\\phi$ is a quasi-isomorphism.  Then\n$\\text{det}(E^\\bullet)$ is an invertible \ndualizing sheaf for $M$. \nBecause of this, \n$\\text{det}(E^\\bullet)$ is called the \\emph{virtual canonical\n  bundle}.  In this section the relations from Section~\\ref{sec-div}\nare used to give a formula for the divisor class of the virtual\ncanonical bundle.  Hypothesis ~\\ref{hyp-GRR} holds for $\\pi$.\n\n\\medskip\\noindent\nDenote by $L_{(\\pi,f)}$ the cotangent complex of the morphism\n$(\\pi,f):C\\rightarrow M\\times X$.  This is a perfect complex of\namplitude $[-1,0]$.  There is a distinguished triangle,\n$$\n\\begin{CD}\nL_\\pi @>>> L_{(\\pi,f)} @>>> f^* \\Omega_X[1] @>>> L_\\pi[1].\n\\end{CD}\n$$\nThere is a slight variation $L_{(\\pi,f,s)}$ taking into account the\nsections which fits into a distinguished triangle,\n$$\n\\begin{CD}\nL_\\pi(s_1(M)+\\dots+s_r(M)) @>>> L_{(\\pi,f,s)} @>>> f^*\\Omega_X[1] @>>>\nL_\\pi(s_1(M)+\\dots+s_r(M))[1]. \n\\end{CD}\n$$\nThe complex $E^\\bullet$ is defined\nto be $(R\\pi_*(L_{(\\pi,f,s)}^\\vee)[1])^\\vee$, where $(F^\\bullet)^\\vee$\nis $R\\textit{Hom}(F^\\bullet,\\mathcal O)$.  In particular,\n$\\text{det}(E^\\bullet)$ is the determinant of\n$R\\pi_*(L_{(\\pi,f,s)}^\\vee)$.  From the distinguished triangle,\n$\\text{det}(E^\\bullet)$ is\n$$\n\\text{det}(R\\pi_*\nR\\textit{Hom}_{\\mathcal O_C}(\\Omega_\\pi(s_1(M)+\\dots+s_r(M)),\\mathcal O_C)) \n\\otimes \\text{det}(R\\pi_* f^*T_X)^\\vee.\n$$\nBy Lemma~\\ref{lem-rel11}, the first term is known.  \nThe second term\nfollows easily from Grothendieck-Riemann-Roch.\n\n\\begin{lem} \\label{lem-TX}\n\\marpar{lem-TX}\nAssume that the relative degree of $f^*C_1(\\Omega_X)$ is nonzero.\nThen $R\\pi_*f^*T_X[-1]$ has rank $\\langle -f^*C_1(\\Omega_X),\\beta\n\\rangle + n$, and up to torsion the first Chern class of the\ndeterminant is,\n$$\n\\begin{array}{c}\n\\frac{1}{2\\langle -f^*C_1(\\Omega_X),\\beta \\rangle} \\left[ 2\\langle\n-f^*C_1(\\Omega_X), \\beta \\rangle \\pi_* f^* C_2(\\Omega_X) \\right. \\\\\n\\\\\n - (\\langle\n-f^* C_1(\\Omega_X),\\beta \\rangle + 1) \\pi_* f^* C_1(\\Omega_X)^2 +\n\\\\ \\\\\n\\left.\n{\\sum}' \\langle -f^*C_1(\\Omega_X),\\beta' \\rangle \\langle\n-f^*C_1(\\Omega_X),\\beta'' \\rangle \\Delta_{\\beta',\\beta''}\\right].\n\\end{array}\n$$\n\\end{lem}\n\n\\begin{proof}\nThe Todd class $\\tau_\\pi$ of $\\pi$ is $1 - \\frac{1}{2}C_1(\\omega_\\pi)\n+ \\tau_2 + \\dots$, \nwhere $\\pi_*\\tau_2 = 0$.  The Chern character of $f^*T_X$ is,\n$$\nn - f^*C_1(\\Omega_X) + \\frac{1}{2}(f^*C_1(\\Omega_X)^2 -\n2f^*C_2(\\Omega_X)) + \\dots\n$$\nTherefore $\\text{ch}(f^*T_X)\\cdot \\tau_\\pi$ equals,\n$$\nn -\n\\left[f^*C_1(\\Omega_X)+\\frac{n}{2}C_1(\\Omega_\\pi) \\right] + \\frac{1}{2}\\left[\nf^*C_1(\\Omega_X)^2 - 2f^*C_2(\\Omega_X) + f^*C_1(\\Omega_X)\\cdot\nC_1(\\omega_\\pi) \\right] + n\\tau_2 + \\dots\n$$\nApplying $\\pi_*$ and using that $\\pi_*\\tau_2 = 0$, the rank is\n$n+\\langle -f^*C_1(\\Omega_X),\\beta \\rangle$, and the determinant has\nfirst Chern class,\n$$\n\\frac{1}{2}\\pi_*\\left[ f^*C_1(\\Omega_X)^2 - 2f^*C_2(\\Omega_X) \\right] +\n\\frac{1}{2} \\pi_*(f^*C_1(\\Omega_X)\\cdot C_1(\\omega_\\pi)).\n$$\nApplying Lemma~\\ref{lem-rel1} and simplifying gives the relation.\n\\end{proof}  \n\n\\begin{prop} \\label{prop-vc}\n\\marpar{prop-vc}\nThe rank of $E^\\bullet$ is $\\langle -f^*C_1(\\Omega_X),\\beta \\rangle +\nn + r - 3.$  The following divisor class reations hold modulo torsion.\nIf $\\langle -f^*C_1(\\Omega_X),\\beta \\rangle\\neq 0$ and $r=0$,\nthe first Chern class of the virtual canonical bundle\nis,\n\\begin{equation}\n\\begin{array}{c}\n\\frac{1}{2\\langle -f^*C_1(\\Omega_X),\\beta \\rangle} \\left[ 2\\langle\n-f^*C_1(\\Omega_X), \\beta \\rangle \\pi_* f^* C_2(\\Omega_X) \\right. \\\\\n\\\\\n- (\\langle\n-f^* C_1(\\Omega_X),\\beta \\rangle + 1) \\pi_* f^* C_1(\\Omega_X)^2 +\n\\\\ \\\\\n\\left.\n{\\sum}' (\\langle -f^*C_1(\\Omega_X),\\beta' \\rangle \\langle\n-f^*C_1(\\Omega_X),\\beta'' \\rangle -4\\langle -f^*C_1(\\Omega_X),\\beta\n\\rangle) \\Delta_{\\beta',\\beta''}\\right].\n\\end{array}\n\\end{equation}\nIf $\\langle -f^*C_1(\\Omega_X),\\beta \\rangle \\neq 0$ and $r=1$, \nthe first Chern class of the virtual canonical bundle is,\n\\begin{equation}\n\\begin{array}{c}\n\\frac{1}{2\\langle -f^*C_1(\\Omega_X),\\beta \\rangle} \\left[ 2\\langle\n-f^*C_1(\\Omega_X), \\beta \\rangle \\pi_* f^* C_2(\\Omega_X) \\right. \\\\\n\\\\\n- (\\langle\n-f^* C_1(\\Omega_X),\\beta \\rangle + 1) \\pi_* f^* C_1(\\Omega_X)^2 +\n\\\\ \\\\\n\\left.\n{\\sum}' (\\langle -f^*C_1(\\Omega_X),\\beta' \\rangle \\langle\n-f^*C_1(\\Omega_X),\\beta'' \\rangle -4\\langle -f^*C_1(\\Omega_X),\\beta\n\\rangle) \\Delta_{\\beta',\\beta''}\\right] \\\\\n\\\\\n-\\pi_*(s_1(M)\\cdot s_1(M)).\n\\end{array}\n\\end{equation}\nIf $\\langle -f^*C_1(\\Omega_X),\\beta \\rangle \\neq 0$ and $r\\geq 2$, \nthe first Chern class of the virtual canonical bundle is,\n\\begin{equation}\n\\begin{array}{c}\n\\frac{1}{2\\langle -f^*C_1(\\Omega_X),\\beta \\rangle} \\left[ 2\\langle\n-f^*C_1(\\Omega_X), \\beta \\rangle \\pi_* f^* C_2(\\Omega_X) \\right. \\\\\n\\\\\n- (\\langle\n-f^* C_1(\\Omega_X),\\beta \\rangle + 1) \\pi_* f^* C_1(\\Omega_X)^2 +\n\\\\ \\\\\n\\left.\n{\\sum}' (\\langle -f^*C_1(\\Omega_X),\\beta' \\rangle \\langle\n-f^*C_1(\\Omega_X),\\beta'' \\rangle -4\\langle -f^*C_1(\\Omega_X),\\beta\n\\rangle) \\Delta_{\\beta',\\beta''}\\right] \\\\\n\\\\\n+ \\frac{1}{r-1} \\sum_{(A,B), 1\\in\n  A} \\#B(r-\\#B)\\Delta_{(A,B)}.\n\\end{array}\n\\end{equation}\n\\end{prop}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Background}\n\nThe Parkes-MIT-NRAO (PMN) 4.85~GHz survey (Griffith~\\&~Wright~1993) covered the entire southern sky to a limiting flux density of 30~mJy (varying slightly with declination); it increased the number of known southern radio sources by a factor of approximately 6. Wright et al. (1996) made simultaneous 4.80\/8.64~GHz Australia Telescope Compact Array (ATCA) measurements of a complete sample of 8068 PMN sources defined by the flux density limits $S_{4.85~\\rm{GHz}} \\geq 70$~mJy $(-73^\\circ < \\delta < -38^\\circ.5)$, $S_{4.85~\\rm{GHz}} \\geq 50$~mJy $(-87^\\circ < \\delta < -73^\\circ)$ and the requirement that galactic latitude $|b| \\geq 2^\\circ$. The Wright et al. ATCA observations resulted in: positions accurate to (standard error) $0.6''$; spectral data and structural information of resolution approximately $1''$. This dataset is the largest deep, complete, unbiased, dual-frequency, interferometric catalogue of southern radio sources. It covers $19~\\%$ of the sky, yet this area contains only 1 of the 39 previously known cases of gravitational lensing classed as \\emph{probable\\ lenses} by the \\emph{CASTLES}\\footnote{$\\rm{http:\/\/cfa-www.harvard.edu\/castles\/castles.edu}$} collaboration (Figure 1). The resolution ($\\sim 1''$) samples the peak of the image separation histogram for the 14 JVAS\/CLASS lenses published prior to these proceedings (Browne~et~al.,~1998, Koopmans~et~al.,~1999, Myers~et~al.,~1999). The 2178 flat--spectrum sources in these data constitute our finding survey.\n\n\nCandidate gravitational lenses were selected using an automated routine, based on objective criteria, designed to reduce sample contamination by radio galaxies and objects within, or behind, the Galactic plane. Radio galaxies are often observed to exhibit steepening radio spectral indices with increasing separation from the core (e.g. Wiita \\& Gopal-Krishna, 1990), objects showing such spectral steepening were not included in the sample.  Simulated ATCA observations of known northern hemisphere lenses (transposed to declinations within the surveyed region) were used to test and refine candidate selection.\n\nThe 110 flat-spectrum objects satisfying our selection criteria of component fluxes $S_{\\nu}> 5 \\sigma~(\\sim~30$~mJy) (at both 4.80 and 8.64~GHz) and $|b| \\geq 5^\\circ$ were chosen for further study at radio, infrared and optical wavelengths. New simultaneous 4.80\/8.64 GHz, dual linear polarisation ATCA observations of each lens candidate have been made. These new data (of resolution $\\sim 1''$ at 8.64~GHz) exhibit a factor 8 improvement in S:N, and vastly improved coverage of the interferometric u-v plane, over data in hand. Total intensity and polarisation maps of all objects have been made and examined. Re-running the automated lens candidate finder rejected half of the observed objects. We have obtained deep near infrared $K_N$-Band images of 35 sources remaining in the lens candidate sample, deep optical B-Band images (and colours) for 19 of these objects and 1 spectrum. \n\n\n\\begin{figure}\n\\caption{\\small{Aitoff projection showing the positions of known gravitational lenses, their discovery method and the sky coverage of the current major surveys for flat radio spectrum gravitational lenses. The hatched region is covered by this survey (the region shaded black represents the un-surveyed area of radius $3^\\circ$ centered on the southern celestial pole). The region bounded by solid black lines is the area ($0^\\circ > \\delta > -40^\\circ$, $|b| > 10^\\circ$) surveyed by the MIT group  (Winn et al., these proceedings), the regions of the northern sky for which $|b| > 10^\\circ$ (dashed line) are covered by CLASS (Browne et al., these proceedings).}}\n\\plotfiddle{oprouton1.eps}{5.15cm}{90}{40}{40}{125}{-50}\n\\end{figure}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\n\nBlandford and Payne \\cite{blandpayne} showed angular momentum transfer via large magnetic stresses, \nin the absence of alpha-viscosity, in the framework of self-similar\nKeplerian disk flows. Here, we show in a simpler 1.5-dimensional, vertically averaged\ndisk model that the Maxwell stresses due to strong magnetic fields are adequate enough \nfor angular momentum transfer even in advective accretion flows, without self-similar assumption, \ndescribing the hard spectral states of black hole sources. \n\nThe idea of exploring magnetic stress in order to explain astrophysical systems is not\nreally new. This was implemented, for example, in the solar wind which was understood to \nhave decreased Sun's angular momentum through the effect\nof magnetic stresses (see, e.g., Ref.~\\refcite{waber}), in the proto-stellar gas clouds \nwhich might have been contracted by magnetic effects \\cite{Mouschovias}. \nOzernoy and Usov \\cite{usov} and Blandford \\cite{bland} showed that the energy is possible to \nextract continuously by electromagnetic torques and\ntwisted field lines in accretion disks. By linear stability analysis of the accretion disks,\nit was shown by Cao and Spruit \\cite{cao} that angular momentum is possible to remove by the \nmagnetic torque exerted by a centrifugally driven wind. \nHowever, by solving the local vertical structure\nof a geometrically thin accretion disk threaded by a poloidal magnetic field, \nOgilvie and Livio \\cite{ogil} showed the shortcoming of launching an outflow and suggested\nfor an existence of additional source of energy for its successful launching.\n\nHere we demonstrate, semi-analytically, the effects of strong magnetic field, stronger than\nthat needed for magnetorotational instability (MRI) \\cite{sujit}, with \nplasma$-\\beta > 1$ yet, on to the vertically averaged advective accretion flows in \nvertical equilibrium in order to transport matter. \nTherefore, we consider the flow variables to depend on the radial coordinate only. \nAlthough, in reality, a non-zero vertical magnetic field should induce a vertical motion,\nin the platform of the present assumption, any vertical motion will be featured as an outward motion. \nIndeed, our aim here is to furnish removal of angular momentum from the flow via\nmagnetic stresses, independent of its vertical or outward transport.\n\n\n\n\n\\section{Basic model equations}\\label{model}\n\nWe describe optically thin, magnetized, viscous, axisymmetric, advective, vertically averaged, steady-state accretion\nflow, in the pseudo-Newtonian framework with the Mukhopadhyay \\cite{m02} potential. Hence, the equation \nof continuity, vertically averaged hydromagnetic equations for energy-momentum balance in different directions are \ngiven by (assuming that the dimensionless variables do not vary significantly in the vertical direction such that \n$\\partial\/ \\partial z \\sim s_i\/h$ and, as a consequence, the vertical component of velocity is zero),\n\\begin{eqnarray}\n\\nonumber\n&&\\dot{M}=4\\pi  x \\rho h \\vartheta,~~~ \n\\vartheta\\frac{d\\vartheta}{dx}+\\frac{1}{\\rho}\\frac{dP}{dx}-\\frac{\\lambda^{2}}{x^3}+F=\\frac{1}{4\\pi \\rho}\\left(B_x\\frac{dB_x}{dx}+s_1\\frac{B_xB_z}{h}-\\frac{B_{\\phi}^2}{x}\\right),\\\\\n\\nonumber\n&&\\vartheta\\frac{d\\lambda}{dx}=\\frac{1}{x\\rho}\\frac{d}{dx}\\left(x^2W_{x\\phi}\\right)+\\frac{x}{4\\pi \\rho}\\left(B_x\\frac{dB_{\\phi}}{dx}+s_2\\frac{B_zB_{\\phi}}{h}+\\frac{B_xB_{\\phi}}{x}\\right),\\\\\n&&\\frac{P}{\\rho h}=\\frac{Fh}{x}-\\frac{1}{4\\pi \\rho}\\left(B_x\\frac{dB_z}{dx}+s_3\\frac{B_z^2}{h}\\right),\\\\\n\\nonumber\n&&\\vartheta T\\frac{ds}{dx}=\\frac{\\vartheta}{\\Gamma_3-1}\\left(\\frac{dP}{dx}-\\frac{\\Gamma_1P}{\\rho}\\frac{d\\rho}{dx}\\right)\n=Q^+-Q^-=Q^+_{vis}+Q^+_{mag}-Q^-_{vis}-Q^-_{mag},\n\\end{eqnarray}\nwhere $W_{x\\phi}=\\alpha(P+\\rho \\vartheta^2)$ and $\\alpha$ the Shakura-Sunyaev viscosity parameter \\cite{ss}.\nNote that $s_1$, $s_2$ and $s_3$ are the degrees of vertical scaling for the radial, azimuthal and vertical components of the \nmagnetic field respectively. \nHere $\\dot{M}$ is the \nconserved mass accretion rate, $\\rho$ the mass density of the flow, $\\vartheta$ the radial velocity, $P$ the total \npressure including the magnetic contribution, $F$ the force corresponding to the pseudo-Newtonian potential for \nrotating black holes \\cite{m02}, $\\lambda$ the angular momentum per unit mass, $W_{x\\phi}$ the viscous shearing stress written \nfollowing the Shakura-Sunyaev prescription \\cite{ss} with appropriate modification \\cite{mg03},\n$h \\sim z$, the half-thickness and $x$ the radial coordinate of the disk, when both of \nthem are expressed in units of $GM\/c^2$, where $G$ the Newton's gravitation constant, $M$ the mass of black hole, \n$c$ the speed of light, $s$ the entropy per unit volume, \n$T$ the (ion) temperature of the flow, $Q^+$ and $Q^-$ are the net rates of energy released and radiated out per unit \nvolume in\/from the flow respectively (when $Q^+_{vis}$, $Q^+_{mag}$, $Q^-_{vis}$, $Q^-_{mag}$ are the respective \ncontributions from viscous and magnetic parts). All the variables are made dimensionless in the spirit of \ndimensionless $x$ and $z$. We further assume, for the present purpose, the heat radiated \nout proportional to the released rate with the proportionality constants $(1 - f_{vis})$ and $(1 - f_m)$, respectively, \nfor viscous and magnetic parts of the radiations. $\\Gamma_1$, $\\Gamma_3$, which are functions of polytropic constant $\\gamma$, \nindicate the polytropic indices depending \non the gas and radiation content in the flow (see, e.g., Ref.~\\refcite{rm1}, for exact expressions) \nand $B_x$, $B_{\\phi}$ and $B_z$ are the components of magnetic field. The model for $Q^+_{vis}$ is taken from the \nprevious work \\cite{rm1}, and the relation for $Q^+_{mag}$ is taken from that by Bisnovatyi-Kogan and \nRuzmaikin \\cite{bis}.