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+{"text":"\\section{Introduction and statement of main results}\n\n\n\\subsection{Notation} Let $\\Phi$ be an irreducible, reduced root\nsystem of rank $n$. We write $Y$, $X$, $X^{\\vee}$, and $Y^{\\vee}$ for\nthe root, weight, coroot, and coweight lattice, respectively. Note\nthat $Y \\subseteq X$ and $X^{\\vee} \\subseteq Y^{\\vee}$. We write\n$\\Delta$ for the set of roots of $\\Phi$.\n\nDenote by $W$ the Weyl group of $\\Phi$. Let $F$ be the Weyl fan in\n$Y^{\\vee} \\otimes_{{\\mathbb Z}} \\mathbb{R}$ and $F_n \\subset F$ the set of chambers\n(i.e., cones of maximal dimension) in $F$. The elements of $F_n$ are\nthe Weyl chambers cut out by the root hyperplanes of $\\Phi$.\n\nWe study the complex toric variety $V$, whose fan is $F$ and \ninitial lattice is $Y^{\\vee}$. The toric variety $V$ has been studied\nby many authors, e.g., \\cite{procesi}, \\cite{klyachko},\n\\cite{dabrowski-normality}, \\cite{carrell-kurth},\n\\cite{carrell-kuttler}, \\cite{qendrim2}, and \\cite{qendrim}. It is a\nsmooth, projective toric variety for the torus $T_1= \\operatorname{Spec}(\\mathbb{C}[Y])\n\\simeq(\\mathbb{C}^{\\times})^n$. \nIt is a well known fact (although we do not use it) that those toric\nvarieties are closures of generic torus orbits in the flag variety\n$G\/B$, where $G$ is the reductive group associated to $\\Phi$ and $B\n\\subseteq G$ is a Borel subgroup.\n\nSince $T_1$ acts on $V$, the torus $T= \\operatorname{Spec}(\\mathbb{C}[X])$ also acts on $V$\nvia the canonical projection $T \\twoheadrightarrow T_1$. Let\n$\\mathcal{L}$ be a $T$-equivariant ample line bundle on $V$. Such line\nbundles (or, more precisely, the isomorphism classes thereof) are in\none-to-one correspondence (see, e.g., \\cite{fulton}) with convex\npolytopes $P \\subset X {\\otimes}_{{\\mathbb Z}} \\mathbb{R}$ satisfying the following\nproperty: The vertices of $P$ are given by a set $\\{ \\mu_{\\sigma} :\n\\sigma \\in F_n\\} \\subset X$, and for any two vertices $\\mu_{\\sigma}$\nand $\\mu_{\\sigma '}$ of $P$, where $\\sigma$ and $\\sigma'$ are adjacent\nchambers, $\\mu_{\\sigma} - \\mu_{\\sigma'} = r_{\\sigma, \\sigma'}\n\\, \\alpha_{\\sigma,\\sigma'}$, for some number $r_{\\sigma, \\sigma'} \\in\n{\\mathbb Z}_{> 0}$, where $\\alpha_{\\sigma,\\sigma'} \\in \\Delta$ is the unique\nroot that is positive on $\\sigma$ and negative on $\\sigma'$. Such\npolytopes are called ``ample.'' (In, e.g., \\cite{arthur}, the \nsets $\\{\\mu_\\sigma\\}$ are called\n``strictly positive orthogonal sets'' in this case.)\n\n\nWe denote by $\\Lambda(P)$ the set of points $x \\in P \\cap X$ whose\nimage in $X\/Y$ coincides with the images in $X\/Y$ of the vertices of\n$P$, i.e., $\\Lambda(P) = P \\cap \\{y+\\mu_\\sigma \\mid y \\in Y\\}$ for any\nchoice of Weyl chamber $\\sigma$. Note that the character $x \\in X$\noccurs in $H^0(X, \\mathcal{L})$ if and only if $x \\in \\Lambda(P)$,\nwhere $P$ is the polytope corresponding to $\\mathcal{L}$ (and then it\noccurs with multiplicity one): see, e.g., \\cite[\\S 23.1,\np.~496]{akot}.\n\n\nTo every chamber $\\sigma \\in F_n$ there corresponds a basis\n$\\{\\alpha_{i, \\sigma}: i \\in I\\} \\subseteq \\Delta$ of $Y$ consisting\nof elements of $\\Delta$, where $I := \\{ 1, \\ldots, n\\}$ (in other\nwords, a choice of simple roots). We say that an element\n$x \\in X \\otimes_{{\\mathbb Z}} \\mathbb{R}$ is $\\sigma$-dominant if $\\langle x,\n\\alpha_{i, \\sigma}^{\\vee} \\rangle \\geq 0$, $\\forall i \\in I$. Here\n$\\langle \\, , \\rangle $ is the usual bilinear pairing $X \\times\nX^{\\vee} \\rightarrow {\\mathbb Z}$, extended to $(X \\otimes_{{\\mathbb Z}} \\mathbb{R}) \\times\nX^{\\vee} \\rightarrow \\mathbb{R}$, and $\\alpha_{i, \\sigma}^{\\vee}$ is the\ncoroot in $\\Phi$ corresponding to $\\alpha_{i, \\sigma}$.\n\nWe impose a restriction on the type of polytopes $P$ that we consider:\n\n\\begin{itemize} \\item[$(\\dagger)$] For every $\\sigma \\in F_n$, the\n  element $\\mu_{\\sigma}$ is $\\sigma$-dominant.\n\\end{itemize}\n\nFollowing Kottwitz (\\emph{op.~cit.}, \\S 12.9, p.~44), we call ample\npolytopes satisfying the property $(\\dagger)$\n\\emph{special}.\\footnote{More generally, Kottwitz defines\n  \\emph{special orthogonal sets}, where an orthogonal set is a\n  collection $\\{\\mu_\\sigma\\}$ where $\\mu_\\sigma - \\mu_\\sigma' =\n  r_{\\sigma, \\sigma'} \\, \\alpha_{\\sigma,\\sigma'}$ for $r_{\\sigma,\n    \\sigma'} \\in {\\mathbb Z}$, not necessarily positive. A special orthogonal\n  set is then one satisfying $(\\dagger)$. They necessarily satisfy\n  $r_{\\sigma, \\sigma'} \\geq 0$ for all adjacent $\\sigma, \\sigma'$\n  (i.e., they are ``positive orthogonal sets''), but the $r_{\\sigma,\n    \\sigma'}$ need not be positive (i.e., $\\{\\mu_\\sigma\\}$ need not be\n  strictly positive, as in the ample case). The associated divisors\n  are in particular a nonnegative linear combination of prime\n  $T$-invariant divisors.}  In what follows we will primarily be\ninterested in special ample polytopes.  Note that the Weil divisors of\nsuch ample polytopes in particular must have strictly positive\ncoefficients of all prime $T$-invariant divisors (but this condition\ndoes not imply speciality).\n\n\n\\subsection{Statement of main results}\n\nOur first main result is the following, which will be proved in \\S\n\\ref{s:main-proof}:\n\n\\begin{thm}\\label{main}\n  Let $P$ be a special ample polytope as above and let $m \\in\n  {\\mathbb Z}_{>0}$. Consider the dilated polytope $m P:=\\{mx : x \\in P\n  \\}$. Then any point $z \\in \\Lambda(mP)$ can be written as a sum\n  $z=z_1+ \\cdots + z_m$, with $z_i \\in \\Lambda(P)$, $\\forall i\n  =1,\\ldots, m$.\n\\end{thm}\n\nThe toric interpretation of the theorem is as follows.  Call an\nequivariant line bundle $\\mathcal{L}$ on $V$ \\emph{special ample} if\nit corresponds to a special ample polytope $P$.\n\n\\begin{cor}\\label{toricmain}\n  Let $\\mathcal{L}$ be a special ample\n  line bundle on $V$.  Then, the canonical\n  map $$H^0(V, \\mathcal{L}) \\otimes H^0(V, \\mathcal{L}) \\otimes \\cdots\n  \\otimes H^0(V, \\mathcal{L}) \\longrightarrow H^0(V, \\mathcal{L}^m)$$\n  is a surjection for all $m \\geq 1$, i.e., $\\mathcal{L}$ is projectively\n  normal.\n\\end{cor}\n\n\\begin{rem}\n  The above corollary is a special case of Oda's Conjecture which\n  claims that the statement of the corollary is true for any ample\n  line bundle on a nonsingular, projective toric variety. In the case\n  of root systems of type $A$, the conjecture, and therefore the\n  corollary, is known to be true (see \\cite{howard}).\n\\end{rem}\n\n\n\nNext, consider the semigroup $S_{P} \\subset X \\times {\\mathbb Z}$ generated by\n$(x, 1)$ for $x \\in \\Lambda(P)$.  Then, the main theorem is equivalent\nto the statement that $S_{P}$ is normal, i.e., it is saturated in $X\n\\times Z$. In other words, it equals its saturation, $\\overline{S_{P}}\n:= \\bigcup_{m \\geq 1} (\\Lambda(mP)) \\times \\{m\\}$, i.e., the\nintersection of the cone $\\mathbb{R}_{> 0} \\cdot (P \\times \\{1\\})$ with the\nlattice $\\{(y + t\\mu_\\sigma, t) \\mid y \\in Y, t \\in {\\mathbb Z}\\}$, for any\nfixed $\\sigma \\in F_n$.\n\n\nIf we instead begin with the semigroup $\\overline{S_{P}}$, then\nTheorem \\ref{main} is equivalent to the statement that this semigroup\nis generated in degree one with respect to the grading $|(x,m)|=m$,\nfor $x \\in X$ and $m \\in {\\mathbb Z}$.\n\nOur second main result is\n\\begin{thm} \\label{main2} The semigroup $S_{P} = \\overline{S_{P}}$ is\n  presented by quadratic relations.  In other words, $S_{P} = \\langle\n  \\Lambda(P) \\times \\{1\\} \\rangle \/ (R)$, where $R$ is spanned by the\n  elements\n\\begin{equation*}\n(x,1) (y,1) - (x', 1) (y', 1),\n\\end{equation*}\nfor $x, y, x', y' \\in \\Lambda(P)$ such that $x+y = x'+y'$.\n\\end{thm}\n\n\\begin{rem} \\label{r:quad-stronger} Put differently, the \n  semigroup ring $\\mathbb{C}[S_{P}] = \\mathbb{C}[\\overline{S_{P}}]$ is quadratic. We\n  will actually prove a stronger version of the above theorem, which\n  roughly says that $(R)$ is spanned by moves which replace $(x_1,\n  \\ldots, x_m) \\in \\Lambda(P)^m$ by $(x_1, \\ldots, x_i + \\alpha,\n  x_{i+1} - \\alpha, \\ldots, x_m)$ for $\\alpha \\in \\Delta$. See \\S\n  \\ref{main2ssec} below for a precise statement.\n\\end{rem}\n\nSince $\\mathbb{C}[\\overline{S_{P}}] \\cong \\bigoplus_{m \\geq 0} H^0(V,\n\\mathcal{L}^{m})$, the toric interpretation of Theorems \\ref{main} and\n\\ref{main2} is\n\\begin{cor} \\label{toricmain2} Let $\\mathcal{L}$ be a special\nample line bundle on $V$.\n Then, the ring $\\bigoplus_{m \\geq 0} H^0(V,\n  \\mathcal{L}^{m})$ is quadratic.\n\\end{cor}\n\\begin{rem} The above corollary is a special case of Sturmfels's\n  conjecture \\cite[Conjecture 13.19]{Stugbcp}, which states that, for\n  any projective nonsingular toric variety $X$ and ample projectively\n  normal line bundle $\\mathcal{L}$, the associated ring $\\bigoplus_{m\n    \\geq 0} H^0(X, \\mathcal{L}^{m})$ is quadratic. (If Oda's conjecture is\n  true, then the projectively normal assumption is automatic.)\n\\end{rem}\n\nThis leaves open the natural\n\\begin{ques} Is the ring $\\mathbb{C}[S_{P}]$ Koszul? \n\\end{ques}\nSee \\cite{payne} and \\S \\ref{ss:diag-split-intro} and\n\\S \\ref{s:not-diag-split} below for such a result in a related situation.\n\n\n\\begin{rem}\n  It is clear that all of the above results remain true if we replace\n  $P$ with the polytope $\\nu + P$, where $\\nu \\in X$ (we still require\n  that $P$ satisfy $(\\dagger)$). Concerning the geometric statements\n  (Corollaries \\ref{toricmain} and \\ref{toricmain2}), the line bundle\n  on $V$ corresponding to the polytope $\\nu + P$ is isomorphic to the\n  line bundle $\\mathcal{L}$ (just equipped with a different\n  equivariant structure, which does not affect these statements).  In\n  other words, the above results can be viewed as applying to\n  nonequivariant ample line bundles which admit a special equivariant\n  structure.\n\\end{rem}\n\n\n\\subsection{Strengthening Theorems \\ref{main} and\n  \\ref{main2}} \\label{main2ssec}\n\nRather than prove Theorem \\ref{main2}, we will prove the following,\nwhich generalizes it and Theorem \\ref{main}. For yet another\nstrengthening, see the appendix.\n\n\\begin{defn}\\label{equiv}\n  Suppose $P_1, \\ldots, P_m$ are special ample polytopes and $(x_1,\n  \\ldots x_m) \\in \\Lambda(P_1) \\times \\cdots \\times \\Lambda(P_m)$.\n  Suppose further that $\\beta \\in \\Delta$ is a root and $i$ and $j$\n  are indices such that $x_i + \\beta \\in \\Lambda(P_i)$ and $x_j -\n  \\beta \\in \\Lambda(P_j)$. Then, we say that\n  \\begin{equation}\\label{eq:equivmove}\n    (x_1, \\ldots, x_m) \\sim (x_1, \\ldots, x_{i-1}, x_i + \\beta, x_{i+1}, \n    \\ldots, x_{j-1}, x_j - \\beta, x_{j+1}, \\ldots, x_m).\n\\end{equation}\nCall this a \\emph{root move}.\nExtend $\\sim$ to the equivalence relation generated by this, i.e.,\n$(x_1, \\ldots, x_m) \\sim (x_1', \\ldots, x_m')$ if the two are related\nby a sequence of root moves.\n\\end{defn}\nNote that, since root moves are reversible, a tuple is related to\nanother tuple by root moves if and only if one can be obtained from\nthe other by a sequence of root moves.\n\nThe following result strengthens Theorems \\ref{main} and \\ref{main2}:\n\\begin{thm}\\label{main2s}\n  If $P_1, \\ldots, P_m$ are special ample polytopes and $x \\in\n  \\Lambda(P_1 + \\cdots + P_m)$, then\n\\begin{enumerate}\n\\item[(i)] There exists a tuple $(x_1, \\ldots, x_m) \\in \\Lambda(P_1)\n  \\times \\cdots \\times \\Lambda(P_m)$ such that $x_1 + \\cdots + x_m =\n  x$;\n\\item[(ii)] If $(x_1, \\ldots, x_m)$ and $(x_1', \\ldots, x_m')$ are two\n  such tuples, then $(x_1, \\ldots, x_m) \\sim (x_1', \\ldots, x_m')$.\n\\end{enumerate}\n\\end{thm}\nSpecializing to the case $m = 2$ and $P = P_1 = P_2$, part (ii)\nimplies that the permutation $(x_1, x_2) \\mapsto (x_2, x_1)$ is a\nseries of root moves inside $\\Lambda(P)^2$.  Therefore, in the case $P\n= P_1 = \\cdots = P_m$ for arbitrary $m$, the relation $\\sim$ is\nactually generated by root moves \\eqref{eq:equivmove} with $j =\ni+1$. This explains Remark \\ref{r:quad-stronger}, and hence Theorem\n\\ref{main2s}.(ii) implies Theorem \\ref{main2}.\n\nOur motivation for allowing $P_1, \\ldots, P_m$ to be distinct\npolytopes is that it allows one to inductively prove the theorem on\n$m$: one deduces the result for $m > 2$ from the pair $(P_1 + \\cdots +\nP_{m-1}, P_m)$.\n\nA toric interpretation of part of the theorem is as follows.\nLet $\\mathcal{L}_1, \\ldots, \\mathcal{L}_m$ be special ample line bundles\n on $V$ and\n $$\\varphi_{\\mathcal{L}_1, \\ldots, \\mathcal{L}_m}: H^0(V,\n  \\mathcal{L}_1) \\otimes \\cdots \\otimes H^0(V, \\mathcal{L}_m)\n  \\longrightarrow H^0(V, \\mathcal{L}_1 \\otimes \\cdots \\otimes\n  \\mathcal{L}_m)$$\nbe the canonical map.\n\\begin{cor}\\label{toricmain2s}\n\\begin{enumerate}\n\\item[(i)] $\\varphi_{\\mathcal{L}_1, \\ldots,\n    \\mathcal{L}_m}$ is surjective.\n\n\\item[(ii)] The kernel of $\\varphi_{\\mathcal{L}_1, \\ldots,\n    \\mathcal{L}_m}$ is spanned by the canonical subspaces\n  $$\\ker(\\varphi_{\\mathcal{L}_i, \\mathcal{L}_j}) \\otimes \\bigotimes_{k\n    \\notin \\{i,j\\}} H^0(V, \\mathcal{L}_k) \\subseteq\n  \\ker(\\varphi_{\\mathcal{L}_1, \\ldots, \\mathcal{L}_m}).$$\n\\end{enumerate}\n\\end{cor}\nSimilarly, we can apply this to the Cayley sum polytope of polytopes $P_1, \\ldots, P_k$.  Recall that this is defined as the polytope inside $(X \\otimes_{\\mathbb Z} \\mathbb{R}) \\times \\mathbb{R}^{k}$ which is the convex hull of $(P_1 \\times \\{e_1\\}) \\cup \\cdots \\cup\n(P_k \\times \\{e_k\\})$, where $e_1, \\ldots, e_k$ are the standard basis of $\\mathbb{R}^k$. The resulting polytope is denoted by $P_1 * P_2 * \\cdots * P_k$ and is considered with respect to the lattice $Y \\times {\\mathbb Z}^k$.\n\\begin{cor}\\footnote{Thanks to S. Payne for observing this corollary.}\n  Let $P_1, \\ldots, P_k$ be special ample polytopes.  Then, the Cayley\n  sum polytope $P = P_1 * \\cdots * P_k$ is normal, and $\\mathbb{C}[S_P]$ is\n  quadratic.\n\\end{cor}\nThe corollary follows from Theorem \\ref{main2s} as follows: for every\n$m_1, \\ldots, m_k \\geq 0$, apply the theorem to the product $\\Lambda(P_1)^{m_1}\n\\times \\cdots \\times \\Lambda(P_k)^{m_k}$, with $m = m_1 + \\cdots + m_k$. Note\nhere that the (degree-one) generators of $\\mathbb{C}[S_P]$ are the elements\n$((y,e_i),1) \\in (\\Lambda(P_i) \\times {\\mathbb Z}^k) \\times {\\mathbb Z}$, where $1 \\leq\ni \\leq k$.\n\nFinally, we give the toric interpretation of the corollary.  Let\n$\\mathcal{L}_1, \\ldots, \\mathcal{L}_k$ be special ample line bundles\non $V$.  Given a vector bundle $\\mathcal{U}$, let\n$\\operatorname{Sym}^m(\\mathcal{U})$ denote its $m$-th symmetric power.\n\\begin{cor}\n  The ring $\\bigoplus_{m \\geq 0} H^0(V, \\operatorname{Sym}^m(\\mathcal{L}_1 \\oplus\n  \\cdots \\oplus \\mathcal{L}_k))$ is quadratic.\n\\end{cor}\n\n\\subsection{Diagonal splitness}\\label{ss:diag-split-intro}\nA closely related toric variety to $V$, studied in, e.g.,\n\\cite{payne}, is the one whose fan is such that its rays (i.e.,\none-dimensional cones) are generated by the elements of $\\Delta$: so,\nits initial lattice is $Y$, dual to the initial lattice of $V$.\nDenote this variety by $U$.\n\nSuppose that $Q$ is an orthogonal polytope corresponding to the fan\nassociated to $U$, i.e., one which describes an equivariant line\nbundle on $U$. Then, the main result of \\emph{op.~cit.} was that, in\nthe case that the root system is of type $A, B, C$, or $D$, the\nsemigroup ring $\\mathbb{C}[S_Q]$ is Koszul (and in particular, $S_Q$ is\nnormal and $\\mathbb{C}[S_Q]$ is quadratic).  This was proved by showing that\n$Q$ is always diagonally split (see, e.g., \\emph{op.~cit.}, or \\S\n\\ref{s:not-diag-split} below).  Note that, unlike in the present\npaper, the arguments did not extend to the exceptional types $E, F$,\nor $G$, and the speciality and ampleness assumptions were not\nrequired.