diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzjsjo" "b/data_all_eng_slimpj/shuffled/split2/finalzzjsjo" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzjsjo" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nMotivated by Platonov's striking work on the reduced Whitehead group $\\SK(D)$\nof\nvalued division algebras $D$, see \\cite{platonov,platsurvey},\nV. Yanchevski\\u\\i, considered the unitary analogue, $\\SK(D, \\tau)$, for a\ndivision algebra~$D$ with unitary (i.e., second kind) involution $\\tau$, see\n\\cite{yin,y,yinverse,yy}.\nWorking with division algebras over a field with henselian discrete (rank $1$)\nvaluation whose residue field also contains\na henselian discrete valuation, and carrying out formidable technical\ncalculations, he produced remarkable analogues to Platonov's results. By relating\n$\\SK(D,\\tau)$ to data over the residue algebra, he showed not only that\n $\\SK(D,\\tau)$ could be nontrivial but that it could be any finite abelian group,\nand he gave a formula in the bicyclic case expressing $\\SK(D,\\tau)$\nas a quotient of relative Brauer groups.\n Over the years since then several approaches have been given to\n understanding and calculating the (nonunitary) group $\\SK$ using different methods,\nnotably by\nErshov~\\cite{ershov}, Suslin~\\cite{sus1,sus2}, Merkurjev and Rost~\\cite{merk}\n(For surveys on the group $\\SK$, see \\cite{platsurvey}, \\cite{gille},\n\\cite{merk} or\n\\cite[\\S6]{wadval}.) However, even after the passage of some 30 years,\nthere does not seem to have\nbeen any improvement in calculating $\\SK$ in the unitary setting.\nThis may be due in part to the complexity of the formulas in Yanchevski\\u\\i's\nwork, and the difficulty in following some of his arguments.\n\nThis paper is a sequel to \\cite{hazwadsworth} where the reduced Whitehead\ngroup $\\SK$ for a graded division algebra was studied. Here we consider the\nreduced unitary Whitehead group of a graded division algebra\nwith unitary graded involution. As in our\nprevious work, we will see that the graded calculus is much easier and more\ntransparent than the non-graded one. We calculate the unitary\n$\\SK$ in several important cases. We also show how this enables one to\ncalculate the unitary $\\SK$ of a tame\ndivision algebra over a henselian field, by passage to the associated\ngraded division algebra.\nThe graded approach allows us not only to recover most of\nYanchevski\\u\\i's results in \\cite{y,yinverse, yy}, with very substantially\nsimplified proofs, but also extend them\nto arbitrary value groups and to calculate the unitary $\\SK$ for\nwider classes of division algebras. There is a significant simplification\ngained by considering arbitrary value groups from the outset, rather than\ntowers of discrete valuations. But the greatest gain comes from passage\nto the graded setting, where the reduction to arithmetic considerations\nin the degree $0$ division subring is quicker and more transparent.\n\nWe briefly describe our principal results. Let $\\mathsf{E}$ be a graded division\nalgebra, with torsion free abelian grade group $\\Gamma_\\mathsf{E}$,\nand let $\\tau$ be a unitary\ngraded involution on $\\mathsf{E}$.\n\\lq\\lq Unitary\" means that\nthe action of $\\tau$ on the center $\\mathsf{T} = Z(\\mathsf{E})$ is nontrivial\n(see~\\S\\ref{unitsk1}).\n The {\\it reduced unitary Whitehead group} for $\\tau$ on~$\\mathsf{E}$ is defined\nas\n$$\n\\SK(\\mathsf{E},\\tau) \\ = \\ \\big\\{a\\in \\mathsf{E}^*\\mid \\Nrd_\\mathsf{E}(a^{1-\\tau})=1\\big \\}\\big\/\n\\big \\langle a\\in \\mathsf{E}^* \\mid a^{1-\\tau}=1 \\big \\rangle,\n$$\nwhere $\\Nrd_\\mathsf{E}$ is the\nreduced norm map $\\Nrd_\\mathsf{E}\\colon\\mathsf{E}^* \\rightarrow \\mathsf{T}^*$\n(see~\\cite[\\S3]{hazwadsworth}). Here, $a^{1-\\tau}$ means $a\\mspace{1mu}\\tau(a)^{-1}$.\nLet $\\mathsf{R}=\\mathsf{T}^\\tau = \\{t\\in \\mathsf{T}\\mid \\tau(t) = t\\}\\subsetneqq\\mathsf{T}$\n(see~\\S\\ref{unitsk1}).\nLet $\\mathsf{E}_0$ be the subring of homogeneous elements of degree~$0$ in $\\mathsf{E}$;\nlikewise for $\\mathsf{T}_0$ and $\\mathsf{R}_0$.\nFor an involution $\\rho$ on $\\mathsf{E}_0$, $S _\\rho(\\mathsf{E}_0)$ denotes\n$\\{a\\in \\mathsf{E}_0 \\mid \\rho(a) = a\\}$ and $\\Sigma_\\rho(\\mathsf{E}_0) = \\langle\nS_\\rho(\\mathsf{E}_0)\\cap \\mathsf{E}_0^*\\rangle$.\nLet $n$ be the index of $\\mathsf{E}$, and $e$ the exponent of the group\n$\\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T}$. Since $[\\mathsf{T}:\\mathsf{R}]=2$, there are just two possible cases:\neither \\ (i) $\\mathsf{T}$ is unramified over $\\mathsf{R}$, i.e.,\n$\\Gamma_\\mathsf{T}=\\Gamma_\\mathsf{R}$; or \\ (ii)~$\\mathsf{T}$~is~totally~ramified over~$\\mathsf{R}$,\ni.e., $|\\Gamma_\\mathsf{T}:\\Gamma_\\mathsf{R}|=2$ . We will prove the following\nformulas for the unitary~$\\SK$:\n\n\\begin{itemize}\n\n\\item[(i)] Suppose $\\mathsf{T}\/\\mathsf{R}$ is unramified:\n\n\\medskip\n\n\\begin{itemize}\n\\item [$\\bullet$] If $\\mathsf{E}\/\\mathsf{T}$ is\nunramified, then $\\SK(\\mathsf{E},\\tau) \\cong \\SK(\\mathsf{E}_0, \\tau|_{\\mathsf{E}_0})$\n(Prop.~\\ref{unramified}).\n\n\\medskip\n\n\\item [$\\bullet$] If $\\mathsf{E}\/\\mathsf{T}$ is totally ramified,\nthen (Th.~\\ref{sktotal}):\n\\begin{align*}\n\\SK(\\mathsf{E},\\tau)\\ & \\cong \\ \\big \\{a\\in \\mathsf{T}_0^*\\mid a^n\\in \\mathsf{R}_0^*\\}\\big \/\n\\{a\\in \\mathsf{T}_0^*\\mid a^e\\in \\mathsf{R}_0^* \\}\n\\\\\n&\\cong \\ \\big \\{\\omega \\in \\mu_n(\\mathsf{T}_0) \\mid \\tau(\\omega)\n=\\omega^{-1}\\big \\}\\big \/\\mu_e.\n\\end{align*}\n\n\n\\item [$\\bullet$] If $\\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T}$ is cyclic,\nand $\\sigma$ is a generator of $\\operatorname{Gal}(Z(E_0)\/T_0)$, then (Prop.~\\ref{cyclic}):\n\\smallskip\n\\begin{itemize}\n\\item [$\\circ$] $\\SK(\\mathsf{E},\\tau) \\ \\cong \\ \\{ a\\in E_0^* \\mid N_{Z(\\mathsf{E}_0)\/\\mathsf{T}_0}\n(\\Nrd_{\\mathsf{E}_0}(a)) \\in \\mathsf{R}_0 \\}\\big \/\\big(\n\\Sigma_\\tau(\\mathsf{E}_0)\\cdot \\Sigma_{\\sigma\\tau}(\\mathsf{E}_0)\\big)$.\n\n\\smallskip\n\n \\item [$\\circ$] If $\\mathsf{E}_0$ is a field, then $\\SK(\\mathsf{E},\\tau)=1$.\n\\end{itemize}\n\n \\medskip\n\n\\item[$\\bullet$] If $\\mathsf{E}$ has a maximal graded subfield\n$\\mathsf{M}$ unramified over $\\mathsf{T}$ and another maximal graded subfield~$\\mathsf{L}$ totally\nramified over $\\mathsf{T}$, with $\\tau(\\mathsf{L} ) =\\mathsf{L}$, then $\\mathsf{E}$ is semiramified and\n(Cor.~\\ref{seses})\n\\begin{equation*}\n\\SK(\\mathsf{E},\\tau) \\ \\cong \\ \\big\\{a \\in \\mathsf{E}_0 \\mid N_{\\mathsf{E}_0\/\\mathsf{T}_0}(a)\\in \\mathsf{R}_0\\big\\}\n \\, {\\big\/} \\, \\textstyle{\\prod\\limits_{h\\in \\operatorname{Gal}(\\mathsf{E}_0\/\\mathsf{T}_0)}}\n\\mathsf{E}_0^{*h \\tau}.\n\\end{equation*}\n\\end{itemize}\n\n\n\\item[(ii)] If $\\mathsf{T}\/\\mathsf{R}$ is totally ramified, then\n$\\SK(\\mathsf{E},\\tau)=1$ (Prop.~\\ref{completely}).\n\\end{itemize}\n\nThe bridge between the graded and the non-graded henselian setting is\nestablished by\nTh.~\\ref{involthm2}, which shows that for a tame division\nalgebra $D$ over a henselian valued field with a unitary involution $\\tau$,\n${\\SK(D,\\tau)\\cong \\SK(\\operatorname{{\\sf gr}}(D),\\widetilde\\tau)}$ where $\\operatorname{{\\sf gr}}(D)$ is the\ngraded division algebra associated to $D$\nby the valuation,\n and $\\widetilde\\tau$\nis the graded involution on $\\operatorname{{\\sf gr}}(D)$ induced by $\\tau$ (see~\\S\\ref{unitary}).\nThus, each of the results listed above for graded division algebras\nyields analogous formulas for valued division algebras over a henselian\nfield, as illustrated in Example~\\ref{toex} and Th.~\\ref{appl}.\nThis recovers existing formulas, which were primarily for the\ncase with value group $\\mathbb Z$ or $\\mathbb Z\\times \\mathbb Z$, but with easier and\nmore transparent proofs than those in the existing literature.\nAdditionally, our results apply for any value groups whatever.\nThe especially simple case where $\\mathsf{E}\/\\mathsf{T}$~is totally ramified\n and $\\mathsf{T}\/\\mathsf{R}$~is unramified is entirely new.\n\nIn the sequel to this paper \\cite{II}, the very interesting special\ncase will be treated where $\\mathsf{E}\/\\mathsf{T}$ is semiramified (and $\\mathsf{T}\/\\mathsf{R}$\nis unramified) and $\\operatorname{Gal}(\\mathsf{E}_0\/\\mathsf{T}_0)$ is bicyclic. This case\nwas the setting of essentially all of Platonov's specifically computed\nexamples with nontrivial $\\SK(D)$ \\cite{platonov,plat76}, and likewise\nYanchevski\\u\\i's unitary examples in \\cite{yinverse}\nwhere the nontrivial $\\SK(D, \\tau)$ was fully computed. This case is\nnot pursued here because it requires some more specialized arguments.\nFor such an $\\mathsf{E}$, it is known that $[\\mathsf{E}]$ decomposes\n(nonuniquely) as $[\\mathsf{I}\\otimes _\\mathsf{T} \\mathsf{N}]$ in the\ngraded Brauer group of $\\mathsf{T}$, where $\\mathsf{I}$ is inertial over $\\mathsf{T}$ and\n$\\mathsf{N}$ is nicely semiramified, i.e., semiramified and containing a\nmaximal graded subfield totally ramified over $\\mathsf{T}$. Then a formula will\nbe given for $\\SK(E)$ as a factor group of the relative Brauer\ngroup $\\operatorname{Br}(\\mathsf{E}_0\/\\mathsf{T}_0)$ modulo other relative Brauer groups and the\nclass of $\\mathsf{I}_0$. An exactly analogous formula will be proved for\n$\\SK(\\mathsf{E}, \\tau)$ in the unitary setting.\n\n\n\\section{Preliminaries}\\label{prel}\n\nThroughout this paper we will be concerned with involutory division algebras and\ninvolutory graded division algebras. In the non-graded setting, we will denote a\ndivision algebra by $D$ and its center by $K$; this~$D$ is equipped with an\ninvolution\n$\\tau$, and we set $F=K^\\tau = \\{a \\in K\\mid \\tau(a) = a\\}$.\n In the graded setting, we will write $\\mathsf{E}$ for a graded division algebra with\ncenter $\\mathsf{T}$, and $\\mathsf{R}=\\mathsf{T}^\\tau$ where $\\tau$ is a graded involution on~$\\mathsf{E}$.\n(This is consistent with the notation used in~\\cite{hazwadsworth}.)\nDepending on\nthe context, we will write $\\tau(a)$~or~$a^\\tau$ for the action of the\ninvolution on an element, and $K^\\tau$ for the set of elements of $K$\ninvariant under $\\tau$. Our convention is that $a^{\\sigma\\tau}$ means\n$\\sigma(\\tau(a))$.\n\n In this section, we recall the notion of graded division algebras\nand collect the facts we need about them in~\\S\\ref{pregda}. We will then\nintroduce the unitary and graded reduced unitary Whitehead groups\nin~\\S\\ref{grinvols} and~\\S\\ref{unitsk1}.\n\n\\subsection{Graded division algebras}\\label{pregda}\n\n\nIn this subsection we establish notation and\nrecall some fundamental facts about graded division\nalgebras indexed by a totally ordered abelian group. For an extensive\nstudy of such graded division algebras\nand their relations with valued division algebras, we refer the reader\nto~\\cite{hwcor}. For generalities on graded rings see~\\cite{vanoy}.\n\n\n\nLet\n$\\mathsf{R} = \\bigoplus_{ \\gamma \\in \\Gamma} \\mathsf{R}_{\\gamma}$ be a\ngraded ring, i.e.,\n $\\Gamma$ is an abelian group, and $\\mathsf{R}$ is a\nunital ring such that each $\\mathsf{R}_{\\gamma}$ is a\nsubgroup of $(\\mathsf{R}, +)$ and\n$\\mathsf{R}_{\\gamma} \\cdot \\mathsf{R}_{\\delta} \\subseteq \\mathsf{R}_{\\gamma +\\delta}$\nfor all $\\gamma, \\delta \\in \\Gamma$. Set\n\\begin{itemize}\n\\item[] $\\Gamma_\\mathsf{R} \\ = \\ \\{\\gamma \\in \\Gamma \\mid \\mathsf{R}_{\\gamma} \\neq 0 \\}$,\n \\ the grade set of $\\mathsf{R}$;\n\\vskip0.05truein\n\\item[] $\\mathsf{R}^h \\ = \\ \\bigcup_{\\gamma \\in\n\\Gamma_{\\mathsf{R}}} \\mathsf{R}_{\\gamma}$, \\ the set of homogeneous elements of $\\mathsf{R}$.\n\\end{itemize}\nFor a homogeneous element of $\\mathsf{R}$ of degree $\\gamma$, i.e., an\n$r \\in \\mathsf{R}_\\gamma\\setminus\\{0\\}$, we write $\\deg(r) = \\gamma$.\nRecall that $\\mathsf{R}_{0}$ is a subring of $\\mathsf{R}$ and that for each $\\gamma \\in\n\\Gamma_{\\mathsf{R}}$, the group $\\mathsf{R}_{\\gamma}$ is a left and right $\\mathsf{R}_{0}$-module.\nA subring $\\mathsf{S}$ of\n$\\mathsf{R}$ is a \\emph{graded subring} if $\\mathsf{S}= \\bigoplus_{ \\gamma \\in\n\\Gamma_{\\mathsf{R}}} (\\mathsf{S} \\cap \\mathsf{R}_{\\gamma})$. For example, the\ncenter of $\\mathsf{R}$, denoted $Z(\\mathsf{R})$, is a graded subring of\n$\\mathsf{R}$.\nIf $\\mathsf{T} = \\bigoplus_{ \\gamma \\in\n\\Gamma} \\mathsf{T}_\\gamma$ is another graded ring,\na {\\it graded ring homomorphism} is a ring homomorphism\n$f\\colon \\mathsf{R}\\to \\mathsf{T}$ with $f(\\mathsf{R}_\\gamma) \\subseteq \\mathsf{T}_\\gamma$\nfor all $\\gamma \\in \\Gamma$. If $f$ is also bijective,\nit is called a graded ring isomorphism; we then write\n$\\mathsf{R}\\cong_{\\mathrm{gr}} \\mathsf{T}$.\n\n\nFor a graded ring $\\mathsf{R}$, a {\\it graded left $\\mathsf{R}$-module} $\\mathsf{M}$ is\na left $\\mathsf{R}$-module with a grading ${\\mathsf{M}=\\bigoplus_{\\gamma \\in \\Gamma'}\n\\mathsf{M}_{\\gamma}}$,\nwhere the $\\mathsf{M}_{\\gamma}$ are all abelian groups and $\\Gamma'$ is a\nabelian group containing $\\Gamma$, such that $\\mathsf{R}_{\\gamma} \\cdot\n\\mathsf{M}_{\\delta} \\subseteq \\mathsf{M}_{\\gamma + \\delta}$ for all $\\gamma \\in \\Gamma_\\mathsf{R},\n\\delta \\in \\Gamma'$.\nThen, $\\Gamma_\\mathsf{M}$ and $\\mathsf{M}^h$ are\ndefined analogously to $\\Gamma_\\mathsf{R}$ and~$\\mathsf{R}^h$. We say that $\\mathsf{M}$ is\na {\\it graded free} $\\mathsf{R}$-module if it has a base as a free\n$\\mathsf{R}$-module consisting of homogeneous elements.\n\n\nA graded ring $\\mathsf{E} = \\bigoplus_{ \\gamma \\in \\Gamma} \\mathsf{E}_{\\gamma}$ is\ncalled a \\emph{graded division ring} if $\\Gamma$ is a\ntorsion-free abelian group and\nevery non-zero homogeneous\nelement of $\\mathsf{E}$ has a multiplicative inverse in $\\mathsf{E}$.\nNote that the grade set $\\Gamma_{\\mathsf{E}}$ is actually a group.\nAlso, $\\mathsf{E}_{0}$ is a division ring,\nand $\\mathsf{E}_\\gamma$ is a $1$-dimensional\nleft and right $\\mathsf{E}_0$ vector space for every $\\gamma\\in \\Gamma_\\mathsf{E}$.\nSet $\\mathsf{E}_\\gamma^* = \\mathsf{E}_\\gamma \\setminus\\{0\\}$.\nThe requirement that $\\Gamma$ be torsion-free is made\nbecause we are interested in graded division rings arising\nfrom valuations on division rings, and all the grade groups\nappearing there are torsion-free. Recall that every\ntorsion-free abelian group $\\Gamma$ admits total orderings compatible\nwith the group structure. (For example, $\\Gamma$ embeds in\n$\\Gamma \\otimes _{\\mathbb Z}\\mathbb Q$ which can be given\na lexicographic total ordering using any base of it as a\n$\\mathbb Q$-vector space.) By using any total ordering on\n$\\Gamma_\\mathsf{E}$, it is easy to see that $\\mathsf{E}$ has no zero divisors\nand that $\\mathsf{E}^*$, the multiplicative group of units of $\\mathsf{E}$,\ncoincides with\n $\\mathsf{E}^{h} \\setminus \\{0\\}$ (cf. \\cite[p.~78]{hwcor}).\nFurthermore, the degree map\n\\begin{equation}\\label{degmap}\n\\deg\\colon \\mathsf{E}^* \\rightarrow \\Gamma_\\mathsf{E}\n\\end{equation} is a group epimorphism with kernel $\\mathsf{E}_0^*$.\n\n\nBy an easy adaptation of the ungraded arguments, one can see\nthat every graded module~$\\mathsf{M}$ over a graded division ring\n$\\mathsf{E}$ is graded free, and every two\nhomogenous bases have the same cardinality.\nWe thus call $\\mathsf{M}$ a \\emph{graded vector space} over $\\mathsf{E}$ and\nwrite $\\dim_\\mathsf{E}(\\mathsf{M})$ for the rank of~$\\mathsf{M}$ as a graded free $\\mathsf{E}$-module.\nLet $\\mathsf{S} \\subseteq \\mathsf{E}$ be a graded subring which is also a graded\ndivision ring. Then we can view $\\mathsf{E}$ as a graded left $\\mathsf{S}$-vector\nspace, and we write $[\\mathsf{E}:\\mathsf{S}]$ for $\\dim_\\mathsf{S}(\\mathsf{E})$. It is easy to\ncheck the \\lq\\lq Fundamental Equality,\"\n\\begin{equation}\\label{fundeq}\n[\\mathsf{E}:\\mathsf{S}] \\ = \\ [\\mathsf{E}_0:\\mathsf{S}_0] \\, |\\Gamma_\\mathsf{E}:\\Gamma_\\mathsf{S}|,\n\\end{equation}\nwhere $[\\mathsf{E}_0:\\mathsf{S}_0]$ is the dimension of $\\mathsf{E}_0$ as a left vector space\nover the division ring $\\mathsf{S}_0$ and $|\\Gamma_\\mathsf{E}:\\Gamma_\\mathsf{S}|$ denotes the\nindex in the group $\\Gamma_\\mathsf{E}$ of its subgroup $\\Gamma_\\mathsf{S}$.\n\nA \\emph{graded field} $\\mathsf{T}$ is a commutative graded division ring.\nSuch a $\\mathsf{T}$ is an integral domain (as $\\Gamma_\\mathsf{T}$ is torsion free),\nso it has a quotient field,\nwhich we denote $q(\\mathsf{T})$. It is known, see \\cite[Cor.~1.3]{hwalg},\nthat $\\mathsf{T}$~is integrally closed in $q(\\mathsf{T})$. An extensive theory of\ngraded algebraic field extensions of graded fields has been developed in\n\\cite{hwalg}.\n\n\n\n\nIf $\\mathsf{E}$ is a graded division ring, then its center $Z(\\mathsf{E})$ is clearly\na graded field. {\\it The graded division rings considered in\nthis paper will always be assumed finite-dimensional over their\ncenters.} The finite-dimensionality assures that $\\mathsf{E}$\nhas a quotient division ring $q(\\mathsf{E})$ obtained by central localization,\ni.e., ${q(\\mathsf{E}) = \\mathsf{E} \\otimes_\\mathsf{T} q(\\mathsf{T})}$, where $\\mathsf{T} = Z(\\mathsf{E})$. Clearly,\n$Z(q(\\mathsf{E})) = q(\\mathsf{T})$ and $\\operatorname{ind}(\\mathsf{E}) = \\operatorname{ind}(q(\\mathsf{E}))$, where\nthe index of $\\mathsf{E}$ is defined by $\\operatorname{ind}(\\mathsf{E})^2 = [\\mathsf{E}:\\mathsf{T}]$\n(see \\cite[p.~89]{hwcor}).\nIf $\\mathsf{S}$ is a graded field which is a graded subring of $Z(\\mathsf{E})$\nand $[\\mathsf{E}:\\mathsf{S}] <\\infty$,\nthen $\\mathsf{E}$ is said to be a {\\it graded division algebra} over~$\\mathsf{S}$.\nWe recall a fundamental connection between $\\Gamma_\\mathsf{E}$ and $Z(\\mathsf{E}_0)$:\nThe field $Z(\\mathsf{E}_0)$\nis Galois over $\\mathsf{T}_0$, and there is a well-defined group epimorphism\n\\begin{equation}\\label{surj}\n\\Theta_\\mathsf{E}\\colon\\Gamma_\\mathsf{E} \\rightarrow \\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{T}_0)\n\\text{\\,\\,\\,\\,\\, given by \\,\\,\\,\\,\\, } \\deg(e)\\mapsto (z\\mapsto eze^{-1}),\n\\end{equation}\nfor any $e\\in \\mathsf{E}^*$.\n(See \\cite[Prop.~2.3]{hwcor} for a proof).\n\n\n\n\n\n\nLet $\\mathsf{E} = \\bigoplus_{\\alpha \\in \\Gamma_\\mathsf{E}} \\mathsf{E}_{\\alpha}$ be a graded division\nalgebra with a graded center $\\mathsf{T}$ (with, as always, $\\Gamma_\\mathsf{E}$ a\ntorsion-free abelian group).\nAfter fixing some total ordering on $\\Gamma_\\mathsf{E}$, define a function\n$$\n\\lambda\\colon \\mathsf{E}\\setminus\\{0\\} \\rightarrow \\mathsf{E}^{\\ast} \\quad\\textrm{ by }\n\\quad\\lambda(\\textstyle\\sum c_{\\gamma}) =\nc_{\\delta}, \\text{ where $\\delta$ is minimal among the $ \\gamma \\in \\Gamma_\\mathsf{E}$ with\n$c_{\\gamma} \\neq0$}.\n$$\n Note that $\\lambda(a) = a$ for $a \\in \\mathsf{E}^{\\ast}$, and\n\\begin{equation}\\label{valhomin}\n \\lambda(ab)= \\lambda(a) \\lambda(b) \\textrm{ for all } a,b \\in \\mathsf{E}\\setminus\\{0\\}.\n\\end{equation}\n\n\nLet $Q = q(\\mathsf{E})= \\mathsf{E} \\otimes_\\mathsf{T} q(\\mathsf{T})$, which is a division ring\nas $\\mathsf{E}$ has no zero divisors and is finite-dimensional over $\\mathsf{T}$.\nWe can extend $\\lambda$ to a map defined on all of $Q^{\\ast} = Q\\setminus\\{0\\}$\n as follows: for $q\n\\in Q^{\\ast}$, write $q = ac^{-1}$ with $a \\in \\mathsf{E}\\setminus\\{0\\}$, $c \\in\nZ(\\mathsf{E})\\setminus\\{0\\}$, and set $\\lambda(q) = \\lambda(a) \\lambda(c)^{-1}$. It follows\n from (\\ref{valhomin}) that $\\lambda\\colon Q^{\\ast} \\rightarrow \\mathsf{E}^{\\ast}$ is\nwell-defined and is\na group homomorphism. Since the composition\n\\begin{equation}\\label{inji}\n\\mathsf{E}^{\\ast} \\hookrightarrow Q^{\\ast}\n\\stackrel{\\lambda}{\\longrightarrow} \\mathsf{E}^{\\ast}\n\\end{equation}\n is the identity, $\\lambda$ is a\nsplitting map for the injection $\\mathsf{E}^{\\ast} \\hookrightarrow\nQ^{\\ast}$.\n\nFor a graded division algebra $\\mathsf{E}$ over its center $\\mathsf{T}$, there is\n a reduced norm map $\\Nrd_\\mathsf{E}\\colon\\mathsf{E}^*\\rightarrow \\mathsf{T}^*$\n(see~\\cite[\\S3]{hazwadsworth}) such that for $a\\in \\mathsf{E}$ one has\n$\\Nrd_\\mathsf{E}(a)=\\Nrd_{q(\\mathsf{E})}(a)$. The {\\it reduced Whitehead group},\n$\\SK(\\mathsf{E})$, is defined as\n$\\mathsf{E}^{(1)}\/\\mathsf{E}'$, where $\\mathsf{E}^{(1)}$ denotes\nthe set of elements of $\\mathsf{E}^*$ with reduced norm 1, and $\\mathsf{E}'$ is the\ncommutator subgroup $[\\mathsf{E}^*,\\mathsf{E}^*]$ of $\\mathsf{E}^*$. This group was studied in detail\nin~\\cite{hazwadsworth}.\nWe will be using the following facts which were established in that paper:\n\\begin{remarks}\\label{grfacts} Let $n = \\operatorname{ind}(\\mathsf{E})$.\n\\begin{enumerate}[\\upshape(i)]\n\\item For $\\gamma\\in \\Gamma_\\mathsf{E}$, if\n$a\\in \\mathsf{E}_\\gamma$ then $\\Nrd_\\mathsf{E}(a)\\in \\mathsf{E}_{n\\gamma}$. In particular,\n$\\mathsf{E}^{(1)}$ is a subset of $\\mathsf{E}_0$.\n\n\\smallskip\n\n\\item If $\\mathsf{S}$ is any graded subfield of $\\mathsf{E}$\ncontaining $\\mathsf{T}$ and $a\\in \\mathsf{S}$, then\n$\\Nrd_\\mathsf{E}(a) = N_{\\mathsf{S}\/\\mathsf{T}}(a)^{n\/[\\mathsf{S}:\\mathsf{T}]}$.\n\n\\smallskip\n\n\\item \\label{rnrd} Set\n\\begin{equation}\\label{dlambda}\n\\partial \\ = \\ \\operatorname{ind}(\\mathsf{E})\\big\/\\big(\\operatorname{ind}(\\mathsf{E}_0) \\,\n[Z(\\mathsf{E}_0) \\!: \\!\\mathsf{T}_0]\\big).\n\\end{equation}\n If $a\\in \\mathsf{E}_0$, then,\n\\begin{equation}\\label{nrddo}\n\\Nrd_\\mathsf{E}(a) \\ = \\ N_{Z(\\mathsf{E}_0)\/\\mathsf{T}_0}\\Nrd_{\\mathsf{E}_0}(a)^\n {\\,\\partial}\n\\in \\mathsf{T}_0.\n\\end{equation}\n\n\\smallskip\n\n\\item \\label{normal} If $N$ is a normal subgroup of\n$\\mathsf{E}^{\\ast}$, then $N^{n} \\subseteq\n\\Nrd_\\mathsf{E}(N)[\\mathsf{E}^{\\ast}, N]$.\n\nFor proofs of (i)-(iv) see \\cite[Prop.~3.2 and~3.3]{hazwadsworth}.\n\n\\smallskip\n\n\\item \\label{torik} $\\SK(\\mathsf{E})$ is $n$-torsion.\n\\begin{proof}\nBy taking $N=\\mathsf{E}^{(1)}$, the assertion follows from (\\ref{normal}).\n\\end{proof}\n\\end{enumerate}\n\\end{remarks}\n\n\nA graded division algebra $\\mathsf{E}$ with center $\\mathsf{T}$ is\nsaid to be {\\it inertial} (or {\\it unramified}) if $\\Gamma_\\mathsf{E}=\\Gamma_\\mathsf{T}$.\nFrom~(\\ref{fundeq}), it then follows that $[\\mathsf{E}:\\mathsf{T}]=[\\mathsf{E}_0:\\mathsf{T}_0]$;\nindeed, $\\mathsf{E}_0$ is central simple over $\\mathsf{T}_0$ and $\\mathsf{E} \\cong_{\\operatorname{{\\sf gr}}} \\mathsf{E}_0 \\otimes _{\\mathsf{T}_0} \\mathsf{T}$.\nAt the other extreme, $\\mathsf{E}$ is said to be {\\it totally ramified}\nif $\\mathsf{E}_0=\\mathsf{T}_0$. In an intermediate case $\\mathsf{E}$ is\nsaid to be {\\it semiramified} if\n $\\mathsf{E}_0$ is a field and ${[\\mathsf{E}_0:\\mathsf{T}_0]=|\\Gamma_\\mathsf{E}:\\Gamma_\\mathsf{T}|=\\operatorname{ind}(\\mathsf{E})}$.\nThese definitions are motivated by analogous definitions for\nvalued division algebras (\\cite{wadval}).\nIndeed, if a tame valued division algebra\nis unramified, semiramified, or totally ramified, then so is its\nassociated graded division algebra.\nLikewise, a graded field extension $\\mathsf{L}$~of~\n$\\mathsf{T}$ is said to be {\\it inertial} (or {\\it unramified})\nif $\\mathsf{L}\\cong_{\\operatorname{{\\sf gr}}} \\mathsf{L}_0 \\otimes_{\\mathsf{T}_0}\\mathsf{T}$ and the field $\\mathsf{L}_0$\nis separable over $\\mathsf{T}_0$. At the other extreme,\n$\\mathsf{L}$ is {\\it totally ramified} over $\\mathsf{T}$ if\n$[\\mathsf{L}:\\mathsf{T}] = |\\Gamma_\\mathsf{L}:\\Gamma_\\mathsf{T}|$. A graded division algebra\n$\\mathsf{E}$ is said to be {\\it inertially split} if $\\mathsf{E}$ has a maximal\ngraded subfield $\\mathsf{L}$ with $\\mathsf{L}$ inertial over $\\mathsf{T}$.\nWhen this occurs, $\\mathsf{E}_0 = \\mathsf{L}_0$, and $\\operatorname{ind}(\\mathsf{E})= \\operatorname{ind}(\\mathsf{E}_0) \\,\n[Z(\\mathsf{E}_0) \\!: \\!\\mathsf{T}_0]$ by Lemma~\\ref{dlambdafacts} below.\nIn particular, if $\\mathsf{E}$ is semiramified then $\\mathsf{E}$ is inertially\nsplit,\n$\\mathsf{E}_0$ is\nabelian Galois over $\\mathsf{T}_0$, and the canonical map\n$\\Theta_\\mathsf{E}\\colon \\Gamma_\\mathsf{E}\\rightarrow \\operatorname{Gal}(\\mathsf{E}_0\/\\mathsf{T}_0)$ has kernel\n$\\Gamma_\\mathsf{T}$ (see~\\eqref{surj} above and\n~\\cite[Prop.~2.3]{hwcor}).\n\n\\begin{lemma}\\label{dlambdafacts}\nLet $\\mathsf{E}$ be a graded division algebra with center $\\mathsf{T}$.\n For the $\\partial$ of~\\eqref{dlambda},\n$\\partial^2 = |\\ker(\\Theta_\\mathsf{E})\/\\Gamma_\\mathsf{T}|$. Also,\n$\\partial = 1$ iff $\\mathsf{E}$ is inertially split.\n\\end{lemma}\n\n\\begin{proof}\nSince $\\Theta_\\mathsf{E}$ is surjective, $\\Gamma_\\mathsf{T}\\subseteq\\ker(\\Theta_\\mathsf{E})$,\nand $Z(\\mathsf{E}_0)$ is Galois over $\\mathsf{T}_0$, we have\n\\begin{align*}\n\\partial^2 \\ &= \\ \\operatorname{ind}(\\mathsf{E})^2 \\, \\big\/ \\,\n\\big(\\operatorname{ind}(\\mathsf{E}_0)^2 \\, [Z(\\mathsf{E}_0):\\mathsf{T}_0]^2\\big)\n \\ = \\ [\\mathsf{E}:\\mathsf{T}] \\, \\big\/\\big([\\mathsf{E}_0:Z(\\mathsf{T}_0)] \\, [Z(\\mathsf{E}_0):\\mathsf{T}_0] \\,\n|\\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{T}_0)|\\big)\\\\\n& = \\ [\\mathsf{E}_0:\\mathsf{T}_0] \\, |\\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T}| \\, \\big\/ \\,\n\\big([\\mathsf{E}_0:\\mathsf{T}_0] \\, |\\operatorname{im}(\\Theta_\\mathsf{E})|\\big) \\ = \\ |\\ker(\\Theta_\\mathsf{E})\/\\Gamma_\\mathsf{T}|.\n\\end{align*}\n\nNow, suppose $M$ is a maximal subfield of $\\mathsf{E}_0$\nwith $M$ separable over $\\mathsf{T}_0$. Then,\n$M\\supseteq Z(\\mathsf{E}_0)$ and $[M:Z(\\mathsf{E}_0)] = \\operatorname{ind}(\\mathsf{E}_0)$. Let\n$\\mathsf{L} = M\\otimes _{\\mathsf{T}_0} \\mathsf{T}$ which is a graded subfield of $\\mathsf{E}$\ninertial over $\\mathsf{T}$, with $\\mathsf{L}_0 = M$. Then,\n$$\n[\\mathsf{L}:\\mathsf{T}] \\ = \\ [\\mathsf{L}_0:\\mathsf{T}_0] \\ = \\ [\\mathsf{L}_0:Z(\\mathsf{E}_0)]\\, [Z(\\mathsf{E}_0):\\mathsf{T}_0]\n \\ = \\ \\operatorname{ind}(\\mathsf{E})\/\\partial.\n$$\nThus, if $\\partial = 1$, then $\\mathsf{E}$ is inertially\nsplit, since $\\mathsf{L}$ is a maximal graded subfield of $\\mathsf{E}$ which is\ninertial over $\\mathsf{T}$. Conversely, suppose $\\mathsf{E}$ is inertially\nsplit, say $\\mathsf{I}$ is a maximal graded subfield of $\\mathsf{E}$ with\n$\\mathsf{I}$ inertial over $\\mathsf{T}$. So, $[\\mathsf{I}_0:\\mathsf{T}_0] = [\\mathsf{I}:\\mathsf{T}] =\n\\operatorname{ind}(\\mathsf{E})$. Since $\\mathsf{I}_0Z(\\mathsf{E}_0)$ is a subfield of $\\mathsf{E}_0$,\nwe have\n\\begin{align*}\n[\\mathsf{I}_0:\\mathsf{T}_0] \\ &\\le \\ [\\mathsf{I}_0Z(\\mathsf{E}_0) :\\mathsf{T}_0] \\ = \\\n[\\mathsf{I}_0 Z(\\mathsf{E}_0):Z(\\mathsf{E}_0)]\\, [Z(\\mathsf{E}_0):\\mathsf{T}_0] \\\\\n&\\le \\\n\\operatorname{ind} (\\mathsf{E}_0)\\, [Z(\\mathsf{E}_0):\\mathsf{T}_0] \\ = \\ \\operatorname{ind}(\\mathsf{E})\/\\partial\n \\ = \\ [\\mathsf{I}_0:\\mathsf{T}_0] \/\\partial.\n\\end{align*}\nSo, as $\\partial$ is a positive integer, $\\partial = 1$.\n\\end{proof}\n\n\n\n\n\\subsection{Unitary $\\SK$ of division algebras}\\label{grinvols}\n\nWe begin with a description of unitary $K_1$ and $\\SK$ for a division\nalgebra with an involution. The analogous definitions for graded\ndivision algebras will be given in ~\\S\\ref{unitsk1}.\n\n\nLet $D$ be a division ring finite-dimensional over its center $K$ of\nindex $n$, and let $\\tau$ be an involution on~$D$, i.e., $\\tau$ is\nan antiautomorphism of $D$ with $\\tau^2 =\\operatorname{id}$.\nLet\n\\begin{itemize}\n\\item[] $\\quad S_\\tau(D) \\, = \\, \\{d \\in D\\mid\\tau(d) = d \\};$\n\\vskip0.05truein\n\\item [] $\\quad \\Sigma_{\\tau}(D) \\, = \\,\n\\left< S_{\\tau}(D) \\cap D^{\\ast} \\right>.$\n\\end{itemize}\n\nNote that $\\Sigma_{\\tau}(D)$ is a normal subgroup of $D^{\\ast}$. For,\nif $a \\in S_{\\tau}(D)$, $a \\neq 0$, and $b \\in D^{\\ast}$, then\n$bab^{-1} = [ba\\tau(b)][b\\tau(b)]^{-1}\\in \\Sigma_\\tau(D)$,\nas $ba\\tau(b), b\\tau(b) \\in S_\\tau(D)$.\n\n\n\nLet $\\varphi$ be an isotropic $m$-dimensional, nondegenerate skew-Hermitian form\nover $D$\nwith respect to an involution $\\tau$ on $D$. Let\n$\\rho$ be the involution on $M_m(D)$ adjoint to $\\varphi$, let\n$U_m(D) = \\{a \\in M_m(D)\\mid a\\rho(a) = 1\\}$ be the unitary group\nassociated to $\\varphi$, and let $\\operatorname{EU}_m(D)$ denote\nthe normal subgroup of $U_m(D)$ generated by the unitary transvections.\nFor $m>2$,\nthe Wall spinor norm map\n$\\Theta\\colon U_m(D) \\rightarrow D^*\/\\Sigma_\\tau(D)D'$ was developed\nin \\cite{wall}, where it was shown that $\\ker(\\Theta) = \\operatorname{EU}_m(D)$.\nHere, $D'$ denotes the multiplicative commutator group $[D^*,D^*]$.\nCombining this\nwith~\\cite[Cor.~1 of~\\S 22]{draxl} one obtains the commutative diagram:\n\\begin{equation}\\label{unnrd}\n\\begin{split}\n\\xymatrix{U_m(D)\\big\/\\operatorname{EU}_m(D) \\ar[r]_-{\\cong}^-\\Theta \\ar[d] &\nD^*\\big\/\\big ( \\Sigma_\\tau(D)D' \\big )\\ar[d]^{1-\\tau}\\\\\n\\operatorname{GL}_m(D)\\big\/E_m(D) \\ar[r]^-{\\det} \\ar[d]_{\\Nrd} & D^*\\big\/D'\n\\ar[d]^{\\Nrd}\\\\\nK^* \\ar[r]^-{\\operatorname{id}} & K^*}\n\\end{split}\n\\end{equation}\nwhere the map $\\det$ is the Dieudonn\\'e determinant and\n$1-\\tau\\colon D^*\/\\big ( \\Sigma_\\tau(D)D' \\big ) \\longrightarrow D^*\/D'$\nis defined as\n$x\\Sigma_\\tau(D)D'\\mapsto x^{1-\\tau}D'$, where $x^{1-\\tau}$ means $x\\mspace{1mu}\\tau(x)^{-1}$\n(see~\\cite[6.4.3]{hahn}).\n\nFrom the diagram, and parallel to the ``absolute'' case, one defines\nthe {\\it unitary Whitehead group},\n$$\nK_1(D,\\tau)\\ = \\ D^*\/\\big (\\Sigma_\\tau(D)D'\\big ).\n$$\n\nFor any involution $\\tau$ on $D$, recall that\n\\begin{equation}\\label{taunrd}\n\\Nrd_D(\\tau(d)) \\ = \\ \\tau(\\Nrd_D(d)),\n\\end{equation} for any $d\\in D$. For, if $p \\in K[x]$ is the\nminimal polynomial of $d$ over $K$,\nthen $\\tau(p)$ is the minimal polynomial of $\\tau(d)$ over $K$\n(see also~\\cite[\\S22, Lemma~5]{draxl}).\n\n\n We consider two cases:\n\n\\subsubsection{ Involutions of the first kind}\\label{firstk}\nIn this case the center $K$ of $D$ is elementwise invariant under the\ninvolution, i.e.,\n$K \\subseteq S_\\tau(D)$. Then $S_\\tau(D)$ is a $K$-vector space. The\n involutions of this kind are further subdivided into two types:\n{\\it orthogonal} and\n{\\it symplectic} involutions ~(see \\cite[Def.~2.5]{kmrt}).\nBy ~(\\cite[Prop.~2.6]{kmrt}), if $\\operatorname{char}(K)\\not = 2$ and $\\tau$ is orthogonal\nthen, $\\dim_K(S_\\tau(D))=n(n+1)\/2$, while if $\\tau$ is symplectic then\n$\\dim_K(S_\\tau(D))=n(n-1)\/2$. However, if $\\operatorname{char}(K)=2$, then\n$\\dim_K(S_\\tau(D))=n(n+1)\/2$ for each type.\n\n\nIf $\\dim_K(S_\\tau(D))=n(n+1)\/2$, then for any\n$x \\in D^*$, we have $xS_\\tau(D) \\cap S_\\tau(D) \\not = 0$\nby dimension count; it then follows that\n$D^*=\\Sigma_\\tau(D)$, and thus $K_1(D,\\tau)=1$. However,\nin the case $\\dim_K(S_\\tau(D))=n(n-1)\/2$, Platonov showed that $K_1(D,\\tau)$ is not in general\ntrivial, settling Dieudonn\\'e's conjecture in negative~\\cite{pld}.\nNote that whenever $\\tau$ is of the first kind we have\n$\\Nrd_D(\\tau(d)) = \\Nrd_D(d)$ for all $d\\in D$, by \\eqref{taunrd}.\nThus, $K_1(D,\\tau)$ is sent to the identity under\nthe composition $\\Nrd \\circ (1-\\tau)$. This explains why\n one does not consider the kernel of this map, i.e., the\nunitary $\\SK$, for involutions of the first kind.\nIf $\\operatorname{char}(K) \\not =2$ and\n$\\tau$~is symplectic,\nthen as the $m$-dimensional form $\\varphi$ over $D$ is skew-Hermitian,\nits associated adjoint involution~$\\rho$ on $M_m(D)$ is\nof orthogonal type, so there is an associated spin group\n$\\operatorname{Spin}(M_m(D), \\rho)$.\nFor any $a\\in S(D)$ one then has $\\Nrd_D(a) \\in K^{*2}$\n(\\cite[Lemma~2.9]{kmrt}).\nOne defines $K_1\\operatorname{Spin}(D,\\tau)=R(D)\/\\big(\\Sigma_\\tau(D)D'\\big)$,\nwhere\n$R(D)=\\{d\\in D^* \\mid \\Nrd_D(d) \\in K^{*2}\\}$. This group\nis related to $\\operatorname{Spin}(M_m(D), \\rho)$, and has been studied\nin~\\cite{monyan}, parallel to the work on absolute $\\SK$ groups and\nunitary $\\SK$ groups for unitary involutions.\n\n \\subsubsection{Involutions of the second kind $($unitary involutions$)$} \\label{secondk}\n In this case $K \\not \\subseteq S_\\tau(D)$.\n Then, let \\break ${F=K^{\\tau}\\ (=K\\cap\nS_\\tau(D)\\,)}$, which is a subfield of $K$ with $[K:F] = 2$.\n It was already observed by Dieudonn\\'e that\n$U_m(D) \\not = \\operatorname{EU}_m(D)$. An important property proved\n by Platonov and Yanchevski\\u\\i, which we will use frequently,\nis that\n\\begin{equation}\\label{primeinsigma}\nD' \\, \\subseteq \\, \\Sigma_\\tau(D).\n\\end{equation}\n(For a proof, see~\\cite[Prop.~17.26]{kmrt}.) Thus\n$K_1(D,\\tau)=D^*\/\\Sigma_\\tau(D)$, which is not trivial in general.\nThe kernel of the\nmap $\\Nrd \\circ (1-\\tau)$ in diagram~\\eqref{unnrd}, is called the\n{\\it reduced unitary Whitehead group}, and denoted by $\\SK(D,\\tau)$.\nUsing~(\\ref{taunrd}), it is straightforward to see that\n$$\n\\SK(D,\\tau) \\ = \\ \\Sigma'_{\\tau}(D) \/ \\Sigma_{\\tau}(D), \\quad \\text{where}\n\\quad\\Sigma'_{\\tau}(D) \\ = \\ \\{ a \\in D^{\\ast} \\mid \\Nrd_D(a) \\in F^{\\ast}\\}.\n$$\nNote that we use the notation $\\SK(D,\\tau)$ for the reduced unitary\n Whitehead group as opposed to Draxl's notation\n$\\operatorname{USK_1}(D,\\tau)$ in~\\cite[p.~172]{draxl} and\nYanchevski\\u\\i's notation\n$\\operatorname{SUK_1}(D,\\tau)$~\\cite{y} and the notation\n$\\operatorname{USK_1}(D)$ in \\cite{kmrt}.\n\n\nBefore we define the corresponding groups in the graded setting, let us\nrecall that all the groups above fit in Tits' framework\n\\cite{tits}\nof the\n{\\it Whitehead group} $W(G,K)=G_K\/G_K^+$ where $G$ is an almost simple,\nsimply connected linear algebraic group defined over an infinite field\n$K$, with $\\operatorname{char}(K)\\not =2$, and $G$ is isotropic over $K$. Here, $G_K$ is the\nset of $K$-rational points of $G$ and $G_K^+$, is the subgroup of $G_K$,\ngenerated by the unipotent radicals of the minimal $K$-parabolic\nsubgroups of $G$. In this setting, for $G_K=\\operatorname{SL}_n(D)$, $n>1$, we have\n$W(G,K)=\\SK(D)$; for $\\tau$ an involution of first or second kind\non $D$ and $F = K^\\tau$, for $G_F=\\operatorname{SL}_n(D,\\tau):=\\operatorname{SL}_n(D)\\cap U_n(D)$\nwe have $W(G,F)=\\SK(D,\\tau)$; and for $\\tau$ a symplectic involution\non~ $D$ and $\\rho$ the adjoint involution of an $m$-dimensional\nisotropic skew-Hermitian\n form over $D$ with $m\\ge 3$,\nfor the spinor group\n$G_K=\\operatorname{Spin}(M_m(D),\\rho)$ we have $W(G,K)$\nis a double cover of $K_1 \\operatorname{Spin}(D,\\tau)$ (see~\\cite{monyan}).\n\n\n\\subsection{Unitary $\\SK$ of graded division algebras}\\label{unitsk1}\nWe will now introduce the unitary $K_1$ and $\\SK$ in the graded setting.\nLet $\\mathsf{E} = \\bigoplus_{\\gamma \\in \\Gamma_\\mathsf{E}} \\mathsf{E}_{\\gamma}$ be a graded division\nring (with $\\Gamma_\\mathsf{E}$ a torsion-free abelian group) such that\n$\\mathsf{E}$ has finite dimension $n^2$ over its center $\\mathsf{T}$, a graded field.\nLet $\\tau$ be a graded involution of~$\\mathsf{E}$, i.e., $\\tau$~is\nan antiautomorphism of $\\mathsf{E}$ with $\\tau^2 =\\operatorname{id}$ and $\\tau(\\mathsf{E}_{\\gamma}) =\n\\mathsf{E}_{\\gamma}$ for each $\\gamma \\in \\Gamma_\\mathsf{E}$. We define $S_{\\tau}(\\mathsf{E})$ and\n$\\Sigma_{\\tau}(\\mathsf{E})$, analogously to the non-graded cases, as the set of\nelements of $\\mathsf{E}$ which are invariant under~$\\tau$, and the\nmultiplicative group\ngenerated by the nonzero homogenous elements of $S_\\tau(\\mathsf{E})$, respectively.\nWe say the involution of the {\\it first kind} if all the elements of the\ncenter $\\mathsf{T}$ are invariant under $\\tau$; it is of the {\\it second kind}\n(or {\\it unitary})\notherwise. If $\\tau$ is of the first kind then, parallel to the\nnon-graded case, either\n$\\dim_\\mathsf{T}(S_\\tau(\\mathsf{E}))=n(n+1)\/2$ or\n$\\dim_\\mathsf{T}(S_\\tau(\\mathsf{E}))=n(n-1)\/2$.\nIndeed, one can show these equalities\nby arguments analogous to the\nnongraded case as in the proof of \\cite [Prop.~2.6(1)]{kmrt}, as $\\mathsf{E}$ is split \nby a graded maximal subfield and the Skolem--Noether theorem is available\nin the graded setting (\\cite[Prop.~1.6]{hwcor}). (These equalities can also\nbe obtained\nby passing to the quotient division algebra as is done in\nLemma~\\ref{invfacts}\\eqref{type} below.)\n\n\n\n Define the {\\it\n unitary Whitehead group}\n$$\nK_1(\\mathsf{E},\\tau) \\ = \\ \\mathsf{E}^*\/\\big(\\Sigma_\\tau(\\mathsf{E})\\mathsf{E}'\\big),\n$$\nwhere $\\mathsf{E}' = [\\mathsf{E}^*,\\mathsf{E}^*]$.\nIf $\\tau$ is of the first kind,\n$\\operatorname{char}(\\mathsf{T})\\not =2$, and $\\dim_\\mathsf{T}(S_\\tau(\\mathsf{E}))=n(n-1)\/2$, a proof similar\nto \\cite[Prop.~2.9]{kmrt}, shows that if $a\n\\in S_\\tau(\\mathsf{E})$ is homogeneous, then $\\Nrd_\\mathsf{E}(a) \\in \\mathsf{T}^{*2}$\n(This can also be verified by passing to the quotient division\nalgebra, then using Lemma~\\ref{invfacts}\\eqref{type} below and invoking the\ncorresponding result for ungraded division algebras.) For\nthis type of involution, define the {\\it spinor Whitehead group}\n$$\nK_1\\operatorname{Spin}(\\mathsf{E},\\tau) \\ = \\ \\{a \\in \\mathsf{E}^*\\mid\n\\Nrd_\\mathsf{E}(a) \\in \\mathsf{T}^{*^2} \\} \\, \/ \\big( \\Sigma_\\tau(\\mathsf{E})\\mathsf{E}'\\big).\n$$\n\nWhen the graded involution $\\tau$ on $\\mathsf{E}$ is unitary, i.e., $\\tau|_\\mathsf{T} \\neq\n\\operatorname{id}$, let $\\mathsf{R}= \\mathsf{T}^{\\tau}$, which is a graded subfield of~$\\mathsf{T}$\nwith $[\\mathsf{T}:\\mathsf{R}]=2$. Furthermore, $\\mathsf{T}$ is Galois over $\\mathsf{R}$, with\n$\\operatorname{Gal} (\\mathsf{T}\/\\mathsf{R}) = \\{\\operatorname{id}, \\tau|_\\mathsf{T}\\}$. (See \\cite{hwalg}\nfor Galois theory for graded field extensions.) Define the\n{\\it reduced unitary Whitehead group}\n\\begin{equation}\\label{skdef}\n\\SK (\\mathsf{E}, \\tau) \\ = \\ \\Sigma'_{\\tau}(\\mathsf{E}) \\, \/ \\,\n(\\Sigma_{\\tau}(\\mathsf{E}) \\, \\mathsf{E}')\n \\ = \\ \\Sigma'_{\\tau}(\\mathsf{E}) \\, \/ \\,\n\\Sigma_{\\tau}(\\mathsf{E}) ,\n\\end{equation}\nwhere\n$$\n\\Sigma'_{\\tau}(\\mathsf{E}) \\ = \\ \\big\\{a\\in \\mathsf{E}^*\\mid\n\\Nrd_\\mathsf{E}(a^{1-\\tau})=1\\big \\} \\ = \\ \\{a\\in \\mathsf{E}^*\\mid \\Nrd_\\mathsf{E}(a) \\in \\mathsf{R}^*\\}\n$$\nand\n$$\n\\Sigma_{\\tau}(\\mathsf{E}) \\ = \\ \\langle a\\in \\mathsf{E}^* \\mid\na^{1-\\tau}=1 \\big \\rangle \\ = \\ \\langle S_\\tau(\\mathsf{E}) \\cap \\mathsf{E}^*\\rangle.\n$$\nHere, $a^{1-\\tau}$ means $a\\tau(a)^{-1}$. See Lemma~\\ref{invfacts}(iv)\nbelow for the second equality in \\eqref{skdef}.\nThe group $\\SK(\\mathsf{E}, \\tau)$ will be the main focus of the rest of the paper.\n\n\nWe will use the following facts repeatedly:\n\n\n\\begin{lemma}\\label{invfacts}\\hfill\n\\begin{enumerate}[\\upshape(i)]\n\\item \\label{type}\nAny graded involution on $\\mathsf{E}$ extends uniquely to\nan involution of the same kind\n$($and type$)$ on $Q=q(\\mathsf{E})$.\n\n\\smallskip\n\n\\item \\label{sigma} For any graded involution $\\tau$ on $\\mathsf{E}$, and its\nextension to $Q=q(\\mathsf{E})$, we have $\\Sigma_\\tau(Q) \\cap \\mathsf{E}^*\n\\subseteq \\Sigma_\\tau(\\mathsf{E})$.\n\n\\smallskip\n\n\\item \\label{firstsig} If $\\tau$ is a graded involution of\nthe first kind on $\\mathsf{E}$ with $\\dim_\\mathsf{T}(S_\\tau(\\mathsf{E}))=n(n+1)\/2$,\n then $\\Sigma_\\tau(\\mathsf{E})=\\mathsf{E}^*$.\n\n\\smallskip\n\n\\item \\label{yan} If $\\tau$ is a unitary graded involution on $\\mathsf{E}$,\nthen $\\mathsf{E}'\\subseteq \\Sigma_\\tau(\\mathsf{E})$.\n\n\n\\smallskip\n\n\\item \\label{tori} If $\\tau$ is a unitary graded involution on $\\mathsf{E}$,\nthen $\\SK(\\mathsf{E},\\tau)$ is a torsion group of bounded exponent\ndividing $n=\\operatorname{ind}(\\mathsf{E})$.\n\n\n\\end{enumerate}\\end{lemma}\n\n\\begin{proof}\\hfill\n\n\n(i) Let $\\tau$ be a graded involution on $\\mathsf{E}$. Then\n$q(\\mathsf{E})=\\mathsf{E}\\otimes_\\mathsf{T} q(\\mathsf{T})=\n\\mathsf{E}\\otimes_\\mathsf{T} (\\mathsf{T}\\otimes_{\\mathsf{T}^\\tau}q(\\mathsf{T}^\\tau))=\n\\mathsf{E}\\otimes_{\\mathsf{T}^\\tau} q(\\mathsf{T}^\\tau)$. The unique extension\nof $\\tau$ to $q(\\mathsf{E})$ is $\\tau \\otimes \\operatorname{id}_{q(\\mathsf{T}^\\tau)}$,\nwhich we denote simply as $\\tau$.\nIt then follows that\n$S_\\tau(q(\\mathsf{E}))=S_\\tau(\\mathsf{E})\\otimes_{\\mathsf{T}^\\tau} q(\\mathsf{T}^\\tau)$.\nSince $q(\\mathsf{T}^\\tau)=q(\\mathsf{T})^\\tau$, the assertion follows.\n\n(ii) Note that for the\n map $\\lambda$ in the sequence (\\ref{inji}) we have\n$\\tau(\\lambda(a))=\\lambda(\\tau(a))$ for all $a\\in Q^*$.\nHence, $\\lambda(\\Sigma_\\tau(Q)) \\subseteq\\Sigma_\\tau(\\mathsf{E})$.\nSince $\\lambda|_{\\mathsf{E}^*}$ is the identity, we have\n$\\Sigma_\\tau(Q) \\cap \\mathsf{E}^* \\subseteq \\Sigma_\\tau(\\mathsf{E})$.\n\n(iii) The extension of the graded involution $\\tau$ to\n$Q=q(\\mathsf{E})$, also denoted $\\tau$, is of the first kind with\n$\\dim_Q(S_\\tau(Q))=n(n+1)\/2$ by (\\ref{type}). Therefore\n$\\Sigma_\\tau(Q)=Q^*$ (see~\\S\\ref{firstk}). Using\n(\\ref{sigma}) now, the assertion follows.\n\n(iv) Since $\\tau$ is a unitary graded involution, its extension to\n$Q=q(\\mathsf{E})$ is also unitary, by (\\ref{type}). But\n${Q' \\subseteq \\Sigma_\\tau(Q)}$,\nas noted in \\eqref{primeinsigma}. From~(\\ref{inji}) it follows that\n$Q' \\cap \\mathsf{E}^* = \\mathsf{E}'$. Hence, using \\eqref{sigma}, \\break\n${\\mathsf{E}' \\subseteq \\mathsf{E}^* \\cap Q' \\subseteq \\mathsf{E}^* \\cap \\Sigma_\\tau(Q)\n\\subseteq \\Sigma_\\tau(\\mathsf{E})}$.\n\n\n\n(v) Setting $N=\\Sigma_\\tau'(\\mathsf{E})$,\nRemark~\\ref{grfacts}(\\ref{normal}) above, coupled with the fact\nthat $\\mathsf{E}' \\subseteq \\Sigma_\\tau(\\mathsf{E})$ (\\ref{yan}), implies that\n$\\SK(\\mathsf{E},\\tau)$ is an $n$-torsion group. This assertion also follows\nby using (\\ref{sigma}) which implies the natural map $\\SK(\\mathsf{E},\\tau)\n\\to \\SK(Q,\\tau)$ is injective and the fact that unitary\n$\\SK$ of a division algebra of index $n$ is\n$n$-torsion~(\\cite[Cor. to~2.5]{y}).\n\\end{proof}\n\n\\subsection{Generalized dihedral groups and field extensions}\n\nThe nontrivial case of $\\SK(\\mathsf{E},\\tau)$ for $\\tau$ a unitary\ngraded involution turns out to be when\n $\\mathsf{T} = Z(\\mathsf{E})$ is unramified over $\\mathsf{R} = \\mathsf{T}^\\tau$ (see \\S\\ref{unram}).\nWhen that occurs, we will see in\nLemma~\\ref{unramfacts}\\eqref{eight} below that $Z(\\mathsf{E}_0)$ is a so-called\ngeneralized dihedral extension over $\\mathsf{R}_0$.\nWe now give the definition and observe a few easy\n facts about generalized dihedral groups and extensions.