\n\nHydromagnetic flow equations must be supplemented by (for the present purpose, steady-state) equations of induction\nand no magnetic monopole at the limit of very large Reynolds number, given by\n\\begin{eqnarray}\n\\nabla \\times \\vec{v} \\times \\vec{B}=0,~~~\n\\frac{d}{dx}(xB_x)+s_3\\frac{B_z}{h}=0,\n\\end{eqnarray}\nwhen $\\vec{v}$ and $\\vec{B}$ are respectively the velocity and magnetic field vectors and $\\nu_m$ is the magnetic \ndiffusivity. \n\n\\section{Solutions and Results}\\label{sol}\n\nWe take into account two situations. (1) Flows with a relatively higher $\\dot{M}$ and, hence, lower $\\gamma$, \nmodelled around stellar mass black holes: such flows may or may not form Keplerian accretion disks. (2) Flows with \na lower $\\dot{M}$ and, hence, higher $\\gamma$, modelled around supermassive black holes: such flows are necessarily \nhot gas dominated advective (or advection dominated) accretion flows.\n\nWe find that the flows with plasma-$\\beta > 1$, but $\\alpha=0$, exhibit adequate matter transport, as efficient as the \n$\\alpha$-viscosity with $\\alpha= 0.08$, but without magnetic stresses, would do. This is interesting as the origin of $\\alpha$ (and the corresponding \ninstability and turbulence) is\nitself not well understood. The maximum required large scale magnetic field is $\\sim 10^5$G in a disk\naround $10M_{\\odot}$ black holes and $\\sim10$G in a disk around $10^7M_{\\odot}$ supermassive black holes,\nwhere $M_\\odot$ is solar mass. \nThe presence of such a field, in particular for a stellar mass black hole disk when the binary companion supplying \nmass is a Sun-like star with\nthe magnetic field on average 1G, may be understood, if the field is approximately frozen with the disk fluids (or the\nsupplied fluids from the companion star remain approximately frozen with the magnetic field) or disk fluids exhibit very\nlarge Reynolds number. Indeed, all the present computations are done at the limit of large Reynolds number, as really\nis the case in accretion flows, such that the term associated with the magnetic diffusivity in the induction equation is\nneglected. The size of a disk around supermassive black holes is proportionately larger compared to that around\na stellar mass black hole. Hence, from the equipartition theory, indeed the magnetic field is expected to be decreased\nhere compared to that around stellar mass black holes. Figure 1 shows a typical set of accretion solutions and confirms that\nflows around a stellar mass black hole with $\\alpha$-viscosity but without large scale magnetic fields (viscous flow)\nshow similar behavior to that with large scale magnetic fields \nwithout $\\alpha$-viscosity (magnetic flow). The flows around a supermassive black hole show very similar features, except\nwith reduced field strength. For other details, see Ref.~\\refcite{kou}.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[angle=0,width=2.5in]{f1}\n\\end{center}\n\\caption{\n(a) Mach number, (b) angular momentum per unit mass in $GM\/c$, (c) inverse of plasma-$\\beta$,\n(d) azimuthal component of magnetic field in G,\nwhen solid and dotted lines are for magnetic flows around\nSchwarzschild ($a=0, \\lambda_c=3.2$) and Kerr ($a=0.998, \\lambda_c=1.8$) black holes respectively,\nand dashed and long-dashed lines are for viscous flows\naround Schwarzschild ($a=0, \\alpha=0.017, \\lambda_c=3.15$) and Kerr ($a=0.998, \\alpha=0.012, \\lambda_c=1.8$)\nblack holes respectively, where $a$ is the dimensionless spin parameter of black hole and $\\lambda_c$ the quantity at critical radius.\nOther parameters are $M=10M_\\odot$, $\\dot{M}=0.1$ Eddington rate,\npolytropic constant $\\gamma=1.335, f_{vis}=f_m=0.5, s_2=-0.5$.\n}\n\\label{stmasm}\n\\end{figure}\n\n\nLet us now explore in more details, how exactly various components of magnetic stress lead to\nangular momentum transfer in the flows. \nFigure \\ref{stmasstr}a shows that the stress component $B_x B_z$ around a Schwarzschild\nblack hole increases almost throughout\nas matter advances towards the black hole. This implies that the flow is prone\nto outflow through the field lines, which effectively helps in\ninfalling matter towards the black hole. However, in the near vicinity of black hole, $B_x B_z$ decreases,\nas indeed outflow is not possible therein. \nIn this flow zone, the angular momentum becomes very small which practically does\nnot affect the infall. The magnitude of $B_\\phi B_z$ decreases till the\ninner region of accretion flow, implying the part of matter to be spiralling out\nand, hence, removing angular momentum leading to infall of matter.\nFinally, the magnitude of $B_x B_\\phi$\nincreases at a large and a small distances from the black hole (except around\nthe transition radius), which helps removing angular momentum and further infall. \nThis is the same as the Shakura-Sunyaev viscous stress would do with the increase of matter pressure.\nHowever, at the intermediate zone,\nthe angular momentum transfer through $B_x B_\\phi$ reverses and a part of\nthe matter outflows. At the Keplerian to sub-Keplerian transition zone,\ndue to the increase of flow thickness, matter is vertically kicked effectively,\nshowing a decrease of $B_x B_\\phi$. Majority of the features remain similar\nfor the flow around a rotating black hole, as shown in Fig. \\ref{stmasstr}b.\nHowever, a rotating black hole reveals a stronger\/efficient outflow\/jet in general. Hence, except at the inner zone,\n$B_x B_\\phi$ decreases throughout, which helps kicking the matter outwards by \ntransferring the angular momentum inwards. \n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[angle=0,width=2.5in]{f2}\n\\end{center}\n\\vskip-4cm\n\\caption{\nComponents of magnetic stress: $B_x B_\\phi$ (solid line),\n$B_x B_z$ (dotted line), $B_\\phi B_z$ (dashed line), for (a) Schwarzschild\nmagnetic flow of Fig. \\ref{stmasm}, (b) Kerr magnetic flow of Fig. \\ref{stmasm}.\n}\n\\label{stmasstr}\n\\end{figure}\n\nDifferent components of the magnetic stress tensor have different roles: $B_xB_{\\phi}$ controls the infall in the disk plane,\nwhereas $B_{\\phi}B_z$ renders the flow to spiral outwards and, hence, outflow. Moreover, $B_xB_z$ helps to kick the \nmatter out vertically. Larger the field strength, larger is the power of magnetic stresses. Interestingly, \nthe magnitude of magnetic\nfield decreases, as the steady-state matter advances towards the black hole. This is primarily because $B_{\\phi}B_z$ \n(and also $B_xB_{\\phi}$ for a rotating black hole) decreases inwards almost entirely in order to induce inflow\nvia angular momentum transfer through outflow. \nThis further reveals\na decreasing $|B_{\\phi}|$ as the output of self-consistent solutions of the coupled set of equations.\n\n\n\\section{Discussion and Conclusions}\\label{last}\n\nIs there any observational support for the existence of such a magnetic field, as required for the magnetic accretion\nflows discussed here? Interestingly, the polarization measurements in the hard state of Cyg X-1 imply that it should\nhave at least 10mG field at the source of emission \\cite{lau}. In order to explain such high polarization, a\njet model was suggested by Zdziarski et al. \\cite{zd}, which requires a magnetic field $\\sim (5 - 10) \\times 10^5$G \nat the base of jet and hence in the underlying accretion disk. \n\nIn the present computations, we have assumed the flow to be vertically averaged without allowing any vertical\ncomponent of the flow velocity. The most self-consistent approach,\nin order to understand vertical transport of matter through the magnetic effects which in turn leads to the radial\ninfall of rest of the matter, is considering the flow to be moving in the vertical direction from the disk plane as well.\nSuch an attempt, in the absence of magnetic and viscous effects, was made earlier by one of the present authors \\cite{deb} in the model\nframework of coupled disk-outflow systems. In such a framework, the authors further showed that the outflow\npower of the correlated disk-outflow systems increases with the increasing spin of black holes. Our future goal is now\nto combine that model with the model of present work, so that the coupled disk-outflow systems can be investigated\nmore self-consistently and rigorously, when the magnetic field plays indispensable role in order to generate vertical\nflux in the three-dimensional flows.\n\n\n\n\n\\section*{Acknowledgments}\nK.C. thanks the Academies' Summer Research Fellowship Programme of India for offering him a Fellowship to\npursue his internship in Indian Institute of Science, Bangalore, \nwhen most of the calculations of this project were done.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecently, high resolution X-ray images using {\\it Chandra} have\nrevealed 49 point sources in the Antennae \\citep{zez02a}.  We will\nassume a distance to the Antennae of 19.3 Mpc (for $H_{0}$=75 km\ns$^{-1}$ Mpc$^{-1}$), which implies 10 sources have X-ray luminosities\ngreater than $10^{39}$ ergs s$^{-1}$.  Considering new observations of\nred giant stars in the Antennae indicate a distance of 13.8 Mpc\n\\citep{sav04}, we point out this ultraluminous X-ray source population\ncould decrease by roughly a half.  Typically, masses of black holes\nproduced from standard stellar evolution are less than $\\sim20$ \n$M_{\\odot}$ \\citep[e.g.,][]{fry01}.  The Eddington luminosity limit\nimplies that X-ray luminosities $>10^{39}$ ergs s$^{-1}$ correspond to\nhigher-mass objects not formed from a typical star.  Several authors\n\\citep[e.g.,][]{fab89,zez99,rob00,mak00} suggest these massive ($10\n\\sbond 1000$ $M_{\\sun}$) compact sources outside galactic nuclei are\nintermediate mass black holes (IMBHs), a new class of BHs.  While\nIMBHs could potentially explain the observed high luminosities, other\ntheories exist as well, including beamed radiation from a stellar mass\nBH \\citep{kin01}; super-Eddington accretion onto lower-mass objects\n\\citep[e.g.,][]{moo03,beg02}; or supernovae exploding in dense\nenvironments \\citep{ple95,fab96}.\n\nCompact objects tend to be associated with massive star formation,\nwhich is strongly suspected to be concentrated in young stellar\nclusters \\citep{lad03}.  Massive stars usually end their lives in\nsupernovae, producing a compact remnant.  This remnant can be kicked\nout of the cluster due to dynamical interactions, stay behind after\nthe cluster evaporates, or remain embedded in its central regions.\nThis last case is of particular interest to us as the compact object\nis still {\\it in situ}, allowing us to investigate its origins via the\nambient cluster population.  The potential for finding such\nassociations is large in the Antennae due to large numbers of both\nX-ray point sources and super star clusters; a further incentive for\nstudying these galaxies.\n\nIn \\citet[henceforth Paper I]{bra05} we presented $J$ and $K_s$\nphotometry of $\\sim220$ clusters in the Antennae.  Analysis of ($J -\nK_s$) colors indicated that many clusters in the overlap region suffer\nfrom 9--10 mag of extinction in the $V$-band.  This result contrasts\nwith previous work by \\citet{whi02} who associated optical sources\nwith radio counterparts in the Antennae \\citep{nef00} and argued that\nextinction is not large in this system.  Here, we continue our\nanalysis of these Antennae IR images by making a frame-tie between the\nIR and {\\it Chandra} X-ray images from \\citet{fab01}.  Utilizing the\nsimilar dust-penetrating properties of these wavelengths, we\ndemonstrate the power of this approach to finding counterparts to\nX-ray sources.  By comparing the photometric properties of clusters\nwith and without X-ray counterparts, we seek to understand the cluster\nenvironments of these X-ray sources.  In \\S2 we discuss the IR\nobservations of the Antennae. \\S3 explains our matching technique and\nthe photometric properties of the IR counterparts.  We conclude with a\nsummary of our results in \\S4.\n\n\\section{Observations and Data Analysis}\n\n\\subsection{Infrared Imaging}\n\nWe obtained near-infrared images of NGC 4038\/9 on 2002 March 22 using\nthe Wide-field InfraRed Camera (WIRC) on the Palomar 5-m telescope.\nAt the time of these observations, WIRC had been commissioned with an\nunder-sized HAWAII-1 array (prior to installation of the full-sized\nHAWAII-2 array in September 2002), providing a $\\sim 4.7 \\times\n4.7$-arcminute field of view with $\\sim0\\farcs25$ pixels (``WIRC-1K''\n-- see \\citet{wil03} for details).  Conditions were non-photometric\ndue to patches of cloud passing through.  Typical seeing-limited\nimages had stellar full-width at half-maximum of $1\\farcs0$ in $K_s$\nand $1\\farcs3$ in $J$.  We obtained images in both the $J$- ($1.25\n\\mu$m) and $K_s$-band ($2.15 \\mu$m) independently.  The details of the\nprocessing used to obtain the final images are given in Paper 1.\n\n\\subsection{Astrometric Frame Ties}\n\nThe relative astrometry between the X-ray sources in NGC 4038\/9 and\nimages at other wavelengths is crucial for successful identification\nof multi-wavelength counterparts.  Previous attempts at this have\nsuffered from the crowded nature of the field and confusion between\npotential counterparts \\citet{zez02b}.  However, the infrared waveband\noffers much better hopes for resolving this issue, due to the similar\ndust-penetrating properties of photons in the {\\it Chandra} and $K_s$\nbands.  (See also \\citet{bra05} for a comparison of IR extinction to\nthe previous optical\/radio extinction work of \\citet{whi02}.)  We thus\nproceeded using the infrared images to establish an astrometric\nframe-tie, i.e. matching {\\it Chandra} coordinates to IR pixel\npositions.\n\nAs demonstrated by \\citet{bau00}, we must take care when searching for\nX-ray source counterparts in crowded regions such as the Antennae.\nTherefore, our astrometric frame-tie used a unique approach based on\nsolving a two-dimensional linear mapping function relating right\nascension and declination coordinates in one image with x and y pixel\npositions in a second image.  The solution is of the form:\n\n\\begin{eqnarray}\nr_1 = ax_1+by_1+c, \\\\\nd_1 = dx_1+dy_1+f\n\\end{eqnarray}\n\nHere $r_1$ and $d_1$ are the right ascension and declination,\nrespectively, for a single source in one frame corresponding to the\n$x_1$ and $y_1$ pixel positions in another frame.  This function\nconsiders both the offset and rotation between each frame.  Since we\nare interested in solving for the coefficients $a$--$f$, elementary\nlinear algebra indicates we need six equations or three separate\nmatches.  Therefore, we need at least three matches to fully describe\nthe rms positional uncertainty of the frame-tie.\n\nWe first used the above method to derive an approximate astrometric\nsolution for the WIRC $K_s$ image utilizing the presence of six\nrelatively bright, compact IR sources which are also present in images\nfrom the 2-Micron All-Sky Survey (2MASS).  We calculated pixel\ncentroids of these objects in both the 2MASS and WIRC images, and used\nthe 2MASS astrometric header information to convert the 2MASS pixel\ncentroids into RA and Dec.  These sources are listed in Table 1.\nApplying these six matches to our fitting function we found a small\nrms positional uncertainty of $0\\farcs2$, which demonstrates an\naccurate frame-tie between the 2MASS and WIRC images.