\n\n  In contrast, in \\S \\ref{s:not-diag-split} below, we show that,\n  except in the cases $A_1, A_2, A_3$, and $B_2$, the ample polytopes\n  $P$ associated to the toric variety $V$ considered in this paper are\n  not diagonally split, and therefore the above argument cannot be\n  applied in our case for \\emph{any} root systems other than these\n  four.\n\n\\subsection{Organization of the paper}\n\nIn \\S \\ref{s:main-proof} we prove Theorem \\ref{main}, where a crucial\nstep involves using a lemma of Stembridge (\\cite[Cor. 2.7]{stem2})\nstating that in the usual partial order of $\\sigma$-dominant weights,\na weight $\\nu$ covers another one $\\nu'$ if and only if the difference\n$\\nu - \\nu'$ is a root that is positive with respect to $\\sigma$. In\n\\S \\ref{s:numbers-game} we recall the numbers game with a cutoff (from\n\\cite{qendrimthesis}; see also \\cite{GS}), which gives a useful\nlanguage to prove Theorem \\ref{main2s}. The proof of the theorem is\nthen given in \\S$\\!$\\S \\ref{s:main2s-proof} and \\ref{s:lemmaproofs}.\nNote that one of our auxiliary results (Lemma \\ref{m2lem1})\ngeneralizes the above lemma of Stembridge. In \\S\n\\ref{s:not-diag-split} we show that ample polytopes for the toric\nvarieties $V$ as above are not diagonally split (with the exception of\nthe cases when the root system $\\Phi$ is of type $A_1, A_2, A_3$, or\n$B_2$).\n\nIn the appendix, we give a generalization of Theorem \\ref{main2s} in\nterms of the numbers game: these allow one to restrict the type of\ntuples needed in the equivalence $\\sim$ above.\n\n\n\n\\subsection{Acknowledgements}\nWe thank Sam Payne for helpful comments.  The first author was\nsupported by an EPDI Fellowship. The second author is an AIM Five-Year\nFellow, and was partially supported by the ARRA-funded NSF grant\nDMS-0900233. We thank IHES and MIT for hospitality.\n\n\n\\section{Proof of Theorem \\ref{main}}\\label{s:main-proof}\n\nRecall that for a cone $\\sigma \\in F_n$, we denote by $\\{ \\alpha_{i,\n  \\sigma}: i \\in I\\} \\subset \\Delta$ the corresponding set of simple\nroots. For a root $\\gamma \\in \\Delta$, we say that it is positive or\nnegative with respect to the chamber $\\sigma$ if $\\gamma$ can be\nwritten as a nonnegative or nonpositive linear combination of the\nelements of $\\{ \\alpha_{i, \\sigma}: i \\in I\\}$, respectively. We write\n$D_{\\sigma}$ for the set of $\\sigma$-dominant elements of $X\n\\otimes_{{\\mathbb Z}} \\mathbb{R}$.\n\nNote that $P$ is the convex hull of the points $\\{ \\mu_{\\sigma} \\in X:\n\\sigma \\in F_n \\}$. The next two lemmas will allow us to better\nunderstand the shape of the polytope $P$. The first one uses the fact\nthat $P$ is ample and the second one that $P$ is\nspecial.\n\\begin{lemma}\\label{one}\n\\emph{(}see, e.g., \\cite[Lemma 12.1, p.~445]{akot}\\emph{)}\n$$P = \\bigcap_{\\sigma \\in F_n} C^*_{\\sigma},$$ where $C^*_{\\sigma} : =\n\\{ \\mu_{\\sigma} - \\sum_{i=1}^{n} t_i \\alpha_{i, \\sigma} : t_i \\in\n\\mathbb{R}_{\\geq 0} \\}$.\n\\end{lemma}\n\n\\begin{lemma}\\label{two}\n  \\emph{(}see, e.g., \\cite[Lemma 12.2, p.~445]{akot}\\emph{)}  $$P\n  \\cap D_{\\sigma} = C^*_{\\sigma} \\cap D_{\\sigma}.$$\n\\end{lemma}\nSpecializing to the points in $\\Lambda(P)$,  we obtain\n\\begin{equation}\\label{e:ltwo-int}\n\\Lambda(P) \\cap D_{\\sigma} = \\{\n\\nu \\in D_{\\sigma} \\cap X : \\nu \\stackrel{\\sigma}{\\preceq}\n\\mu_{\\sigma}\\},\n\\end{equation}\nwhere $\\stackrel{\\sigma}{\\preceq}$ stands for the\npartial order in $X$ determined by the chamber $\\sigma$, i.e., $\\nu\n\\stackrel{\\sigma}{\\preceq} \\mu_{\\sigma}$ if $\\mu_{\\sigma} - \\nu$ is a\nnonnegative integral linear combination of the roots $\\{ \\alpha_{i,\n  \\sigma}: i \\in I \\}$.\n\n\nFix a chamber $\\sigma \\in F_{n}$. Since $W$ acts simply\ntransitively on $\\{ D_{\\tau}$: $\\tau \\in F_n \\}$ and since\n$\\Lambda(mP) = \\bigcup_{w \\in W} (\\Lambda(mP) \\cap (w D_{\\sigma}))$,\nit suffices to prove the statement of Theorem \\ref{main} for $z \\in\n\\Lambda(m P) \\cap D_{\\sigma} $.\n\nBy \\eqref{e:ltwo-int}, every element $z\n\\in \\Lambda(m P) \\cap D_{\\sigma}$ satisfies $z\n\\stackrel{\\sigma}{\\preceq} m \\mu_{\\sigma}$. Clearly, for $z = m\n\\mu_{\\sigma}$ the assertion of Theorem \\ref{main} is true. So, to\nprove the theorem, it suffices to show that, whenever it holds for $x\n\\in \\Lambda(m P) \\cap D_{\\sigma}$, then it also holds for every $z\n\\in \\Lambda(m P) \\cap D_{\\sigma}$ such that $x$ covers $z$.  Here,\n$x$ \\emph{covers} $z$ means that $z \\stackrel{\\sigma}{\\preceq} t\n\\stackrel{\\sigma}{\\preceq} x$ and $t \\in D_{\\sigma} \\cap X$\nimplies that $t = z$ or $t = x$.\n\nSo, assume that the statement of the theorem is true for $x \\in\n\\Lambda(m P) \\cap D_{\\sigma}$ and that $x$ covers $z \\in \\Lambda(m P)\n\\cap D_{\\sigma}$. By a lemma of Stembridge (\\cite[Cor. 2.7]{stem2};\nsee also \\cite[Lemma 2.3]{rapoport} and Remark \\ref{strem} below),\nthere exists a $\\sigma$-positive root $\\beta$ such that $x- z =\n\\beta$. Since $z$ is $\\sigma$-dominant and $\\beta$ is\n$\\sigma$-positive, $\\langle z ,\\beta^{\\vee} \\rangle \\geq 0$, and thus\n$\\langle x -\\beta ,\\beta^{\\vee} \\rangle \\geq 0$, i.e., $$\\langle x\n,\\beta^{\\vee} \\rangle \\geq 2.$$\n\nBy assumption, $x$ can be written as a sum $x=x_1 +\n\\cdots + x_m$, with $x_i \\in \\Lambda(P)$, $\\forall i =1,\\ldots,\nm$. The last inequality guarantees that $\\langle x_j ,\\beta^{\\vee}\n\\rangle \\geq 1$ for at least one $j \\in \\{1,\\ldots,m\\}$. The\nproposition below then ensures that $x_j-\\beta \\in P$:\n\n\\begin{prop}\\label{mainclaim}\n  Let $y \\in \\Lambda(P)$ and $\\beta \\in \\Delta$. If $\\langle y,\n  \\beta^{\\vee} \\rangle \\geq 1$, then $y-\\beta \\in \\Lambda(P)$.\n\\end{prop}\n\nNow we put $z_i = x_i, \\forall i \\neq j$, and $z_j = x_j - \\beta$, and\nthen $z = z_1 + \\cdots + z_m$, which verifies the theorem.  This\nconcludes the proof of Theorem \\ref{main}, and it only remains to prove\nthe last proposition.\n\n\\subsection{Proof of Proposition \\ref{mainclaim}}\n\nLet $y \\in \\Lambda(P)$ and $\\beta \\in \\Delta$ be\nsuch that $\\langle y ,\\beta^{\\vee} \\rangle \\geq 1$.\n\nWe must show that, for every $\\sigma \\in F_n$, $y-\\beta\n\\stackrel{\\sigma}{\\preceq} \\mu_{\\sigma}$. Note that, since $y \\in\n\\Lambda(P)$, there exist nonnegative integers $h_{i, \\sigma}$,\n$i=1,\\ldots, n$, such that\n\\begin{equation}\\label{h}\n\\mu_{\\sigma} - y = \\sum_{i=1}^{n} h_{i, \\sigma} \\alpha_{i, \\sigma}.\n\\end{equation}\nLet $\\beta = \\sum_{i=1}^{n} b_{i, \\sigma} \\alpha_{i, \\sigma}$. Then\n$\\mu_{\\sigma} - (y - \\beta) = \\sum_{i=1}^{n} (h_{i, \\sigma} + b_{i,\n  \\sigma}) \\alpha_{i, \\sigma}$. If the chamber $\\sigma$ is such that\n$\\beta$ is positive with respect to it, then clearly $y-\\beta\n\\stackrel{\\sigma}{\\preceq} \\mu_{\\sigma}$.\n\nWe are therefore left to consider only the chambers with respect to\nwhich $\\beta$ is negative. Denote the set of such chambers by\n$F_{-}$. Then we can write $F_{-}$ as a disjoint union $F_{-} = F_{-}'\n\\cup F_{-}''$, where $$F_{-}' = \\{ \\tau \\in F_{-} : \\tau \\, \\text{is\n  adjacent to a chamber with respect to which}\\, \\beta\\, \\text{is\n  positive}\\},$$ and $F_{-}'' = F_{-} \\setminus F_{-}'$.\n\nSince $P$ is special, $\\langle \\mu_{\\sigma}, \\beta^{\\vee} \\rangle \\leq\n0, \\forall \\sigma \\in F_{-}$. Moreover, we claim that $\\langle\n\\mu_{\\sigma} , \\beta^{\\vee} \\rangle \\leq -1, \\forall \\sigma \\in\nF_{-}''$. This follows from the previous statement because $P$ is\nample: indeed, if $\\sigma \\in F_-''$, then by definition $-\\beta$ is\npositive but not simple for $\\sigma$, so there exists $\\alpha_{i,\n  \\sigma}$ (necessarily not equal to $-\\beta$) such that $\\langle\n\\alpha_{i, \\sigma}, -\\beta^\\vee \\rangle \\geq 1$.  Therefore, if\n$\\langle \\mu_\\sigma, \\beta^\\vee \\rangle = 0$, then if $\\sigma'$ is the\nchamber adjacent to $\\sigma$ corresponding to $\\alpha_{i, \\sigma}$, it\nfollows that $\\langle \\mu_{\\sigma'}, \\beta^\\vee \\rangle = \\langle\n\\mu_\\sigma-r_{\\sigma, \\sigma'} \\alpha_{i, \\sigma}, \\beta^\\vee \\rangle\n\\geq r_{\\sigma,\\sigma'} > 0$.  However, $\\sigma' \\in F_-$ by\ndefinition, which furnishes a contradiction.\n\nTo prove the proposition, we claim that it suffices to show\nthat $y-\\beta \\stackrel{\\sigma}{\\preceq} \\mu_{\\sigma}$ when $\\sigma\n\\in F_{-}'$. Since $\\langle y - \\beta, \\beta^{\\vee}\\rangle \\geq -1$ by\nassumption, it will then follow that $y-\\beta$ lies in all of the\nhalf-spaces whose intersection defines $P$ (whose boundaries are\nmaximal-dimensional facets of $P$), except possibly for those whose\nboundary planes meet vertices of $P$ only in $F_-''$. Suppose, for\nsake of contradiction, that $y-\\beta \\notin P$.  Let $0 \\leq t < 1$ be\nmaximal such that $y-t\\beta \\in P$. Then $y-t\\beta$ lies on a boundary\nplane meeting vertices of $P$ only in $F_-''$.  Since $\\langle\n\\mu_{\\sigma}, \\beta^{\\vee} \\rangle \\leq -1, \\forall \\sigma \\in\nF_{-}''$, it follows that $\\langle y-\\beta, \\beta^\\vee \\rangle <\n\\langle y-t\\beta, \\beta^\\vee \\rangle \\leq -1$. This is impossible,\nsince $\\langle y, \\beta^\\vee \\rangle \\geq 1$.  Thus, $y-\\beta \\in P$,\nas desired.\n\nThus, take $\\sigma \\in F_{-}'$. In the remainder of the proof, we show\nthat $y-\\beta \\stackrel{\\sigma}{\\preceq} \\mu_{\\sigma}$.  \nDenote by $\\tau$ a chamber in\n$F$ that is adjacent to $\\sigma$ and such that $\\beta$ is positive\nwith respect to $\\tau$. We write $\\alpha_i$ instead of $\\alpha_{i,\n  \\tau}$ for the simple roots corresponding to $\\tau$. Since $\\beta$\nis negative with respect to $\\sigma$, there exists $j \\in I$ such\nthat $\\beta = \\alpha_{j} = -\\alpha_{j,\\sigma}$. Moreover,\n\\begin{equation}\\label{alpha}\n\\alpha_{i} = \\alpha_{i,\\sigma} + \\langle \\alpha_i , \\beta^{\\vee} \\rangle \\beta.\n\\end{equation}\n\nSince $P$ is ample, $\\mu_{\\sigma} = - r_{\\tau,\\sigma} \\beta + \\mu_{\\tau}$,\nfor $r_{\\tau, \\sigma} > 0$.\nThus, using (\\ref{h}) and applying (\\ref{alpha}), we get \n$$\\sum_{i=1}^{n} h_{i, \\sigma}\\alpha_{i,\\sigma}\n= - r_{\\tau, \\sigma} \\beta + \\sum_{i=1}^{n} h_{i, \\tau}\n(\\alpha_{i,\\sigma} + \\langle \\alpha_i , \\beta^{\\vee} \\rangle \\beta).$$\nSince $\\{\\alpha_{i,\\sigma}: i \\in I\\}$ is a basis for $\\Delta$ and\n$\\alpha_{j,\\sigma}=-\\beta$, from the last identity we deduce that\n\\begin{gather}\n  h_{i, \\sigma} = h_{i, \\tau}, \\forall i \\in I \\setminus \\{\\,j\\}, \n\\quad \\text{and} \\notag \\\\\n\\label{hqk}\nh_{j, \\sigma} = r_{\\tau, \\sigma} - h_{j, \\tau} - \\sum_{i \\in I\n  \\setminus \\{\\,j \\}} h_{i, \\tau} \\langle \\alpha_i, \\beta^{\\vee}\n\\rangle.\n\\end{gather}\n\nNow, $\\mu_{\\sigma} - (y-\\beta) = \\left( \\sum_{i=1}^{n} h_{i, \\sigma}\n  \\alpha_{i, \\sigma} \\right) - \\alpha_{j, \\sigma}$, so in order to\nprove that $y-\\beta \\stackrel{\\sigma}{\\preceq} \\mu_{\\sigma}$, it\nsuffices to show that $h_{j, \\sigma} \\geq 1$. For a contradiction,\nassume $h_{j, \\sigma} = 0$.  From (\\ref{hqk}) we then get\n\\begin{equation}\\label{q1}\n  h_{j,\\tau} - r_{\\tau, \\sigma} = - \\sum_{i \\in I \\setminus \\{\\,j \\}} \n  h_{i, \\tau} \\langle \\alpha_i, \\beta^{\\vee} \\rangle.\n\\end{equation}\nNext, recall that $\\langle y, \\beta^{\\vee} \\rangle \\geq 1$, so\n\\begin{multline*}\n  \\langle \\mu_{\\sigma}, \\beta^{\\vee} \\rangle = \\langle \\mu_{\\tau} -\n  r_{\\tau, \\sigma} \\beta, \\beta^{\\vee} \\rangle = \\langle y +\n  \\sum_{i=1}^{n}\n  h_{i, \\tau} \\alpha_{i}\\, ,\\beta^{\\vee} \\rangle - 2r_{\\tau, \\sigma} \\\\\n  \\geq 1 + 2h_{j, \\tau} - 2r_{\\tau, \\sigma} + \\sum_{i \\in I \\setminus\n    \\{\\,j \\}} h_{i, \\tau} \\langle \\alpha_i, \\beta^{\\vee} \\rangle = 1 -\n  \\sum_{i \\in I \\setminus \\{\\,j \\}} h_{i, \\tau} \\langle \\alpha_i,\n  \\beta^{\\vee} \\rangle,\n\\end{multline*}\nwhere to get the last equality we used (\\ref{q1}). But, the last\nexpression is strictly positive (since $\\langle \\alpha_i, \\beta^\\vee\n\\rangle = \\langle \\alpha_i, \\alpha_{j}^\\vee \\rangle \\leq 0$ for all $i\n\\neq j$), and, since the polytope $P$ is special, $\\langle\n\\mu_{\\sigma}, \\alpha_{j}^{\\vee} \\rangle \\leq 0$, a contradiction. This\nends the proof that $y-\\beta \\stackrel{\\sigma}{\\preceq} \\mu_{\\sigma}$\nand concludes the proof of Proposition \\ref{mainclaim}.\n\n\n\n\\section{The numbers game with a cutoff}\\label{s:numbers-game}\nIn order to prove Theorem \\ref{main2}, we use the language of\n\\emph{the numbers game with a cutoff}, from \\cite{qendrim3} (see also\n\\cite{GS}). In this section we recall what we will need.\n\n\\subsection{The usual numbers game}\nWe first recall Mozes's numbers game \\cite{mozes}, which has been\nwidely studied (e.g., in \\cite{Pro-bru, Pro-min, DE, Erik-no1,\n  Erik-no2, erikconf, Erik-no3, Erik-no4, eriksson, Wild-no1,\n  Wild-no2}). Fix an unoriented, finite graph with no loops and no\nmultiple edges. Let $I$ be the set of vertices. Fix also a Cartan\nmatrix $C = (c_{ij})_{i,j \\in I} \\in \\mathbb{R}^I \\times \\mathbb{R}^I$, such that\n$c_{ii} = 2$ for all $i$, $c_{ij} = 0$ whenever $i$ and $j$ are not\nadjacent, and otherwise $c_{ij}, c_{ji} < 0$, and either $c_{ij}\nc_{ji} = 4 \\cos^2(\\frac{\\pi}{n_{ij}})$ (when $n_{ij}$ is finite) or\n$c_{ij} c_{ji} \\geq 4$ (when $n_{ij} = \\infty$).\n\nWe will only need to consider the case where our graph is the\nunderlying graph of a Dynkin diagram (undirected and without multiple\nedges), and $C$ is the standard Cartan matrix for the diagram, i.e.,\n$c_{ij} = \\langle \\alpha_i, \\alpha_j^\\vee \\rangle$. In particular,\n$c_{ij} \\in {\\mathbb Z}$ for all $i, j$.  Hence, the reader may assume this if\ndesired.\n\nThe \\emph{configurations} of the game consist of vectors from\n$\\mathbb{R}^I$. The moves of the game are as follows: For any vector ${v} \\in\n\\mathbb{R}^I$ and any vertex $i \\in I$ such that ${v}_i < 0$, one may perform\nthe following move, called \\emph{firing the vertex $i$}: ${v}$ is\nreplaced by the new configuration $f_i({v})$, defined by\n\\begin{equation*}\nf_i({v})_j = v_j - c_{ij} v_i.\n\\end{equation*}\nThe entries ${v}_i$ of the vector ${v}$ are called\n\\emph{amplitudes}. The game terminates if all the amplitudes are\nnonnegative.  Let us emphasize that \\emph{only negative-amplitude\n  vertices may be fired}.\\footnote{In some of the literature, the\n  opposite convention is used, i.e., only positive-amplitude vertices\n  may be fired.}\n\n\\begin{prop}\\cite{eriksson}\n  The numbers game is \\emph{strongly convergent}: if the game can\n  terminate, then it must terminate, and in exactly the same number of\n  moves and arriving at the same configuration, regardless of the\n  choices made.\n\\end{prop}\n\n\\subsection{The numbers game with a cutoff}\n\nIn \\cite{qendrimthesis}, the numbers game \\emph{with a cutoff} was\ndefined: The moves are the same as in the ordinary numbers game, but\nthe game continues (and in fact starts) only as long as all amplitudes\nremain greater than or equal to $-1$.  Such configurations are called\n\\emph{allowed}. Every configuration which does not have this property\nis called \\emph{forbidden}, and upon reaching such a configuration the\ngame terminates (we lose). We call a configuration \\emph{winning} if\nit is possible, by playing the numbers game with a cutoff, to reach a\nconfiguration with all nonnegative amplitudes.\n\nIn \\cite{GS}, a simple criterion was given to determine when the\nnumbers game with a cutoff is winning.  We will restrict to the Dynkin\ncase, with $C$ the standard Cartan matrix. Let $\\Delta$ be the set of\nroots. Pick simple roots corresponding to the vertices of the Dynkin\ndiagram, and write $\\Delta = \\Delta_+ \\sqcup -\\Delta_+$, where\n$\\Delta_+$ is the set of positive roots.\n\nWe can view $\\Delta \\subseteq {\\mathbb Z}^I$ and $\\Delta_+ \\subseteq {\\mathbb Z}_{\\geq\n  0}^I$.  