\n\n\\begin{deff}\\label{gendi}\\hfill\n\\begin{enumerate}[\\upshape(i)]\n\\item \\label{deffo1} A group $G$ is said to be {\\it generalized dihedral} if $G$ has a\nsubgroup $H$ such that $[G:H]=2$ and every $\\tau \\in G \\backslash H$\nsatisfies $\\tau^2=\\operatorname{id}$.\n\nNote that if $G$ is generalized dihedral and $H$ the distinguished subgroup,\nthen $H$ is abelian and $(h\\tau)^2=\\operatorname{id}$, for all $\\tau \\in G\\backslash H$\nand $h\\in H$. Thus, $\\tau^2=\\operatorname{id}$ and $\\tau h \\tau ^{-1}=h^{-1}$ for all\n$\\tau \\in G\\backslash H, h \\in H$. Furthermore, every subgroup of $H$ is\nnormal in $G$. Clearly every dihedral group is generalized dihedral, as\nis every elementary abelian $2$-group. More generally, if $H$ is any\nabelian group and $\\chi \\in \\operatorname{Aut}(H)$ is the map $h \\mapsto h^{-1}$, then\nthe semi-direct product\n$H \\rtimes_i \\langle \\chi \\rangle$ is a generalized dihedral group,\nwhere $i\\colon\n\\langle \\chi \\rangle \\rightarrow \\operatorname{Aut}(H)$ is the inclusion map.\nIt is easy to check that every generalized dihedral group is isomorphic to\nsuch a semi-direct product.\n\n\\item \\label{deffo2} Let $F \\subseteq K \\subseteq L$ be fields with $[L:F]<\\infty$ and\n$[K:F]=2$. We say that $L$ is {\\it generalized dihedral for $K\/F$} if $L$ is\nGalois over $F$ and every element of $\\operatorname{Gal}(L\/F) \\backslash \\operatorname{Gal}(L\/K)$\nhas order $2$, i.e., $\\operatorname{Gal}(L\/F)$ is a generalized dihedral group.\nNote that when this occurs, $L$ is compositum of fields $L_i$\n containing~$K$ with each\n$L_i$ generalized dihedral for $K\/F$ with $\\operatorname{Gal}(L_i\/K)$ cyclic, i.e.,\n$L_i$ is Galois over $F$ with $\\operatorname{Gal}(L_i\/F)$ dihedral (or a Klein $4$-group).\nConversely, if $L$ and $M$ are generalized dihedral for $K\/F$ then so is\ntheir compositum.\n\n\\end{enumerate}\n\\end{deff}\n\\begin{example} Let $n\\in \\mathbb N$, $n \\geq 3$, and let $F\\subseteq K$ be fields with\n$[K:F]=2$ and $K=F(\\omega)$, where $\\omega$ is a primitive $n$-th root of unity\n(so $\\operatorname{char}(F) \\nmid n$). Suppose the non-identity element of $\\operatorname{Gal}(K\/F)$ maps\n$\\omega$ to~$\\omega^{-1}$. For any $c_1,\\dots,c_k \\in F^*$, if\n$\\omega \\not \\in F(\\sqrt[n]{c_1},\\dots,\\sqrt[n]{c_k})$, then\n$K(\\sqrt[n]{c_1},\\dots,\\sqrt[n]{c_k})$ is generalized dihedral for $K\/F$.\n\\end{example}\n\n\n\n\\section{Henselian to graded reduction}\\label{unitary}\n\nThe main goal of this section is to prove an isomorphism\nbetween the unitary $\\SK$ of a valued division\nalgebra with involution over a henselian field\n and the graded $\\SK$ of its associated graded division algebra.\nWe first recall how to associate a graded division algebra to a valued\ndivision algebra.\n\nLet $D$ be a division algebra finite dimensional over its\ncenter $K$, with a valuation\n$v\\colon D^{\\ast} \\rightarrow\n\\Gamma$. So, $\\Gamma$ is a totally ordered abelian group,\nand $v$ satisifies the conditions that for all\n$a, b \\in D^{\\ast}$,\n\\begin{enumerate}\n\\item $ \\qquad \\, v(ab) \\, = \\, v(a) + v(b)$;\n\n\\item $\\quad v(a+b) \\, \\geq \\, \\min \\{v(a),v(b) \\}\\;\\;\\;\\;\\; (b \\neq -a).$\n\\end{enumerate}\nLet\n\\begin{align*}\nV_D \\ &= \\ \\{ a \\in D^{\\ast} \\mid v(a) \\geq 0 \\}\\cup\\{0\\},\n\\text{ the\nvaluation ring of $v$};\\\\\nM_D \\ &= \\ \\{ a \\in D^{\\ast} \\mid v(a) > 0\n\\}\\cup\\{0\\}, \\text{ the unique maximal left (and right) ideal\n of $V_D$}; \\\\\n\\overline{D} \\ &= \\ V_D \/ M_D, \\text{ the residue\ndivision ring of $v$ on $D$; and} \\\\\n\\Gamma_D \\ &= \\ \\mathrm{im}(v), \\text{ the value\ngroup of the valuation}.\n\\end{align*}\n\nNow let $K$ be a field with a valuation $v$,\nand suppose $v$ is {\\it henselian}; that is,\n $v$ has a unique extension to every algebraic\nfield extension of $K$. Recall that a\n field extension $L$ of $K$ of degree~$n<\\infty$ is\nsaid to be {\\it tamely ramified} or {\\it tame} over $K$\nif, with respect to the unique extension of $v$ to $L$,\nthe residue field\n$\\overline L$ is separable over\n$\\overline K$ and $\\operatorname{char}({\\overline K})\\nmid\n\\big ( n\\big\/[\\overline L:\\overline K] \\big )$. Such an $L$~is\nnecessarily {\\it defectless} over $K$,\ni.e., $[L:K] = [\\overline L:\\overline K] \\, |\\Gamma_L:\\Gamma_K|$,\nby \\cite[Th.~3.3.3]{EP} (applied to $N\/K$ and $N\/L$, where\n$N$ is a normal closure of $L$ over $K$).\nAlong the same lines, let $D$ be a\ndivision algebra with center $K$ (so, by convention,\n$[D:K] < \\infty$); then the henselian valuation $v$ on $K$ extends uniquely\nto a valuation\non $D$ ~(\\cite{wad87}). With respect to this valuation, $D$\nis said to be\n{\\it tamely ramified} or {\\it tame} if the center $Z(\\overline D)$ is\n separable over $\\overline K$ and ${\\operatorname{char}({\\overline K}) \\nmid\n\\big[\\operatorname{ind}(D)\\big\/\\big(\\operatorname{ind}(\\overline D)[Z(\\overline D):\\overline K]\\big)}\\big]$.\nRecall from \\cite[Prop.~1.7]{jw}, that whenever the field extension\n$Z(\\overline D)\/\\overline K$ is separable, it is abelian Galois.\nIt is known that\n$D$ is tame if and only if $D$~is split by the\nmaximal tamely ramified field extension of~$K$, if and only if\n$\\operatorname{char}(\\overline K) = 0$\nor $\\operatorname{char}(\\overline K) = p\\ne 0$ and the $p$-primary component of~\n$D$ is inertially split, i.e., split by the maximal unramified\nextension of~$K$~(\\cite[Lemma~6.1]{jw}).\nWe say $D$ is \\emph{strongly tame}\nif $\\operatorname{char}(\\overline K)\\nmid\\operatorname{ind}(D)$.\nNote that strong tameness implies tameness.\nThis is clear from the last characterization of tameness,\nor from \\eqref{Ostrowski} below.\n Recall also from \\cite[Th.~3]{M}, that for a valued\ndivision algebra $D$ finite dimensional over its center $K$ (here not necessarily\nhenselian), we have the \\lq\\lq Ostrowski theorem\"\n\\begin{equation}\\label{Ostrowski}\n[D:K]\n\\ = \\ q^k\\,[\\overline D:\\overline K] \\,|\\Gamma_D:\\Gamma_K|,\n\\end{equation}\nwhere $q=\\operatorname{char}({\\overline D})$ and $k \\in \\mathbb Z$ with $k\\geq 0$\n(and $q^k = 1$ if $\\operatorname{char}(\\overline D) = 0$).\nIf $q^k = 1$ in equation~ \\eqref{Ostrowski}, then $D$~is\nsaid to be {\\it defectless} over $K$.\nFor background on valued division algebras,\nsee~\\cite{jw} or the survey paper~\\cite{wadval}.\n\n\\begin{remark}\\label{shensel}\nIf a field $K$ has a henselian valuation $v$ and $L$ is a subfield of\n$K$ with $[K:L] <\\infty$, then the restriction $w = v|_L$ need not be\nhenselian. But it is easy to see that $w$ is then \\lq\\lq semihenselian,\"\ni.e., $w$~has more than one but only finitely many different extensions to\na separable closure $L_{\\textrm{sep}}$ of $L$.\nSee \\cite{engler} for a thorough analysis of\nsemihenselian valuations. Notably, Engler shows\nthat $w$~is semihenselian iff the residue\nfield $\\overline L_w$ is algebraically closed but\nthere is a henselian valuation $u$ on $L$ such that $u$\n is a proper coarsening of $w$ and the residue field $\\overline L_u$ is\nreal closed. When this occurs, $\\operatorname{char}(L) = 0$, $L$ is formally real,\n$w$ has exactly two extensions to\n$L_{\\textrm{sep}}$, the value group $\\Gamma_{L,w}$\nhas a nontrivial divisible subgroup, and the henselization\nof $L$ re $w$ is $L(\\sqrt{-1})$, which lies in $K$.\nFor example, if we take any prime number $p$,\nlet $w_p$ be the\n$p$-adic discrete valuation on $\\mathbb Q$, and let $L = \\{r\\in \\mathbb R\\,|\\ r\n\\text{ is algebraic over $\\mathbb Q$}\\}$; then any extension of $w_p$ to\n$L$ is a semihenselian valuation.\nNote that if $v$ on $K$ is discrete, i.e., $\\Gamma_K\\cong \\mathbb Z$,\nthen $w$ on $L$ cannot be semihenselian, since $\\Gamma_L$ has no\nnontrivial divisible subgroup; so, $w$ on $L$ must be henselian.\nThis preservation of the henselian property for discrete valuations\nwas asserted in \\cite[Lemma, p.~195]{y}, but the proof given there is invalid.\n\\end{remark}\n\nOne associates to a valued division algebra $D$ a graded division algebra\nas follows:\nFor each $\\gamma\\in \\Gamma_D$, let\n\\begin{align*}\n D^{\\ge\\gamma} \\ &= \\\n\\{ d \\in D^{\\ast} \\mid v(d) \\geq \\gamma \\}\\cup\\{0\\}, \\text{ an additive\nsubgroup of $D$}; \\qquad \\qquad\\qquad\\qquad\\qquad \\ \\\\\nD^{>\\gamma} \\ &= \\ \\{ d \\in D^{\\ast} \\mid v(d) > \\gamma \\}\\cup\\{0\\},\n\\text{ a subgroup\nof $D^{\\ge\\gamma}$}; \\text{ and}\\\\\n \\operatorname{{\\sf gr}}(D)_\\gamma \\ &= \\\nD^{\\ge\\gamma}\\big\/D^{>\\gamma}.\n\\end{align*}\nThen define\n$$\n \\operatorname{{\\sf gr}}(D) \\ = \\ \\textstyle\\bigoplus\\limits_{\\gamma \\in \\Gamma_D}\n\\operatorname{{\\sf gr}}(D)_\\gamma. \\ \\\n$$\nBecause $D^{>\\gamma}D^{\\ge\\delta} \\,+\\, D^{\\ge\\gamma}D^{>\\delta}\n\\subseteq D^{>(\\gamma +\n\\delta)}$ for all $\\gamma , \\delta \\in \\Gamma_D$, the multiplication on\n$\\operatorname{{\\sf gr}}(D)$ induced by multiplication on $D$ is\nwell-defined, giving that $\\operatorname{{\\sf gr}}(D)$ is a graded ring, called the\n{\\it associated graded ring} of $D$. The\nmultiplicative property\n(1) of the valuation $v$ implies that $\\operatorname{{\\sf gr}}(D)$ is a graded\ndivision ring.\nClearly,\nwe have ${\\operatorname{{\\sf gr}}(D)}_0 = \\overline{D}$ and $\\Gamma_{\\operatorname{{\\sf gr}}(D)} = \\Gamma_D$.\nFor $d\\in D^*$, we write $\\widetilde d$ for the image\n$d + D^{>v(d)}$ of $d$ in $\\operatorname{{\\sf gr}}(D)_{v(d)}$. Thus,\nthe map given by $d\\mapsto \\widetilde d$ is\na group epimorphism $\\rho: D^* \\rightarrow {\\operatorname{{\\sf gr}}(D)^*}$ with\nkernel~$1+M_D$, giving us the short exact sequence\n\\begin{equation}\\label{grmap}\n1 \\ \\longrightarrow 1+M_D \\ \\longrightarrow D^* \\ \\longrightarrow \\ \\operatorname{{\\sf gr}}(D)^*\n \\ \\longrightarrow \\ 1,\n\\end{equation} which will be used throughout. For a detailed\nstudy of the associated graded algebra of a valued\ndivision algebra\nrefer to \\cite[\\S4]{hwcor}. As shown in \\cite[Cor.~4.4]{hazwadsworth},\nthe reduced norm maps\nfor $D$ and $\\operatorname{{\\sf gr}}(D)$ are related by\n\\begin{equation}\\label{nrdrel}\n\\widetilde{\\Nrd_D(a)} \\ = \\ \\Nrd_{\\operatorname{{\\sf gr}}(D)}(\\widetilde a) \\ \\ \\ \\text{for all }\na\\in D^*.\n\\end{equation}\n\n\nNow let $K$ be a field with a henselian valuation $v$ and, as before,\nlet $D$ be a division algebra with center~$K$. Then $v$ extends uniquely\nto a valuation on $D$, also denoted $v$,\n and one obtains associated to~$D$ the graded division algebra $\\,\\operatorname{{\\sf gr}}(D)=\n\\bigoplus_{\\gamma \\in \\Gamma_D} D_{\\gamma}$. Further, suppose $D$ is\ntame with respect\nto $v$. This implies that $[\\operatorname{{\\sf gr}}(D):\\operatorname{{\\sf gr}}(K)] = [D:K]$, $\\operatorname{{\\sf gr}}(K) = Z(\\operatorname{{\\sf gr}}(D))$ and\n$D$ has a maximal subfield $L$ with $L$ tamely ramified over\n$K$~(\\cite[Prop.~4.3]{hwcor}).\nWe can then associate to an\ninvolution $\\tau$ on $D$, a graded involution $\\widetilde \\tau$ on $\\operatorname{{\\sf gr}}(D)$.\nFirst, suppose\n$\\tau$ is of the first kind on $D$. Then $v \\circ \\tau$ is\nalso a valuation on $D$ which restricts to $v$ on $K$;\nthen, $v \\circ \\tau = v$\n since $v$ has a\nunique extension to $D$. So,\n$\\tau$ induces a well-defined map $\\widetilde{\\tau}\\colon \\operatorname{{\\sf gr}}(D) \\rightarrow \\operatorname{{\\sf gr}}(D)$,\ndefined on homogeneous elements by $\\widetilde{\\tau}(\\widetilde{a}) =\n\\widetilde{\\tau(a)}$ for all $a \\in D^{\\ast}$. Clearly,\n$\\widetilde{\\tau}$ is a well-defined\ngraded involution on $\\operatorname{{\\sf gr}}(D)$; it is of the first kind, as\nit leaves $Z(\\operatorname{{\\sf gr}}(D))=\\operatorname{{\\sf gr}}(K)$ invariant.\n\nIf $\\tau$ is a unitary involution on $D$, let $F=K^\\tau$. In\nthis case, we need to assume that the restriction of the valuation $v$\nfrom $K$ to $F$\ninduces a henselian valuation on $F$, and that $K$ is tamely ramified\nover $F$. Since $(v\\circ\\tau)|_F = v|_F$, an argument similar to the\none above\n shows that $v \\circ \\tau$ coincides with $v$ on $K$ and\nthus on~$D$, and the induced map $\\widetilde \\tau$ on $\\operatorname{{\\sf gr}}(D)$\nas above is a graded\ninvolution. That $K$ is tamely ramified over $F$\nmeans that $[K:F] = [\\operatorname{{\\sf gr}}(K):\\operatorname{{\\sf gr}}(F)]$, $\\overline{K}$ is separable\nover $\\overline{F}$, and $\\operatorname{char}(\\overline F)\n\\nmid |\\Gamma_K : \\Gamma_F|$. Since $[K:F]=2$, $K$ is always tamely\nramified over $F$ if $\\operatorname{char}(\\overline F) \\not = 2$. But if\n$\\operatorname{char}(\\overline{F})=2$, $K$ is tamely ramified over $F$ if and\nonly if $[\\overline{K}: \\overline{F}]= 2$, $\\Gamma_K = \\Gamma_F$, and\n$\\overline{K}$ is separable (so Galois) over $\\overline{F}$.\nSince $K$ is Galois over $F$, the canonical map $\\operatorname{Gal}(K\/F)\\to\n\\operatorname{Gal}(\\overline K\/\\overline F)$ is surjective, by \\cite[pp.~123--124, proof of\nLemma~5.2.6(1)]{EP}.\nHence, $\\tau$~induces the\nnonidentity $\\overline{F}$-automorphism $\\overline \\tau$ of $\\overline{K}$.\nAlso $\\widetilde \\tau$ is unitary, i.e.,\n$\\widetilde{\\tau}|_{\\operatorname{{\\sf gr}}(K)}\n\\neq \\operatorname{id}$. This is obvious if $\\operatorname{char} (\\overline{F}) \\neq 2$,\nsince then $K = F(\\sqrt c)$ for some $c \\in F^{\\ast}$, and\n$\\widetilde{\\tau}(\\widetilde{\\sqrt c}) =\n\\widetilde{\\tau (\\sqrt c)}=\n-\\widetilde{ \\sqrt c} \\neq \\widetilde{\\sqrt c}$.\nIf $\\operatorname{char}(\\overline{F})= 2$, then $K$ is unramified over $F$ and\n$\\widetilde{\\tau}|_{\\operatorname{{\\sf gr}}(K)_0}= \\overline{\\tau}$ (the automorphism\nof $\\overline{K}$ induced by $\\tau|_K$) which is nontrivial as\n$\\operatorname{Gal}(K\/F)$ maps onto $\\operatorname{Gal}(\\overline{K}\/ \\overline{F})$; so\nagain $\\widetilde{\\tau}|_{\\operatorname{{\\sf gr}}(K)} \\neq \\operatorname{id}$.\nThus, $\\widetilde{\\tau}$ is a unitary graded involution\nin any characteristic.\nMoreover, for the graded fixed field $\\operatorname{{\\sf gr}}(K)^{\\widetilde{\\tau}}$ we have\n$\\operatorname{{\\sf gr}}(F) \\subseteq \\operatorname{{\\sf gr}}(K)^{\\widetilde{\\tau}} \\subsetneqq \\operatorname{{\\sf gr}}(K)$ and\n$[\\operatorname{{\\sf gr}}(K):\\operatorname{{\\sf gr}}(F)] =2$, so $\\operatorname{{\\sf gr}}(K)^{\\widetilde{\\tau}} = \\operatorname{{\\sf gr}}(F)$.\n\n\n\n\n\n\\begin{thm}\\label{involthm1}\nLet $(D,v)$ be a tame valued division algebra over a\nhenselian field $K$,\nwith $\\operatorname{char}(\\overline K)\\not = 2$.\n If $\\tau$ is an involution of the first kind on $D$, then\n$$\nK_1(D,\\tau) \\ \\cong \\ K_1(\\operatorname{{\\sf gr}}(D),\\widetilde \\tau),\n$$\nand if $\\tau$ is symplectic, then\n$$\nK_1\\operatorname{Spin}(D,\\tau) \\ \\cong \\ K_1\\operatorname{Spin}(\\operatorname{{\\sf gr}}(D),\\widetilde \\tau).\n$$\n\\end{thm}\n\n\n\n\\begin{proof}\n\n Let $\\rho\\colon D^* \\rightarrow \\operatorname{{\\sf gr}}(D)^*$ be the\ngroup epimorphism given in (\\ref{grmap}). Clearly\n$\\rho(S_{\\tau}(D))\n\\subseteq S_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D))$, so\n$\\rho (\\Sigma_{\\tau}(D))\n\\subseteq \\Sigma_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D))$.\nConsider the following diagram:\n\n\\begin{equation} \\label{exactdiagramfirst}\n\\begin{split}\n\\xymatrix{\n1 \\ar[r] & (1+M_D) \\cap \\Sigma_{\\tau}(D)D' \\ar[r] \\ar[d]&\n\\Sigma_{\\tau}(D)D' \\ar[r]^-{\\rho} \\ar[d] &\n\\Sigma_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D))\\operatorname{{\\sf gr}}(D)' \\ar@{.>}[r] \\ar[d]& 1 \\\\\n1 \\ar[r] & (1+M_D) \\ar[r] &\nD^* \\ar[r]^-{\\rho} & \\operatorname{{\\sf gr}}(D)^*\n \\ar[r] & 1. }\n\\end{split}\n\\end{equation}\n\nThe top row of the diagram is exact. To see this, note that\n$\\rho(D')=\\operatorname{{\\sf gr}}(D)'$. Thus, it suffices to show that $\\rho$ maps\n$S_{\\tau}(D) \\cap D^{\\ast}$ onto $S_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D)) \\cap\n\\operatorname{{\\sf gr}}(D)^{\\ast}$. For this, take any $d \\in D^{\\ast}$ with $\\widetilde{d} =\n\\widetilde{\\tau}(\\widetilde{d})$. Let ${b = \\frac{1}{2} (d+ \\tau(d)) \\in\nS_{\\tau}(D)}$. Since $v(b) = v(\\tau(b))$ and $\\widetilde{d} +\n\\widetilde{\\tau(d)} = 2 \\widetilde{d} \\neq 0$, ${\\widetilde{b} = \\frac{1}{2}\n(\\widetilde{d+ \\tau(d)}) = \\frac{1}{2} (\\widetilde{d}+\n\\widetilde{\\tau(d)}) = \\widetilde{d}}$. Since $\\tau$ on $D$ is an\ninvolution of the first kind, the index of $D$ is a power of $2$\n(\\cite[Th.~1, \\S 16]{draxl}). As ${\\operatorname{char}(\\overline K)\\not = 2}$, it follows\nthat the valuation is strongly tame, and by \\cite[Lemma~2.1]{haz},\n$$\n{1+M_D \\ = \\ (1+M_K)[D^*,1+M_D] \\ \\subseteq \\ \\Sigma_{\\tau}(D)D'}.\n$$\nTherefore,\nthe left vertical map is the identity map. It follows (for example\nusing the snake lemma) that $K_1(D,\\tau) \\cong K_1(\\operatorname{{\\sf gr}}(D),\\widetilde\n\\tau)$. The proof for $K_1\\operatorname{Spin}$ when $\\tau$ is of symplectic type\nis similar.\n\\end{proof}\n\nThe key to proving the corresponding result for unitary\ninvolutions is the Congruence Theorem:\n\n\\begin{thm}[Congruence Theorem]\\label{congthm}\nLet $D$ be a tame division algebra over a\nfield $K$ with henselian valuation $v$. Let\n$D^{(1)} = \\{ a\\in D^*\\mid \\Nrd_D(a) = 1\\}$.\nThen,\n$$\nD^{(1)} \\, \\cap \\, (1+M_D) \\ \\subseteq \\ [D^*,D^*].\n$$\n\\end{thm}\n\nThis theorem was proved by Platonov in \\cite{platonov} for\n$v$ a complete discrete valuation, and it was an essential tool\nin all his calculations of $\\SK$ for division rings.\nThe Congruence Theorem was asserted by\nErshov in \\cite{ershov} in the generality given here.\nA full proof is given in \\cite[Th.~ B.1]{hazwadsworth}.\n\n\n\n\\begin{prop}[Unitary Congruence Theorem]\\label{unitarycongthm}\nLet $D$ be a tame division algebra over a\nfield~$K$ with henselian valuation $v$, and let\n $\\tau$ be a unitary involution on $D$. Let $F=K^\\tau$.\nIf $F$ is henselian with respect to $v|_F$\nand $K$ is tamely ramified over $F$, then\n$$\n(1+M_D) \\, \\cap \\,\n\\Sigma'_{\\tau}(D) \\ \\subseteq \\ \\Sigma_{\\tau}(D).\n$$\n\\end{prop}\n\\begin{proof} The only published proof of this we know is\n\\cite[Th.~4.9]{y}, which is just for the case $v$ discrete rank~1;\nthat proof is rather hard to follow, and appears to apply for other\n valuations only if $D$ is inertially split. Here we provide another\nproof, in full generality.\n\nWe use the well-known facts that\n\\begin{equation}\\label{nrd1+m}\n\\Nrd_D(1+M_D) \\ = \\ 1+M_K \\text{\\ \\ \\ and \\ \\ \\ }\nN_{K\/F}(1+M_K) \\ = \\ 1+M_F.\n\\end{equation}\n(The second equation holds as $K$ is tamely ramified over $F$.)\nSee \\cite[Prop.~2]{ershov} or \\cite[Prop.~4.6, Cor.~4.7]{hazwadsworth}\nfor a proof.\n\nNow, take $m \\in M_D$ with $\\Nrd_D(1 +m) \\in F$. Then $\\Nrd_D(1+m) \\in F\n\\cap(1+M_K) = 1 +M_F$. By \\eqref{nrd1+m} there is $c \\in 1 +M_K$ with\n$\\Nrd_D(1 +m) = N_{K\/F}(c) = c \\tau(c)$, and there is $b \\in 1+M_D$\nwith $\\Nrd_D(b) = c$. Then,\n$$\n\\Nrd_D(b \\tau(b)) \\ = \\ c \\tau(c) \\ = \\ N_{K\/F}(c) \\ = \\ \\Nrd_D(1+m).\n$$\nLet $s = (1+m)(b \\tau(b))^{-1} \\in 1+M_D$. Since $\\Nrd_D(s) = 1$, by\nthe Congruence Theorem for $\\SK$, Th.~\\ref{congthm} above,\n$s \\in [D^{\\ast}, D^{\\ast}]\n\\subseteq \\Sigma_{\\tau}(D)$, (recall \\eqref{primeinsigma})\n. Since $b \\tau(b) \\in\nS_{\\tau}(D)$, we have $1+m = s(b \\tau(b)) \\in \\Sigma_{\\tau}(D)$.\n\\end{proof}\n\n\n\n\\begin{thm}\\label{involthm2}\nLet $D$ be a tame division algebra over a\n field $K$ with henselian valuation $v$.\nLet $\\tau$ be a unitary involution on $D$, and let $F=K^\\tau$.\nIf $F$ is henselian with respect to $v|_F$\nand $K$ is tamely ramified over $F$, then $\\tau$ induces a\nunitary graded involution $\\widetilde{\\tau}$ of $\\operatorname{{\\sf gr}}(D)$ with\n$\\operatorname{{\\sf gr}}(F)=\\operatorname{{\\sf gr}}(K)^{\\widetilde{\\tau}}$, and\n$$\n\\SK(D, \\tau) \\ \\cong \\ \\SK(\\operatorname{{\\sf gr}}(D), \\widetilde{\\tau}).\n$$\n\\end{thm}\n\n\\begin{proof}\nThat\n$\\widetilde{\\tau}$ is a unitary graded involution\non $\\operatorname{{\\sf gr}}(D)$ and $\\operatorname{{\\sf gr}}(F)=\\operatorname{{\\sf gr}}(K)^{\\widetilde \\tau}$ was\nalready observed (see the discussion before Th.~\\ref{involthm1}).\nFor the canonical\nepimorphism $\\rho\\colon D^{\\ast} \\rightarrow \\operatorname{{\\sf gr}}(D)^{\\ast}$,\n$a \\mapsto \\widetilde{a}$, it follows from \\eqref{nrdrel} that\n$\\rho( \\Sigma'_{\\tau}(D) \\subseteq\n\\Sigma'_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D))$. Also, clearly $\\rho(S_{\\tau}(D))\n\\subseteq S_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D))$, so $\\rho (\\Sigma_{\\tau}(D))\n\\subseteq \\Sigma_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D))$. Thus, there is a\ncommutative diagram\n\\begin{equation} \\label{exactdiagram}\n\\begin{split}\n\\xymatrix{1 \\ar[r] & (1+M_D) \\cap \\Sigma_{\\tau}(D) \\ar[d] \\ar[r] &\n\\Sigma_{\\tau}(D) \\ar[d] \\ar[r]^-{\\rho} &\n\\Sigma_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D))\n\\ar[d] \\ar@{.>}[r] & 1 \\\\\n1 \\ar[r] & (1+M_D) \\cap \\Sigma'_{\\tau}(D) \\ar[r] &\n\\Sigma'_{\\tau}(D) \\ar[r]^-{\\rho} &\n\\Sigma'_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D)) \\ar@{.>}[r] & 1,}\n\\end{split}\n\\end{equation}\nwhere the vertical maps are inclusions, and the left vertical map is\nbijective, by Prop.~\\ref{unitarycongthm} above.\n\n\nTo see that the bottom row of diagram~\\eqref{exactdiagram} is exact at\n$\\Sigma'_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D))$, take $b \\in D$ with\n$\\Nrd_{\\operatorname{{\\sf gr}}(D)}(\\widetilde{b}) \\in \\operatorname{{\\sf gr}}(F)$. Let $c = \\Nrd_D(b) \\in\nK^{\\ast}$. Then $\\widetilde{c} = \\Nrd_{\\operatorname{{\\sf gr}}(D)} (\\widetilde{b}) \\in \\operatorname{{\\sf gr}}(F)$, so\n$\\widetilde{c} = \\widetilde{t}$ for some $t \\in F^{\\ast}$. Let $ u = c^{-1}\nt \\in 1 +M_K$. By \\eqref{nrd1+m} above, there is $d \\in 1+M_D$ with $\\Nrd_D(d)\n= u$. So, $\\Nrd_D(bd) = cu = t \\in F^{\\ast}$. Thus, $bd \\in\n\\Sigma'_{\\tau}(D)$ and $\\rho(bd) = \\widetilde{\\mspace{1mu} bd\\mspace{1mu} } = \\widetilde{b}$. This\ngives the claimed exactness, and shows that the bottom row of\ndiagram~\\eqref{exactdiagram} is exact.\n\nTo see that the top row of diagram~\\eqref{exactdiagram} is exact at\n$\\Sigma_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D))$, it suffices to show that $\\rho$ maps\n${S_{\\tau}(D) \\cap D^{\\ast}}$ onto $S_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D)) \\cap\n\\operatorname{{\\sf gr}}(D)^{\\ast}$. For this, take any $d \\in D^{\\ast}$ with $\\widetilde{d} =\n\\widetilde{\\tau}(\\widetilde{d})$. If $\\operatorname{char}(\\overline{F}) \\neq 2$, as\nin the proof of Th.~\\ref{involthm1}, let $b =\n\\frac{1}{2} (d+ \\tau(d)) \\in S_{\\tau}(D)$. Since $v(b) =\nv(\\tau(b))$ and $\\widetilde{d} + \\widetilde{\\tau(d)} = 2 \\widetilde{d} \\neq\n0$, we have ${\\widetilde{b} = \\frac{1}{2} (\\widetilde{d+ \\tau(d)}) = \\frac{1}{2}\n(\\widetilde{d}+ \\widetilde{\\tau(d)}) = \\widetilde{d}}$. If $\\operatorname{char}(\\overline{F})\n= 2$, then $K$ is unramified over $F$, so $\\overline{K}$ is Galois over~\n$\\overline{F}$ with $[\\overline{K}: \\overline{F}]=2$, and the map\n$\\overline{\\tau}\\colon\n\\overline{K} \\rightarrow \\overline{K}$ induced by $\\tau$ is the nonidentity\n$\\overline{F}$-automorphism of $\\overline{K}$. Of course,\n$\\overline{K} = \\operatorname{{\\sf gr}}(K)_0$ and\n$\\overline{\\tau} = \\widetilde{\\tau}|_{\\operatorname{{\\sf gr}}(K)_0}$. Because $\\overline{K}$\nis separable\nover $\\overline{F}$, the trace $\\textrm{tr}_{\\overline{K}\/\\overline{F}}$ is\nsurjective, so\nthere is $r \\in V_K$ with $\\widetilde{r} + \\widetilde{\\tau}(\\widetilde{r}) = 1\n\\in \\operatorname{{\\sf gr}}(F)_0$. Let $c = rd + \\tau(rd) \\in S_{\\tau}(D)$. We have\n$\\widetilde{\\mspace{1mu} rd\\mspace{1mu} } = \\widetilde{r}\\widetilde{d}$ and\n$$\n\\widetilde{\\tau(rd)} \\, = \\,\n\\widetilde{\\tau}(\\widetilde{\\mspace{1mu} rd\\mspace{1mu}})\n \\, = \\, \\widetilde{\\tau}(\\widetilde{r}\\widetilde{d}) \\, = \\,\n\\widetilde{\\tau}(\\widetilde{d})\\widetilde{\\tau}(\\widetilde{r}) \\, = \\,\n\\widetilde{\\tau}(\\widetilde{r})\\widetilde{d}.\n$$\n Since $v(rd) = v(\\tau(rd))$ and\n$\\widetilde{\\mspace{1mu} rd\\mspace{1mu} } + \\widetilde{\\tau(rd)} = \\widetilde{r}\\widetilde{d}\n+\\widetilde{\\tau}(\\widetilde{r})\\widetilde{d} = \\widetilde{d} \\neq 0$, we\nhave\n$\\widetilde{c} = \\widetilde{\\mspace{1mu} rd\\mspace{1mu} } + \\widetilde{\\tau(rd)} = \\widetilde{d}$. So,\nin all cases $\\rho(S_{\\tau}(D) \\cap D^{\\ast}) = S_{\\widetilde \\tau}(\\operatorname{{\\sf gr}}(D)) \\cap\n\\operatorname{{\\sf gr}}(D)^{\\ast}$, from which it follows that the bottom row of\ndiagram~\\eqref{exactdiagram}\nis exact. Since each row of~(\\ref{exactdiagram}) is exact,\nwe have a right exact sequence of\ncokernels of the vertical maps, which yields the isomorphism of the\ntheorem.\n\\end{proof}\n\nHaving established the bridge between the unitary $K$-groups in\nthe graded setting and the non-graded henselian case (Th.~\\ref{involthm1},\nTh.~\\ref{involthm2}), we\ncan deduce\nknown formulas in the literature for the unitary Whitehead\ngroup of certain valued division algebras, by passing to\nthe graded setting. The proofs are much easier than those\n previously available.\nWe will do this systematically for unitary involutions\nin Section~\\ref{grsec}. Before we turn to that, here is an example\nwith an involution of the first kind:\n\n\\begin{example} Let $\\mathsf{E}$ be a graded division algebra over\nits center $\\mathsf{T}$ with an involution $\\tau$ of the first kind.\nIf $\\mathsf{E}$~is unramified over $\\mathsf{T}$, then, by using $\\mathsf{E}^*=\\mathsf{E}_0^*\\mathsf{T}^*$, it\nfollows easily that\n\\begin{equation}\\label{kuni}\nK_1(\\mathsf{E},\\tau) \\ \\cong \\ K_1(\\mathsf{E}_0,\\tau|_{\\mathsf{E}_0}),\n\\end{equation} and, if $\\operatorname{char}(\\mathsf{E}) \\ne 2$ and $\\tau$ is symplectic,\n\\begin{equation}\nK_1\\operatorname{Spin}(\\mathsf{E}, \\tau) \\ \\cong \\ K_1\\operatorname{Spin}(\\mathsf{E}_0,\\tau|_{\\mathsf{E}_0}).\n\\end{equation}\nNow if $D$ is a tame and unramified division algebra\nover a henselian valued field and $D$ has an\n involution $\\tau$ of the first kind, then the associated\ngraded division ring $\\operatorname{{\\sf gr}}(D)$ is also unramified with the\ncorresponding graded involution $\\widetilde \\tau$ of the first kind;\nthen Th.~\\ref{involthm1} and (\\ref{kuni}) above show that\n$$\nK_1(D,\\tau) \\ \\cong \\ K_1(\\operatorname{{\\sf gr}}(D),\\widetilde{\\tau})\n \\ \\cong \\ K_1(\\operatorname{{\\sf gr}}(D)_0,\\tau|_{\\operatorname{{\\sf gr}}(D)_0}) \\ = \\ K_1(\\overline{D},\n\\overline{\\tau}),\n$$\nyielding a theorem of Platonov-Yanchevski\\u\\i~\\cite[Th. 5.11]{py85}\n(that $K_1(D,\\tau)\\cong K_1(\\overline{D},\n\\overline{\\tau})$ when $D$ is unramified over $K$ and\nthe valuation is henselian and discrete rank $1$.)\nSimilarly, when $\\operatorname{char}(\\overline D) \\ne 2$ and $\\tau$ is symplectic,\n$$\nK_1\\operatorname{Spin}(D,\\tau) \\ \\cong \\ K_1\\operatorname{Spin}(\\operatorname{{\\sf gr}}(D),\\widetilde{\\tau})\n \\ \\cong \\ K_1\\operatorname{Spin}(\\operatorname{{\\sf gr}}(D)_0,\\tau|_{\\operatorname{{\\sf gr}}(D)_0}) \\ = \\ K_1\\operatorname{Spin}(\\overline{D},\n\\overline{\\tau}).\n$$\n\\end{example}\n\n\\begin{remark}\\label{goodrem}\nWe have the following commutative diagram connecting unitary $\\SK$ to\nnon-unitary $\\SK$,\nwhere $\\SH(D,\\tau)$ and $\\SH(D)$ are the cokernels of $\\Nrd\\circ(1-\\tau)$\nand $\\Nrd$ respectively (see diagram~\\eqref{unnrd}).\n\n\n\\begin{equation}\\label{goodd}\n\\begin{split}\n\\xymatrix{\n1 \\ar[r] & \\SK(D,\\tau) \\ar[r] \\ar[d] & D^*\/\\Sigma(D)\n\\ar[rr]^-{\\Nrd\\circ(1-\\tau)} \\ar[d]^{1-\\tau} &\n& K^* \\ar[r] \\ar[d]^{\\operatorname{id}} & \\SH(D,\\tau) \\ar[r] \\ar[d]& 1\\\\\n1 \\ar[r] & \\SK(D) \\ar[r] & D^*\/D' \\ar[rr]^-{\\Nrd} &\n& K^* \\ar[r] & \\SH(D) \\ar[r] & 1.\n}\n\\end{split}\n\\end{equation}\n\nNow, let $D$ be a tame valued division algebra with center $K$ and with\na unitary involution $\\tau$, such that the valuation\nrestricts to a henselian valuation on $F=K^\\tau$. By\nTh.~\\ref{involthm2}, $\\SK(D,\\tau)\\cong \\SK(\\operatorname{{\\sf gr}}(D),\\widetilde \\tau)$\nand by \\cite[Th.~4.8, Th.~4.12]{hazwadsworth}, $\\SK(D)\\cong \\SK(\\operatorname{{\\sf gr}}(D))$\nand $\\SH(D)\\cong\\SH(\\operatorname{{\\sf gr}}(D))$. However, $\\SH(D,\\tau)$ is not stable\nunder ``valued filtration'', i.e.,\n$\\SH(D,\\tau)\\not \\cong \\SH(\\operatorname{{\\sf gr}}(D),\\widetilde \\tau)$. In fact\nusing~(\\ref{grmap}), we can build a commutative diagram with exact rows,\n\\begin{equation*}\n\\begin{split}\n\\xymatrix{\n1 \\ar[r] & (1+M_K)\\cap \\Nrd(D^*)^{1-\\tau} \\ar[r] \\ar@{^{(}->}[d] & \\Nrd(D^*)^{1-\\tau} \\ar[r] \\ar@{^{(}->}[d]& \\Nrd\\big(\\operatorname{{\\sf gr}}(D)^*\\big)^{1-\\widetilde \\tau} \\ar[r]\\ar@{^{(}->}[d] & 1\\\\\n1 \\ar[r] & 1+M_K \\ar[r] & K^* \\ar[r] & \\operatorname{{\\sf gr}}(K)^* \\ar[r] & 1,\n}\n\\end{split}\n\\end{equation*}\nwhich induces the exact sequence\n$$\n1\\longrightarrow (1+M_K)\\big\/\\big( (1+M_K)\\cap\\Nrd(D^*)^{1-\\tau}\\big)\n\\longrightarrow\n\\SH(D,\\tau) \\longrightarrow \\SH(\\operatorname{{\\sf gr}}(D),\\widetilde \\tau) \\longrightarrow 1.\n$$\nBy considering the norm $N_{K\/F}\\colon K^* \\rightarrow F^*$, we clearly have\n$\\Nrd(D^*)^{1-\\tau}\n\\subseteq \\ker N_{K\/F}$. However, by \\eqref{nrd1+m},\n${N_{K\/F}\\colon 1+M_K\\rightarrow 1+M_F}$ is surjective, which shows that\n$1+M_K\\big\/\\big( (1+M_K)\\cap\\Nrd(D^*)^{1-\\tau}\\big)$ is not trivial and thus\n${\\SH(D,\\tau)\\not \\cong \\SH(\\operatorname{{\\sf gr}}(D),\\widetilde \\tau)}$.\n\\end{remark}\n\n\\section{Graded Unitary $\\SK$ Calculus}\\label{grsec}\n\nLet $\\mathsf{E}$ be a graded division algebra over its center $\\mathsf{T}$\nwith a unitary graded involution $\\tau$, and\nlet $\\mathsf{R}=\\mathsf{T}^\\tau$.\nSince $[\\mathsf{T}:\\mathsf{R}]=2=[\\mathsf{T}_0:\\mathsf{R}_0]\\,|\\Gamma_\\mathsf{T}:\\Gamma_\\mathsf{R}|$,\nthere are just two possible cases:\n\n\n\\begin{itemize}\n\\item[$\\bullet$] $\\mathsf{T}$ is totally ramified over\n$\\mathsf{R}$, i.e., $|\\Gamma_\\mathsf{T}:\\Gamma_\\mathsf{R}|=2$\n\n\\item [$\\bullet$] $\\mathsf{T}$ is unramfied over\n$\\mathsf{R}$, i.e., $|\\Gamma_\\mathsf{T}:\\Gamma_\\mathsf{R}|=1$.\n\\end{itemize}\n\nWe will consider $\\SK(\\mathsf{E},\\tau)$ in these two cases separately\nin \\S\\ref{totram} and \\S\\ref{unram}.\n\nThe following notation will be used throughout this section and the next:\nLet $\\tau'$ be another involution on $\\mathsf{E}$. We write\n$\\tau' \\sim \\tau$ if\n$\\tau'|_{Z(\\mathsf{E})}=\\tau|_{Z(\\mathsf{E})}$. For $t \\in \\mathsf{E}^*$,\nlet $\\varphi_t$ denote the map from $E$ to $E$ given by conjugation by $t$,\ni.e., $\\varphi_t(x)=txt^{-1}$.\nLet $ \\Sigma_0=\\Sigma_\\tau \\cap \\mathsf{E}^*_0$ and\n$\\Sigma'_0=\\Sigma_\\tau'\\cap \\mathsf{E}_0^*$.\n\n\nWe first collect some facts which will be used below.\nThey all follow by easy calculations.\n\n\n\\begin{remarks}\\hfill\n\\begin{enumerate}[\\upshape(i)]\\label{easyob}\n\\item \\label{one} We have $\\tau' \\sim \\tau$ if and only if there is a\n$t\\in \\mathsf{E}^*$ with $\\tau(t)=t$ and $\\tau'=\\tau \\varphi_t$.\n(The proof is analogous to the ungraded version given, e.g. in\n\\cite[ Prop.~2.18]{kmrt}.)\n\n\\smallskip\n\n\\item \\label{two} If $\\tau'\\sim \\tau$, then $\\Sigma_{\\tau'}=\\Sigma_{\\tau}$\nand $\\Sigma'_{\\tau'}=\\Sigma'_{\\tau}$; thus $\\SK(\\mathsf{E},\\tau')=\\SK(\\mathsf{E},\\tau)$.\n(See~\\cite[Lemma~1]{yin} for the analogous ungraded result.)\n\\smallskip\n\\item \\label{three} For any $s \\in \\mathsf{E}^*$, we have\n$\\tau \\varphi_s=\\varphi_{\\tau(s)^{-1}}\\tau$. Hence, $\\tau \\varphi_s$ is an involution\n(necessarily $\\sim \\tau$) if and only if $\\tau \\varphi_s=\\varphi_{s^{-1}}\\tau$\nif and only if $\\tau(s)\/s \\in \\mathsf{T}$.\n\\smallskip\n\\item \\label{four} If $s \\in \\mathsf{E}^*_\\gamma$ and $\\tau(s)=s$, then\n$\\Sigma_\\tau' \\cap \\mathsf{E}_\\gamma= s \\Sigma'_0$ and\n$S_\\tau\\cap \\mathsf{E}_\\gamma=s(S_{\\tau_s}\\cap \\mathsf{E}_0)$ where $\\tau_s=\\tau \\varphi_s$.\n\\smallskip\n\\end{enumerate}\n\\end{remarks}\n\n\n\\subsection{$\\mathsf{T}\/\\mathsf{R}$ totally ramified} \\label{totram}\nLet $\\mathsf{E}$ be a graded division algebra with a unitary graded involution $\\tau$\n such that $\\mathsf{T}=Z(\\mathsf{E})$ is totally\nramified over $\\mathsf{R}=\\mathsf{T}^\\tau$. In this section we will show\nthat $\\SK(\\mathsf{E},\\tau)=1$. Note that the assumption that $\\mathsf{T}\/\\mathsf{R}$ is\ntotally ramified implies that $\\operatorname{char}(\\mathsf{T}) \\ne 2$. For, if\n$\\operatorname{char}(\\mathsf{T}) = 2$ and $\\mathsf{T}$ is totally ramified over a graded\nsubfield $\\mathsf{R}$ with $[\\mathsf{T}:\\mathsf{R}] = 2$, then for any $x\\in \\mathsf{T}^*\\setminus\n\\mathsf{R}^*$, we have $\\deg(x^2) \\in \\Gamma_R$, so $x^2\\in \\mathsf{R}$; thus, $\\mathsf{T}$ is\npurely inseparable over $\\mathsf{R}$. That cannot happen here, as\n$\\tau|_\\mathsf{T}$ is a nontrivial $\\mathsf{R}$-automorphism of~$\\mathsf{T}$.\n\n\\begin{lemma}\\label{six}\nIf $\\mathsf{T}$ is totally ramified over $\\mathsf{R}$, then\n$\\tau\\sim\\tau'$ for some graded involution $\\tau'$, where\n$\\tau'|_{\\mathsf{E}_0}$ is of the first kind.\n\\end{lemma}\n\n\\begin{proof}\nLet $Z_0=Z(\\mathsf{E}_0)$.\nSince $\\mathsf{T}$ is totally ramified over $\\mathsf{R}$, $\\mathsf{T}_0=\\mathsf{R}_0$, so $\\tau|_{Z_0}\n\\in \\operatorname{Gal}(Z_0\/\\mathsf{T}_0)$. Since the map\n$\\Theta_\\mathsf{E}\\colon\\Gamma_\\mathsf{E} \\rightarrow \\operatorname{Gal}(Z_0\/\\mathsf{T}_0)$ is surjective\n(see \\eqref{surj}), there is $ \\gamma\\in \\Gamma_\\mathsf{E}$\nwith $\\Theta_\\mathsf{E}(\\gamma)=\\tau|_{Z_0} $. Choose $y \\in \\mathsf{E}^*_\\gamma$ with\n$\\tau(y)=\\pm y$. Then set $\\tau'=\\tau\\varphi_{y^{-1}}$.\n\\end{proof}\n\n\\begin{example}\\label{trex} Here is a construction of examples of\n graded division algebras\n$\\mathsf{E}$ with unitary graded involution~$\\tau$ with $E$ totally ramified over\n$Z(\\mathsf{E})^\\tau$. We will see below that these are all such examples.\nLet $\\mathsf{R}$ be any graded field\nwith $\\operatorname{char}(\\mathsf{R})\\ne 2$, and let $\\mathsf{A}$ be a graded division algebra\nwith center $\\mathsf{R}$, such that $\\mathsf{A}$~is totally ramified over $\\mathsf{R}$ with\n$\\exp(\\Gamma_\\mathsf{A}\/\\Gamma_\\mathsf{R}) = 2$. Let $\\mathsf{T}$ be a graded field\nextension of $\\mathsf{R}$ with $[\\mathsf{T}:\\mathsf{R}] = 2$, $\\mathsf{T}$ totally ramified over $\\mathsf{R}$,\nand $\\Gamma_\\mathsf{T}\\cap \\Gamma_\\mathsf{A} = \\Gamma_\\mathsf{R}$. Let $\\mathsf{E} = \\mathsf{A}\\otimes_\\mathsf{R}\n\\mathsf{T}$, which is a graded central simple algebra over $\\mathsf{T}$, as\n$\\mathsf{A}$ is graded central simple over $\\mathsf{R}$, by \\cite[Prop.~1.1]{hwcor}.\nBut because $\\Gamma_\\mathsf{T}\\cap \\Gamma_\\mathsf{A} = \\Gamma_\\mathsf{R}$, we have\n$\\mathsf{E}_0 = \\mathsf{A}_0 \\otimes_{\\mathsf{R}_0} \\mathsf{T}_0 = \\mathsf{R}_0 \\otimes_{\\mathsf{R}_0} \\mathsf{R}_0 = \\mathsf{R}_0$.\nSince $\\mathsf{E}_0$ is a division ring, $\\mathsf{E}$ must be a graded division ring, which is\ntotally ramified over $\\mathsf{R}$, as $\\mathsf{E}_0 = \\mathsf{R}_0$. Now, because $\\mathsf{A}$ is\ntotally ramified over $\\mathsf{R}$, we have $\\exp(\\mathsf{A}) = \\exp(\\Gamma_\\mathsf{A}\/\n\\Gamma_\\mathsf{R}) = 2$, and $\\mathsf{A}= \\mathsf{Q}_1\\otimes_\\mathsf{R} \\ldots \\otimes_\\mathsf{R}\\mathsf{Q}_m$, where\neach $\\mathsf{Q}_i$ is a graded symbol algebra of degree at most $2$, i.e., a\ngraded quaternion algebra. Let $\\sigma_i$ be a graded involution of the\nfirst kind on $\\mathsf{Q}_i$ (e.g., the canonical symplectic graded involution),\n and let $\\rho$ be the nonidentity $\\mathsf{R}$-automorphism of $\\mathsf{T}$.\nThen, $\\sigma = \\sigma_1\\otimes \\ldots \\otimes \\sigma_m$ is a graded involution\nof the first kind on $\\mathsf{A}$, so $\\sigma\\otimes \\rho$ is a unitary graded\ninvolution on~$\\mathsf{E}$, with $\\mathsf{T}^\\tau = \\mathsf{R}$.\n\n\n\\end{example}\n\n\\begin{prop}\\label{total}\nIf $\\mathsf{E}$ is totally ramified over $\\mathsf{R}$, and $\\mathsf{E} \\ne \\mathsf{T}$,\nthen $\\Sigma_{\\tau}=\\mathsf{E}^*$, so\n$\\SK(\\mathsf{E},\\tau)=1$. Furthermore, $\\mathsf{E}$ and $\\tau$ are as described\nin Ex.~\\ref{trex}.\n\\end{prop}\n\n\\begin{proof}\nWe have $\\mathsf{E}_0=\\mathsf{T}_0=\\mathsf{R}_0$.\nFor any $\\gamma \\in \\Gamma_\\mathsf{E}$, there is a nonzero $a\\in \\mathsf{E}_\\gamma$\nwith $\\tau(a) = \\epsilon a$ where $\\epsilon = \\pm 1$. Then,\nfor any $b\\in \\mathsf{E}_\\gamma$, $b = ra$ for some $r\\in \\mathsf{E}_0 = \\mathsf{R}_0$.\nSince $r$ is central and symmetric, $\\tau(b) = \\epsilon b$.\nThus, every element of $\\mathsf{E}^*$ is symmetric or\nskew-symmetric. Indeed, fix any $t\\in \\mathsf{T}^*\\setminus \\mathsf{R}^*$.\nThen $\\tau(t) \\ne t$, as $t\\notin \\mathsf{R}^*$. Hence, $\\tau(t) =\n-t$. Since $t$ is central and skew-symmetric, every\n$a\\in \\mathsf{E}^* $ is symmetric iff $ta$ is skew-symmetric.\nThus, $\\mathsf{E}^* = S_\\tau^* \\cup tS_\\tau^*$. To see that\n$\\Sigma_\\tau = \\mathsf{E}^*$, it suffices to show that $t\\in \\Sigma_\\tau$.\nTo see this, take any $c,d\\in \\mathsf{E}^*$ with $dc\\ne cd$. (They exist, as\n$\\mathsf{E} \\ne \\mathsf{T}$.) By replacing $c$ (resp.~$d$) if necessary by\n$tc$ (resp.~$td$), we may assume that $\\tau(c) = c$ and $\\tau(d) = d$.\nThen, $dc = \\tau(cd) = \\epsilon cd$, where $\\epsilon = \\pm 1$;\nsince $dc\\ne cd$, $\\epsilon = -1$; hence $\\tau(tcd) = tcd$.\nThus, $t = (tcd) c^{-1} d^{-1} \\in \\Sigma_\\tau(\\mathsf{E})$, completing the proof\nthat $\\Sigma_\\tau(\\mathsf{E}) = \\mathsf{E}^*$.\n\nFor $\\gamma \\in \\Gamma_\\mathsf{E}$, let $\\overline \\gamma = \\gamma+\\Gamma_\\mathsf{T} \\in\n\\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T}$.\nTo see the structure of $\\mathsf{E}$, recall that as $\\mathsf{E}$ is totally ramified\nover~$\\mathsf{T}$ there is a well-defined nondegenerate $\\mathbb Z$-bilinear symplectic\npairing $\\beta\\colon (\\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T}) \\times\\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T})\n\\to \\mathsf{E}_0^*$ given by $\\beta(\\overline \\gamma, \\overline \\delta) = y_\\gamma y_\\delta\ny_\\gamma^{-1}y_\\delta^{-1}$ for any nonzero $y_\\gamma\\in \\mathsf{E}_\\gamma$,\n$y_\\delta\\in \\mathsf{E}_\\delta$. The computation above for $c$ and $d$ shows\nthat $\\operatorname{im} (\\beta) = \\{\\pm 1\\}$. Since the pairing $\\beta$ is nondegenerate\nby \\cite[Prop.~2.1]{hwcor} there is a symplectic base of $\\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T}$, i.e., a subset\n$\\{\\overline \\gamma_1, \\overline \\delta_1, \\ldots , \\overline \\gamma_m, \\overline \\delta_m\\}$ of\n$\\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T}$ such that $\\beta(\\overline \\gamma_i, \\overline \\delta_i) = -1$\nwhile $\\beta(\\overline \\gamma_i, \\overline \\gamma_j) = \\beta(\\overline \\delta_i, \\overline \\delta_j)\n=1$ for all $i,j$, and $\\beta(\\overline \\gamma_i, \\overline \\delta_j) = 1$ whenever\n$i\\ne j$, and $\\Gamma_\\mathsf{E} = \\langle \\gamma_1, \\delta_1,\\ldots,\n\\gamma_m, \\delta_m\\rangle + \\Gamma_\\mathsf{T}$. Choose any nonzero $\\mathbf{i}_i\\in\n\\mathsf{E}_{\\gamma_i}$ and $\\mathbf{j}_i\\in \\mathsf{E}_{\\delta_i}$. The properties of the\n$\\overline\\gamma_i, \\overline \\delta_i$ under $\\beta$ translate to:\n$\\mathbf{i}_i \\mathbf{j}_i = - \\mathbf{j}_i \\mathbf{i}_i$ while $\\mathbf{i}_i \\mathbf{i}_j = \\mathbf{i}_j \\mathbf{i}_i$\nand $\\mathbf{j}_i \\mathbf{j}_j = \\mathbf{j}_j \\mathbf{j}_i$ for all $i,j$, and\n$\\mathbf{i}_i \\mathbf{j}_j = \\mathbf{j}_j \\mathbf{i}_i$ whenever $i \\ne j$. Since $\\beta(\n\\overline {2\\gamma_i}, \\overline \\eta) = 1$ for all $i$ and all $\\eta\\in \\Gamma_\\mathsf{E}$,\neach~$\\mathbf{i}_i^2$ is central in $\\mathsf{E}$. But also $\\tau(\\mathbf{i}_i^2) = \\mathbf{i}_i^2$,\nas $\\tau(\\mathbf{i}_i) = \\pm \\mathbf{i}_i$. So, each $\\mathbf{i}_i^2 \\in \\mathsf{R}^*$, and likewise each\n$\\mathbf{j}_i^2\\in \\mathsf{R}^*$. Let\n$\\mathsf{Q}_i = \\mathsf{R}\\text{-span}\\{1, \\mathbf{i}_i, \\mathbf{j}_i, \\mathbf{i}_i\\mathbf{j}_i\\}$ in~$\\mathsf{E}$.\nThe relations on the $\\mathbf{i}_i, \\mathbf{j}_i$ show that each $\\mathsf{Q}_i$ is a\ngraded quaternion algebra over $\\mathsf{R}$, and the distinct~$\\mathsf{Q}_i$ centralize\neach other in $\\mathsf{E}$. Since each $\\mathsf{Q}_i$ is graded central simple over $\\mathsf{R}$,\n$\\mathsf{Q}_1\\otimes_\\mathsf{R} \\ldots \\otimes_\\mathsf{R}\\mathsf{Q}_m$ is graded central simple over\n$\\mathsf{R}$\nby \\cite[Prop.~1.1]{hwcor}. Let $\\mathsf{A} = \\mathsf{Q}_1\\ldots \\mathsf{Q}_m \\subseteq \\mathsf{E}$. The graded\n$\\mathsf{R}$-algebra epimorphism $\\mathsf{Q}_1\\otimes_\\mathsf{R} \\ldots \\otimes_\\mathsf{R}\\mathsf{Q}_m \\to\n\\mathsf{A}$ must be an isomorphism, as the domain is graded simple.\nIf $\\Gamma_\\mathsf{T} \\subseteq \\Gamma_\\mathsf{A}$, then $\\mathsf{T} \\subseteq \\mathsf{A}$,\nsince $\\mathsf{E}$ is totally ramified over $\\mathsf{R}$. But this cannot occur, as\n$\\mathsf{T}$ centralizes $\\mathsf{A}$ but $\\mathsf{T}\\supsetneqq \\mathsf{R} = Z(\\mathsf{A})$. Hence,\nas $|\\Gamma_\\mathsf{T}:\\Gamma_\\mathsf{R}| = 2$, we must have $\\Gamma_\\mathsf{T} \\cap \\Gamma_\\mathsf{A}\n = \\Gamma_\\mathsf{R}$.\nThe graded $\\mathsf{R}$-algebra homomorphism $\\mathsf{A} \\otimes_\\mathsf{R} \\mathsf{T} \\to\n\\mathsf{E}$ is injective since its domain is graded simple, by \\cite[Prop.~1.1]{hwcor};\nit is also surjective, since $\\mathsf{E}_0 = \\mathsf{R}_0 \\subseteq \\mathsf{A}\\otimes_\\mathsf{R} \\mathsf{T}$\nand $\\Gamma_{\\mathsf{A} \\otimes_\\mathsf{R} \\mathsf{T}} \\supseteq \\langle \\gamma_1, \\delta_1,\\ldots,\n\\gamma_m, \\delta_m\\rangle +\\Gamma_\\mathsf{T} = \\Gamma_\\mathsf{E}$. Clearly,\n$\\tau = \\tau|_\\mathsf{A} \\otimes \\tau|_\\mathsf{T}$.\n\\end{proof}\n\n\n\n\n\n\n\\begin{prop}\\label{completely}\nIf $\\mathsf{E} \\ne \\mathsf{T}$ and $\\mathsf{T}$ is totally ramified over $\\mathsf{R}$, then\n$\\Sigma_{\\tau}=\\mathsf{E}^*$, so $\\SK(\\mathsf{E},\\tau)=1$.\n\\end{prop}\n\n\n\\begin{proof}\nThe case where $\\mathsf{E}_0 = \\mathsf{T}_0$ was covered by Prop.~\\ref{total}.\nThus, we may assume that $\\mathsf{E}_0 \\supsetneqq\\mathsf{T}_0$.\nBy Lemma~\\ref{six} and Remark~\\ref{easyob}(\\ref{two}), we can assume\nthat $\\tau|_{\\mathsf{E}_0}$ is of the first kind. Further, we can assume that\n$\\mathsf{E}_0^*=\\Sigma_{\\tau|_{\\mathsf{E}_0}}(\\mathsf{E}_0).$\nFor, if $\\tau|_{\\mathsf{E}_0}$\nis symplectic, take any $a\\in \\mathsf{E}_0^*$ with $\\tau(a)=-a$, and let\n$\\tau'=\\tau\\varphi_a$.