\n\nUsing the 2MASS astrometric solution as a baseline, we identified\nseven clear matches between {\\it Chandra} and WIRC sources, which had\nbright compact IR counterparts with no potentially confusing sources\nnearby (listed in Table 2).  We then applied the procedure described\nabove, using the {\\it Chandra} coordinates listed in Table 1 of\n\\citet{zez02a} (see that reference for details on the {\\it Chandra}\nastrometry) and the WIRC pixel centroids, and derived the astrometric\nsolution for the IR images in the X-ray coordinate frame.  For the 7\nmatches, we find an rms residual positional uncertainty of $\\sim\n0\\farcs5$ which we adopt as our $1 \\sigma$ position uncertainty.  We\nnote that the positional uncertainty is an entirely {\\it empirical}\nquantity.  It shows the achieved uncertainty in mapping a target from\none image reference frame to the reference frame in another band, and\nautomatically incorporates all contributing sources of uncertainty in\nit.  These include, but are not limited to, systematic uncertainties\n(i.e. field distortion, PSF variation, etc. in both {\\it Chandra} and\nWIRC) and random uncertainties (i.e. centroid shifts induced by photon\nnoise, flatfield noise, etc. in both {\\it Chandra} and WIRC).  Thus,\ngiven the empirical nature of this uncertainty, we expect it to\nprovide a robust measure of the actual mapping error -- an expectation\nwhich seems to be borne out by the counterpart identification in the\nfollowing section.\n\nTo further test the accuracy of our astrometric solution we explored the range\nin rms positional uncertainties for several different frame-ties.\nSpecifically we picked ten IR\/X-ray matches separated by\n$<$1$\\arcsec$, which are listed in Table 3 (see \\S3.1).  Of these ten\nwe chose 24 different combinations of seven matches resulting in 24\nunique frame-ties.  Computing the rms positional uncertainty for each\nwe found a mean of 0$\\farcs$4 with a 1$\\sigma$ uncertainty of\n0$\\farcs$1.  Considering the rms positional uncertainty for the\nframe-tie used in our analysis falls within 1$\\sigma$ of this mean\nrms, this indicates we made an accurate astrometric match between the\nIR and X-ray.\n\n\n\\subsection{Infrared Photometry}\n\nWe performed aperture photometry on 222 clusters in the $J$-band and\n221 clusters in the $K_s$-band (see also Paper 1).  We found that the\nfull width at half maximum (FWHM) was 3.5 pixels ($0\\farcs9$) in the\n$K_s$ image and 4.6 pixels ($1\\farcs2$) in the $J$ band.  We used a\nphotometric aperture of 5-pixel radius in $K_s$ band, and 6-pixel\nradius in $J$ band, corresponding to $\\sim 3\\sigma$ of the Gaussian\nPSF.\n\nBackground subtraction is both very important and very difficult in an\nenvironment such as the Antennae due to the brightness and complex\nstructure of the underlying galaxies and the plethora of nearby\nclusters.  In order to address the uncertainties in background\nsubtraction, we measured the background in two separate annuli around\neach source: one from 9 to 12 pixels and another from 12 to 15 pixels.\nDue to the high concentration of clusters, crowding became an issue.\nTo circumvent this problem, we employed the use of sky background arcs\ninstead of annuli for some sources.  These were defined by a position\nangle and opening angle with respect to the source center.  All radii\nwere kept constant to ensure consistency.  In addition, nearby bright\nsources could shift the computed central peak position by as much as a\npixel or two.  If the centroid position determined for a given source\ndiffered significantly ($>1$ pixel) from the apparent brightness peak\ndue to such contamination, we forced the center of all photometric\napertures to be at the apparent brightness peak.  For both annular\nregions, we calculated the mean and median backgrounds per pixel.\n\nMultiplying these by the area of the central aperture, these values\nwere subtracted from the flux measurement of the central aperture to\nyield 4 flux values for the source in terms of DN.  Averaging these\nfour values provided us with a flux value for each cluster.  We\ncomputed errors by considering both variations in sky background,\n$\\sigma_{sky}$, and Poisson noise, $\\sigma_{adu}$.  We computed\n$\\sigma_{sky}$ by taking the standard deviation of the four measured\nflux values.  We then calculated the expected Poisson noise by scaling\nDN to $e^-$ using the known gain of WIRC \\citep[2 $e^{-1}$\nDN$^{-1}$,][]{wil03} and taking the square root of this value.  We\nadded both terms in quadrature to find the total estimated error in\nphotometry.\n\nWe then calibrated our photometry using the bright 2MASS star in Table 1.\n\n\\section{Results and Discussion}\n\n\\subsection{Identification of IR Counterparts to {\\it Chandra} Sources}\n\nWe used the astrometric frame-tie described above to identify IR\ncounterpart candidates to {\\it Chandra} X-ray sources in the WIRC\n$K_s$ image.  We restricted our analysis to sources brighter than\n$K_s\\sim19.4$ mag.  This is our $K_s$ sensitivity limit which we\ndefine in our photometric analysis below (see \\S3.2.1).  Using the\n$0\\farcs5$ rms positional uncertainty of our frame-tie, we defined\ncircles with 2$\\sigma$ and 3$\\sigma$ radii around each {\\it Chandra}\nX-ray source where we searched for IR counterparts (Figure 1).  If an\nIR source lay within a $1\\farcs0$ radius ($2\\sigma$) of an X-ray\nsource, we labeled these counterparts as ``strong''.  Those IR sources\nbetween $1\\farcs0$ and $1\\farcs5$ ($2-3 \\sigma$) from an X-ray source\nwe labeled as ``possible'' counterparts.  We found a total of 13\nstrong and 6 possible counterparts to X-ray sources in the Antennae.\nThese sources are listed in Table 3 and shown in Figure 3.  Of the 19\nX-ray sources with counterparts, two are the nuclei \\citep{zez02a},\none is a background quasar \\citep{cla05}, and two share the same IR\ncounterpart.  Therefore, in our analysis of cluster properties, we\nonly consider the 15 IR counterparts that are clusters.  (While\nX-42 has two IR counterparts, we chose the closer, fainter cluster for\nour analysis.)\n\nWe then attempted to estimate the level of ``contamination'' of these\nsamples due to chance superposition of unrelated X-ray sources and IR\nclusters.  This estimation can be significantly complicated by the\ncomplex structure and non-uniform distribution of both X-ray sources\nand IR clusters in the Antennae, so we developed a simple, practical\napproach.  Given the $<0\\farcs5$ rms residuals in our relative\nastrometry for sources in Table 2, we assume that any IR clusters\nlying in a background annulus with radial size of\n$2\\farcs0$--$3\\farcs0$ ($4-6\\sigma$) centered on all X-ray source\npositions are chance alignments, with no real physical connection (see\nFigure 1).  Dividing the total number of IR sources within the\nbackground annuli of the 49 X-ray source positions by the total area\nof these annuli, we find a background IR source surface density of\n0.02 arcsecond$^{-2}$ near {\\it Chandra} X-ray sources.  Multiplying\nthis surface density by the total area of all ``strong'' regions\n($1\\farcs0$ radius circles) and ``possible'' regions ($1\\farcs0$ --\n$1\\farcs5$ annuli) around the 49 X-ray source positions, we estimated\nthe level of source contamination contributing to our ``strong'' and\n``possible'' IR counterpart candidates.  We expect two with a\n$1\\sigma$ uncertainty of +0.2\/-0.1\\footnote{Found using confidence\nlevels for small number statistics listed in Tables 1 and 2 of\n\\citet{geh86}.} of the 13 ``strong'' counterparts to be due to chance\nsuperpositions, and three with a $1\\sigma$ uncertainty of\n+0.5\/-0.3\\footnotemark[\\value{footnote}] of the six ``possible''\ncounterparts to be chance superpositions.\n\nThis result has several important implications.  First of all, it is\nclear that we have a significant excess of IR counterparts within\n$1\\farcs0$ of the X-ray sources -- 13, where we expect only two in the\nnull hypothesis of no physical counterparts.  Even including the\n``possible'' counterparts out to $1\\farcs5$, we have a total of 19\ncounterparts, where we expect only five from chance superposition.\nSecondly, this implies that for any given ``strong'' IR counterpart,\nwe have a probability of $\\sim$ 85\\% ($11\/13$ with a $1\\sigma$\nuncertainty of 0.3\\footnotemark[\\value{footnote}]) that the\nassociation with an X-ray source is real.  Even for the ``possible''\ncounterparts, the probability of true association is $\\sim$50\\%. These\nlevels of certainty are a tremendous improvement over the\nX-ray\/optical associations provided by \\citet{zez02b}, and are strong\nmotivators for follow-up multi-wavelength studies of the IR\ncounterparts.  Finally, we can also conclude from strong concentration\nof IR counterparts within $\\sim 1 \\arcsec$ of X-ray sources that the\nframe tie uncertainty estimates described above are reasonable.\n\nFigure 2 is a $4\\farcm3\\times4\\farcm3$ $K_s$ image of the Antennae\nwith X-ray source positions overlaid.  We designate those X-ray\nsources with counterparts using red circles.  Notice that those\nsources with counterparts lie in the spiral arms and bridge region of\nthe Antennae.  Since these regions are abundant in star formation,\nthis seems to indicate many of the X-ray sources in the Antennae are\ntied to star formation in these galaxies.\n\n\\subsection{Photometric Properties of the IR Counterparts}\n\n\\subsubsection{Color Magnitude Diagrams}\n\nUsing the 219 clusters that had both $J$ and $K_s$ photometry, we made\n$(J-K_s)$ versus $K_s$ color magnitude diagrams (Figure 4).  We\nestimated our sensitivity limit by first finding all clusters with\nsignal-to-noise $\\sim5\\sigma$.  The mean $J$ and $K_s$ magnitudes for\nthese clusters were computed separately and defined as cutoff values\nfor statistical analyzes.  This yielded 19.0 mag in $J$ and 19.4 mag\nin $K_s$.  We note that the X-ray clusters are generally bright in the\nIR compared to the general population of clusters.  While the IR\ncounterpart for one X-ray source (X-32) falls below our $J$-band\nsensitivity limit, its $K_s$ magnitude is still above our $K_s$\ncutoff.  Therefore, we retained this source in our analysis.\n\nWe then broke down the X-ray sources into three luminosity classes\n(Figure 4).  We took the absorption-corrected X-ray luminosities,\n$L_X$, as listed in Table 1 of \\citet{zez02a} for all sources of\ninterest.  These luminosities assumed a distance to the Antennae of 29\nMpc.  We used 19.3 Mpc (for $H_{0}$=75 km s$^{-1}$ Mpc$^{-1}$) instead\nand so divided these values by 2.25 as suggested in \\citet{zez02a}.\nWe defined the three X-ray luminosities as follows: Low Luminosity\nX-ray sources (LLX's) had $L_{X}$ $<$ 3$\\times$$10^{38}$ ergs\ns$^{-1}$, High Luminosity X-ray sources (HLX's) were between $L_{X}$\nof 3$\\times10^{38}$ergs s$^{-1}$ and 1$\\times10^{39}$ ergs\ns$^{-1}$, while $L_{X}$ $>$ 1$\\times10^{39}$ ergs s$^{-1}$ were\nUltra-Luminous X-ray Sources (ULX's).  In Figure 4 we designate each\nIR counterpart according to the luminosity class of its corresponding\nX-ray source.  There does not appear to be a noticeable trend in the\nIR cluster counterparts between these different groupings.\n\n\\subsubsection{Absolute K Magnitudes}\n\nTo further study the properties of these IR counterparts, we\ncalculated $M_{K_s}$ for all IR clusters.  We calculated reddening\nusing the observed colors, $(J-K_s)_{obs}$, (hence forth the ``color\nmethod'').  Assuming all clusters are dominated by O and B stars,\ntheir intrinsic $(J-K_s)$ colors are $\\sim$0.2 mag.  Approximating\nthis value as 0 mag, this allowed us to estimate $A_{K_s}$ as $\\simeq$\n$(J-K_s)_{obs}$\/1.33 using the extinction law defined in\n\\citet[hereafter CCM]{car89}.  Since these derived reddenings are\nbiased towards young clusters, they will lead to an overestimate of\n$M_{K_s}$ for older clusters.\n\nFor IR counterparts to X-ray sources, we also computed X-ray-estimated\n$A_{K_s}$ using the column densities, $N_{H}$, listed in Table 5 of\n\\citet{zez02a}.  Here, $N_{H}$ is derived by fitting both a Power Law\n(PL) and Raymond-Smith \\citep[RS][]{ray77} model to the X-ray spectra.\nUsing the CCM law, $A_{K_s}$ is defined as 0.12$A_{V}$.  Taking\n$A_{V}$ = $5\\times10^{-22}$ mag cm$^2$ $N_{H}$, we could then derive\n$A_{K_s}$.\n\nThen we compared $A_{K_s}$ calculated using the ``color method'' to\n$A_{K_s}$ found using the above two $N_{H}$ models.  We found the\n``color method'' matched closest to $N_{H}(PL)$ for all except one\n(the cluster associated with {\\it Chandra} source 32 as designated in\n\\citet{zez02b}).\n\nIn Figure 5, we plot histograms of the distribution of $K_s$-band\nluminosity, $M_{K_s}$.  Figure 5 displays all clusters as well as over\nplotting only those with X-ray counterparts.  Notice that the clusters\nwith associated X-ray sources look more luminous.  To study whether\nthis apparent trend in luminosity is real, we compared these two\ndistributions using a two-sided Kolmogorov-Smirnov (KS) test.  In our\nanalysis, we only included clusters below $M_{K_s}$ $<$ -13.2 mag.\nRestricting our study to sources with ``good'' photometry, we first\ndefined a limit in $K_s$, 18.2 mag, using the limiting $J$ magnitude,\n19.0 mag as stated above, and, since the limit in $K_s$ is a function\nof cluster color, the median $(J-K_s)$ of 0.8 mag.  Subtracting the\ndistance modulus to the Antennae, 31.4 mag, from this $K_s$ limit, we\ncomputed our cutoff in $M_{K_s}$.  Since all clusters with X-ray\nsources fall below this cutoff, our subsample is sufficient to perform\na statistical comparison.\n\nThe KS test yielded a D-statistic of 0.37 with a probability of\n$3.2\\times10^{-2}$ that the two distributions of clusters with and\nwithout associated X-ray sources are related.  Considering the\nseparate cluster populations as two probability distributions, each\ncan be expressed as a cumulative distribution.  The D-statistic is\nthen the absolute value of the maximum difference between each\ncumulative distribution.  This test indicates that those clusters with\nX-ray counterparts are more luminous than most clusters in the\nAntennae.\n\n\\subsubsection{Cluster Mass Estimates}\n\n\\citet{whi99} found 70\\% of the bright clusters observed with the {\\it\nHubble Space Telescope} have ages $<$20 Myr.  Therefore, in this study\nwe will assume all clusters are typically the same age, $\\sim$20 Myr.\nThis allows us to make the simplifying assumption that cluster mass is\nproportional to luminosity and ask: Does the cluster mass affect the\npropensity for a given progenitor star to produce an X-ray binary?  We\nestimated cluster mass using $K_s$ luminosity ($M_{K_s}$).  Since\ncluster mass increases linearly with flux (for an assumed constant age\nof all clusters), we converted $M_{K_s}$ to flux.  Using the data as\nbinned in the $M_{K_s}$ histogram (Figure 5), we calculated an average\nflux per bin.  By computing the fraction of number of clusters per\naverage flux, we are in essence asking what is the probability of\nfinding a cluster with a specific mass.  Since those clusters with\nX-ray sources are more luminous, we expect a higher probability of\nfinding an X-ray source in a more massive cluster.  As seen in Figure\n6, this trend does seem to be true.  Applying a KS-test between the\ndistributions for all clusters and those associated with X-ray sources\nfor clusters below the $M_{K_s}$ completeness limit defined in the\nprevious section, we find a D-statistic of 0.66 and a probability of\n$7.2\\times10^{-3}$ that they are the same.  