For $i \\in I$, let $\\alpha_i$ be the simple root\ncorresponding to $i$, which as an element of ${\\mathbb Z}^I$ is the elementary\nvector $(\\alpha_i)_j = \\delta_{ij}$. Note that, since $\\alpha_i$\nrefers to a vector in ${\\mathbb Z}^I$, in the case that $\\alpha \\in {\\mathbb Z}^I$, we\nwill \\emph{never} use $\\alpha_i$ to refer to a component of $\\alpha$,\nreserving it exclusively for the elementary vector $\\alpha_i \\in\n{\\mathbb Z}^I$.\n\nA useful description of $\\Delta$ is\n\\begin{equation*}\n\\Delta = \\bigcup_{i \\in I} W \\cdot \\alpha_i,\n\\end{equation*}\nwhere $W$ is the Weyl group generated by the simple reflections $s_i:\n\\mathbb{R}^I \\rightarrow \\mathbb{R}^I$, for all $i \\in I$, given by\n\\begin{equation*}\ns_i(\\beta)_j =\n\\begin{cases} \\beta_j, & \\text{if $j \\neq i$}, \\\\\n  -\\beta_i - \\sum_{k \\neq i} c_{ki} \\beta_k, & \\text{if $j = i$}.\n\\end{cases}\n\\end{equation*}\n\n\\begin{prop}\\label{gsprop}\\cite[Theorem 3.1, Corollary 5.10.(a)]{GS}\n  Fix a Dynkin diagram with standard Cartan matrix $C$. Beginning with\n  a configuration $v \\in \\mathbb{R}^I$, the numbers game with a cutoff is\n  winning if and only if\n\\begin{equation} \\label{gscond}\nv \\cdot \\alpha \\geq -1, \\forall \\alpha \\in \\Delta_+,\n\\end{equation}\nand in this case, one always wins the numbers game with a cutoff, no\nmatter which moves are made, and arrives at the same final\nconfiguration in the same total number of moves.\n\\end{prop}\nHere, $\\cdot$ is the dot product of $v \\in \\mathbb{R}^I$ with $\\alpha \\in\n{\\mathbb Z}^I$, i.e., $v \\cdot \\bigl(\\sum_i c_i \\alpha_i\\bigr) = \\sum_i c_i\nv_i$, for $c_i \\in \\mathbb{R}$.\n\n\n\\subsection{Relation to the polytope $P$}\n\nProposition \\ref{mainclaim} has the following consequence in terms of\nthe numbers game with a cutoff.  We consider the embedding\n\\begin{equation*}\n  \\iota: X \\hookrightarrow \\mathbb{R}^I, \\quad x \\mapsto \\iota(x), \n  \\quad \\iota(x)_i := \\langle x, \\alpha_i^\\vee \\rangle.\n\\end{equation*}\nIn this language, the condition \\eqref{gscond} translates for $x \\in\nX$ as follows: The configuration $\\iota(x)$ is winning if and only if\n\\begin{equation}\\label{gscond2}\n\\langle x, \\alpha^\\vee \\rangle \\geq -1, \\forall \\alpha \\in \\Delta_+.\n\\end{equation}\nThen, Proposition \\ref{mainclaim} implies\n\\begin{cor}\\label{ngcor}\n  If $x, y \\in X$ and $\\iota(y)$ can be obtained from $\\iota(x)$ by\n  playing the numbers game with a cutoff, then $x \\in \\Lambda(P)$ if\n  and only if $y \\in \\Lambda(P)$.\n\\end{cor}\n\\begin{proof}\n  Suppose that $u \\in X$ and $\\iota(u) \\in {\\mathbb Z}^I$ is obtained along the\n  way from $\\iota(x)$ to $\\iota(y)$. From $u$, any move in the numbers\n  game with a cutoff is of the form $u \\mapsto u + \\iota(\\alpha_i)$\n  for some $i \\in I$ such that $u_i = -1$. Hence, $\\langle u,\n  \\alpha_i^\\vee \\rangle = -1$ and $(u + \\iota(\\alpha_i))_i = \\langle u\n  + \\alpha_i, \\alpha_i^\\vee \\rangle = 1$.  We therefore conclude from\n  Proposition \\ref{mainclaim} that $u \\in \\Lambda(P)$ if and only if\n  $u + \\alpha_i \\in \\Lambda(P)$. The corollary follows.\n\\end{proof}\nNote that the choice of simple roots was arbitrary, so the corollary\nin fact holds for any choice of simple roots (equivalently, any choice\nof dominant chamber).\n\n\\begin{rem}\nThe corollary extends to the case where $y$ is\nobtained from $x$ in the usual numbers game by firing vertices only of\namplitude $-1$, i.e., we can continue the numbers game even if there\nis an amplitude less than $-1$, as long as we never fire such\nvertices. (This seems to be a reasonable variation on the numbers game\nwith a cutoff.)\n\\end{rem}\n\n\\section{Proof of Theorem \\ref{main2s}}\\label{s:main2s-proof}\n\nIt is convenient to abuse notation slightly, by omitting the map $\\iota$:\n\\begin{ntn} If $x \\in X$ and $\\iota(x)$ is winning, we say also that\n  $x$ is winning. Moreover, if $x, y \\in X$ and $\\iota(y)$ is obtained\n  from $\\iota(x)$ by playing the numbers game (with or without a\n  cutoff), we also say that $y$ is obtained from $x$ by playing the\n  numbers game (with or without a cutoff, respectively).\n\\end{ntn}\n\nFix once and for all a dominant chamber $\\sigma$, and write $D$,\n$\\prec$, $\\preceq$, and $\\mu$, instead of $D_{\\sigma}$,\n$\\stackrel{\\sigma}{\\prec}$, $\\stackrel{\\sigma}{\\preceq}$, and\n$\\mu_{\\sigma}$, respectively. We omit $\\sigma$ from now on, and by a\ndominant element we always mean an element of $D$.\n\nNext, given special ample polytopes $P_1, \\ldots, P_m$, we let $\\mu_1,\n\\ldots, \\mu_m$ denote the vertices $\\mu_1, \\ldots, \\mu_m \\in D$ of\neach corresponding to the dominant chamber.\n\n\n\nFinally, we recall the notion of \\emph{length} of roots.  For\nsimply-laced root systems (i.e., types $A_n$, $D_n$, and $E_n$, since\nwe only consider the Dynkin case), we say that all roots have the same\nlength. For the other root systems, the set of roots $\\Delta$ is\npartitioned into the subsets of \\emph{short} and \\emph{long} roots,\nand we say that the long roots are \\emph{longer} than the short roots.\nOne way to define the partition (which will be useful to us) is that,\nif $\\beta \\in \\Delta$ is at least as long as $\\alpha \\in \\Delta$ and\n$\\alpha \\neq \\pm \\beta$, then $\\langle \\alpha, \\beta^\\vee \\rangle \\in\n\\{-1, 0, 1\\}$. Recall also that the partition is preserved by the Weyl\ngroup action.  We emphasize that, for us, $\\langle \\alpha, \\alpha^\\vee\n\\rangle = 2$ for all $\\alpha \\in \\Delta$, long or short; the\nterminology of length comes from the norm under the symmetrized Cartan\nform, which we will not use.\n\n\\subsection{Outline of the proof}\nFirst, Theorem \\ref{main2s}.(i) follows in exactly the same manner as\nTheorem \\ref{main}. We give a short proof in the spirit of this\nsection, based on Proposition \\ref{mainclaim}, in \\S\n\\ref{ss:main2smgen} below.\n\nOur strategy underlying the proof of Theorem \\ref{main2s}.(ii) is to\nperform induction on the sum $x_1 + \\cdots + x_m$, which we can assume\nis winning (in fact we could assume it is dominant using the action of\nthe Weyl group).  The induction will be over a certain partial order\non the sum polytope $P = P_1 + \\cdots + P_m$.\n\nThe proof is broken into three parts: first we prove results about the\npartial order on the winning locus of $P$, which boil down to a\nstrengthening of the lemma of Stembridge mentioned earlier.  Second,\nwe prove the theorem in the case $m=2$.  Third, we inductively deduce\nthe theorem for general $m$. In what follows, we will explain the\nproof modulo some lemmas whose proofs will be provided in \\S\n\\ref{s:lemmaproofs}.\n\n\\subsection{Partial ordering on the winning locus of $P$}\nLet $P$ be a special ample polytope.\n\\begin{defn}\n  Suppose $x \\in \\Lambda(P)$.  If $x \\neq \\mu$, then a simple root\n  $\\alpha$ is \\emph{$P$-progressive} for $x$ if either $x$ is dominant\n  and $\\alpha$ has minimum length such that $x+\\alpha \\preceq \\mu$, or\n  else $\\langle x, \\alpha^\\vee \\rangle \\leq -1$.\n\\end{defn}\nIt is immediate that, for all $x \\neq \\mu$, there exists a simple root\nwhich is $P$-progressive for $x$.\n\nThis subsection is devoted to the proof of\n\\begin{prop} \\label{p:powin} If $\\alpha$ is $P$-progressive for $x$,\n  then $x +\\alpha \\in \\Lambda(P)$. Moreover, if $x$ is winning, then\n  so is $x+\\alpha$.\n\\end{prop}\n\\begin{proof}\n  First, suppose that $\\alpha$ is a simple\n  root such that $\\langle x, \\alpha^\\vee \\rangle \\leq-1$. By\n  Proposition \\ref{mainclaim}, $x+\\alpha \\in \\Lambda(P)$.  If $x$ is\n  winning, then $x+\\alpha$ is obtained from $x$ by a move of the\n  numbers game, and hence it is winning.\n\n  If $x$ is dominant and $\\alpha$ is a simple root of minimum length\n  such that $x+\\alpha \\preceq \\mu$, the result follows from the case\n  $y=\\mu$ of the Lemma \\ref{m2lem1} below.  Namely, by Corollary\n  \\ref{ngcor}, to show that $x+\\alpha \\in \\Lambda(P)$, it suffices to\n  show that $z \\in \\Lambda(P)$, where $z$ is the result of playing the\n  numbers game with a cutoff beginning with $x+\\alpha$.  Next, if\n  $\\beta \\in \\Delta_+$ is any positive root such that $x+\\beta \\preceq\n  \\mu$, then $\\beta$ must be at least as long as $\\alpha$; otherwise\n  $\\beta$ would be short and $\\alpha$ long, and there would exist a\n  short simple root $\\gamma$ such that $\\gamma \\preceq \\beta$.  In the\n  latter case, $x + \\gamma \\preceq x+\\beta \\preceq \\mu$, contradicting\n  our assumption that $\\alpha$ has minimum length such that $x+\\alpha\n  \\preceq \\mu$.  Therefore, we may apply Lemma \\ref{m2lem1} with $y =\n  \\mu$.  We conclude that $x+\\alpha$ is winning, i.e., $z$ is\n  dominant, and also $z \\preceq \\mu$.  Then, $z \\in \\Lambda(P)$ by\n  \\eqref{e:ltwo-int}.\n\\end{proof}\n\\begin{lemma}\\label{m2lem1}\n  Suppose $x \\prec y$ and $x, y \\in X \\cap D$.  Let $\\alpha \\in\n  \\Delta_+$ be a positive root of minimum length such that $x + \\alpha\n  \\preceq y$. Then, $x+\\alpha$ is winning, and if $z$ is the result of\n  playing the numbers game with a cutoff, then $x+\\alpha \\preceq z\n  \\preceq y$.\n\\end{lemma}\nThe lemma will be proved in \\S \\ref{ss:m2lem1-proof}.\n\\begin{rem} \\label{strem} Lemma \\ref{m2lem1} strengthens the\n    aforementioned result of Stembridge.  Specifically, if $y$ covers\n    $x$, then $y=z$, i.e., $y$ is obtainable from $x+\\alpha$ by\n    playing the numbers game with a cutoff.  In this case, $y = x +\n    \\beta$, where $\\beta \\in \\Delta_+$ is obtained from $\\alpha$ by\n    playing the numbers game (using the same firing sequence as\n    for $x+\\alpha \\mapsto x+\\beta$, which involves firing only\n    vertices of amplitude $-1$).  This was our motivation for\n    replacing $\\mu$ by $y$ in the statement of the lemma.\n\\end{rem}\n\n\n\n\\subsection{The case $m = 2$ of Theorem\n  \\ref{main2s}.(ii)} \\label{ss:main2smeq2} The heart of the proof of\nTheorem \\ref{main2s}.(ii) is contained in the case $m=2$. Then,\ngeneral $m$ will be a straightforward generalization.  In turn, the\ncase $m=2$ is based on the following lemma.\n\\begin{lemma} \\label{m2lem2} Let $P_1$ and $P_2$ be special ample\n  polytopes, $(x_1, x_2) \\in \\Lambda(P_1) \\times \\Lambda(P_2)$, $P =\n  P_1 + P_2$, and $x = x_1 + x_2 \\in \\Lambda(P)$.  If $\\alpha$ is\n  $P$-progressive for $x$, then there exists $(x_1', x_2')$ and an\n  index $i\\in\\{1,2\\}$ such that $(x_1, x_2) \\sim (x_1', x_2')$ and\n  $\\alpha$ is $P_i$-progressive for $x_i'$.\n\\end{lemma}\nThis will be proved in \\S \\ref{ss:m2lem2-proof} below. Here, we explain\nhow it implies Theorem \\ref{main2s}.(ii) in the case $m=2$.\n\\begin{proof}[Proof of Theorem \\ref{main2s}.(ii) for $m=2$] As remarked\n  earlier, it is enough to prove the theorem in the case that $x$ is\n  winning. Let $\\mu = \\mu_1 + \\mu_2$ (where by convention $\\mu_i$ is\n  the vertex of $P_i$ corresponding to the dominant chamber).  The\n  theorem is immediate in the case that $x = \\mu$, since then $x = x_1\n  + x_2$ implies that $x_1 = \\mu_1$ and $x_2 = \\mu_2$ (and\n  vice-versa).  Inductively, suppose that $x \\in \\Lambda(P)$ is\n  winning, and for some $P$-progressive $\\alpha$, the theorem holds\n  for $x+\\alpha$.\n\n \n \n \n \n \n \n \n\n  Suppose that $(x_1, x_2), (x_1', x_2') \\in \\Lambda(P_1) \\times\n  \\Lambda(P_2)$ are pairs such that $x_1 + x_2 = x = x_1'+x_2'$.  Let\n  $\\alpha$ be $P$-progressive for $x$. By Lemma \\ref{m2lem2} (applied\n  to both $(x_1, x_2)$ and $(x_1', x_2')$ separately), it is enough to\n  assume that there exist indices $i$ and $j$ such that $\\alpha$ is\n  $P_i$-progressive for $x_i$ and $P_j$-progressive for $x_j'$.\n  Without loss of generality, suppose that $i = 1$.  Let $(y_1, y_2)$\n  and $(y_1', y_2')$ be given by $y_1 = x_1+\\alpha$, $y_2 = x_2$,\n  $y_j' = x_j' + \\alpha$, and $y_k' = x_k'$ for $k \\neq j$.  Since\n  $x_1+x_2+\\alpha = x + \\alpha = x_1' + x_2'+\\alpha$, by hypothesis,\n  $(y_1, y_2) \\sim (y_1', y_2')$.  By induction on the number of root\n  moves \\eqref{eq:equivmove} required to realize the latter\n  equivalence, Lemma \\ref{m2lem3} below then implies that either\n  $(x_1,x_2) \\sim (y_1'-\\alpha, y_2') \\in \\Lambda(P_1) \\times\n  \\Lambda(P_2)$ or $(x_1, x_2) \\sim (y_1', y_2' - \\alpha) \\in\n  \\Lambda(P_1) \\times \\Lambda(P_2)$ (where, for the purposes of\n  induction, we drop the assumption that $\\alpha$ is $P_1$-progressive\n  for $x_1$ and assume only that $x_1+\\alpha \\in \\Lambda(P_1)$, and\n  similarly for $x_j'$ and $P_j$). If the result is $(x_1', x_2')$, we\n  are done. If not, the result must be $(x_1' \\pm \\alpha, x_2' \\mp\n  \\alpha)$, which is related to $(x_1', x_2')$ by a single root move.\n\\end{proof}\nAbove we needed the following lemma, whose proof will be given in \\S\n\\ref{ss:m2lem3-proof}.\n\\begin{lemma} \\label{m2lem3} Suppose that $(x_1, x_2) \\in \\Lambda(P_1)\n  \\times \\Lambda(P_2)$, and $\\alpha$ is a simple root such that\n  $x = x_1+x_2$ satisfies \n  $\\langle x, \\alpha^\\vee \\rangle \\geq -1$, \n \n \n  and such that $x_1 + \\alpha \\in \\Lambda(P_1)$.  If $\\beta \\in\n  \\Delta$ is such that $(x_1+\\alpha + \\beta, x_2-\\beta) \\in\n  \\Lambda(P_1) \\times \\Lambda(P_2)$, then either $(x_1 + \\beta, x_2 -\n  \\beta) \\in \\Lambda(P_1) \\times \\Lambda(P_2)$ or $(x_1 + \\alpha +\n  \\beta, x_2 - \\alpha - \\beta) \\in \\Lambda(P_1) \\times \\Lambda(P_2)$.\n  In the latter case, either $\\alpha + \\beta \\in \\Delta$, or $(x_1 +\n  \\alpha, x_2 - \\alpha) \\in \\Lambda(P_1) \\times \\Lambda(P_2)$.\n\\end{lemma}\n\\subsection{Proof of Theorem \\ref{main2s} for general\n  $m$}\\label{ss:main2smgen}\nLet $\\mu = \\mu_1 +\n  \\cdots + \\mu_m$, where $\\mu_i$ is the vertex of $P_i$ corresponding\n  to the dominant chamber. The theorem is immediate in the case\n  $x=\\mu$.  It is enough to prove the theorem when $x$ is winning,\n  under the inductive hypothesis that the theorem holds for $x +\n  \\alpha$ where $\\alpha$ is $P$-progressive for $x$.\n\n  To prove part (i), suppose that $(y_1, \\ldots, y_m) \\in \\Lambda(P_1)\n  \\times \\cdots \\times \\Lambda(P_m)$ with $y_1 + \\cdots + y_m =\n  x+\\alpha$. Then for some index $i$, $\\langle y_i, \\alpha^\\vee \\rangle\n  \\geq 1$, and by Proposition \\ref{mainclaim}, $(y_1, \\ldots, y_{i-1},\n  y_i-\\alpha, y_{i+1}, \\ldots, y_m) \\in \\Lambda(P_1) \\times \\cdots\n  \\times \\Lambda(P_m)$, with the desired sum $x$.\n\n  For part (ii), we will additionally induct on $m$, i.e., we assume\n  that the theorem holds for smaller values of $m$.  Let $Q := P_1 +\n  \\cdots + P_{m-1}$, so that $P = Q + P_m$. Let $y = x_1 + \\cdots +\n  x_{m-1}$ and $y' = x_1' + \\cdots + x_{m-1}'$.  Then, by the previous\n  subsection, there exist root moves relating $(y, x_m)$ to $(y',\n  x_m')$.  To turn this into root moves relating $(x_1, \\ldots, x_m)$\n  and $(x_1', \\ldots, x_m')$, it is enough to apply the theorem for\n  the case $m-1$ (i.e., for $(P_1, \\ldots, P_{m-1})$) together with\n  the following lemma.\n  \\begin{lemma} \\label{l:main2s-indlem} Suppose that $y \\in Q = P_1 +\n    \\cdots + P_{m-1}$, $\\beta \\in \\Delta$, and $y + \\beta \\in Q$.\n    Assume Theorem \\ref{main2s}.(i) holds for $(P_1, \\ldots,\n    P_{m-1})$. Then, there exists a tuple $(y_1, \\ldots, y_{m-1}) \\in\n    \\Lambda(P_1) \\times \\cdots \\times \\Lambda(P_{m-1})$ such that $y =\n    y_1 + \\cdots + y_{m-1}$ and an index $j$ such that $y_j + \\beta\n    \\in \\Lambda(P_j)$.\n\\end{lemma}\nThe lemma will be proved in \\S \\ref{ss:l:main2s-indlem-proof}.\n\n\n\\section{Proof of lemmas} \\label{s:lemmaproofs}\n\\subsection{Proof of Lemma \\ref{m2lem1}} \\label{ss:m2lem1-proof} We\nwill use the following general result:\n\\begin{claim}\\label{baswincl}\n  If $x \\in {\\mathbb Z}^I$ is dominant and $\\alpha \\in \\Delta_+$ is any\n  positive root, and the usual numbers game on $x+\\alpha$ does not\n  involve firing any vertices corresponding to simple roots shorter\n  than $\\alpha$, then $x+\\alpha$ is winning.