\nThen, $\\tau' \\sim \\tau$ (see Remark~\\ref{easyob}(\\ref{three})).\n Also, $\\tau'|_{Z(\\mathsf{E}_0)}=\\tau|_{Z(\\mathsf{E}_0)}$, as $a \\in \\mathsf{E}_0$ and so\n$\\varphi_a |_{Z(\\mathsf{E}_0)}=\\operatorname{id}$. Therefore, $\\tau'|_{\\mathsf{E}_0}$ is of the first kind.\nBut as $\\tau(a)=-a$, $\\tau'|_{\\mathsf{E}_0}$ is orthogonal.\nThus $\\mathsf{E}_0^*=\\Sigma_{\\tau'|_{\\mathsf{E}_0}}(\\mathsf{E}_0)$, as noted at the beginning of\n\\S\\ref{firstk}. Now replace $\\tau$ by $\\tau'$.\n\nWe consider two cases.\n\n{Case I.} Suppose for each $\\gamma\\in \\Gamma_\\mathsf{E}$ there is\n$x_\\gamma \\in \\mathsf{E}^*_\\gamma$ such that $\\tau(x_\\gamma)=x_\\gamma$.\n Then, $\\mathsf{E}^* = \\bigcup_{\\gamma \\in \\Gamma_\\mathsf{E}}\\mathsf{E}_0^*x_\\gamma\n\\subseteq \\Sigma_\\tau(\\mathsf{E})$, as desired.\n\n{Case II.} Suppose there is $\\gamma\\in \\Gamma_\\mathsf{E}$ with\n$\\mathsf{E}_\\gamma \\cap S_\\tau=0$. Then $\\tau(d)=-d$ for each $d\\in \\mathsf{E}_\\gamma$.\nFix $t \\in \\mathsf{E}^*_\\gamma$. For any $a \\in \\mathsf{E}_0$, we have $ta \\in \\mathsf{E}_\\gamma$;\n so,\n$-ta=\\tau(ta)=\\tau(a)\\tau(t)=-\\tau(a)t$. That is,\n\\begin{equation}\\label{localeq}\n\\tau(a) \\, = \\, \\varphi_t(a)\\quad\\text{for all }a \\in \\mathsf{E}_0.\n\\end{equation}\nLet $\\tau''=\\tau\\varphi_t$, which is a unitary involution on $\\mathsf{E}$ with\n$\\tau''\\sim\\tau$ (see Remark~\\ref{easyob}(\\ref{three})). But,\n$\\tau''(a)=a$ for all\n$a \\in \\mathsf{E}_0$, i.e., $\\tau''|_{\\mathsf{E}_0}=\\operatorname{id}$. This implies that $\\mathsf{E}_0$ is\na field. Replace $\\tau$ by $\\tau''$. The rest of the argument uses this\nnew $\\tau$.\nSo $\\tau|_{\\mathsf{E}_0}=\\operatorname{id}$. If we are now in\nCase I for this $\\tau$, then we are done by Case I. So, assume we are\nin Case II.\nTake any $\\gamma\\in \\Gamma_\\mathsf{E}$ with\n$\\mathsf{E}_\\gamma \\cap S_\\tau=0$. For any nonzero $t\\in \\mathsf{E}_\\gamma$,\nequation~(\\ref{localeq}) applies to $t$, showing $\\varphi_t(a)=\\tau(a)=a$\nfor all $a \\in \\mathsf{E}_0$; hence for the map $\\Theta_\\mathsf{E}$ of\n\\eqref{surj}, $\\Theta_\\mathsf{E}(\\gamma) = \\operatorname{id}_{\\mathsf{E}_0}$. But recall that\n$\\mathsf{E}_0$~is Galois over $\\mathsf{T}_0$ and $\\Theta_\\mathsf{E}\\colon \\Gamma_\\mathsf{E}\\to\n\\operatorname{Gal}(\\mathsf{E}_0\/\\mathsf{T}_0)$ is surjective. Since $\\mathsf{E}_0 \\ne \\mathsf{T}_0$,\n there is $\\delta \\in \\Gamma_\\mathsf{E}$ with\n$\\Theta_\\mathsf{E}(\\delta) \\ne \\operatorname{id}$. Hence, there must be some\n$s\\in \\mathsf{E}_\\delta^*\\cap S_\\tau$. Likewise, since $\\Theta_\\mathsf{E}(\\gamma-\\delta)\n= \\Theta_\\mathsf{E}(\\gamma) \\Theta_\\mathsf{E}(\\delta)^{-1} \\ne \\operatorname{id}$, there\nis some $r\\in \\mathsf{E}_{\\gamma-\\delta}^*\\cap S_\\tau$. Then,\nas $rs\\in \\mathsf{E}_\\gamma^*$, we have $\\mathsf{E}_\\gamma^* = \\mathsf{E}_0^*rs \\subseteq\n\\Sigma_\\tau$. This is true for every $\\gamma$ with $\\mathsf{E}_\\gamma\\cap S_\\tau\n= 0$. But for any other $\\gamma\\in \\Gamma_\\mathsf{E}$, there is an $x_\\gamma$\nin $\\mathsf{E}_\\gamma^* \\cap S_\\tau$; then $\\mathsf{E}_\\gamma^* = \\mathsf{E}_0^*x_\\gamma \\subseteq\n\\Sigma_\\tau$. Thus, $\\mathsf{E}^* = \\bigcup_{\\gamma\\in \\Gamma_\\mathsf{E}}\\mathsf{E}_\\gamma^*\n\\subseteq \\Sigma_\\tau$.\n \\end{proof}\n\n\n\\subsection{$\\mathsf{T}\/\\mathsf{R}$ unramified} \\label{unram}\n\nLet $\\mathsf{E}$ be a graded division algebra with a unitary involution\n $\\tau$ such that $\\mathsf{T}=Z(\\mathsf{E})$ is unramified over $\\mathsf{R}=\\mathsf{T}^\\tau$.\nIn this subsection, we will give a general formula for $\\SK(\\mathsf{E},\\tau)$ in\nterms of data in $\\mathsf{E}_0$.\n\n\\begin{lemma}\\label{unramfacts}\nSuppose $\\mathsf{T}$ is unramified over $\\mathsf{R}$. Then,\n\\begin{enumerate}[\\upshape (i)]\n \\item \\label{seven}\nEvery $\\mathsf{E}_\\gamma$ contains both nonzero symmetric\nand skew symmetric elements.\n\n\\smallskip\n\n \\item \\label{eight}\n$Z(\\mathsf{E}_0)$ is a generalized dihedral extension for\n$\\mathsf{T}_0$ over\n$\\mathsf{R}_0$ $($see Def.~\\ref{gendi}$)$.\n\n\\smallskip\n\n \\item \\label{nine}\nIf $\\mathsf{T}$ is unramified over $\\mathsf{R}$, then\n$\\SK(\\mathsf{E},\\tau)=\\Sigma'_0\/\\Sigma_0$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\\hfill\n\n(i) If $\\operatorname{char}(\\mathsf{E})=2$, it is easy to see that every\n$\\mathsf{E}_\\gamma$ contains a symmetric element (which is also skew\nsymmetric) regardless of any assumption on $\\mathsf{T}\/\\mathsf{R}$.\nLet $\\operatorname{char}(\\mathsf{E})\\not=2$.\nSince $[\\mathsf{T}_0:\\mathsf{R}_0]=2$ and $\\mathsf{R}_0=\\mathsf{T}_0^\\tau$, there is\n$ c\\in \\mathsf{T}_0$ with $\\tau(c)=-c$. Now there is\n$t\\in \\mathsf{E}_\\gamma$, $t\\not =0$, with $\\tau(t)=\\epsilon t$ where\n$\\epsilon=\\pm 1$. Then $\\tau(c t)=-\\epsilon c t $.\n\n(ii) Let $G=\\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{R}_0)$ and $H=\\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{T}_0)$.\nNote that $[G:H]=2$. Since $\\tau$ is unitary,\n$\\tau |_{Z(\\mathsf{E}_0)} \\in G\\setminus H$. We will denote\n$\\tau |_{Z(\\mathsf{E}_0)}$ by $\\overline \\tau$ and will show that for\nany $h\\in H$, $(\\overline \\tau h)^2=1$. By (\\ref{surj}),\n$\\Theta_\\mathsf{E}\\colon \\Gamma_\\mathsf{E} \\rightarrow \\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{T}_0)$ is onto,\nso there is $\\gamma \\in \\Gamma_\\mathsf{E}$, such that $\\Theta_\\mathsf{E}(\\gamma)=h$.\nAlso by (\\ref{seven}),\nthere is an $x\\in \\mathsf{E}_\\gamma^*$ with $\\tau(x)=x$.\nThen $\\tau\\varphi_x$ is an involution, where $\\varphi_x$ is conjugation by $x$;\ntherefore, $\\tau \\varphi_x|_{Z(\\mathsf{E}_0)} \\in G$ has order $2$.\nBut $\\varphi_x|_{Z(\\mathsf{E}_0)}=\\Theta_\\mathsf{E}(\\gamma)=h$. Thus\n$(\\overline \\tau h)^2=1$.\n\n(iii) By~(\\ref{seven}), for each $\\gamma \\in \\Gamma_\\mathsf{E}$, there is\n$s_\\gamma \\in \\mathsf{E}_\\gamma$, $s_\\gamma\\not =0$, with\n$\\tau(s_\\gamma)=s_\\gamma$. By Remark~\\ref{easyob}\\eqref{four},\n${\\Sigma_\\tau'=\\bigcup_{\\gamma \\in \\Gamma_\\mathsf{E}} s_\\gamma \\Sigma'_0}$.\nSince each $s_\\gamma \\in S_\\tau \\subseteq \\Sigma_\\tau$, the injective map\n$\\Sigma'_0\/\\Sigma_0 \\rightarrow \\Sigma_\\tau'\/\\Sigma_\\tau$ is an isomorphism.\n\\end{proof}\n\nTo simplify notation in the next theorem, let $\\overline \\tau =\n\\tau|_{Z(\\mathsf{E}_0)} \\in \\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{R}_0)$, and\nfor any $h \\in \\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{T}_0)$,\n write $\\Sigma_{h \\overline \\tau}(\\mathsf{E}_0)$ for $\\Sigma_\\rho(\\mathsf{E}_0)$ for any unitary\ninvolution $\\rho$ on $\\mathsf{E}_0$ such that $\\rho|_{Z(\\mathsf{E}_0)}=h\\overline \\tau$.\nThis is well-defined, independent of the choice of $\\rho$, by the ungraded\nanalogue of Remark\\eqref{easyob}\\eqref{two}.\n\n\\begin{theorem}\\label{msem}\nLet $\\mathsf{E}$ be a graded division algebra with center $\\mathsf{T}$, with a unitary\ngraded involution $\\tau$, such that $\\mathsf{T}$~is unramified over $\\mathsf{R}=\\mathsf{T}^\\tau$.\nFor each $\\gamma \\in \\Gamma_\\mathsf{E}$ choose a nonzero\n$x_\\gamma \\in S_\\tau \\cap \\mathsf{E}_\\gamma$. Let $H=\\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{T}_0)$.\nThen,\n$$\n\\SK(\\mathsf{E},\\tau) \\ \\cong \\ (\\Sigma'_\\tau \\cap \\mathsf{E}_0) \\big \/\n(\\Sigma_\\tau \\cap \\mathsf{E}_0),\n$$\nwith\n\\begin{equation}\\label{ersh}\n\\Sigma'_\\tau \\cap \\mathsf{E}_0 \\ = \\\n\\big \\{ a \\in \\mathsf{E}_0^*\\mid\nN_{Z(\\mathsf{E}_0)\/\\mathsf{T}_0}\\Nrd_{\\mathsf{E}_0}(a)^\\partial\\in \\mathsf{R}_0 \\big \\},\n\\textrm{ \\ \\ \\ where \\ \\ \\ } \\partial \\, = \\\n\\operatorname{ind}(\\mathsf{E})\/\\big(\\operatorname{ind}(\\mathsf{E}_0) \\, [Z(\\mathsf{E}_0):\\mathsf{T}_0]\\big)\n\\end{equation}\nand\n\\begin{equation}\\label{ersh1}\n\\Sigma_\\tau\\cap\\mathsf{E}_0 \\ = \\ P\\cdot X, \\textrm{\\ \\ \\ where\\ \\ \\ }\nP\\, =\\ \\textstyle{\\prod}_{h \\in H}\\Sigma_{h\\overline \\tau}(\\mathsf{E}_0)\n\\textrm{ \\ \\ and \\ \\ }\nX\\, = \\ \\langle x_\\gamma x_\\delta x_{\\gamma+\\delta}^{-1} \\mid\n\\gamma,\\delta \\in \\Gamma_\\mathsf{E} \\rangle \\, \\subseteq \\, \\mathsf{E}_0^*.\n\\end{equation}\nFurthermore, if $H=\\langle h_1,\\dots,h_m \\rangle$, then\n$P \\, = \\ \\prod_{(\\varepsilon_1,\\dots,\\varepsilon_m)\\in \\{0,1\\}^m}\n\\Sigma_{h_1^{\\varepsilon_1}\\dots h_m^{\\varepsilon_m} \\overline \\tau}(\\mathsf{E}_0)$.\n\\end{theorem}\n\nBefore proving the theorem, we record the following:\n\n\\begin{lemma}\\label{lem5}\nLet $A$ be a central simple algebra over a field $K$, with an\ninvolution $\\tau$ and an automorphism or anti-automorphism $\\sigma$.\nThen,\n\n\\begin{enumerate}[\\upshape(i)]\n\\item \\label{hyh1} $\\sigma \\tau \\sigma^{-1}$ is an involution of $A$ of the same\n kind as $\\tau$, and\n$$\nS_{\\sigma \\tau \\sigma^{-1}} \\, = \\, \\sigma(S_\\tau),\n\\textrm{ \\ \\ so \\ \\ } \\Sigma_{\\sigma \\tau \\sigma^{-1}} \\, = \\,\n\\sigma(\\Sigma_\\tau).\n$$\n\n\\item \\label{hyh2} Suppose $A$ is a division ring.\nIf $\\sigma$ and $\\tau$ are each unitary involutions, then\n$($writing {${S_\\tau^*=S_\\tau\\cap A^*}$}$)$,\n$$\nS_\\tau^* \\ \\subseteq \\ S_\\sigma^* \\cdot \\sigma(S_\\tau^*)\n \\ = \\ S_\\sigma^* \\cdot S_{\\sigma \\tau \\sigma^{-1}}^*, \\textrm{ \\ \\ so \\ \\ ~~}\n\\Sigma_\\tau \\ \\subseteq \\ \\Sigma_\\sigma\\cdot \\Sigma_{\\sigma \\tau \\sigma^{-1}}.\n$$\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\\hfill\n\n(i) This follows by easy calculations.\n\n(ii) Observe that if $a \\in S_\\tau^*$,\nthen $a=\\big (a\\sigma(a)\\big)\\sigma(a^{-1})$ with ${a\\sigma(a) \\in S_\\sigma^*}$\nand $\\sigma(a^{-1}) \\in \\sigma(S_\\tau^*)=S_{\\sigma \\tau \\sigma^{-1}}^*$ by~(\\ref{hyh1}).\nThus,~(\\ref{hyh2}) follows from~(\\ref{hyh1}) and the fact\nthat ${A' \\subseteq \\Sigma_\\tau \\cap \\Sigma_\\sigma}$\n(see \\eqref{primeinsigma}).\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{msem}]\n\nFirst note that by Lemma~\\ref{unramfacts}\\eqref{nine} the canonical map\n$$\n(\\Sigma'_\\tau \\cap \\mathsf{E}_0) \\, \\big\/ \\, (\\Sigma_\\tau \\cap \\mathsf{E}_0)\n \\ \\longrightarrow \\ \\Sigma'_\\tau\/\\Sigma_\\tau \\ = \\ \\SK(\\mathsf{E},\\tau)\n$$\nis an isomorphism.\nThe description of $\\Sigma'_\\tau \\cap \\mathsf{E}_0$\nin~(\\ref{ersh}) is immediate from the fact that\nfor $a \\in \\mathsf{E}_0$,\n$\\Nrd_{\\mathsf{E}}(a)=N_{Z(\\mathsf{E}_0)\/\\mathsf{T}_0}\\Nrd_{\\mathsf{E}_0}(a)^\\partial\\in \\mathsf{T}_0$\n(see Remark~\\ref{grfacts}\\eqref{rnrd}).\n\nFor $\\Sigma_\\tau \\cap \\mathsf{E}_0$, note that for each\n$\\gamma \\in \\Gamma_\\mathsf{E}$, if $a\\in \\mathsf{E}_0$, then\n$a x_\\gamma \\in S_\\tau$ if and only if\n $x_\\gamma \\tau(a) x_\\gamma^{-1}=a$. That is,\n$S_\\tau \\cap \\mathsf{E}_\\gamma = S(\\varphi_{x_\\gamma} \\tau;\\mathsf{E}_0)x_\\gamma$,\nwhere $S(\\varphi_{x_\\gamma} \\tau;\\mathsf{E}_0)$ denotes the set of\nsymmetric elements in $\\mathsf{E}_0$ for the unitary involution\n$\\varphi_{x_\\gamma} \\tau|_{\\mathsf{E}_0}$.\nTherefore,\n$$\n\\Sigma_\\tau \\cap \\mathsf{E}_0 \\ = \\\n\\big \\langle S(\\varphi_{x_\\gamma} \\tau;\\mathsf{E}_0)^* x_\\gamma \\mid\n\\gamma \\in \\Gamma_\\mathsf{E} \\big \\rangle \\, \\cap \\, \\mathsf{E}_0.\n$$\nTake a product $a_1x_1\\ldots a_kx_k$ in $\\Sigma_\\tau \\cap \\mathsf{E}_0$\nwhere each $x_i=x_{\\gamma_i}$ for some $\\gamma_i \\in \\Gamma_\\mathsf{E}$\nand $a_i \\in S(\\varphi_{x_i} \\tau;\\mathsf{E}_0)^*$. Then,\n\\begin{equation}\\label{pet}\na_1x_1\\dots a_k x_k \\ = \\\na_1\\varphi_{x_1}(a_2)\\ldots\\varphi_{x_1\\ldots x_{i-1}}(a_i)\\ldots\n\\varphi_{x_1\\ldots x_{k-1}}(a_k)x_1\\ldots x_k \\, \\in \\,\n\\mathsf{E}_{\\gamma_1+\\ldots+\\gamma_k}.\n\\end{equation}\nSo, $\\gamma_1+\\ldots+\\gamma_k=0$. Now, as $a_i \\in S(\\varphi_{x_i} \\tau;\\mathsf{E}_0)$\nand $\\tau \\varphi_{x_j}^{-1}=\\varphi_{x_j}\\tau$ for all $j$,\nby Lemma~\\ref{lem5}(\\ref{hyh1}) we obtain\n\\begin{equation}\\label{longd}\n\\varphi_{x_1\\ldots x_{i-1}}(a_i)\\, \\in \\,\nS(\\varphi_{x_1}\\ldots \\varphi_{x_{i-1}}(\\varphi_{x_i}\\tau)\n\\varphi_{x_{i-1}}^{-1}\\ldots\\varphi_{x_1}^{-1};\n\\mathsf{E}_0)^* \\ = \\ S(\\varphi_{x_1\\ldots x_{i-1}x_ix_{i-1}\\ldots x_1} \\tau;\\mathsf{E}_0)^*\n \\ \\subseteq \\ \\Sigma_{h\\overline \\tau}(\\mathsf{E}_0) \\ \\subseteq \\, P,\n\\end{equation}\nwhere $h=\\varphi_{x_1\\dots x_{i-1}x_ix_{i-1}\\dots x_1}|_{Z(\\mathsf{E}_0)} \\in H$.\nNote also that if $k=1$, then\n$x_1 \\in S_\\tau \\cap \\mathsf{E}_0^* \\subseteq \\Sigma_{\\overline \\tau}(\\mathsf{E}_0)\n\\subseteq P.$\n\nIf $k>1$, then\n$$\nx_1\\ldots x_k \\ = \\ x_{\\gamma_1}\\ldots x_{\\gamma_k} \\\n= \\ (x_{\\gamma_1} x_{\\gamma_2}\nx_{\\gamma_1+\\gamma_2}^{-1})(x_{\\gamma_1+\\gamma_2}x_{\\gamma_3}\n\\ldots x_{\\gamma_k}),\n$$\nwith\n$(\\gamma_1+\\gamma_2)+\\gamma_3+\\ldots+\\gamma_k=0$. It follows by induction\non $k$ that $x_1 \\dots x_k \\in X$. With this and ~(\\ref{pet})\nand ~(\\ref{longd}), we have $a_1x_1\\dots a_kx_k \\in P\\cdot X$\n(which is a group, as $\\mathsf{E}_0' \\subseteq \\Sigma_{\\overline \\tau}(\\mathsf{E}_0)\n\\subseteq P$\nby\n\\eqref{primeinsigma}), showing that $\\Sigma_\\tau \\cap \\mathsf{E}_0\n\\subseteq P \\cdot X$. For the reverse inclusion, take any\n$h \\in H$ and choose $\\gamma \\in \\Gamma_\\mathsf{E}$ with\n$\\varphi_{x_\\gamma}|_{Z(\\mathsf{E}_0)}=h$. Then, $x_\\gamma \\in S_\\tau^*\n\\subseteq \\Sigma_\\tau$ and $S(\\varphi_{x_\\gamma}\\tau;\\mathsf{E}_0)^*x_\\gamma=\nS_\\tau^*\\cap \\mathsf{E}_\\gamma \\subseteq \\Sigma_\\tau$, so\n$\\Sigma_{h\\overline \\tau}(\\mathsf{E}_0)=\\Sigma_{\\varphi_{x_\\gamma}\\tau}(\\mathsf{E}_0)=\n\\langle S(\\varphi_{x_\\gamma}\\tau;\\mathsf{E}_0)^* \\rangle \\subseteq\n\\Sigma_\\tau \\cap \\mathsf{E}_0$. Thus, $P \\subseteq \\Sigma_\\tau \\cap \\mathsf{E}_0$,\nand clearly also $X \\subseteq \\Sigma_\\tau \\cap \\mathsf{E}_0$. Hence,\n$\\Sigma_\\tau \\cap \\mathsf{E}_0 =P\\cdot X$.\n\nThe final equality for $P$ in the Theorem follows from\nLemma~\\ref{lembe} below by taking $U=\\mathsf{E}_0^*$, $A=H$, and\n$W_h=\\Sigma_{h\\overline \\tau}(\\mathsf{E}_0)$ for $h \\in H$. To see that\nthe lemma applies, note that each $\\Sigma_{h\\overline \\tau}(\\mathsf{E}_0)$\ncontains $\\mathsf{E}_0'$ by \\eqref{primeinsigma}. Furthermore, take any $h,\\ell \\in H$, and choose\n$x,y \\in E^* \\cap S_\\tau$ with $\\varphi_x|_{Z(\\mathsf{E}_0)}=h$ and\n$\\varphi_y|_{Z(\\mathsf{E}_0)}=\\ell$. Then,\n$$\n(\\varphi_y\\tau)(\\varphi_x\\tau)(\\varphi_y\\tau)^{-1} \\ = \\\n\\varphi_y\\tau\\varphi_x\\varphi_y^{-1} \\ = \\ \\varphi_{yx^{-1}y}\\tau,\n$$\nand $\\varphi_{yx^{-1}y}|_{Z(\\mathsf{E}_0)}=\\ell h^{-1}\\ell=\\ell^2h^{-1}$. Hence, by\nLemma~\\ref{lem5}(\\ref{hyh2}),\n$\\Sigma_{h\\overline \\tau}(\\mathsf{E}_0)\\subseteq\n\\Sigma_{\\ell\\overline \\tau}(\\mathsf{E}_0)\\Sigma_{\\ell^2h^{-1}\\overline \\tau}(\\mathsf{E}_0)$.\nThis shows that hypothesis~(\\ref{hyp}) of Lemma~\\ref{lembe} below is\nsatisfied here.\n\\end{proof}\n\n\\begin{lemma}\\label{lembe}\nLet $U$ be a group, $A$ an abelian group, and $\\{W_a\\mid a\\in A\\}$ a family\nof subgroups of $U$ with each $W_a \\supseteq [U,U]$. Suppose\n\\begin{equation}\\label{hyp}\nW_a \\, \\subseteq \\, W_bW_{2b-a} \\textrm{ \\ \\ for~all } a,b \\in A.\n\\end{equation}\nIf $A=\\langle a_1,\\dots,a_m \\rangle$, then\n$$\n\\textstyle{\\prod\\limits_{a\\in A}} W_a \\ = \\\n\\textstyle{\\prod\\limits_{(\\varepsilon_1,\\dots,\\varepsilon_m)\\in \\{0,1\\}^m}}\nW_{\\varepsilon_1a_1+\\ldots+\\varepsilon_ma_m}. $$\n\\end{lemma}\n\\begin{proof}\nSince each $W_a \\supseteq [U,U]$, we have $W_aW_b=W_bW_a$, and this is a\nsubgroup of $U$, for all $ a,b \\in A$.\nLet\n$$\nQ \\, = \\textstyle{\\prod\\limits_{(\\varepsilon_1,\\dots,\\varepsilon_m)\\in \\{0,1\\}^m}}\n W_{\\varepsilon_1a_1+\\ldots+\\varepsilon_ma_m}.\n$$\nWe prove by induction on $m$ that each $W_a\\subseteq Q$. The lemma then\nfollows, as $Q$ is a subgroup of $U$.\nNote that condition~(\\ref{hyp}) can be conveniently restated,\n\\begin{equation}\\label{hyp1}\n\\textrm{ if \\ } a+b \\, = \\, 2d \\in A, \\textrm{ \\ then \\ }\nW_a \\, \\subseteq \\, W_dW_b.\n\\end{equation}\nTake any $c\\in A$. Then, (\\ref{hyp1}) shows that $W_{-c}\\subseteq W_0W_c$.\nTake any $i \\in \\mathbb Z$, and suppose $W_{ic}\\subseteq W_0W_c$. Then by\n(\\ref{hyp1}) $W_{-ic}\\subseteq W_0W_{ic} \\subseteq W_0W_c$. So, by\n(\\ref{hyp1}) again, $W_{(i+2)c}\\subseteq W_cW_{-ic} \\subseteq W_0W_c$ and\n${W_{(i-2)c}\\subseteq W_{-c}W_{-ic} \\subseteq W_0W_c}$. Hence, by induction\n(starting with $j=0$ and $j=1$), $W_{jc}\\subseteq W_0W_c$ for every\n$j \\in \\mathbb Z$. This proves the lemma when $m=1$.\n\nNow assume $m>1$ and let $B=\\langle a_1,\\dots,a_{m-1} \\rangle \\subseteq A$.\nBy induction, for all\n$b\\in B$,\n$$\nW_b \\ \\subseteq\n\\textstyle{\\prod\\limits_{(\\varepsilon_1,\\dots,\\varepsilon_{m-1})\\in \\{0,1\\}^{m-1}}}\nW_{\\varepsilon_1a_1+\\ldots+\\varepsilon_{m-1}a_{m-1}} \\ \\subseteq \\, Q.\n$$\nAlso, by the cyclic case done above,\n$W_{j a_m} \\subseteq W_0W_{a_m} \\subseteq Q$\nfor all $j \\in \\mathbb Z$. So, for any $b \\in B, j \\in \\mathbb Z$,\nusing~(\\ref{hyp1}),\n\\begin{equation}\\label{thte}\nW_{2b+ja_m} \\, \\subseteq \\, W_b W_{-ja_m} \\, \\subseteq \\, Q,\n\\end{equation}\nand\n\\begin{equation}\\label{thte2}\nW_{b+2ja_m} \\, \\subseteq \\, W_{ja_m} W_{-b} \\, \\subseteq \\, Q.\n\\end{equation}\nLet $d=a_{i_1}+\\ldots+a_{i_\\ell}$ for any indices\n$1\\leq i_1< i_2 <\\ldots 2$ then $\\mathsf{T}_0=\\mathsf{R}_0(\\mu_e)$, and $\\tau$ acts on $\\mu_e$ by\n$\\omega\\mapsto \\omega^{-1}$.\n\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nSince $\\mathsf{T}$ is unramified over $\\mathsf{R}$ and $\\mathsf{E}_0=\\mathsf{T}_0$, the formulas of\nTh.~\\ref{msem} for $\\SK(\\mathsf{E},\\tau)$ reduce to $\\partial = n$ and\n\\begin{equation}\\label{gendesb}\n\\SK(\\mathsf{E},\\tau) \\ \\cong \\ \\{a\\in \\mathsf{T}_0^*\\mid a^n\\in \\mathsf{R}_0^* \\} \\,\n\\big \/ \\, \\big(\\mathsf{R}_0^* \\, \\langle x_\\gamma x_\\delta x_{\\gamma+\\delta}^{-1}\n\\mid \\gamma, \\delta \\in \\Gamma_\\mathsf{E} \\rangle\\big),\n\\end{equation}\nwhere each $x_\\gamma\\in \\mathsf{E}_\\gamma^*$ with $\\tau(x_\\gamma) = x_\\gamma$.\nRecall that as $\\mathsf{E}\/\\mathsf{T}$ is totally ramified,\nthe canonical pairing $\\mathsf{E}^*\\times \\mathsf{E}^* \\rightarrow \\mu_e(\\mathsf{T}_0)$\ngiven by $(s,t)\\mapsto [s,t]$ is surjective\n(\\cite[Prop.~2.1]{hwcor}), and $\\mu_e(\\mathsf{T}_0) = \\mu_e$,\ni.e., $\\mathsf{T}_0$ contains all $e$-th roots of unity.\nSince each $\\mathsf{E}_\\gamma = \\mathsf{T}_0 x_\\gamma$ with $\\mathsf{T}_0$ central,\nit follows that\n$\\{[x_\\delta,x_\\gamma] \\mid \\gamma, \\delta \\in \\Gamma_\\mathsf{E}\\} = \\mu_e$.\nNow consider $c=x_\\gamma x_\\delta x_{\\gamma+\\delta}^{-1}$ for any $\\gamma,\n\\delta\\in\\Gamma_\\mathsf{E}$. Then, $\\tau(c)=x_{\\gamma+\\delta}^{-1}x_\\delta x_\\gamma$.\nNote that $x_\\delta x_\\gamma$ and $x_{\\gamma+\\delta}$\neach lie in $\\mathsf{E}_{\\gamma+\\delta} = \\mathsf{T}_0x_{\\gamma+\\delta}$,\nso they commute. Hence,\n\\begin{equation}\\label{c}\n\\tau(c) c^{-1} \\, = \\ x_{\\gamma+\\delta}^{-1} (x_\\delta x_\\gamma)\nx_{\\gamma+\\delta}x_\\delta^{-1}x_\\gamma^{-1}\n \\ = \\ [x_\\delta,x_\\gamma].\n\\end{equation}\n Since $[x_\\delta,x_\\gamma] \\in \\mu_e$,\nthis shows that\n$c \\in \\big \\{a\\in \\mathsf{T}_0^*\\mid a^e\\in \\mathsf{R}_0^*\\big \\}$. For the reverse\ninclusion, take any\n$d$~in~$\\mathsf{T}_0^*$ such that $d^e\\in \\mathsf{R}_0^*$.\nSo $\\tau(d)d^{-1} \\in \\mu_e$. Thus,\n$\\tau(d) d^{-1}= [x_\\delta,x_\\gamma]$, for some $\\gamma, \\delta\\in \\Gamma_\\mathsf{E}$.\n Taking ${c=x_\\gamma x_\\delta x_{\\gamma+\\delta}^{-1}}$, we have\n$\\tau(d) d^{-1}=\\tau(c) c^{-1}$\nby \\eqref{c}, which implies that $dc^{-1}$ is\n$\\tau$-stable, so lies in $\\mathsf{R}_0^*$;\n thus, ${d \\in \\mathsf{R}_0^* \\, \\langle x_\\gamma x_\\delta x_{\\gamma+\\delta}^{-1}\n\\mid \\gamma, \\delta \\in \\Gamma_\\mathsf{E} \\rangle}$. Therefore,\n$\\mathsf{R}_0^* \\, \\langle x_\\gamma x_\\delta x_{\\gamma+\\delta}^{-1} \\mid\n\\gamma, \\delta \\in \\Gamma_\\mathsf{E} \\rangle\n=\\{a\\in \\mathsf{T}_0^*\\mid a^e\\in \\mathsf{R}_0^* \\}$. Inserting this in ~(\\ref{gendesb}) we\nobtain (\\ref{genesa}).\n\n(i) Consider the well-defined map\n$\\alpha\\colon\\SK(\\mathsf{E},\\tau)\\rightarrow \\SK(\\mathsf{E})$ given by\n$a\\Sigma_\\tau\\mapsto a^{1-\\tau}\\mathsf{E}'$ (see diagram~\\eqref{goodd} for the\nnon-graded version).\nBy~\\cite[Cor.~3.6(ii)]{hazwadsworth},\n$\\SK(\\mathsf{E})\\cong \\mu_n(\\mathsf{T}_0)\/\\mu_e$. Taking into account\nformula ~\\eqref{genesa} for $\\SK(\\mathsf{E},\\tau)$,\nit is easy to see that $\\alpha$ is injective.\n\nWe now verify that\n\\begin{equation}\\label{imalpha}\n\\operatorname{im}(\\alpha) \\ = \\ \\big \\{\\omega \\in \\mu_n(\\mathsf{T}_0) \\mid\n\\tau(\\omega)=\\omega^{-1}\\big \\} \\, \\big \/\\mu_e,\n\\end{equation}\nand thus obtain ~(\\ref{genesa1}).\nIndeed, since $\\mu_e = \\{[x_\\delta,x_\\gamma] \\mid \\gamma, \\delta\n\\in \\Gamma_\\mathsf{E}\\}$, by setting\n$c=x_\\gamma x_\\delta x_{\\gamma+\\delta}^{-1}$ we have\n${[x_\\delta,x_\\gamma]=\\tau(c) c^{-1}}$ by \\eqref{c}. This shows that $\\mu_e \\subseteq \\big \\{\\omega \\in \\mu_n(\\mathsf{T}_0) \\mid \\tau(\\omega)=\\omega^{-1}\\big \\}$. Now for\nany $\\omega \\in \\mu_n(\\mathsf{T}_0)$ with $\\tau(\\omega)=\\omega^{-1}$,\nwe have $N_{\\mathsf{T}_0\/\\mathsf{R}_0}(\\omega)=1$, so Hilbert~90 guarantees that\n$\\omega=c^{1-\\tau}$ for some $c\\in \\mathsf{T}_0^*$. Then,\n$(c^n)^{1-\\tau} = \\omega^n = 1$, so $c^n\\in \\mathsf{R}_0^*$.\nThus,\n$c\\in \\Sigma_\\tau'$, and clearly $\\alpha(c\\Sigma_\\tau)=\\omega\\mu_e$.\nThis shows $\\supseteq$ in \\eqref{imalpha}; the reverse inclusion is clear\nfrom the definition of $\\alpha$.\n\n\n(ii)\nSuppose $e$ is odd. Let $m = |\\mu_n(\\mathsf{T}_0)|$. So, $\\mu_n(\\mathsf{T}_0) = \\mu_m$,\nwith $m \\,|\\, n$. Also, $e\\,|\\, m$, as $\\mu_e\\subseteq \\mathsf{T}_0$. Since $e$ and $n$\nhave the same prime factors, this is also true for $e$ and $m$. Recall that\n$\\operatorname{Aut}(\\mu_m)\\cong (\\mathbb Z\/m\\mathbb Z)^*$, the multiplicative group of units of the ring\n$\\mathbb Z\/m\\mathbb Z$; so, $|\\operatorname{Aut}(\\mu_m)| = \\varphi(m)$, where $\\varphi$ is Euler's\n$\\varphi$-function. Since $e\\,|\\, m$ and $e$ and $m$ have the same prime\nfactors (all odd), the canonical map $\\psi\\colon \\operatorname{Aut}(\\mu_m) \\to\n\\operatorname{Aut}(\\mu_e)$ given by restriction is surjective with kernel of order\n$\\varphi(m)\/\\varphi(e) = m\/e$, which is odd. Therefore,\n$\\psi$ induces an isomorphism on the $2$-torsion subgroups,\n$_2\\!\\operatorname{Aut}(\\mu_m) \\cong \\ _2\\!\\operatorname{Aut}(\\mu_e)$. Now, $\\tau|_{\\mu_m}\n\\in \\, _2\\!\\operatorname{Aut}(\\mu_m)$ and we saw for~(i) that $\\tau|_{\\mu_e}$\nis the inverse map $\\omega \\mapsto \\omega^{-1}$. The inverse map\non $\\mu_m$ also lies in $_2\\!\\operatorname{Aut}(\\mu_m)$ and has the same restriction\nto $\\mu_e$ as $\\tau$. Hence, $\\tau|_{\\mu_m}$ must be the inverse map.\nThat is, $\\{\\omega \\in \\mu_n(\\mathsf{T}_0) \\mid\n\\tau(\\omega)=\\omega^{-1} \\} = \\mu_n(\\mathsf{T}_0)$. Therefore,\n\\eqref{imalpha} above shows that $\\operatorname{im}(\\alpha) = \\mu_n(\\mathsf{T}_0)\/\\mu_e$,\nwhich we noted above is isomorphic to $\\SK(\\mathsf{E})$.\n\n(iii) We saw in the proof of part (i) that\n$\\tau$ acts on $\\mu_e$ by the inverse map. So,\nif $e>2$, then $\\mu_e \\not \\subseteq \\mathsf{R}_0$.\nSince $[\\mathsf{T}_0:\\mathsf{R}_0]=2$, it then follows that\n$\\mathsf{T}_0=\\mathsf{R}_0(\\mu_e)$.\n\\end{proof}\n\n\\begin{remark}\nThe isomorphism $\\SK(\\mathsf{E},\\tau)\\cong \\SK(\\mathsf{E})$ of part (\\ref{hhh}) of the\nabove theorem can be obtained under the milder condition that\n$\\mathsf{E}_0=\\mathsf{T}_0\\mathsf{E}'$ provided that the exponent of $\\mathsf{E}$ is a prime power. The\nproof is similar.\n\\end{remark}\n\n\n\n\n\\begin{example} \\label{toex}\nLet $r_1, \\ldots, r_m$ be integers with each\n$r_i \\ge 2$. Let $e = \\text{lcm}(r_1, \\ldots, r_m)$, and let\n$n = r_1\\ldots r_m$. Let $C$ be any field such that $\\mu_e\n\\subseteq C$ and $C$ has an automorphism $\\theta$\nof order $2$ such that\n$\\theta(\\omega) = \\omega^{-1}$ for all $\\omega\\in \\mu_e$.\nLet $R$ be the fixed field $C^\\theta$.\nLet $x_1,\\dots, x_{2m}$ be $2m$ independent indeterminates, and let\n$K$ be the iterated Laurent power series field $C((x_1))\\dots ((x_{2m}))$.\nThis $K$~is equipped with its standard valuation\n$v\\colon K^* \\rightarrow \\mathbb Z^{2m}$ where $\\mathbb Z^{2m}$ is given\nthe right-to-left lexicographical ordering. With this valuation $K$ is\nhenselian (see~\\cite[p.~397]{wadval}). Consider the tensor product of\nsymbol algebras\n\\begin{equation*}\nD\\ = \\ \\Big (\\frac{x_{1},x_{2}}{K}\\Big )_{\\omega_1} \\otimes_K\n\\ldots \\otimes_K\n\\Big (\\frac{x_{2m-1},x_{2m}}{K}\\Big )_{\\omega_m},\n\\end{equation*}\nwhere for $1\\leq i \\leq m$, $\\omega_i$ is a primitive $r_i$-th root of unity\nin $C$.\nUsing the valuation theory developed for division algebras, it is known that\n$D$ is a division algebra, the valuation $v$ extends to $D$, and $D$ is\ntotally ramified over $K$ (see \\cite[Ex.~4.4(ii)]{wadval} and\n\\cite[Ex.~3.6]{tw}) with\n$$\n\\Gamma_D\/\\Gamma_K \\ \\cong \\\n\\textstyle{\\prod\\limits_{i=1}^m} (\\mathbb Z\/r_i \\mathbb Z)\\times\n(\\mathbb Z\/r_i\\mathbb Z),\n$$\nand $\\overline D = \\overline K\\cong C$.\nExtend $\\theta$ to an automorphism $\\theta'$ of order $2$ on $K$ in the obvious\nway, i.e., acting by~$\\theta$ on the coefficients of a Laurent series, and with\n$\\theta'(x_i) = x_i$ for $1\\le i\\le 2m$.\nOn each of the symbol algebras\n$\\Big (\\frac{x_{2i-1},x_{2i}}{K}\\Big )_{\\omega_i}$ with its generators $\\mathbf{i}_i$\nand $\\mathbf{j}_i$ such that $\\mathbf{i}_i^{r_i}=x_{2i-1}$, $\\mathbf{j}_i^{r_i}=x_{2i}$, and\n${\\mathbf{i}_i\\mathbf{j}_i=\\omega_i\\mathbf{j}_i\\mathbf{i}_i}$, define an involution $\\tau_i$ as follows:\n$\\tau_i(c \\, \\mathbf{i}_i^k \\mathbf{j}_i^l)= \\theta'(c) \\, \\mathbf{j}_i^l \\mathbf{i}_i^k$, where $c\\in K$ and $0\\leq l,k < r_i$.\n Clearly\n${K^{\\tau_i}= K^{\\theta'} = R((x_1))\\dots ((x_{2m}))}$, and therefore $\\tau_i$~is a\nunitary involution. Since the $\\tau_i$ agree\non $K$ for $1\\le i\\le m$, they yield a unitary involution\n$\\tau=\\otimes_{i=1}^m\\tau_i$\non $D$. Now by Th.~\\ref{involthm2},\n$\\SK(D,\\tau)\\cong\\SK(\\operatorname{{\\sf gr}}(D),\\widetilde \\tau)$. Since\n$D$ is totally ramified over $K$, which is unramified over $K^{\\tau}$,\nwe have correspondingly that $\\operatorname{{\\sf gr}}(D)$ is totally ramified over~$\\operatorname{{\\sf gr}}(K)$,\nwhich is unramified over $\\operatorname{{\\sf gr}}(K)^{\\widetilde \\tau}$.\nAlso, $\\operatorname{{\\sf gr}}(K)_0 \\cong \\overline K \\cong C$.\nWe have ${\\exp(\\operatorname{{\\sf gr}}(D)) = \\exp(D) = \\exp(\\Gamma_D\/\\Gamma_K) =\n\\text{lcm}(r_1, \\ldots, r_m) = e}$ and\n$ {\\operatorname{ind}}(\\operatorname{{\\sf gr}}(D)) = \\operatorname{ind}(D) = r_1\\ldots r_m = n$.\nBy Th.~\\ref{sktotal},\n$$\n\\SK(D,\\tau) \\ \\cong \\ \\SK(\\operatorname{{\\sf gr}}(D),\\widetilde \\tau)\\ \\cong \\\n\\{ \\omega\\in \\mu_n(C) \\mid \\theta(\\omega) = \\omega^{-1}\\}\\big\/ \\mu_e,\n$$\nwhile by \\cite[Th.~4.8, Cor.~3.6(ii)]{hazwadsworth},\n$$\n\\SK(D) \\ \\cong \\ \\SK(\\operatorname{{\\sf gr}}(D))\\ \\cong \\mu_n(C)\/\\mu_e .\n$$\nHere are some more specific examples:\n\n(i) Let $C = \\mathbb C$, the complex numbers, and let\n$\\theta$ be complex conjugation, which maps every root of\nunity to its inverse. So, $R = C^\\theta = \\mathbb R$.\nThen, $\\SK(D, \\tau) \\cong \\SK(D) \\cong \\mu_n\/\\mu_e \\cong\\mathbb Z\n\/(n\/e)\\mathbb Z$.\n\n(ii) Let $r_1 = r_2 = 4$, so $e = 4$ and $n = 16$. Let\n$\\omega_{16}$ be a primitive sixteenth root of unity in $\\mathbb C$,\nand let\n${C = \\mathbb Q(\\omega_{16})}$, the sixteenth\ncyclotomic extension of $\\mathbb Q$. Recall that ${\\operatorname{Gal}(C\/\\mathbb Q)\n \\cong \\operatorname{Aut}(\\mu_{16}) \\cong (\\mathbb Z\/4\\mathbb Z) \\times (\\mathbb Z\/2\\mathbb Z)}$,\nLet $\\theta\\colon C \\to C$ be the automorphism which\nmaps $\\omega_{16} \\mapsto (\\omega_{16})^7$. Then,\n$\\theta^2 = \\operatorname{id}_C$, as $7^2 \\equiv 1 \\ (\\text{mod }16)$,\nand $\\{\\omega \\in \\mu_{16}\\mid \\theta(\\omega) = \\omega^{-1}\\}\n = \\mu_8$. Thus, $\\SK(D, \\tau) \\cong \\mu_8\/\\mu_4 \\cong\n\\mathbb Z\/2\\mathbb Z$, while $\\SK(D) \\cong \\mu_{16}\/\\mu_4 \\cong \\mathbb Z\/4\\mathbb Z$.\nSo, here the injection $\\SK(D,\\tau) \\to \\SK(D)$ is not surjective.\n\n(iii) Let $r_1 = \\ldots = r_m = 2$, so $e = 2$ and $n = 2^m$.\nHere, $C$ could be any quadratic extension of any field~\n$R$ with $\\operatorname{char}(R) \\ne 2$. Take $\\theta$ to be the unique\nnonidentity $R$-automorphism of $C$. The resulting\n$D$ is a tensor product of\n$m$ quaternion algebras over $C((x_1))\\ldots((x_{2m}))$,\nand ${\\SK(D,\\tau) \\cong\n\\{\\omega\\in \\mu_{2^m}(C)\\mid \\theta(\\omega) = \\omega^{-1}\\}\\big\/\n\\mu_2}$, while $\\SK(D) \\cong \\mu_{2^m}(C)\/\\mu_2$.\n\\end{example}\n\nEx.~\\ref{toex} gives an indication how to use the graded approach to\nrecover results in the literature on\nthe unitary $\\SK$ in a unified\nmanner and to extend them from division algebras with discrete valued groups to\n arbitrary valued groups. While $\\SK(D)$ has long been known for the $D$\nof Ex.~\\ref{toex},\nthe formula for $\\SK(D,\\tau)$ is new.\n\nHere is a more complete statement of what the results in the preceding sections\n yield for\n$\\SK(D, \\tau)$ for valued division algebras $D$.\n\n\n\\begin{theorem}\\label{appl}\nLet $(D,v)$ be a tame valued division algebra over a field $K$\nwith $v|_K$ henselian,\nwith a unitary involution $\\tau$; let $F=K^\\tau$, and suppose $v|_F$ is\nhenselian and that $K$ is tamely\nramified over $F$. Let $\\overline \\tau$ be the involution on $\\overline D$\ninduced by $\\tau$.\nThen,\n\\begin{enumerate}\n \\item[$(1)$] Suppose $K$ is unramified over $F$.\n\\begin{enumerate}[\\upshape(i)]\n \\item \\label{apun} If $D$ is unramified over $K$, then\n$\\SK(D,\\tau)\\cong \\SK(\\overline D,\\overline \\tau)$.\n\\smallskip\n \\item \\label{tolast} If $D$ is totally ramified over $K$, let $e = \\exp(D)$\nand $n = \\operatorname{ind}(D)$; then,\n $$\n\\SK(D,\\tau) \\ \\cong \\ \\{\\omega \\in \\mu_n(\\overline K) \\mid \\tau(\\omega) =\n\\omega^{-1}\\}\\big \/\\mu_e,\n$$\nwhile $\\SK(D) \\cong \\mu_n(\\overline K)\/\\mu_e$.\n\\smallskip\n \\item \\label{tirl} If $D$ has a maximal graded subfield\n$M$ unramified over $K$ and another maximal graded subfield $L$ totally\nramified over $K$, with $\\tau(L ) =L$, then $D$ is semiramified and\n\\begin{equation*}\\label{semirsk3}\n\\SK(D,\\tau) \\ = \\ \\big\\{a \\in \\overline D^* \\mid N_{\\overline D\/\\overline K}(a)\\in \\overline F\\big\\}\n \\, {\\big\/} \\, \\textstyle{\\prod\\limits_{h\\in \\operatorname{Gal}(\\overline D\/\\overline K)}}\n\\overline F^{*h\\overline \\tau}.\n\\end{equation*}\n \\item\\label{gamcyclic} Suppose $\\Gamma_D\/\\Gamma_K$ is cyclic. Let\n$\\sigma$ be a generator of $\\operatorname{Gal}(Z(\\overline D)\/\\overline K)$. Then,\n$$\n\\SK(D, \\tau) \\ \\cong \\ \\{ a\\in \\overline D^*\\mid N_{Z(\\overline D)\/\\overline K}(\\Nrd_{\\overline D}(a))\n\\in \\overline F\\} \\, \\big\/ \\, \\big(\\Sigma_{\\overline \\tau}(\\overline D) \\cdot\n\\Sigma_{\\sigma\\overline \\tau}(\\overline D)\\big).\n$$\n \\item \\label{apsem} If $D$ is inertially split, $\\overline D$ is a field and\n$\\operatorname{Gal}(\\overline D\/\\overline K)$ is cyclic, then $\\SK(D,\\tau)=1$.\n\\end{enumerate}\n\\medskip\n \\item[$(2)$] \\label{apto} If $K$ is totally ramified over $F$, then\n$\\SK(D,\\tau)=1$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nLet $\\operatorname{{\\sf gr}}(D)$ be the associated graded division algebra of $D$.\nThe tameness assumptions assure that $\\operatorname{{\\sf gr}}(K)$ is the center of\n$\\operatorname{{\\sf gr}}(D)$ with $[\\operatorname{{\\sf gr}}(D):\\operatorname{{\\sf gr}}(K)] = [D:K]$ and that the graded involution\n$\\widetilde\\tau$ on $\\operatorname{{\\sf gr}}(D)$ induced by $\\tau$ is unitary with\n$\\operatorname{{\\sf gr}}(K)^{\\widetilde \\tau} = \\operatorname{{\\sf gr}}(K^\\tau)$. In each case of Th.~\\ref{appl},\nthe conditions on $D$ yield analogous conditions on $\\operatorname{{\\sf gr}}(D)$.\nSince by Th.~\\ref{involthm2}, $\\SK(D,\\tau)\\cong \\SK(\\operatorname{{\\sf gr}}(D),\\widetilde \\tau)$,\n(2) and (1)(v) follow immediately from Prop.~\\ref{completely}\nand Prop.~\\ref{cyclic}(\\ref{unrkl}), respectively. Part (1)(i),\nalso follows from\nTh.~\\ref{involthm2}, and Cor.~\\ref{unramified} as follows:\n$$\n\\SK(D,\\tau) \\ \\cong \\ \\SK(\\operatorname{{\\sf gr}}(D),\\widetilde{\\tau})\n \\ \\cong \\ \\SK(\\operatorname{{\\sf gr}}(D)_0,\\tau|_{\\operatorname{{\\sf gr}}(D)_0}) \\ = \\\n \\SK(\\overline{D},\\overline{\\tau}).\n$$\nParts (1)(ii), (1)(iii), and (1)(iv) follow similarly using\nTh.~\\ref{sktotal}, Cor.~\\ref{seses}, and Prop.~\\ref{cyclic}\\eqref{rh2}\nrespectively.\n\\end{proof}\n\nIn the special case that the henselian valuation\non $K$ is discrete (rank $1$),\nTh.~\\ref{appl}~(1)(i), (iii), (iv), (v) and (2) were obtained by\nYanchevski\\u\\i~ \\cite{y}. In this discrete case, the assumption\nthat $v$ on $K$ is henselian already implies that $v|_F$ is henselian\n(see Remark~\\ref{shensel}).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nObservations of high-redshift quasars reveal that supermassive\nblack holes (SMBHs) with mass of $\\ga 10^9~M_\\odot$ already exist\nas early as redshifts $z\\ga 6$\n\\citep{2006NewAR..50..665F,2007AJ....134.2435W,2011Natur.474..616M}.\nGas accretion and mergers of the remnant black holes (BHs) formed by the collapse of\nfirst-generation stars ($\\sim 100~M_\\odot$) have been considered for\nproducing such SMBHs\n\\citep[e.g.][]{2001ApJ...552..459H,2003ApJ...582..559V,2007ApJ...665..187L}.\nHowever, various forms of radiative feedback can prevent efficient BH\ngrowth, making it difficult to reach $\\ga 10^9~M_\\odot$ within the age\nof the high-redshift Universe\n\\citep{2007MNRAS.374.1557J, 2009ApJ...701L.133A, 2009ApJ...696L.146M,2012ApJ...747....9P,2012MNRAS.425.2974T}.\n\nOne possible alternative solution is the rapid formation of\nsupermassive stars (SMSs; $\\ga 10^5~M_\\odot$) and their subsequent\ncollapse directly to massive BHs in the first galaxies\n\\citep[e.g.,][]{1994ApJ...432...52L,2006MNRAS.370..289B,2006MNRAS.371.1813L}.\nThe massive seed BHs, formed by direct collapse, shorten the total\nSMBH growth time sufficiently, even in the presence of subsequent\nradiative feedback (e.g. \\citealt{th2009,2012ApJ...745L..29D}).\n\nSMSs can form out of primordial gas in massive `atomic cooling'\nhaloes with virial temperatures $T_{\\rm vir}\\ga 10^4$K, if H$_2$\nformation and line cooling are prohibited through the pre-stellar\ncollapse. Possible mechanisms to suppress H$_2$ formation are\nphoto dissociation by far-ultraviolet (FUV) radiation\n\\citep{2001ApJ...546..635O,2003ApJ...596...34B,2008MNRAS.391.1961D,\n2009MNRAS.396..343R,2010MNRAS.402.1249S,\n2011MNRAS.416.2748I,2011MNRAS.418..838W,2013MNRAS.428.1857J} and\ncollisional dissociation\n\\citep{2012MNRAS.422.2539I,2014MNRAS.439.3798F}. In the absence of\n${\\rm H_2}$, the primordial gas remains warm ($\\sim8,000$K) and may\ncollapse monolithically without strong fragmentation\n\\citep{2003ApJ...596...34B,2010MNRAS.402.1249S,\n 2013MNRAS.433.1607L,2014arXiv1404.4630I}. The resulting central\nprotostar can grow via accretion at a high rate of $\\ga 0.1~M_\\odot~{\\rm yr}^{-1}$.\nIf the accretion rate maintains such a high value, the embryonic\nprotostar evolves to a SMS with mass of $\\ga 10^5~M_\\odot$. The SMS\ncollapses as a whole to a single BH either directly\n\\citep{2002ApJ...572L..39S}, or via the intermediate stage of a close\nbinary BH \\citep{2013PhRvL.111o1101R}, through a general relativistic\ninstability \\citep[e.g.,][]{1971reas.book.....Z,1983bhwd.book.....S}.\nThe massive remnant BH is then a promising seed that can grow into one\nof the observed SMBHs at $z\\approx 6-7$.\n\n\nIn the scenario above, a major unresolved question is whether rapid\naccretion will continue unabated through the protostellar evolution.\nRecent high-resolution simulations by \\cite{2014MNRAS.439.1160R}\nsuggest that the compact nuclear accretion disc, surrounding a central\nembryonic protostar, is gravitationally unstable. Thus,\nself-gravitating clumps are expected to form during the early stages\nof the accretion phase. Such efficient fragmentation could prevent\nthe rapid growth of the central protostar, and preclude eventual SMS\nformation. This is analogous to the fragmentation of gravitationally\nunstable discs in the cores of lower mass `minihaloes', which had\nbeen suggested to reduce the characteristic masses of Population III (Pop III)\nstars \\citep{2010MNRAS.403...45S, greif+11}. Moreover, if the gas is slightly polluted by\nheavy elements ($Z\\la 10^{-4}~Z_\\odot$), dust cooling can decrease the\ntemperature and induce the fragmentation \\citep{2008ApJ...686..801O},\nwhich would be a further obstacle to SMS formation. Because of this,\nthe existence of the pristine, metal-free gas has often been\nconsidered as a necessary condition of forming a SMS in\nsemi-analytical models \\citep[e.g.,][]{2012MNRAS.425.2854A,2014MNRAS.442.2036D}.\nWe note that even very efficient fragmentation may not prevent accretion \non to a central point source \\citep[e.g.,][]{2007ApJ...671.1264E} and\nthe rapid inward migration of the fragments, on a time-scale comparable \nto the orbital period, may help its growth further \n\\citep[e.g.,][]{2010MNRAS.404.2151C}.\nIn our context, however, even a slow down of the accretion rate could bring it \nbelow the critical value required for SMS formation.\n\nIn this paper, motivated by the above, we discuss the expected properties of a compact,\nmarginally unstable nuclear protogalactic disc, and the fate of the\nclumps formed in the disc by gravitational instability. Using\nanalytical models, we argue that despite fragmentation, the growth of\na SMS remains the most likely outcome. The reason for\nthis conclusion is the rapid inward migration of the fragments, and\ntheir merger with the central protostar.\n\nThe rest of this paper is organized as follows. In \\S~\\ref{sec:2}, we\ndescribe the basic model of the fragmenting disc and of the clumps\nformed in the disc. In \\S~\\ref{sec:fate}, we discuss the fate of the\nclumps, considering various important processes: migration, accretion,\ncontraction, and star-formation by reaching the zero-age main sequence\n(ZAMS). We also consider the possibility that radiative feedback from\nmassive stars, formed from the clumps, may halt gas accretion on to the\nwhole system. In \\S~\\ref{sec:metal}, we discuss whether\nmetal pollution and associated dust cooling could prohibit SMS\nformation. Finally, we discuss our results and summarize our\nconclusions in \\S~\\ref{sec:conc}.\n\n\n\n\n\n\\section{Fragmentation of the accretion disc around a supermassive star}\n\\label{sec:2}\n\n\\subsection{Basic equations}\n\nWe consider the properties of a disc formed after the collapse of\nprimordial gas inside an atomic cooling halo,\nwhen molecular hydrogen formation is suppressed.\nSince the parent gas\ncloud is hot ($T\\sim 8000$ K), the accretion rate on to the disc is\nhigh:\n\\begin{equation}\n\\dot M_{\\rm tot} \\sim \\frac{c_{\\rm s}^3}{G}\\sim 0.1~M_\\odot~{\\rm yr}^{-1} \\left(\\frac{T}{8000~{\\rm K}}\\right)^{3\/2},\n\\end{equation}\nwhere $c_{\\rm s}$ is the sound speed and $G$ is the gravitational\nconstant \\citep{1977ApJ...214..488S, 1986ApJ...302..590S}. The\naccretion disc around the central protostar can become unstable\nagainst self-gravity. To understand the stability of the disc,\nToomre's parameter \\citep{1964ApJ...139.1217T} defined by\n\\begin{equation}\nQ=\\frac{c_{\\rm s}\\Omega}{\\pi G\\Sigma}\n\\label{eq:Q}\n\\end{equation}\nis often useful, where $\\Omega$ is the orbital frequency and $\\Sigma$\nis the surface density of the disc. In the marginal case of $1\\la Q\n\\la 2$, strong spiral arms are formed in the disc. The spiral arms\nredistribute angular momentum and heat the disc by forming shocks\n\\citep[e.g.,][]{2005ApJ...633L.137V,2006ApJ...650..956V}.\nThe resulting disc is self-regulated to the marginal state. \nThus, we here assume $Q\\simeq 1$.\n\n\nSince the gas temperature in the atomic cooling halo is kept near\n$\\sim 8000$ K, the external accretion rate on to the disc, from larger\nradii, is also nearly constant. We estimate the surface density of\nthe self-regulated disc by assuming that it is in steady state, and\nthat it has an effective viscosity $\\nu$ (arising from gravitational\ntorques),\n\\begin{equation}\n\\Sigma=\\frac{\\dot M_{\\rm tot}}{3\\pi \\nu}.\n\\label{eq:continue}\n\\end{equation}\nThe scale-height of the disc is estimated from vertical hydrostatic\nbalance,\n\\begin{equation}\nH=\\frac{c_{\\rm s}}{\\Omega}. \n\\label{eq:scaleheight}\n\\end{equation}\nWe define the particle number density as $n\\equiv\\Sigma \/(2m_{\\rm\n p}H)$, where $m_{\\rm p}$ is the proton mass.\n\n\\begin{figure*}\n\\vspace{1mm}\n\\begin{center}\n\\begin{tabular}{ccc}\n \\begin{minipage}{0.33\\hsize}\n \\begin{center}\n \\vspace{-1mm}\n \\includegraphics[width=54mm]{temp_Hm_1.eps}\n \\end{center}\n \\end{minipage}\n\\begin{minipage}{0.32\\hsize}\n \\begin{center}\n \\includegraphics[width=54.5mm]{Sigma_d_Hm_1.eps}\n \\end{center}\n \\end{minipage}\n\\begin{minipage}{0.32\\hsize}\n \\begin{center}\n \\includegraphics[width=55mm]{alpha_Hm_1.eps}\n \\end{center}\n \\end{minipage}\n\\end{tabular}\n\\caption{Radial profiles of disc quantities, shown as a function of\n the orbital period: (a) temperature, (b) surface density, and (c)\n viscous parameter. The solid curves show profiles with H$^-$\n free-bound emission as the only radiative cooling process. The\n dashed curves additionally include Ly$\\alpha$ emission. The\n horizontal dot-dashed line in panel (c) indicates the critical value\n of the viscous parameter for fragmentation, $\\alpha_{\\rm f}=1$. The\n shaded bar marks the location of the fragmenting radius at\n $P_{\\rm f}=2\\pi\/\\Omega_{\\rm f}=10^4$ yr (or $\\Omega_{\\rm f}=2\\times 10^{-11}$\n s$^{-1}$). }\n \\label{fig:profile}\n\\end{center}\n\\end{figure*}\n\n\nTo determine the thermal state of the disc we assume that it is in\nequilibrium, i.e. we balance heating and radiative cooling\n($Q_+=Q_-$). In particular, we consider viscous heating by turbulence\nand spiral shocks,\n\\begin{align}\nQ_+&=\\frac{9}{4}\\nu \\Sigma \\Omega ^2,\n\\end{align}\nand radiative cooling\n\\begin{align}\nQ_-&=2H\\Lambda,\n\\end{align}\nwhere $\\Lambda$ is the cooling rate in units of erg s$^{-1}$\ncm$^{-3}$. For the cooling process, we consider H$^-$ free-bound\nemission, which is the dominant channel in the thermal evolution at\n$n\\ga 10^8~{\\rm cm}^{-3}$ for a warm primordial gas in an atomic cooling halo.