Hence, the two\ndistributions are distinct, indicating it is statistically significant\nthat more massive clusters tend to contain X-ray sources.\n\nWhile we assume all clusters are $\\sim$20 Myr above, we note that the\nactual range in ages is $\\sim$1--100 Myr \\citep{whi99}.\nBruzual-Charlot spectral photometric models \\citep{bru03} indicate\nthat clusters in this age range could vary by a factor of roughly 100\nin mass for a given $K_s$ luminosity.  Thus, we emphasize that the\nanalysis above should be taken as suggestive rather than conclusive\nevidence, and note that in a future paper (Clark, et al. 2006, in\npreparation) we explore this line of investigation and the impacts of\nage variations on the result in depth.\n\n\\subsubsection{Non-detections of IR Counterparts to X-ray Sources}\n\nTo assess whether our counterpart detections were dependent of\nreddening or their intrinsic brightness, we found limiting values for\n$M_{K_s}$ for those X-ray sources without detected IR counterparts.\nWe achieved this by setting all clusters $K_s$ magnitudes equal to our\ncompleteness limit defined for the CMDs (19.4 mag; see \\S3.2.1) and\nthen finding $M_{K_s(lim)}$ for each using $A_{K_s}$ calculated for\nthat cluster.  Since $M_{K_s(lim)}$ is theoretical and only depends on\nreddening, we could now find this limit for all X-ray sources using an\n$A_{K_s}$ estimated from the observed $N_{H}$ values.  Thus we\nconsidered all IR counterparts (detections) and those X-ray sources\nwithout a counterpart (nondetections).  If nondetections are due to\nreddening there should not exist a difference in $M_{K_s(lim)}$\nbetween detections and nondetections.  In contrast, if nondetections\nare intrinsically fainter, we expect a higher $M_{K_s(lim)}$ for these\nsources.  In the case of detections, we considered reddening from both\nthe ``color method'' and the $N_{H}(PL)$ separately.  We could only\nderive nondetections using $N_{H}(PL)$ reddening.  Figure 7 shows\n$M_{K_s(lim)}$ appears higher for all nondetections. To test if this\nobservation is significant, we applied a KS-test to investigate\nwhether detections and nondetections are separate distributions.  We\nfind a D-statistic of 0.82 and probability of $8.8\\times10^{-6}$ that\nthese two distributions are the same using the ``color method'' for\ndetections. Considering the $N_{H}(PL)$ reddening method for\ndetections instead, the D-statistic drops to 0.48 and the probability\nincreases to $3.9\\times10^{-2}$.  Since both tests indicate these\ndistributions are distinct, the observed high $M_{K_s(lim)}$ for\nnondetections seems to be real.  This leads to the conclusion that\nthese sources were undetected because they are intrinsically IR-faint,\nand that reddening does not play the dominant role in nondetections.\n\nWe summarize these statistics in Table 4.  Here we calculated the mean\n$K_s$, $(J-K_s)$, and $M_{K_s}$ for three different categories: 1) all\nclusters, 2) clusters only connected with X-ray sources, and 3) these\nclusters broken down by luminosity class.  We also include\nuncertainties in each quantity.  Notice that the IR counterparts\nappear brighter in $K_s$ and intrinsically more luminous than most\nclusters in the Antennae, although there is no significant trend in\ncolor.  We also summarize the above KS-test results in Table 5.\n\n\\section{Conclusions}\n\nWe have demonstrated a successful method for finding counterparts to\nX-ray sources in the Antennae using IR wavelengths.  We mapped {\\it\nChandra} X-ray coordinates to WIRC pixel positions with a positional\nuncertainty of $\\sim 0\\farcs5$.  Using this precise frame-tie we\nfound 13 ``strong'' matches ($< 1\\farcs0$ separation) and 5\n``possible'' matches ($1 - 1\\farcs5$ separation) between X-ray\nsources and IR counterparts.  After performing a spatial and\nphotometric analysis of these counterparts, we reached the following\nconclusions:\n\n1.  We expect only 2 of the 13 ``strong'' IR counterparts to be chance\nsuperpositions.  Including all 19 IR counterparts, we estimated 5 are\nunrelated associations.  Clearly, a large majority of the X-ray\/IR\nassociations are real.\n\n2.  The IR counterparts tend to reside in the spiral arms and bridge\nregion between these interacting galaxies.  Since these regions\ncontain the heaviest amounts of star formation, it seems evident that\nmany of the X-ray sources are closely tied to star formation in this\npair of galaxies.\n\n3.  A $K_s$ vs. $(J - K_s)$ CMD reveal those clusters associated with\nX-ray sources are brighter in $K_s$ but there does not seem to be a\ntrend in color.  Separating clusters by the X-ray luminosity classes\nof their X-ray counterpart does not reveal any significant trends.\n\n4.  Using reddening derived $(J - K_s)$ colors as well as from\nX-ray-derived $N_H$, we found $K_s$-band luminosity for all clusters.\nA comparison reveals those clusters associated with X-ray sources are\nmore luminous than most clusters in the Antennae.  A KS-test indicates\na significant difference between X-ray counterpart clusters and the\ngeneral population of clusters.\n\n5.  By relating flux to cluster luminosity, simplistically assuming a\nconstant age for all clusters, we estimated cluster mass.  Computing\nthe fraction of number of clusters per average flux, we estimated the\nprobability of finding a cluster with a specific mass.  We find more\nmassive clusters are more likely to contain X-ray sources, even after\nwe normalize by mass.\n\n6.  We computed a theoretical, limiting $M_{K_s}$ for all counterparts\nto X-ray sources in the Antennae using X-ray-derived reddenings.\nComparing detections to non-detections, we found those clusters with\nX-ray source are intrinsically more luminous in the IR.\n\nIn a future paper exploring the effects of cluster mass on XRB\nformation rate (Clark, et al. 2006a, in preparation), we will\ninvestigate the effects of age on cluster luminosity and hence our\ncluster mass estimates.  Another paper will extend our study of the\nAntennae to optical wavelengths (Clark, et al. 2006b, in preparation).\nThrough an in depth, multi-wavelength investigation we hope to achieve\na more complete picture of counterparts to several X-ray sources in\nthese colliding galaxies.\n\n\\acknowledgments\n\nThe authors thank the staff of Palomar Observatory for their excellent\nassistance in commissioning WIRC and obtaining these data.  WIRC was\nmade possible by support from the NSF (NSF-AST0328522), the Norris\nFoundation, and Cornell University.  S.S.E. and D.M.C. are supported\nin part by an NSF CAREER award (NSF-9983830).  We also thank\nJ.R. Houck for his support of the WIRC instrument project.\n\n\\vfill \\eject\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWe consider realizations of linear dynamical systems that are variously characterized as positive real, passive, or port-Hamiltonian.\nWe restrict ourselves to \\emph{linear time-invariant systems} represented as\n\\begin{equation} \\label{statespace}\n \\begin{array}{rcl} \\dot x & = & Ax + B u \\quad \\mbox{with}\\ x(0)=0,\\\\\ny&=& Cx+Du,\n\\end{array}\n\\end{equation}\nwhere $u:\\mathbb R\\to\\mathbb{C}^m$,   $x:\\mathbb R\\to \\mathbb{C}^n$,  and  $y:\\mathbb R\\to\\mathbb{C}^m$  are vector-valued functions denoting, respectively, the \\emph{input}, \\emph{state},\nand \\emph{output} of the system. The coefficient matrices $A\\in \\mathbb{C}^{n \\times n}$,   $B\\in \\mathbb{C}^{n \\times m}$, $C\\in \\mathbb{C}^{m \\times n}$, and  $D\\in \\mathbb{C}^{m \\times m}$ are constant.\nReal and complex $n$-vectors ($n\\times m$ matrices) are denoted by $\\mathbb R^n$, $\\mathbb C^{n}$\n($\\mathbb R^{n \\times m}$, $\\mathbb{C}^{n \\times m}$), respectively.\nWe refer to (\\ref{statespace}) concisely in terms of a four-tuple of matrices describing the realization ${\\mathcal M}:=\\left\\{A,B,C,D\\right\\}$.\n\nOur principal focus is on the structure of \\emph{passive} systems and relationships with positive-real transfer functions and port-Hamiltonian system representations.\nA nice introduction to passive systems can be found in the seminal papers of Willems (\\cite{Wil71},  \\cite{Wil72a}, \\cite{Wil72b}), where a general notion of system passivity is introduced and linked to related system-theoretic notions such as positive realness and stability. Willems refers to the earlier works of Kalman \\cite{Kal63}, Popov \\cite{Pop73}, and Yakubovich \\cite{Yak62}, where versions of what are now called Kalman-Yakubovich-Popov (KYP) conditions were derived.\nRenewed interest in these ideas came from the study of port-Hamiltonian (pH) systems, which may be viewed as particular parameterizations of passive systems that arise from certain energy-based modeling frameworks (see e.g. \\cite{Sch04}, \\cite{Sch13}, \\cite{SchJ14}). The KYP conditions lead to characterizations of system passivity through the solution set of an associated \\emph{linear matrix inequality (LMI)}.  The convexity of this solution set has led to extensive use of\nconvex optimization techniques in systems and control theory (see e.g. \\cite{BoyB90}, \\cite{NesN94}).\n\nThe solution set of the KYP-LMI leads to a natural parametrization of families of pH realizations for a given passive system.\nWith this observation, it is not surprising that some pH realizations of a given system reflect well the underlying robustness of passivity to system perturbations and that some pH realizations will do this better than others.  Our main result shows that the analytic center of certain barrier functions associated with the KYP-LMI leads to favorable pH realizations in this sense; we derive computable bounds for the passivity\nradii for these realizations.\n\nThe paper is organized as follows. In Section~\\ref{sec:prelim}, we recall the KYP conditions and link the solution set of the KYP-LMI to different system realizations that reflect passivity. In Section~\\ref{sec:analytic} we review some basic concepts in convex analysis\n and introduce the concept of an \\emph{analytic center} associated with a barrier function for the KYP-LMI.\nIn Section~\\ref{sec:passrad} we define the passivity radius for a model realization ${\\mathcal M}$, measuring its robustness to system perturbations that may lead to a loss of passivity.  We show that the analytic center of the KYP-LMI yields a model representation with good robustness properties. In Section~\\ref{sec:criteria} we consider other measures that could also serve as criteria for robustness of model passivity.\nIn Section~\\ref{sec:num} we illustrate our analytic results with a few numerical examples. We conclude with Section~\\ref{sec:conclusion},\noffering also some points for further research.\n\n\\section{Positive-realness, passivity, and\n\\hfill \\break \\mbox{port-Hamiltonian systems} }\\label{sec:prelim}\n\n\nWe restrict ourselves to linear time-invariant systems as in \\eqref{statespace} which are \\emph{minimal}, that is, the pair $(A,B)$ is \\emph{controllable}  (for all $s\\in \\mathbb C$, $\\rank \\mbox{\\small $[\\,s I-A \\quad B\\,]$} =n$ ), and the pair $(A,C)$ is \\emph{observable} ($(A^\\mathsf{H},C^\\mathsf{H})$ is controllable). Here, the Hermitian (or conjugate) transpose (transpose) of a vector or matrix $V$ is denoted by\n$V^{\\mathsf{H}}$ ($V^{mathsf{T}}$).  We also assume that $\\rank B=\\rank C=m$.\nWe require input\/output port dimensions to be equal  ($m$) and for convenience we assume that the system is initially in a quiescent state, $x(0)=0$. By applying the Laplace transform to \\eqref{statespace} and eliminating the state, we obtain the  \\emph{transfer function}\n\\begin{equation} \\label{ABCD}\n\\mathcal T(s):=D+C(sI_n-A)^{-1}B,\n\\end{equation}\nmapping the Laplace transform of $u$ to the Laplace transform of $y$.\n$I_n$ is the identity matrix in $\\mathbb{C}^{n \\times n}$\n(subsequently, the subscript may be omitted if the dimension is clear from the context).\nOn the imaginary axis $\\imath \\mathbb R$, $\\mathcal T(\\imath\\omega)$\ndescribes the \\emph{frequency response} of the system.\n\n\nWe denote the set of Hermitian matrices in $\\mathbb{C}^{n \\times n}$ by $\\Hn$.\nPositive definiteness (semidefiniteness) of  $A\\in \\Hn$ is denoted by $A>0$ ($A\\geq 0$).\nThe set of all positive definite (positive semidefinite) matrices in $\\Hn$ is denoted $\\Hnpd$ ($\\Hnpsd$).  The real and imaginary parts of a complex matrix, $Z$, are written as ${\\mathfrak{Re}} (Z)$ and ${\\mathfrak{Im}} (Z)$, respectively.\n\nWe proceed to review briefly some representations of linear systems associated with the notion of passivity.\n\\subsection{Positive-real systems}\\label{sec:posreal}\nConsider a system ${\\mathcal M} $ as in (\\ref{statespace}) and its transfer function $\\mathcal T$ as in \\eqref{ABCD}.\n\\begin{definition}\\label{def:posreal}\nA transfer function $\\mathcal T(s)$ is {\\em positive real}\nif the matrix-valued rational function\n\\begin{equation}\\label{defphi}\n\\Phi(s):= \\mathcal T^{\\mathsf{H}}(-s) + \\mathcal T(s)\n\\end{equation}\nis positive semidefinite for $s$ on the imaginary axis:\n$$\n\\Phi(\\imath\\omega)\\in \\Hmpsd \\quad \\mbox{ for all }\\omega\\in \\mathbb{R}.\n$$\n$\\mathcal T(s)$ is \\emph{strictly positive real} if $\\Phi(\\imath \\omega)\\in \\Hmpd $ for all $\\ \\omega\\in \\mathbb{R}$.\n\\end{definition}\nFor any $X \\in \\Hn$, define the matrix function\n\\begin{eqnarray*} \\label{prls}\nW(X) &:=& \\left[\n\\begin{array}{cc}\n-X\\,A - A^{\\mathsf{H}}X & C^{\\mathsf{H}} - X\\,B \\\\\nC- B^{\\mathsf{H}}X & D+D^{\\mathsf{H}}\n\\end{array}\n\\right]\\\\\n& =& W(0)- \\left[\n\\begin{array}{cc}\nX\\,A + A^{\\mathsf{H}}X &  X\\,B \\\\\nB^{\\mathsf{H}}X & 0\n\\end{array}\n\\right].\n\\end{eqnarray*}\n From \\eqref{ABCD} and (\\ref{defphi}), simple manipulations produce\n\\begin{eqnarray*}\n\\Phi\\,(s) &= &\n\\left[ \\begin{array}{cc} B^{\\mathsf{H}}(-s\\,I_n - A^{\\mathsf{H}})^{-1} & I_m  \\end{array} \\right]\n\\, W(0) \\left[ \\begin{array}{c} (s\\,I_n -A)^{-1}B \\\\ I_m \\end{array} \\right] \\nonumber \\\\\n & =&\n\\left[ \\begin{array}{cc} B^{\\mathsf{H}}(-s\\,I_n - A^{\\mathsf{H}})^{-1} & I_m  \\end{array} \\right]\n\\, W(X) \\left[ \\begin{array}{c} (s\\,I_n -A)^{-1}B \\\\ I_m \\end{array} \\right]. \\label{popovs}\n\\end{eqnarray*}\nDefine the \\emph{matrix pencils}:\n\\[\n\\mathcal L_{0}(s) =\ns \\left[ \\begin{array}{cc|c} 0 & I_n & 0\\\\[1mm]\n\t-I_n & 0 & 0\\\\[1mm]\n\t\\hline \\rule{0mm}{1.1em}\n        0 & 0 & 0 \\end{array} \\right]\n\t-\\left[ \\begin{array}{cc|c} 0 & A & B \\\\[1mm]\n\tA^{\\mathsf{H}} & 0 & C^{\\mathsf{H}} \\\\[1mm]\n        \\hline \\rule{0mm}{1.1em}\n         B^{\\mathsf{H}} & C & D+D^{\\mathsf{H}}  \\end{array} \\right],\n\\]\nand, for any $X\\in \\Hn$,\n\\[\n\\mathcal L_{X}(s) =\n\\left[ \\begin{array}{cc|c} I_n & 0 & 0 \\\\\n-X & I_n & 0 \\\\ \\hline 0 & 0 & I_m  \\end{array} \\right] \\mathcal L_{0}(s)\n\\left[ \\begin{array}{cc|c} I_n & -X & 0 \\\\\n0 & I_n & 0 \\\\ \\hline 0 & 0 & I_m  \\end{array} \\right],\n\\]\nObserve for any $X\\in \\Hn$, $\\mathcal L_{X}(s)$ and $\\mathcal L_{0}(s)$ are equivalent pencils, since they are related\nvia a congruence transformation. Note also that $\\Phi(s)$ is\nthe \\emph{Schur complement} associated with the $(3,3)$ block of $\\mathcal L_{0}(s)$\n(and hence, also of $\\mathcal L_{X}(s)$ for any $X\\in \\Hn$).