\n\\end{claim}\nIn particular, if $\\alpha$ is a short positive root and $x \\in {\\mathbb Z}^I$\nis dominant then $x+\\alpha$ is winning (equivalently, all short\npositive roots are themselves winning).\n\\begin{proof}\n  Let us play the usual numbers game on $x + \\alpha$.  If we fire a\n  vertex $i$ corresponding to a simple root $\\beta$ whose length is at\n  least that of $\\alpha$, then since $\\langle \\alpha, \\beta^\\vee\n  \\rangle \\geq -1$, the amplitude at vertex $i$ is $-1$. Since the\n  length of $\\alpha + \\beta$ is equal to that of $\\alpha$, we can\n  replace $\\alpha$ with $\\alpha+\\beta$, and then $x+(\\alpha+\\beta)$\n  takes one fewer move under the numbers game to reach a dominant\n  configuration. By induction on the number of moves required to play\n  the numbers game on $x + \\alpha$, we see that all vertices fired\n  have amplitude $-1$, and hence $x + \\alpha$ is winning.\n\\end{proof}\nSuppose that $y \\in X \\cap D$ and $\\alpha$ is a positive root of\nminimum length such that $x+\\alpha \\preceq y$.  Let us play the\nnumbers game with a cutoff on $x+\\alpha$.  We claim that this only\ninvolves firing vertices corresponding to simple roots of length at\nleast $\\alpha$.  Then, by Claim \\ref{baswincl}, $x+\\alpha$ is winning.\nMoreover, the result $z$ of playing the numbers game with a cutoff is\nthe dominant configuration obtained from $x+\\alpha$ by adding the\nminimum positive combination of simple roots.  Since $y$ is dominant\nand $x+\\alpha \\preceq y$, $y$ is also such a configuration, and it\nfollows that $z \\preceq y$.\n\nIt remains to show that playing the numbers game beginning with\n$x+\\alpha$ does not involve firing a vertex corresponding to a simple\nroot of length shorter than $\\alpha$. For a contradiction, suppose\nnot, and consider the first vertex fired corresponding to a shorter\nsimple root. Call this simple root $\\gamma$.  It follows as above that\nevery dominant configuration is obtainable from $x+\\alpha$ by adding\nsimple roots adds $\\gamma$ as well.  Therefore, since $x+ \\alpha\n\\preceq y$ and $y$ is dominant, also $x +\\alpha+\\gamma \\preceq y$ and\nhence $x+\\gamma \\preceq y$, which is a contradiction.\n\n\n\n\\subsection{Proof of Lemma \\ref{m2lem2}} \\label{ss:m2lem2-proof}\nFirst, if $x$ is not dominant, then $\\langle x, \\alpha^\\vee \\rangle\n\\leq -1$, and hence $\\langle x_i, \\alpha^\\vee \\rangle \\leq -1$ for\nsome $i$, which shows that $\\alpha$ is $P_i$-progressive for $x_i$.  So\nwe can restrict to the dominant case.  Thus, $\\alpha$ has minimal\nlength among simple roots such that $x+\\alpha \\preceq \\mu$.\n\n\nGiven a simple root $\\alpha$, let $P^\\alpha$ denote the\nmaximum-dimensional boundary facet of $P$ meeting $\\mu$ which is\nparallel to the span of all simple roots other than $\\alpha$.  In\nother words (using Lemma \\ref{one}), $x \\in \\Lambda(P^\\alpha)$ if and\nonly if $x \\in \\Lambda(P)$ but $x + \\alpha \\not \\preceq \\mu$.\n\\begin{claim}\\label{m2lem2cl}\nIf any element $x_i$ of the pair $(x_1, x_2)$ is dominant,\nthen either $x_i \\in \\Lambda(P^\\alpha_i)$, or else $\\alpha$ is\n$P_i$-progressive for $x_i$.\n\\end{claim}\n\\begin{proof}\n  If $x_i \\notin \\Lambda(P^\\alpha_i)$, then $\\alpha$ must be of\n  minimal length with this property, since $x_i \\notin\n  \\Lambda(P^\\beta_i)$ implies that $x_1 + x_2 \\notin\n  \\Lambda(P^\\beta)$, which implies by assumption that $\\beta$ is at\n  least as long as $\\alpha$.\n\\end{proof}\nNow, we prove the lemma. If, for any simple root $\\beta$, $\\langle\nx_1, \\beta^\\vee \\rangle \\leq -1$ but $\\langle x_2, \\beta^\\vee \\rangle\n\\geq 1$, we can perform a move $(x_1, x_2) \\mapsto (x_1 + \\beta, x_2 -\n\\beta)$.  So, after performing such moves, we can assume that this\ndoes not happen. Since $x = x_1 + x_2$ is dominant, this implies that\n$x_1$ is dominant.  By Claim \\ref{m2lem2cl}, we are done unless $x_1\n\\in P_1^\\alpha$. So assume this is the case. By performing moves of\nthe form $(x_1, x_2) \\mapsto (x_1 - \\beta, x_2 + \\beta)$ for simple\nroots $\\beta \\neq \\alpha$ (which may make $x_1$ no longer dominant,\nbut preserves the property that $x_1 \\in P_1^\\alpha$), we can assume\nthat $\\langle x_2, \\beta^\\vee \\rangle \\geq 0$ for all simple roots\n$\\beta \\neq \\alpha$, without changing the assumption that $x_1 \\in\nP_1^\\alpha$.  Then, either $\\langle x_2, \\alpha^\\vee \\rangle \\leq -1$,\nor $x_2$ is dominant. In the former case, $\\alpha$ is\n$P_2$-progressive for $x_2$, as desired. In the latter case, by Claim\n\\ref{m2lem2cl}, it is enough to suppose that $x_2 \\in\nP_2^\\alpha$. However, in this case, $x=x_1 + x_2 \\in P_1^\\alpha +\nP_2^\\alpha = P^\\alpha$, contradicting our assumption that $x + \\alpha\n\\preceq \\mu$.\n\n\n\n\n\\subsection{Proof of Lemma \\ref{m2lem3}} \\label{ss:m2lem3-proof}\nFirst, if $\\langle x_1 + \\alpha + \\beta, \\alpha^\\vee \\rangle \\geq 1$,\nthen $x_1 + \\beta\\in \\Lambda(P_1)$ by Proposition \\ref{mainclaim}.\nSince $x_2 -\\beta \\in \\Lambda(P_2)$ by assumption, this proves the\nlemma.  Next, suppose that $\\langle x_1 + \\alpha + \\beta, \\alpha^\\vee\n\\rangle \\leq 0$, i.e., $\\langle x_1 + \\beta, \\alpha^\\vee \\rangle \\leq\n-2$. Since $x = (x_1 + \\beta) + (x_2 - \\beta)$ satisfies\n$\\langle x, \\alpha^\\vee \\rangle \\geq -1$, it follows that $\\langle\nx_2 - \\beta, \\alpha^\\vee \\rangle \\geq 1$.  By Proposition\n\\ref{mainclaim}, $x_2 - \\beta - \\alpha \\in \\Lambda(P_2)$. Since $x_1 +\n\\alpha + \\beta \\in \\Lambda(P_1)$ by assumption, this proves that $(x_1\n+ \\alpha + \\beta, x_2 - \\beta - \\alpha) \\in \\Lambda(P_1) \\times\n\\Lambda(P_2)$. It remains to prove the final assertion. Suppose that\n$(x_1 + \\alpha, x_2 - \\alpha) \\notin \\Lambda(P_1) \\times\n\\Lambda(P_2)$. By assumption $x_1 + \\alpha \\in \\Lambda(P_1)$, so $x_2\n- \\alpha \\notin \\Lambda(P_2)$.  In view of Proposition\n\\ref{mainclaim}, $\\langle x_2, \\alpha^{\\vee}\\rangle \\leq 0$.  Since\n$\\langle x_2 - \\beta, \\alpha^{\\vee} \\rangle \\geq 1$ (as observed\nabove), this implies that $\\langle \\beta, \\alpha^\\vee \\rangle \\leq\n-1$. In this case, $\\alpha + \\beta$ must be a root.\n\n\\subsection{Proof of Lemma\n  \\ref{l:main2s-indlem}}\\label{ss:l:main2s-indlem-proof}\nFirst, consider the case that $\\langle y, \\beta^\\vee \\rangle <\n0$. Then, we can let $(y_1, \\ldots, y_{m-1}) \\in \\Lambda(P_1) \\times\n\\cdots \\times \\Lambda(P_{m-1})$ be arbitrary such that $y = y_1 +\n\\cdots + y_{m-1}$, and then for some $j$ one must have $\\langle y_j,\n\\beta^\\vee \\rangle < 0$ as well, so that $y_j + \\beta \\in\n\\Lambda(P_j)$ by Proposition \\ref{mainclaim}. Similarly, if $\\langle\ny, \\beta^\\vee \\rangle \\geq 0$, then $\\langle y + \\beta, \\beta^\\vee\n\\rangle > 0$, and we can take any $(z_1, \\ldots, z_{m-1}) \\in\n\\Lambda(P_1) \\times \\cdots \\times \\Lambda(P_{m-1})$ such that $y +\n\\beta = z_1 + \\cdots + z_{m-1}$. Then, there exists some $j$ such that\n$\\langle z_j, \\beta^\\vee \\rangle > 0$, so again by Proposition\n\\ref{mainclaim}, $z_j - \\beta \\in \\Lambda(P_j)$. Hence, the tuple\n$(z_1, \\ldots, z_{j-1}, z_j - \\beta, z_{j+1}, \\ldots, z_m)$ satisfies\nthe needed conditions.\n\n\n\n\n\n\n\n\n\n\\section{Ample polytopes are not diagonally split, after\n  Payne}\\label{s:not-diag-split}\n\nAs mentioned in the introduction, Payne (in \\cite{payne}) considers a\ntoric variety, $U$, similar to the one we consider, $V$, but for which\nthe rays of the fan are $\\mathbb{R}_{\\geq 0} \\cdot \\alpha$, for all $\\alpha\n\\in \\Delta$.  He proves that, in types $A, B, C$, and $D$, for all\nlattice polytopes $P$ corresponding to a torus-equivariant line bundle\non $U$ (even if not ample), the corresponding semigroup\n$S_P$ is normal, and the ring $\\mathbb{C}[S_P]$ is\nKoszul.  This follows from the fact, that he proves, that such lattice\npolytopes are \\emph{diagonally split} for some integer $q \\geq 2$.\n\nHere, we show that ample polytopes for the varieties $V$ considered in\nthis paper are diagonally split for some integer $q \\geq 2$ only in\nthe cases $A_1, A_2, A_3$, and $B_2 (= C_2)$.\n\nRecall from \\cite{payne} the following definition.  Let $\\Gamma$ be a\nlattice with dual lattice $\\Gamma^\\vee$, and let $\\Gamma_\\mathbb{R} := \\Gamma\n\\otimes_{\\mathbb Z} \\mathbb{R}$ and $\\Gamma^\\vee_\\mathbb{R} := \\Gamma^\\vee \\otimes_{\\mathbb Z} \\mathbb{R}$. Let\n$P \\subseteq \\Gamma \\otimes_{\\mathbb Z} \\mathbb{R}$ be a lattice polytope (with\nvertices in $\\Gamma$). Let $v_1, \\ldots, v_k \\in \\Gamma^\\vee$ be the\nprimitive lattice generators of the inward normal rays of the facets\nof $P$. Define\n\n\\begin{equation}\n  \\mathbb{F}_P^{\\circ} := \\{u \\in \\Gamma_\\mathbb{R}  \\mid -1 < \\langle u, \n  v_i \\rangle  < 1, \\forall i =1, \\ldots, k\\}.\n\\end{equation}\nLet $q \\geq 2$ be an integer.  Then, $P$ is \\emph{diagonally split}\nfor $q$ if and only if every element $z \\in (\\frac{1}{q}\n\\Gamma)\/\\Gamma$ has a representative $\\tilde z \\in\n\\mathbb{F}_P^{\\circ} \\cap \\frac{1}{q} \\Gamma$.\n\nNote that, in our case, $\\Gamma = Y$. It is clear that all lattice\npolytopes corresponding to equivariant line bundles on a toric variety\nare diagonally split if and only if the polytopes corresponding to\nample bundles are diagonally split. Moreover, such polytopes are\ndiagonally split if and only if any one such polytope is diagonally\nsplit.\n\n\\begin{prop}\n  An ample polytope \\emph{(}in $Y_\\mathbb{R}$, with vertices in\n  $Y$\\emph{)} is diagonally split for some $q \\geq 2$ if and only if\n  the root system is of type $A_1, A_2, A_3$, or $B_2 (= C_2)$.  For\n  $A_1$ and $A_2$, ample polytopes are diagonally split for all\n  $q \\geq 2$, and for $A_3$ and $B_2$, ample polytopes are\n  diagonally split for odd but not even $q \\geq 2$.\n\\end{prop}\n\n\\begin{proof}\n  The inward primitive normal vectors for an ample polytope\n  are the images under the Weyl group of the fundamental coweights\n  $\\omega_i, i \\in I$.  Hence, the polytope is diagonally split for\n  $q$ if and only if, for all $z \\in \\frac{1}{q} Y \/ Y$, there is a\n  representative $\\tilde z \\in \\frac{1}{q} Y$ such that $-1 < \\langle\n  w \\tilde z, \\omega_i \\rangle < 1$ for all $i \\in I$ and all $w \\in\n  W$.\n\n  We first prove that such polytopes are not diagonally split if the\n  root system is not listed above. Such root systems contain, as a\n  subsystem, either a root system of type $A_4$, $D_4$, $B_3$, $C_3$,\n  or $G_2$.  It is clear that, for this direction, it suffices to show\n  that, for every $q \\geq 2$, ample polytopes for these\n  four root systems are not diagonally split.  To do so, it suffices\n  to exhibit in each of these cases a particular element $z \\in\n  \\frac{1}{q} Y \/ Y$ such that, for all representatives $\\tilde z \\in\n  \\frac{1}{q} Y$, there exists $w \\in W$ and $i \\in I$ such that $|\n  \\langle w \\tilde z, \\omega_i \\rangle | \\geq 1$.\n\nWe use the standard labeling of roots as in \\cite[\\S VI.4]{bourbaki} (which we\nwill also recall). Also, for\nevery $i \\in I$, we denote by $s_i$ the simple reflection\ncorresponding to the simple root $\\alpha_i$.\n\nFirst let us consider a root system of type $A_3$ and even $q$, and\nshow that $P$ is not diagonally split. Recall that, for $A_n$ type,\nthe simple roots $\\alpha_1, \\ldots, \\alpha_n$ are linearly ordered\nalong a line segment. We consider the element $z := \\frac{1}{2}\n\\alpha_1 + \\frac{1}{2} \\alpha_3$.  Then, for every $\\tilde z \\in z +\nY$, either $|\\langle \\tilde z, \\omega_i \\rangle| \\geq 1$ for some $i$,\nor $\\tilde z$ is in the same Weyl orbit as $z$.  But, $\\langle s_2 z,\n\\omega_2 \\rangle = 1$, which yields the desired inequality.  In\nparticular, ample polytopes for any root system containing $A_3$ are\nnot diagonally split for even $q$. Also, the same is true for root\nsystems containing $B_3$ or $C_3$.  Thus, for the cases $A_4, D_4,\nB_3$, and $C_3$, it suffices to restrict our attention to the case\nwhere $q$ is odd.\n\nFrom now on, fix an odd integer $q \\geq 3$ and set $p :=\n\\frac{q-1}{2}$.  Suppose that the root system is of type $A_4$. Then,\nwe consider the element\n\\begin{equation}\n  z := \\frac{p+1}{q} \\alpha_1 + \\frac{p+1}{q} \\alpha_3 + \n  \\frac{1}{q} \\alpha_4.\n\\end{equation}\nThe only elements $\\tilde z \\in z + \\frac{1}{q} Y$ that we need to\nconsider are the eight elements\n\\begin{equation}\n  \\tilde z = z - \\delta_1 \\alpha_1 -\\delta_3 \\alpha_3 - \\delta_4 \\alpha_4, \n  \\quad \\delta_i \\in \\{0,1\\}.\n\\end{equation}\nFirst, consider the case that $(\\delta_1, \\delta_3) \\neq (1,1)$. If\nalso $(\\delta_1, \\delta_3) \\neq (0,0)$, then $|\\langle s_2 s_1\n\\widetilde{z}, \\omega_2 \\rangle| = 1$.  If $(\\delta_1, \\delta_3) =\n(0,0)$, then $\\langle s_2 \\widetilde{z}, \\omega_2 \\rangle =\n\\frac{q+1}{q} > 1$.\n\n\n\nNext, consider the case that $\\delta_1=\\delta_3=1$ and\n$\\delta_4=0$. Then, $s_3 \\tilde z = -\\frac{p}{q} \\alpha_1 +\n\\frac{p+1}{q} \\alpha_3 + \\frac{1}{q} \\alpha_4$, which is a case we\nalready considered in the preceding paragraph.\n\nThus, it remains to consider the case\n$\\delta_1=\\delta_3=\\delta_4=1$. Then, $s_4 \\tilde z =\n-\\frac{p}{q} \\alpha_1 -\\frac{p}{q} \\alpha_3 + \\frac{p}{q} \\alpha_4$.\nHence, $s_3 s_4 \\tilde z = -\\frac{p}{q} \\alpha_1 + \\frac{q-1}{q}\n\\alpha_3 + \\frac{p}{q} \\alpha_4$.  Finally, $s_2 s_1 s_3 s_4 \\tilde z\n= \\frac{p}{q} \\alpha_1 + \\frac{p+q-1}{q} \\alpha_2 + \\frac{q-1}{q}\n\\alpha_3 + \\frac{p}{q} \\alpha_4$, and hence $\\langle s_2 s_1 s_3 s_4\n\\tilde z , \\omega_2 \\rangle \\geq 1$.  \n\nHence, ample polytopes for root systems containing $A_4$ are not\ndiagonally split for odd $q \\geq 3$, and together with the even case\nabove, they are not diagonally split for any $q \\geq 2$.\n\nNext, consider the root system $D_4$.  As in \\cite[\\S VI.4]{bourbaki},\n$\\alpha_2$ is the simple root corresponding to the node, and\n$\\alpha_1, \\alpha_3$, and $\\alpha_4$ are the other simple\nroots. Define the element\n\\begin{equation}\nz = \\frac{p}{q} (\\alpha_1 + \\alpha_3 + \\alpha_4).\n\\end{equation}\nSimilarly to the $A_4$ case, we only need to consider the elements\n\\begin{equation}\n  \\tilde z = z - \\delta_1 \\alpha_1 - \\delta_3 \\alpha_3 - \\delta_4 \\alpha_4, \n  \\quad \\delta_i \\in \\{0,1\\}.\n\\end{equation}\nIf $\\delta_1=\\delta_3=\\delta_4$ then we see that $|\\langle s_2 \\tilde\nz, \\omega_2 \\rangle| \\geq 1$.  For the other cases, using symmetry, we\nmay assume that $\\delta_1 = \\delta_3 = 0$ and $\\delta_4 = 1$.  Then,\n$|\\langle s_2 s_4 \\tilde z, \\omega_2 \\rangle | > 1$. Hence, ample\npolytopes containing $D_4$ are not diagonally split.\n\nConsider now the root system $B_3$, with simple roots $\\alpha_1,\n\\alpha_2, \\alpha_3$, so that $\\alpha_2$ corresponds to the central\nvertex and $\\alpha_3$ is the short simple root. Let $z = \\frac{p}{q}\n(\\alpha_1 + \\alpha_3)$. Then, $|\\langle s_3 s_2 z, \\omega_3 \\rangle|\n\\geq 1$, and the same is true if we replace $z$ by $z - (\\alpha_1 +\n\\alpha_3)$, $s_1 (z - \\alpha_1)$, or $s_3 (z - \\alpha_3)$.  This\nproves the desired inequality, so that ample polytopes containing\n$B_3$ are not diagonally split.\n\nSimilarly, consider the root system $C_3$, with simple roots\n$\\alpha_1, \\alpha_2, \\alpha_3$ such that $\\alpha_2$ corresponds to the\ncentral vertex and $\\alpha_3$ is the long simple root. Let $z :=\n\\frac{p}{q} (\\alpha_1 + \\alpha_3)$. Then, $|\\langle s_2 z, \\omega_2\n\\rangle | \\geq 1$, and the same is true if we replace $z$ by $z -\n(\\alpha_1 + \\alpha_3)$, $s_1 (z - \\alpha_1)$, or $s_3 (z - \\alpha_3)$.\n\nFinally, consider the root system $G_2$, and now allow $q \\geq 2$ to\nbe any integer. Let $p = \\lfloor \\frac{q}{2} \\rfloor$.  Let $\\alpha_1$\nbe the short simple root and $\\alpha_2$ be the long simple root.\nConsider $z := \\frac{p}{q} \\alpha_2$.  