\nThe form of the cooling rate is given in \\S~\\ref{sec:chem}.\n\nFinally, we adopt the standard $\\alpha$-prescription\n\\citep{1973A&A....24..337S} as a model for the viscosity by\ngravitational torques:\n\\begin{equation}\n\\nu = \\alpha c_{\\rm s}H,\n\\label{eq:viscous}\n\\end{equation}\nwhere $\\alpha$ is the viscous parameter. From hydrodynamical\nsimulations of a self-gravitating disc, the disc is found to be\nsusceptible to fragmentation if the viscous parameter exceeds a\ncritical threshold value, $\\alpha \\ga \\alpha_{\\rm f}$\n\\citep[e.g.,][]{2001ApJ...553..174G}. \\cite{2005MNRAS.364L..56R} have\nstudied fragmentation conditions for various specific heat ratios, and\nobtained $\\alpha_{\\rm f}\\sim 0.06$. However, the critical value\ndepends on both initial conditions and on numerical resolution\n\\citep{2011MNRAS.410..559M, 2011MNRAS.411L...1M}. Recently,\n\\cite{2012ApJ...746..110Z} have considered mass loading from an\ninfalling envelope, realistic radiative cooling, and radiative\ntrapping of energy inside clumps, and suggested that the critical\nvalue for fragmentation is around $\\alpha_{\\rm f}\\sim 1$. Moreover,\nnumerical simulations of fragmentation of discs with relatively high\naccretion rates ($\\dot M_{\\rm tot}\\sim 10^{-4}-10^{-2}~M_\\odot~{\\rm yr}^{-1}$), in\nthe context of present-day massive star formation \\citep[e.g.,\n][]{2007ApJ...656..959K} and Pop III star formation\n\\citep{2011Sci...331.1040C} have shown that the effective viscous\nparameter arising from gravitational torques is $\\sim 0.1-1$. Based\non these results, we here set $\\alpha _{\\rm f}=1$ as our fiducial\nvalue, and analyze disc fragmentation using the condition of $\\alpha\n\\geq \\alpha_{\\rm f}$ following previous analytical works\n\\citep[e.g.,][]{2005ApJ...621L..69R, 2007MNRAS.374..515L}.\n\n\n\\subsection{Radiative cooling and chemistry}\n\\label{sec:chem}\n\nIn dense ($n\\ga 10^8~{\\rm cm}^{-3}$) and warm ($3000\\la T\\la 8000$ K) primordial\ngas, the dominant cooling processes is the free-bound emission of\nH$^-$ ions (H + e$^-$ $\\rightarrow$ H$^-$ + $\\gamma$;\n\\citealt{2001ApJ...546..635O,2014arXiv1404.4630I}). The cooling rate,\nin units of erg s$^{-1}$ cm$^{-3}$, is given by\n\\begin{equation}\n\\Lambda =\\lambda(T)n^2x_{\\rm e},\n\\end{equation}\nwhere $x_{\\rm e}(\\ll1)$ is the free-electron fraction and\n$\\lambda(T)\\simeq 10^{-30}~ T$ (with $T$ in units of Kelvin). \nIn such a warm gas, prior to protostar formation, \nthe electron fraction is determined by the balance between\nrecombination (H$^+$ + e$^-$ $\\rightarrow$ H + $\\gamma$) and\ncollisional ionization (2H $\\rightarrow$ H$_2^+$ + e$^-$).\nThe evolution of the electron fraction follows the equation\n\\begin{equation}\n\\frac{dx_{\\rm e}}{dt}=-\\alpha_{\\rm rec}nx_{\\rm e}^2 + \\alpha_{\\rm ci}n.\n\\end{equation}\nAt the densities of interest here, the electron fraction is in equilibrium,\nand given simply by \n\\begin{equation}\nx_{\\rm e}=\\sqrt{\\frac{\\alpha_{\\rm ci}}{\\alpha_{\\rm rec}}}.\n\\end{equation}\nFrom Appendix \\ref{sec:appA}, we obtain the specific value of\n\\begin{align}\nx_{\\rm e}&\\simeq f(T)~n^{1\/2} \\exp(-\\epsilon\/2T)\\label{eq:x_e}\\\\\\nonumber\n&\\simeq 5.0\\times 10^{-11}~T^{1.2}~n^{1\/2} \\exp(-\\epsilon\/2T).\n\\end{align}\n\n\n\\subsection{Radial profile of the disc}\n\\label{sec:disc}\n\nWe next obtain the properties of the marginally fragmenting disc,\nusing the energy conservation equation ($Q_+=Q_-$). Using\nEq.~(\\ref{eq:Q}), the particle density can be written by\n\\begin{equation}\nn=\\frac{\\Sigma \\Omega}{2m_{\\rm p}c_{\\rm s}}=\\frac{\\Omega ^2}{2m_{\\rm p}\\pi G}.\n\\end{equation}\nThen, we can solve the energy conservation equation with respect to $\\Omega$:\n\\begin{equation}\n\\Omega ^2=\n\\frac{3\\dot M_{\\rm tot}}{8\\pi}\n\\frac{(2\\pi G m_{\\rm p})^{5\/2}}{c_{\\rm s} \\lambda(T)f(T)}\n\\exp(\\epsilon\/2T).\n\\end{equation}\n\n\nWe present the profiles of the temperature, surface density, and\nviscous parameter as a function of $\\Omega$ for the case of\n$\\dot M_{\\rm tot} =0.1~M_\\odot~{\\rm yr}^{-1}$ in Figure~\\ref{fig:profile}. As\ndescribed above, we assume that fragmentation occurs efficiently at\nthe radii where $\\alpha \\ga \\alpha_{\\rm f}\\simeq 1$. From\nFigure~\\ref{fig:profile}(c), we find that fragmentations occur in the\ncentral regions where the orbital period is shorter than $10^4$ yr.\nWithin the fragmenting region, the surface density is approximately\ngiven by $\\Sigma =\\Sigma_{\\rm f}(\\Omega\/\\Omega_{\\rm f})$, where\n$\\Sigma_{\\rm f}=50$ g cm$^{-2}$ and $\\Omega_{\\rm f}=2\\times 10^{-11}$\ns$^{-1}$, respectively. \nDuring the earliest stages, i.e. prior to the formation of any central\nprotostar embryo, or immediately following it, the disc mass dominates\nthe central protostellar mass. We can then obtain the radius within\nwhich the disc fragments effectively,\n\\begin{equation}\nR_{\\rm f}=\\frac{2\\pi G\\Sigma_{\\rm f}}{\\Omega_{\\rm f}^2}\\simeq 2\\times 10^{-2}~{\\rm pc}.\n\\label{eq:r_f_early}\n\\end{equation}\nMoreover, during this stage, the profiles of the surface density, disc\nmass, and number density can be written as functions of the radial\ndistance $R$ from the central protostar:\n\\begin{equation}\n\\Sigma = \\Sigma_{\\rm f}\\left(\\frac{R}{R_{\\rm f}}\\right)^{-1},\n\\label{eq:sig0}\n\\end{equation}\n\\begin{equation}\nM_d \\simeq 430~M_\\odot \\left(\\frac{R}{R_{\\rm f}}\\right),\n\\label{eq:md0}\n\\end{equation}\nand\n\\begin{equation}\nn \\simeq 6\\times 10^8~{\\rm cm}^{-3} \\left(\\frac{R}{R_{\\rm f}}\\right)^{-2},\n\\label{eq:numdens}\n\\end{equation}\nrespectively.\nThe typical mass of the clumps formed at $\\simeq R_{\\rm f}$\nis estimated as\n\\begin{equation}\nM_{\\rm c}\\simeq \\Sigma_{\\rm f} H_{\\rm f}^2\\simeq 30~M_\\odot.\n\\label{eq:M_c}\n\\end{equation}\nThese disc profiles and the clump mass are in good agreement with the\nhighest resolution results in the numerical simulations by\n\\cite{2014MNRAS.439.1160R}, lending credence to our simplified `toy'\nmodel.\n\nIn the above, we have assumed that the mass of the central protostar\nis negligible. However, the central protostar grows via rapid\naccretion and having a central point source can then modify the disc\nstructure. After $\\ga 10^4$ yr, the gas within $R_{\\rm f}$ accretes\non to the central protostar, and the protostellar mass exceeds the disc\nmass within $R_{\\rm f}$. The fragmentation radius (defined by\n$\\alpha(R_{\\rm f})=1$)\nremains roughly constant for the first $\\sim {\\rm a~few}~\\times 10^4$ yr, after which it\nbegins to move outward slowly, according to\n\\begin{align}\nR_{\\rm f}&=\\left(\\frac{GM_\\ast}{\\Omega_{\\rm f}^2}\\right)^{1\/3}\\label{eq:rf_late}\n\\\\\\nonumber\n&\\simeq 5\\times 10^{-2}~{\\rm pc}\\left(\\frac{M_\\ast}{10^4~M_\\odot}\\right)^{1\/3}\n\\propto t^{1\/3},\n\\end{align}\nwhere $M_\\ast$ is the mass of the protostellar embryo at the center,\nand we have assumed that the central accretion rate remains constant\nover time.\nIn the regions where $M_\\ast>M_{\\rm d}$, the orbital frequency is\nproportional to $R^{-3\/2}$ and thus the surface density and mass of\nthe disc are given by\n\\begin{equation}\n\\Sigma =\\frac{\\Sigma_{\\rm f}}{\\Omega_{\\rm f}}\\sqrt{\\frac{GM_\\ast}{R^3}}\n\\label{eq:sig1}\n\\end{equation}\nand\n\\begin{equation}\nM_d =4\\pi \\frac{\\Sigma_{\\rm f}}{\\Omega_{\\rm f}}\\sqrt{GM_\\ast R},\n\\label{eq:md1}\n\\end{equation}\nrespectively. The radius where $M_\\ast=M_{\\rm d}$ is given by\n\\begin{equation}\nR_{\\rm g\\ast} =\\frac{M_\\ast \\Omega_{\\rm f}^2}{16\\pi^2 G\\Sigma_{\\rm f}^2}\\propto t.\n\\end{equation}\nThe condition $R_{\\rm f}\\la R_{\\rm g\\ast}$ is satisfied when the\nprotostellar mass exceeds $\\simeq 3\\times 10^3~M_\\odot$, which corresponds to \n$t\\simeq 3\\times 10^4~{\\rm yr}~(M_\\ast\/3\\times 10^3~M_\\odot )(\\dot M_{\\rm tot}\/0.1~M_\\odot~{\\rm yr}^{-1} )^{-1}$.\nFor convenience in the following section (\\S~\\ref{sec:fate}), we define the transition epoch as $t_{\\rm g\\ast}=10^5$ yr\nneglecting the difference of the factor of 3 because the ratio of $R_{\\rm g\\ast}\/R_{\\rm f}\\propto t^{2\/3}$ \ndoes not change significantly in the range $3\\times 10^4\n2\\pi\/\\Omega$ because the torques exerted on the clump are assumed to\nbe averaged over an orbit (at a fixed radius). Despite this\ncomplication, we can still safely conclude $t_{\\rm mig}\\la\n2\\pi\/\\Omega$. This is because assuming a slower migration ($t_{\\rm\n mig}\\ga 2\\pi\/\\Omega$) would justify the use of orbit-averaged\nType~II torques, which would then yield the contradiction $t_{\\rm\n mig}< 2\\pi\/\\Omega$.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=58mm,width=82mm]{mig_time.eps}\n\\end{center}\n\\caption{The decay time of the clump's orbit, for the case of $M_\\ast\n <10^4~M_\\odot$ ($M_\\astM_{\\rm d}$; $t>t_{\\rm g\\ast}$), respectively. In each case, the initial\n position of the clump is set to the fragmentation radius $R_{\\rm\n f}$. In the first ($M_\\ast<10^4~M_\\odot$) case, we show the\n slow-down of the migration expected if the clump grows by accretion\n at a rate of $f\\dot M_{\\rm tot}$ with $f=0$ (solid), $0.2$ (dashed),\n and $0.5$ (dotted).}\n\\label{fig:mig}\n\\end{figure}\n\nWhen the clump migrates to $\\simeq 0.1~R_{\\rm f}$, its mass becomes\ncomparable to the local disc mass (neglecting for now any growth of\neither the disc or the clump during the migration).\nWithin $\\sim 0.1~R_{\\rm f}$, the migration speed slows down because\nthe disc outside the clump can no longer absorb the orbital angular\nmomentum of the clump \\citep{1995MNRAS.277..758S}.\nIn the clump-dominated case, the migration time is somewhat modified\nas\n\\begin{equation}\nt_{\\rm mig}\\simeq q_{\\rm B}^{-k}t_{\\rm vis},\n\\end{equation}\nwhere\n\\begin{align}\nq_{\\rm B}&=\\frac{(1+q)\\dot M_{\\rm tot}}{M_{\\rm c}}t_{\\rm vis}\\\\\\nonumber\n&\\simeq 1.1~\\left(\\frac{1+q}{1.1}\\right)\\left(\\frac{M_{\\rm c}}{30~M_\\odot}\\right)^{-1}\n\\left(\\frac{\\dot M_{\\rm tot}}{0.1~M_\\odot~{\\rm yr}^{-1}}\\right)\\left(\\frac{t_{\\rm vis}}{300~{\\rm yr}}\\right)\n\\end{align}\nand\n\\begin{align}\nk=1-\\left(1+\\frac{\\partial \\ln \\Sigma}{\\partial \\ln \\dot M_{\\rm tot}}\\right)^{-1}\\simeq 0.4.\n\\end{align}\n\nIn Figure~\\ref{fig:mig}, we show the evolution of the orbital radius\nof the clump (black lines: $M_\\ast<10^4~M_\\odot$). The horizontal axis\nis the time from the onset of the orbital decay at $R_{\\rm f}$. To see the\neffect of the slow down, we assume that the clump grows at the\naccretion rate of $f\\dot M_{\\rm tot}$ with $f=0$ (solid curve), $0.2$\n(dashed), and $0.5$ (dotted), respectively. In each case, the clump's\norbit decays within $10^4$ yr, even if we consider the slow down of\nthe migration.\n\nNext, we consider the case that the protostellar gravity exceeds the\nself-gravity of the disc within $R_{\\rm f}$ (i.e. $t\\geq t_{\\rm g\\ast}$). \nIn Figure~\\ref{fig:mig}, we show the corresponding\norbital evolution for $M_\\ast =10^4$ (blue middle curve) and\n$10^5~M_\\odot$ (red right most curve) as examples of this stage.\nIn both the cases, we do not consider the clump growth because\nthe initial clump mass is much smaller than the disc mass within \n$R_{\\rm f}$ and thus the slow-down effect works after the orbit\nhas decayed by more than two orders of magnitude.\nFor $M_\\ast =10^5~M_\\odot$, the fragmentation radius moves out to $\\sim 0.1$ pc. \nIn this case, the decay time becomes longer than the orbital\nperiod, and the clump could evolve into a normal massive star, rather\nthan an SMS. However, this scenario is realized only by assuming that\na SMS with $M_\\ast \\sim 10^5~M_\\odot$ has already grown at the center of\nthe disc.\n\n\n\n\\subsection{Clump accretion and evolution}\n\\label{sec:kh}\n\nIn the fragmenting disc, many clumps are formed within $R_{\\rm f}$ and\ntheir orbits decay through interaction with the accretion disc. Since\nthe typical density of the clumps is $\\sim 10^9~{\\rm cm}^{-3}$ (Eq.~\\ref{eq:numdens})\nand since they are optically thin to the H$^-$ free-bound emission,\ntiny protostars are formed in the clumps within their free-fall time\n$\\sim 10^3$ yr, which is an order of magnitude shorter than the\norbital decay time. In this section, we simply estimate the accretion\nrate on to the clumps (which we assume quickly evolve to protostars),\nand then discuss the possibility of forming ZAMS stars from the\nclumps.\n\nThe accretion rate on to a point-like clump in a Keplerian disc,\nwhere we assume a rotationally supported disc \\citep{2014MNRAS.439.1160R},\nis estimated as\n\\begin{equation}\n\\dot M_{\\rm c}=\\frac{3}{2}\\Sigma \\Omega (f_{\\rm H}R_{\\rm H})^2,\n\\end{equation}\nwhere $R_{\\rm H}$ is the Hill radius defined by $R(M_{\\rm\n c}\/3M_\\ast)^{1\/3}$ and $f_{\\rm H}\\sim O(1)$\n\\citep[e.g.,][]{2004ApJ...608..108G}. We have also assumed that the\nclump accretes gas orbiting in the nearby disc, within an impact\nparameter $f_{\\rm H}R_{\\rm H}$.\nAt the fragmentation radius, the accretion rate is given by\n\\begin{align}\n\\dot M_{\\rm c}|_{R_{\\rm f}}&=\\frac{3}{2}\\Sigma_{\\rm f} \\Omega_{\\rm f} f_{\\rm H}^2 R_{\\rm f}^2\\left(\\frac{M_c}{3M_\\ast}\\right)^{2\/3}\\label{eq:clump_acc}\\\\\\nonumber\n&\\simeq 1.1\\times 10^{-2}~M_\\odot~{\\rm yr}^{-1}~\\left(\\frac{f_{\\rm H}}{1.5}\\right)^2\\left(\\frac{M_{\\rm c}}{30~M_\\odot}\\right)^{2\/3},\n\\end{align}\nwhich is similar to the critical rate ($\\approx4\\times\n10^{-3}~M_\\odot~{\\rm yr}^{-1}$) at which the evolution of a protostar changes\nqualitatively \\citep{2001ApJ...561L..55O, 2003ApJ...589..677O}. Below\nthe critical rate, the protostar grows to a usual ZAMS star. Above\nthe critical rate, the protostar evolves instead to a structure with a\nbloated envelope resembling a giant star \\citep{2012ApJ...756...93H, 2013ApJ...778..178H}.\nThe accretion rate given by Eq.~(\\ref{eq:clump_acc}) \nis only a factor of $\\approx 2$ above the critical value. We\ntherefore consider the case in which some of the clumps grow at a\nsub-critical rate and form massive ZAMS stars.\n\nThe protostar embedded in the clump begins to undergo Kelvin-Helmholtz\n(KH) contraction, loosing energy by radiative diffusion. After the\ncontraction, the star reaches a ZAMS star when the central temperature\nincreases to $\\sim 10^8$K and hydrogen burning begins\n\\citep{2001ApJ...561L..55O, 2003ApJ...589..677O}. \nBelow the critical rate, the time-scale of the KH contraction is estimated as \n\\begin{align}\nt_{\\rm KH}&\\simeq \\frac{M_{\\rm c}}{\\dot M_{\\rm c}}\n\\ga 10^4~{\\rm yr}\\left(\\frac{M_{\\rm c}}{30~M_\\odot }\\right)\n\\left(\\frac{\\dot M_{\\rm c}}{3\\times 10^{-3}~M_\\odot~{\\rm yr}^{-1}}\\right)^{-1}.\n\\end{align}\nThus, it is longer than\nboth the migration time for $M_\\ast \\leq 10^4~M_\\odot$ and the orbital\nperiod at the fragmentation radius.\nTherefore, most clumps formed\npromptly in the early (disc-dominated) phase are expected to migrate\nand merge with the central protostar before reaching the massive ZAMS\nstars ($M_\\ast \\leq 10^4~M_\\odot$). On the other hand, clumps formed in\nthe late (central protostar-dominated) phase ($M_\\ast >10^4~M_\\odot$)\ncould survive and evolve to ZAMS stars.\n\n\n\\subsection{Radiative feedback}\n\nAs we have seen in \\S~\\ref{sec:kh}, the decay time of the clump orbit\nbecomes longer than the KH time during the late stage for\n$M_\\ast>10^4~M_\\odot$. In this case, the clump can contract by loosing\nenergy and reach the ZAMS star before migrating towards the central\nprotostar. This, however, requires a central SMS to be already present.\nOn the other hand, even during the early stage for $M_\\ast <10^4~M_\\odot$, \nsome clumps could survive for their KH times, because\nclumps interact with each other, as well as with the disc, within the\nfragmentation radius. In a clumpy disc, these gravitational\ninteractions will make the orbital decay time longer or shorter\nstochastically, and can also cause clumps to be temporarily ejected\nfrom the disc\n\\citep{2005ApJ...630..152E,2012MNRAS.424..399G,2012ApJ...746..110Z,2013ApJ...777L..14F}.\nSome of these clumps can evolve to ZAMS stars and emit strong UV\nradiation, which could prevent the accretion on to the protostar and\non to the disc as a while.\n\nWe briefly estimate the number of the clumps which exist in the disc at a moment.\nThe corresponding range where the fragmentation occurs is $\\Delta R\\simeq R_{\\rm f}$. \nThe Hill radius of the clump at $R_{\\rm f}$ is written by\n\\begin{equation}\nR_{\\rm H}=1.5\\times 10^{16}~{\\rm cm}\\left(\\frac{M_{\\rm c}}{30~M_\\odot}\\right)^{1\/3}.\n\\label{eq:hill}\n\\end{equation}\nThus, we roughly estimate the maximum number of the clumps which survives as\n\\begin{equation}\nN_{\\rm c}\\sim \\frac{\\Delta R}{R_{\\rm H}}\\simeq 20\\left(\\frac{M_\\ast}{10^5~M_\\odot}\\right)^{1\/3}\n\\left(\\frac{M_{\\rm c}}{30~M_\\odot}\\right)^{-1\/3}.\n\\label{eq:nc}\n\\end{equation}\nIf some clumps grow at rates higher than $\\sim 10^{-3}~M_\\odot~{\\rm yr}^{-1}$ and\ntheir masses increase within $t_{\\rm KH}$, the maximum number\ndecreases. \nWe here consider $N_{\\rm c}\\approx10-20$ as a conservative value.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=36mm,width=83mm]{disk.eps}\n\\end{center}\n\\caption{Schematic figure of the accretion disc and a small\n circum clump disc, showing the distance of the clump from the\n central star $R$, the Hill radius of the clump $R_{\\rm H}$ (Eq.~\\ref{eq:hill}), \n the size of the circum clump disc $R_{\\rm d,c}$, the\n size of the H$_{\\rm II}$ region around the clump $R_{\\rm ion}$\n (Eq.~\\ref{eq:ion}), the gravitational radius of the clump\n $R_{\\rm g,c}$ (Eq.~\\ref{eq:grav}), and the size of the whole\n nuclear disc $R_{\\rm d}$ (Eq.~\\ref{eq:rd}). }\n\\label{fig:clump_disc}\n\\end{figure}\n\n\nFirst, let us consider radiative feedback on the accretion on to the\nZAMS star (clump) itself. Figure~\\ref{fig:clump_disc} illustrates the\ngas structure around the clumps and the characteristic radii. The gas\naccreting on to the ZAMS star makes a small circum-clump disc. The\ndisc size is roughly estimated as $R_{\\rm d,c}\\simeq R_{\\rm H}\/3$\n\\citep{1998ApJ...508..707Q}, based on numerical simulations\n\\citep{2009MNRAS.397..657A, 2012ApJ...747...47T}. The ZAMS star emits\nUV radiation and an H$_{\\rm II}$ region is formed above and below the\ndisc. The size of the H$_{\\rm II}$ region depends on the ionizing\nluminosity of the ZAMS star and on the density profile of the\nin-falling material. For simplicity, we estimate the size $R_{\\rm\n ion}$ in the polar direction using Eq.~(36) of\n\\cite{2008ApJ...681..771M} as\n\\begin{equation}\n\\frac{R_{\\rm ion}}{R_{\\rm d,c}}\\simeq 0.16\\left(\\frac{M_{\\rm c}}{30~M_\\odot}\\right)^{1.25}\n\\left(\\frac{\\dot M_{\\rm c}}{3\\times 10^{-3}~M_\\odot~{\\rm yr}^{-1}}\\right)^{-1},\n\\end{equation}\nwhere we assume that the disc mass is equal to the clump mass and we\ntake the temperature of the H$_{\\rm II}$ region to be $3\\times 10^4$ K\n\\citep{2011Sci...334.1250H}. The sound speed in the H$_{\\rm II}$\nregion is $c_{\\rm s,ion}\\simeq 20$ km s$^{-1}$. For a ZAMS star\nlocated at $R_{\\rm f}$, we obtain\n\\begin{equation}\nR_{\\rm ion}\\simeq 8.0\\times 10^{14}~{\\rm cm} \\left(\\frac{M_{\\rm c}}{30~M_\\odot}\\right)^{19\/12}\n\\left(\\frac{\\dot M_{\\rm c}}{3\\times 10^{-3}~M_\\odot~{\\rm yr}^{-1}}\\right)^{-1}.\n\\label{eq:ion}\n\\end{equation}\nSince the sound-crossing time within $R_{\\rm ion}$ is much shorter\n($\\sim13$ yr), the ionization front expands rapidly. When the front\nreaches the gravitational radius\n\\begin{equation}\nR_{\\rm g,c}=\\frac{GM_{\\rm c}}{c_{\\rm s,ion}^2}\n\\simeq 1.0\\times 10^{15}~{\\rm cm} \\left(\\frac{M_{\\rm c}}{30~M_\\odot}\\right),\n\\label{eq:grav}\n\\end{equation}\nthe ionized gas breaks out through the neutral infalling gas. The\nionizing photons subsequently heat the disc surface, and thus\nphoto-evaporation begins to suppress the accretion rate. The\nphoto-evaporation rate can be expressed as\n\\begin{align}\n\\dot M_{\\rm PE}\\simeq 3.8\\times 10^{-4}\\left(\\frac{\\Phi_{\\rm EUV}}{10^{50}~{\\rm s}^{-1}}\\right)^{1\/2}\n\\left(\\frac{R_{\\rm d,c}}{10^{16}~{\\rm cm}}\\right)^{1\/2}~M_\\odot~{\\rm yr}^{-1},\n\\end{align}\nwhere $\\Phi_{\\rm EUV}$ is the ionizing photon number flux\n\\citep{2013ApJ...773..155T}. Using the relation $\\Phi_{\\rm\n EUV}=3.7\\times 10^{49}N_{\\rm c}$ s$^{-1}(M_{\\rm c}\/60~M_\\odot)^{3\/2}$\nin the mass range $60\\la M_{\\rm c} \\la 300~M_\\odot$, we obtain\n\\begin{align}\n\\dot M_{\\rm PE}\\simeq 7.4\\times 10^{-4}\\left(\\frac{M_\\ast}{10^5~M_\\odot}\\right)^{1\/6}\n\\left(\\frac{M_{\\rm c}}{60~M_\\odot}\\right)^{3\/4}\n~M_\\odot~{\\rm yr}^{-1},\n\\end{align}\nwhich is smaller than $\\dot M_{\\rm c}\\sim {\\rm a~few}\\times\n10^{-3}~M_\\odot~{\\rm yr}^{-1}$ by approximately an order of magnitude. We therefore\nconclude that accretion on to the clump could not be suppressed by its\nown photoionization heating. After the H$_{\\rm II}$ region breaks out\nof the disc, however, the accretion proceeds only from the shadow of\nthe disc and thus the accretion rate is reduced. Moreover,\n\\cite{2011Sci...334.1250H} suggest that the gas behind the disc is\nshocked and accelerated outward by the pressure gradient after the\nexpansion of the H$_{\\rm II}$ region. This process halts the\naccretion at $M_{\\rm c} \\approx 40~M_\\odot$. \n\\citet{2013ApJ...773..155T}\nhave shown that photo-evaporation starts to suppress the accretion \nwhen $\\dot M_{\\rm PE}\/\\dot M_{\\rm c}\\ga 0.2$.\nNote that accretion on to a clump inside $R_{\\rm f}$ cannot be\nsuppressed, even if the H$_{\\rm II}$ region expands by a large factor.\nIn particular, a clump growing at a super critical rate, $\\dot M_{\\rm\n c}\\ga 10^{-2}~M_\\odot~{\\rm yr}^{-1}$, is not affected by the radiative feedback\nbecause $\\dot M_{\\rm PE}\/\\dot M_{\\rm c} \\la 0.08$.\n\nNext, we discuss the possibility that the ionizing photons from the\ncollection of all ZAMS stars together suppress the accretion (with\n$\\dot M_{\\rm tot}\\simeq 0.1~M_\\odot~{\\rm yr}^{-1}$) from the parent cloud on to the\ndisc as a whole. To make our discussion conservative, we assume that\nthe clumps which emit strong UV radiation grow at the rate just below\nthe critical accretion rate. For $M_\\ast \\sim 10^5~M_\\odot$, the\norbital decay time is $\\sim 3\\times 10^4$ yr and so each clump then\ngrows to $\\sim 100~M_\\odot$. The corresponding total ionizing photon\nrate is $\\sim 1.1\\times 10^{51}$~s$^{-1}$. From numerical simulations\nof the gravitational collapse of an atomic-cooling cloud\n\\citep{2014arXiv1404.4630I}, the rotational velocity is proportional\nto the Keplerian velocity, $v_{\\rm rot}=f_{\\rm Kep}v_{\\rm Kep}$, with\n$f_{\\rm Kep}\\simeq 0.5$. From this relation, the size of the whole\ndisc around the central protostar is given by\n\\begin{align}\nR_{\\rm d}&=f_{\\rm Kep}^2 R_{\\rm env},\\label{eq:rd}\\\\\\nonumber\n&\\simeq 2.5~{\\rm pc}\\left(\\frac{f_{\\rm Kep}}{0.5}\\right)^{2}\n\\left(\\frac{R_{\\rm env}}{10~{\\rm pc}}\\right),\n\\end{align}\nwhere $R_{\\rm env}$ is the size of the quasi-spherical parent cloud\nfuelling the inner disc. We approximate $R_{\\rm env}$ using a critical\nBonnor-Ebert sphere with a temperature of $8,000$ K and a central\ndensity of $10^4~{\\rm cm}^{-3}$, \nwhich corresponds to that of the gravitationally unstable core \nin the atomic cooling haloes without H$_2$ molecules \\citep[e.g.,][]{2008ApJ...682..745W}.\nTherefore, we obtain $\\dot M_{\\rm PE}=3.5\\times 10^{-2}~M_\\odot~{\\rm yr}^{-1}~(\\Phi_{\\rm EUV}\/1.1\\times 10^{51}~{\\rm s}^{-1})^{1\/2}\n(R_{\\rm d}\/2.5~{\\rm pc})^{1\/2}$ and thus $\\dot M_{\\rm PE}\/\\dot M_{\\rm tot}\\simeq 0.35$,\nwhich is comparable or only slightly above the critical value.\nTherefore, even with conservative assumptions, the strong UV radiation\ncannot deplete the gas supply on to the disc from the parent cloud, and\nthe central protostar can grow to a SMS star with $M_\\ast \\simeq 10^5~M_\\odot$.\n\n\n\n\\section{fragmentation by metals: dust cooling}\n\\label{sec:metal}\n\nThe site envisioned for forming a SMS is atomic cooling gas in a halo\nwith virial temperature $>10^4$ K, in which H$_2$ cooling is\nprohibited prior to and throughout the protostellar collapse. Recent\nnumerical simulations suggest that the majority of the atomic cooling\nhaloes are polluted by heavy elements due to Pop III supernovae from\nprior star formation \\citep{2010ApJ...716..510G,2012ApJ...745...50W}.\nThe resulting metallicity is $Z\\la 10^{-4}~Z_\\odot$ ($\\la\n10^{-3}~Z_\\odot$) if the Pop III supernovae are core-collapse type\n(pair-instability type). If the gas is slightly polluted with $Z\\ga\n5\\times 10^{-6}~Z_\\odot$, the temperature decreases below $\\sim 500$ K\nby dust cooling \\citep{2008ApJ...686..801O}, which could promote\nefficient fragmentation. The details of this fragmentation, and\nwhether dust cooling ultimately prevents SMS formation, are not yet\nunderstood.\n\nThe temperature of the gas with dust grains begins to decrease through\nheat exchange with cool dust grains ($T_{\\rm gr}\\ll T$) when collisions\nbetween gas particles and dust are sufficiently frequent. Collisional\ncooling of the gas is efficient until $T\\sim 500$ K, where the gas and\ndust is thermally coupled ($T\\simeq T_{\\rm gr}$). At the cooling\nphase, the compressional heating and collisional cooling are balanced\nas\n\\begin{equation}\n\\frac{\\rho c_{\\rm s}^2}{t_{\\rm ff}}\\simeq H_{\\rm gr},\n\\label{eq:d1}\n\\end{equation}\nwhere $t_{\\rm ff}=\\sqrt{3\\pi \/(32G\\rho)}$ is the free-fall time and\n$H_{\\rm gr}$ is the energy exchange rate (in erg s$^{-1}$ cm$^{-3}$)\nbetween the gas and dust, defined by\n\\begin{equation}\nH_{\\rm gr}\\simeq 2k_{\\rm B}Tn_{\\rm H}n_{\\rm gr}\\sigma_{\\rm gr}c_{\\rm s},\n\\label{eq:d2}\n\\end{equation}\nfor $T\\gg T_{\\rm gr}$ \\citep{2006MNRAS.369.1437S,2012MNRAS.419.1566S}.\nWe adopt the number density and cross-section of the dust particles\nfrom \\cite{2003A&A...410..611S},\n\\begin{equation}\n\\frac{n_{\\rm gr}\\sigma_{\\rm gr}}{\\rho}=4.7\\times 10^{-2}\\left(\\frac{Z}{10^{-4}~Z_\\odot}\\right)~{\\rm cm^2~g^{-1}},\n\\label{eq:d3}\n\\end{equation}\nwhere the depletion factor of metals to dusts is assumed as high as the present-day Galactic value \n$f_{\\rm dep}\\simeq 0.5$.\nFrom Eqs.~(\\ref{eq:d1}), (\\ref{eq:d2}), and (\\ref{eq:d3}), we obtain \n\\begin{equation}\nn\\simeq 6.8\\times 10^7~{\\rm cm}^{-3} \\left(\\frac{T}{6000~{\\rm K}}\\right)^{-1}\\left(\\frac{Z}{10^{-4}~Z_\\odot}\\right)^{-2},\n\\label{eq:d4}\n\\end{equation}\nabove which the temperature begins to decrease compared to the\nzero-metallicity case. The density given by Eq.~(\\ref{eq:d4}) is\ncomparable to that at the fragmenting radius. This implies that the\ndiscussion in \\S~\\ref{sec:fate} remains valid in metal-polluted gas,\nas long as the metallicity remains below $Z\\la 10^{-4}~Z_\\odot$. We\nconclude that disc fragmentation cannot prevent SMS formation, as long\nas the metallicity is below this value. \nOn the other hand, at\nhigher metallicity ($Z> 3\\times10^{-4}~Z_\\odot$),\nthe gas temperature rapidly decreases by metal-line cooling at\ndensities below $\\sim 10^4~{\\rm cm}^{-3}$ \\citep{2008ApJ...686..801O,2012MNRAS.422.2539I}. \nNumerical simulations of metal-enriched collapse into atomic cooling haloes\nalso capture the character of gas fragmentation for $Z\\ga 10^{-3}~Z_\\odot$\n\\citep{2014MNRAS.438.1669S,2014MNRAS.440L..76S}.\nThe outcome would be\na compact star-cluster which consists of many low-mass stars with\n$\\sim 1~M_\\odot$ \\citep{2014MNRAS.439.1884T}.\n\n\n\n\n\n\n\n\n\\section{Discussion and conclusions}\n\\label{sec:conc}\n\nIn this paper, we discussed the properties of a disc around the embryo\nof a SMS ($\\ga 10^5~M_\\odot$), expected to be present\nin a primordial gas without H$_2$ molecules in massive haloes with\nvirial temperature $\\ga 10^4$ K. A high accretion rate $\\ga\n0.1~M_\\odot~{\\rm yr}^{-1}$, sustained for $\\ga 10^5$yr, is required to form a SMS.\nThe inner region of such a disc is gravitationally unstable, and\nfragments into $O(10)$ clumps with characteristic mass of $\\sim\n30~M_\\odot$. We discuss the possibility that this fragmentation\nprevents SMS formation. We argue that most of the clumps formed in\nthe disc rapidly migrate towards the central protostar and merge with\nit. The orbital decay time is shorter than or comparable to the\norbital period of the clumps ($\\la 10^4$ yr). Some of the clumps can\ngrow via accretion and evolve to ZAMS stars within\ntheir KH time, either because they survive longer\ndue to the stochasticity of the migration process, or because they\nform later, further out in the disc.\n\nOur toy model, on which these conclusions are based,\n can be tested against simulations of Pop III star formation in\n lower mass minihaloes. In this case, the dominant cooling process is\n H$_2$ line emission ($T\\sim 10^3$ K) and the resulting accretion\n rate is $\\sim 10^{-3}-10^{-2}~M_\\odot~{\\rm yr}^{-1}$, which is smaller than that\n in the SMS formation case. According to high-resolution numerical\n simulations by \\cite{2012MNRAS.424..399G}, clump formation occurs at\n $\\sim 10$ AU ($\\Sigma \\sim 5\\times 10^3$ g cm$^{-2}$, $\\Omega \\sim\n 0.1$ yr$^{-1}$, $c_{\\rm s}\\sim 2.5$ km s$^{-1}$ and $M_{\\rm d}\\sim\n 1~M_\\odot$ in their Fig.~2) in the gravitationally unstable disc at\n the early stage $\\la 10$ yr. From Eq.~(\\ref{eq:r_f_early}), we can\n estimate the fragmentation radius as $\\sim 20$ AU, which is\n consistent with the fragmentation radii in the simulations. At this\n stage, the average accretion rate is high, $\\dot M_{\\rm tot}\\simeq\n 10^{-2}~M_\\odot~{\\rm yr}^{-1}$ in their Fig.~8. From Eq.~(\\ref{eq:continue}),\n (\\ref{eq:scaleheight}), (\\ref{eq:viscous}), and $Q\\simeq 1$,\n\\begin{equation}\n\\dot M_{\\rm tot}\\simeq 3\\alpha \\frac{c_{\\rm s}^3}{G}.\n\\end{equation}\nUsing these equations, the sound speed at the fragmentation radius\n($\\alpha _{\\rm f}=1$) is estimated as $c_{\\rm s}=2.5$ km s$^{-1}$, which\nagrees well with the numerical result. The viscous time at $\\sim 10$\nAU is roughly estimated as $t_{\\rm vis}\\sim M_{\\rm d}\/\\dot M_{\\rm\n tot}\\sim 10^2$ yr. Since some clumps migrate inward within $10$ yr\n($< t_{\\rm vis}$) in simulation, other effects (Type I migration and\ninteraction between clumps) could accelerate the clumps migration.\nTherefore, our results from the toy model are expected to be\nconservative, and appear to capture the essential features of disc\nfragmentation and clump migration.\n\nThe gas in most of the massive haloes is likely to be polluted by heavy\nelements due to prior star formation and Pop III supernovae. With the\nmetallicity of $5\\times 10^{-6} \\la Z \\la 10^{-4}~Z_\\odot$, the dust\ncooling decreases the temperature rapidly and could promote the\nfragmentation at $n> 10^{8}~{\\rm cm}^{-3}$ \\citep{2008ApJ...686..801O}. As a\nresult, the existence of dusts has been considered as one of the most\nsevere obstacles to the SMS formation. However, the fragmentation\ninduced by the dust cooling is expected to occur only within the\nfragmenting radius we estimated. \nThis means that the clumps formed by \nthe dust cooling can migrate towards the center on a time-scale comparable to the \norbital period, in the same way as the primordial case. We conclude\nthat the SMS formation is not prevented, unless the metal-line cooling\ndominates for $Z\\ga 3\\times 10^{-4}~Z_\\odot$. Our results therefore\nremove a significant obstacle for SMS formation.\n\nOur estimates for the orbital decay time of the clumps are based on\nthe well-known formulae of Type~I and II planetary migration.\nAlthough some numerical simulations find that these formulae remain\napproximately valid in gravitationally unstable and clumpy discs,\nthere are many uncertainties regarding the migration rate in this\ncase. For example, we neglect the orbital eccentricities of the\nclumps. The clumps formed by the disc fragmentation are expected to\nhave eccentric orbits and thus the angular momentum transfer between\nthe disc and clumps could change from the case of the circular orbits.\nMoreover, in such a situation, the interaction with eccentric clumps\nretard the orbital decay of the other clumps\n\\citep{2013ApJ...777L..14F}. To estimate the migration time within\nthe clumpy disc around the SMS star, more sophisticated numerical\nsimulations are necessary.\n\nIt is worth further emphasizing the limitations and uncertainties of\nour disc model. In Figure~\\ref{fig:profile}, we present the disc\nprofile based on cooling by free-bound emission of H$^-$ and thermal\nequilibrium. Hydrogen atomic cooling (Ly$\\alpha$ and two-photon\nemission) dominates at $T\\ga 8000$ K ($P>10^4$ yr). To quantify the\neffect, we include Ly$\\alpha$ cooling ($\\Lambda =\\lambda_{\\rm\n Ly\\alpha}n^2x_{\\rm e}$; \\citealt{2007ApJ...666....1G}) in the\nlimiting optically thin case (dashed lines in Fig.~\\ref{fig:profile}).\nSince the gas is in fact optically thick to the Ly$\\alpha$ photons at\n$n>10^5~{\\rm cm}^{-3}$, and the cooling efficiency of the two-photon channel is\nsmaller than the Ly$\\alpha$ emission, the dashed lines overestimate\nthe additional atomic cooling rate. Nevertheless, the profiles\ndeviate from our fiducial disc model only modestly, beginning at\n$P>10^4$ yr and thus the critical period changes at most within the\nshaded region in Figure~\\ref{fig:profile} (c).\n\nThe crucial ingredient of this paper is the critical value of the\neffective viscous parameter $\\alpha_{\\rm f}$ for fragmentation. In\nour model, the fragmentation radius depends sensitively on the value\nof $\\alpha_{\\rm f}$. For example, if we chose $\\alpha_{\\rm f}=0.5$,\nwe obtain $\\Omega_{\\rm f}=2\\times 10^{-12}$ s$^{-1}$ ($P=10^5$ yr),\n$\\Sigma_{\\rm f}=6$ g cm$^{-2}$, \n$M_{\\rm c}\\simeq 400~M_\\odot$ and $R_{\\rm f}\\simeq 0.2$ pc. In this\ncase, the orbital decay time becomes longer than the KH time of the\nclumps by a factor of 4, and most of the clumps may evolve to\nmassive stars and emit strong UV radiation. \nNevertheless, the number\nof clumps decreases by a factor of 2 from the case of $\\alpha_{\\rm f}=1.0$\n(see Eq. \\ref{eq:nc}).\nTherefore, we expect that our main conclusion also does not change. \nHowever, if $\\alpha_{\\rm f}\\la 0.1$, the fragmenting radius moves outside the size of the\nwhole disc given by Eq.~(\\ref{eq:rd}), which means that the disc\ncannot exist, i.e., our model is no longer self-consistent. To\nexplore the precise value of $\\alpha_{\\rm f}$ is left for future\ninvestigations. In particular, our results motivate high-resolution\nnumerical simulations similar to \\cite{2014MNRAS.439.1160R}.\nMoreover, more realistic treatments of radiative cooling and chemical\nreactions at densities higher than $\\sim 10^8~{\\rm cm}^{-3}$\n\\citep{2014arXiv1404.4630I} should be included since the fragmentation\nefficiency strongly depends on the equation of state of the gas.\n\nDespite these caveats, we expect that SMS formation in metal-poor gas\nin atomic-cooling haloes is difficult to avoid, as it requires that\nthe high mass inflow rate, $\\ga 0.1~M_\\odot~{\\rm yr}^{-1}$, is democratically\ndistributed among $O(100)$ fragments (so that {\\em none} of them\naccretes at a super critical rate for SMS formation) and that most of\nthese fragments survive rapid migration and avoid coalescence with the\ngrowing central protostar for several orbital times. The toy models\npresented in this paper disfavor this scenario.\n\n\n\\section*{Acknowledgements} \nWe thank Greg Bryan, Kazuyuki Omukai, Takashi Hosokawa, Kei Tanaka and\nEli Visbal for fruitful discussions. This work is supported by the\nGrants-in-Aid by the Ministry of Education, Culture, and Science of\nJapan (KI), and by NASA grant NNX11AE05G (ZH).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nDuring recent years, there has been a substantial increase of research activity in the field of medical image reconstruction. One particular application area is the acceleration of Magnetic Resonance Imaging (MRI) scans. This is an area of high impact, because MRI is the leading diagnostic modality for a wide range of exams, but the physics of its data acquisition process make it inherently slower than modalities like X-Ray, Computed Tomography or Ultrasound. Therefore, the shortening of scan times has been a major driving factor for routine clinical application of MRI.\n\nOne of the most important and successful technical developments to decrease MRI scan time in the last 20 years was parallel imaging ~\\cite{Sodickson1997,Pruessmann1999,Griswold2005}. Currently, essentially all clinical MRI scanners from all vendors are equipped with parallel imaging technology, and it is the default option for a large number of scan protocols. As a consequence, there is a substantial benefit of using multi-coil data for machine learning based image reconstruction. Not only does it provide a complementary source of acceleration that is unavailable when operating on single channel data, or on the level of image enhancement and post-processing, it also is the scenario that ultimately defines the use-case for accelerated clinical MRI, which makes it a requirement for clinical translation of new reconstruction approaches. The drawback is that working with multi-coil data adds a layer of complexity that creates a gap between cutting edge developments in deep learning~\\cite{LeCun2015} and computer vision, where the default data type are images. The goal of this manuscript is to bridge this gap by providing both a comprehensive review of the properties of parallel MRI, together with an introduction how current machine learning methods can be used for this particular application.\n\t\t\n\\begin{figure*}[!t]\n\t\\centering\n\t\\includegraphics[width = 6.5in]{.\/Picture1_v2}\n \t\\caption{In k-space based parallel imaging methods, missing data is recovered first in k-space, followed by an inverse Fourier transform and combination of the individual coil elements. In image space based parallel imaging, the Fourier transform is performed as the first step, followed by coil sensitivity based removal of the aliasing artifacts from the reconstructed image by solving an inverse problem.}\n \t\\vspace{-.3cm}\n \t\\label{fig:recon_PI_image_kspace}\n\\end{figure*}\t\t\n\t\t\n\\subsection{Background on multi-coil acquisitions in MRI}\nThe original motivation behind phased array receive coils~\\cite{Roemer1990} was to increase the SNR of MR measurements. These arrays consist of $n_c$ multiple small coil elements, where an individual coil element covers only a part of the imaging field of view. These individual signals are then combined to form a single image of the complete field of view. The central idea of all parallel imaging methods is to complement spatial signal encoding of gradient fields with information about the spatial position of these multiple coil elements. For multiple receiver coils, the MR signal equation can be written as follows\n\\begin{equation}\n\\label{eq:signal_multiplecoils}\nf_j(k_x,k_y) = \\int \\limits_{-\\infty}^{\\infty}\t\\int \\limits_{-\\infty}^{\\infty} u(x,y) c_j(x,y) e^{-i (k_x x + k_y y)}\\dd{x} \\dd{y}.\n\\end{equation}\nIn \\Cref{eq:signal_multiplecoils}, $f_j$ is the MR signal of coil $j=1,\\ldots,n_c$, $u$ is the target image to be reconstructed, and $c_j$ is the corresponding coil sensitivity. Parallel imaging methods use the redundancies in these multi-coil acquisitions to reconstruct undersampled k-space data. After discretization, this undersampling is described in matrix-vector notation by\n \\begin{equation}\n \\mathbf{f}_j = \\mathbf{F}_{\\Omega}\\mathbf{C}_j\\mathbf{u} + \\mathbf{n}_j,\n \\end{equation}\n where $\\mathbf{u}$ is the image to be reconstructed, $\\mathbf{f}_j$ is the acquired k-space data in the $j^\\textrm{th}$ coil, $\\mathbf{C}_j$ is a diagonal matrix containing \n the sensitivity profile of the receiver coil \\cite{Pruessmann1999}, $\\mathbf{F}_{\\Omega}$ is a partial Fourier sampling operator that samples locations ${\\Omega},$ and $\\mathbf{n}_j$ is measurement noise in the $j^{\\textrm{th}}$ coil.\n\nHistorically, parallel imaging methods were put in two categories: Approaches that operate in image domain, inspired by the sensitivity encoding (SENSE) method~\\cite{Pruessmann1999} and approaches that operated in k-space, inspired by simultaneous acquisition of spatial harmonics (SMASH)~\\cite{Sodickson1997} and generalized autocalibrating partial parallel acquisition (GRAPPA)~\\cite{Griswold2002}. This is conceptually illustrated in \\Cref{fig:recon_PI_image_kspace}. While these two schools of thought are closely related~\\cite{Kholmovski,Uecker2014}, we organized this document according to these classic criteria for historical reasons.\n\n\n\n\n\n\n\n\\section{Classical parallel imaging in image space}\n\\label{sec:imagespace_PI}\nClassical parallel imaging in image space follows the SENSE method~\\cite{Pruessmann1999}, which can be identified by two key features. First, the elimination of the aliasing artifacts is performed in image space after the application of an inverse Fourier transform. Second, information about receive coil sensitivities is obtained via precomputed, explicit coil sensitivity maps from either a separate reference scan or from a fully sampled block of data at the center of k-space (all didactic experiments that are shown in this manuscript follow the latter approach). More recent approaches jointly estimate coil sensitivity profiles during the image reconstruction process~\\cite{Ying2007,Uecker2008}, but for the rest of this manuscript, we assume that sensitivity maps were precomputed. The reconstruction in image domain in \\Cref{fig:recon_PI_image_kspace} shows three example undersampled coil images, corresponding coil sensitivity maps and the final reconstructed images from a brain MRI dataset. The coil sensitivities were estimated using ESPIRiT~\\cite{Uecker2014}.