\n\nIf $\\mathcal T(s)$ is  positive real,\nthen it is known \\cite{Wil71} that there exists $X\\in \\Hn$ such that the KYP-LMI holds, namely\n\\begin{equation} \\label{KYP-LMI}\nW(X) \\geq 0,\n\\end{equation}\nand so a factorization must exist:\n\\begin{equation} \\label{lw}\nW(X) =\n\\left[ \\begin{array}{c} L^{\\mathsf{H}} \\\\ M^{\\mathsf{H}} \\end{array} \\right]\\,\n\\left[ \\begin{array}{cc} L &  M \\end{array} \\right]\n\\end{equation}\nfor $L\\in \\mathbb{C}^{r \\times n}$ and $M \\in \\mathbb{C}^{r \\times m} $,\n where $r=\\rank W(X)$.\n  Introducing $\\mathcal G(s)= L(s\\,I_n - A)^{-1}B + M,$~one may then define the \\emph{spectral factorization} of $\\Phi(s)$ as\n\\begin{equation} \\label{spectral}\n\\Phi(s)= \\mathcal G^{\\mathsf{H}}(- s) \\mathcal G(s).\n\\end{equation}\nDefine the solution set and subsets to the KYP-LMI (\\ref{KYP-LMI}):\n\\begin{subequations}\\label{LMIsolnsets}\n\\begin{align}\n&{\\mathbb X}:=\\left\\{ X\\in \\Hn \\left|\\  W(X) \\geq 0 \\right.\\right\\}, \\label{XsolnWpsd} \\\\[1mm]\n&\\XWpd :=\\left\\{ X\\in \\Hn \\left|   W(X) \\geq 0,\\ X >0 \\right.\\right\\} = \\Hnpd \\cap {\\mathbb X}, \\label{XpdsolnWpsd} \\\\[1mm]\n&\\XWpdpd :=\\left\\{ X\\in \\Hn \\left|   W(X) > 0,\\ X >0 \\right.\\right\\}, \\label{XpdsolWpd}\n\\end{align}\n\\end{subequations}\nFor each $X\\in{\\mathbb X}$, there is a factorization of $W(X)$ of the form (\\ref{lw}) leading to a spectral factorization (\\ref{spectral}) of $\\Phi(s)$. We are mainly interested in $\\XWpd$ and $\\XWpdpd$, which are\nrespectively,  the set of positive-definite solutions to the KYP-LMI (\\ref{KYP-LMI}), and the subset of those solutions for which the KYP-LMI (\\ref{KYP-LMI}) holds \\emph{strictly}.\n\nAn important subset of ${\\mathbb X}$ are those solutions to (\\ref{KYP-LMI}) for which the\nrank $r$ of $W(X)$ is minimal ({i.e.}, for which $r=\\rank\\Phi(s)$). Let $S:= D+D^{\\mathsf{H}}=\\lim_{s\\rightarrow\\infty}\\Phi(s)$.\nIf $S$ is nonsingular, then\nthe minimum rank solutions in $\\XWpd$\nare those for which $\\rank W(X) = \\rank S  = m$, which in turn is the case\nif and only if the Schur complement of $S$ in $W(X)$ is zero.  This Schur\ncomplement is associated with the \\emph{algebraic Riccati equation (ARE)}:\n\\begin{multline}\n\\mathsf{Ricc}(X) := -XA-A^{\\mathsf{H}}X  \\\\ -(C^{\\mathsf{H}}-XB)S^{-1}(C-B^{\\mathsf{H}}X)=0.\\label{riccati}\n\\end{multline}\nSolutions to (\\ref{riccati}) produce directly a spectral factorization of $\\Phi(s)$,\nIndeed,  each solution $X$ of~\\eqref{riccati} corresponds to a\n\\emph{Lagrangian invariant subspace} spanned by the columns of $U:=\\mat{cc} I_n & -X^{\\mathsf{T}} \\rix^{\\mathsf{T}} $\nthat remains invariant under the action of the Hamiltonian matrix\n\\begin{equation}\\label{HamMatrix}\nH:=\\mat{cc} A-B S^{-1} C & - B S^{-1} B^{\\mathsf{H}} \\\\\nC^{\\mathsf{H}} S^{-1} C & -(A-B S^{-1} C)^{\\mathsf{H}} \\rix.\n\\end{equation}\n$U$ satisfies $HU=U A_F$ for a \\emph{closed loop matrix} $A_F=A-BF$ with $F := S^{-1}(C-B^{\\mathsf{H}}X)$ (see e.g., \\cite{FreMX02}).\nEach solution $X$ of~\\eqref{riccati}  could also be associated with an \\emph{extended Lagrangian invariant subspace}\nfor the pencil $\\mathcal{L}_{0}(s)$ (see \\cite{BenLMV15}), spanned by the columns of\n$ \\widehat{U}:=\\mat{ccc} -X^{\\mathsf{T}}\n& I_n & -F^{\\mathsf{T}}  \\rix^{\\mathsf{T}}$.\n In particular, $\\widehat{U}$ satisfies\n\\[\n\\left[ \\begin{array}{ccc} 0 & A & B \\\\\n\tA^{\\mathsf{H}} & 0 & C^{\\mathsf{H}} \\\\  B^{\\mathsf{H}} & C & S  \\end{array} \\right] \\widehat{U}\n  =\\left[ \\begin{array}{ccc} 0 & I_n & 0\\\\\n\t-I_n & 0 & 0\\\\ 0 & 0 & 0 \\end{array} \\right] \\widehat{U} A_F,\n\\]\nsee also {e.g.} \\cite{IonOW98,Wil71}.   The condition that $S$ is invertible is equivalent\nto the condition that the pencil $\\mathcal{L}_{0}(s)$ has \\emph{differentiation index one},\n{i.e.}, all eigenvalues at $\\infty$ are semi-simple, \\cite{KunM06}.\nIf $\\mathcal{L}_{0}(s)$ has no purely imaginary eigenvalues, then there are ${2n}\\choose{n}$ solutions $X\\in \\Hn$ of (\\ref{riccati}), each associated with an appropriate choice of a Lagrangian invariant subspace for $\\mathcal{L}_{0}(s)$. Every choice leads to different spectra for the closed loop matrix, $A_F$ (see \\cite{FreMX02,Wil71} for a parametrization of all possible Lagrangian subspaces).\nAmong the possible solutions of (\\ref{riccati}) there are two extremal solutions, $X_-$ and $X_+$.  $X_-$ leads to a closed loop matrix, $A_F$, with spectra in the (open) left half-plane; $X_+$ leads to $A_F$ with spectra in the (open) right half-plane.  All solutions $X$ of (\\ref{riccati}) are bracketed by $X_-$ and $X_+$:\n\\begin{equation} \\label{Xbounded}\n0\\leq X_- \\leq X \\leq X_+\n\\end{equation}\nand so, in this special case the set ${\\mathbb X}$ is bounded, but it may be empty or the solution may be unique if $X_-=X_+$, see Section~\\ref{sec:analytic}.\n\n\\subsection{Passive systems}\\label{sec:DisSys}\n\\begin{definition}\\label{def:passive}\nA system ${\\mathcal M} :=\\left\\{A,B,C,D\\right\\}$ is \\emph{passive} if there exists a state-dependent\n\\emph{storage function}, $\\mathcal H(x)\\geq 0$, such that for any $\\mu,t_0\\in \\mathbb R$ with $\\mu>t_0$,\n the \\emph{dissipation inequality} holds:\n\\begin{equation} \\label{supply} \\mathcal H(x(\\mu))-\\mathcal H(x(t_0)) \\le \\int_{t_0}^{\\mu} {\\mathfrak{Re}} (y(t)^{\\mathsf{H}}u(t)) \\, dt\n\\end{equation}\nIf for all $\\mu>t_0$, the inequality in \\eqref{supply}\nis strict then the system is \\emph{strictly passive}.\n\\end{definition}\nIn the terminology of (\\cite{Wil72a}),\n${\\mathfrak{Re}} (y(t)^{\\mathsf{H}}u(t))$ is the \\emph{supply rate} of the system.\nA general theory of dissipative systems (of which passive systems are a special case) was developed in the seminal papers \\cite{Wil71,Wil72a,Wil72b}, where links to earlier work by Kalman, Popov, and Yakubovich and the KYP-LMI (\\ref{KYP-LMI}) are given.  Note that the original definition of passivity given by Willems was for real systems; we reformulate it here for complex systems.\n\\begin{theorem}[\\cite{Wil71}]\\label{LMIWil}\nSuppose the system ${\\mathcal M}$ of (\\ref{statespace}) is minimal. Then the KYP-LMI (\\ref{KYP-LMI}): $W(X)\\geq 0$,\nhas a solution $X\\in \\Hnpd$ if and only if ${\\mathcal M}$ is a passive system. If this is the case, then\n\\begin{itemize}\n\\item $\\mathcal H(x):=\\frac{1}{2}x^{\\mathsf{H}}Xx$ defines a storage function associated with the supply rate ${\\mathfrak{Re}}(y^\\mathsf{H}u)$ satisfying the dissipation inequality \\eqref{supply};\\\\\n\\item there exist maximal and minimal solutions $X_- \\leq X_+$  in $\\Hnpd$ of \\eqref{KYP-LMI},\nsuch that for all solutions, $X$, of \\eqref{KYP-LMI}:\n    %\n\\[\n 0 < X_- \\leq X \\leq X_+.\n\\]\n\\end{itemize}\n\\end{theorem}\nRecall that a matrix $A\\in \\mathbb{C}^{n \\times n}$  is \\emph{asymptotically stable} if all its eigenvalues are in the open left half plane and \\emph{(Lyapunov) stable} if all eigenvalues are in the closed left half plane with any eigenvalues occurring on the imaginary axis being semisimple.\nTheorem~\\ref{LMIWil} asserts that if $X>0$  is a solution of $W(X)\\geq 0$, then the system ${\\mathcal M}$ of \\eqref{statespace} is stable and if it satisfies $W(X)> 0$, then it is asymptotically stable, since $\\mathcal H(x)$ is a Lyapunov function for ${\\mathcal M}$ which is strict if $W(X)> 0$, (see {e.g.} \\cite{LanT85}). Note, however, that  for (asymptotic) stability of $A$ it is sufficient if the $(1,1)$ block of $W(X)$ is (positive definite) positive semidefinite.\n\\begin{corollary}\\label{cor:pr}\nConsider a minimal system ${\\mathcal M} $ as in (\\ref{statespace}). ${\\mathcal M}$ is passive if and only if it is positive real\nand stable. It is strictly passive if and only if it is strictly positive real and asymptotically stable. In the latter case, $X_+ -X_- >0$ (\\cite{Wil71}).\n\\end{corollary}\nNote that minimality is not necessary for passivity.  For example, the system\n$\\dot{x}=-x,y = u$  is both stable and passive but not minimal.  In this case, the KYP-LMI (\\ref{KYP-LMI}) is satisfied with any (scalar) $X>0$, the Hamiltonian may be defined as\n$\\mathcal H(x)=\\frac{X}{2} x(t)^2$, and the dissipation inequality evidently holds since for $t_1\\geq t_0$,\n{\\small\n\\vspace{-5mm}\\begin{align*}\n\\mathcal{H}(x(t_1))-\\mathcal{H}(x(t_0)) & = \\frac{X}{2} (x(t_0) e^{-(t_1-t_0)})^2 -\\frac{X}{2} (x(t_0))^2 \\\\\n = \\frac{X}{2} (x(t_0))^2& (e^{-2(t_1-t_0)}-1) \\leq 0 \\leq \\int_{t_0}^{t_1} y(t) \\, u(t) \\, dt\n\\end{align*} }\n\\subsection{Port-Hamiltonian systems}\\label{sec:PH}\n\\begin{definition}\\label{def:ph}\nA linear time-invariant \\emph{port-Hamiltonian (pH) system} is one for which the following realization is possible:\n\\begin{equation} \\label{pH}\n \\begin{array}{rcl} \\dot x  & = & (J-R)Q x + (G-K) u,\\\\\ny&=& (G+K)^{\\mathsf{H}}Q x+Du,\n\\end{array}\n\\end{equation}\nwhere $Q=Q^{\\mathsf{H}} >0$, $J=-J^{\\mathsf{H}}$, and\n\\[\n\\left[ \\begin{array}{lc} R & K \\\\ K^{\\mathsf{H}} & \\mathsf{sym}(D) \\end{array} \\right] \\geq 0 \\quad \\mbox{with}\n\\quad \\mathsf{sym}(D)=\\frac12(D+D^{\\mathsf{H}})\n\\]\n\\end{definition}\nPort-Hamiltonian systems were introduced  in \\cite{Sch04} as a tool for energy-based modeling.  An energy storage function\n$\\mathcal{H}(x)=\\frac12x^{\\mathsf{H}}Qx$ plays a central role and under the conditions given, the dissipation inequality (\\ref{supply}) holds and so pH systems are always passive.  Conversely, any passive system may be represented as a pH system ({\\ref{pH}), see e.g.,  \\cite{BeaMX15_ppt}.\nWe briefly describe the construction of such a representation: Suppose the model ${\\mathcal M}$ of \\eqref{statespace} is minimal and passive and let $X=Q\\in \\XWpd$ be a solution of the KYP-LMI \\eqref{KYP-LMI}.\nFor this $Q$, define $J:=\\frac12 (AQ^{-1}- Q^{-1}A^{\\mathsf{H}})$, $R:=-\\frac 12 (AQ^{-1}+ Q^{-1}A^{\\mathsf{H}})$,\n$K:= \\frac12\\left(Q^{-1}C^{\\mathsf{H}}-B\\right)$, and $G:= \\frac12\\left(Q^{-1}C^{\\mathsf{H}}+B\\right)$.\nDirect substitution shows that (\\ref{statespace}) may be written in the form of (\\ref{pH}), with $J=-J^{\\mathsf{H}}$, and\n\\[\n\\left[ \\begin{array}{lc} R & K \\\\ K^{\\mathsf{H}} & \\mathsf{sym}(D) \\end{array} \\right] =\n\\frac12 \\left[ \\begin{array}{lc} Q^{-1} & 0 \\\\ 0  & I \\end{array} \\right] \\ W(Q)\\\n\\left[ \\begin{array}{lc} Q^{-1} & 0 \\\\ 0  & I \\end{array} \\right]  \\geq 0.\n\\] \n\nAnother possible representation of a passive system as a standard pH system can be obtained by using a symmetric factorization of a solution $X$ of\n\\eqref{KYP-LMI}:  $X=T^{\\mathsf{H}}T$ with $T\\in \\mathbb{C}^{n \\times n}$ (e.g., the Hermitian square root of $X$ or the Cholesky factorization of $X$ are two possibilities).\nOne defines a state-space transformation, $x_T=Tx$, leading to an equivalent realization in $T$-coordinates:\n\\[\n\\{A_T,B_T,C_T,D_T\\} := \\{TAT^{-1}, TB, CT^{-1}, D \\}\n\\]\nThe associated KYP-LMI \\eqref{KYP-LMI} with respect to the new coordinate system can be written as\n\\begin{eqnarray*} \\label{prls}\nW_T(\\hat{X}) &:=& \\left[\n\\begin{array}{cc}\n-\\hat{X}\\,A_T - A_T^{\\mathsf{H}} \\hat{X} & C_T^{\\mathsf{H}} - \\hat{X}\\,B_T \\\\\nC_T- B_T^{\\mathsf{H}} \\hat{X} & D+D^{\\mathsf{H}}\n\\end{array}\n\\right] \\geq 0,\n\\end{eqnarray*}\nbut since $X=T^{\\mathsf{H}}T$ is a solution to the KYP-LMI \\eqref{KYP-LMI} in the original state-space coordinates, we have $\\hat{X}=I$ and\n\\begin{eqnarray}  \\nonumber\nW_T(I)&=&\\left[ \\begin{array}{cccc} T^{-H} & 0\\\\ 0 & I_m\n\\end{array}\n\\right] W(X)\n\\left[ \\begin{array}{cccc} T^{-1} & 0\\\\ 0 & I_m\n\\end{array}\n\\right] \\geq 0.  \\label{PH}\n\\end{eqnarray}\nWe can then use the Hermitian and skew-Hermitian part of $A_T$ to obtain a pH representation in $T$-coordinates:\n$J_T:=\\frac12 (A_T-A_T^{\\mathsf{H}})$, $R_T:=-\\frac 12 (A_T+ A_T^{\\mathsf{H}})$,\n$K_T:= \\frac12\\left(C_T^{\\mathsf{H}}-B_T\\right)$, $G_T:= \\frac12\\left(C_T^{\\mathsf{H}}+B_T\\right)$, and $Q_T=I$, so that\n\\begin{equation} \\label{pHalt}\n \\begin{array}{rcl} \\dot{x}_T  & = & (J_T-R_T)x_T + (G_T-K_T) u,\\\\\ny&=& (G_T+K_T)^{\\mathsf{H}} x_T+Du,\n\\end{array}\n\\end{equation}\nis a valid pH representation in $T$ state-space coordinates.\n\n\nWe have briefly presented three closely related concepts for a minimal linear time-invariant system of the form \\eqref{statespace}, positive realness, passivity, and that the system has pH structure. All three properties can be characterized algebraically via the solutions of linear matrix inequalities, invariant subspaces of special even pencils, or solutions of Riccati inequalities. However, there typically is a lot of freedom in the representation of such systems. This freedom, which results from particular choices of solutions to the KYP-LMI as well as  subsequent  state space transformations, may be used to make the representation more robust\nto perturbations. In many ways the pH representation seems to be the most robust representation \\cite{MehMS16,MehMS17} and it also has many other advantages: it encodes the geometric and algebraic properties directly in the properties of the coefficients \\cite{SchJ14}; it allows easy ways for structure preserving model reduction \\cite{GugPBS12,PolV10}; it easily extends to descriptor systems \\cite{BeaMXZ17_ppt,Sch13}; and it greatly simplifies optimization methods for computing stability and passivity radii \\cite{GilMS18,GilS16,GilS17,OveV05}.\n\n\nThe remainder of this paper will deal with the question of how to make use of this freedom in the state space transformation to determine a 'good' or even `optimal' representation as a pH system. To do this we study in the next section the set of solutions of the KYP-LMI \\eqref{KYP-LMI} and in particular its \\emph{analytic center}.\n\n\\section{The analytic center of the solution set $\\mathbb{X}^>$} \\label{sec:analytic}\nSolutions of the KYP-LMI  \\eqref{KYP-LMI} and of linear matrix inequalities\nare usually obtained via optimization algorithms, see {e.g.} \\cite{BoyEFB94}.  A common approach involves defining a\na \\emph{barrier function} $b:\\mathbb{C}^{n \\times n} \\to \\mathbb C$ that is defined and finite throughout the interior of the constraint set becoming infinite as the boundary is approached, and\nthen using this function in the optimization scheme. The minimum of the barrier function\nitself is of independent interest and is called the \\emph{analytic center} of the constraint set \\cite{GenNV99}.\n\nWe have seen in Section~\\ref{sec:DisSys} that for a system ${\\mathcal M} $ that is minimal and strictly passive there  exists a (possibly large) class of state space transformations that  transform the system to pH form. This class is characterized by the set $\\XWpd$ of positive definite solutions of (the strict version) of the linear matrix inequality \\eqref{KYP-LMI}.\nIf the set $\\XWpd$ is non-empty and bounded, then the \\emph{barrier function}\n\\begin{equation*}\nb(X) := - \\log \\det W(X),\n\\end{equation*}\nis bounded from below, but becomes infinitely large when $W(X)$ becomes singular.\nThe analytic center of the domain $\\XWpd$ is\nthe minimizer of this barrier function. To characterize the analytic center, we analyze the \\emph{interior}  of the set $\\XWpd$\ngiven by\n\\begin{eqnarray*\n\\mathrm{Int} \\, \\XWpd &:=& \\left\\{ X\\in \\XWpd \\; |\n\\mbox{ there exists } \\delta>0 \\mbox{ such that }\\right .