Then, $|\\langle s_1 z, \\omega_1\n\\rangle | \\geq 1$. The same is true if we replace $z$ by $s_2(z -\n\\alpha_2)$.  This proves that ample polytopes are not diagonally split\nfor $G_2$.\n\n\nThis completes the proof that ample polytopes for root systems other\nthan $A_1, A_2, A_3$, and $B_2$ are not diagonally split for any $q\n\\geq 2$.  We claim also that ample polytopes are not diagonally split\nin the case where $q$ is even and the root system is of type $B_2$.\nFor this, let $\\alpha_1$ be the long simple root and $\\alpha_2$ be the\nshort simple root.  Consider the element $z = \\frac{1}{2} \\alpha_1$.\nThen, the same argument as in the case $G_2$ applies.\n\nIt remains to prove the claims that ample polytopes are diagonally\nsplit for odd $q$ in the cases $A_1, A_2, A_3$, and $B_2$, and in the\ncase of $A_1$ and $A_2$, also for even $q$. For the case $A_1$, this\nis clear, and in the case $A_2$, it follows by choosing, for any $z\n\\in \\frac{1}{q} Y \/ Y$, the representative $\\tilde z \\in \\frac{1}{q}\nY$ such that $\\langle \\tilde z, \\omega_i \\rangle \\in [0,1)$ for both\nfundamental coweights $\\omega_i$.  Next, consider the case $B_2$, and\nlet $q \\geq 3$ be odd.  Let $\\alpha_1$ be the long root and $\\alpha_2$\nbe the short root.  Then, for any $z \\in \\frac{1}{q} Y \/ Y$, choose\nthe representative $\\tilde z \\in \\frac{1}{q} Y$ such that $|\\langle\n\\tilde z, \\omega_1 \\rangle| < \\frac{1}{2}$, $|\\langle \\tilde z,\n\\omega_2 \\rangle| < 1$, and $\\langle \\tilde z, \\omega_1 \\rangle$ and\n$\\langle \\tilde z, \\omega_2 \\rangle$ are either both nonnegative or\nboth nonpositive.  It is easy to verify that $\\tilde z \\in\n\\mathbb{F}_P^{\\circ} \\cap \\frac{1}{q} \\Gamma$, as required.\n\nFinally, consider the case $A_3$ with $q$ odd. Let\n$\\alpha_1, \\alpha_2$, and $\\alpha_3$ be the simple roots, with\n$\\alpha_2$ corresponding to the central vertex.  Then, for any $z \\in\n\\frac{1}{q} Y \/ Y$, first suppose that $\\langle \\tilde z, \\omega_2\n\\rangle$ is integral for all representatives $\\tilde z$ of $z$. In\nthis case, choose the representative $\\tilde z$ so that $\\langle\n\\tilde z, \\omega_2 \\rangle = 0$ and $|\\langle \\tilde z, \\omega_i\n\\rangle| < \\frac{1}{2}$ for $i \\in \\{1,3\\}$.  Otherwise, if $\\langle\n\\tilde z, \\omega_2 \\rangle$ is not integral for any representative\n$\\tilde z$ of $z$, choose $\\tilde z$ such that $|\\langle \\tilde z,\n\\omega_i \\rangle | < 1$ for all $i$, either all $\\langle \\tilde z,\n\\omega_i \\rangle$ are nonnegative or all are nonpositive, and such\nthat $|\\langle \\tilde z, \\omega_1 + \\omega_3 \\rangle | \\leq 1$ (where\n$\\alpha_2$ corresponds to the central vertex).  A straightforward\ncomputation verifies that this yields a diagonal splitting.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{figure*}[!t]\n\\plotone{f1}\n\\caption{The co-added COS data, plotted versus observed wavelength, and binned by 3 raw COS pixels to optimal sampling of 2 bins per resolution element. The Lyman limit system analyzed in this paper and the DLA from \\cite{Meiring:11:1} are marked at the positions of Lyman series lines (green), metal lines (blue), and \\ion{O}{6} (red). The Lyman limit of the present system is clearly visible as a complete absorption of the QSO spectrum at $\\sim 1240$ \\AA. Geocoronal Ly$\\alpha$ emission has been excised near $1216$ \\AA. The solid and dashed curves mark the opacity of the Lyman limit system using total column densities of $\\log N$(\\ion{H}{1}) = 17.8 and 18.0, respectively, as described in \\S~3.5. \\label{wholespecfig}}\n\\vspace{0.05in}\n\\end{figure*}\n\nA full understanding of how galaxies acquire their gas from the IGM and return it there in the form of chemical and kinematic feedback will likely form an important part of any complete picture of galaxy formation. Motivated by questions of how galaxies obtain their observed stellar masses and morphology, theorists have developed a picture in which gas enters dark matter halos and galaxies by either cold flows along filaments \\citep{Keres:05:2a}, accretion of hot material that has passed through a shock on its way in from the IGM \\citep{Dekel:06:2}, or in some complex, multiphase mixture of the two \\citep{Maller:04:694}. Mass loss by feedback is driven in ``superwind'' ouflows fueled by many correlated supernovae, by radiation pressure on dust \\citep{Murray:05:569}, or in AGN-triggered flows in those galaxies that possess active nuclei. Tidal or ram-pressure stripping of gas from satellite galaxies during gravitational encounters may also provide a gas supply to larger galaxies, or at least to their halos. All these gas processes may bear on such observational features of galaxies as their stellar masses, morphologies, and colors, but to be effective they must act across the 100 kpc scales of galaxy halos. Unfortunately the proposed accretion, feedback, and stripping processes are difficult to test directly because diffuse, ionized gas in the immediate vicinity of galaxies is difficult to detect. We therefore have an incomplete empirical picture of how accreting gas is distributed around galaxies of all types, what its temperature, density, and metallicity configurations look like, and how these features influence or are influenced by the host galaxy. \n\nThe classic quasar absorption-line technique provides a method for studying intergalactic and circumgalactic medium gas, even to very low density and metallicity. This technique has been used effectively to probe diffuse ionized gas in galaxy halos,  over long pathlengths through the IGM, and throughout the halo of the Milky Way. The IGM samples reveal the statistical correlations of galaxies with intervening absorbers, and in some cases show physical properties of gas that is well within the virial radius. However, most such information comes from {\\it post-facto} galaxy surveys that do not uniformly sample the range of galaxy properties that may be related to the gas properties. While extensive observational information exists about the quantity and physical state of multiphase high-velocity cloud (HVC) gas in the Milky Way halo \\citep{Sembach:03:165, Fox:05:332, Collins:05:196, Shull:09:754}, and gas stripping from dwarf galaxies is clearly demonstrated in the case of the Magellanic Clouds and Stream, it is not known how typical the Milky Way is in this regard. \n\nTo address the problem of how halo gas in other galaxies is distributed and how it compares to the halo gas of the Milky Way, we have begun an effort to systematically survey the gaseous halos of low-redshift galaxies using the Cosmic Origins Spectrograph (COS) aboard the {\\it Hubble Space Telescope}. Our survey chose galaxies with sightlines to UV-bright QSOs passing through their halos, with redshifts tuned to place the $\\lambda\\lambda$1032,1038 doublet of \\ion{O}{6} near the peak of COS sensitivity at $1250-1350$ \\AA. This survey design allows us access to all the key far-UV ionization diagnostics from which gas budgets, metallicities, and kinematics can be inferred. As part of this survey we are also obtaining galaxy spectra and high-resolution optical spectra of the QSOs. The main results of this survey will be published elsewhere; here we report on a serendipitous discovery from this program that nevertheless addresses its main goal: to examine the gaseous fuel and\/or waste residing in galactic halos. \n\nThis paper reports new HST\/COS and HST\/WFC3 and Keck HIRES and LRIS data for the sightline to \\objectname{J1009+0713}. This sightline exhibits a strong Lyman-limit system (LLS) bearing a wide range of ionic absorption from multiple ionization stages and in several distinct kinematic components. The system was discovered serendipitously in the data on this quasar obtained as part of our survey; the absorption-line system associated with the targeted galaxy along this sightline will be published separately. This paper first describes the data collection and analysis for the HST\/COS and Keck\/HIRES data on the sightline to the QSO, and on our HST\/WFC3 and Keck\/LRIS data on the galaxies in the field (\\S~2). We then report the details of our line identification and analysis (\\S~\\ref{lls-section}), and on the analysis of the galaxy images and spectra (\\S~\\ref{gal-section}). Section 5 describes physical models for the absorber. In Section~\\ref{interp-section} we describe our general results and their significance for the larger picture of gas in galaxy halos. We adopt the WMAP7 cosmological parameters $H_0 = 100h= 71$ km s$^{-1}$\\ Mpc$^{-1}$, $\\Omega _{\\Lambda} = 0.734$, and $ \\Omega _{m} h^2= 0.1334$ \\citep{Komatsu:10:4538}. \n\n\\section{Observations}\n\n\\begin{figure*}\n\\begin{center} \n\\includegraphics[width=5.0 in]{f2}\n\\caption{The galaxy field, aligned with north at top and east to the left. The SDSS Skyserver image and our WFC3 image are shown for contrast.  The larger panels are approximately 25 arcsec across. Galaxies 170\\_9 and 86\\_4 are at $z = 0.356$, the redshift of the LLS. At the redshift of the system, the $5\"$ range bar subtends $25h^{-1}$ kpc. Precise impact parameters for the galaxies are given in Table~2. The insets show zoomed images of the three galaxies of interest and the QSO itself. \\label{fieldfig}}\n\\end{center} \n\\vspace{0.05in}\n\\end{figure*}\n\n\n\\subsection{COS Data} \n\nData on the SDSSJ100902.06+071343.8 sightline (hereafter J1009+0713) were obtained by COS over 3 orbits on 29-30 Mar 2010 as Visit 13 in Program GO11598. Two exposures were obtained with the FUV G130M grating at central wavelength settings 1291 and 1309, with total exposure times of 1497 and 2191 sec, respectively. Two exposures were obtained with the FUV G160M grating at central wavelengths settings 1577 and 1600, with exposure times of 2002 and 2007 sec, respectively. All exposures were taken in TIME-TAG readout mode with contemporaneous wavelength stim pulses and the default FP-POS position. The data were processed with the standard CALCOS data pipeline (v2.12) on retrieval from the archive. More details on the performance of COS can be found in the COS Instrument Handbook \\citep{Dixon:10:202} and at the STScI website. \n\nWe began our analysis with the {\\tt x1d} files provided by CALCOS. These extracted 1D spectra were then coadded to combine the multiple exposures for each grating, and in turn the two gratings, into a single spectrum. To align the four exposures as closely as possible in wavelength space, small ($\\lesssim10$ pixel) shifts were derived for each exposure individually from the profiles of strong Galactic ISM absorption lines.  This coaddition was performed on the ``gross counts'' vector from the CALCOS {\\tt x1d} files. The co-addition process sums counts from separate exposures for each pixel, keeping track of the effective exposure time for each pixel correctly. That is, if a given pixel is ignored in a particular exposure because of data quality flags, that exposure time is not summed for that pixel. Thus we are effectively co-adding {\\em count rates} rather than counts. Co-addition in count space allows for a simple derivation of the correct error vector in the Poisson limit of low total counts \\citep{1986ApJ...303..336G}. The final coadded spectrum possesses a signal-to-noise ratio $S\/N = 7 - 15$ per resolution element. The resolving power of COS is approximately $R = \\lambda \/ \\Delta \\lambda = 16-18,000$ over 1140 - 1750 \\AA, and the resolution element is sampled by approximately six (analog) pixels in the 1D extraction. We binned our final, coadded spectrum into 3-pixel bins for analysis. This binning leaves approximately 2 bins per resolution element, close to optimal sampling. \n\nWe note that the adopted version of CALCOS did not apply a correction for fixed pattern noise, but that the coaddition of exposures taken with different central wavelength settings mitigates the regular shadowing pattern of the detector quantum-efficiency (DQE) grid wires lying above the face of the COS micro-channel plate detectors. During the coaddition of the data we applied a correction for these grid wire shadows provided to us by the COS instrument team. This correction removes the grid-wire features, which otherwise result in $\\sim 15$\\% less counts recorded over a few pixels and can mimic absorption lines. \n\nThe final, reduced COS data appear in Figure~\\ref{wholespecfig}. The spectrum reveals at least two absorption line systems of significance along this sightline. First, we discovered the $z_{abs} = 0.356$ Lyman-limit system (LLS) that is the main focus of this paper. This system presents an obvious Lyman-limit break at 1240 \\AA\\ and a host of other multiphase ionization stages that will be analyzed below. A damped-Lyman-$\\alpha$ system (DLA) was discovered at $z_{abs} = 0.114$ and has been analyzed by \\cite{Meiring:11:1}. Both of these latter systems were serendipitous discoveries along a sightline that was selected to pass through the halo of an unrelated galaxy at $z < 0.25$, and we have no reason to suspect that the selection of the QSO itself or the targeted galaxy introduced a bias in favor of these two extraordinary intervening systems. The properties of the targeted galaxy and its associated absorber will be presented as part of the main survey. \n\n\n\\subsection{WFC3 Data}\n\nBased on the SDSS images, we identified two galaxies that could be associated with the LLS at $z = 0.356$ and\/or the DLA at $z = 0.114$. These are labeled ``80\\_5+86\\_4'' and ``170\\_9'' in Figure~\\ref{fieldfig} (left panel)\\footnote{We label our galaxies with a position angle and angular separation in arcseconds with respect to the target QSO; thus galaxy 170\\_9 has position angle of 170$^{\\circ}$ degrees (N through E) and is separated by 9$\"$ from J1009+0713.}. The indistinct profile near the QSO is partially hidden from view by the ground-based PSF of J1009+0713 in the SDSS image. Any attempt to relate the LLS to the galaxy properties, or even to obtain an accurate measurement of the impact parameter of this galaxy to the QSO sightline, was significantly hindered by this source confusion. In an attempt to conclusively identify the LLS-associated galaxies and to possibly discover a DLA host galaxy close to the sightline (fully within the SDSS PSF of J1009+0713), we obtained an image of this field with the Wide Field Camera 3 aboard {\\it HST}.\n\nThe WFC3\/UVIS data on this field were obtained on 24 June 2010 as Visit 44 in program GO11598. We obtained images in two broadband filters, F390W and F625W. The QSO was placed at the ``UVIS1'' aperture position, and the WFC3-UVIS-MOS-DITH-LINE dither pattern was used with an exposure time of 376 sec in F625W and 395 sec in F390W. This pattern resulted in total exposure times of 2256 sec and 2370 sec in F625W and F390W, respectively. The six individual exposures in each filter were first processed by the default CALWFC3 pipeline. Each exposure then had the QSO subtracted off using a TinyTim model of the telescope PSF before the six exposures were added together using MultiDrizzle. More details about the processing of the WFC3 image can be found in the paper by \\cite{Meiring:11:1}. \n\nEven without coaddition or QSO PSF subtraction, the WFC3 image revealed that the candidate galaxy nearest the QSO in the SDSS image resolves into two galaxies, labeled 80\\_5 and 86\\_4 in Figure~\\ref{fieldfig}, that are separated by only $\\sim 1''$. The image itself does not tell us whether one or more of these galaxies is associated with the DLA or LLS. We have attempted to constrain their redshifts separately with followup LRIS data, as described in the next section. \n\n\\subsection{Keck LRIS}\n\nAfter the COS data revealed two unexpected and interesting absorbers in this sightline, we re-examined the imaging data from which we had selected the target QSO. To determine which of these galaxies, if any, were associated with our newly detected absorbers, we obtained 1\\arcsec-wide longslit spectra of these two galaxy candidates with the Keck\/Low-Resolution Imaging Spectrometer (LRIS) on 5 April 2010. These LRIS data were taken using the D560 dichroic with the 600\/4000 l\/mm grism (blue side) and 600\/7500 l\/mm grating (red side) which gives a spectral coverage between 3000 and 5500 \\AA\\ (blue side), and 5600 to 8200 \\AA\\ (red side). On the blue side, binning the data 2 $\\times$ 2 resulted in a dispersion of 1.2 \\AA\\ per pixel and a FWHM resolution of $\\sim$280 km\/s. On the red side, the data were binned 1 $\\times$ 2, resulting in a dispersion of 2.3 \\AA\\ per pixel and a FWHM resolution of  $\\sim$200 km\/s. Exposure times were 800s in the blue and 2 $\\times$ 360 s in the red, which resulted in signal-to-noise ratios of at least 3 per pixel for strong nebular emission lines in the galaxy spectra. \n\nData reduction and calibration were carried out using the LowRedux\\footnote{http:\/\/www.ucolick.org\/$\\sim$xavier\/LowRedux\/index.html} IDL software package, which includes flat fielding to correct for pixel-to-pixel response variations and larger scale illumination variations, wavelength calibration, sky subtraction, and flux calibration using the spectrophotometric standard star G191B2B.  