\n\n\n\nMRI reconstruction in general and parallel imaging in particular can be formulated as an inverse problem. This provides a general framework that allows easy integration of the concepts of regularized and constrained image reconstruction as well as machine learning that are discussed in more detail in later sections. \\Cref{eq:signal_multiplecoils} can be discretized and then written in matrix-vector form:\n\\begin{equation}\n\t\\label{eq:forward_problem}\n\t{\\mathbf{f} = \\mathbf{E}\\mathbf{u} +\\mathbf{n}},\n\\end{equation}\nwhere $\\mathbf{f}$ contains all k-space measurement data points and $\\mathbf{E}$ is the forward encoding operator that includes information about the sampling trajectory and the receive coil sensitivities and $\\mathbf{n}$ is measurement noise. The task of image reconstruction is to recover the image $\\mathbf{u}$. In classic parallel imaging one generally operates under the condition that the number of receive elements is larger than the acceleration factor. Therefore, \\Cref{eq:forward_problem} corresponds to an over-determined system of equations. However, the rows of $\\mathbf{E}$ are linearly dependent because individual coil elements do not measure completely independent information. Therefore the inversion of $\\mathbf{E}$ is an ill-posed problem, which can lead to severe noise amplification, described via the g-factor in the original SENSE paper~\\cite{Pruessmann1999}. \\Cref{eq:forward_problem} is usually solved in an iterative manner, which is the topic of the following sections.\n \n\\subsection{Overview of conjugate gradient SENSE (CG-SENSE)}\nThe original SENSE approach is based on equidistant or uniform Cartesian k-space sampling, where the aliasing pattern is defined by a point spread function that has a small number of sharp equidistant peaks. This property leads to a small number of pixels that are folded on top of each other, which allows a very efficient implementation~\\cite{Pruessmann1999}. When using alternative k-space sampling strategies like non-Cartesian acquisitions or random undersampling, this is no longer possible and image reconstruction requires a full inversion of the encoding matrix in \\Cref{eq:forward_problem}. This operation is demanding both in terms of compute and memory requirements (the dimensions of $\\mathbf{E}$ are the total number of acquired k-space points times $N^2$ where $N$ is the size of the image matrix that is to be reconstructed), which lead to the development of iterative methods, in particular the CG-SENSE method introduced by Pruessmann et al. as a follow up of the original SENSE paper~\\cite{Pruessmann2001}. In iterative image reconstruction the goal is to find a $\\hat{\\mathbf{u}}$ that is a minimizer of the following cost function, which corresponds to the quadratic form of the system in \\Cref{eq:forward_problem}:\n\\begin{equation}\n \\label{eq:inverse_leastSquares}\n \\hat{\\mathbf{u}} \\in \\argmin\\limits_{\\mathbf{u}} \\frac{1}{2} \\| \\mathbf{E} \\mathbf{u} - \\mathbf{f} \\|_2^2.\n\\end{equation}\nIn standard parallel imaging, $\\mathbf{E}$ is linear and \\Cref{eq:inverse_leastSquares} is a convex optimization problem that can be solved with a large number of numerical algorithms like gradient descent, Landweber iterations~\\cite{Landweber1951}, primal-dual methods~\\cite{Pock_PD_2010} or the alternating direction method of multipliers (ADMM) algorithm~\\cite{Boyd2011} (a detailed review of numerical methods is outside the scope of this article). \nIn the original version of CG-SENSE~\\cite{Pruessmann2001}, the conjugate gradient method~\\cite{Hestenes1952} is employed. However, since MR k-space data are corrupted by noise, it is common practice stop iterating before theoretical convergence is reached, which can be seen as a form of regularization.\nRegularization can be also incorportated via additional constraints in \\Cref{eq:inverse_leastSquares}, which will be covered in the next section.\n\nAs a didactic example for this manuscript, we will use a single slice of a 2D coronal knee exam to illustrate various reconstruction approaches. This data were acquired on a clinical 3T system (Siemens Skyra) using a 15 channel phased array knee coil. A turbo spin echo sequence was used with the following sequence parameters: TR=2750ms, TE=27m, echo train length=4, field of view 160mm$^2$ in-plane resolution 0.5mm$^2$, slice thickness 3mm.\nReadout oversampling with a factor of 2 was used, and all images were cropped in the frequency encoding direction (superior-inferior) for display purposes. In the spirit of reproducible research, data, sampling masks and coil sensitivity estimations that were used for the numerical results in this manuscript are available online\\footnote{\\url{https:\/\/app.globus.org\/}: Endpoint: NYULH Radiology Reconstruction Data, coronal pd data, subject 17, slice 25.}. \n\\Cref{fig:knee_reconstructions} shows an example of a retrospectively undersampled CG-SENSE reconstruction with an acceleration factor of 4 for the data from \\Cref{fig:knee_reconstructions}. Early stopping was employed by setting the numeric tolerance of the iteration to $5\\cdot10^{-5}$, which resulted in the the algorithm stopping after 14 CG iterations.\n\n\\subsection{Nonlinear regularization and compressed sensing} \\label{sec:2c}\n\n\\Cref{eq:inverse_leastSquares} can be extended by including a-priori knowledge via additional penalty terms, which results in a constrained optimization problem defined in \\Cref{eq:inverse_regularization}, which forms the cornerstone of almost all modern MRI reconstruction methods\n\\begin{equation}\n \\label{eq:inverse_regularization}\n \\hat{\\mathbf{u}} \\in \\argmin\\limits_{\\mathbf{u}} \\frac{1}{2} \\| \\mathbf{E} \\mathbf{u} - \\mathbf{f} \\|_2^2 + \\sum_{i} \\lambda_i \\Psi_i(\\mathbf{u}).\n\\end{equation}\nHere, $\\Psi_i$ are dedicated regularization terms and $\\lambda_i$ are regularization parameters that balance the trade-off between data fidelity and prior. Since the introduction of compressed sensing~\\cite{Candes2006, Donoho2006} and its adoption for MRI~\\cite{Lustig2007,Block2007,Lustig2008a}, nonlinear regularization terms, in particular $\\ell_1$-norm based ones, are popular in image reconstruction and are commonly used in parallel imaging~\\cite{Block2007,Lustig2010,Knoll2011,Knoll2012, Akcakaya2011, Akcakaya2014, Jung2009}. The goal of these regularization terms is to provide a separation between the target image that is to be reconstructed from the aliasing artifacts that are introduced due to an undersampled acquisition. Therefore, they are usually designed in conjunction with a particular data sampling strategy. The classic formulation of compressed sensing in MRI~\\cite{Lustig2007} is based on sparsity of the image in a transform domain (Wavelets are a popular choice for static images) in combination with pseudo-random sampling, which introduces aliasing artifacts that are incoherent in the respective domain. For dynamic acquisitions where periodic motion is encountered, sparsity in the Fourier domain common choice~\\cite{Gamper2008}. Total Variation based methods have been used successfully in combination with radial~\\cite{Block2007} and spiral~\\cite{Valvano2016} acquisitions as well as in dynamic imaging~\\cite{Feng2013}. More advanced regularizers based on low-rank properties have also been utilized \\cite{Lingala2011}.\n\n\\Cref{fig:knee_reconstructions} shows an example of a nonlinear combined parallel imaging and compressed sensing reconstruction with a Total Generalized Variation~\\cite{Knoll2011} constraint. The regularization parameter $\\lambda$ was set to $2.5\\cdot10^{-5}$ \nand the reconstruction was using 1000 primal-dual~\\cite{Pock_PD_2010} iterations. The used equidistant sampling was chosen for consistency with the other reconstruction methods is not optimal for the incoherence condition in compressed sensing. Nevertheless, the nonlinear regularization still provides a superior reduction of aliasing artifacts and noise suppression in comparison to the CG-SENSE reconstruction from the last section. \n\t\t\t\t\t\t\n\\section{Classical parallel imaging in k-space}\n\nParallel imaging reconstruction can also be formulated in k-space as an interpolation procedure. The initial connections between the SENSE-type image-domain inverse problem approach and k-space interpolation has been made more than a decade ago \\cite{Kholmovski}, where it was noted that the forward model in \\Cref{eq:forward_problem} can be restated in terms of the Fourier transform, ${\\bm \\kappa}$ of the combined image, $\\mathbf{u}$ as \n\\begin{equation}\n \\mathbf{f} = \\mathbf{A}\\mathbf{F}^*{\\bm \\kappa} \\triangleq \\mathbf{G}_{\\text{acq}} {\\bm \\kappa},\n\\end{equation}\nwhere $\\mathbf{f}$ corresponds to the acquired k-space lines across all coils, and $\\mathbf{G}_{\\text{acq}}$ is a linear operator. Similarly, the unacquired k-space lines across all coils can be formulated using \n\\begin{equation}\n \\mathbf{f}_{\\text{unacq}} = \\mathbf{G}_{\\text{unacq}} {\\bm \\kappa}\n\\end{equation}\nCombining these two equations yield\n\\begin{equation}\n \\mathbf{f}_{\\text{unacq}} = \\mathbf{G}_{\\text{unacq}} \\mathbf{G}_{\\text{acq}}^{\\dagger} \\mathbf{f}.\n\\end{equation}\nThus, the unaccquired k-space lines across all coils can be interpolated based on the acquired lines across all coils, assuming the pseudo-inverse, $\\mathbf{G}_{\\text{acq}}^{\\dagger}$, of $\\mathbf{G}_{\\text{acq}}$ exists \\cite{Kholmovski}. Thus, the main difference between the k-space parallel imaging methods and the aforementioned image domain parallel imaging techniques is that the former produces k-space data across all coils at the output, whereas the latter typically produces one image that combines the information from all coils.\n\n\\subsection{Linear k-space interpolation in GRAPPA}\nThe most clinically used k-space reconstruction method for parallel imaging is GRAPPA, which uses linear shift-invariant convolutional kernels to interpolate missing k-space lines using uniformly-spaced acquired k-space lines \\cite{Griswold2002}. For the $j^\\textrm{th}$ coil k-space data, ${\\kappa}_j$, we have\n\\begin{align}\n {\\kappa}_j&(k_x, k_y - m\\Delta k_y) \\nonumber \\\\\n &= \\sum_{c=1}^{n_c} \\sum_{b_x = -B_x}^{B_x} \\sum_{b_y = -B_y}^{B_y} g_{j,m} (b_x,b_y,c) \\nonumber \\\\\n &\\quad\\quad \\quad \\quad \\kappa_c(k_x - b_x \\Delta k_x,k_y - Rb_y \\Delta k_y),\n\\end{align}\nwhere $R$ is the acceleration rate of the uniformly undersamped acquisition; $m \\in \\{1, \\dots, R-1\\}$; $g_{j,m}(b_x,b_y,c)$ are the linear convolutional kernels for estimating the $m^\\textrm{th}$ spacing location in $j^\\textrm{th}$ coil; $n_c$ is the number of coils; and $B_x$, $B_y$ are parameters determined from the convolutional kernel size. A high-level overview of such interpolation is shown in the reconstruction in k-space section of \\Cref{fig:recon_PI_image_kspace}.\n\nSimilar to the coil sensitivity estimation in SENSE-type reconstruction, the convolutional kernels, $g_{j,m}(b_x,b_y,c)$ are estimated for each subject, from either a separate reference scan or from a fully-sampled block of data at the center of k-space, called autocalibrating signal (ACS) \\cite{Griswold2002}. A sliding window approach is used in this calibration region to identify the fully-sampled acquisition locations specified by the kernel size and the corresponding missing entries. The former, taken across all coils, is used as rows of a calibration matrix, $\\mathbf{A}$; while the latter, for a specific coil, yields a single entry in the target vector, $\\mathbf{b}$. Thus for each coil $j$ and missing location $m \\in \\{1, \\dots, R-1\\}$, a set of linear equations are formed, from which the vectorized kernel weights $g_{j,m}(b_x,b_y,c)$, denoted $\\mathbf{g}_{j,m}$, are estimated via least squares, as $\\mathbf{g}_{j,m}\\in \\arg \\min_{\\mathbf{g}} ||{\\mathbf{b}- \\mathbf{Ag}}||_2^2$. GRAPPA has been shown to have several favorable properties compared to SENSE, including lower g-factors, sometimes even less than unity at certain parts of the image \\cite{Robson2008}, and more smoothly varying g-factor maps \\cite{Breuer2009}. Furthermore, k-space interpolation is often less sensitive to motion \\cite{Breuer2005}. Due to these favorable properties, GRAPPA has found utility in multiple large-scale projects, such as the Human Connectome Project \\cite{Ugurbil2013}.\n\n\n\n\\subsection{Advances in k-space interpolation methods}\n\nThough GRAPPA is widely used in clinical practice, it is a linear method that suffers from noise amplification based on the coil geometry and the acceleration rate \\cite{Pruessmann1999}. Therefore, several alternative strategies have been proposed in the literature to reduce the noise in reconstruction.\n\nIterative self-consistent parallel imaging reconstruction (SPIRiT) is a strategy for enforcing self-consistency among the k-space data in multiple receiver coils by exploiting correlations between neighboring k-space points \\cite{Lustig2010}. Similar to GRAPPA, SPIRiT also estimates a linear shift-invariant convolutional kernel from ACS data. In GRAPPA, this convolutional kernel used information from acquired lines in a neighborhood to estimate a missing k-space point. In SPIRiT, the kernel includes contributions from all points, both acquired and missing, across all coils for a neighborhood around a given k-space point. \nThe self-consistency idea suggests that the full k-space data should remain unchanged under this convolution operation. SPIRiT objective function also includes a term that enforces consistency with the acquired data, where the undersampling can be performed with arbitrary patterns, including random patterns that are typically employed in compressed sensing \\cite{Block2007, Lustig2007}.\nAdditionally, this formulation allows incorporation of regularizers, for instance based on transform-domain sparsity, in the objective function \nto reduce reconstruction noise via non-linear processing \\cite{Lustig2010\n. Furthermore, SPIRiT has facilitated the connection between coil sensitivities used in image-domain parallel imaging methods and the convolutional kernels used in k-space methods via a subspace analysis \\cite{Uecker2014}.\n\nAn alternative line of work utilizes non-linear k-space interpolation for estimating missing k-space points for uniformly undersampled parallel imaging acquisitions \\cite{Chang2012}. It was noted that during GRAPPA calibration, both the regressand and the regressor have errors in them due to measurement noise in the acquisition of calibration data, which leads to a non-linear relationship in the estimation. Thus, the reconstruction method, called non-linear GRAPPA, uses a kernel approach to map the data to a higher-dimensional feature space, where linear interpolation is performed, which also corresponds to a non-linear interpolation in the original data space. The interpolation function is estimated from the ACS data, although this approach typically required more ACS data than GRAPPA \\cite{Chang2012}. This method was shown to reduce reconstruction noise compared to GRAPPA. Note that non-linear GRAPPA, through its use of the kernel approach, is a type of machine learning approach, though the non-linear kernel functions were empirically fixed a-priori and not learned from data.\n\n\\subsection{k-space reconstruction via low-rank matrix completion}\nWhile k-space interpolation methods remain the prevalent method for k-space parallel imaging reconstruction, there has been recent efforts on recasting this type of reconstruction as a matrix completion problem. Simultaneous autocalibrating and k-space estimation (SAKE) is an early work in this direction, where local neighborhoods in k-space across all coils are restructured into a matrix with block Hankel form \\cite{Shin2014}. Then low-rank matrix completion is performed on this matrix, subject to consistency with acquired data, enabling k-space parallel imaging reconstruction without additional calibration data acquisition. Low-rank matrix modeling of local k-space neighborhoods (LORAKS) is another method exploiting similar ideas, where the motivation is based on utilizing finite image support and image phase constraints instead of correlations across multiple coils \\cite{Haldar2014}. This method was later extended to parallel imaging to further include the similarities between image supports and phase constraints across coils \\cite{Haldar2016}. A further generalization to LORAKS is annihilating filter-based low rank Hankel matrix approach (ALOHA), which extends the finite support constraint to transform domains \\cite{Jin2016a}. By relating transform domain sparsity to the existence of annihilating filters in a weighted k-space, where the weighting is determined by the choice of transform domain, ALOHA recasts the reconstruction problem as the low-rank recovery of the associated Hankel matrix.\n\n\n\\section{Machine learning methods for parallel imaging in image space}\n\\label{sec:imagespace_learning}\nThe use of machine learning for image-based parallel MR imaging evolves naturally from \\Cref{eq:inverse_regularization} based on the following key insights. First, in classic compressed sensing, $\\Psi$ are a general regularizers like the image gradient or wavelet transforms, which were not designed specifically with undersampled parallel MRI acquisitions in mind. These regularizers can be generalized to models that have a higher computational complexity. $\\Psi$ can be formulated as a convolutional neural network (CNN)~\\cite{LeCun1989}, where the model parameters can be learned from training data inspired by the concepts of deep learning~\\cite{LeCun2015}. This was already demonstrated earlier in the context of computer vision by Roth and Black~\\cite{Roth2009}. They proposed a non-convex regularizer of the following form:\n\\begin{equation}\n \\label{eq:foe_model}\n \\Psi{(\\mathbf{u})} = \\sum_{i=1}^{N_k} \\langle \\rho_i(\\mathbf{K}_i\\mathbf{u}),\\mathbf{1} \\rangle.\n\\end{equation}\nThe regularizer in \\Cref{eq:foe_model} consists of $N_k$ terms of non-linear potential functions $\\rho_i$, $\\mathbf{K}_i$ are convolution operators. $\\mathbf{1}$ indicates a vector of ones. \nThe parameters of the convolution operators and the parametrization of the non-linear potential functions, form the free parameters of the model, which are learned from training data.\n\n\\begin{figure}[!t]\n\n\t\\includegraphics[width = 1 \\columnwidth]{.\/figure_learning_image_space_illustration_150dpi}\n \t\\caption{Illustration of machine learning-based image reconstruction. The network architecture consists of $T$ stages that perform the equivalent of gradient descent steps in a classic iterative algorithm. Each stage consists of a regularizer and a data consistency layer. Training the network parameters $\\Theta$ is performed by retrospectively undersampling fully sampled multi-coil raw k-space data and comparing the output of the network $\\mathbf{u}^T_s(\\Theta)$ to a target reference reconstruction $\\mathbf{u}_{\\text{ref}}$ obtained from the fully sampled data.}\n \t\\label{fig:learning_image_space_illustration}\n\\end{figure}\n\n\n\\begin{figure*}[!t]\n\\begin{center}\n\n \\includegraphics[width = 1 \\textwidth]{.\/figure_knee_reconstructions_150dpi}\n \t\\caption{Comparison of image-domain based parallel imaging reconstructions of a retrospectively accelerated coronal knee acquisition.\n \tThe used sampling pattern, zero-filling, CG-SENSE, combined parallel imaging and compressed sensing with a TGV constrained and a learned reconstruction are shown together with their SSIM values to the fully sampled reference. See text in the respective sections for details on the individual experiments.}\n \t\\label{fig:knee_reconstructions}\n \t\\vspace{-.3cm}\n\\end{center}\n\\end{figure*}\n\nThe second insight is that the iterative algorithm that is used to solve \\Cref{eq:inverse_regularization} naturally maps to the structure of a neural network, where every layer in the network represents an iteration step of a classic algorithm \\cite{Gregor2010}.\nThis follows naturally from gradient descent for the least squares problem in \\Cref{eq:inverse_leastSquares} that leads to the iterative Landweber method~\\cite{Landweber1951}. After choosing an initial $\\mathbf{u}^0$, the iteration scheme is given by \\Cref{eq:landweber}:\n\\begin{equation}\n \\label{eq:landweber}\n\t\\mathbf{u}^{t} = \\mathbf{u}^{t-1} - \\alpha^t \\mathbf{E}^*(\\mathbf{E}\\mathbf{u}^{t-1} - \\mathbf{f}), \\quad t > 0.\n\\end{equation}\n$\\mathbf{E}^*$ is the adjoint of the encoding operator and $\\alpha^t$ is the step size of iteration $t$. Using this iteration scheme to solve the reconstruction problem in \\Cref{eq:inverse_regularization} with the regularizer defined in \\Cref{eq:foe_model} leads to the update scheme defined in \\Cref{eq:landweber_foe}, which forms the basis of recently proposed image space based machine learning methods for parallel MRI:\n\\begin{equation}\n \\label{eq:landweber_foe}\n\t\\mathbf{u}^{t} = \\mathbf{u}^{t-1} - \\alpha^t \\left( \\sum_{i=1}^{N_k} (\\mathbf{K}_i)^\\top \\rho_i'(\\mathbf{K}_i \\mathbf{u}^{t-1}) + \\lambda^t \\mathbf{E}^*(\\mathbf{E}\\mathbf{u}^{t-1} - \\mathbf{f}) \\right).\n\\end{equation}\nThis update scheme can then be represented as a neural network with $T$ stages corresponding to $T$ iteration steps \\Cref{eq:landweber_foe}. $\\rho_i'$ are the first derivatives of the nonlinear potential functions $\\rho_i$, which are represented as activation functions in the neural network. The transposed convolution operations ${\\mathbf{K}}_i^\\top$ correspond to convolutions with filter kernels rotated by 180 degrees. Most recently proposed approaches follow this structure, and their difference mainly lies is the used model architecture. The idea of the variational network~\\cite{Hammernik2018} follows the structure of classic variational methods and gradient-based optimization, and the network architecture is designed to mimic a classic iterative image reconstruction. The approach from Aggarwal et al.~\\cite{Aggarwal2019} follows a similar design concept, while using convolutional neural networks (CNNs), but shares the same set of parameters for all stages of the network, thus reducing the number of free parameters. It also uses an unrolled conjugate-gradient step for data consistency instead of the gradient based on in \\Cref{eq:landweber}. \n\nAn illustration of image space based machine learning for parallel MRI along the lines of~\\cite{Hammernik2018,Aggarwal2019} is shown in \\Cref{fig:learning_image_space_illustration}. \nTo determine the model parameters of the network that will perform the parallel imaging reconstruction task, an optimization problem needs to be defined that minimizes a training objective. In general, this can be formulated in a supervised or unsupervised manner. Supervised approaches are predominantly used while unsupervised approaches are still a topic of ongoing research (an approach for low-dose CT reconstruction was presented in~\\cite{Wu2017}). Therefore, we will focus on supervised approaches for the remainder of this section. We define the number of stages, corresponding to gradient steps in the network, as $T$. $s$ is the current training image out of the complete set of training data $S$. The variable $\\Theta$ contains all trainable parameters of the reconstruction model. The training objective then takes the following form:\n\\begin{equation}\n \\label{eq:image_space_learning_training}\n L(\\Theta) = \\min_{\\Theta} \\frac{1}{2S} \\sum_{s=1}^{S} \\| \\mathbf{u}^T_s(\\Theta) - \\mathbf{u}_{\\text{ref},s} \\|_2^2.\n\\end{equation}\n\n\n\nAs it is common in deep learning, \\Cref{eq:image_space_learning_training} is a non-convex optimization problem that is solved with standard numerical optimizers like stochastic gradient descent or ADAM~\\cite{Kingma2014}. This requires the computation of the gradient of the training objective with respect to the model parameters $\\Theta$. This gradient can be computed via backpropagation~\\cite{LeCun2012}:\n\\begin{equation}\n\t\\frac{\\partial L(\\Theta)}{\\partial \\Theta^t} = \\frac{\\partial \\mathbf{u}^{t+1}}{\\partial \\Theta^t} \\cdot \\frac{\\partial \\mathbf{u}^{t+2}}{\\partial \\mathbf{u}^{t+1}} \\hdots \\cdot \\frac{\\partial \\mathbf{u}^T}{\\partial \\mathbf{u}^{T-1}} \\cdot \\frac{\\partial L(\\Theta)}{\\partial \\mathbf{u}^T}.\n\\end{equation}\n\n\nThe basis of supervised approaches is the availability of a target reference reconstruction $\\mathbf{u}_{\\text{ref}}$. This requires the availability of a fully-sampled set of raw phased array coil k-space data. This data is then retrospectively undersampled by removing k-space data points as defined by the sampling trajectory in the forward operator $\\mathbf{E}$ and serves as the input of the reconstruction network. The current output of the network $\\mathbf{u}^T_s(\\Theta)$ is then compared to the reference $\\mathbf{u}_{\\text{ref}}$ via an error metric. The choice of this error metric has an influence on the properties of the trained network, which is a topic of currently ongoing work. A popular choice is the mean squared error (MSE), which was also used in \\Cref{eq:image_space_learning_training}. Other choices are the $\\ell_1$ norm of the difference~\\cite{Hammernik2017b} and the structural similarity index (SSIM)~\\cite{Wang2004}. Research on generative adversarial networks~\\cite{Goodfellow2014} and learned content loss functions is currently in progress. The current literature in this area is further noted in the discussion section. \n\n\nAn example reconstruction that compares the variational network learning approach from~\\cite{Hammernik2018} to CG-SENSE and constrained reconstructions from the previous sections is shown in \\Cref{fig:knee_reconstructions} together with the SSIM to the fully sampled reference. It can be observed that the learned reconstruction outperforms the other approaches in terms of artifact removal and preservation of small image features, which is also reflected in the highest SSIM. All source code\\footnote{\\url{https:\/\/github.com\/VLOGroup\/mri-variationalnetwork}} for this method is available online. \n\n\n\n\\section{Machine learning methods for parallel imaging in k-space} \\label{sec:kspace_learning}\n\nThere has been a recent interest in using neural network to improve the k-space interpolation techniques using non-linear approaches in a data-driven manner. These newer approaches can be divided into two groups based on how the interpolation functions are trained. The first group uses scan-specific ACS lines to train neural networks for interpolation, similar to existing interpolation approaches, such as GRAPPA or non-linear GRAPPA. The second group uses training databases, similar to the machine learning methods discussed in image domain parallel imaging. \n\nRobust artificial-neural-networks for k-space interpolation (RAKI) is a scan-specific machine learning approach for improved k-space interpolation \\cite{Akcakaya2019}. This approach trains CNNs on ACS data, and uses these for interpolating missing k-space points from acquired ones. The interpolation function can be represented by\n\\begin{align}\n {\\kappa}&_j(k_x, k_y - m\\Delta k_y) = f_{j,m}(\\{{\\kappa}_c(k_x - b_x \\Delta_x, \n \\nonumber \\\\\n &\\quad k_y - R b_y \\Delta_y)\\}\n _{b_x \\in [-B_x,B_x], b_y \\in [-B_y, B_y], c \\in [1, n_c]}),\n\\end{align}\nwhere $f_{j,m}$ is the interpolation rule implemented via a multi-layer CNN for outputting the k-space of the $m^\\textrm{th}$ set of uniformly spaced missing lines in the $j^\\textrm{th}$ coil, $R$ is the undersampling rate, $B_x, B_y$ are parameters specified by the receptive field of the CNN, $n_c$ is the number of coils. \nThus, the premise of RAKI is similar to GRAPPA, while the interpolation function is implemented using CNNs, whose parameters are learned from ACS data with an MSE loss function. The scan-specific nature of this method is attractive since it requires no training databases, and can be applied to scenarios where a fully-sampled gold reference cannot be acquired, for instance in perfusion or real-time cardiac MRI, or high-resolution brain imaging. \n\\begin{figure}\n\\centering\n\t\\includegraphics[width = \\columnwidth]{.\/RAKI_7T}\n\t \\caption{A slice from a high-resolution (0.6 mm isotropic) 7T brain acquisition, where all acquisitions were performed with prospective acceleration. It is difficult to acquire fully-sampled reference datasets for training for such acquisitions, thus two scan-specific k-space methods were compared. The CNN-based RAKI method visibly reduced noise amplification compared to the linear GRAPPA reconstruction.}\n \n \t\\label{fig:raki7T}\n \t\\vspace{-.2cm}\n\\end{figure}\nExample RAKI and GRAPPA reconstructions for such high-resolution brain imaging datasets, which were acquired with prospective undersampling are shown in \\Cref{fig:raki7T}. These data were acquired on a 7T system (Siemens Magnex Scientific) with 0.6 mm isotropic resolution. $R = 5,6$ data were acquired with two averages for improved SNR to facilitate visualization of any residual artifacts. Other imaging parameters are available in \\cite{Akcakaya2019}. For these datasets, RAKI leads to a reduction in noise amplification compared to GRAPPA. Note the noise reduction here is based on exploiting properties of the coil geometry, and not on assumptions about image structure, as in traditional regularized inverse problem approaches, as in \\Cref{sec:2c}. \nHowever, the scan-specificity also comes with downsides, such as the computational burden of training for each scan \\cite{Zhang2018a}, as well as the requirement for typically more calibration data. In \\Cref{fig:knee_kspace}, reconstructions of the knee dataset from \\Cref{fig:knee_reconstructions} are shown, where all methods, which rely on subject-specific calibration data, exhibit a degree of artifacts, due to the small size of the ACS region, while RAKI has the highest SSIM among these.\n\nWhile originally designed for uniform undersampling patterns, this method has been extended to arbitrary sampling, building on the self-consistency approach of SPIRiT \\cite{Hosseini2019}. Additionally, recent work has also reformulated this interpolation procedure as a residual CNN, with residual defined based on a GRAPPA interpolation kernel \\cite{Zhang2019}. Thus, in this approach called residual RAKI (rRAKI), the CNN effectively learns to remove the noise amplification and artifacts associated with GRAPPA, giving a physical interpretation to the CNN output, which is similar to the use of residual networks in image denoising \\cite{Zhang2017denoise}. An example application of the rRAKI approach in simultaneous multi-slice imaging \\cite{Zhang2018b} is shown in \\Cref{fig:rraki}.\n\\begin{figure}[!t]\n\\centering\n\t\\includegraphics[width =\\columnwidth]{.\/knee_kspace}\n \t\\caption{Comparison of k-space parallel imaging reconstructions of a retrospectively accelerated coronal knee acquisition, as in \\Cref{fig:knee_reconstructions}. Due to the small size of the ACS data relative to the acceleration rate, the methods, none of which utilizes training databases, exhibit artifacts. GRAPPA has residual aliasing, whereas SPIRiT shows noise amplification. These are reduced in RAKI, though the residual artifacts remain. Respective SSIM values reflect these visual assessment.}\n \t\\label{fig:knee_kspace}\n \t\\vspace{-.2cm}\n\\end{figure}\n\n\\begin{figure*}[!t]\n\\centering\n\t\\includegraphics[width = \\textwidth]{.\/fmri_mid8slice}\n \t\\caption{Reconstruction results of simultaneous multi-slice imaging of 16 slices in fMRI, where the central 8 slices are shown. GRAPPA method exhibits noise amplification at this high acceleration rate. The rRAKI method, whose linear and residual components are depicted by $G^C$ and $F^C$ respectively, exhibits reduced noise. Due to imperfections in the ACS data for this application, the residual component includes both noise amplification and residual artifacts.}\n \t\\label{fig:rraki}\n\\end{figure*}\n\nA different line of work, called DeepSPIRiT, explores using CNNs trained on large databases for k-space interpolation with a SPIRiT-type approach \\cite{Cheng2018}. Since sensitivity profiles and number of coils vary for different anatomies and hardware configurations, k-space data in the database were normalized using coil compression to yield the same number of channels \\cite{Buehrer2007, Huang2008}. Coil compression methods effectively capture most of the energy across coils in a few virtual channels, with the first virtual channel containing most of the energy, the second being the second most dominant, and so on, in a manner reminiscent of principal component analysis. After this normalization of the k-space database, CNNs are trained for different regions of k-space, which are subsequently applied in a multi-resolution approach, successively improving the resolution of the reconstructions, as illustrated in \\Cref{fig:deepspirit}. The method was shown to remove aliasing artifacts, though difficulty with high-resolution content was noted. Since DeepSPIRiT trains interpolation kernels on a database, it does not require calibration data for a given scan, potentially reducing acquisition time further. \n\n\\begin{figure}[!b]\n\\centering\n\n\t\\includegraphics[width =\\columnwidth]{.\/deepspirit_small}\n\t\\caption{The multi-resolution k-space interpolation in DeepSPIRiT uses distinct CNNs for diffrent regions of k-space, successively refining the resolution of the reconstructed k-space.}\n \n \t\\label{fig:deepspirit}\n\\end{figure}\n\nNeural networks have also been applied to the Hankel matrix based approaches in k-space \\cite{Han2018a}. Specifically, the completion of the weighted k-space in ALOHA method has been replaced with a CNN, trained with an MSE loss function. The method was shown to not only improve the computational time, but also the reconstruction quality compared to original ALOHA by exploiting structures beyond low-rankness of Hankel matrices.\n\n\n\n\n\\section{Discussion}\n\\subsection{Issues and open problems}\nSeveral advantages of machine learning approaches over classic constrained reconstruction using predefined regularizers have been proposed in the literature. First, the regularizer is tailored to a specific image reconstruction task, which improves the removal of residual artifacts. This becomes particularly relevant in situations where the used sampling trajectory does not fulfill the incoherence requirements of compressed sensing, which is often the case for clinical parallel imaging protocols. Second, machine learning approaches decouple the compute-heavy training step from a lean inference step. In medical image reconstruction, it is critical to have images available immediately after the scan, while prolonged training procedures that can be done on specialized computing hardware, are generally acceptable. The training for the experiment in \\Cref{fig:knee_reconstructions} took 40 hours for 150 epochs with 200 slices of training data on a single NVIDIA M40 GPU with 12GB of memory. Training data, model and training parameters exactly follow the training from~\\cite{Knoll2019}. Reconstruction of one slice then took 200ms, in comparison to 10ms for zero filling, 150ms for CG-SENSE and 10000ms for the PI-CS TGV constrained reconstruction.\n\nThe focus in \\Cref{sec:imagespace_learning} and \\Cref{sec:kspace_learning} were on methods that were developed specifically in the context of parallel imaging. Some architectures for image domain machine learning have been designed specifically towards a target application, for example dynamic imaging~\\cite{Schlemper2018,Qin2019}. In their current form, these were not yet demonstrated in the context of multi-coil data. The approach recently proposed by Zhu et al. learns the complete mapping from k-space raw data to the reconstructed image~\\cite{Zhu2018a}. The proposed advantage is that since no information about the acquisition is included in the forward operator $A$, it is more robust against systematic calibration errors during the acquisition. This comes at the price of a significantly higher number of model parameters. The corresponding memory requirements make it challenging use this model for matrix sizes that are currently used in clinical applications. \nWe also note that there are fewer works in k-space machine learning methods for MRI reconstruction. This may be due to the different nature of k-space signal that usually has very different intensity characteristics in the center versus the outer k-space, which makes it difficult to generalize the plethora of techniques developed in computer vision and image processing that exploit properties of natural images.\n\nMachine learning reconstruction approaches also come with a number of drawbacks when compared to classic constrained parallel imaging. First, they require the availability of a curated training data set that is representative so that the trained model generalized towards new unseen test data. While recent approaches from the literature~\\cite{Hammernik2018,Aggarwal2019,Schlemper2018,Qin2019,Chen2018} have either been trained with hundreds of examples rather than millions of examples as it is common in deep learning for computer vision~\\cite{Deng2009,LeCun2015}, or trained on synthetic non-medical data~\\cite{Zhu2018a} that is publicly available from existing databases. However, this is still a challenge that will potentially limit the use of machine learning to certain applications. Several applications in imaging of moving organs, such as the heart, or in imaging of the brain connectivity, such as diffusion MRI, cannot be acquired with fully-sampled data due to constraints on spatio-temporal resolutions. This hinders the use of fully-sampled training labels for such datasets, highlighting applications for scan-specific approaches or unsupervised training strategies.\n\nThese reconstruction methods also require the availability of computing resources during the training stage. This is a less critical issue due to the increased availability and reduced prices of GPUs. The experiments in this paper were made with computing resources that are available for less than 10,000 USD, which are usually available in academic institutions. In addition, the availability of on-demand cloud based machine learning solutions is constantly increasing. \n\nA more severe issue is that in contrast to conventional parallel imaging and compressed sensing, machine learning models are mostly non-convex. Their properties, especially regarding their failure modes and generalization potential for daily clinical use, are understood less well than conventional iterative approaches based on convex optimization. For example, it was recently shown that while reconstructions generalize well with respect to changes in image contrast between training and test data, they are susceptible towards systematic deviations in SNR~\\cite{Knoll2019}. It is also still an open question how specific trained models have to be. \nIs it sufficient to train a single model for all types of MR exams, or are separate models required for scans of different anatomical areas, pulse sequences, acquisition trajectories and acceleration factors as well as scanner manufacturers, field strengths and receive coils? \\cite{icassp2017_open_problems} \nWhile pre-training a large number of separate models for different exams would be feasible in clinical practice, if certain models do not generalize with respect to scan parameter settings that are usually tailored to the specific anatomy of an individual patient by the MR technologist, this will severely impact their translational potential and ultimately their clinical use.\n\nFinally, the choice of the loss function that is used during the training has an impact on the properties of the trained network, and a particular ongoing research direction is the use of GANs~\\cite{Goodfellow2014,Arjovsky2017,Gulrajani2017,Mao2017}. This is an interesting direction because these models have the potential to create images that are visually indistinguishable from fully sampled reference images, where features in the images that are not supported by the amount of acquired data, are hallucinated by the network. This situation obviously must be avoided in any application in medical imaging. Strategies to mitigate this effect are the combination of GANs with conventional error metrics like MSE~\\cite{Isola2016image,Ledig2017c}. Comparable approaches were used in the context of MRI reconstruction~\\cite{Quan2017,Shitrit2017,Yang2018,Hammernik2018a,Kim2018a,Mardani2018}. \n\n\\subsection{Availability of training databases and community challenges}\nAs mentioned in the previous section, one open issue in the field of machine learning reconstruction for parallel imaging is the lack of publicly available databases of multi-channel raw k-space data. This restricts the number of researchers who can work in this field to those who are based at large academic medical centers where this data is available, and for the most part excludes the core machine learning community that has the necessary theoretical and algorithmic background to advance the field. In addition, since the used training data becomes an essential part of the performance of a certain model, it is currently almost impossible to compare new approaches that are proposed in the literature with each other if the training data is not shared when publishing the manuscript. While the momentum in initiatives for public releases of raw k-space data is growing~\\cite{fastMRIcommunity}, the number of available data sets is still on the order of hundreds and limited to very specific types of exams. Examples of publicly available rawdata sets are mridata.org\\footnote{\\url{https:\/\/mridata.org}} and the fastMRI dataset\\footnote{\\url{https:\/\/fastmri.med.nyu.edu\/}}.\n\n\\section{Conclusion}\n\nMachine learning methods have recently been proposed to improve the reconstruction quality in parallel imaging MRI. These techniques include both image domain approaches for better image regularization and k-space approaches for better k-space completion. While the field is still in its development, there are many open problems and high-impact applications, which are likely to be of interest to the broader signal processing community.\n\n\\bibliographystyle{IEEEbib}\n\\input{output.bbl}\n\n\n\n\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\n\\IEEEPARstart{R}{ecent} work on visual recognition focuses on \nthe importance of obtaining large labeled datasets \nsuch as ImageNet~\\cite{deng2009imagenet}. \nLarge-scale datasets when used for \ntraining deep neural network models tend to produce \nstate-of-the-art results on visual recognition~\\cite{krizhevsky2012imagenet}. \nHowever, in some cases, it may be difficult to obtain a large number of samples \nfor certain rare or fine-grained categories. \nHence, recognizing these rare categories become difficult. \nHumans, on the other hand, can easily recognize these rare categories \nby identifying the semantic description of the new category \nand how it is related to the seen categories. \nFor example, a person can identify a new animal zebra by \nidentifying the semantic description of a zebra having \nblack and white stripes and looking like a horse.