\\\\ && \\left .X+\\Delta_X \\in \\XWpd\n \\mbox{for all}\\;  \\Delta_X \\in \\Hn \\mbox{ with } \\|\\Delta_X\\|\\le \\delta  \\right\\}.\n\\end{eqnarray*\nHere $\\|\\Delta_X\\|$ is the spectral norm of $\\Delta_X$ given by the maximal singular value of $\\Delta_X$.\nWe compare $\\mathrm{Int} \\, \\XWpd$ with the open set\n\\\n\\XWpdpd = \\left\\{ X\\in \\XWpd \\; | \\; W(X)> 0 \\right\\}.\n\\\nSince $b(X)$ is finite for all points in $\\XWpdpd$, there is an open neighborhood where it stays bounded, and this implies that $\\XWpdpd \\subseteq \\mathrm{Int} \\, \\XWpd$.\nThe converse inclusion is not necessarily true. For example, consider a $2\\times 2$ transfer function having the form, $\\mathcal T(s)=\\mathrm{diag} (t(s), 0 )$,  where $t(s)$ is a scalar-valued, strictly passive transfer function.  The LMI is rank deficient for all $X\\in \\Hn$ (hence $\\XWpdpd=\\emptyset$) but there is a relative interior, since $t(s)$ is strictly passive. The characterization when both sets are equal is given by the following theorem.\n\\begin{theorem}\\label{thm:interior}\nSuppose the system ${\\mathcal M}$ of (\\ref{statespace}) is passive and $\\rank(B)=m$.  Then $\\XWpdpd \\equiv \\mathrm{Int}\\,\\XWpd$.\n\\end{theorem}\n{\\bf Proof:} If $\\XWpd=\\emptyset$  then $\\XWpdpd=\\emptyset$  as well.\nOtherwise, pick an $X\\in \\mathrm{Int}\\,\\XWpd$ and suppose\nthat $W(X)$ is  positive semidefinite and singular.  Then there exists a nontrivial $0\\neq [z_1^\\mathsf{T},z_2^\\mathsf{T}]^\\mathsf{T}\\in\\mathsf{Ker}\\, W(X)$ and\nan $\\varepsilon>0$ sufficiently small so that\nif $\\Delta X \\in \\Hn$ with $\\|\\Delta X\\|_F\\leq \\varepsilon$ then  $X+\\Delta X\\in \\XWpd$.\nObserve that for all such $\\Delta X $, we have $W(X+\\Delta X)=W(X)+\\Gamma(\\Delta X)\\geq 0$, where\n $ \\Gamma(\\Delta X)=-\\left[\\begin{array}{cc} \\Delta X A+ A^{\\mathsf{H}} \\Delta X &  \\Delta X B \\\\\n  B^{\\mathsf{H}}\\Delta X  & 0 \\end{array} \\right] $,\n and so\n %\n\\begin{equation}\n0\\leq \\left[\\begin{array}{c} z_1\\\\ z_2 \\end{array} \\right]^{\\mathsf{H}}\nW(X+\\Delta X)\\ \\left[\\begin{array}{c} z_1\\\\ z_2 \\end{array} \\right]\n =\n\\left[\\begin{array}{c} z_1\\\\ z_2 \\end{array} \\right]^{\\mathsf{H}}\n\\Gamma(\\Delta X)\\ \\left[\\begin{array}{c} z_1\\\\z_2 \\end{array} \\right] \\label{nonstrict}\n \\end{equation}\n %\nIf there was a choice for $\\Delta X \\in \\Hn$ with $\\|\\Delta X\\|_F\\leq \\varepsilon$ producing strict inequality in \\eqref{nonstrict},\nthen we would arrive at a contradiction, since the choice $-\\Delta X$ satisfies the same requirements yet violates the inequality.  Thus, we must have equality in \\eqref{nonstrict}\nfor all $\\Delta X \\in \\Hn$,\nwhich in turn implies\n\\[\nW(X+\\Delta X)\\left[\\begin{array}{c} z_1\\\\ z_2 \\end{array} \\right]= \\Gamma(\\Delta X)\\left[\\begin{array}{c} z_1\\\\ z_2 \\end{array} \\right]=0.\n\\]\nThis\nmeans that\n $B^{\\mathsf{H}}\\Delta X\\, u=0$ for all $\\Delta X \\in \\Hn$.\n If $z_1=0$, then we find that $\\Delta X B\\, z_2=0$ for all $\\Delta X \\in \\Hn$ which in turn means $B z_2=0$ and so, in light of the initial hypothesis on $B$, means that $z_2=0$ which is a contradiction, and thus,\n \nwe must conclude that $W(X)$ is nonsingular after all, hence positive definite.\n To eliminate the last remaining case, suppose that $z_1\\neq 0$.  Choosing first $\\Delta X=I$, we find that $z_1\\perp \\mathsf{Ran}(B)$.\n Pick $0\\neq b\\in  \\mathsf{Ran}(B)$ and define  $\\Delta X =I-2ww^{\\mathsf{H}}$ with\n $w=\\frac{1}{\\sqrt{2}}(\\frac{z_1}{\\|z_1\\|}-\\frac{b}{\\|b\\|})$.  Then\n $B^{\\mathsf{H}}\\Delta X\\, z_1 = \\frac{\\|z_1\\|}{\\|b\\|}B^{\\mathsf{H}}b =0$\n which implies that $z_1=0$, and so, $z=0$,\n $W(X)>0$, and again the assertion holds.\n\\hfill $\\Box$\n\nTo characterize when $\\XWpdpd \\equiv \\mathrm{Int}\\,\\XWpd\\neq \\emptyset$ is complicated and there is some confusion in the literature, because several factors may influence the solvability of the KYP-LMI.\nIt is clear that $S=D+D^{H}$ must be positive definite for a solution to be in $\\XWpdpd$, but clearly this is not sufficient as is demonstrated by the simple system $\\dot x=u$, $y=x+du$ (with $d>0$) which is minimal and has $X=1\\in  \\XWpd$ as only solution of the KYP-LMI, so  $\\mathrm{Int} \\, \\XWpd=\\emptyset$.\nIn this case the associated pencil $\\mathcal L_0$ has one eigenvalue $\\infty$ and the  purely imaginary eigenvalues $\\pm i  d$. It is passive, but not strictly passive, and stable (but not asymptotically stable), which is in contradiction to many statements in the literature, see e.~g.~\\cite{Gri04}, where unfortunately no distinction between passivity and strict passivity is made. The system is, furthermore, port-Hamiltonian with $J=0$, $R=0$, $B,C=1$, $Q=1$, $P=0$ and $D=1$, and satisfies the dissipation inequality. An analogous example is obtained with $A= \\left [ \\begin{array}{cc} 0 & 1 \\\\ -1 & 0 \\end{array} \\right ]$, $B^{\\mathsf{H}}=C=\\left [ \\begin{array}{cc} 1 & 0  \\end{array} \\right ]$, and $D=1$.\nThen, $X=I_2$ is the unique positive definite solution, $\\mathrm{Int} \\, \\XWpd=\\emptyset$, and there are double eigenvalues of $\\mathcal L_0$ at $\\pm i$. The system is not asymptotically stable and not strictly passive, but stable, passive and pH. If in this example we choose $D=0$, then still $X=I$ is the unique positive definite solution of the KYP-LMI, but now $\\mathcal L_0$ has the only purely imaginary eigenvalue $0$ and two Kronecker blocks of size $2$ for the eigenvalue $\\infty$. In this case the Riccati equation \\eqref{riccati} cannot be be formed and there does not exist a two-dimensional extended Lagrangian invariant space associated with the stable eigenvalues.\n\n\n\\begin{remark}{\\rm\nNote that the solutions $X_+$ and $X_-$ of the Riccati equation \\eqref{riccati} yield singular $W(X_+)$ and $W(X_-)$, and are thus  on the boundary of $\\XWpd$,\neven though they are positive definite, as was pointed out in Theorem \\ref{LMIWil}.\n}\\end{remark}\n\n\nIn the sequel, we  assume that $\\XWpdpd\\neq \\emptyset$, so that the analytic center of $\\XWpd$ is well-defined, see also \\cite{NesN94}, and we can compute it as a candidate for a `good' solution to the LMI (\\ref{KYP-LMI}) yielding a robust representation. This requires the solution of an optimization problem.\nKeep in mind that we have the following the set inclusions\n\\[\n\\mathbb{X}_{W} \\subset   \\mathrm{Int}\\XWpd=\\XWpdpd  \\subset \\XWpd \\subset   \\Hnpd    \\subset \\Hn.\n\\]\nFor $X,Y \\in \\Hn $ we define the \\emph{Frobenius inner product}\n\\[\n \\langle X,Y\\rangle  := \\mathsf{trace}\\!\\left({\\mathfrak{Re}}(Y)^T{\\mathfrak{Re}}(X)+{\\mathfrak{Im}}(Y)^T{\\mathfrak{Im}}(X)\\right),\n\\]\nwhich has the properties $\\langle X,Y \\rangle= \\langle Y,X \\rangle$, $\\|X\\|_F= \\langle X,X \\rangle^{\\frac{1}{2}}$, and $YZ \\rangle= \\langle YX,Z \\rangle = \\langle XZ,Y \\rangle$.\n\nThe \\emph{gradient} of the barrier function $b(X)$ with respect to $W$ is given by\n\\[\n\\partial b(X) \/ \\partial W =  -W(X)^{-1}.\n\\]\nUsing the chain rule and the Frobenius inner product, it follows from \\cite{NesN94}\nthat $X\\in \\mathbb{C}^{n \\times n}$ is an extremal point of $b(X)$ if and only if\n\\[\n\\langle \\partial b(X) \/ \\partial W,  \\Delta W(X)[\\Delta_X] \\rangle = 0 \\quad \\mbox{for all} \\; \\Delta_X \\in \\Hn,\n\\]\nwhere $\\Delta W(X)[\\Delta_X]$ is the incremental step in the direction $\\Delta_X$ given by\n\\[\n\\Delta W(X)[\\Delta_X] = -\\left[ \\begin{array}{cc}\nA^{\\mathsf{H}}\\Delta_X+\\Delta_X A & \\Delta_X B \\\\ B^{\\mathsf{H}}\\Delta_X & 0 \\end{array} \\right].\n\\]\nFor an extremal point it is then necessary that\n\\begin{equation}\\label{orth}\n\\langle W(X)^{-1} , \\left[ \\begin{array}{cc}\nA^{\\mathsf{H}}\\Delta_X + \\Delta_X A & \\Delta_X B \\\\ B^{\\mathsf{H}}\\Delta_X & 0 \\end{array} \\right] \\rangle \\ =0\n\\end{equation}\nfor all $\\Delta_X\\in \\Hn$.\nDefining $F := S^{-1}(C-B^HX)$,\n$P :=-A^HX-XA-F^{\\mathsf{H}}SF$, and $A_F := A-BF$, it has been shown in \\cite{GenNV99} that \\eqref{orth} holds if and only if $P$ is invertible and\n\\begin{equation} \\label{skew}\nA_F^{\\mathsf{H}}P +PA_F =0.\n\\end{equation}\nNote that $P$ is nothing but the evaluation of the Riccati operator \\eqref{riccati} at $X$, and that $A_F$ is the corresponding closed loop matrix. For the solutions of the Riccati equation we have\n$P=\\mathsf{Ricc}(X)=0$ (so $P$ is not invertible) and the corresponding closed loop matrix\nhas all its eigenvalues equal to an adequate   subset of the eigenvalues of the Hamiltonian matrix $H$ in (\\ref{HamMatrix}). For an interior point of $\\XWpd$\nwe have $P=\\mathsf{Ricc}(X)> 0$, and hence $P$ has an invertible\nsquare root $P^{\\frac{1}{2}}\\in \\Hnpd$. Multiplying\n(\\ref{skew}) on both sides with  $P^{-\\frac{1}{2}}$\nwe obtain that\n\\[\nP^{-\\frac{1}{2}}A_F^{\\mathsf{H}}P^{\\frac{1}{2}} + P^{\\frac{1}{2}}A_FP^{-\\frac{1}{2}}=0.\n\\]\nThus, $\\hat A_F:=P^{\\frac{1}{2}}A_F P^{-\\frac{1}{2}}$ is skew-Hermitian and therefore  $\\hat A_F$ as well as $A_F$ have all its eigenvalues on the imaginary axis. Hence the closed loop matrix $A_F$ of the analytic center has a spectrum that is also 'central' in a certain sense.\n\nIt is important to note that\n\\[\n \\det (W(X)) = \\det (\\mathsf{Ricc}(X)) \\det S,\n\\]\nwhich implies that we are also finding a stationary point of\n$\\det(\\mathsf{Ricc}(X))$, since $S$\nis constant and invertible. Since  $P\\in \\Hnpd$, we can rewrite the equations defining the analytic center of\n$\\XWpd$ as the solutions $X\\in \\Hn$, $P\\in \\Hnpd$, $F\\in \\mathbb C^{m,n}$ of the system of matrix equations\n\\begin{eqnarray} \\nonumber\nS F  &=& C-B^{\\mathsf{H}}X,\\\\\n P   &=& -A^{\\mathsf{H}}X-XA-F^{\\mathsf{H}}SF,  \\label{FXP}\\\\ \\nonumber\n 0&=& P(A-BF)+(A^{\\mathsf{H}}-F^{\\mathsf{H}}B^{\\mathsf{H}})P\\\\\n &=&PA_F+A_F^{\\mathsf{H}}P.\\nonumber\n\\end{eqnarray}\nSystem (\\ref{FXP}) can be used to determine a solution via an iterative method, that uses a starting value $X_0$  to compute $P_0,F_0$ and then consecutively solutions $X_i$, $i=1,2,\\ldots$\nfollowed by computing a new $P_i$ and $F_i$.\n\n\\begin{remark}\\label{rem:evp}{\\rm\nFor given matrices $P,F$ the solution $X$ of \\eqref{riccati} can be obtained via\nthe invariant subspace\n\\[\n\\left[ \\begin{array}{ccc} 0 & I_n & 0\\\\\n\t-I_n & 0 & 0\\\\ 0 & 0 & 0 \\end{array} \\right]\\mat{c} -X \\\\ \\phantom{-}I_n \\\\ -F \\rix Z=\n\\left[ \\begin{array}{ccc} 0 & A & B \\\\\n\tA^{\\mathsf{H}} & -P & C^{\\mathsf{H}} \\\\  B^{\\mathsf{H}} & C & S \\end{array} \\right]\\mat{c} -X \\\\ \\phantom{-}I_n \\\\ -F \\rix,\n\\]\nComputing this subspace for $P=\\mathsf{Ricc}(X)=0$ allows to compute the extremal solutions $X_+$ and $X_-$ of \\eqref{riccati}, which then can be used to compute a starting point for an optimization scheme, see \\cite{BanMVN17_ppt} for details.\n}\n\\end{remark}\n\n\n\n\\section{The passivity radius}\\label{sec:passrad}\nOur goal to achieve `good' or even `optimal' pH representations of a passive system can be realized in different ways. A natural measure for optimality is a large \\emph{passivity radius}\n$\\rho_{{\\mathcal M}}$, which is the smallest perturbation (in an appropriate norm) to the coefficients of a model ${\\mathcal M} $ that makes the system non-passive. Computational methods to determine $\\rho_{{\\mathcal M}}$ were introduced in \\cite{OveV05}, while the converse question, what is the nearest passive system to a non-passive system has recently been discussed in \\cite{GilMS18,GilS17}.\n\nOnce we have determined a solution $X\\in \\XWpd$ to the LMI~(\\ref{KYP-LMI}), we can determine the representation \\eqref{pH}\nas in Section~\\ref{sec:PH} and the system is automatically passive (but not necessarily strictly passive). For each such representation we can determine the passivity radius and then choose the most robust solution $X\\in \\XWpd$ under perturbations by maximizing the passivity radius or by minimizing the condition number of $X^{\\frac 12}$, which is the transformation matrix to pH form, see Section~\\ref{sec:cond}.\n\n\\subsection{The $X$-passivity radius}\n\nAlternatively,  for $X\\in \\mathrm{Int}\\XWpd$ we can determine the smallest perturbation ${\\Delta_{\\mathcal M}}$ of the system matrices $A,B,C,D$ of the model ${\\mathcal M} $ that leads to a loss of positive definiteness of $W(X)$, because then we are on the boundary of the set of passive systems. This is a very suitable approach for perturbation analysis, since for fixed $X$ the matrix\n\\begin{equation*} \\label{hx}\nW(X) = \\left[\n\\begin{array}{cc}\n0 & C^{\\mathsf{H}} \\\\\nC & D+D^{\\mathsf{H}}\n\\end{array}\n\\right]\n- \\left[\\begin{array}{cc}\nA^{\\mathsf{H}}X + X\\,A & X\\,B \\\\\nB^{\\mathsf{H}}X & 0\n\\end{array}\n\\right]\n\\end{equation*}\nis linear in the unknowns $A,B,C,D$ and when we perturb the coefficients, then we preserve strict passivity as long as\n\\begin{eqnarray*}\n&& W_{\\Delta_{\\mathcal M}} (X) := \\left[\n\\begin{array}{cc}\n 0 & (C+\\Delta_C)^{\\mathsf{H}} \\\\\n(C+\\Delta_C) & (D+\\Delta_D)+(D+\\Delta_D)^{\\mathsf{H}}\n\\end{array}\n\\right]\\\\\n&& \\qquad - \\left[\\begin{array}{cc}\n(A+\\Delta_A)^{\\mathsf{H}}X + X\\,(A+\\Delta_A) & X\\,(B+\\Delta_B) \\\\\n(B+\\Delta_B)^{\\mathsf{H}}X & 0\n\\end{array}\n\\right]>0.\n\\end{eqnarray*}\nHence, given $X\\in \\mathrm{Int}\\XWpd$, we can look for the smallest perturbation $\\Delta_{\\mathcal M}$ to the model ${\\mathcal M} $ that makes $\\det (W_{\\Delta_{\\mathcal M}}(X)=0$. To measure the size of the perturbation $\\Delta_{\\mathcal M}$ of a state space model ${\\mathcal M} $ ,we use the Frobenius norm\n\\[\n \\|\\Delta_{\\mathcal M} \\| := \\left \\|\\left[\\begin{array}{ccc}\n\\Delta_A & \\Delta_B \\\\\n\\Delta_C & \\Delta_D\n\\end{array}\\right] \\right \\|_F.\n\\]\nDefining for $X\\in \\mathrm{Int}\\XWpd$ the  \\emph{$X$-passivity radius} as\n\\[\n\t\\rho_{\\mathcal M}(X):= \\inf_{\\Delta_{\\mathcal M}\\in \\mathbb C^{n+m,n+m}}\\left\\{ \\| \\Delta_{\\mathcal M} \\| \\; | \\; \\det W_{\\Delta_{\\mathcal M}}(X) = 0\\right\\}.\n\\]\nNote that in order to compute $\\rho_{\\mathcal M}(X)$ for the model ${\\mathcal M} $, we must first find a point $X\\in \\mathrm{Int}\\XWpd$, since $W(X)$  must be positive definite to start with and also $X$ should be positive definite to obtain a state-space transformation to pH form.\n\nWe have the following relation between the $X$-passivity radius and the usual passivity radius.\n\\begin{lemma}\\label{passbound}\nConsider a given model ${\\mathcal M}$ . Then the passivity radius is given by\n\\begin{eqnarray*}\n\\nonumber\n\t\\rho_{{\\mathcal M}}&=& \\sup_{X\\in \\mathrm{Int}\\XWpd}\\inf_{\\Delta_{\\mathcal M}\\in \\mathbb C^{n+m,n+m}}\\{\\| \\Delta_{\\mathcal M} \\| | \\det W_{\\Delta_{\\mathcal M}}(X)=0\\}\\\\ &=& \\sup_{X\\in \\mathrm{Int}\\XWpd} \\rho_{{\\mathcal M}}(X).\\label{passive}\n\t\\end{eqnarray*}\n\\end{lemma}\n{\\bf Proof:}\n\tIf for any given $X \\in \\mathrm{Int} \\XWpd$ we have that  $\\| \\Delta_{\\mathcal M} \\|< \\rho_{{\\mathcal M}}(X)$, then all systems ${\\mathcal M} +\\Delta_{\\mathcal M}$ with $\\| \\Delta_{\\mathcal M} \\| < \\rho_{\\mathcal M}(X)$ are strictly passive.