Precise and accurate systemic redshifts were obtained for both galaxies using a modified version of the SDSS IDL code ``zfind'', which works by fitting smoothed template SDSS eigenspectra to the galaxy emission-line spectra on both red and blue sides. The resultant weighted-mean redshift for 170\\_9 is $z = 0.355687 \\pm 0.00001$. The long slit used to obtain this first set of Keck\/LRIS observations was oriented to run from galaxy 170\\_9 through the midpoint of the faint profile seen for 80\\_5+86\\_4 in the SDSS image (the WFC3 image was not yet available). The slit was 1$''$ wide, so the recorded spectrum most likely includes contributions from both galaxies 80\\_5 and 86\\_4. For this spectrum, we obtain $z = 0.355574 \\pm 0.00002$, with error bars calculated from statistical noise only. To account for systematic uncertainties in the absolute wavelength calibration and instrument flexure during the LRIS exposures, we adopt a larger 25 km s$^{-1}$\\ error on the galaxy redshifts. The redshifts of the two galaxies are marked with ticks and $25$ km s$^{-1}$\\ errors at the top of Figures~\\ref{s0.355_ly}, \\ref{metal_stack}, and \\ref{kinplotfig} after translation into the rest frame of the absorber ($z_{abs} = 0.3558$). The galaxies appear approximately 25 km s$^{-1}$\\ apart. Neither appears to line up exactly with any of the strongest absorption components, but their coincidence is enough to give us strong evidence of their kinematic association. \n\nAdditional LRIS exposures of this field were obtained with LRIS in January 2011 in an attempt to separately constrain the redshifts of 80\\_5 and 86\\_4, which were blended together in the first set of LRIS exposures. LRIS was configured using the 400\/3400 grism on the blue side, the 600\/7500 grating on the red side, and the d650 dichroic.  This setup provided full coverage of wavelengths $3000 < \\lambda < 8700$ \\AA.  Three exposures, each lasting 600 seconds, were obtained in $\\sim 1$ arcsecond seeing.  Each exposure had a different position angle, with the goal of obtaining spectra of 80\\_5, 86\\_4, and other sources, while minimizing contaminating flux from the QSO.  This observation confirmed the detection of the [\\ion{O}{3}] and H$\\beta$ lines at $z = 0.355$ for 86\\_4. With the separation of 80\\_5 and 86\\_4 measured from the WFC3 image, we were able to marginally separate the spectral traces of 80\\_5 and 86\\_4. We do not find the same pattern of emission lines at the position of 80\\_5, so we conclude that this galaxy is not at the same redshift as 86\\_4. The best solution for the weak emission line detections at this position gives a solution of $z = 0.879$.  We therefore disregard this galaxy in our subsequent analysis. \n\n\\subsection{Keck HIRES}\n\nUsing the Keck I  High Resolution Echelle Spectrometer with the blue collimator (HIRESb), we obtained a high dispersion spectrum of the J1009+0713 on 26 March 2010.  These data were taken through the C1 decker, resulting in a FWHM resolution of $\\approx 6$ km s$^{-1}$. The grating angles were set to provide wavelength coverage from $3050-5870$ \\AA, with two small gaps related to the separation of the three CCD mosaic.  We integrated for two 1800s exposures under good conditions.  The data were reduced with the HIRedux pipeline\\footnote{http:\/\/www.ucolick.org\/$\\sim$xavier\/HIRedux\/index.html} to flatfield, sky subtract, wavelength calibrate, and extract the spectra.  The data were optimally coadded and normalized to unit flux by fitting a series of Chebyshev polynomials.  The final spectrum has a S\/N of 15 per 1.3 km s$^{-1}$\\ pixel redwards of 3800\\AA\\ decreasing to S\/N=5 at 3200 \\AA.\n\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{f3}\n\\caption{Lyman series lines for the $z = 0.355$ LLS. The velocity zero point is set to $z = 0.3558$. The points with error bars at the top mark the measured velocities of the galaxies with 25 km s$^{-1}$\\ uncertainty. \\label{s0.355_ly}}\n\\end{figure}\n\n\n\\begin{figure*}\n\\epsscale{1}\n\\plotone{f4}\n\\caption{Metal absorption lines from the $z = 0.3558$ LLS. Vertical dashed lines mark the centroid velocities of components A - D. Transitions are ordered by atomic element and, within that, by increasing ionization potential. The points with error bars at the top mark the measured velocities of the two associated galaxies with 25 km s$^{-1}$\\ uncertainty.  \\label{metal_stack}}\n\\end{figure*}\n\n\\section{Absorption Line Identification and Analysis}\n\\label{lls-section}\n\n\\subsection{General Approach}\n\nTo identify and fit absorption lines in the spectrum of J1009+0713 we have adopted the vacuum wavelengths and atomic data from the compilation by \\cite{Morton:03:205}. Line measurements appear in Table~\\ref{comp-list}. Most column densities were obtained by direct integration over the line profile, except in the case of the optical lines,  some saturated UV lines, and the Lyman series lines in the outlying components, where profile fits were used. \n\nThe line profile fits use a custom Voigt-profile fitting software that attempts to optimize a model of the line column densities, doppler $b$ parameters, and velocities for an arbitrary input set of lines and atomic data, by minimizing the $\\chi ^2$. This code was first used to analyze the PG1211+143 sightline by \\cite{Tumlinson:05:95}. This software first constructs intrinsic line profiles and then convolves these with the appropriate non-Gaussian COS line-spread function (LSF) as described by \\cite{Ghavamian:09:1} and available on the STScI COS website. These tabulated LSFs are specified for 50 \\AA\\ intervals through the range of the G130M and G160M gratings. For line fits we adopt the nearest LSF grid point based on the observed wavelength of the lines being fitted. \n\n\n\\subsection{The Lyman Limit System}\n\nThe $z = 0.3558$ absorber exhibits a line-black Lyman limit at 1236 \\AA\\ in the observed frame. This system also appears in at least 14 distinct Lyman series transitions, the first twelve of which appear in Figure~\\ref{s0.355_ly}. Inspection of the strong Ly$\\alpha$ profile reveals absorption extending over $\\sim 400$ km s$^{-1}$. No damping wings are apparent, which limits the total column density to $\\log N$(\\ion{H}{1}) $\\lesssim 19$. At Ly$\\beta$ - $\\epsilon$, a separate component at $+180$ km s$^{-1}$\\ becomes distinct, which we term component D. By Ly$\\eta$, a separate component at $-95$ km s$^{-1}$\\ is visible and distinct until blending between the Lyman series lines themselves becomes severe at Ly$\\mu$. The two innermost components labeled B and C in Figure~\\ref{s0.355_ly} never separate in the Lyman series lines and are defined by their distinct profiles in the metal lines, particularly \\ion{Mg}{2}, as shown below. Component groups A, B, and C all break further into multiple blended components in the HIRES data. \n\nWe note that these components are defined, wherever possible, by the \\ion{H}{1} absorption or by the metal lines, as noted in Table 1. The number of apparent components is larger in the higher resolution optical data covering \\ion{Mg}{2} and \\ion{Fe}{2}. This finding suggests that there may be further component structure in the \\ion{H}{1} and metal lines that we cannot resolve at the R = 18,000 resolution of COS. In what follows, we analyze the major component groups as coherent objects with the proviso that they may in fact consistent of subcomponents below the level of our spectral resolution. \n\n\\input t1.tex\n\n\\subsection{Metal lines and ionization}\n\nThe COS data on the LLS reveal absorption from a wide range of ionization stages, as shown in Figure~\\ref{metal_stack}. We detect absorption by C, N, O, Mg, Ca, Si, S, and Fe in the COS and\/or HIRES data, and in ionization stages from \\ion{Mg}{1} to \\ion{O}{6}. However, the overall impression is of an absorption-line system that possesses a high degree of ionization. The species \\ion{Mg}{1} and \\ion{O}{1} are the only neutral atoms that are convincingly detected; \\ion{Ca}{2} is usually detected in mostly neutral gas but is very weak here. Limits are placed on the neutrals of C, N, Si, and Fe. By contrast, strong absorption is detected in \\ion{C}{3}, \\ion{N}{3}, \\ion{Si}{3}, and \\ion{Fe}{3}.  Our COS data do not cover \\ion{C}{4} or \\ion{Si}{4}, but \\ion{N}{5} $\\lambda\\lambda 1238,1242$, typically weak, is not detected in the S\/N  $\\sim 5$ data at that wavelength. These detections and limits, apart from any detailed analysis, point to gas that is substantially ionized in all components. \n\n\\subsection{The O VI Absorption}\n\n\n\\begin{figure}[!t]\n\\epsscale{1}\n\\plotone{f5}\n\\caption{Kinematic structure in apparent column densities \\citep{Savage:91:245} for low ions and \\ion{O}{6}.  Major components A, B, and C are broken down further according to their appearance in the Mg II and Fe II data obtained by HIRES. Note the wide range of apparent column density ratios between the low ions and \\ion{O}{6}, particularly in components A and D. Flat tops arise from the truncation of the flux at 0.01 in the normalized data to avoid unphysical values in the optical depth for saturated lines. \\label{kinplotfig}}\n\\end{figure}\n\nThe strength and kinematics of the \\ion{O}{6}\\ in this absorber are extraordinary. Integrating over the full profile, this is among the strongest intervening intergalactic \\ion{O}{6}\\ absorbers yet detected, with a rest-frame equivalent width $W_r = 835 \\pm 49$ m\\AA\\ in the $\\lambda$1032 profile \\citep[cf.][]{Tripp:08:39, Thom:08:22, Danforth:08:194}\\footnote{The reported values for some of the other $W_r > 300$ m\\AA\\ systems vary from study to study, depending on how each system was broken down into components. Our reported value takes in all the identifiable components. This pattern emphasizes the difficulty of interpreting kinematically complex absorbers in terms of simple models, which we will take up below.}. As shown in Figure~\\ref{metal_stack}, this absorption spreads over the full 400 km s$^{-1}$\\ range of the system, in a nearly flat, complex profile. Figure~\\ref{kinplotfig} shows the \\ion{O}{6}\\ profile after it has been converted into apparent column densities using the method of \\cite{Savage:91:245}. Accounting for a modest degree of noise in the data, the $\\lambda$1032 and $\\lambda$1038 profiles agree well over the full velocity range, supporting the case that all this absorption is truly O VI, and that it covers the full kinematic range of absorption by the other detected ionization stages. While component groups A and D appear distinctly in the \\ion{O}{6}\\ profiles, B and C are not clearly separated; they are both blended together and possibly too broad to show individual peaks in the column density plot (Figure~\\ref{kinplotfig}). We conclude that \\ion{O}{6}\\ is associated with all four identified component groups in this system.  \n\nAnother notable feature of the \\ion{O}{6}\\ is its relationship to the neutral and low-ionization gas in the various components. In component A, strong \\ion{O}{6}\\ at $\\log N$(\\ion{O}{6}) = 14.6 coincides with strong \\ion{C}{3}, \\ion{N}{3}, and \\ion{Si}{3}, but also \\ion{Mg}{2} and \\ion{C}{2}. In components B and C, somewhat weaker \\ion{O}{6}\\ is associated with much stronger low ions, such as \\ion{C}{2}, \\ion{Mg}{2}, \\ion{Si}{2}, and \\ion{Fe}{2}. In component D, only \\ion{H}{1}, \\ion{C}{3}, \\ion{N}{3}, \\ion{Si}{3}, and \\ion{O}{6}\\ are detected, with no corresponding \\ion{Mg}{2}. Clearly the state of ionization varies across the absorber, which has an intrinsically multiphase structure. \n\n\n\\subsection{HI Solution}\n\nWith the four components defined either by their \\ion{H}{1} or metal-line profiles (see notes to Table \\ref{comp-list}), we can now derive a solution for the \\ion{H}{1} column densities in this absorber. Components A and D both appear in 3 or 4 higher Lyman series profiles, and can be profile-fitted directly. However, both B and C are strongly saturated and blended together, so simple line-profile fitting cannot disentangle them and measure their column densities without additional assumptions. The two straightforward components are taken up first. All relative velocities are given with respect to the adopted systemic redshift of $z_{abs} = 0.3558$. \n\n{\\it Component A:} This component is defined at $-95$ km s$^{-1}$\\ by Ly$\\eta$, $\\theta$, $\\kappa$, and $\\lambda$, where it appears distinctly but is blended with the blue wing of component B. It appears to be contaminated with unrelated absorption at Ly$\\iota$. Profile fits to Ly$\\eta$, $\\theta$, and $\\kappa$ were attempted, with the results appearing in Table~\\ref{comp-list}. Consistent results are obtained with $\\log N$(\\ion{H}{1}) $= 16.42\\pm 0.08$, $b =  20 \\pm 3$ km s$^{-1}$, at $-94 \\pm 1$ km s$^{-1}$. These values are adopted for further analysis. \n\n{\\it Component D:}  This component is defined at $+180$ km s$^{-1}$\\ by Ly$\\beta$, $\\gamma$, and perhaps $\\delta$. It is inseparably blended with component C of Ly$\\alpha$, contaminated at Ly$\\epsilon$, and absent in higher Lyman series lines. Only Ly$\\beta$ and Ly$\\gamma$ give acceptable fits, with $\\log$ N(\\ion{H}{1}) $= 14.8$, $b = 16 \\pm 3$ km s$^{-1}$\\ at 180 km s$^{-1}$. This fit is adopted for further analysis. \n\n{\\it Components B and C:} These two components are both strongly saturated and severely blended with one another at a velocity separation of 60-80 km s$^{-1}$\\ (as shown by their metal lines). Because of this blending and saturation we cannot, even in principle, fit them and constrain their column densities separately. Instead, we attempt to impose firm limits on the {\\it total} $N$(\\ion{H}{1}), based on the Lyman limit opacity and observed Lyman series line profiles.\n\nA firm lower limit to the total $N$(\\ion{H}{1}) comes from the observed Lyman-limit opacity. For closely-spaced components, this limit is insensitive to the separations and column density ratios and traces mainly the total column density of \\ion{H}{1}. We display two limiting cases in Figure~\\ref{wholespecfig}. We adopt the Lyman continuum opacity as function of wavelength given by \\cite{1989agna.book.....O}. The red solid curve shows $\\log N$(\\ion{H}{1}) = 17.8, which leaves $\\sim 5$\\% transmission across the detected range of the Lyman limit system. By $\\log N$(\\ion{H}{1}) = 18.0, the transmission is no longer visible and matches the absence of source flux in this region of the spectrum; higher column densities are also consistent with the data. We therefore regard $\\log N$(\\ion{H}{1}) = 17.8 as a firm, conservative lower limit to the total column density of the LLS. Given the Lyman-limit opacity, 17.8 seems to have a low probability, so we also adopt a slightly higher value of $\\log N$(\\ion{H}{1}) to bound the lower end of a 95\\% confidence interval.