\nA similar approach is undertaken for \nlearning models to recognize unseen and rare categories. \nThis learning scenario is known as zero-shot learning (ZSL) \nbecause zero labeled samples of the unseen categories \nare available for the training stage. ZSL has promising ramifications in autonomous vehicles, medical imaging, robotics, etc., where it is difficult to annotate images of novel categories but high-level semantic descriptions of classes can be obtained easily.\n\nTo be able to recognize unseen categories, \nwe usually train a learning model using a large collection of labeled samples \nfrom the seen categories and then adapt it to unseen categories. \nFor zero-shot recognition, the seen and the unseen categories \nare related through a high-dimensional vector space \nknown as semantic-descriptor space. \nEach category is assigned a unique semantic descriptor. \nExamples of semantic descriptor can be manually \ndefined attributes~\\cite{lampert2014attribute} \nor automatically extracted word vectors~\\cite{word2vec}. \nFigure~\\ref{fig:zsl} depicts the ZSL problem in terms of \nhow much information is available during training and testing.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=8cm]{img\/zsl.png}\n\\vspace*{-0.1in}\n\\caption{Depiction of the zero-shot learning problem. \nDuring training, we have lots of labeled images from seen classes (cat, dog, elephant) \nbut no labeled images from unseen classes. \nWe do have semantic descriptors of all the classes available. \nUsing all the information, the goal is to recognize the unseen classes.}\n\\label{fig:zsl}\n\\vspace*{-0.2in}\n\\end{figure}\n\nMost ZSL methods involve mapping from the visual feature space \nto the semantic-descriptor space or \nvice versa~\\cite{Zhang_2017_CVPR,akata2016label,frome2013devise,socher2013zero}. \nSometimes, both the visual features and the semantic descriptors \nare mapped to a common feature space~\\cite{zhang2016zero,changpinyo2016synthesized}. \nMost of these mapping-based approaches \nlearn an embedding function for samples and semantic descriptors. \nThe embedding is learned by minimizing a similarity function \nbetween the embedded samples and the corresponding embedded semantic descriptors. \nThus, most ZSL methods differ in the choice of the embedding and similarity functions. \nLampert et. al~\\cite{lampert2014attribute} used linear classifiers, \nidentity function and Euclidean distance for the sample embedding, \nsemantic embedding and similarity metric, respectively. \nRomera-Paredes et al.~\\cite{romera2015embarrassingly} \nused linear projection, identity function and dot product. \nALE~\\cite{akata2016label}, DEVISE~\\cite{frome2013devise}, \nSJE~\\cite{akata2015evaluation} all used a bilinear compatibility framework, \nwhere the projection was linear and the similarity metric was a dot product. \nThey used different variations of pairwise ranking objective to train the model. \nLATEM~\\cite{xian2016latent} was an extension of the above method, \nwhich used piecewise linear projections to account for the non-linearity. \nCMT~\\cite{socher2013zero} used a neural network to map image features \nto semantic descriptors with an additional novelty detection stage to detect unseen categories.\nSAE~\\cite{kodirov2017semantic} used an auto-encoder-based approach, \nwhere the image feature is linearly mapped to a semantic descriptor \nas well as being reconstructed from the semantic-descriptor space. \nDEM~\\cite{Zhang_2017_CVPR} used a neural network to map \nfrom a semantic-descriptor space to an image-feature space. \n \nAfter the embedding is carried out, \nclassification is performed using the nearest-neighbor search. \nAn earlier study~\\cite{radovanovic2010hubs} showed that the nearest-neighbor search \nin such a high-dimensional space suffers from the \\emph{hubness phenomenon} \nbecause only a certain number of data-points becomes nearest neighbor \nor hubs for almost all the query points, resulting in erroneous classification results. \nHowever, Shigeto et al.~\\cite{shigeto2015ridge} showed that \nmapping from a semantic-descriptor space \nto a visual-feature space does not aggravate the hubness problem. \nThus, in this paper, we pursue a semantic-descriptor-space\nto a visual-feature-space mapping approach. \nWe further introduce the concept of relative features \nthat uses pairwise relations between data-points. \nThis not only provides additional structural information about the data \nbut also reduces the dimensionality of the feature space implicitly~\\cite{2stage}, \nthus alleviating the hubness problem.\n\nZero-shot learning further suffers from a projection-domain-shift problem \nbecause the mapping from the semantic-descriptor space \nto the visual-feature space is learned from \nthe data belonging to only the seen categories. \nAs a result, the projected semantic descriptors of \nthe unseen categories are misplaced from the unseen test-data distribution. \nFu et al.~\\cite{fu2015transductive} identified the domain-shift problem \nand used multiple semantic information sources and label propagation \non unlabeled data from the unseen categories to counter the problem. \nKodirov et al.~\\cite{kodirov2015unsupervised} cast ZSL \nas a dictionary-learning problem and constrained the dictionary \nof the seen and unseen data to be close to each other. \nThis transductive approach is unrealistic \nas it assumes access to the unlabeled test data from unseen categories \nduring the training stage. \nAt the very least, we could carry out the test-time post-processing \nof the semantic descriptors. \nFor test-time adaptation, we propose to find correspondences \nbetween the projected semantic descriptors and the unlabeled test data \nafter which the descriptors are further mapped to the corresponding data-points. \nThis is inspired by recent work on local correspondence-based approach \nto unsupervised domain adaptation~\\cite{das2018sample}, \nwhich produces better results than global domain-adaptation methods.\n\nAnother problem with ZSL is that models \nare generally evaluated only on unseen categories. \nIn a real-world scenario, we expect the seen categories to appear \nmore frequently compared to the unseen categories. \nAs a result, it is appropriate to test our model on both seen and unseen categories. \nThis evaluation setting is known as Generalized Zero-Shot Learning (GZSL) \nand was initially introduced by Chao, et al.~\\cite{chao2016empirical}. \nThey found that the performance of unseen categories \nin the GZSL setting was poor and proposed a shifted-calibration mechanism \nto improve the performance. \nThis shifted-calibration mechanism lowers the classification scores of the seen categories. \nWe propose to develop a scaled-calibration mechanism to study the effect on recognition performance. \nThis has an effect of changing \nthe effective variance of a class and is therefore more interpretable.\n\nOther methods for ZSL include hybrid and synthesized methods. \nHybrid models expressed image features or semantic embeddings \nas a combination\/mixture of existing seen features or semantic embeddings. \nSemantic Similarity Embedding (SSE)~\\cite{zhang2015zero} \nexploits class relationship at both the image-feature and semantic-descriptor spaces \nto map them into a common embedding space. \nOur proposed ZSL method also exploits pairwise relationships \nbetween classes by minimizing the discrepancy between \nthe projected semantic descriptors and the corresponding class prototype \nobtained from the image features. \nCONSE~\\cite{norouzi2013zero} learns the probability of a seen sample \nbelonging to a seen class and uses the probability of an unseen sample \nbelonging to seen classes to relate to the semantic-descriptor space. \nSynthetic Classifiers (SYNC)~\\cite{changpinyo2016synthesized} \nlearn a mapping between the semantic-embedding space \nand the model-parameter space. \nThe model parameters of the classes are represented as a combination of phantom classes, \nthe relationship with which is encoded through a weighted bipartite graph. \nSynthesized methods generally convert ZSL into \na standard supervised-learning problem \nby generating samples for the unseen categories. \nSome of these methods\ninclude~\\cite{verma2018generalized,guo2017synthesizing}. \nThe limitations of these methods lie in not being able \nto generate samples very close to the true distribution. \nA more comprehensive overview of recent work on ZSL can be found in~\\cite{xian2018zero}.\n\nTo summarize, we propose a three-step approach to zero-shot learning. \nFirstly, to prevent aggravating the hubness problem, \na mapping is learned from the semantic-descriptor space to the image-feature space that minimizes both one-to-one and pairwise distances \nbetween semantic embeddings and the image features. \nSecondly, to alleviate the domain-shift problem at test time, \nwe propose a domain-adaptation method that finds\ncorrespondences between the semantic descriptors and the image features of test data.\nThirdly, to reduce biased-ness in the GZSL setting, \nwe propose scaled calibration on the classification scores of the seen classes\nto balance the performance on the seen and unseen categories.\nFinally, we evaluated our proposed approach on four standard ZSL datasets \nand compared our approach against state-of-the art methods \nfollowed by further analyzing the contribution of each component of our approach.\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=16cm]{img\/approach.png}\n\\vspace*{-0.1in}\n\\caption{The semantic descriptors are mapped to the image-feature space through the multi-layer perceptron. Then the semantic embeddings are regressed to the corresponding features through one-to-one and pairwise relations. After that, the semantic embeddings of the unseen classes are adapted to the unseen test data. This is followed by scaled calibration during testing when classification scores of seen classes are modified.}\n\\label{fig:approach}\n\\vspace*{-0.2in}\n\\end{figure*}\n\n\\section{Methodology}\n\\subsection{Problem Description}\nLet the training dataset $\\mathcal{D}_{tr}$ \nconsist of $N_{tr}$ samples such that \n$\\mathcal{D}_{tr}=\\{(\\mathbf{x}_i, \\mathbf{a}_i, y_i), i=1,2,\\ldots,N_{tr}\\}$. \nHere, $\\mathbf{x}_i \\in \\mathbb{R}^{m\\! \\times \\! n \\! \\times \\!c}$ \nis an image sample ($m\\! \\times \\!n$ is the image size and $c$ is the number of channels) \nand $\\mathbf{a}_i \\in \\mathbb{R}^{s}$ is the semantic descriptor \nof the sample's class. \nEach semantic descriptor $\\mathbf{a}_i$ is uniquely associated with \na class label $y_i \\in \\mathcal{Y}_{tr}$. \nThe goal of ZSL is to predict the class label $y_j \\in \\mathcal{Y}_{te}$ \nfor the $j^{th}$ test sample $\\mathbf{x}_j$. \nIn the traditional ZSL setting, \nwe assume that $\\mathcal{Y}_{tr} \\cap \\mathcal{Y}_{te} = \\varnothing$; \nthat is, the seen (training) and the unseen (testing) classes are disjoint. \nHowever, in the GZSL setting, both seen and unseen classes \ncan be used for testing; \nthat is, $\\mathcal{Y}_{tr} \\subset \\mathcal{Y}_{te}$. \nIn the training stage, we have the semantic descriptors \nof both the seen and unseen classes available \nbut no labeled training data of the unseen classes are available. \nThe overall framework of our proposed ZSL approach is shown in Fig.~\\ref{fig:approach}.\n\n\\subsection{Relational Matching}\nOur goal is to learn a mapping $\\mathbf{f}(\\cdot)$ \nthat maps a semantic descriptor $\\mathbf{a}_i$ \nto its corresponding image feature $\\mathbf{\\phi}(\\mathbf{x}_i)$. \nHere, $\\mathbf{x}_i$ is an image and $\\mathbf{\\phi}(\\cdot)$ \nrepresents a CNN architecture that extracts a high-dimensional feature map. \nThe mapping ${\\bf f}(\\cdot)$ is a fully-connected neural network. \nSince our goal is to make the embedded semantic descriptor \nclose to the corresponding image feature, \nwe use a least square loss function to minimize the difference. \nWe also need to regularize the parameters of $\\mathbf{f}(\\cdot)$. \nIncluding these costs and averaging over all the instances, \nour initial objective function $\\mathcal{L}_1$ is as follows:\n\\begin{equation}\n\\mathcal{L}_1 = \\frac{1}{N_{tr}}\\sum_{i=1}^{N_{tr}}||\\mathbf{f}({\\bf a}_i)-\\mathbf{\\phi}(\\mathbf{x}_i)||_2^2 + \\lambda_rg({\\bf f})~,\n\\end{equation}\nwhere $g(\\cdot)$ is the regularization loss for the mapping function. \nThe loss function $\\mathcal{L}_1$ minimizes the point-to-point discrepancy \nbetween the semantic descriptors and the image features. \nTo account for the structural matching between \nthe semantic-descriptor space and the image-feature space, \nwe try to minimize the inter-class pairwise relations in these two spaces. \nThus, we construct relational matrices for both the semantic descriptors and image features. \nThe semantic relational matrix $\\mathbf{D}_a$ is established \nsuch that each element, \n$[\\mathbf{D}_a]_{uv} = ||{\\bf f}({\\bf a}^u)-{\\bf f}({\\bf a}^v)||_2^2$, \nwhere ${\\bf a}^u$ and ${\\bf a}^v$ are semantic descriptors \nof seen categories $u$ and $v$, respectively. \nThe image feature relational matrix $\\mathbf{D}_\\phi$ \nis constructed such that each element, \n$[\\mathbf{D}_\\phi]_{uv} = ||\\overline{\\mathbf{\\phi}}^u-\\overline{\\mathbf{\\phi}}^v||_2^2$, \nwhere $\\overline{\\mathbf{\\phi}}^u$ and $\\overline{\\mathbf{\\phi}}^v$ \nare mean representations of the categories $u$ and $v$, respectively. \n$\\overline{\\mathbf{\\phi}}^u$ can be represented as \n\\begin{equation}\n\\overline{\\mathbf{\\phi}}^u=\\frac{1}{|\\mathcal{Y}^{u}_{tr}|}\\sum_{y_i \\in \\mathcal{Y}^{u}_{tr}}^{}\\mathbf{\\phi}({\\mbox{\\boldmath $x$}}_i)~,\n\\end{equation}\nwhere the summation is over the representations of class $u$, \nand $|\\mathcal{Y}^{u}_{tr}|$ is the cardinality of the training set of class $u$. \nA similar formula holds for class $v$. \nFor structural alignment, we want the two relational matrices, \n$\\mathbf{D}_a$ and $\\mathbf{D}_{\\phi}$, \nto be close to one another. \nHence, we want to minimize the structural alignment loss function $\\mathcal{L}_2$,\n\\begin{equation}\n\\mathcal{L}_2=||\\mathbf{D}_a-\\mathbf{D}_{\\phi}||_F^2~,\n\\end{equation}\nwhere $||\\cdot||_F^2$ stands for the Frobenius norm. \nCombining the loss functions $\\mathcal{L}_1$ and $\\mathcal{L}_2$, \nwe have the total loss $\\mathcal{L}_{total}$,\n\\begin{equation}\n\\mathcal{L}_{total} = \\mathcal{L}_1 + \\rho \\mathcal{L}_2~,\n\\end{equation}\nwhere $\\rho \\geq 0$ weighs the loss contribution of $\\mathcal{L}_2$.\n$\\mathcal{L}_{total}$ is to be optimized with respect to \nthe parameters of the semantic-descriptor-to-visual-feature-space mapping ${\\bf f}(\\cdot)$.\n\n\\subsection{Domain Adaptation}\nAfter the training is carried out, \ndomain discrepancy may be present between \nthe mapped semantic descriptors and the image features of unseen categories. \nThis is because the unseen data has not been used in the training \nand our regularized model does not \ngeneralize well for the unseen categories. \nHence, we need to adapt the mapped semantic descriptors \nfor the unseen categories using the test data from the unseen categories. \nLet the mapped descriptors for the unseen categories be stacked vertically \nin the form of a matrix ${\\bf A} \\in \\mathbb{R}^{n_u\\!\\! \\times \\!\\!d}$, \nwhere $n_u$ is the number of unseen categories \nand $d$ is the dimension of the mapped semantic-descriptor space, \nand therefore it is also the dimension of the image-feature space. \nLet ${\\bf U} \\in \\mathbb{R}^{o_u\\!\\! \\times \\!\\!d}$ be the unseen test dataset, \nwhere $o_u$ is the number of test instances from the unseen categories. \nFor adapting the mapped descriptors, \nwe propose to find the point-to-point correspondence \nbetween the descriptors and the test data. \nLet the correspondence be represented as a matrix \n${\\bf C} \\in \\mathbb{R}^{n_u\\! \\times \\!o_u}$. \nWe want to rearrange the rows of ${\\bf U}$ such that \neach row of the modified matrix corresponds to the row in ${\\bf A}$. \nThis is done by minimizing the following loss function $\\mathcal{L}_3$,\n\\begin{equation}\n\\mathcal{L}_3=||{\\bf C}{\\bf U}-{\\bf A}||_F^2~.\n\\end{equation}\n\nThis loss function enforces that ${\\bf C}{\\bf U}$ produces the adapted semantic descriptors. \nHowever, a problem may exist that an instance in ${\\bf U}$ \ncorresponds to more than one descriptor in ${\\bf A}$. \nThis would essentially result in a test sample corresponding to more than one category. \nTo avoid that, we use an additional group-based regularization function \n$\\mathcal{L}_4$ using Group-Lasso,\n\\begin{equation}\n\\mathcal{L}_4=\\sum_{j}\\sum_{c}||[{\\bf C}]_{I_cj}||_2~,\n\\end{equation}\nwhere $I_c$ corresponds to the indices of those rows in ${\\bf A}$ \nthat belong to the unseen class $c$. \nTherefore, $[{\\bf C}]_{I_cj}$ is the vector consisting of \nthe row indices from $I_c$ and the $j^{th}$ column. \nSince ${\\bf C}$ is a correspondence matrix, \nsome constraints should be enforced such as ${\\bf C} \\geq \\mathbf{0}$, \n${\\bf C}\\mathbf{1}_{o_u}=\\mathbf{1}_{n_u}$ \nand ${\\bf C}^{T}\\mathbf{1}_{n_u}=\\frac{n_u}{o_u}\\mathbf{1}_{o_u}$, \nwhere $\\mathbf{1}_{n}$ is an $n\\! \\times \\!1$ vector of one's. \nThe second equality constraint is scaled by the factor $\\frac{n_u}{o_u}$ \nto account for the difference in the number of instances \nin the mapped semantic-descriptor space and the image-feature space \nfor the unseen categories. \nHence, the domain adaptation optimization problem becomes \n\\begin{equation}\n\\underset{{\\bf C}}{\\text{min}} \\hspace{0.1em} \\left\\{\\mathcal{L}_3 + \\lambda_g \\mathcal{L}_4\\right\\} \\hspace{0.5em} s.t. \\hspace{0.5em} {\\bf C} \\geq \\mathbf{0}, {\\bf C}\\mathbf{1}_{o_u}=\\mathbf{1}_{n_u},{\\bf C}^{T}\\mathbf{1}_{n_u}=\\frac{n_u}{o_u}\\mathbf{1}_{o_u}~,\n\\label{eq7}\n\\end{equation}\nwhere $\\lambda_g$ weighs the loss function $\\mathcal{L}_4$.\n\nThe above optimization problem is convex and can be efficiently solved \nusing the conditional gradient method~\\cite{frank1956algorithm}. \nThe conditional gradient method requires solving \na linear program as an intermediate step over the constraints \n${\\bf C} \\in \\mathcal{D}=\\{{\\bf C}: {\\bf C} \\geq {\\bf 0}, {\\bf C}\\mathbf{1}_{o_u}=\\mathbf{1}_{n_u}, \n{\\bf C}^{T}\\mathbf{1}_{n_u}=\\frac{n_u}{o_u}\\mathbf{1}_{o_u}\\}$ \nas shown in Algorithm 1. \nThe linear program of finding the intermediate variable \n${\\bf C}_d$ in Algorithm 1 can be easily solved \nusing a network simplex formulation \nof the earth-mover's distance problem~\\cite{bonneel2011displacement}.\n\\begin{algorithm}[]\n\\SetAlgoLined\n \\textbf{Intitialize :} ${\\bf C}_0 = \\frac{1}{(n_uo_u)}\\mathbf{1}_{n_u\\! \\times \\!o_u}$, $t=1$\\\\\n \\textbf{Repeat}\\\\\n \\quad ${\\bf C}_d = \\underset{{\\bf C}}{\\text{argmin}} \\hspace{0.5em} \\text{Tr}(\\nabla_{{\\bf C}={\\bf C}_0} (\\mathcal{L}_3 + \\lambda_g \\mathcal{L}_4)^{T}{\\bf C}), \\hspace{0.5em} s.t. \\hspace{0.5em} {\\bf C} \\in \\mathcal{D}$ \\\\\n \\quad ${\\bf C}_1 = {\\bf C}_0 + \\alpha({\\bf C}_d - {\\bf C}_0), \\quad \\text{for} \\quad \\alpha = \\frac{2}{t+2}$ \\\\\n \\quad ${\\bf C}_0={\\bf C}_1$ \\quad \\text{and} \\quad $t=t+1$ \\\\\n \\textbf{Until} Convergence \\\\\n \\textbf{Output :} ${\\bf C}_0 = \\text{arg}\\min \\limits_{{\\bf C}}\\{\\mathcal{L}_3 + \\lambda_g \\mathcal{L}_4\\} \\quad s.t. \\quad {\\bf C} \\in \\mathcal{D}$\n\\caption{Conditional Gradient Method (CG)}\n\\end{algorithm}\n\nOnce the final solution of the correspondence matrix ${\\bf C}_0$ \nin Algorithm 1 is obtained, \nwe inspect ${\\bf C}_0$. \nFor each test instance, \nwe assign the class correspondence to the highest value of the correspondence variable. \nThis is done for all the test instances. \nThe new semantic descriptors are obtained by taking the mean \nof the feature instances belonging to the corresponding class. \nThe adapted semantic descriptors are then stacked vertically in the matrix ${\\bf A}'$.\n\n\\subsection{Scaled Calibration}\nIn the GZSL setting, \nit is known that the classification results are biased \ntowards the seen categories~\\cite{chao2016empirical}. \nTo counteract the bias, \nwe propose the use of multiplicative calibration on the classification scores. \nIn our case, we use 1-Nearest Neighbor (1-NN) \nwith the Euclidean distance metric as the classifier. \nThe classification score for a test point \nis given by the Euclidean distance of the test image feature \nto the mapped semantic descriptor of a category. \nFor a test point ${\\mbox{\\boldmath $x$}}$, \nwe adjust the classification scores on the seen categories as follows\n\\begin{equation}\n\\hat{y}=\\underset{c \\in \\mathcal{T}}{\\text{argmin}}\n\\hspace{0.5em}||{\\mbox{\\boldmath $x$}}-{\\bf f}({\\bf a}^c)||_2\\cdot\\mathbb{I}[c \\in \\mathcal{S}]~,\n\\end{equation} \nwhere $\\mathbb{I}[\\cdot]=\\gamma$ if $c \\in \\mathcal{S}$ \nand 1 if $c \\in \\mathcal{U}$ and $\\mathcal{S} \\cup \\mathcal{U} = \\mathcal{T}$. \nHere, $\\mathcal{S}, \\mathcal{U}$ and $\\mathcal{T}$ \nrepresent the sets of seen, unseen and all categories, respectively. \nThe effect of scaling is to change the effective variance of the seen categories. \nWhen the nearest-neighbor classification is carried out \nwith the Euclidean distance metric, \nit assumes that all classes have equal variance. \nBut since the unseen categories are not used \nfor learning the embedding space, \nthe variance of the unseen-category features is not accounted for. \nThat is why the Euclidean distance metric \nfor the seen categories needs to be adjusted for. \nFor $\\gamma > 1$, if we obtain a balanced performance \nbetween the seen and unseen classes, \nit implies that the variance of the seen classes has been overestimated. \nSimilarly, if we obtain a balanced performance for $\\gamma < 1$, \nit means that the variance of the seen classes has been underestimated. \nThe overall procedure of our proposed zero-shot learning method \nfrom training to testing is given in Algorithm 2.\n\\begin{algorithm}[]\n\\SetAlgoLined\n\\textbf{Input:} Training Dataset $\\{(\\mathbf{x}_i, \\mathbf{a}_i, y_i)\\}_{i=1}^{N_{tr}}$\\\\\n\\textbf{Parameters:} $\\lambda_r$, $\\rho$, $\\lambda_g$, $\\gamma$\\\\\n\\textbf{Repeat} (Training)\\\\\n\\quad Sample Minibatch of $\\{({\\mbox{\\boldmath $x$}}_i,{\\bf a}_i)\\}$ pairs\\\\\n\\quad Gradient descent $\\mathcal{L}_1 + \\rho \\mathcal{L}_2$ w.r.t parameters of ${\\bf f}(\\cdot)$\\\\\n\\textbf{Until} Convergence \\\\\n\\textbf{Input:} Test Dataset $\\{(\\mathbf{x}_i)\\}_{i=1}^{N_{te}}$\\\\\n\\quad Apply Algorithm 1 to obtain adapted descriptors \\\\\n\\quad of unseen classes ${\\bf A}'$ (Adaptation)\\\\\n\\textbf{Repeat} for each test point ${\\mbox{\\boldmath $x$}}$ (Testing)\\\\\n\\quad $\\hat{y}=\\underset{c \\in \\mathcal{T}}{\\text{argmin}}\\hspace{0.5em}||{\\mbox{\\boldmath $x$}}-{\\bf f}({\\bf a}^c)||_2\\cdot\\mathbb{I}[c \\in \\mathcal{S}]$ (Calibration)\\\\\n\\textbf{Until} all test points covered\n\\caption{Proposed Zero-shot Learning Algorithm}\n\\end{algorithm}\n\n\\section{Experimental Results}\nFollowing the previous experimental settings~\\cite{xian2018zero}, \nwe used the following four datasets for evaluation: \n\\textbf{AwA2}~\\cite{lampert2014attribute} (Animal with Attributes) \ncontains 37,322 images of 50 classes of animals. \n40 classes of animals are considered to be the seen categories \nwhile 10 classes of animals are considered to be the unseen categories. \nEach class is associated with a 85-dimensional continuous semantic descriptor. \n\\textbf{aPY}~\\cite{farhadi2009describing} (attribute Pascal and Yahoo) \nconsists of 20 seen categories and 12 unseen categories. \nEach category has an associated 64-dimensional semantic descriptor. \n\\textbf{CUB}~\\cite{welinder2010caltech} (Caltech-UCSD Birds-200-2011) \nis a fine-grained dataset consisting of 11,788 images of birds. \nFor evaluation, all the bird categories are split into 150 seen classes and 50 unseen classes. \nEach class is associated with a 312-dimensional continuous semantic descriptor. \n\\textbf{SUN}~\\cite{patterson2012sun} (Scene UNderstanding database) \nconsists of 14340 scene images. \nAmong these, 645 scene categories are selected as seen categories \nwhile 72 categories are selected as unseen categories \nand it consists of a 102-dimensional semantic descriptor.\n\nFor the purpose of evaluation, \nwe used class-wise accuracy because it prevents dense-sampled classes \nfrom dominating the performance. \nAccordingly, class-wise accuracy is averaged as follows\n\\begin{equation}\nacc=\\frac{1}{|\\mathcal{Y}|}\\sum_{y=1}^{|\\mathcal{Y}|}\\frac{\\text{No. of correct predictions in class}\\hspace{0.5em} y}{\\text{No. of samples in class}\\hspace{0.5em} y},\n\\end{equation}\nwhere $|\\mathcal{Y}|$ is the number of testing classes.\nIn the GZSL case, class-wise accuracy of both seen and unseen classes \nare obtained separately and then averaged using harmonic mean $H$~\\cite{xian2018zero}. \nThis is done so that the performance on seen classes \ndoes not dominate the overall accuracy,\n\\begin{equation}\nH=\\frac{2\\times acc_s\\! \\times \\!acc_u}{acc_s + acc_u},\n\\end{equation}\nwhere $acc_s$ and $acc_u$ are the class-wise accuracy \non seen and unseen categories, respectively. \nIn the GZSL classification setting, \nthe search space of predicted categories consists of both seen and unseen categories. Based on \\cite{xian2018zero} and for fair comparison, a single trial of experimental results on a large batch of training and testing dataset is reported.\n\nFor the experiments, \nwe used a two-layer feedforward neural network \nfor the semantic embedding ${\\bf f}(\\cdot)$. \nThe dimensionality of the hidden layer was chosen as 1600, 1600, 1200 \nand 1600 for the \\textbf{AwA2}, \\textbf{aPY}, \\textbf{CUB} \nand \\textbf{SUN} datasets, respectively. \nThe activation used was ReLU. \nThe image features used were the ResNet-101. \nWe compared different variations of our proposed method with previous approaches. \nOURS-R variation is with the training stage \nincluding the structural loss $\\mathcal{L}_2$. \nOURS-RA includes the structural loss as well as the domain adaptation stage \nincluding the loss functions $\\mathcal{L}_3$ and $\\mathcal{L}_4$. \nOURS-RC includes the structural loss as well as the calibrated testing stage. \nOURS-RAC includes all the components of structural loss, \ndomain adaptation and calibrated testing. \nWithout all these components, the proposed method reduces to \nthe Deep Embedding Model (DEM)~\\cite{Zhang_2017_CVPR} baseline. \nThe parameters $(\\lambda_r, \\rho, \\lambda_g, \\gamma)$ \nfor the \\textbf{AwA2}, \\textbf{aPY}, \\textbf{CUB} and \\textbf{SUN} datasets \nare set as $(10^{-3}, 10^{-1}, 10^{-1}, 1.1)$, \n$(10^{-4}, 10^{-1}, 10^{-1}, 1.1)$, $(10^{-2}, 0, 10^{-1}, 1.1)$ \nand $(10^{-5}, 10^{-1}, 10^{-1}, 1.1)$, respectively. \nFor the OURS-RAC variation, \nwe used different calibration parameter values of $0.98, 1.1,0.97, 0.999$ \nfor the \\textbf{AwA2}, \\textbf{aPY}, \\textbf{CUB} \nand \\textbf{SUN} datasets, respectively. \n$\\rho$ was set to $0$ for the \\textbf{CUB} dataset \nbecause it is a fine-grained dataset and since the categories \nare very close to each other in the feature space, \nstructural matching does not provide additional information. \nIn Table~\\ref{table:compare}, \nwe reported class-wise accuracy results \nfor the conventional unseen classes setting (\\textbf{tr}), \ngeneralized unseen classes setting (\\textbf{u}), \ngeneralized seen classes setting (\\textbf{s}), \nand the Harmonic mean (\\textbf{H}) of the generalized accuracies.\n\n\\begin{table*}[]\n\\caption{Results of variations of our proposed approach in comparison \nwith previous methods on the \\textbf{AwA2}, \\textbf{aPY}, \\textbf{CUB} \nand \\textbf{SUN} datasets. \nThe best results of each setting in each dataset are shown in boldface.}\n\\label{table:compare}\n\\centering\n\\scalebox{1.0}{\\begin{tabular}{l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}\n\\hline\n & \\multicolumn{4}{c|}{\\textbf{AwA2}} & \\multicolumn{4}{c|}{\\textbf{aPY}} & \\multicolumn{4}{c|}{\\textbf{CUB}} & \\multicolumn{4}{c}{\\textbf{SUN}} \\\\ \\hline\nMethod & \\textbf{tr} & \\textbf{u} & \\textbf{s} & \\textbf{H} \n& \\textbf{tr} & \\textbf{u} & \\textbf{s} & \\textbf{H} \n& \\textbf{tr} & \\textbf{u} & \\textbf{s} & \\textbf{H} \n& \\textbf{tr} & \\textbf{u} & \\textbf{s} & \\textbf{H} \n\\\\ \\hline\nDAP~\\cite{lampert2014attribute} &46.1 &0.0 &84.7 &0.0 &33.8 &4.8 &78.3 &9.0 &40.0 &1.7 &67.9 &3.3 &39.9 &4.2 &25.1 &7.2 \\\\ \nIAP~\\cite{lampert2014attribute} &35.9 &0.9 &87.6 &1.8 &36.6 &5.7 &65.6 &10.4 &24.0 &0.2 &\\textbf{72.8} &0.4 &19.4 &1.0 &37.8 &1.8 \\\\ \nCONSE~\\cite{norouzi2013zero} &44.5 &0.5 &\\textbf{90.6} &1.0 &26.9 &0.0 &\\textbf{91.2} &0.0 &34.3 &1.6 &72.2 &3.1 &38.8 &6.8 &39.9 &11.6 \\\\ \nCMT~\\cite{socher2013zero} &37.9 &0.5 &90.0 &1.0 &28.0 &1.4 &85.2 &2.8 &34.6 &7.2 &49.8 &12.6 &39.9 &8.1 &21.8 &11.8 \\\\ \nSSE~\\cite{zhang2015zero} &61.0 &8.1 &82.5 &14.8 &34.0 &0.2 &78.9 &0.4 &43.9 &8.5 &46.9 &14.4 &51.5 &2.1 &36.4 &4.0 \\\\ \nLATEM~\\cite{xian2016latent} &55.8 &11.5 &77.3 &20.0 &35.2 &0.1 &73.0 &0.2 &49.3 &15.2 &57.3 &24.0 &55.3 &14.7 &28.8 &19.5 \\\\ \nALE~\\cite{akata2016label} &62.5 &14.0 &81.8 &23.9 &39.7 &4.6 &73.7 &8.7 &54.9 &23.7 &62.8 &34.4 &58.1 &21.8 &33.1 &26.3 \\\\ \nDEVISE~\\cite{frome2013devise} &59.7 &17.1 &74.7 &27.8 &\\textbf{39.8} &4.9 &76.9 &9.2 &52.0 &23.8 &53.0 &32.8 &56.5 &16.9 &27.4 &20.9 \\\\ \nSJE~\\cite{akata2015evaluation} &61.9 &8.0 &73.9 &14.4 &32.9 &3.7 &55.7 &6.9 &53.9 &23.5 &59.2 &33.6 &53.7 &14.7 &30.5 &19.8 \\\\ \nESZSL~\\cite{romera2015embarrassingly} &58.6 &5.9 &77.8 &11.0 &38.3 &2.4 &70.1 &4.6 &53.9 &12.6 &63.8 &21.0 &54.5 &11.0 &27.9 &15.8 \\\\ \nSYNC~\\cite{changpinyo2016synthesized} &46.6 &10.0 &90.5 &18.0 &23.9 &7.4 &66.3 &13.3 &55.6 &11.5 &70.9 &19.8 &56.3 &7.9 &\\textbf{43.3} &13.4 \\\\ \nSAE~\\cite{kodirov2017semantic} &54.1 &1.1 &82.2 &2.2 &8.3 &0.4 &80.9 &0.9 &33.3 &7.8 &54.0 &13.6 &40.3 &8.8 &18.0 &11.8 \\\\ \nGFZSL~\\cite{verma2017simple} &63.8 &2.5 &80.1 &4.8 &38.4 &0.0 &83.3 &0.0 &49.3 &0.0 &45.7 &0.0 &60.6 &0.0 &39.6 &0.0 \\\\ \nSR~\\cite{annadani2018preserving} &63.8 &20.7 &73.8 &32.3 &38.4 &13.5 &51.4 &21.4 &\\textbf{56.0} &24.6 &54.3 &33.9 &61.4 &20.8 &37.2 &26.7 \\\\ \nDEM~\\cite{Zhang_2017_CVPR} &\\textbf{67.1} &30.5 &86.4 &45.1 &35.0 &11.1 &75.1 &19.4 &51.7 &19.6 &57.9 &29.2 &40.3 &20.5 &34.3 &25.6 \\\\ \n\\hline\nOURS-R &63.4 &36.5 &80.6 &50.3 &29.9 &15.3 &71.4 &25.2 &46.6 &20.2 &48.6 &28.6 &59.9 &21.7 &{38.1} &27.6 \\\\ \nOURS-RA &64.4 &\\textbf{61.8} &69.9 &65.6 &35.4 &30.4 &72.9 &42.9 &52.6 &\\textbf{47.6} &41.0 &44.1 &\\textbf{67.5} &\\textbf{54.4} &36.6 &\\textbf{43.7} \\\\ \nOURS-RC &63.4 &57.9 &72.0 &64.2 &29.9 &26.4 &53.3 &35.3 &46.6 &27.2 &43.9 &33.6 &59.9 &42.4 &32.6 &36.8 \\\\ \nOURS-RAC &64.4 &60.6 &72.3 &\\textbf{65.9} &35.4 &\\textbf{34.1} &63.5 &\\textbf{44.4} &52.6 &44.0 &45.1 &\\textbf{44.6} &\\textbf{67.5} &54.1 &36.6 &\\textbf{43.7} \\\\ \\hline\n\\end{tabular}}\n\\vspace*{-0.2in}\n\\end{table*}\n\nFrom the table, we observed that our proposed approach \noutperforms previous methods \nby a large margin in the generalized harmonic mean setting. \nTo be more specific, our proposed method produces \nan improvement of around 20\\%, 23\\%, 10\\% and 16\\% \nharmonic mean accuracy over the previous best approach for the \\textbf{AwA2}, \\textbf{aPY}, \\textbf{CUB} and \\textbf{SUN} datasets, respectively. \nThe large improvement in performance can be attributed \nto our three-step procedure for improvement. \nUsing only the structural matching (OURS-R), \nwe produced better results than previous approaches \nexcept for the \\textbf{CUB} dataset, \nwhere it produces a harmonic mean accuracy of about 28\\%. \nThis is because \\textbf{CUB} requires minute fine-grained feature extraction. \nAdditional usage of domain adaptation (OURS-RA) \nand calibrated testing (OURS-RC) produced \nmuch better results than OURS-R for all the datasets. \nHowever, domain adaptation produced better result than the calibration procedure. \nThis is because our correspondence-based approach \nproduced class-specific adaptation of the unseen class semantic embeddings. \nThe scaled-calibration procedure is not class-specific \nand just differentiates between seen and unseen classes. \nIt also does not adapt to the test data. \n\nIt is to be noted that the difference in performance \nbetween OURS-RA and OURS-RAC is negligible. \nThis is because the domain adaptation step transforms \nthe unseen semantic embeddings away \nfrom the seen categories towards the unseen categories, \nthus reducing the bias towards the seen categories \nand rendering further calibration ineffective. \nThe effect of domain adaptation is visualized in \nFig.~\\ref{fig:tsnezsl1} for the \\textbf{AwA2} dataset \nusing t-SNE~\\cite{maaten2008visualizing}. \nIn Fig.~\\ref{fig:tsnezsl1}(a), \nthe unseen class semantic embeddings (blue) remained very close to \nthe seen class features (maroon). \nHowever, with the domain adaptation step, \nthe unseen class semantic embeddings get transformed to \nnear the centre of unseen class feature clusters (green) \nas shown in Fig.~\\ref{fig:tsnezsl1}(b).\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=8cm]{img\/tsnezsl1.png}\n\\vspace*{-0.1in}\n\\caption{2D t-SNE map of the embedded instances. (a) Without domain adaptation and \n(b) with domain adaptation for the \\textbf{AwA2} dataset. \nHere, the seen and unseen image features are shown in maroon and green, respectively. \nThe embedded semantic descriptors for the seen and unseen classes \nare shown in red and blue color, respectively. }\n\\label{fig:tsnezsl1}\n\\vspace*{-0.2in}\n\\end{figure}\n\nWe also analyzed the effect of the structural matching \nby varying $\\rho \\in \\{10^{-3}, 10^{-2}, 10^{-1}, 10^{0}, 10^{1}, 10^{2}\\}$ \nand observed how the class-wise accuracy changes. \nWe carried out experiments using the \\textbf{AwA2} and \\textbf{SUN} datasets, \nthe results of which are reported in Fig.~\\ref{fig:rhosens}. \nWe also reported the DEM baseline ($\\rho=0$) in dotted lines. \nFrom the plots, the Conventional Unseen and the Generalized Seen accuracies \nare better than or equal to the baseline for only a small range of $\\rho$. \nOn the other hand, the Generalized Unseen accuracy is greater than the baseline \nover a large range of $\\rho$ for the \\textbf{AwA2} dataset \nwhile it oscillated about the baseline for the \\textbf{SUN} dataset. \nFor the \\textbf{SUN} dataset, \nwe do not have a significant gain over the baseline \nbecause \\textbf{SUN} is a fine-grained dataset \nwhere structural matching does not carry additional information. \nThe goal of structural regularization is to exploit \nthe pairwise relations among classes so as to generalize better to novel classes. \nTherefore, we did not see huge difference in performance \nfrom the baseline for the Generalized Seen accuracy. \nSurprisingly, there was a drop in conventional unseen accuracy \nas $\\rho$ was increased. \nThis might be probably because there was no overlap \nbetween the classes used for testing and the classes used for structural matching. \nThis is not the case though in the generalized setting. \n\\begin{figure}[]\n\\centering\n\\includegraphics[width=8cm]{img\/rhosens.png}\n\\vspace*{-0.1in}\n\\caption{Results of class-wise accuracy as $\\rho$ is varied for different settings \non the \\textbf{AwA2} and the \\textbf{SUN} datasets. \nThe baseline used is DEM. The different performance settings \nare Conventional Unseen Accuracy (Left Top), \nGeneralized Seen Accuracy (Right Top), Generalized Unseen Accuracy (Left Bottom) \nand Generalized Harmonic Mean Accuracy (Right Bottom).}\n\\label{fig:rhosens}\n\\vspace*{-0.2in}\n\\end{figure}\n\nWe also studied the effect of varying the calibration parameter $\\gamma$ \non the generalized accuracy for the \\textbf{AwA2} and \\textbf{SUN} datasets. \nThe results are shown in Fig.~\\ref{fig:calib}. \nAs expected, the generalized unseen accuracy increases \nand the generalized seen accuracy decreases with increasing $\\gamma$. \nThe peak of the harmonic mean accuracy was observed close to \nwhen the seen and unseen accuracies became equal. \nThe maximum unseen accuracy is less than the maximum seen accuracy \nfor the \\textbf{AwA2} dataset because the unseen classes \nare less separated and therefore more difficult to classify. \nThe situation is reversed for the \\textbf{SUN} dataset \nwhere the maximum unseen accuracy is more than the maximum seen accuracy.\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=8cm]{img\/calib.png}\n\\vspace*{-0.1in}\n\\caption{Results of Generalized Seen Accuracy (Red), \nGeneralized Unseen Accuracy (Green) \nand Generalized Harmonic Mean Accuracy (Blue) \nas the calibration parameter $\\gamma$ is varied \non the \\textbf{AwA2} and \\textbf{SUN} datasets.}\n\\label{fig:calib}\n\\vspace*{-0.2in}\n\\end{figure}\n\nWe also reported convergence results of the test accuracy \nwith respect to the number of epochs for both the \\textbf{AwA2} \nand the \\textbf{SUN} datasets in Figs.~\\ref{fig:epAwA2} and~\\ref{fig:epSUN}, respectively.\nWe used the OURS-R variation with $\\rho=0.1$ to compare with the DEM baseline. \nThe convergence rate for the baseline and OURS-R variation seems \nto be similar in all the settings for both datasets. \nHowever, our steady-state values were higher \nfor the generalized unseen and generalized harmonic mean setting. \nFor the conventional unseen and generalized seen setting, \nour steady-state value was less than the baseline. \nThe reason is explained previously \nwhile describing performance sensitivity to $\\rho$. \n\\begin{figure}[]\n\\centering\n\\includegraphics[width=8cm]{img\/epAwA2.png}\n\\vspace*{-0.1in}\n\\caption{Convergence results of test accuracy with respect to the number of epochs \nunder different settings for the \\textbf{AwA2} dataset. \nOURS-R results are shown in red color while the DEM baseline is shown in blue color.}\n\\label{fig:epAwA2}\n\\vspace*{-0.2in}\n\\end{figure}\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=8cm]{img\/epSUN.png}\n\\vspace*{-0.1in}\n\\caption{Convergence results of test accuracy \nwith respect to the number of epochs under different settings \nfor the \\textbf{SUN} dataset. \nOURS-R results are shown in red color while the DEM baseline is shown in blue color.}\n\\label{fig:epSUN}\n\\vspace*{-0.2in}\n\\end{figure}\n\nWe also studied the effect of varying the number of test unseen samples per class \non the generalized harmonic mean accuracy. \nWe used OURS-RA variation of our model for this study. \n$\\rho=0.1$ was set for the experiments on the \\textbf{AwA2} (blue color) \nand the \\textbf{SUN} (yellow color) datasets \nand the result was reported in Fig.~\\ref{fig:nsamp}. \nWhen the fraction is 0.01 for the \\textbf{SUN} dataset, \nthe number of samples in some classes becomes zero \nand therefore the performance is not reported. \nFrom the results, it is seen that the test accuracy was stable \nwith change in the fraction of total number of samples used for testing. \nThere is a slight increase in accuracy with decreasing number of samples, \nwhich is surprising because domain adaptation would perform poorly \nwith less number of samples. \nHowever, this effect is nullified since the probability \nof including challenging examples is reduced \nand so we observed a slight improvement in performance.\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=6cm]{img\/nsamp.png}\n\\vspace*{-0.1in}\n\\caption{Generalized Harmonic Mean Accuracy results \nas the number of test samples per class is varied for the \\textbf{AwA2} (blue) \nand \\textbf{SUN} (yellow) datasets.}\n\\label{fig:nsamp}\n\\vspace*{-0.2in}\n\\end{figure}\n\nWe also studied how the test performance varies \nas the number of seen classes for training is reduced \nfor the \\textbf{AwA2} dataset using OURS-R model. \nWe set $\\rho=0.1$ and reported results over 5 trials in Fig.~\\ref{fig:nclass}. \nWe observed that the change in the seen-class accuracy \nis not much because the training and testing distributions are the same. \nThe conventional unseen-class accuracy dips by a large amount \nas the number of training classes decreases because \nthere is less representative information to be transferred to novel categories. \nHowever, we obtained a peak for the generalized unseen accuracy results \nat a fraction of 0.4 of the number of seen classes. \nThis is because as the number of training classes decreases, \nthe amount of representative information decreases, causing decrease in performance. \nOn the other hand, less number of seen classes implies \nless bias towards seen categories and improvement of unseen-class accuracies. \nAlso, there is large performance variation for unseen-class accuracy \nbecause training and testing distributions are different \nand the performance can vary depending on \nhow related are the training classes to the unseen classes in a trial.\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=6cm]{img\/nclass.png}\n\\vspace*{-0.1in}\n\\caption{Test accuracy results as the number of seen classes \nused for training is varied for the \\textbf{AwA2} dataset.}\n\\label{fig:nclass}\n\\vspace*{-0.2in}\n\\end{figure}\n\nWe also performed experiments to find whether the OURS-R variant \nreduces hubness compared to DEM. \nThe hubness of a set of predictions is measured \nusing the skewness of the 1-Nearest-Neighbor histogram ($N_1$). \nThe $N_1$ histogram is a frequency plot for $N_1[i]$ \nof the number of times a search solution $i$ (in our case a class attribute) \nis found as the Nearest Neighbor for the test samples. \nLess skewness of $N_1$ histogram implies less hubness of the predictions. \nWe used the test samples of the unseen classes in the generalized setting \nfor both DEM and OURS-R on the \\textbf{AwA2} and the \\textbf{aPY} datasets. \nWe used $\\rho=0.1$ and reported results averaging over 5 trials \nin Table~\\ref{table:hubness}. \nFrom the results, \nOURS-R method produced less skewness of the $N_1$ histogram on both the datasets. \nThis implies that using the additional structural term reduces \nhubness and therefore the curse of dimensionality is reduced.\n\n\\begin{table}[]\n\\center\n\\caption{Hubness comparison using skewness for DEM and OURS-R methods \non the \\textbf{AwA2} and \\textbf{aPY} datasets}\n\\label{table:hubness}\n\\begin{tabular}{@{}ccc@{}}\n\\toprule\nSkewness & AwA2 & aPY \\\\ \\midrule\nDEM & 3.39 & 1.85 \\\\\nOURS-R & 2.41 & 1.33 \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\section{Conclusion}\nThis paper proposed a three-step approach to improve \nthe performance of zero-shot learning for image classification. \nThe three-step approach involved exploiting structural information in data, \ndomain adaptation to unseen test samples and calibration of classification scores. \nWhen the proposed method was applied to \nstandard datasets of zero-shot image classification, \nit outperformed previous methods by a large margin, where\nthe most effective component was the domain adaptation step. \n\n\\iffalse\n\\section*{Acknowledgement}\nThis work was supported in part by the National\nScience Foundation under Grant IIS-1813935.