\n\tTherefore $\\rho_{\\mathcal M} \\ge \\sup_{\\mathrm{Int}\\XWpd} \\rho_{\\mathcal M}(X)$. Equality follows, since there exists a perturbation $\\Delta_{\\mathcal M}$ of norm $\\rho_{\\mathcal M}$ for which there does not exist a point $X\\in \\mathrm{Int}\\XWpd$ for which $W_{\\Delta_{\\mathcal M}}(X)> 0$, hence, this system is not strictly passive anymore.\n\\hfill $\\Box$\n\nWe derive an explicit formula for the $X$-passivity radius based on a one parameter  optimization problem. For this, we rewrite the condition $W_{\\Delta_{\\mathcal M}} (X)>0$ as\n\\begin{eqnarray} \\nonumber\n&& \\left[\\begin{array}{cc} -X & 0 \\\\ 0 & I_m \\end{array}\\right]\n\\left[\\begin{array}{cc} A + \\Delta_A & B+\\Delta_B \\\\ C+\\Delta_C & D+\\Delta_D \\end{array}\\right] \\\\\n&& \\label{wdelta} \\qquad +\n\\left[\\begin{array}{cc} A^{\\mathsf{H}}+\\Delta_A^{\\mathsf{H}} & C^{\\mathsf{H}}+\\Delta_C^{\\mathsf{H}} \\\\ B^{\\mathsf{H}}+\\Delta_B^{\\mathsf{H}} & D^{\\mathsf{H}}+\\Delta_D^{\\mathsf{H}} \\end{array}\\right]\n\\left[\\begin{array}{cc} -X & 0 \\\\ 0 & I_m \\end{array}\\right] >0.\n\\end{eqnarray}\nSetting\n\\begin{equation} \\label{defWhatX}\n\\hat W:=W(X), \\; \\hat X := \\left[\\begin{array}{cc} X & 0 \\\\ 0 & I_m \\end{array}\\right], \\; \\Delta_T := \\left[\\begin{array}{cc} -\\Delta_A & -\\Delta_B \\\\ \\Delta_C & \\Delta_D \\end{array}\\right],\n\\end{equation}\ninequality (\\ref{wdelta}) can be written as the LMI\n\\begin{equation} \\label{WDelta}\nW_{\\Delta_{\\mathcal M}}  = \\hat W+\\hat X \\Delta_T + \\Delta_T^{\\mathsf{H}} \\hat X >0\n\\end{equation}\nas long as the system is still passive. In order to violate this condition, we need to find the smallest $\\Delta_T$ such that $\\det W_{\\Delta_{\\mathcal M}} =0$. Factoring out $\\hat W^{-\\frac{1}{2}}$ on both sides of (\\ref{WDelta})  yields the characterization\n\\begin{eqnarray}\n&&\\det\\left(I_{n+m} +\n\\hat W^{-\\frac{1}{2}}\\hat X \\Delta_T \\hat W^{-\\frac{1}{2}}+\n\\hat W^{-\\frac{1}{2}}\\Delta_T^{\\mathsf{H}} \\hat X \\hat W^{-\\frac{1}{2}}\\right) \\nonumber \\\\\n&&\\det\\left(I_{n+m} +\n\\left[ \\begin{array}{cc} \\hat W^{-\\frac{1}{2}}\\hat X & \\hat W^{-\\frac{1}{2}}\\end{array}\\right]\n\\left[ \\begin{array}{cc} 0 & \\Delta_T \\\\ \\Delta_T^{\\mathsf{H}} & 0 \\end{array}\\right]\n\\left[ \\begin{array}{cc} \\hat X\\hat W^{-\\frac{1}{2}} \\\\ \\hat W^{-\\frac{1}{2}}\\end{array}\\right] \\right)\n\\nonumber  \\\\ \\nonumber\n&&\\det\\left(I_{2(n+m)} +\n\\left[ \\begin{array}{cc} 0 & \\Delta_T \\\\ \\Delta_T^{\\mathsf{H}} & 0 \\end{array}\\right]\n\\left[ \\begin{array}{cc} \\hat X \\hat W^{-\\frac{1}{2}} \\\\ \\hat W^{-\\frac{1}{2}}\\end{array}\\right]\n\\left[ \\begin{array}{cc} \\hat W^{-\\frac{1}{2}}\\hat X & \\hat W^{-\\frac{1}{2}}\\end{array}\\right] \\right)\\\\\n&& =0.\\label{rhoX}\n\\end{eqnarray}\nThe minimal perturbation $\\Delta_T$ for which this is the case was described in \\cite{OveV05}\nusing the following theorem, which we have slightly modified in order to take into account the positive semi-definiteness of the considered matrix.\n\\begin{theorem} \\label{thm:OveV} Consider the matrices $\\hat X, \\hat W$ in (\\ref{defWhatX}) and the pointwise positive semidefinite matrix function\n\\begin{equation}\\label{defmgamma}\nM(\\gamma):= \\left[ \\begin{array}{cc} \\gamma \\hat X \\hat W^{-\\frac{1}{2}} \\\\ \\hat W^{-\\frac{1}{2}} \/ \\gamma\\end{array}\\right]\n\\left[ \\begin{array}{cc} \\gamma \\hat W^{-\\frac{1}{2}} \\hat X & \\hat W^{-\\frac{1}{2}} \/ \\gamma \\end{array}\\right]\n\\end{equation}\nin the real parameter $\\gamma$. Then the largest eigenvalue $\\lambda_{\\max}(M(\\gamma))$ is a \\emph{unimodal function} of $\\gamma$ ({i.e.} it is first monotonically decreasing and then monotonically increasing with growing $\\gamma$). At the minimizing value $\\underline \\gamma$,  $M(\\underline{\\gamma})$ has an eigenvector $z$, {i.e.}\n\\[\n M(\\underline{\\gamma}) z = \\underline\\lambda_{\\max} z, \\quad z:=\\left[ \\begin{array}{cc} u \\\\ v \\end{array}\\right],\n \\]\nwhere\n$  \\|u\\|_2^2=\\|v\\|_2^2=\\frac{1}{2}$.\nThe minimum norm perturbation $\\Delta_T$ is of rank $1$ and is given by $\\Delta_T=2uv^{\\mathsf{H}}\/\\underline{\\lambda}_{\\max}$. It has norm $1\/\\underline{\\lambda}_{\\max}$\nboth in the spectral norm and in the Frobenius norm.\n\\end{theorem}\n{\\bf Proof}\nThe proof for a slightly different formulation was presented in \\cite{OveV05}. Here we therefore just present the basic idea in our formulation. Let $\\Gamma:=\\mathrm{diag}(\\gamma I_{m+n}, \\frac{1}{\\gamma} I_{m+n})$, then $M(\\gamma)=\\Gamma H_1 \\Gamma$, while\n\\[\n\\Gamma^{-1}\\left[ \\begin{array}{cc} 0 & \\Delta_T \\\\ \\Delta_T^{\\mathsf{H}} & 0 \\end{array}\\right]\\Gamma^{-1}=\\left[ \\begin{array}{cc} 0 & \\Delta_T \\\\ \\Delta_T^{\\mathsf{H}} & 0 \\end{array}\\right].\n\\]\nSetting\n\\\nK(\\gamma):=\\left( I_{2(n+m)} + \\left[ \\begin{array}{cc} 0 & \\Delta_T \\\\ \\Delta_T^{\\mathsf{H}} & 0 \\end{array}\\right] M(\\gamma) \\right),\n\\\nthen $\\det(K(\\gamma))$ is independent of $\\gamma$. But the vector $z$ is in the kernel of $K(\\gamma)$, which implies that also $K(1)$ is singular. The value of the norm of the constructed\n$\\Delta_T$ follows from the fact that the subvectors $u$ and $v$ must have equal norm at the minimum.\n\\hfill $\\Box$\n\nA bound for $\\underline{\\lambda}_{\\max}$ in Theorem~\\ref{thm:OveV} is obtained by the following result.\n\\begin{corollary}\\label{cor:lev} Consider the matrices $\\hat X, \\hat W$ in (\\ref{defWhatX}) and the pointwise matrix function $M(\\gamma)$ as in (\\ref{defmgamma}). The largest eigenvalue of $M(\\gamma)$ is also the largest eigenvalue of\n\\[\n\\gamma^2 \\hat W^{-\\frac{1}{2}} \\hat X^2 \\hat W^{-\\frac{1}{2}} + \\hat W^{-1}\/\\gamma^2.\n\\]\nA simple upper bound for $\\underline{\\lambda}_{\\max}$ is given by $\\underline{\\lambda}_{\\max}\\le \\frac{2}{\\alpha\\beta}$ where $\\alpha^2:=\\lambda_{\\min}(\\hat W)$ and $\\beta^2=\\lambda_{\\min}(\\hat X^{-1}\\hat W\\hat X^{-1})$. The corresponding lower bound for $\\| \\Delta_T \\|_F$ then becomes\n\\[\n  \\rho_{\\mathcal M}(X) = \\min_{\\gamma} \\| \\Delta_T \\|_F \\ge \\alpha\\beta\/2.\n\\]\n\\end{corollary}\n{\\bf Proof} Clearly $\\|\\hat W^{-1}\\|_2\\le \\frac{1}{\\alpha^2}$ and $\\|\\hat W^{-\\frac{1}{2}} \\hat X^2 \\hat W^{-\\frac{1}{2}} \\|_2\\le \\frac{1}{\\beta^2}$. So if we choose $\\gamma^2=\\frac{\\beta}{\\alpha}$\n\tthen\n\\begin{eqnarray*}\n&& \\min_{\\gamma} \\|\\gamma^2 \\hat W^{-\\frac{1}{2}} \\hat X^2 \\hat W^{-\\frac{1}{2}} + \\hat W^{-1}\/\\gamma^2\\|\\\\\n && \\qquad\\le  \\| (\\beta\/\\alpha)\\hat W^{-\\frac{1}{2}} \\hat X^2 \\hat W^{-\\frac{1}{2}} + (\\alpha\/\\beta)\\hat W^{-1}\\|\\\\\t &&\\qquad \\le  \\frac{2}{\\alpha\\beta} . \\qquad \\Box\n\\end{eqnarray*}\n\n\tWe can construct a perturbation $\\Delta_T=\\epsilon (\\alpha\\beta)vu^{\\mathsf{H}}$ of norm $|\\epsilon|(\\alpha\\beta)$ which makes the matrix  $W_{\\Delta_{\\mathcal M}}$ singular and therefore gives an upper bound for $\\rho_M(X)$. To compute this perturbation, let $u$, $v$ and $w$ be vectors of norm $1$, satisfying\n$\\hat W^{-\\frac{1}{2}}u=u\/\\alpha$, $\\hat W^{-\\frac{1}{2}} \\hat X v=w\/\\beta$, $\\Delta_T=\\epsilon(\\alpha\\beta)vu^{\\mathsf{H}}$, and $\\epsilon u^{\\mathsf{H}}w=-|\\epsilon u^{\\mathsf{H}}w|$,\n{i.e.},  $u$, $v$ and $w$ are the singular vectors associated with the largest singular values\n$1\/\\alpha$ of\t$\\hat W^{-\\frac{1}{2}}$ and $1\/\\beta$ of $\\hat W^{-\\frac{1}{2}}\\hat X$. Inserting these values in (\\ref{rhoX}), it follows that\n\\begin{eqnarray*}\n&& \\det\\left(I_{n+m} +\n\t\\left[ \\begin{array}{cc} \\hat W^{-\\frac{1}{2}}\\hat X & \\hat W^{-\\frac{1}{2}}\\end{array}\\right]\n\t\\left[ \\begin{array}{cc} 0 & \\Delta_T \\\\ \\Delta_T^{\\mathsf{H}} & 0 \\end{array}\\right]\n\t\\left[ \\begin{array}{cc} \\hat X\\hat W^{-\\frac{1}{2}} \\\\ \\hat W^{-\\frac{1}{2}}\\end{array}\\right] \\right) \\\\\n&& \\qquad =\\det\\left(I_{n+m} +\n\t\\left[ \\begin{array}{cc} w & u \\end{array}\\right]\n\t\\left[ \\begin{array}{cc} 0 & \\epsilon \\\\ \\overline \\epsilon & 0 \\end{array}\\right]\n\t\\left[ \\begin{array}{cc} w^{\\mathsf{H}} \\\\ u^{\\mathsf{H}} \\end{array}\\right] \\right) \\\\\n&& \\qquad =\n\t\\det\\left(I_{2} + \\left[ \\begin{array}{cc} 0 & \\epsilon \\\\ \\overline \\epsilon & 0 \\end{array}\\right]\n\t\\left[ \\begin{array}{cc} w^{\\mathsf{H}} \\\\ u^{\\mathsf{H}} \\end{array}\\right] \\left[ \\begin{array}{cc} w &  u \\end{array}\\right]\\right).\n\\end{eqnarray*}\nIf we now choose the argument of the complex number $\\epsilon$ such that $\\epsilon u^{\\mathsf{H}}w$ is real and negative and the amplitude of $\\epsilon$ such that $1=|\\epsilon u^{\\mathsf{H}}w|+ |\\epsilon|$, \tthen\n\\[\n\\det \\left(I_{2} + \\left[ \\begin{array}{cc} \\epsilon u^{\\mathsf{H}}w & \\epsilon\\\\ \\overline \\epsilon & \\overline \\epsilon w^{\\mathsf{H}}u \\end{array}\\right]\\right) = (1-|\\epsilon u^{\\mathsf{H}}w|)^2-|\\epsilon|^2=0.\n\\]\nSince $0\\le |u^{\\mathsf{H}}w| \\le 1$, we have that $\\frac{1}{2} \\le |\\epsilon| \\le 1$ and thus we have the interval $\\alpha\\beta\/2 \\le \\rho_{\\mathcal M}(X) \\le |\\epsilon| \\alpha\\beta$, $\\frac{1}{2} \\le |\\epsilon| \\le 1$ in which the  $X$-passivity radius is located.\nIf $u$ and $w$ are linearly dependent, then this interval shrinks to a point and the estimate is exact.\nWe summarize these observations in the following theorem.\n\\begin{theorem}\\label{thm:Xpassivity}\nLet ${\\mathcal M}=\\{A,B,C,D\\}$ be a given model and let $X\\in \\mathrm{Int}\\XWpd$. Then the $X$-passivity radius $\\rho_{\\mathcal M}(X)$ at this $X$ is bounded by\n\\[\n \\alpha\\beta\/2 \\le \\rho_{\\mathcal M}(X) \\le  \\alpha\\beta\/(1+|u^{\\mathsf{H}}w|),\n \\]\n %\nwhere\n$\\alpha^2:=\\lambda_{\\min}(\\hat W)$, $\\beta^2=\\lambda_{\\min}(\\hat X^{-1}\\hat W\\hat X^{-1})$, $\\hat W^{-\\frac{1}{2}}u=u\/\\alpha$, $\\hat W^{-\\frac{1}{2}} \\hat X v=w\/\\beta$.\nMoreover, if $u$ and $w$ are linear dependent, then $\\rho_{\\mathcal M}(X)=\\alpha\\beta\/2$.\n\\end{theorem}\nIf we use these bounds for the passivity radius in a pH system, we have the following corollary.\n\\begin{corollary}\\label{cor:xeqI} If for a given system $\\mathcal M$ we have that $X=I_n\\in \\mathrm{Int}\\XWpd$   then\nthe corresponding representation of the system is port-Hamiltonian, {i.e.}, it has the representation ${\\mathcal M}:= \\{J-R,G-K,G^{\\mathsf{H}}+K^{\\mathsf{H}},S+N\\}$  and the X-passivity radius\nis given by\n \\[\n \\rho_{\\mathcal M}(I)=\\lambda_{\\min}W(I)=\\lambda_{\\min}\\left[\\begin{array}{cc} R & K \\\\ K^{\\mathsf{H}} & S\t \\end{array}\\right].\n \\]\n %\nMoreover, if $X=I_n$ is the analytic center of $\\mathrm{Int}\\XWpd$, then\n$\\rho_{\\mathcal M}(I)$ equals the passivity radius $\\rho_{\\mathcal M}$ of ${\\mathcal M}$.\n\\end{corollary}\n{\\bf Proof}\n    This follows directly from Theorem~\\ref{thm:Xpassivity}, since then  $\\alpha=\\beta$ and we can choose $u=w$.\n\\hfill $\\Box$\n\\begin{remark}\\label{rem:alphabeta}{\\rm Considering a pH representation,\nthe conditions $\\hat W\\geq \\alpha^2 I_{n+m}$ and $\\hat X^{-1} \\hat W \\hat X^{-1}\\geq \\beta^2 I_{n+m}$ yield the necessary (but not sufficient) condition for passivity that\n\\[\n\\left[ \\begin{array}{cc} \\hat W & \\alpha \\beta I_{n+m} \\\\\n    \t\\alpha \\beta I_{n+m} &  \\hat X^{-1} \\hat W \\hat X^{-1}\t\n    \t\\end{array} \\right] \\geq 0,\n\\]  \t\nUsing the square root $\\hat T:= \\hat X^{\\frac12}$ of  $\\hat X>0$ and a congruence transformation, one finds that this is equivalent to\n\\[\n \\left[ \\begin{array}{cc}  \\hat T^{-1} \\hat W  \\hat T^{-1} & \\alpha \\beta I_{n+m} \\\\\n    \t\\alpha \\beta I_{n+m} &  \\hat T^{-1} \\hat W \\hat T^{-1}\t\n    \t\\end{array} \\right] \\geq 0,\n\\]\nwhich implies that $\\hat T^{-1} \\hat W  \\hat T^{-1} \\geq \\alpha\\beta I_{n+m}$.\n\nDefining $\\xi:=\\lambda_{\\min}(\\hat T^{-1} \\hat W  \\hat T^{-1})$, we then also obtain the inequality $\\xi \\ge \\alpha\\beta$. Because of \\eqref{PH}, this is also equal to\n\\[\n \\xi =\\lambda_{\\min} \\left[\\begin{array}{cc}\nR & K \\\\ K^{\\mathsf{H}} & S\n\\end{array}\\right]\n\\]\nwhich suggests that pH representations\nare likely to guarantee a good passivity margin. In order to compute the optimal product $\\xi=\\alpha\\beta$, we could maximize $\\xi$ under the constraint\n\\[\n\\hat W-\\xi\\hat X = \\left[ \\begin{array}{cc} - XA- A^{\\mathsf{H}} X - \\xi X & C^{\\mathsf{H}}- XB \\\\ C-B^{\\mathsf{H}} X & S-\\xi I_m \\end{array}\\right]> 0.\n\\]\n}\n\\end{remark}\n\n\\subsection{A port-Hamiltonian barrier function}\n\nFrom our previous discussions it appears that if we want to make sure that a state-space representation has a large passivity radius, we should not maximize the determinant of $W(X)$, but maximize instead\n\\begin{equation} \\label{tilde}\n\\det \\left ( \\left[ \\begin{array}{cc} X^{-\\frac12} & 0 \\\\ 0 & I_m \\end{array}\\right]  W(X)\\left[ \\begin{array}{cc} X^{-\\frac12} & 0 \\\\ 0 & I_m \\end{array}\\right] \\right )\n\\end{equation}\nunder the constraint $X>0$ so that its square root $T=X^\\frac12$ exists.\nEquivalently, if $X>0$, we can maximize  the determinant of\n\\begin{eqnarray*} \\tilde W(X)&:=& W(X)\\left[ \\begin{array}{cc} X^{-1} & 0 \\\\ 0 & I_m \\end{array}\\right]\\\\\n&=&  \\left[ \\begin{array}{cc} -XAX^{-1}-A^{\\mathsf{H}} & C^{\\mathsf{H}}-XB \\\\ CX^{-1}-B^{\\mathsf{H}} & S \\end{array}\\right]\n\\end{eqnarray*}\nwhich has the same eigenvalues and the same determinant, but is expressed in terms of the variable $X$.\n\nWith this modified barrier function $\\tilde b(X):=- \\log \\det \\tilde W(X)$ we obtain the following formulas for the gradient of the barrier with respect to $\\tilde W$ and the incremental step of $\\tilde W(X)$ in the direction $\\Delta_X$.\n\\begin{eqnarray*}\n\\partial \\tilde b(X\/\\partial \\tilde W) &=& -\\tilde W(X)^{-{\\mathsf{H}}} = -W(X)^{-1}\\left[ \\begin{array}{cc} X & 0 \\\\ 0 & I_m \\end{array}\\right], \\\\\n\\Delta \\tilde W(X)[\\Delta_X] &=&  \\left[ \\begin{array}{cc}\nXAX^{-1}\\Delta_X  - \\Delta_X A & -\\Delta_X B \\\\ -CX^{-1}\\Delta_X & 0 \\end{array} \\right]\\left[ \\begin{array}{cc} X^{-1} & 0 \\\\ 0 & I_m \\end{array}\\right].\n\\end{eqnarray*}\nUsing again the chain rule, the necessary condition for an extremal point is then that for all $\\Delta_X \\in \\Hn$, $<\\partial \\tilde b(X) \/ \\partial \\tilde W,  \\Delta \\tilde W(X)[\\Delta_X]> = 0$, or equivalently\n\\begin{equation} \\label{orthtilde}\n< W(X)^{-1} , \\left[ \\begin{array}{cc} XAX^{-1}\\Delta_X -\\Delta_X A  & -\\Delta_X B \\\\ -CX^{-1}\\Delta_X & 0\n\\end{array} \\right] > \\ =0 \\; .\n\\end{equation}\nDefining $P$ and $F$ as before, and using that\n\\[\n W(X)^{-1} = \\left[ \\begin{array}{cc} I_n & 0 \\\\ -F & I_m \\end{array}\\right]\n \\left[ \\begin{array}{cc} P^{-1} & 0 \\\\ 0 & S^{-1} \\end{array}\\right]\n  \\left[ \\begin{array}{cc} I_n & -F^{\\mathsf{H}} \\\\ 0 & I_m \\end{array}\\right],\n\\]\nit then follows that \\eqref{orthtilde} holds if and only if $P$ is invertible and for all $\\Delta_X \\in \\Hn$\nwe have\n\\[\n<  P^{-1} , (XAX^{-1}+F^{\\mathsf{H}}CX^{-1})\\Delta_X  - \\Delta_X (A-BF) > \\ =0,\n\\]\nor equivalently\n\\[\n< \\Delta_X , P^{-1} (XAX^{-1}+F^{\\mathsf{H}}CX^{-1}) - (A-BF)P^{-1}   > \\ =0,\n\\]\nwhich can be expressed as\n\\begin{eqnarray*}\n&& P[(X^{-1}A^{\\mathsf{H}}X+X^{-1}C^{\\mathsf{H}}F)-(A-BF)]\\\\\n&& \\qquad + [(XAX^{-1}+F^HCX^{-1})-(A-BF)^{\\mathsf{H}}]P=0.