\n\nUpper limits to total $N$(\\ion{H}{1}) come from the absence of damping wings from the Ly$\\alpha$ profile of the LLS. This limit is somewhat less firm than the limit from continuum opacity, since it varies slightly with the line broadening and number of components. Two identical components at the positions of the B and C components with $b = 20$ km s$^{-1}$\\ and $\\log N$(\\ion{H}{1}) = 18.5 show damping wings that are only marginally consistent with the data (Figure~\\ref{lya_stack_fig}). When each component is increased to $\\log N$(\\ion{H}{1}) = 18.7 (for a total of 19.0), the damping wings are clearly excluded by the data. These profiles and limits strictly depend on the number of assumed components, which we can infer from the HIRES data. If we assume components at each of the three strongest \\ion{Mg}{1} and \\ion{Mg}{2} components observed in the HIRES optical data, which are likely to trace the majority of the total \\ion{H}{1}, we find that by a total of $\\log N$(\\ion{H}{1}) $=18.5$, the damping wings appear, and higher values are increasingly poor fits. This is consistent with the two-component analysis. As the number of assumed components within the inner 80 km s$^{-1}$\\ increases, the column density per component increases. By $\\log N$(\\ion{H}{1}) = 19, no tuning of $b$ and number of components (even one) can hide the damping wings, so we regard this value as a firm upper limit to the total N(\\ion{H}{1}). The weak presence of damping wings at a total $\\log N$(\\ion{H}{1}) = 18.8 suggest that the value is likely to be lower than that, so we adopt this as an upper bound to a 95\\% confidence interval. For further analysis, we adopt at 95\\% confidence interval of $\\log N$(\\ion{H}{1}) = 18.0 - 18.8, with a flat probability density within that range, since we have no information about which values in this range are more likely. \n\n\n\\begin{figure}[!t]\n\\plotone{f6}\n\\caption{The LLS Ly$\\alpha$ profile with model lines overlaid, showing the effects of line saturation. The top panel shows a model with $\\log N$(\\ion{H}{1}) = 18.5 in each of the components B and C, for a total of $\\log N$(\\ion{H}{1}) = 18.8. The combined profile, shown with the heavy black line, is still a marginally acceptable fit to the data. The lower panel increases the individual components to $\\log N$(\\ion{H}{1}) = 18.7 and the total to 19.0; by this point damping wings no longer fit the data. \\label{lya_stack_fig}}\n\\end{figure}\n\n\\section{Metallicity and Star Formation of the Nearby Galaxies}\n\\label{gal-section}\n\nThe Keck\/LRIS and WFC3\/UVIS data on the galaxies in the field provide useful information about their redshift, luminosity, star formation rate, and metallicity. The reduced Keck\/LRIS spectra appear in Figure~\\ref{galspec} and the calibrated and PSF-subtracted images appear in Figure~\\ref{fieldfig}.\n\nAt $z = 0.3558$, [OII] $\\lambda\\lambda$ 3727, H$\\beta$, and [OIII]$\\lambda4959$$\\lambda5007$ are readily detected. The strengths of these emission lines, with appropriate corrections, provide constraints on the star formation rate and metallicity of the galaxies \\citep{Levesque:10:712}. Accordingly, the accuracy of these quantities depends upon the accuracy of the relative emission line strengths and on the degree to which the flux-calibration of the spectra is ``absolute.\"   We therefore try to ensure that the flux calibrations of the  blue and red sides of the LRIS spectra are consistent with each other, and attempt to correct for light losses in the 1\\arcsec\\  slit by comparing our spectra with SDSS photometry if it is available for our galaxies. We accomplish this absolute flux correction by convolving the LRIS spectra with SDSS $ugriz$ filters (see \\cite{Silva:11:1111} for a description of the IDL code ``spec2mag\") and comparing the observed spectral apparent magnitudes with the SDSS catalog apparent magnitudes (see Werk et al., in prep, for details).\n\nOnly galaxy 170\\_9 has available SDSS photometry that enables us to perform an absolute flux correction.  We do not correct the spectral fluxes of objects 80\\_5 and 86\\_4 since they do not have SDSS photometry (they are not even detected as one galaxy). The resultant corrective flux factor for 170\\_9 is 1.58 (0.494 magnitudes in the $g-$band) on the blue side and 1.65 (0.543 magnitudes in the $i-$band)  on the red side.   Since we have $ugriz$ photometry available to us for object 170\\_9, we are able to use the IDL-code $kcorrect$ \\citep[][v4\\_2]{Blanton:07:734} to obtain an estimate of its stellar mass, approximately 5.5$\\times 10^{9}$ M$_{\\odot}$.\n\nWe estimate the SFR from the H$\\beta$ emission line flux using the calibration of \\cite{Calzetti:10:1256}, such that SFR (H$\\beta$) = SFR (H$\\alpha$) \/  2.86, where SFR (H$\\alpha$) (M$_{\\odot}$ yr$^{-1}$) = 5.45 $\\times$ 10$^{-42}$ L(H$\\alpha$) erg s$^{-1}$. The factor of 2.86 represents the intrinsic ratio of H$\\alpha$ to H$\\beta$ for case B recombination at an effective temperature of 10,000 K and electron density of 100 cm$^{-3}$ \\citep{Hummer:87:801}.  The SFRs appear in Table 2. For 86\\_4,  the SFR is a formal lower limit, since we assume there was significant light loss from using a 1\\arcsec\\ slit (on the order of $\\sim$38\\%). The errors given in the table account for only the RMS of the line measurements; the flux calibration uncertainty, read noise, sky noise, and flat-fielding errors, are approximately $\\sim$1\\% for the emission lines in 170\\_9 and $\\sim$7\\% for the lower S\/N emission lines in 86\\_4.\n\n\n\n\\begin{figure}[!t]\n\\epsscale{1}\n\\plotone{f7a}\n\\plotone{f7b}\n\\caption{Keck\/LRIS spectra of galaxies in the field for 86\\_4 (top), and 170\\_9 (bottom). The break in the wavelength range is placed to avoid LRIS dichroic near 5500 \\AA. Detected emission lines are labeled. The red and blue sides for 86\\_4 are not normalized as they are for 170\\_9 (see \\S~2.3). \\label{galspec}}\n\\end{figure}\n\n\nWe obtain the nebular oxygen abundances for galaxies 86\\_4 and 170\\_9 using the strong line R23 method originally presented by \\cite{Pagel:79:95}, according to the calibration of \\cite{McGaugh:91:140}.  R23 is defined as log [([OII] $\\lambda\\lambda3727$ + [OIII] $\\lambda4959$ + [OIII] $\\lambda5007$)\/H$\\beta$]. The drawbacks of the R23 method include a well known degeneracy and turnover at $\\sim0.3$Z$_{\\odot}$ and large systematic errors on the order of 0.25 dex due to age effects and stellar distributions \\citep{Ercolano:07:945}.  To correct the emission lines for foreground reddening, we use a reddening function normalized at H$\\beta$ from the Galactic reddening law of \\cite{Cardelli:89:245} assuming R$_{v}$ = A$_{v}$\/E(B$-$V) = 3.1, and using an E(B$-$V) value of 0.013 \\citep{Schlegel:98:525}. We cannot break the R23 degeneracy using any of the known methods (e.g. [NII]\/[OII]) because our LRIS data do not cover lines redward of $\\sim$6500 \\AA\\ in the rest frame of the galaxies. The upper and lower branch metallicities are presented in Table~2. Based on the well-known mass-metallicity relation of galaxies \\citep{Tremonti:04:898}, it is reasonable to assume the upper-branch metallicity for 170\\_9 given its estimated stellar mass of $5.5 \\times 10^{9}$ M$_{\\odot}$. The upper branch value is consistent with solar metallicity according to the updated solar oxygen abundance of  $12 + \\log (O\/H) = 8.69$ \\citep{Prieto:01:L63, Asplund:09:481}.  The spectrum of 86\\_4 indicates a slightly lower upper-branch metallicity than the brighter galaxy 170\\_9, but given the errors in the measurements, all the upper branch measurements are consistent with solar metallicity and with one another. \n\nAs described below, we unfortunately have no direct measurement of the gas metallicity in the strongest components of the LLS. However, the indirectly indicated metallicity from photoionization modeling is $Z = 0.05 - 0.5 Z_{\\odot}$. If this is correct, the gas would appear to have a low metallicity with respect to the galaxies and would favor an explanation in which the gas has entered this system from some other source - the IGM or a stripped dwarf galaxy. However, given the uncertainty in the metallicity of the strong LLS components, and the upper\/lower branch ambiguity for the galaxies, we cannot draw firm conclusions from this metallicity comparison. \n\n\\begin{deluxetable}{lccc}[!t]\n\\tablewidth{0pt}\n\\tablenum{2} \n\\tablecaption{Galaxy Properties} \n\\tablehead{\n\\colhead{Property}&\n\\colhead{80\\_5}&\n\\colhead{86\\_4}&\n\\colhead{170\\_9}}\n\\startdata\nRedshift\\tablenotemark{a} \t\t\t& 0.88(1) \t\t\t& \t0.355574(2)    \t& 0.355687(1) \t    \\\\\n$\\rho$ [kpc]\t\t\t\t& \\nodata\t\t\t&      14.25\t\t& 46.48                  \\\\\n$m_{390W}$\\tablenotemark{b}\t\t& $23.1\\pm0.2$\t& $24.1\\pm0.2$\t& $22.2\\pm0.1$    \\\\\n$m_{625W}$\t\t\t\t& $22.5\\pm0.1$\t& $23.2\\pm0.2$\t& $21.3\\pm0.1$    \\\\\n$[$O\/H$]$ (upper)\\tablenotemark{c}\t& \\nodata\t\t\t& $-0.2$\t        \t& $0.0$ \\\\\n$[$O\/H$]$ (lower)\\tablenotemark{c}          & \\nodata                   & $-0.6$                      &  $-0.9$   \\\\\nSFR [$M_\\odot$ yr$^{-1}$]\t& \\nodata\t\t\t& $>0.2$ \t& $2.1 \\pm 0.02$\n\\enddata\n\\label{galtable}\n\\tablenotetext{a}{To account for systematic effects in addition to formal fitting errors, we adopt a $\\pm 25$ km s$^{-1}$\\ uncertainty for these redshifts.}\n\\tablenotetext{b}{Broadband AB magnitudes derived from the imaging analysis of \\cite{Meiring:11:1}.}\n\\tablenotetext{c}{These estimates are equivalent to [O\/H] = $\\log$ (O\/H) $- \\log$ (O\/H)$_\\odot$, where the solar abundance corresponds to $[$O\/H$]$ = 0 or 12 + log(O\/H) = 8.69 \\citep{Prieto:01:L63,Asplund:09:481}. These estimates carry an error of $\\pm 0.15$ dex.}\n\\end{deluxetable}\n\n\n\n\n\\section{Physical Modeling and Interpretation}\n\nIn this section we move from measuring column densities and kinematics to interpreting these quantities in terms of physical models, focusing on what we can learn about the ionization, metallicity, and total gas budgets in these components from the detected absorption. \n\nFrom the detections of the various ionization stages of C, N, O, Mg, S, Si, and Fe alone it is evident that at least some of the gas in this absorber possesses a high degree of ionization, even in the higher column density component groups B and C. These two components exhibit absorption in the second ions of C, N, S, Si, and Fe, the third ion of S, \\ion{O}{6}, and relatively little absorption from the neutrals and first ions of these elements. Apart from any detailed modeling of ionization, the ratios \\ion{N}{2} \/ \\ion{N}{1} and \\ion{Fe}{3} \/ \\ion{Fe}{2} indicate a significant ionization fraction. The solid detection of \\ion{O}{6} in all components further indicates a highly ionized fraction of the gas, whether or or not it arises from the same ionization mechanism as the lower ions. \n\nTo estimate the degree of ionization in the absorbing clouds and from that the total budget of gas that has been detected, we produced photoionzation models with the Cloudy modeling code  (version 08.00) last described by \\cite{Ferland:98:761}. This effort is hindered by several limitations that must be acknowledged at the outset. Photoionization models described by a density, ionization parameter, incident spectrum, metallicity, and total gas content require a larger number of uncertain parameters than are formally constrained by the data, so any results must be considered indicative rather than conclusive. The shape and orientation of the absorbing cloud with respect to the nearby galaxies and to the observer are both unknown; we can only assume the simplest cases. While these two factors are generic to absorption systems in QSO sightlines, this particular system presents two additional features that complicate modeling. Its two strongest components, labeled B and C, are inseparably blended with one another in all available lines of \\ion{H}{1}, so models are not robustly constrained as usual by a definite N(\\ion{H}{1}); we have only a loose constraint on the {\\it sum} of the neutral hydrogen column in B and C (see \\S~3.5). Furthermore, many of the most important lines of intermediate ions, such as \\ion{C}{3} and \\ion{Si}{3}, are strongly saturated and blended across all the component groups. \n\nWith these caveats in mind, we model the absorbing clouds with a single plane-parallel slab model illuminated on one side by a uniform incident radiation field. The model field uses two components. First, we attempted models with the composite extragalactic radiation field of Haardt \\& Madau (HM05, as implemented within Cloudy). This spectrum yields $\\Phi _{HM} = 3.2 \\times 10^4$ photons cm s$^{-1}$ for $>1$ Ryd photons at each point in space. By itself, the composite HM field generally fails to achieve the observed ratios of second to first ions of C, N, O, Si, and Fe until the ionization parameter is increased by a low density and the clouds are more than 1 Mpc in size, too large to reside in galaxy halos. It is perhaps not surprising that this spectrum fails, since it represents the average ionizing background of all galaxies and QSOs as transmitted by the IGM. But our absorbers are not located at a random point in space, but within 15 - 50 kpc of galaxies with star formation rates of $\\sim 0.2 - 2$ M$_{\\odot}$ yr$^{-1}$. Models of the Milky Way halo that attempt to account for the ionization of the Galactic high-velocity clouds typically find that the Milky Way may contribute $10^5$ ionizing photons cm$^{-2}$ s$^{-1}$ even $\\sim 100$ kpc into the halo in the direction normal to the disk \\citep{Bland-Hawthorn:99:212,Bland-Hawthorn:99:L33, Fox:05:332}, and nearly $10^6$ photons cm$^{-2}$ s$^{-1}$ at 40 kpc\\footnote{These models are consistent with the H$\\alpha$ emission from the HVCs \\citep{Weiner:02:256,Putman:03:948}, but not the Magellanic Stream, which is hypothesized to have at least some of its gas ionized by shocks \\citep{Bland-Hawthorn:07:L109}.}. This flux exceeds the contribution of $\\Phi _{HM}$ by a significant margin. Thus even a modest degree of star formation, if its UV radiation propagates efficiently into the halo, can exceed the extragalactic background at physical separations like those observed in the present system. \n\nWe therefore add in a second ionizing field component, modeled as a galaxy undergoing a recent episode of star formation, drawn from the Starburst99 libraries \\citep{Leitherer:99:3} and corresponding to a continuous star-formation history over 10 Myr with a Salpeter IMF from 1 - 100 $M_{\\odot}$ and solar metallicity (the ionizing photon output at $0.5 Z_{\\odot}$ is only 15\\% higher). For a star formation history extending far into the past, the instantaneous spectrum reaches a steady state beyond a few million years and changes negligibly with the assumed age. The galaxy metallicities (see \\S~4) suggest that we should adopt $Z_{\\odot}$ for the starburst. This spectrum generates a total of $Q_* = 10^{53.14}$ (SFR \/ $M_{\\odot}$ yr$^{-1}$) ionizing photons s$^{-1}$. We assume that a uniform, wavelength-independent fraction $f_{esc}$ of these ionizing photons escape the ISM of the galaxy to illuminate material in the halo.  