\nAny opinion, findings, and conclusions or recommendations \nexpressed in this material are those of the authors \nand do not necessarily reflect the views of the National Science Foundation. \nWe also gratefully acknowledge the support of NVIDIA Corporation \nwith the donation of an TITAN XP GPU used for this research.\n\\fi\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nFrom isotopic studies of meteorites it is known that the solar nebula\ncontained at least a dozen different short-lived radionuclides, or SLRs\n\\citep[see reviews by][]{MckeD03, MeyeZ06, Wadh07}.\nIdentification of the sources of these SLRs could greatly constrain the \nSun's birth environment and processes acting during star formation. \nThe half-lives of some of these isotopes are shorter than the timescales\n$\\sim 10^{6} - 10^{7} \\, {\\rm yr}$ typically associated with star formation, \nso they must have been produced near the time and place of the Sun's formation.\nThe SLRs $\\mbox{${}^{26}{\\rm Al}$}$, $\\mbox{${}^{41}{\\rm Ca}$}$ and $\\mbox{${}^{60}{\\rm Fe}$}$, in particular, cannot have been inherited from\nthe Sun's molecular cloud in abundances consistent with ongoing Galactic nucleosynthesis,\nand must have been late additions \\citep{Jaco05}. \nA leading candidate for the source of these and other SLRs is one or more core-collapse \nsupernovae in the Sun's birth environment, contaminating either its molecular cloud \n\\citep{CameT77, VanhB02, GounK09}, or\nits protoplanetary disk \\citep{Chev00, OuelD05, Ouel07, OuelDH10, LoonT06}.\nAnother leading candidate is production of SLRs by irradiation (by solar cosmic \nrays, essentially), within the solar nebula \\citep{Goun01}.\nBecause this latter mechanism by itself is inadequate to explain the abundance of \n$\\mbox{${}^{60}{\\rm Fe}$}$ in the early solar system \\citep{LeyaH03, Goun06}, it is generally\naccepted that the source of $\\mbox{${}^{60}{\\rm Fe}$}$ is core-collapse supernovae \\citep[e.g.][]{Wadh07}, \nalthough it is not clear whether the source of $\\mbox{${}^{60}{\\rm Fe}$}$ is a single, nearby \nsupernova, or many (possibly distant) supernovae \\citep[as in][]{GounK09}.\n\nThe origins of the other SLRs are also debated.\nA correlation between $\\mbox{${}^{26}{\\rm Al}$}$ and $\\mbox{${}^{41}{\\rm Ca}$}$ has been observed in meteorites, demanding\na common source for these two isotopes {\\it after} the formation of the solar nebula,\nas in the so called ``late-injection\" hypothesis of \\citet{SahiG98}.\nIt is not yet clear whether these two SLRs are correlated with $\\mbox{${}^{60}{\\rm Fe}$}$.\nIf evidence for a corelation could be found, this would strongly suggest that $\\mbox{${}^{26}{\\rm Al}$}$ \nand $\\mbox{${}^{41}{\\rm Ca}$}$ were injected by the same supernova or supernovae that injected $\\mbox{${}^{60}{\\rm Fe}$}$.\nThe lack of such evidence, though, leaves open the possibility that $\\mbox{${}^{26}{\\rm Al}$}$ and $\\mbox{${}^{41}{\\rm Ca}$}$ \nwere created by irradiation within the solar nebula while $\\mbox{${}^{60}{\\rm Fe}$}$ was injected separately\nby one or more supernovae, into the Sun's molecular cloud or protoplanetary disk. \n\nThe abundances of the SLRs alone have not yet enabled a discrimination between \nthese possibilities, but \\citet[][hereafter GM07]{Goun07} have proposed \nthat the oxygen isotopic ratios of early solar system materials may be used to rule \nout certain hypotheses. \nSpecifically, they argue that if $\\mbox{${}^{26}{\\rm Al}$}$ and $\\mbox{${}^{41}{\\rm Ca}$}$ were injected by a nearby\nsupernova into the Sun's protoplanetary disk, sufficient to produce the observed \nmeteoritic ratio $\\alratio \\approx 5 \\times 10^{-5}$ \\citep{MacPD95}, then \nthe oxygen isotopic ratio of the solar nebula would be considerably altered:\nsolar nebula materials formed before the injection would have oxygen isotopic \nratios {\\it significantly} different from later-formed materials. \nGM07 calculated the shifts in oxygen isotopic ratios accompanying injection of \nsupernova $\\mbox{${}^{26}{\\rm Al}$}$ into the Sun's protoplanetary disk, using the isotopic yields in\nbulk supernova ejecta calculated by \\citet{Raus02}.\nA robust prediction of the GM07 models is that ${}^{17}{\\rm O} \/ {}^{16}{\\rm O}$ \nof pre-injection materials should be significantly higher, by several percent,\nthan post-injection materials.\nExamples of pre-injection materials may exist in meteorites, or especially in the \nsolar wind sample returned by the ${\\it Genesis}$ mission \\citep{Burn03}. \nSince preliminary results from {\\it Genesis} suggest the Sun is {\\it not} \nisotopically heavy in oxygen \\citep{Mcke09}, and because no such \n${}^{17}{\\rm O}$-rich (or ${}^{16}{\\rm O}$-poor) components have been discovered \nin meteorites, GM07 rule out a supernova origin for the $\\mbox{${}^{26}{\\rm Al}$}$ and $\\mbox{${}^{41}{\\rm Ca}$}$ in meteorites.\n\nThe purpose of this paper is to reproduce and refine the method pioneered by GM07, \nand to test the conclusion that $\\mbox{${}^{26}{\\rm Al}$}$ cannot have a supernova origin.\nGM07 originally considered only bulk ejecta of spherically symmetric supernova \nexplosions.\nWe begin our analysis with this case, but make necessary refinements to the method,\nand use our current nucleosynthesis models to predict the isotopic yields.\nWe then expand on the analysis of GM07, calculating the isotopic yields by allowing \nthe disk to intercept ejecta from different \nparts of the supernova explosion rather than a uniformly mixed total yield, and by examining \nanisotropic explosions.\nWe also simultaneously consider the injection of ${}^{41}{\\rm Ca}$ into the disk.\n\nThe paper is organized as follows. \nIn \\S 2, we outline the method used to calculate shifts in oxygen isotopic\ncomposition due to supernova injection of $\\mbox{${}^{26}{\\rm Al}$}$ and $\\mbox{${}^{41}{\\rm Ca}$}$, including updates to the method of GM07.\nIn \\S 3 we describe the results of nucleosynthesis simulations we have carried out,\nto determine the isotopic yields in supernova ejecta under various explosion scenarios.\nWe determine the inputs needed to compute the shifts in solar nebula oxygen isotopic \ncomposition. \nThese shifts in oxygen isotope before and after injection are presented in \\S 4, and \nin \\S 5 we draw conclusions. \n\n\n\n\\section{Method}\n\n\\subsection{Calculation of Isotopic Shifts} \n\nThe method of GM07 is fairly straightforward. \nThey assume that meteoritic components that sample the solar nebula's starting composition,\n{\\it before} the acquisition of ${}^{26}{\\rm Al}$, can be identified and measured. \nLikewise, they assume samples {\\it after} the acquisition of ${}^{26}{\\rm Al}$ can be \nidentified and measured. \nAny difference in the oxygen isotopic content between samples of those two \ngroups would then constitute a shift in oxygen isotopes brought about by the injection of \nsupernova material.\nThe preditced shift in oxygen isotopes due to injection of supernova material into the \nprotoplanetary disk can then be compared to the actual difference in oxygen isotopes\nbefore and after.\nIn practice, because the vast majority of meteoritic components sample the solar nebula\nafter injection, GM07 assumed a ``final\" value for the solar nebula oxygen isotopes, and\nused the isotopic yields in supernova ejecta to predict the initial composition.\nTesting the supernova injection hypothesis thus amounts to finding meteoritic inclusions \nwith this initial oxygen isotopic composition. \nSuch inclusions should have evidence for no live ${}^{26}{\\rm Al}$ at the time of their\nformation, and should be among the oldest meteoritic inclusions. \n\nThe earliest-formed solids in the solar system are widely accepted to be the calcium-rich, \naluminum-rich inclusions (CAIs), both because they contain minerals that are the first \nsolids expected to condense in a cooling solar nebula \\citep{Gros72}, and because their\nPb-Pb ages are the oldest measured, at 4568.6 Myr \\citep{BouvW09}.\nIt is worth noting that because many of the minerals in CAIs are condensates, their\nisotopic composition should reflect that of the solar nebula gas. \nThe vast majority of CAIs have inferred initial ratios $\\alratio \\approx 5 \\times 10^{-5}$\nor appear to have been isotopically reset at a later date \\citep{MacPD95}.\nOnly in a handful of CAIs known as ``FUN\" CAIs (fractionation with unknown nuclear\neffects) has it been possible to set firm upper limits on the initial $\\alratio$ ratio\nand show these CAIs did not contain live $\\mbox{${}^{26}{\\rm Al}$}$ when they formed \\citep{FaheG87, MacPD95}.\nThus, CAIs overall reflect the composition of the solar nebula at an early time, and \nFUN CAIs possibly record the oxygen isotopic abundance before the solar nebula acquired $\\mbox{${}^{26}{\\rm Al}$}$.\n\nTo make more precise statements, it is necessary to quantify the oxygen isotopic composition\nof the nebula and various components. \nThe molar fraction of oxygen in gas and rock can vary, so the relevant quantities are the \nratios of the stable oxygen isotopes, $\\mbox{$^{17}{\\rm O}$} \/ \\mbox{$^{16}{\\rm O}$}$ and $\\mbox{$^{18}{\\rm O}$} \/ \\mbox{$^{16}{\\rm O}$}$.\nIn the field of cosmochemistry, these ratios are commonly expressed as deviations from a \nstandard, in this case Standard Mean Ocean Water (SMOW), which has \n$\\mbox{$^{17}{\\rm O}$} \/ \\mbox{$^{16}{\\rm O}$} = 3.8288 \\times 10^{-4}$ and $\\mbox{$^{18}{\\rm O}$} \/ \\mbox{$^{16}{\\rm O}$} = 2.0052 \\times 10^{-3}$\n\\citep{oneil86}. \nThe fractional deviations of the isotopic ratios from these standard values are\n$\\dxvii$ and $\\mbox{$\\delta^{18}{\\rm O}$}$, and are measured in parts per thousand, or ``permil\" ($\\promille$). \n[That is, $\\dxvii = 1000 \\times \\left( (\\mbox{$^{17}{\\rm O}$} \/ \\mbox{$^{16}{\\rm O}$}) \/ (\\mbox{$^{17}{\\rm O}$} \/ \\mbox{$^{16}{\\rm O}$})_{\\rm SMOW} - 1 \\right)$.]\nIt is also standard to report the quantity $\\Delta^{17}{\\rm O} \\approx \\dxvii - 0.52 \\, \\mbox{$\\delta^{18}{\\rm O}$}$,\nbecause this quantity is conserved during almost all chemical fractionation processes. \n[More precisely, $\\Delta\\mbox{$^{17}{\\rm O}$} \\equiv \\ln(1+\\dxvii) - 0.5247 \\ln(1+\\mbox{$\\delta^{18}{\\rm O}$})$ \\citep{Mill02}.]\n\nIt is clear that the final oxygen isotopic composition of the nebula, $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})'$, \nwill depend on its starting composition $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})_{0}$, the composition of the \nsupernova material, $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})_{\\rm SN}$, and the mass of supernova material injected\n(relative to the mass of the disk). \nIt is straightforward to show that \n\\begin{equation}\n\\delta^{17}{\\rm O}' - \\delta^{17}{\\rm O}_{0} =\n \\frac{ x }{ 1 + x } \\, \\left( \\delta^{17}{\\rm O}_{\\rm SN} - \\delta^{17}{\\rm O}_{0} \\right),\n \\label{eq:shift}\n\\end{equation}\nwhere $\\delta^{17}{\\rm O}_{\\rm SN}$ is the isotopic ratio of the supernova material\ninjected into the disk, and $x \\equiv M({}^{16}{\\rm O})_{\\rm SN} \/ M({}^{16}{\\rm O})_{\\rm disk}$\nmeasures the mass of injected oxygen relative to the oxygen present in the disk (with a similar\nformula applying to $\\mbox{$\\delta^{18}{\\rm O}$}'$).\nIn terms of the masses involved, \n\\begin{equation}\nx = \\frac{ M({}^{16}{\\rm O})_{\\rm SN} }{ M({}^{26}{\\rm Al})_{\\rm SN} } \\times\n \\frac{ M({}^{26}{\\rm Al})_{\\rm SN} }{ M({}^{26}{\\rm Al})_{\\rm disk} } \\times\n \\frac{ M({}^{26}{\\rm Al})_{\\rm disk}}{ M({}^{27}{\\rm Al})_{\\rm disk} } \\times\n \\frac{ M({}^{27}{\\rm Al})_{\\rm disk}}{ M({}^{16}{\\rm O})_{\\rm disk} }. \n \\label{eq:x}\n\\end{equation}\nMost of these terms are defineable. \nFirst, $M({}^{26}{\\rm Al})_{\\rm SN} \/ M({}^{26}{\\rm Al})_{\\rm disk} \\equiv \\exp( +\\Delta t \/ \\tau)$,\nwhere $\\Delta t$ is the time delay between supernova injection and isotopic closure of the\nmeteoritic materials, and $\\tau = 1.03 \\, {\\rm Myr}$ is the mean lifetime of $\\mbox{${}^{26}{\\rm Al}$}$.\nBy definition, $M({}^{26}{\\rm Al})_{\\rm disk} \/ M({}^{27}{\\rm Al})_{\\rm disk} \\equiv$\n$(26\/27) \\times (5 \\times 10^{-5})$, because sufficient $\\mbox{${}^{26}{\\rm Al}$}$ must be injected to\nyield the meteoritic ratio. \nFinally, the isotopic abundances in the solar nebula are known (the ratio \n${}^{27}{\\rm Al} \/ {}^{16}{\\rm O}$ is taken from \\citet{Lodd03}), so \nwe derive \n\\begin{equation}\nx = \\frac{ 4.846 \\times 10^{-7} }{ [ M({}^{26}{\\rm Al}) \/ M({}^{16}{\\rm O})]_{\\rm SN} } \\, \\exp( +\\Delta t \/ \\tau).\n\\end{equation}\nNote that $x$ is independent of the mass of the disk, but it increases with $\\Delta t$, since \nlarger values of $\\Delta t$ imply that more supernova material had to be injected to yield the\nsame $\\alratio$ ratio, thereby implying larger isotopic shifts in oxygen associated with this\ninjection.\n\nBesides the time delay $\\Delta t$, the major inputs needed to infer $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})_{0}$ are\nthe isotopic composition $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})_{\\rm SN}$ and ratio of ${}^{16}{\\rm O}$ to ${}^{26}{\\rm Al}$ \nin the supernova ejecta, and the oxygen isotopic composition of the post-injection solar nebula.\nGM07 used bulk abundances of supernova ejecta calculated by \\citet{Raus02}\nfor the first set of quantities.\nThey also assumed that the oxygen isotopic ratios of the post-injection solar nebula matched the SMOW \nvalues of the present-day Earth: $(\\dxvii',\\mbox{$\\delta^{18}{\\rm O}$}') = (0\\, \\promille, 0\\, \\promille)$. \nThis assumption is the main reason why they concluded that the pre-injection solar nebula had to be \n${}^{17}{\\rm O}$-rich, as we now demonstrate. \nRearranging equation~\\ref{eq:shift} yields\n\\begin{equation}\n\\delta^{17}{\\rm O}_{0} = \\delta^{17}{\\rm O}' \n +x \\left( \\delta^{17}{\\rm O}' - \\delta^{17}{\\rm O}_{\\rm SN} \\right). \n\\end{equation} \nSupernova ejecta tend to be ${}^{16}{\\rm O}$-rich; in the extreme limit, \n$\\delta^{17}{\\rm O}_{\\rm SN} \\approx -1000\\, \\promille$. \nIf $\\delta^{17}{\\rm O}' \\approx 0\\, \\promille$ also, then \n$\\delta^{17}{\\rm O}_{0} \\approx +(1000 x)\\, \\promille$.\nThat is, $\\delta^{17}{\\rm O}_{0}$ is inferred to have been positive and potentially\nquite large if $x > 10^{-2}$. \nThe isotopic yields of the supernova ejecta computed by \\citet{Raus02}\nwere consistent with such large values of $x$ and $\\delta^{17}{\\rm O}_{\\rm SN} < 0$, \nleading GM07 to conclude that generally $\\delta^{17}{\\rm O}_{0} > 0\\, \\promille$.\nIndeed, for progenitor masses $15 - 25 \\, M_{\\odot}$, GM07 inferred\n$\\delta^{17}{\\rm O}_{0} \\approx +35$ to $+220\\, \\promille$.\nSince there are no early-formed meteoritic components with $\\delta^{17}{\\rm O}$ this\nhigh, and because the oxygen isotopic composition of the Sun appears to be consistent\nwith $\\delta^{17}{\\rm O} \\approx -60\\, \\promille$ \\citep{Mcke09}, GM07 ruled\nout supernova injection of $\\mbox{${}^{26}{\\rm Al}$}$ and $\\mbox{${}^{41}{\\rm Ca}$}$.\nThis conclusion depends on a few key assumptions that we update below.\nWe consider the starting composition of the solar nebula, and take into account the \nnon-homogeneity of supernova ejecta. \n\n\\subsection{Solar Nebula Oxygen Isotopic Composition}\n\nOxygen isotopic ratios potentially can test or rule out the supernova injection \nhypothesis, but several caveats must be applied to the method of GM07.\nThe first and most important correction involves the oxygen isotopic composition of the \nsolar nebula immediately before and after the injection of supernova material.\nGM07 assumed the post-injection composition was equal to SMOW; \nhowever, SMOW is widely understood {\\it not} to reflect the oxygen \nisotopic ratios of the solar nebula immediately after injection.\nOn a three-isotope diagram of $\\dxvii$ versus $\\mbox{$\\delta^{18}{\\rm O}$}$, the oxygen isotopes of planetary and\nmeteoritic materials are arrayed along a mixing line called the Carbonaceous Chondrite Anhydrous\nMineral (CCAM) line discovered by \\citet{Clay73}.\nAfter correcting for isotopic fractionation by thermal and chemical processes, \\citet{YounR98}\ninferred a mixing line with slope 1.0 in the three-isotope diagram, and so we will\nrefer to this mixing line as the ``slope-1\" line.\nToday the oxygen isotopic composition of the Earth (SMOW) is widely recognized to reflect a mixture\nof an isotopically lighter rocky component (to which CAIs belong), and an isotopically heavy reservoir\n\\citep[e.g.][]{Clay03,YounR98}.\nIt is very likely that this component is isotopically heavy water, with\n$\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$} > +30\\, \\promille$\n\\citep{Clay84,LyonY05,Lyonea09}.\nThe existence of isotopically heavy water is supported by the discovery (in the primitive\ncarbonaceous chondrite Acfer 094) of a poorly characterized product of aqueous alteration, with\n$\\dxvii \\approx \\mbox{$\\delta^{18}{\\rm O}$} \\approx +180\\, \\promille$ \\citep{SakaS07}.\nQuite possibly this heavy water is the result of a mass-dependent photodissociation of CO in the outer\nsolar nebula by an external ultraviolet source \\citep{LyonY05,Lyonea09}.\nThe photodissociation can be isotopically selective because the different isotopologues of CO\nmolecules can self-shield; ${\\rm C}^{17}{\\rm O}$ and ${\\rm C}^{18}{\\rm O}$ are optically thin and \ndissociate more completely, releasing ${}^{17}{\\rm O}$ and ${}^{18}{\\rm O}$ atoms that react with \n${\\rm H}_{2}$ to form isotopically heavy water, while the abundant molecule ${\\rm C}^{16}{\\rm O}$ \nis more optically thick and does not as completely dissociate.\nThe light CO molecule is eventually lost with the nebular gas.\nWhatever the source of the isotopically heavy component, \nSMOW only represents a late stage in nebular evolution, and does not represent\nthe state of the nebula immediately after injection of supernova material. \n\nApplying the same reasoning, it is likely that the starting composition of the solar nebula\nwas lower (more ${}^{16}{\\rm O}$-enriched) on the slope-1 line than most CAIs. \nThe majority of CAIs tend to cluster near $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$}) \\approx$ \n$(-41\\, \\promille,-40\\, \\promille)$, i.e., $\\Delta^{17}{\\rm O} \\approx -20.2\\, \\promille$\n\\citep[see][and references therein]{Clay03};\nbut many of the most primitive and unaltered CAIs cluster near \n$(\\dxvii,\\mbox{$^{18}{\\rm O}$}) \\approx$ $(-50\\, \\promille,-50\\, \\promille)$,\nor $\\Delta\\mbox{$^{17}{\\rm O}$} \\approx -24\\, \\promille$ \\citep{ScotK01}.\nLikewise, \\citet{MakiN09} report $\\Delta\\mbox{$^{17}{\\rm O}$} = -23.3 \\pm 1.9\\, \\promille$ for\n``mineralogically pristine\" CAIs. \nCAIs also contain grains of hibonite, spinel, and corundum, which are among the first \nminerals expected to condense from a cooling gas of solar composition \\citep{EbelG00},\nand which are presumably even more primitive than CAIs themselves. \n\\citet{ScotK01} report that hibonite grains are also found to cluster near\n$(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$}) \\approx (-50\\, \\promille,-50\\, \\promille)$,\nor $\\Delta^{17}{\\rm O} = -24 \\,\\promille$, while \\citet{MakiN09b}\nobserved 4 hibonite grains from Allende and Semarkona to have oxygen isotopes\nin the range $\\Delta^{17}{\\rm O} = -32$ to $-17\\, \\promille$. \nThey also found that spinel grains from the CV chondrite Allende had \n$\\Delta^{17}{\\rm O} = -25 \\pm 5\\,\\promille$, and that corundum grains from the CM chondrite \nSemarkona clustered strongly in the range $\\Delta^{17}{\\rm O} = -24 \\pm 2\\, \\promille$. \n\\citet{KrotN10} likewise report $\\Delta^{17}{\\rm O} = -24 \\pm 2\\, \\promille$ for primitive CAIs \nand amoeboid olivine aggregates, which are also believed to have condensed from solar nebula gas. \nFrom these results we infer that $(\\dxvii',\\mbox{$\\delta^{18}{\\rm O}$}') \\approx (-50\\, \\promille,-50\\, \\promille)$\nin the solar nebula immediately after the injection of supernova material. \n\nMeteoritic and other samples also constrain the initial (pre-injection) oxygen isotopic \ncomposition of the solar nebula, and find it to be very similar. \nAs described above, very firm and low upper limits to initial $\\alratio$ exist for\nFUN CAIs that mark them as having formed before the injection of $\\mbox{${}^{26}{\\rm Al}$}$ and $\\mbox{${}^{41}{\\rm Ca}$}$\n\\citep{SahiG98}. \n\\citet{KrotC08} have identified a fractionation line associated with the FUN CAIs\nwith $\\Delta^{17}{\\rm O} = -24.1\\, \\promille$ that passes through\n$(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$}) \\approx (-51\\, \\promille,-52\\, \\promille)$.\nPresumably the original isotopic composition of the nebula matched that of the Sun,\nwhich might therefore be measured by {\\it Genesis} mission \\citep{Burn03}.\nPreliminary measurements can be interpreted as clustering on a fractionation line\nwith $\\Delta^{17}{\\rm O} \\approx = -26.5 \\pm 5.6\\, \\promille$ \\citep{Mcke09}, \nwhich would intersect the slope-1 line at \n$(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$}) \\approx (-56\\, \\promille, -57\\, \\promille)$, and other analyses\nsuggest $\\Delta^{17}{\\rm O} \\approx -33 \\pm 8 \\, \\promille$ [$2\\sigma$ errors]\n\\citep{McKeea10}. \n\nFrom these results it seems likely that the original solar nebula oxygen isotopic\ncomposition was near the {\\it Genesis} preliminary result of\n$(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$}) \\approx (-54\\, \\promille,-53\\, \\promille)$, or possibly much lower\nalong the slope-1 line. \nSubsequent reaction of rock with a ${}^{16}{\\rm O}$-depleted reservoir \nthen moved material along the slope-1 line to $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$}) \\approx$ \n$(-41\\, \\promille,-40\\, \\promille)$,\nwhere most CAIs are found \\citep[][and references therein]{Clay03}.\nFUN CAIs appear to represent an intermediate stage in this process, only partially \nevolved along the slope-1 line. \nTo fix values, we will simply assume the solar system protoplanetary disk isotopic\nratios started as $(\\delta\\mbox{$^{17}{\\rm O}$},\\delta\\mbox{$^{18}{\\rm O}$}) = (-60\\,\\promille,-60\\,\\promille)$.\n\nThe above discussion changes the criterion by which one can reject the supernova\ninjection hypothesis. \nBecause GM07 assumed an initial solar nebula composition near SMOW, \nthey concluded that pre-injection samples necessarily would have had $\\dxvii > 0$, \nand the lack of such samples in meteorites ruled out the hypothesis.\nBut we assert that the supernova injection hypothesis can be ruled out only if \ninjection of supernova material necessarily shift the oxygen isotopic composition of \nthe solar nebula from a composition near $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$}) \\approx (-60\\, \\promille,-60\\, \\promille)$ \nto one far off the slope-1 line, or one on the slope-1 line but with $\\dxvii > -50 \\, \\promille$.\nIn this way, the GM07 method of using oxygen isotopic constraints might still allow\na test of the supernova injection hypothesis. \n\n\\subsection{Magnitude of Isotopic Shift \\label{magofshifts}}\n\nThere are at least three scenarios wherein the shift in oxygen isotopes following injection of \nsupernova material can be consistent with the above constraints.\nFrom equation~\\ref{eq:shift}, it is seen that even if the supernova ejecta and the protoplanetary \ndisk differ in oxygen isotopic composition by hundreds of permil, the shift in oxygen isotopes \nmay be small ($< 1$ permil) if the injected mass is small, so that $x < 10^{-2}$. \nMore precisely, if $\\mbox{${}^{26}{\\rm Al}$}$ and the other SLRs are injected by a supernova into the solar nebula\ndisk, then the magnitude of the shift in oxygen isotopes will depend on the fraction of \nejecta oxygen that accompanies Al. \nIn this first scenario, O and Al may be significantly fractionated during delivery of the ejecta \nto the solar nebula.\nFor example, \\citet{Ouel07} find that effectively only material condensed from the supernova \nejecta into large ($> 1 \\, \\mu{\\rm m}$ radius) grains can be injected directly into a protoplanetary \ndisk.\nIn the extreme event that the only grains that entered the protoplanetary disk were corundum\n(${\\rm Al}_{2}{\\rm O}_{3}$) grains, the isotopic shifts in oxygen would be negligible\n($< 0.001\\,\\promille$).\nOr, if only 10\\% of the oxygen in the ejecta condensed into grains, and 90\\% remained in gas\nthat was excluded from the disk, then the isotopic shifts in oxygen isotope (for a given amount\nof injected $\\mbox{${}^{26}{\\rm Al}$}$) would be 10 times smaller than predicted by GM07. \nIt is therefore not possible to determine the shifts in oxygen isotopes following injection into \na disk without quantifying the degree to which O and Al are fractionated between gas and solids. \nIn what follows, we assume no fractionation, as such a calculation is beyond the scope of the \npresent investigation; but we consider dust condensation in supernova ejecta to be a very important \neffect, one that potentially could significantly reduce the predicted isotopic shifts. \n\nIn the second scenario, the shifts in oxygen isotopes could also remain small if the injected \nmaterial was simply higher than expected in $\\mbox{${}^{26}{\\rm Al}$}$ (or lower in O), so that again $x < 10^{-2}$. \nThe calculations of GM07 relied on the {\\it bulk} abundances calculated by \\citet{Raus02}.\nThat is, GM07 assumed that the injected material uniformly sampled the entirety of the supernova\nejecta.\nSuch a uniform sampling is unlikely, as supernovae often do explode in \na clumpy fashion and asymmetrically. It has long been understood that asymmetries or hydrodynamic \ninstabilities may disrupt the stratification of the progenitor star, but they do not result in large scale \ncompositional mixing \\citep[e.g.][]{joggerst08, hftm05, fam91}.\nThe X-ray elemental maps of the Cassiopeia A supernova remnant \\citep{Hwan04} dramatically \ndemonstrate that massive stars are likely to explode as thousands of clumps of material, each sampling\ndifferent burning zones within the progenitor.\n\\citet{OuelDH10} have argued that this may be a near-universal feature of core-collapse\nsupernovae; at the very least, observations do not rule out this possibility. \nSo it is more than possible that the solar nebula received materials from only limited regions \nwithin the ejecta in which the $\\mbox{${}^{26}{\\rm Al}$} \/ {}^{16}{\\rm O}$ ratio could have varied considerably from\nthe average value for the ejecta. \nThe non-uniformity of the $\\mbox{${}^{26}{\\rm Al}$} \/ {}^{16}{\\rm O}$ ratio may be magnified if the star explodes\nasymmetrically, allowing explosive nucleosynthesis to proceed differently even in parcels of \ngas in the same burning zone. \n\nFinally, in the third scenario by which isotopic shifts may conform to measurements, $x$ need not\nbe small, and the isotopic shifts may approach $10\\, \\promille$ in magnitude,\nso long as the injection moved the composition {\\it up} the slope-1 line by $\\approx 10\\, \\promille$ \n(i.e., the change in $\\dxvii$ equalled the change in $\\mbox{$\\delta^{18}{\\rm O}$}$, both being $< 10\\, \\promille$), \nor {\\it down} the slope-1 line by a comparable or even larger amount.\nA shift from an initial composition $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})_{0} \\approx (-60\\, \\promille, -60\\, \\promille)$,\nconsistent with ${\\it Genesis}$ measurements of the Sun's composition, to \n$(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})' \\approx (-50\\, \\promille, -50\\, \\promille)$, consistent with primitive meteoritic\ncomponents, would not conflict with the data.\nAlternatively, a shift from an initial composition $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})_{0} \\approx (-60\\, \\promille, -60\\, \\promille)$,\nto $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})' \\approx (-70\\, \\promille, -70\\, \\promille)$, \nor even $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})' \\approx (-80\\, \\promille, -80\\, \\promille)$, \nfollowed by mixing with the ${}^{16}{\\rm O}$-poor reservoir that moves solar nebula solids up the \nslope-1 line, would also conform to the data. \n\nIn the next section we compute the isotopic yields in core-collapse supernovae of various progenitor\nmasses, both in spherically symmetric explosions \\citep[as considered by][]{Raus02}\nand asymmetric explosions.\nThese calculations allow us to predict the oxygen isotopic composition $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})_{\\rm SN}$ \nand the ratio $x$ of the supernova material at various locations within the explosion, to assess the \nrange of possible isotopic shifts under the second and third scenarios. A supernova injection scenario\nwould be ruled out, {\\it unless} either the injection of material results in small overall shifts (i.e. the \ninjected material contains a high \\mbox{${}^{26}{\\rm Al}$}~ abundance relative to oxygen, or vice versa, a low oxygen \nabundance relative to Al), or the oxygen isotopes are shifted along the \n'slope-1 line', in which case shifts of up to $\\sim$10 permil in either direction are allowed.\n\n\n\n\\section{Isotope Production within Supernovae}\n\n\\begin{deluxetable}{lcccc}\n\\tablewidth{0pt}\n\\tablecaption{Explosion Simulations\\label{tab1}}\n\\tablehead{\n \\colhead{Simulation}\n& \\colhead{progenitor}\n& \\colhead{Energy}\n& \\colhead{M$_{\\rm Remn}$}\n& \\colhead{Delay} \\\\\n \\colhead{}\n& \\colhead{}\n& \\colhead{$10^{51}$\\,erg}\n& \\colhead{(\\Msol)}\n& \\colhead{ms}\n}\n\\startdata\n23e-0.8 & 23 \\Msol\\, single & 0.8 & 5.7 & 20 \\\\\n23e-1.2 & 23 \\Msol\\, single & 1.2 & 4.1& 20 \\\\\n23e-1.5 & 23 \\Msol\\, single & 1.5 & 3.2 & 20 \\\\\n23e-0.7-0.8 & 23 \\Msol\\, single & 0.8 & 3.2 & 700 \\\\\n23e-0.7-1.5 & 23 \\Msol\\, single & 1.5 & 2.3 & 700 \\\\\n16m-run1 & 16 \\Msol\\, binary & 1.5 & 2.06 & 20 \\\\\n16m-run2 & 16 \\Msol\\, binary & 0.8 & 2.43 & 20 \\\\\n16m-run4 & 16 \\Msol\\, binary & 5.9 & 1.53 & 20 \\\\\n23m-run1 & 23 \\Msol\\, binary & 1.9 & 3.57 & 20 \\\\\n23m-run2 & 23 \\Msol\\, binary & 1.2 & 4.03 & 20 \\\\\n23m-run5 & 23 \\Msol\\, binary & 6.6 & 1.73 & 20 \\\\\n40m-run1 & 40 \\Msol\\, single & 10 & 1.75 & 20 \\\\\n40m-run5 & 40 \\Msol\\, single & 1.8 & 4.51 & 20 \\\\\n40m-run9 & 40 \\Msol\\, single & 2.4 & 6.02 & \\,20\n\\enddata\n\\label{tb:explsim}\n\\end{deluxetable}\n\n\\subsection{Numerical Methods}\n\nWe calculate the yields of oxygen isotopes, \\mbox{${}^{41}{\\rm Ca}$}, and \\mbox{${}^{26}{\\rm Al}$}\\, in several core-collapse\nsupernova scenarios, listed in Table \\ref{tb:explsim}.\nThese calculations explore four different progenitor models, with a range of explosion scenarios\nfor each.\nWe use a large set of thermally driven 1D explosions with varying kinetic energies and delays,\nfor a star of initial mass 23 $\\Msol$ and a more restricted range of explosions for a 16 $\\Msol$, \nand 23 $\\Msol$, with the hydrogen envelope stripped in a case B binary scenario, and a single 40 \n$\\Msol$ progenitor that ends its life as a type WC\/O after extensive mass loss.\nWe also examine a 3D explosion of the 23 $\\Msol$ binary progenitor.\nDetails of the simulations can be found in \\citet{yca09}.\nThe set of progenitor models we selected by no means samples the entire diversity of supernovae, but it\nrepresents a variety of cases across a large range of progenitor masses and explosion parameters.\nIt is sufficiently diverse to make generalizations for behaviors that appear across all models.\n\nProgenitor models were produced with the TYCHO stellar evolution code\n\\citep{YouA05}.\nTo model collapse and explosion, we use a 1-dimensional Lagrangian code developed by \\citet{her94}\nto follow the collapse through core bounce.\nThis code includes 3-flavor neutrino transport using a flux-limited diffusion calculation and a\ncoupled set of equations of state to model the wide range of densities in the collapse phase\n\\citep[see][for details]{her94,fryer99a}.\nIt includes a 14-element nuclear network \\citep{bth89}\nto follow the energy generation.\nTo get a range of explosion energies, we opted to remove the neutron star and drive an explosion by\ninjecting energy in the innermost 15 zones (roughly 0.035 $\\Msol$).\nThe duration and magnitude of energy injection of these artificial explosions were altered to produce the\ndifferent explosion energies.\nOur 3-dimensional simulation uses the output of the 1-dimensional explosion (23 $\\Msol$ binary star, 23m-run5)\nwhen the shock has reached 10$^9$\\,cm.\nWe then map the structure of this explosion into our 3D Smooth Particle Hydrodynamics code SNSPH\n\\citep{frw06}\nby placing shells of particles whose properties are determined by\nthe structure of our 1-dimensional explosion \\citep[see][for details]{hun05, frw06}.\nThe 3D simulation uses 1 million SPH particles and is followed for 800 seconds after collapse.\n\nThe initial intent of the 3D simulation was to create a fully 3-dimensional\ncalculation of an explosion with a moderate bipolar asymmetry \\citep{Younea06}.\nThe interesting behavior of \\mbox{${}^{26}{\\rm Al}$}\\, in the explosion then prompted us to \nconsider the composition of this material in relation to an isotopic enrichment scenario of\nthe solar system.\nBoth observational and theoretical evidence indicate that asymmetry is strong and ubiquitous in supernovae \n\\citep[e.g.][]{grb07,yf07,hun05, Lopeea09}. In the situation of supernova injection the asymmetric \nmodel's primary utility lies not in modeling a specific event, but rather sampling a wide range of \nthermodynamic histories for material capable of producing \\mbox{${}^{26}{\\rm Al}$}. Injected material is likely to sample \nonly a small region of the supernova, meaning we can treat each SPH particle as an isolated trajectory \nwhose further evolution is not dependent upon the progenitor star or or global parameters of the \nexplosion. Any explosion that produces a similar thermodynamic trajectory will end up with similar \nyields. We can thus probe a large variety of explosion condition not accessible to 1D calculations \nwithout a prohibitive investment of computational time. Therefore we consider this asymmetric\nsimulation sufficiently generic to justify its usage for this investigation.\nWe create an asymmetric explosion with a geometric aspect ratio and final kinetic energy axis ratio \ndesigned to be roughly consistent with the degree of asymmetry implied by supernova polarization \nmeasurements. To simulate an asymmetric explosion, we modify the velocities within each shell by \nincreasing those of\nparticles within 30$^\\circ$ of the z-axis by a factor of 6; we will refer to\nthese parts of the supernova as high velocity structures (HVSs). The velocities of the remaining particles were\ndecreased by a factor of 1.2, roughly conserving the explosion energy. This results in a 2:1 morphology\nbetween the semimajor to semiminor axes ratio by the end of the simulation.\nWe did not introduce any angular dependence in the thermal energy.\nAt these early times in the explosion, much of the explosion energy remains in thermal energy, so the total\nasymmetry in the explosion is not as extreme as our velocity modifications suggest.\nFor a detailed discussion on choosing asymmetry parameters for 3D explosions see\n\\citet{hun03,hftm05,fw04}.\n\n\\begin{figure}[tbh]\n\\centering \n\\subfloat[]{\\label{fig:dens}\\includegraphics[angle=90,width=0.49\\columnwidth]{temperature.eps}}\\\\\n\n\\subfloat[]{\\label{fig:temp}\\includegraphics[angle=90,width=0.49\\columnwidth]{density.eps}}\\\\\n\n\\subfloat[]{\\label{fig:entr}\\includegraphics[angle=90,width=0.49\\columnwidth]{entropy.eps}}\n\\caption{Shown is the time evolution of temperature, density, and entropy \nfor representative examples of particles from\nthe Ring and Bubble region of the 3D calculation. Each line is labelled with the abundance of \n\\mbox{${}^{26}{\\rm Al}$}, \\mbox{$^{18}{\\rm O}$}, and \\mbox{$^{18}{\\rm F}$}~(above each graph) in mass fraction per particle at the end of \nthe simulation.}\n\\label{fig:traj}\n\\end{figure}\n\nAs noted above, although we are only considering one 3D model, the results we get from that \nmodel are representative of a range of nucleosynthetic conditions that may occur in multiple \nexplosion\/progenitor scenarios. Each parcel of gas follows its \nown density and temperature evolution, which is determined by the \\emph{local} velocity of the \nparcel of gas. It is the local conditions of the gas that matter; it is unaware of the global evolution. \nAn example of the trajectories from the Ring and Bubble regions are shown in Figure \n\\ref{fig:traj}. The Figure shows the temperature, density, and radiation entropy evolution for \nrepresentative $\\mbox{${}^{26}{\\rm Al}$}$-rich particles in the explosion. The lines are labeled with $\\mbox{${}^{26}{\\rm Al}$}$, $^{18}$F, \nand $\\mbox{$^{18}{\\rm O}$}$ abundances at the emd of the simulation (before complete radioactive decay of $^{18}$F). \nWe see three classes of trajectories: high temperature and high entropy, high temperature and \nlow entropy, and low temperature, high entropy. Predictably, the high temperature, low entropy \ntrajectories tend to have low $^{18}$F (and therefore $\\mbox{$^{18}{\\rm O}$}$) abundances due to the \nphotodisintegration of $^{18}$F into $^{14}$N $+\\ \\alpha$. High temperature, high entropy \ntrajectories have a higher reverse rate for that reaction, preserving slightly more $^{18}$F. The low \ntemperature particles have the highest $^{18}$F abundance at the end of burning.\n\nThe asymmetric explosion samples trajectories with a large span \nof velocity evolutions reasonable for plausible asymmetries. As we demonstrate with the 1D models and in \n\\citet{yca09}, the sites of production for \\mbox{${}^{26}{\\rm Al}$}~ are similar across a wide range of stellar \nmasses, so as long as we sample the particle trajectories well in a single asymmetric explosion, the \nresults are robust to very large changes in progenitor mass and explosion asymmetry. \nThis assumption is valid, since we are \nlooking for regions in the explosion that produce plausible abundances, not a bulk yield.\n\nThe network in the explosion code terminates at $^{56}$Ni and cannot follow neutron excess, so to\naccurately calculate the yields from these models we turn to a post-process step.\nNucleosynthesis post-processing was performed with the Burn code \\citep{yf07},\nusing a 524 element network terminating at $^{99}$Tc.\nThe initial abundances in each SPH particle are the 177 nuclei in the initial stellar model.\nThe network machinery is identical to that in TYCHO \\citep[for details of the simulations see][]{yca09}.\n\n\n\\begin{figure}[ptbh]\n\\centering\n\\includegraphics[angle=0,width=0.69\\columnwidth]{figure1.eps}\n\\\\\n\\includegraphics[angle=0,width=0.69\\columnwidth]{figure2.eps}\n\\caption{Shown is a $2.0\\times10^{11}$ cm thick slice in the x-z plane of\nthe 3D simulation. \\mbox{${}^{26}{\\rm Al}$}\\, abundances are in red-tones (amount per particle in \\Msol) and number\nratio of $^{17}$O\/$^{16}$O (top panel) or $^{18}$O\/$^{16}$O (bottom panel)\nper particle in blue-tones. The sizes of the data points are arbitrarily chosen, but\nscale with their values. A color gradient was also used to visualize the different\nabundances per particle. The ligher colors\/bigger data points correspond to\nhigher abundances.\nApparent is a Ring on either side of the center along the axis of symmetry,\nand further out from them the Bubbles, where the highest \\mbox{${}^{26}{\\rm Al}$}\\, abundance is found.}\n\\label{fig:al}\n\\end{figure}\n\n\\begin{figure}[tbh]\n\\centering\n\\includegraphics[angle=0,width=0.7\\columnwidth]{figure3.eps}\n\\caption{Same as Figure \\ref{fig:al} but with $^{41}$Ca shown in green. The highest \\mbox{${}^{41}{\\rm Ca}$}\\, abundance\nis adjacent to the Rings, and only partially overlaps with the Ring- regions.}\n\\label{fig:ca}\n\\end{figure}\n\n\n\\subsection{Production of ${}^{26}{\\rm Al}$ in 1D and 3D explosions}\n\nThere are three primary sites for production of \\mbox{${}^{26}{\\rm Al}$}\\, in a massive star and its accompanying\nsupernova.\nIt can be produced by hydrogen burning at high temperatures in the shell-burning regions of\nmassive stars or evolved AGB stars. But neither of these production sites\nis important to the supernova injection scenario, as discussed in \\citet{yca09}.\nIn the 1D simulations, the two dominant production sites are two peaks in $^{26}{\\rm Al}$\nabundances that coincide with peak temperatures\nin the explosion of $2.2 \\times 10^9 \\, {\\rm K}$ and $1.5 \\times 10^9 \\, {\\rm K}$, in material\nthat has undergone hydrostatic C burning in the progenitor.\nThe higher of the two temperatures is sufficient for explosive C and Ne burning, the lower of the\ntemperatures is near explosive C burning.\nThe production of \\mbox{${}^{26}{\\rm Al}$}\\, in both regions is due to a significant increase in the\nflux of free p, n, and $\\alpha$-particles.\nAt higher temperatures, characteristic of O burning, $^{26}{\\rm Al}$ is quickly destroyed.\n\nWithin the 3D calculations, $^{26}{\\rm Al}$ is produced in two main regions\n(see Figure \\ref{fig:al}), similar to the 1D results.\nThe first is a ring-like structure and an associated small bubble where the two\nHVSs emerge into a lower density region, and which we denote the ``Ring\".\nMaterial in the Ring has undergone explosive Ne and C burning during the explosion at\ntemperatures slightly above $2 \\times 10^9 \\, {\\rm K}$, and\ncorresponds to the explosive C and Ne region in the 1D simulations identified above.\nThe second region, further out at the terminal end of the HVSs, is denoted the ``Bubble\"\n(see Figure \\ref{fig:al}).\nWithin the Bubble, material has undergone hydrostatic C burning and then experienced\npeak shock temperatures $\\sim 1.5 \\times 10^9 \\, {\\rm K}$ during the explosion,\nand corresponds to the second of the \\mbox{${}^{26}{\\rm Al}$}\\, peaks in the 1D\nsimulations identified above (which we will refer to as sub-explosive C burning region).