\n\\end{eqnarray*}\nNote that if one performs the coordinate transformation\n\\[\n\\{A_T,B_T,C_T,D_T\\} := \\{TAT^{-1}, TB, CT^{-1}, D \\}\n\\]\nwhere $T^2=X$, then  $P_T=T^{-1}PT^{-1}$ and $F_T=FT^{-1}$, which yields\nthe equivalent condition\n\\begin{eqnarray*}\n&& P_T[(A_T^{\\mathsf{H}}-A_T)+(C^{\\mathsf{H}}_T+B_T)F_T]\\\\\n && \\qquad +[(A_T^{\\mathsf{H}}-A_T)+(B_T+C_T^{\\mathsf{H}})F_T]^{\\mathsf{H}}P_T=0.\n\\end{eqnarray*}\nMoreover, we have that\n\\[\n F_T = S^{-1}(C_T-B_T^{\\mathsf{H}}), \\quad P_T= -A_T-A_T^{\\mathsf{H}}-F_T^{\\mathsf{H}}SF_T.\n\\]\nThus, if we use a pH representation ${\\mathcal M}=\\{A_T,B_T,C_T,D_T\\}=\\{J-R,G-K,(G+K)^{\\mathsf{H}},D\\}$, then at the analytic center of the modified barrier function, we have\n\\begin{eqnarray*}\nSF_T&=&K^{\\mathsf{H}}, \\ P_T= R -F_T^{\\mathsf{H}}SF_T, \\\\\n0&=&  P_T(J-GF_T)+(J-GF_T)^{\\mathsf{H}}P_T,\n\\end{eqnarray*}\nand of course $X_T=I$, which implies that the passivity radius is given by\n\\[\n \\lambda_{\\min} \\left[ \\begin{array}{cc} R & K \\\\ K^{\\mathsf{H}} & S \\end{array}\\right].\n\\]\nOn the other hand, since\nwe have optimized the determinant of $\\tilde W(X)$ which has the same determinant as \\eqref{tilde}, it follows that\n$$  \\det \\tilde W(X)= \\det \\left[ \\begin{array}{cc} R & K \\\\ K^{\\mathsf{H}} & S \\end{array}\\right]\n$$\nand we can expect to have obtained a nearly optimal passivity margin as well.\n\n\n\n\\section{Other Radii} \\label{sec:criteria}\n\t\nEven though in this paper we are focusing on the passivity, we point out that pH representations also have other properties that are important to consider. In this section, we consider two such properties.\n\n\\subsection{The Stability Radius}\n\n\nIf a positive definite solution of the LMI (\\ref{KYP-LMI}) exists, then it follows from the positive definiteness of the  $(1,1)$ block that  the system is asymptotically stable. Hence we can employ the same technique that we have used for the characterization of the passivity radius to bound the  \\emph{stability radius}, {i.e.}, the smallest perturbation $\\Delta_A$ that makes the system loose its asymptotic stability.  Introducing the positive definite matrices\n\\begin{eqnarray}\\nonumber\nV(X)&:=& -XA-A^{\\mathsf{H}}X , \\\\ V_{\\Delta_A}(X)&:=& V(X) - X\\Delta_A-\\Delta_A^{\\mathsf{H}}X,\n\\label{VDelta}\n\\end{eqnarray}\nwe define the \\emph{$X$-stability radius} as the smallest $\\Delta_A$, for which $V_{\\Delta_A}(X)$ looses its positive definiteness, {i.e.} for\n$X\\in {\\mathcal X}^{>0}$ with $V(X)> 0$, the  $X$-stability radius is defined as\n\t%\n\t\\[\n\t\\rho_A(X):= \\inf_{\\Delta_A\\in \\mathbb{C}^{n \\times n}}\\left\\{ \\| \\Delta_A \\| \\; | \\; \\det V_{\\Delta_A}(X) = 0\\right\\}.\n\t\\]\n\t%\nNote that \\eqref{VDelta} is defined similar to  \\eqref{WDelta}, except for a sign change and therefore as for the passivity radius we obtain the bound\n\\[\n\\alpha\\beta\/2 \\le \\rho_A(X) \\le \\alpha\\beta\/(1+|u^{\\mathsf{H}}w|),\n\\]\nwhere $\\alpha^2 = \\lambda_{\\min}(V)$, $\\beta^2 = \\lambda_{\\min}(X^{-1}VX^{-1})$, and where $u$, $v$ and $w$ are normalized vectors satisfying\n\\[\nV^{-\\frac{1}{2}}u=u\/\\alpha, \\quad V^{-\\frac{1}{2}} X v=w\/\\beta.\n\\]\n\\begin{corollary}\\label{cor:xeqIstab} If for a given system $\\mathcal M$ we have that $X=I_n\\in \\mathrm{Int}\\XWpd$   then at this point the corresponding representation of the system is port-Hamiltonian, {i.e.}, it has the representation ${\\mathcal M}:= \\{J-R,G-K,G^{\\mathsf{H}}+K^{\\mathsf{H}},D\\}$  and the X-stability radius of this model is given by\n\\[\n\t\\rho_A(I)=\\lambda_{\\min}V(I)=\\lambda_{\\min} (R).\n\\]\n\\end{corollary}\n{\\bf Proof}\n\tThis follows directly from Theorem~\\ref{thm:Xpassivity}, since then  $\\alpha=\\beta$ and we can choose $u=w$.\n\\hfill $\\Box$\n\n\\begin{remark}\\label{rem:stabrad}{\\rm\n\t\tIt follows from the conditions $V\\geq \\alpha^2 I_{n+m}$ and $X^{-1} V X^{-1}\\geq \\beta^2 I_{n+m}$ that a necessary (but not sufficient) condition for stability is given by $T^{-1}VT^{-1}\\geq \\alpha\\beta I_n$, where $T=X^{\\frac12}$.\n\t}\n\\end{remark}\nAnother robustness measure for the transformation $T$ is to require that the \\emph{field of values} $\\{ x^{\\mathsf{H}}A_Tx | x\\in \\mathbb C^n\\}$ of the transformed matrix $A_T$ is as far left as possible into the left half plane. In other words, we want to minimize the real part of the right most \\emph{Rayleigh quotient} of $A_T$ given by\n\\[\n\\min_{T s.t. T^2\\in \\XWpd} \\{ \\max_{x\\neq 0, x\\in \\mathbb{C}^n} {\\mathfrak{Re}}(\\frac{x^{\\mathsf{H}}A_Tx}{x^Hx}) \\}.\n\\]\nWriting $A_T=J_T-R_T$ with $J_T=-J_T^{\\mathsf{H}}$ and $R_T=R_T^{\\mathsf{H}}$  we clearly only need to $x^{\\mathsf{H}}R_Tx$, since ${\\mathfrak{Re}} (x^{\\mathsf{H}}J_Tx)=0$. In other words, we want to determine\n\\[\n\\min_{T\\in \\XWpd} \\{ \\max_{x\\neq 0, x\\in \\mathbb{C}^n} (\\frac{x^{\\mathsf{H}}R_Tx}{x^{\\mathsf{H}}x}) \\},\n\\]\nwhich amounts to maximizing the smallest eigenvalue of the $(1,1)$ block of the LMI~(\\ref{KYP-LMI}).\nIt therefore is an alternative to maximize the determinant of $W_T(X)$, since this will tend to maximize all of its eigenvalues, including those of the principal submatrices.\n\nWe are not advocating here to use either of these two approaches to compute the stability radius of our system, since we know that it is given explicitly by the formula\n\n\\[ \\rho_A =  \\min_{\\omega\\in \\mathbb{R} } \\sigma_{\\min}(A-\\imath \\omega I_n).\n\\]\nWe just want to stress here that using a pH realization based on the analytic center will also yield a robust stability margin.\n\n\n\\subsection{Well conditioned state-space transformations}\\label{sec:cond}\n\nSince for any solution $X\\in \\XWpd$, $T=X^{\\frac 12}$  yields a state space transformation to pH form, we can also try to optimize the condition number of $T$ or directly the condition number of $X=T^2\\in \\XWpd$ within the set described by $ X_- \\leq X \\leq X_+ $.\n\nLet us first consider the special case that $X_+$ and $X_-$ commute. In this  case there exists a unitary transformation $U$ that  simultaneously diagonalizes both $X_-$ and $X_+$, {i.e.}, $\nU^{\\mathsf{H}}X_-U=\\mathrm{diag}\\{\\lambda^{(-)}_1, \\ldots, \\lambda^{(-)}_n\\}$ and  $U^{\\mathsf{H}}X_+U=\\mathrm{diag}\\{\\lambda^{(+)}_1, \\ldots, \\lambda^{(+)}_n\\}$.\nSince $ X_- \\leq X \\leq X_+ $, it follows that each eigenvalue $\\lambda_i$ of $X$, $i=1,\\ldots,n$ must lie in the closed interval\n$\\lambda_i \\in [\\lambda^{(-)}_i , \\lambda^{(+)}_i]$, and that these intervals are nonempty. If there exists a point $\\lambda$\nin the intersection of all these intervals, then $X=\\lambda I_n$ is an optimal choice and it has condition number $\\kappa(X)=1$. If not, then there are at least two non-intersecting intervals, which implies that\n$\\lambda^{(+)}_{\\min} < \\lambda^{(-)}_{\\max}$ and hence that the closed interval $[\\lambda^{(+)}_{\\min} , \\lambda^{(-)}_{\\max}]$ must then be non-empty. Moreover, it must also intersect each of the intervals $[\\lambda^{(-)}_i , \\lambda^{(+)}_i]$ in at least one point.\nThus, if we choose for any $i=1,\\ldots,n$\n\\[\n \\lambda_i \\in [\\lambda^{(-)}_i , \\lambda^{(+)}_i] \\cap [\\lambda^{(+)}_{\\min} , \\lambda^{(-)}_{\\max}],\n\\]\nthen the resulting matrix will have optimal condition number $\\kappa(X)=\n\\frac{\\lambda^{(-)}_{\\max}}{\\lambda^{(+)}_{\\min}}$, and hence $\\kappa(T)=\\sqrt{\\frac{\\lambda^{(-)}_{\\max}}{\\lambda^{(+)}_{\\min}}}$.\nThe proof that this is optimal follows from the Loewner ordering of positive semidefinite matrices. The largest eigenvalue of $X$ must be larger or equal to $\\lambda_{\\max}^{(-)}$ and the smallest eigenvalue of $X$ must be smaller or equal to  $\\lambda_{\\min}^{(+)}$.\n\nIf  $X_+$ and $X_-$ do not commute, then there still exists a (non-unitary) congruence transformation $L$ which simultaneously diagonalizes both $D^{(-)}:=L^{\\mathsf{H}}X_-L$ and $D^{(+)}:=L^{\\mathsf{H}}X_+L$ but the resulting diagonal elements $d^{(-)}_i$\nand $d^{(+)}_i$ are not the eigenvalues anymore. Nevertheless, the same construction holds for any matrix $X$, but we cannot prove optimality anymore. On the other hand, we can guarantee the bound\n\\[\n\\kappa(T)\\le \\max \\left (\\kappa(L), \\kappa(L) \\sqrt{\\frac{d^{(-)}_{\\max}}{d^{(+)}_{\\min}}}\\right ).\n\\]\n\n\\section{Numerical examples}\\label{sec:num}\nIn this section we present a few numerical examples for realizations that are construct on the basis of the analytic center.\n\nWe first look at a real scalar transfer function of first degree ($m$=$n$=1) because in this case both analytic centers that we proposed earlier, are easy to compute analytically. The transfer functions of interest are given by\n\\begin{eqnarray*} T(s)&=&d + \\frac{cb}{s-a}, \\\\\n \\Phi(s)&=&\\left[ \\begin{array}{cc} b\/(-s-a) & 1  \\end{array} \\right]\n\\left[\n\\begin{array}{cc}\n0 & c \\\\\nc & 2d\n\\end{array}\n\\right].\\left[ \\begin{array}{c} b\/(s-a) \\\\ 1 \\end{array} \\right].\n\\end{eqnarray*}\nIf we assume that it the system is strictly passive, then\n\\[\nW(x) = \\left[\\begin{array}{cc} -2ax & c-bx \\\\ c-bx & 2d \\end{array}\\right]\n\\]\nmust be positive definite for some value of $x$. This implies that $d>0$ and that\n$\\det W(x)=-4adx-(c-bx)^2=-b^2x^2-2(2ad-cb)x-c^2 $ is positive for some value of $x$.\nThis implies that the discriminant $(2ad-cb)^2-b^2c^2 = 4a^2d^2-4abcd = 4ad(ad-bc)$ must be positive. Since the system is also stable, we finally have the following necessary and sufficient conditions for strict passivity of a real first degree scalar function:\n\\[\n  a<0, \\quad d>0, \\quad \\det  \\left[\\begin{array}{cc}\na & b \\\\ c & d \\end{array}\\right] < 0.\n\\]\nIt is interesting to point out that the transfer function $\\Phi(\\jmath\\omega)$ is a non-negative and unimodal function of $\\omega$ with extrema at $0$ and $\\infty$.\nWe thus can check strict passivity by verifying the positivity of $\\Phi(\\jmath\\omega)$ at these two values, and the stability of $a$~:\n\\[ \\Phi(\\infty) = d >0 , \\ \\Phi(0)= \\frac{2(ad-cb)}{a} >0, \\ a<0.\n\\]\nSince the determinant is quadratic in $x$, it is easy to determine the analytic center  $x_*$ of the linear matrix inequality $W(x)>0$ and the corresponding feedback and Riccati operator:\n\\begin{eqnarray*} x_*&=&\\frac{c}{b}-2d\\frac{a}{b^2}, \\quad\nf=\\frac{a}{b}, \\quad  p=2d\\frac{a^2}{b^2}-2c\\frac{a}{b},\\\\\nW(x) &=& \\left[\\begin{array}{cc}\n4d\\frac{a^2}{b^2}-2c\\frac{a}{b} & 2d\\frac{a}{b} \\\\ 2d\\frac{a}{b} & 2d \\end{array}\\right]\n= \\left[\\begin{array}{cc} 1 & \\frac{a}{b} \\\\ 0 & 1 \\end{array}\\right].\n\\left[\\begin{array}{cc}p & 0 \\\\ 0  & 2d \\end{array}\\right]\n\\left[\\begin{array}{cc} 1 & 0 \\\\ \\frac{a}{b} & 1\\end{array}\\right],\n\\end{eqnarray*}\nwhich implies $\\det H(x)=2d\\cdot p$ and the strict passivity condition\n\\[ a < 0, \\quad r > 0 \\quad \\mathrm{and}  \\quad p= \\frac{2a}{b^2}(ad-bc) >0.\n\\]\nThe strict passivity is lost when either one of the following happens\n\\[ d+\\delta_d=0, \\quad a+\\delta_a = 0, \\quad \\det \\left[ \\begin{array}{cc} a+\\delta_a & b+\\delta_b \\\\ c+\\delta_c & d+\\delta_d \\end{array} \\right]=0.\n\\]\nTherefore, it follows that\n\\[\n\\rho = \\min(d, a, \\sigma_2 \\left[ \\begin{array}{ccc} a & b \\\\ c & d  \\end{array} \\right]) =  \\sigma_2 \\left[ \\begin{array}{ccc} a & b \\\\ c & d  \\end{array} \\right].\n\\]\nBut at the analytic center $x_*=(2da-cb)\/b^2$, we have\n\\[\n  \\det\n2d\\left[ \\begin{array}{cc} 2\\frac{a^2}{b^2}-\\frac{ac}{bd} & \\frac{a}{b} \\\\ \\frac{a}{b} & 1 \\end{array} \\right]=2\\frac{da}{b^2}(\\frac{a}{b}-\\frac{c}{d})\n\\]\nwhich shows that the positivity at the analytic center yields the correct condition for strict passivity of the model.\n\nIf we use the port-Hamiltonian barrier function, we have to consider only $x>0$\nsince  $\\det \\tilde W(x)=\\det W(x) \/x$. One easily checks that the derivative of\n$\\det \\tilde W(x)$ has a positive zero at $x_*=|\\frac{c}{b}|$, which eventually yields a balanced realization ${\\mathcal M}_T=\\{a,\\sqrt{|bc|}, bc\/\\sqrt{|bc|},d\\}$, and an improved passivity radius\n\\[\n \\sigma_2 \\left[ \\begin{array}{ccc} a & bc\/\\sqrt{|bc|} \\\\ \\sqrt{|bc|} & d  \\end{array} \\right].\n\\]\n\n\nAs second test case we look at a random numerical  model  $\\{A,B,C,D\\}$ in pH form of state dimension $n=6$ and input\/output dimension $m=3$  via\n\\[\n  \\left[ \\begin{array}{ccc} R & K \\\\ K^{\\mathsf{H}} & S  \\end{array} \\right] := M M^{\\mathsf{H}} ,\n\\]\nwhere $M$ is a random $(n+m)\\times(n+m)$ matrix generated in MATLAB. From this we then identified the model $A:=-R\/2, B:=-C^{\\mathsf{H}}:=-K\/2$ and $D:=S\/2$.\nThis construction guarantees us that $X_0=I_n$ satisfies the LMI positivity constraint for the model ${\\mathcal M}:=\\{A,B,C,D\\}$. We then used the Newton iteration developed in \\cite{BanMVN17_ppt} to compute the analytic center $X_c$ of the LMI\n\\[ W(X):= \\left[\n\\begin{array}{cc}\n-X\\,A - A^{\\mathsf{H}}X & C^{\\mathsf{H}} - X\\,B \\\\\nC- B^{\\mathsf{H}}X & S\n\\end{array}\n\\right] >0,\n\\]\n\nusing the barrier function $b(X):=-\\ln \\det W(X)$. We then determined the quantities $\\alpha^2 := \\lambda_{\\min}()\\hat W)$, $\\beta^2 :=  \\lambda_{\\min}(\\hat X^{-1}\\hat W \\hat X^{-1})$, and $\\xi := \\lambda_{\\min}(\\hat X^{-\\frac12}\\hat W \\hat X^{-\\frac12})$,\nwhere $\\tilde W:=W(X_c)$ and $\\tilde X := \\mathrm{diag} \\{X_c,I_m\\}$. The constructed matrix\n\\[\n\\hat X^{-\\frac12}\\hat W \\hat X^{-\\frac12}=\\left[ \\begin{array}{ccc} R_c & K_c \\\\ K_c^{\\mathsf{H}} & S_c  \\end{array} \\right]\n\\]\nalso contains the parameters of the port-Hamiltonian realization at the analytic center $X_c$. The results are given in the table below\n\\[\n\\begin{array}{c|c|c|c|c|c}\n\t \\alpha^2 & \\beta^2 & \\xi & \\alpha\\beta & \\lambda_{\\min}(R_c)  & \\rho_{stab} \\\\\n\t \\hline\n\t 0.002366 & 0.001065 &  0.002381 & 0.001587 & 0.1254 & 0.1035\n\\end{array}\n\\]\nThey indicate that $\\lambda_{\\min}(R_c)$ at the analytic center is a good approximation of the true stability radius, and that $\\lambda_{\\min}(\\hat X^{-\\frac12}\\hat W \\hat X^{-\\frac12})$ at the analytic center is a good approximation of the $X_c$-passivity radius estimate $\\alpha\\beta$.\n\n\n\\section{Conclusion} \\label{sec:conclusion}\n\nIn this paper we have introduced the concept of the analytic center for a barrier function derived from the KYP LMI $W(X)>0$ for passive systems.\nWe have shown that the analytic center yields very promising results for choosing the  coordinate transformation  to pH form for a given passive system.\n\nSeveral important issues have not been addressed yet. Can we also apply these ideas to the limiting situations where $W(X)$ is singular and\/or $X$ is singular, or when the given system is not minimal.  More importantly, one should also analyze if these ideas can also be adapted to models\nrepresented in descriptor form.\n\nAnother interesting issue is that of finding the nearest passive system to a given system that is not passive. This has an important application in identification,\nwhere may loose the property of passivity, due to computational and round-off errors incurred during the identification.\n\\section*{Acknowledgment}\nThe authors greatly appreciate the help of Daniel Bankmann from TU Berlin for the computation of the numerical example.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}