\nWhile the production sites of \\mbox{${}^{26}{\\rm Al}$}\\, in the simulation in 3D occur in zones of about the same\ntemperatures as in the 1D cases, the peak in \\mbox{${}^{26}{\\rm Al}$}\\, abundance in those regions are reversed\nfrom the corresponding regions in 1D (i.e. the peak that is higher in \\mbox{${}^{26}{\\rm Al}$}\\, in 1D is lower in \\mbox{${}^{26}{\\rm Al}$}\\,\nin 3D, and vice versa). An important aspect of the Bubble is that due to the rapid expansion\nof the HVSs, its density drops rapidly, quenching some of the nuclear reactions.\nThe decrease in density in the 1D simulations occurred at\na slower rate, conversely more of the \\mbox{${}^{26}{\\rm Al}$}\\, was able to be processed into other species.\nThis freezeout of nuclear reactions (suppressing subsequent destruction of \\mbox{${}^{26}{\\rm Al}$})\nin the 3D simulation is the reason for the higher production of $^{26}{\\rm Al}$\nin the Bubble, as compared to the Ring.\n\nIn 3D, the \\mbox{${}^{41}{\\rm Ca}$}\\, production occurs in only one main production site, in a region adjacent to and\npartly overlapping the Ring in the 3D simulation (see Figure \\ref{fig:ca}), and in both the \nexplosive C\/Ne and sub-explosive C burning regions in the 1D calculations. The production of \n\\mbox{${}^{41}{\\rm Ca}$}~ requires a high $^{40}$Ca abundance as the seed \nnucleus, and the main production channel is p- and n-capture onto $^{40}$Ca. \nThe \nslower drop in density and temperature in the 1D calculations tended to favor a low level production of \n\\mbox{${}^{41}{\\rm Ca}$}, which is why its abundance is slightly higher as compared to the 3D calculation. In the 3D \ncalculation, \\mbox{${}^{41}{\\rm Ca}$}~ is produced in the Ring, but the faster expansion of the material there due to the \nvelocity asymmetry shuts off the reactions faster than in the 1D models, and the final \\mbox{${}^{41}{\\rm Ca}$}~ abundance\nis lower than in 1D. In the bubble region, the lower temperature and rapid density falloff preclude \nany significant \\mbox{${}^{41}{\\rm Ca}$}~ production. \n\nWithin the zones where \\mbox{${}^{26}{\\rm Al}$}\\, is produced, the M$({}^{26}{\\rm Al})$\/M(${}^{16}{\\rm O})$\nratios can differ significantly from the bulk abundances.\nFor the 1D models, these ratios can vary by a factor of $\\sim 1$ up to a factor of\n$\\sim 100$ between the\nexplosive C\/Ne burning region and the sub-explosive C burning region, with a typical\nvariation of a factor of $\\sim 2-3$.\nFor example, in model 23e-1.5 the\nM(\\mbox{${}^{26}{\\rm Al}$})\/M(\\mbox{$^{16}{\\rm O}$}) ratio varies from $(6.7-8.4)\\times10^{-6}$, and\nfor model 16m-run2 varies from $1.6\\times10^{-7}$ to $1.7\\times10^{-5}$.\nThese are to be compared to the abundances in the bulk of the ejecta, which are\nM(\\mbox{${}^{26}{\\rm Al}$})\/M(\\mbox{$^{16}{\\rm O}$}) $\\approx 6.7 \\times 10^{-6}$ for model 23e-1.5,\n$\\approx 1.7 \\times 10^{-6}$ for model 16m-run2, and varies between\n$4.0 \\times 10^{-7}$ to $1.9 \\times 10^{-4}$\nacross all 1D explosions. In the 3D model, the ratios are\nM(\\mbox{${}^{26}{\\rm Al}$})\/M(\\mbox{$^{16}{\\rm O}$}) $\\approx 1 - 4 \\times 10^{-4}$ in the SPH particles\nin the Bubble, $\\approx 1 - 4 \\times 10^{-5}$ in the SPH particles\nin the Ring, and\n$2.88 \\times 10^{-5}$ for the bulk supernova abundances.\nThus we see that injection of material from the Bubble brings in an order of\nmagnitude less oxygen (essentially all ${}^{16}{\\rm O}$) per \\mbox{${}^{26}{\\rm Al}$}\\, atom than injection\nof material from the supernova overall or from the Ring.\n\n\n\\subsection{Production of O isotopes in ${}^{26}$Al-producing regions}\n\nThe abundances and isotopic compositions of oxygen within a localized region of the\nsupernova can vary significantly from the bulk values, as their production is sensitive to\nthe density, temperature, and composition, and to their variations with time in that region.\nIn the 1D explosions, density falls off roughly as a power law \\citep{arnett96}.\nBecause this maintains a high density in the region where\n\\mbox{${}^{26}{\\rm Al}$}\\, forms by explosive C and Ne burning,\n\\mbox{$^{18}{\\rm O}$}\\, is effectively synthesized into heavier species.\n\\mbox{$^{17}{\\rm O}$}\\, is also synthesized into heavier species but is also created by neutron\ncaptures onto \\mbox{$^{16}{\\rm O}$}.\nThe net effect is that both \\mbox{$^{17}{\\rm O}$}\\, and \\mbox{$^{18}{\\rm O}$}\\, are reduced relative to \\mbox{$^{16}{\\rm O}$}, and\nthe \\mbox{$^{18}{\\rm O}$}\/\\mbox{$^{17}{\\rm O}$}\\, ratio is reduced.\nIn the 23e-1.5 model, their mass fractions in the \\mbox{${}^{26}{\\rm Al}$}\\, rich zones never exceed\n$\\sim 10^{-5}$, and the isotopic composition in nearly all our 1D cases\napproaches $(-1000\\,\\promille,-1000\\,\\promille)$, effectively pure \\mbox{$^{16}{\\rm O}$}.\n\nThe details of oxygen isotopic abundances in the \\mbox{${}^{26}{\\rm Al}$}\\, rich zones of the 3D explosion\ndiffer.\nThe more rapid expansion of the material in the 3D calculation limits the processing of \n\\mbox{$^{17}{\\rm O}$}~ and other isotopes into heavier species, so the yield of those is higher than in the 1D calculations.\nAs one of the main burning products of explosive Ne burning, $^{16}$O is\nquite abundant in the Ring.\nHowever, some of the free p and n produced during explosive burning capture onto\n\\mbox{$^{16}{\\rm O}$}, producing $^{17}$O, so the Ring (as Figure \\ref{fig:al} shows) is quite enriched\nin \\mbox{$^{17}{\\rm O}$}\\, relative to the rest of the explosion. \nIn the Bubble, \\mbox{$^{16}{\\rm O}$}\\, is not produced explosively, and is mostly\nleft over from the progenitor.\nThe increased flux of free particles also burns some of the $^{16}$O there to $^{17}$O\nand $^{18}$O.\nThe freezeout from the expansion limits the processing of\nthese isotopes into heavier species, so the 3D explosion is richer in these isotopes than\nthe 1D simulation.\n\nPart of the reason for the large variation in ${}^{18}{\\rm O}$ isotopic yields is that\nmost of it is produced by decay of $^{18}{\\rm F}$ ($t_{1\/2} = 110$ minutes), which was\nco-produced with \\mbox{$^{17}{\\rm O}$}\\, and \\mbox{$^{18}{\\rm O}$}. \nThus it depends sensitively on how much \\mbox{$^{18}{\\rm F}$}~ is \npresent once nuclear burning shuts off, which in turn depends sensitively on the \ntrajectories taken by the gas.\nAt low temperatures the classical decay reaction $^{18}{\\rm F} \\, \\rightarrow \\, {}^{18}{\\rm O} + e^{+}$ \ncompletely dominates, but at high temperatures (above $\\sim 10^{9} \\, {\\rm K}$), \nand low proton density, another decay channel opens up for\n${}^{18}{\\rm F}$, and it\ncan decay also via \n$^{18}{\\rm F} \\, \\rightarrow \\, {}^{14}{\\rm N} + \\alpha$ \\citep{ga00}.\nThe branching ratio of these two reactions is very sensitive to temperature at around $1\\times10^9$ K,\nwith higher temperatures overwhelmingly favoring the decay to ${}^{14}{\\rm N} + \\alpha$.\n\nThe amount of \\mbox{$^{18}{\\rm F}$}~ remaining at the end of burning is highly dependent on the time taken \nto drop below that temperature, and the density evolution, as high entropies favor the destruction over \nthe synthesis. \nBecause of the power law drop off, the density in the 1D calculations stayed higher for a longer period of time,\nas compared to the 3D calculation, thus isotopes had a longer time window in which they could be processed\nto higher species. The density of the 3D calculation dropped faster due to the \nincreased velocities of particles to create the asymmetry, so the nuclear burning shut off earlier, \nand more isotopes like \\mbox{$^{17}{\\rm O}$}, \\mbox{$^{18}{\\rm O}$}, or \\mbox{$^{18}{\\rm F}$}~ survived the nucleosynthesis of the explosion.\nThis results in a substantial variation in the $^{18}{\\rm F}$ abundance\nbetween the 1D and the 3D calculation, and that same effect (i.e. how quickly the density drops) \nis also responsible for the variation in abundance \nof particles in the Ring and the Bubble by the end of the 3D simulation.\n\nIn the 1D simulations the full decay of all \\mbox{$^{18}{\\rm F}$}\\, after the explosion was calculated in the reaction network.\nThe 3D simulation was terminated earlier in its evolution before complete decay of the \\mbox{$^{18}{\\rm F}$}. As the\n temperature at that point in the explosion was well below $10^9$ K,\nwe assumed that any \\mbox{$^{18}{\\rm F}$}\\, still present would decay into \\mbox{$^{18}{\\rm O}$}, as this is the only significant \nchannel at these lower temperatures.\n\n\n\n\\section{Solar system oxygen isotopic shifts accompanying ${}^{26}{\\rm Al}$ delivery}\n\n\\begin{figure}[tbh]\n\\centering\n\\includegraphics[angle=90,width=0.8\\columnwidth]{figure5.eps}\n\\caption{Three isotope plot showing the shifts in the oxygen isotopes we calculate following\ninjection of supernova material for different scenarios. The shifts from he 3D \ncases are plotted in green, those from the 1D bulk cases are plotted in blue, the 1D\nexplosive C\/Ne burning cases are plotted in orange, and the 1D sub-explosive C burning cases \nare plotted in cyan. Indicated by the bigger black dots are SMOW at \n($0\\, \\promille$, $0\\, \\promille$) and our assumed pre-injection composition at \n($-60\\, \\promille$, $-60\\, \\promille$). The very large shift of the 3D bulk scenario was omitted for clarity.}\n\\label{fig:3isob}\n\\end{figure}\n\n\\subsection{Spherically Symmetric Supernova Explosions}\n\nIt is now possible to calculate the shifts in oxygen isotopic abundances before and after\nthe injection of supernova material, using equation~\\ref{eq:shift}.\nAs described in \\S 2, the initial composition of the solar nebula was probably close to \n$(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})_{0} \\approx (-60\\, \\promille,-60\\, \\promille)$, and we adopt this as \nthe starting value.\nBased on the numerical simulations of \\S 3, we have calculated the ratios $x$ and the isotopic \nabundances $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})_{\\rm SN}$ within the ejecta overall, and within the regions where \n$\\mbox{${}^{26}{\\rm Al}$}$ is produced.\n\nWe begin with the case of the 1D explosions.\nThe regions where $\\mbox{${}^{26}{\\rm Al}$}$ is produced are those we identified as the C\/Ne explosive burning region,\nand the sub-explosive C burning region.\nFor our purposes, these regions were defined based on the $\\mbox{${}^{26}{\\rm Al}$}$ content. \nThe exact amount of $\\mbox{${}^{26}{\\rm Al}$}$ produced varied among the simulations, but the (radial) abundance \ndistribution of $\\mbox{${}^{26}{\\rm Al}$}$ in each simulation showed two distinct peaks that were at least one order \nof magnitude higher than the average $\\mbox{${}^{26}{\\rm Al}$}$ mass fraction. \nThus the $\\mbox{${}^{26}{\\rm Al}$}$-rich regions in the 1D simulations were defined to be at least one order of \nmagnitude higher in mass fraction than the average distribution.\nThe final isotopic composition of the solar nebula following injection of supernova material\nhas been calculated first assuming the material had the average (bulk) composition of the ejecta for comparison with \\citet{Goun07},\nand then that of one of these $\\mbox{${}^{26}{\\rm Al}$}$-rich regions.\nThe results are presented in Tables~\\ref{tb:1Da}- \\ref{tb:1Dc} and in Figure \\ref{fig:3isob}.\nOur results for injection of bulk ejecta from 1D explosions conforms closely to the findings \nof GM07 using the 1D models of \\citet{Raus02}.\nThe ejecta are generally very ${}^{16}{\\rm O}$-rich, with $\\dxvii$ and $\\mbox{$\\delta^{18}{\\rm O}$}$ that are \nlarge and negative.\n\n\\begin{deluxetable}{lrrrrr}\n \\tablewidth{0pt}\n \\tablecaption{Oxygen isotopic shifts following injection from a 1D supernova}\n \\tablehead{\n \\colhead{$\\,$} & \\colhead{16m-} & \\colhead{23m-} & \\colhead{40m-} & \\colhead{23e-} & \\colhead{23e-0.7-} \\\\\n \\colhead{$\\,$} & \\colhead{average} & \\colhead{average} & \\colhead{average} & \\colhead{average} & \\colhead{average} }\n \\startdata\n \\multicolumn{6}{c}{\\underline{Bulk}}\\\\\n \\\\\n \\mbox{$^{16}{\\rm O}$} & $1.97$ \\Msol & $1.48$ \\Msol & $3.29$ \\Msol & $2.93$ \\Msol & $5.44$ \\Msol \\\\\n \\mbox{$^{17}{\\rm O}$} & $2.24\\times10^{-4}$ \\Msol & $7.67\\times10^{-6}$ \\Msol & $1.80\\times10^{-4}$ \\Msol & $3.6\\times10^{-4}$ \\Msol & $2.94\\times10^{-4}$ \\Msol \\\\\n \\mbox{$^{18}{\\rm O}$} & $1.85\\times10^{-3}$ \\Msol & $1.53\\times10^{-5}$ \\Msol & $3.29\\times10^{-5}$ \\Msol & $6.63\\times10^{-5}$ \\Msol & $1.27\\times10^{-4}$ \\Msol \\\\\n \\mbox{${}^{26}{\\rm Al}$} & $3.86\\times10^{-6}$ \\Msol & $1.91\\times10^{-4}$ \\Msol & $1.52\\times10^{-5}$ \\Msol & $2.17\\times10^{-5}$ \\Msol & $2.15\\times10^{-5}$ \\Msol \\\\\n \\mbox{${}^{41}{\\rm Ca}$} & $4.06\\times10^{-6}$ \\Msol & $2.63\\times10^{-6}$ \\Msol & $1.21\\times10^{-5}$ \\Msol & $1.06\\times10^{-5}$ \\Msol & $1.49\\times10^{-4}$ \\Msol \\\\\n \\\\\n \\dxvii & $-719\\,\\promille$ & $-987\\,\\promille$ & $-899\\,\\promille$ & $-788\\,\\promille$ & $-854\\,\\promille$ \\\\\n \\mbox{$\\delta^{18}{\\rm O}$} & $-581\\,\\promille$ & $-996\\,\\promille$ & $-997\\,\\promille$ & $-993\\,\\promille$ & $-990\\,\\promille$ \\\\\n $x$ & 0.247 & 0.00375 & 0.105 & 0.0655 & 0.123 \\\\\n \\\\\n Final \\dxvii & $-191\\,\\promille$ & $-63.5\\,\\promille$ & $-139\\,\\promille$ & $-105\\,\\promille$ & $-147\\,\\promille$ \\\\\n Final \\mbox{$\\delta^{18}{\\rm O}$} & $-163\\,\\promille$ & $-63.5\\,\\promille$ & $-149\\,\\promille$ & $-117\\,\\promille$ & $-162\\,\\promille$ \\\\\n Final $\\Delta\\mbox{$^{17}{\\rm O}$}$ & $-118\\,\\promille$ & $-31.1\\,\\promille$ & $-65.6\\,\\promille$& $-45.1\\,\\promille$ & $-66.2\\,\\promille$ \\\\\n $\\Delta$t \t\t& 1.41 Myr\t\t& 0.65 Myr\t\t& 1.36 Myr\t\t& 1.31 Myr\t\t& 1.73 Myr \\\\\n\\enddata\n\\label{tb:1Da}\n\\end{deluxetable}\n\n\\begin{deluxetable}{lrrrrr}\n \\tablewidth{0pt}\n \\tablecaption{Oxygen isotopic shifts following injection from a 1D supernova}\n \\tablehead{\n \\colhead{$\\,$} & \\colhead{16m-} & \\colhead{23m-} & \\colhead{40m-} & \\colhead{23e-} & \\colhead{23e-0.7-} \\\\\n \\colhead{$\\,$} & \\colhead{average} & \\colhead{average} & \\colhead{average} & \\colhead{average} & \\colhead{average} }\n \\startdata\n \\multicolumn{6}{c}{\\underline{explosive C\/Ne burning}}\\\\\n \\\\\n \\mbox{$^{16}{\\rm O}$} & $5.19\\times10^{-1}$ \\Msol & $5.14\\times10^{-1}$ \\Msol & $4.83\\times10^{-1}$ \\Msol & $1.35$ \\Msol & $1.86$ \\Msol \\\\\n \\mbox{$^{17}{\\rm O}$} & $1.47\\times10^{-7}$ \\Msol & $2.89\\times10^{-7}$ \\Msol & $2.76\\times10^{-8}$ \\Msol & $8.79\\times10^{-7}$ \\Msol & $4.96\\times10^{-6}$ \\Msol \\\\\n \\mbox{$^{18}{\\rm O}$} & $4.41\\times10^{-7}$ \\Msol & $4.02\\times10^{-6}$ \\Msol & $1.70\\times10^{-8}$ \\Msol & $1.64\\times10^{-8}$ \\Msol & $3.21\\times10^{-8}$ \\Msol \\\\\n \\mbox{${}^{26}{\\rm Al}$} & $8.98\\times10^{-7}$ \\Msol & $1.04\\times10^{-4}$ \\Msol & $4.55\\times10^{-7}$ \\Msol & $9.37\\times10^{-6}$ \\Msol & $1.53\\times10^{-5}$ \\Msol \\\\\n \\mbox{${}^{41}{\\rm Ca}$} & $1.74\\times10^{-6}$ \\Msol & $2.74\\times10^{-6}$ \\Msol & $5.59\\times10^{-8}$ \\Msol & $6.70\\times10^{-6}$ \\Msol & $1.16\\times10^{-4}$ \\Msol \\\\\n \\\\\n \\dxvii & $-999\\,\\promille$ & $-998\\,\\promille$ & $-999\\,\\promille$ & $-997\\,\\promille$ & $-994\\,\\promille$ \\\\\n \\mbox{$\\delta^{18}{\\rm O}$} & $-1000\\,\\promille$ & $-996\\,\\promille$ & $-1000\\,\\promille$ & $-1000\\,\\promille$ & $-1000\\,\\promille$ \\\\\n $x$ & 0.280 & 0.00241 & 0.514 & 0.0698 & 0.0587 \\\\\n \\\\\n Final \\dxvii & $-266\\,\\promille$ & $-62.3\\,\\promille$ & $-379\\,\\promille$ & $-121\\,\\promille$ & $-112\\,\\promille$ \\\\\n Final \\mbox{$\\delta^{18}{\\rm O}$} & $-266\\,\\promille$ & $-62.2\\,\\promille$ & $-379\\,\\promille$ & $-121\\,\\promille$ & $-112\\,\\promille$ \\\\\n Final $\\Delta\\mbox{$^{17}{\\rm O}$}$ & $-147\\,\\promille$ & $-30.6\\,\\promille$ & $-227\\,\\promille$ & $-61.3\\,\\promille$ & $-56.1\\,\\promille$ \\\\\n $\\Delta$t \t\t& 1.51 Myr\t\t& 0.77 Myr\t\t& 1.03 Myr\t\t& 1.34 Myr\t\t& 1.75 Myr \\\\\n\\enddata\n\\label{tb:1Db}\n\\end{deluxetable}\n\n\n \\begin{deluxetable}{lrrrrr}\n \\tablewidth{0pt}\n \\tablecaption{Oxygen isotopic shifts following injection from a 1D supernova}\n \\tablehead{\n \\colhead{$\\,$} & \\colhead{16m-} & \\colhead{23m-} & \\colhead{40m-} & \\colhead{23e-} & \\colhead{23e-0.7-} \\\\\n \\colhead{$\\,$} & \\colhead{average} & \\colhead{average} & \\colhead{average} & \\colhead{average} & \\colhead{average} }\n \\startdata\n \\multicolumn{6}{c}{\\underline{sub-explosive C burning }}\\\\\n \\\\\n \\mbox{$^{16}{\\rm O}$} & $1.78\\times10^{-1}$ \\Msol & $3.85\\times10^{-1}$ \\Msol & $2.47$ \\Msol & $1.84$ \\Msol & $1.91$ \\Msol \\\\\n \\mbox{$^{17}{\\rm O}$} & $1.81\\times10^{-4}$ \\Msol & $1.03\\times10^{-6}$ \\Msol & $1.88\\times10^{-7}$ \\Msol & $4.18\\times10^{-5}$ \\Msol & $4.86\\times10^{-5}$ \\Msol \\\\\n \\mbox{$^{18}{\\rm O}$} & $3.12\\times10^{-4}$ \\Msol & $7.32\\times10^{-6}$ \\Msol & $6.51\\times10^{-8}$ \\Msol & $8.31\\times10^{-5}$ \\Msol & $9.14\\times10^{-5}$ \\Msol \\\\\n \\mbox{${}^{26}{\\rm Al}$} & $2.95\\times10^{-6}$ \\Msol & $6.77\\times10^{-5}$ \\Msol & $1.54\\times10^{-5}$ \\Msol & $1.42\\times10^{-5}$ \\Msol & $8.75\\times10^{-6}$ \\Msol \\\\\n \\mbox{${}^{41}{\\rm Ca}$} & $1.65\\times10^{-7}$ \\Msol & $1.35\\times10^{-6}$ \\Msol & $8.42\\times10^{-6}$ \\Msol & $7.78\\times10^{-6}$ \\Msol & $8.44\\times10^{-6}$ \\Msol \\\\\n \\\\\n \\dxvii & $1503\\,\\promille$ & $-990\\,\\promille$ & $-1000\\,\\promille$ & $-944\\,\\promille$ & $-938\\,\\promille$ \\\\\n \\mbox{$\\delta^{18}{\\rm O}$} & $-221\\,\\promille$ & $-995\\,\\promille$ & $-1000\\,\\promille$ & $-980\\,\\promille$ & $-979\\,\\promille$ \\\\\n $x$ & 0.0292 & 0.00276 & 0.0775 & 0.0628 & 0.106 \\\\\n \\\\\n Final \\dxvii & $-15.7\\,\\promille$ & $-62.6\\,\\promille$ & $-128\\,\\promille$ & $-112\\,\\promille$ & $-144\\,\\promille$ \\\\\n Final \\mbox{$\\delta^{18}{\\rm O}$} & $-64.6\\,\\promille$ & $-62.6\\,\\promille$ & $-128\\,\\promille$ & $-114\\,\\promille$ & $-148\\,\\promille$ \\\\\n Final $\\Delta\\mbox{$^{17}{\\rm O}$}$ & $+19.2\\,\\promille$ & $-30.7\\,\\promille$ & $-64.9\\,\\promille$ & $-55.3\\,\\promille$ & $-71.4\\,\\promille$ \\\\\n $\\Delta$t \t\t& 0.90 Myr\t\t& 0.72 Myr\t\t& 1.29 Myr\t\t& 1.29 Myr\t\t& 1.39 Myr \\\\\n\\enddata\n\\label{tb:1Dc}\n\\end{deluxetable}\n\nGM07 likewise found, using the 1D models of \\citet{Raus02}, that the ejecta are depleted \nin $\\mbox{$^{17}{\\rm O}$}$, and in most cases $\\mbox{$^{18}{\\rm O}$}$ as well. (Their $15 \\, M_{\\odot}$\ncase is enriched in $\\mbox{$^{18}{\\rm O}$}$, in contrast to our $16 \\, M_{\\odot}$ case).\nSimilarly to our $23 \\Msol$ models, both the $21 \\Msol$ and the \n$25 \\Msol$ model of \\citet{Raus02} are very depleted in \\mbox{$^{17}{\\rm O}$}~ and \\mbox{$^{18}{\\rm O}$}, and the composition of \nthose ejecta approach $-1000\\, \\promille$ for both \\dxvii~ and \\mbox{$\\delta^{18}{\\rm O}$}, although the \\citet{Raus02}\nmodels tend to be slightly richer in \\mbox{$^{17}{\\rm O}$}~ and \\mbox{$^{18}{\\rm O}$}~ than ours. The similarity between the \\citet{Raus02} \nbulk abundances and those calculated for this work are unsurprising. Post-He burning stages rapidly \ndestroy $\\mbox{$^{17}{\\rm O}$}$ and $\\mbox{$^{18}{\\rm O}$}$, resulting in the oxygen in interior zones being nearly pure $\\mbox{$^{16}{\\rm O}$}$. \nA 20-25 \\Msol model has a very large oxygen mantle. The slightly higher fraction of heavy isotopes \nin the \\citet{Raus02} arise in a somewhat larger He shell that results from a less accurate treatment \nof mixing in their earlier stellar models. \n\nThese differences reflect the variability inherent in calculations of nucleosynthesis in \nmassive stars, especially where small shifts in stable isotopes are concerned.\nWe also find, as did GM07, that the isotopic shifts associated with injection from ejecta from \n1D supernova explosions {\\it tend} to be large (tens of permil), but not in all cases.\nIn the 23m runs, $\\mbox{${}^{26}{\\rm Al}$}$ is produced more abundantly, and the ${}^{26}{\\rm Al} \/ {}^{16}{\\rm O}$\nratios yield $x < 10^{-2}$ in these explosions.\nIn the 23m cases, the ejecta are particularly ${}^{16}{\\rm O}$-rich, but relatively less oxygen\nneeds to be injected per $\\mbox{${}^{26}{\\rm Al}$}$ because more $\\mbox{${}^{26}{\\rm Al}$}$ is produced. (It should be remembered that 23m \nis a binary case, where a significant fraction of the $\\mbox{$^{17}{\\rm O}$}$ and $\\mbox{$^{18}{\\rm O}$}$ have been removed by \nmass loss from the He shell, and production of $\\mbox{${}^{26}{\\rm Al}$}$ has been enhanced by higher peak shock \ntemperatures relative to the 23 $\\Msol$ single star models.)\nThe isotopic shifts associated with injection from 23m 1D explosions are typically\n$< 3\\,\\promille$.\n\nFor all the cases the shifts in both $\\dxvii$ and $\\mbox{$\\delta^{18}{\\rm O}$}$ are negative and similar in magnitude; that is, \nthe injection of material from this supernova moves the composition down\nthe slope-1 line.\nLater nebula evolution would produce materials that move back up this line. The most favorable case \nis the 23m supernova.\nThe magnitude and direction of the isotopic shifts associated with the 23m case\nare such that current measurements of solar nebula materials do not rule out this possibility, even \nfor bulk abundances. As we argue below, however, bulk abundances are not the best representation\n of the abundances of injected material.\n\n\nTables~\\ref{tb:1Da}, \\ref{tb:1Db}, and~\\ref{tb:1Dc} and Figure \\ref{fig:3isob} \nalso show the isotopic shifts associated with \ninjection from only $\\mbox{${}^{26}{\\rm Al}$}$-rich regions within the supernova.\nEven when considering injection of $\\mbox{${}^{26}{\\rm Al}$}$-rich regions only, the conclusions are not much changed:\nthe isotopic shifts in oxygen generally are many tens of permil, and make the solar nebula more\n${}^{16}{\\rm O}$-rich. The sub-explosive C burning region of the 16m model is the only case that does \nnot move the nebular composition along the slope 1 line. In general the sub-explosive C burning in the \nhigher mass models provide the best results, as they produce the highest ratio of $\\mbox{${}^{26}{\\rm Al}$}$ to oxygen. \n\n\nTables~\\ref{tb:1Da}, \\ref{tb:1Db}, and~\\ref{tb:1Dc} also give the yields of $\\mbox{${}^{41}{\\rm Ca}$}$ produced in\neach of the 1D explosion scenarios.\nThe post-injection ${}^{41}{\\rm Ca} \/ {}^{40}{\\rm Ca}$ ratio is generally more than sufficient \nto match the meteoritic ratio, and a time delay is implied before isotopic closure, so that \n$\\mbox{${}^{41}{\\rm Ca}$}$ can decay. \nFor the ejecta from the $23 \\, M_{\\odot}$ progenitors, the implied time delay (for $\\mbox{${}^{41}{\\rm Ca}$}$ to \ndecay to a level ${}^{41}{\\rm Ca} \/ {}^{40}{\\rm Ca} = 1.4 \\times 10^{-8}$) for the 23m cases is\n$\\sim$ 0.7 Myr for all three regions considered (i.e. bulk, explosive C\/Ne burning, and sub-explosive \nC burning).\nThe implied time delay for injection from the 23e cases is $\\sim$ 1.3 Myr for all three regions, \nand from the 23e-0.7 cases is 1.4 Myr (for the sub-explosive C burning region) -- 1.7 Myr (for the other \ntwo regions). The time delays for the \nother progenitor cases are all within those ranges. The range of these time delays are very similar to \nthe range of 1.0 -- 1.8 Myr calculated by \\citet{Goun07}.\nThe effect of this time delay is to cause $\\mbox{${}^{26}{\\rm Al}$}$ to decay, too, before isotopic closure, and \nto increase the isotopic shifts in oxygen.\nThe shifts are increased by factors of 2 (for the 23m) at the low end to 5.2 (for the 23e-0.7 cases) \nat the high end.\nIf isotopic closure is to be achieved in a few $\\times (10^{5} - 10^6) \\, {\\rm yr}$ \\citep{MacPD95, KitaH05},\nthen injection from the 23e and 23e-0.7 would seem to introduce too much \n$\\mbox{${}^{41}{\\rm Ca}$}$ to match constraints.\nInjection from the more energetic 23m progenitor cases are consistent with a small shift in oxygen isotopes downward along the slope-1 line,\nas well as the final ${}^{41}{\\rm Ca} \/ {}^{40}{\\rm Ca}$ ratio of the solar nebula. \n\n\n\\subsection{Asymmetric Supernova Explosions}\n\nIn Table~\\ref{tb:3D} and Figure \\ref{fig:3isob}, \nwe present the yields of $\\mbox{${}^{26}{\\rm Al}$}$, $\\mbox{${}^{41}{\\rm Ca}$}$, and oxygen isotopes in various \nregions of the ejecta in our simulation of the 3D explosion.\nWe calculate the isotopic shifts if the injection uniformly samples all of the ejecta (bulk),\nif it samples the Ring material, and if it samples the Bubble material.\nBy design, membership in the Bubble and Ring material is defined by high $\\mbox{${}^{26}{\\rm Al}$}$ content.\nThese $\\mbox{${}^{26}{\\rm Al}$}$-rich regions do not have well-defined edges, instead fading out monotonically\nin $\\mbox{${}^{26}{\\rm Al}$}$-abundance as one moves out into the surrounding ejecta (see Figure \\ref{fig:al}).\nIn order to not impose an arbitrary geometry on these regions we determined\nmembership by $\\mbox{${}^{26}{\\rm Al}$}$ amount per SNSPH particle. \nWe used two different lower limits or thresholds for inclusion -- $1.5 \\times 10^{-13} \\, \\Msol$ \nof $\\mbox{${}^{26}{\\rm Al}$}$ per particle for a maximum extent of the \\mbox{${}^{26}{\\rm Al}$}\\, rich region, and \n$1.5 \\times 10^{-11} \\, \\Msol$ per particle for a minimum extent of the $\\mbox{${}^{26}{\\rm Al}$}$-rich region \n(``high \\mbox{${}^{26}{\\rm Al}$}\" case).\nThe most $\\mbox{${}^{26}{\\rm Al}$}$-rich SPH particles in the Ring and Bubble had $1.5 \\times 10^{-10} \\, \\Msol$\nand $4.8 \\times 10^{-10} \\, \\Msol$ of $\\mbox{${}^{26}{\\rm Al}$}$, respectively. \nEach threshold picked out {\\it all} SPH particles in the respective regions that it identified.\n\n\\begin{deluxetable}{lrrrrr} \n \\tablewidth{0pt}\n \\tablecaption{\\mbox{${}^{26}{\\rm Al}$}\\, and O in the 3D explosion \\label{tb:3D}} \n \\tablehead{\n \\colhead{$\\,$} & \\colhead{Bulk} & \\colhead{Ring} & \\colhead{Ring} & \\colhead{Bubble} & \\colhead{Bubble} }\n \\startdata \n & & & high Al26 & & \nhigh Al26 \\\\\n \\mbox{${}^{26}{\\rm Al}$}\\,(\\Msol) & $1.474\\times10^{-6}$ & $3.510\\times10^{-7}$ & $2.420\\times10^{-7}$ & $9.996\\times10^{-7}$ & $9.585\\times10^{-7}$ \\\\\n \\mbox{$^{16}{\\rm O}$}\\,($\\Msol$) & 0.511 & $2.888\\times10^{-2}$ & $5.949\\times10^{-3}$ & $7.025\\times10^{-3}$ & $2.550\\times10^{-3}$ \\\\\n \\mbox{$^{17}{\\rm O}$}\\,($\\Msol$) & $2.170\\times10^{-4}$ & $3.400\\times10^{-5}$ & $2.241\\times10^{-5}$ & $1.021\\times10^{-5}$ & $1.738\\times10^{-9}$ \\\\ \n \\mbox{$^{18}{\\rm O}$}\\,($\\Msol$) & $1.509\\times10^{-1}$ & $3.553\\times10^{-3}$ & $1.018\\times10^{-4}$ & $7.314\\times10^{-3}$ & $5.635\\times10^{-3}$ \\\\ \n \\mbox{${}^{41}{\\rm Ca}$}\\,(\\Msol) & $2.132\\times10^{-8}$ & $2.039\\times10^{-8}$ & $4.312\\times10^{-9}$ & $1.827\\times10^{-11}$ & $1.239\\times10^{-11}$ \\\\\n \\\\\n $\\delta\\mbox{$^{17}{\\rm O}$}$ & $+43.9\\,\\promille$ & $+1894\\,\\promille$ & $+8258\\,\\promille$ & $+2573\\,\\promille$ & $-998.3\\,\\promille$ \\\\\n $\\delta\\mbox{$^{18}{\\rm O}$}$ & $+129887\\,\\promille$ & $+53546\\,\\promille$ & $+6582\\,\\promille$ & $+460545\\,\\promille$ & $+978646\\,\\promille$ \\\\ \n $x$ & 0.16808 & 0.03987 & 0.01192 & 0.003406 & 0\n.001289 \\\\\n \\\\\n Final $\\delta\\mbox{$^{17}{\\rm O}$}$ & $-45.1\\,\\promille$ & $+14.8\\,\\promille$ & $+37.9\\,\\promille$ & $-51.1\\,\\promille$ & $-61.2\\,\\promille$ \\\\\n Final $\\delta\\mbox{$^{18}{\\rm O}$}$& $+18638\\,\\promille$ & $+1995\\,\\promille$ & $+18.2\\,\\promille$ & $+1503\\,\\promille$ & $+1200\\,\\promille$ \\\\\n Final $\\Delta\\mbox{$^{17}{\\rm O}$}$ & $-1608\\,\\promille$ & $-561\\,\\promille$ & $+27.8\\,\\promille$ & $-534\\,\\promille$ & $-477\\,\\promille$ \\\\\n $\\Delta$t \t\t& 0.66 Myr\t\t& 0.90 Myr\t\t& 0.70 Myr\t\t& --\t\t& -- \\\\\n\n\\enddata\n\\end{deluxetable}\n\nOverall, the ejecta of the 3D simulation are much richer in \\mbox{$^{17}{\\rm O}$}\\, and \\mbox{$^{18}{\\rm O}$}\\, than the 1D\nsimulations, but also contain two regions (the Ring and the Bubble) in which the $\\mbox{${}^{26}{\\rm Al}$}$\nproduction is increased over the 1D calculations.\nAs we have previously discussed, the production of \\mbox{$^{18}{\\rm O}$}\\, is significantly altered\nfrom the 1D results. \nThe added yield from the decay of \\mbox{$^{18}{\\rm F}$}\\, to \\mbox{$^{18}{\\rm O}$}\\, makes the ejecta significantly richer \nin this isotope, and results in large (tens of permil) to very large\n(hundreds of permil) and positive shifts in \\mbox{$\\delta^{18}{\\rm O}$}\\, for material from both the Ring and\nBubble, and the bulk. \nThis is in stark contrast to the \\mbox{$^{18}{\\rm O}$}\\, poor ejecta produced in the 1D simulations, and emphasizes \nthe sensitive dependence on the prevailing thermodynamic conditions of \\mbox{$^{18}{\\rm O}$}\\, production. \nThe production of \\mbox{$^{17}{\\rm O}$}\\, is much less sensitive to the thermodynamic conditions.\nIn the 3D simulation we also see an increase over the 1D cases in the production of \\mbox{$^{17}{\\rm O}$}\\, in the \n\\mbox{${}^{26}{\\rm Al}$}\\, rich regions and the bulk; however the change is not as drastic as in \\mbox{$^{18}{\\rm O}$}. \nThis again differs from the 1D calculations, and the more $\\mbox{$^{17}{\\rm O}$}$-rich ejecta result in positive \nshifts in \\dxvii, on the order of $-1 \\, \\rm{to }+15\\,\\promille$ for the Bubble and bulk,\nand close to $+100\\,\\promille$ for the Ring.\n\nTable~\\ref{tb:3D} also shows the ${}^{41}{\\rm Ca} \/ {}^{40}{\\rm Ca}$ ratio following injection\nof material from the 3D supernova into the solar nebula.\nIf injection comes from the Bubble region only, the amount of $\\mbox{${}^{41}{\\rm Ca}$}$ injected is too low to \nconform to meteoritic ratios, and injection from the Bubble can be ruled out on these grounds.\nInjection of material from the Ring or bulk regions, in contrast, imply reasonable time delays \n$\\approx 0.66 - 0.90 \\, {\\rm Myr}$. \nThis implies an increase in oxygen isotopic shifts of $< 2.5$ over what is presented in Table~\\ref{tb:3D}. \nThese time delays are just below the ones \\citet{Goun07} calculate, which again is explained by \nthe faster density- drop off in the 3D calculation producing slightly less \\mbox{${}^{41}{\\rm Ca}$}~ than in 1D.\n\n\nWhen an explosion samples a variety of thermodynamic trajectories through asymmetry, including those \nthat result in freeze-out conditions due to rapid expansion, the overriding conclusion to be derived is that a very large range in \noxygen isotopic shifts is allowed. \nIt would seem extremely unlikely that conditions in an asymmetric explosion would conspire to \nyield a small isotopic shift consistent with the meteoritic constraints, though more``normal\" trajectories that \ndo not experience this freeze-out process are still candidate production sites, as wee see in 1D. \n\n\n\\section{Discussion} \n\nAs \\citet{NichP99} strongly advocated, injection of supernova ejecta can produce measurable\n``collateral damage\" to stable isotope systems in protoplanetary disks.\nGM07 in particular point to the role of oxygen isotopes in constraining this process.\nThe point of that paper was that the injection of $\\mbox{${}^{26}{\\rm Al}$}$ (and $\\mbox{${}^{41}{\\rm Ca}$}$) from a single nearby \nsupernova necessarily would have brought in significant levels of oxygen isotopically distinct\nfrom the pre-injection solar nebula.\nThe solar nebula after injection, they argued, would differ in its oxygen isotopes\nby several tens of permil from the pre-injection values, which they robustly predicted\nwould be more ${}^{17}{\\rm O}$-rich than the solar nebula.\nThey cited the {\\it Genesis} measurements of solar wind oxygen as those most likely to\nsample the pre-injection solar nebula.\nSince preliminary results from {\\it Genesis} \\citep{Mcke09, }(McKeegan et al.\\ 2009, 2010) \nare revealing the Sun to be ${}^{16}{\\rm O}$-rich, GM07 would rule out injection\nof $\\mbox{${}^{26}{\\rm Al}$}$ and $\\mbox{${}^{41}{\\rm Ca}$}$ from a single supernova. \n\nIn this paper, we attempted to reproduce the calculations of GM07, to apply\ntheir method of using oxygen isotopes to test the supernvoa injection hypothesis.\nWe made necessary corrections to their method, mostly in regard to the presumed\noxygen isotopic composition of the (post-injection) solar nebula.\nGM07 assumed this was identical to SMOW, meaning the pre-injection \nsolar nebula had to be more $\\mbox{$^{17}{\\rm O}$}$-rich than almost any known inclusions.\nWe presented considerable evidence that the post-injection composition was in fact \nmuch more ${}^{16}{\\rm O}$-rich than that, closer to $(-60\\, \\promille,-60\\, \\promille)$. \nWe carried out stellar nucleosynthesis calculations, to calculate the isotopic yields\nof $\\mbox{${}^{26}{\\rm Al}$}$, $\\mbox{${}^{41}{\\rm Ca}$}$ and oxygen isotopes in a variety of supernova explosion scenarios, \nincluding the 1D (spherically symmetric) cases \nas well as 3D (asymmetric) explosions. \nBecause $\\mbox{${}^{26}{\\rm Al}$}$ and $\\mbox{${}^{41}{\\rm Ca}$}$ are observed to be correlated \\citep{SahiG98}, \nwe also simultaneously considered injection of $\\mbox{${}^{41}{\\rm Ca}$}$ into the solar nebula. \nWe then computed the shifts in oxygen isotopes and the final ${}^{41}{\\rm Ca} \/ {}^{40}{\\rm Ca}$\nratio in the solar nebula following injection of sufficient supernova material to produce\nthe meteoritic ratio ${}^{26}{\\rm Al} \/ {}^{27}{\\rm Al} = 5 \\times 10^{-5}$. \n\nOur 1D simulations largely confirm the results of GM07, that isotopic shifts are likely\nto be tens of permil and to make the solar nebula more ${}^{16}{\\rm O}$-rich than before\nthe injection.\nWe found that injection of material from either the bulk or the explosive C\/Ne burning and sub-explosive C burning\nregions of supernovae moved the composition of the solar nebula down the slope 1 line. Our \n$23 \\Msol$ progenitors led to isotopic shifts in oxygen\n which moved the composition of the solar nebula down the slope-1 line, with the less energetic \nexplosions producing larger shifts and time delays.\nThe 23m progenitors, which were the most energetic of the $23 \\Msol$ cases and especially effective in\nproducing $\\mbox{${}^{26}{\\rm Al}$}$, generated shifts that amounted to only $< 6\\, \\promille$, including a time delay of \n0.7 Myr for $\\mbox{${}^{41}{\\rm Ca}$}$ to decay to its meteoritic value. \nThis scenario, at least, is consistent with all the applied meteoritic constraints. If less than 100\\% of the\n oxygen penetrated the solar nebula material due to, for example, dust condensation, all but one of the \n 1D cases are consistent with the evidence from the early solar system.\n\nWe note that this conclusion differs from what GM07 infer for injection of bulk ejecta from\n21 and $25 \\, M_{\\odot}$ progenitors. \nGM07 likewise found isotopic shifts downward along the slope-1 line, but with a magnitude\nof 40 to 50 permil.\nIt is worth noting that had GM07 assumed the same starting composition for the solar nebula,\n$(-60\\, \\promille,-60\\, \\promille)$, \nthat we do, \nthen they would have found the solar nebula oxygen isotopic composition to be \n$(-82\\, \\promille,-82\\, \\promille)$ after injection of supernova material from an\n$25 \\, M_{\\odot}$ progenitor, and $(-81\\, \\promille,-74\\, \\promille)$ after injection of supernova material from an\n$21 \\, M_{\\odot}$ progenitor. Although these shifts are moderately large, they are {\\it down} the slope-1 line.\nAs we established in \\S \\ref{magofshifts}, this would not have been {\\it in}compatible with the meteoritic constraints, as some \nvery ${}^{16}{\\rm O}$-rich meteoritic samples in this range are known, including CAIs in \nIsheyevo, at $\\approx (-68\\,\\promille,-66\\,\\promille)$ \\citep{GounK09},\nand a ferromagnesian cryptocrystalline chondrule in the CH chondrite Acfer 214,\nat $\\approx (-75\\,\\promille,-75\\,\\promille)$ \\citep{KobaI03}.\nSubsequently the mass-independent fractionation process would have shifted the nebula upward\nalong the slope-1 line, erasing this isotopic shift and eventually producing the composition\n$(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$}) \\approx (-40\\,\\promille,-40\\,\\promille)$ common to most CAIs\n\\citep[e.g.][]{ItohK04}.\nWe conclude that the supernova injection hypothesis cannot be rejected based\non 1D models. \n\nOur investigation of other parameters suggest that it is even more difficult to be conclusive\nabout supernova injection.\nWe have considered a small number of progenitor masses undergoing spherically symmetric collapse;\nin a few cases we varied other parameters, such as varying the explosion energy, or allowing for loss of a\nhydrogen envelope in a binary scenario, or allowing an asymmetrical explosion.\nIn most of these cases the isotopic shifts in oxygen were large.\nAmong the cases considered here, the final $\\dxvii$ values in the solar nebula varied from\n$-379\\,\\promille$ to $+15\\,\\promille$, and the final $\\mbox{$\\delta^{18}{\\rm O}$}$ values varied from\n$-379\\,\\promille$ to $+18000\\,\\promille$.\nAs GM07 found, most of the cases where meteoritic abundances of \\mbox{${}^{26}{\\rm Al}$}\\, are injected lead to \nlarge ($>10\\,\\promille$) shifts in oxygen isotopes. \nWe also considered the yields in a 3D anisotropic explosion of a $23 \\, M_{\\odot}$\nprogenitor, in the bulk ejecta and two $\\mbox{${}^{26}{\\rm Al}$}$-rich zones analogous to those in the 1D\nexplosions.\nWe find that a wide range of outcomes is possible, with oxygen isotopic shifts as large\nas hundreds of permil, or as low as $< 3\\,\\promille$.\nThe fact that ${}^{18}{\\rm F}$ can decay to ${}^{14}{\\rm N}$ instead of ${}^{18}{\\rm O}$\nat high temperatures makes the yield of ${}^{18}{\\rm O}$ especially sensitive to the\nthermodynamic trajectory of the ejecta, which partially accounts for the spread in the\n$\\mbox{$^{18}{\\rm O}$}$ yields.\nOn the one hand, the wide range of possible outcomes makes it nearly impossible to state \nconclusively that all supernova injection scenarios can be ruled out.\nOn the other hand, the wide range of possible outcomes seems to imply a degree of fine\ntuning so that the oxygen isotopic shifts in the solar nebula were not large, especially\nfor the 3D case.\n\nWe conclude that the hypothesis, that the $\\mbox{${}^{26}{\\rm Al}$}$ in the solar nebula was due to supernova \nmaterial injected into the Sun's protoplanetary disk, can still be made compatible with\nmeteoritic constraints, under two scenarios.\nThe first is that the injected supernova material came from either the bulk ejecta, or\nfrom \na region in a supernova that experienced thermodynamic conditions like the \nsubexplosive C burning zone. The latter is more physically likely.\nWith a $0.7$ Myr time delay, the injection would have moved the solar nebula oxygen\nisotopic composition from \n$(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$}) \\approx (-60\\, \\promille,-60\\, \\promille)$ to a more $\\mbox{$^{16}{\\rm O}$}$-rich value along the slope one \nline. All but one of our explosions produce movement along the slope 1 line. We produce shifts as small as \n$\\approx (-63\\, \\promille,-63\\, \\promille)$, which would have produced an accompanying meteoritic ratio \n${}^{41}{\\rm Ca} \/ {}^{40}{\\rm Ca} = 1.4 \\times 10^{-8}$.\nSubsequent mixing of rocky material with a ${}^{16}{\\rm O}$-depleted reservoir would have\nthen moved the composition of meteoritic inclusions up the slope-1 line to values \n$(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$}) \\approx (-50\\, \\promille,-50\\, \\promille)$, consistent with primitive\nCAIs, and further up the slope-1 line with time.\n\nThe second scenario is one in which only dust grains are injected in to the protoplanetary\ndisk, and very little of the supernova oxygen condenses into dust grains. \nIf only the most refractory grains such as corundum were injected, then potentially\n$x \\ll 10^{-5}$, and the isotopic shifts would be negligible ($\\ll 1\\, \\promille$),\nfor nearly all the cases considered here. \nIt is worth noting that Ca is equally refractory to Al and is likely to condense from\nsupernova ejecta under the same conditions that Al condenses, so the meteoritic abundance\nof $\\mbox{${}^{41}{\\rm Ca}$}$ could still be matched following injection of $\\mbox{${}^{26}{\\rm Al}$}$.\n\\citet{Ouel07} have calculated that only $1\\%$ of gas-phase ejecta are injected \ninto a disk.\nIf almost all of the Ca and Al in the ejecta are locked up in large grains \n(radii $> 0.1 \\, \\mu{\\rm m}$) that are efficiently injected \\citep{OuelDH10}, but\nless than a few percent of the oxygen is, then \npotentially all of the isotopic shifts in oxygen calculated here should be reduced by a \nfactor of about 100. \nEssentially all of the 1D cases considered here would then conform with the meteoritic \nconstraints, and even some of the 3D cases as well. \n\n\nTo summarize, we agree with GM07 that oxygen isotopes can be a powerful constraint on \nsupernova injection models.\nOur calculations of oxygen isotopic shifts following injection from the bulk ejecta\nof 1D supernovae broadly match the results of GM07.\nHad GM07 assumed the same starting composition of the solar nebula that we did, and\nconsidered a smaller time delay between injection and isotopic closure, they would\nhave found isotopic shifts for $20 - 25 \\, M_{\\odot}$ progenitors that would not\nbe inconsistent with meteoritic constraints.\nOur own calculations of the same case predict shifts that are similar, although smaller\nin magnitude, and which are also consistent with meteoritic constraints. \nThe existence of an example that is consistent with the oxygen isotopic composition and\nthe ${}^{41}{\\rm Ca} \/ {}^{40}{\\rm Ca}$ ratio of the solar nebula means that the \nsupernova injection hypothesis cannot be ruled out. \nBecause the nucleosynthesis of oxygen differs in asymmetric explosions, a much wider\nrange of oxygen isotopes is possible in 3D explosions. \nBecause of the contingent nature of the injection it becomes difficult to make any statement \nabout the possibility that the solar nebula acquired\n$\\mbox{${}^{26}{\\rm Al}$}$ from such an asymmetric explosion.\nFinally, all oxygen isotopic shifts are reduced if only large grains are injected\ninto the protoplanetary disk, and only a small fraction of oxygen condenses into\nlarge grains.\nQuantifying the fractionation of Al and O during injection into a protoplanetary disk\nis the focus of ongoing work by this research group.\nIf only a few percent of the total oxygen is injected, then nearly all the 1D explosions \nconsidered here could be consistent with the meteoritic constraints on oxygen isotopes\nand $\\mbox{${}^{41}{\\rm Ca}$}$ abundances. \nWe therefore conclude it is premature to rule out the supernova injection hypothesis\nbased on oxygen isotopes. \n\n\n\n\n\n\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}