At a distance $d$ in kpc, this ionizing field will impart to a cloud of uniform density $n_H$ a field characterized by the dimensionless ionization parameter $U$:  \n\\begin{align*}\n  U &=  \\frac{1}{c n_H}  \\left( \\frac{Q _* f_{esc}}{4 \\pi d^2}  + \\Phi _{HM} \\right) \\\\\n       &=   \\frac{0.01\\, {\\rm cm}^{-3}}{n_H} \\left[ 10^{-3}  \\frac{SFR}{M_{\\odot} {\\rm yr}^{-1}}  \\frac{f_{esc}}{0.1}  \\left( \\frac{20\\, {\\rm kpc}}{d} \\right) ^2 + 10^{-4} \\right]\n\\end{align*}\nwhere in the final equation the quantities have been scaled to typical values for halo clouds. This relation does not account for radiative transfer in the Lyman continuum, and assumes normally incident radiation, and so it is only an approximation for real clouds. For $d = 20$ kpc and $SFR = 1$ M$_{\\odot}$ yr$^{-1}$, the ionization parameter $\\log U \\sim -3$. The constants $10^{-3}$ and $10^{-4}$ are convenient approximations good to better than $10$\\%. In these typical conditions, the stellar ionizing flux exceeds the HM spectrum by a order of magnitude, consistent with results from the Milky Way. At all higher SFRs, higher $f_{esc}$, and lower distances, the stellar ionizing field is even more dominant. Since we do not know the cloud distance to the galaxy, the density, or the $f_{esc}$, we cannot use this relation to specify $U$ for a given absorber. However, once a photoionization model has been found and $U$ constrained by observations, along with SFR for the galaxies, we can use this relation to obtain estimates of the other parameters. \n\nThe sum of these HM05 and starburst radiation field components is normalized within Cloudy to achieve a specific ionization parameter at the face of the model cloud. The model clouds are calculated from their sharp illuminated face to a stopping column density of $\\log N$(\\ion{H}{1}) = 19.5, to cover components B and C; these models are valid for any lower total $N$(\\ion{H}{1}) if their results are truncated at that column density so are applicable to A and D as well. The model clouds are assumed to have a uniform metallicity of appropriately scaled solar relative abundances; models with total metallicity from solar to 1\\% solar were explored. \n\n\\subsection{Components B and C}\n\n\\subsubsection{Metallicity}\n\nIn B and C we detect weak absorption from neutral oxygen in two transitions: \\ion{O}{1} $\\lambda$1302 and $\\lambda$988. These lines are likely unsaturated, as the apparent column density profiles shown in Figure~\\ref{kinplotfig} (middle panel)  are consistent in the strong component groups B and C. The B component of the $\\lambda$1302 line appears mildly contaminated; we adopt the value $\\log N$(\\ion{O}{1})$ = 14.5 \\pm 0.2$, boosting the error from 0.1 to 0.2 dex to account for the uncertainty in the contamination. For component C, we obtain consistent measurements from both lines, $\\log N$(\\ion{O}{1}) $= 14.2 \\pm 0.2$. Significant absorption is not detected in components A or D, and direct integrations over their velocity ranges yield upper limits of $\\log N$(OI) $< 13.9$ ($2\\sigma$). \n\nNeutral oxygen is a useful metallicity indicator in moderately ionized gas, since its ionization potential of 13.6 eV and charge exchange reactions with H lock its ionization fraction to that of hydrogen in typical conditions. Unfortunately these well-detected \\ion{O}{1} lines must be compared to a highly uncertain total \\ion{H}{1} column, ranging anywhere from $\\log N$(\\ion{H}{1}) = 18.0 - 18.8 (95\\% confidence).  With this range, metallicities from $0.1 - 1.0 Z_{\\odot}$ are all permitted in comparison with the updated solar oxygen abundance $12 + \\log (O\/H) = 8.69$ \\citep{Prieto:01:L63, Asplund:09:481}. We therefore cannot obtain a robust, direct measurement of gas metallicity in this system, owing to the systematic error in $N$(\\ion{H}{1}) introduced by the strong saturation and blending of the two inner components. However, we can impose upper and lower limits, $Z = 0.15 - 1.0 Z_{\\odot}$ based on the 95\\% confidence range derived for $N$(\\ion{H}{1}) above (or $0.1 - 1.6 Z_{\\odot}$ with the firm limits on $N$(\\ion{H}{1})). This circumstance somewhat limits our ability to interpret this material as either infall or outflow from the associated galaxies. However, modeling of absorption from ionized gas indirectly indicates sub-solar metallicity, as shown in the next section. \n\n\n\\subsubsection{Ionization}\n\n\\begin{figure}\n\\epsscale{1}\n\\plotone{f8}\n\\caption{Photoionization modeling for the B and C velocity components of the LLS. This model assumes the Starburst99 spectrum described in the text with $\\log U = -3.5$ and $Z = 0.1 Z_\\odot$. The left column shows column density ratios of key ions, while the right shows the cumulative column densities. Both are plotted with respect to cumulative N(\\ion{H}{1}). The ranges permitted by the data for column densities and ratios are marked, coded in color to match their model curve. In the right column, the species are color coded according to their stage of ionization, e.g. 1 equals \\ion{C}{1} in orange, 2 equals \\ion{C}{2} in blue. \\label{allions}}\n\\end{figure}\n\nThe results of our Cloudy photoionization model with this Starburst99 input spectrum appear in Figure~\\ref{allions}, specifically for the parameters $\\log U = -3.5$, and 10\\% solar metallicity in the gas. The left column of this figure shows the column density ratios of the key ions, and the right column shows the cumulative column densities, $N$(X), as a function of \\ion{H}{1} column density into the cloud (note that this is not physical depth, or total H column, which is proportional to depth for constant density). These quantities are plotted with respect to a varying $N$(\\ion{H}{1}) since we must account for the uncertain total \\ion{H}{1} column density. If that column density were well-constrained by measurement, we could choose one vertical slice though this space with a fixed $N$(\\ion{H}{1}) and analyze models of varying $U$ and $Z$, as is typically done for photoionization models. Instead, we analyze these families of models with a range of $N$(\\ion{H}{1}) and proceed in two steps. First, we adjust the ionization parameter $U$ to place the ion ratios in the left column within their permitted ranges. Then, holding $U$ fixed, we vary the metallicity until the cumulative column densities in the right column fall into the observed ranges for themselves individually and also for \\ion{H}{1}. Where these curves lie inside the boxes that mark data, both the total $N$(X) and the $N$(\\ion{H}{1}) are reproduced by that particular model. Thus, the ion ratios constrain the ionization parameter while the column densities constrain the metallicity. \n\nThe model with $\\log U = -3.5$ provides an acceptable fit to the ratios of the observed ions, except that it underproduces \\ion{N}{3} and produces essentially no \\ion{O}{6} or \\ion{S}{4}. However, the detected \\ion{N}{3} is likely contaminated by \\ion{Si}{2} or other absorption and so the observed ratio \\ion{N}{3} \/ \\ion{N}{2} is artificially high; we discount this ratio to discriminate models. Otherwise, these parameters are a good model for the components B and C. At the depth into the cloud at which this model has achieved $N$(\\ion{H}{1}) $\\simeq 18 - 18.8$ the clouds are still predominantly ionized. As shown in Figure 8, the lower $N$(\\ion{H}{1}) models within the permitted range have lower neutral fractions, while the higher $N$(\\ion{H}{1}) have higher neutral fractions, so that the typical total N(H) range of these models is $N$(H) $\\simeq 19.5 - 20$. Thus in models with pure photoionization, these clouds are nearly all ionized. \n\nThe column densities of well-measured ionization stages (right column) also indicate a subsolar metallicity. This constraint on the metallicity should be considered indirect, because it depends on the assumption of pure photoionization. Yet under this assumption, models with an ionization parameter, $\\log U \\simeq -3.5$,  that matches the observed ratios of ions from the same element fail to match the column densities of all these ionization stages unless the metallicity is $5 - 50$\\%, as shown in Figure~\\ref{allions}. This indirect measure of the gas metallicity is affected less by the saturation of \\ion{H}{1} than might be expected. As shown in the right column of Figure~\\ref{allions}, the column densities of the first and second ions increase slowly with respect to $N$(\\ion{H}{1}) through the range where it increases from 18.0 to 18.8, because over this range the H is becoming increasingly neutral and is added to the line integral through the cloud at a much higher rate, per pathlength, than species such as \\ion{Mg}{2} or \\ion{Fe}{2} for which the ionization fractions change little over this range. For a given $U$, these cumulative column density curves shift up and down in those panels of the figure in direct proportion to the assumed cloud metallicity, which has been adjusted to best match the greatest number of column densities. The best fitting model has $0.1 Z_{\\odot}$, but values from $0.05 - 0.5 Z_{\\odot}$ still provide decent matches for most ions. Even though the metallicities estimated directly from the detection of \\ion{O}{1} are highly uncertain because of the saturation of the \\ion{H}{1}, they are consistent with these models at the higher end of the permitted range of $N$(\\ion{H}{1}). \n\nFinally, we have assumed a nominal density of $n_H = 0.01$ cm$^{-3}$, which yields cloud sizes of $1 - 3$ kpc. We therefore have a model in which the two strongest components in the LLS absorber are consistent with diffuse, possibly low-metallicity clouds of total gas column $\\log N \\simeq 19-20$ and with kpc scale that are ionized by the stars in their nearby galaxies. These absorbers suggest that star-forming galaxies with $f_{esc} = 0.1$ may be able to photoionize some of their own halo gas under the right conditions. \n\nIn light of the appearance of many ionization stages in this absorber, e.g. both \\ion{O}{1} and \\ion{O}{6}, we must consider that some or all of the gas arises in warm or hot collisionally ionized material, such as might arise in a shock or conductive interface between hot and cold gas. For purely collisionally ionized gas without a photoionizing field, we can use individual ionization ratios to constrain the temperature. The ratios of the second ions to the first ions of C, N, Si, and Fe  (e.g. \\ion{N}{3} to \\ion{N}{2}) all prefer a temperature of between $20000 - 80000$ K. Over this range the neutral fraction of H drops from $\\sim 0.1$ to 10$^{-4}$, so if collisional ionization dominates in these absorbers, the total column density of H will exceed $\\log N$(\\ion{H}{1}) = 19.5 and could be substantially higher. However, the observed ratio of \\ion{Mg}{2} \/  \\ion{Mg}{1}  = 1.8 dex prefers a temperature $T < 20000$ K, and is one of the better determined ratios in components B and C. Thus, no single temperature in CIE describes all the detections. By contrast, the photoionization model above appears to be able to account for these ratios altogether, with the significant exception of the \\ion{O}{6}. \n\nAn important caveat to this modeling is that we have considered ionization mechanisms in isolation, rather than possible combinations of them. Such a model might explain the co-existence of \\ion{Mg}{2} and \\ion{O}{6}, for instance, if the latter is collisionally produced while the former arises in a photoionized cloud. Such a model will require more parameters than either model on its own, but may be tenable. We will discuss the possible ``multiphase'' nature of the \\ion{O}{6} absorber below. \n\nIndeed it is impossible to reconcile \\ion{O}{6} and low ions such as \\ion{Mg}{1} into the same CIE or photoionization model, and both of these appear in components B and C. Since the \\ion{O}{6} is not readily reproduced by either of the photoionization models considered above, it probably does arise in collisionally ionized gas. If so, the corresponding \\ion{H}{1} may be too weak to separate from the stronger \\ion{H}{1} from the photoionized component. If the \\ion{O}{6} exists at its peak ionization fraction $f_{OVI} = 0.2$ in CIE at $T = 10^{5.5}$ K, the corresponding \\ion{H}{1} has N(\\ion{H}{1}) $\\simeq (2.5\/Z) \\times 10^{12}$ cm$^{-2}$ where $Z$ is the metallicity relative to solar (thus $2.5 \\times 10^{13}$ cm$^{-2}$ at 0.1$Z_{\\odot}$). Only if the assumed temperature drops to $100,000$ K, where $f_{OVI} \\simeq 10^{-5}$, does the associated \\ion{H}{1} exceed $10^{18}$ cm$^{-2}$ and so become comparable to the observed values. However, at this temperature the H neutral fraction is also $\\simeq 10^{-5}$ and the total H column is an implausibly large $10^{23}$ cm$^{-2}$. We therefore conclude that a collisional origin for the \\ion{O}{6} (and possibly other high ions for which we do not have coverage) is possible, and that the corresponding \\ion{H}{1} would go undetected under most circumstances. Such a case might occur if the \\ion{O}{6} arises in the interface layer between a photoionized cloud and a hot, diffuse halo, as has been proposed for the high-velocity cloud population of the Milky Way \\citep{Sembach:03:165, Fox:04:738}. As these interfaces lie at intermediate temperatures between hot and cold material, they may account for some portion of the intermediate ionization stages as well. Were we able to account for this properly, it might change our conclusion about the photoionization models described above. However, these interface models typically produce less \\ion{O}{6} than is observed, as discussed more below.  \n\nWe are therefore forced to conclude that these components consist of complex multiphase material, and that they may arise from some combination of photo- and collisional-ionization. The essential point of these ionization modeling exercises is not that a particular model gives a perfect fit to the observed ion ratios; the uncertain N(\\ion{H}{1}), component blending, and line saturation ensure that no perfectly robust modeling is tenable. However, ionization models with a range of plausible conditions all point in the same direction; that the bulk of the gas traced by the \\ion{H}{1} and the metal-line absorption is highly ionized, with a total H column density exceeding that seen in \\ion{H}{1} by a factor of 10 - 100. There is still more highly ionized gas, traced by \\ion{O}{6} in all the components, that may indicate interfaces between this photoionized halo gas and its environment. While we cannot completely rule out collisional ionization for the bulk of the strong components B and C, a photoionization model incorporating UV light from star forming regions in the nearby galaxies appears plausible if the cloud has density $n_H \\simeq 10^{-2}$ cm$^{-3}$, lies 20 - 50 kpc from the associated galaxies,  and if those galaxies propagate $\\sim 10$\\% of their ionizing photons into the halo. \n\n\\subsection{Component A}\n\n\\begin{figure}[!t]\n\\epsscale{1.2}\n\\plotone{f9}\n\\caption{Parameter space for photoionization models corresponding to Component A. The most informative constraints are imposed by the column densities of \\ion{C}{2} (blue), \\ion{Mg}{2} (orange), \\ion{S}{3} (black), and \\ion{S}{4} (red).   \\label{compAfig}}\n\\end{figure}\n\nThis component exhibits a wide range of ionization stages, from \\ion{Mg}{2} to \\ion{O}{6}. This range of species cannot be accommodated within a single photoionization or collisional ionization model, for the same reasons that applied to B and C above; it is clearly ``multiphase''. Following the same procedure as above, we attempted to fit as many species as possible into a single model to estimate what properties the cloud might have under various physical scenarios. \n\nWe first ask whether the combination of a galactic star-forming spectrum with the diffuse extragalactic background will provide a suitable model, as it did for components B and C. Since we have a robust measurement for $\\log N$(\\ion{H}{1}) = 16.5 in this case, we can restrict our attention to models that give that value. These results appear in Figure~\\ref{compAfig}. Most regions of this parameter space with $\\log U \\gtrsim -3$ (from the combination of the HM and S99 spectra) satisfy the lower limits on \\ion{C}{3} and \\ion{Si}{3}. The measured column densities of \\ion{C}{2},  \\ion{Mg}{2},  \\ion{S}{3}, and  \\ion{S}{4} are constraining, and mutually consistent in a small region centered around the filled circle at $\\log U = -2$, $Z = 0.2$ $Z_{\\odot}$. This model has a total H column of $\\log N$(H) $\\simeq 19$, and thus a neutral fraction of only $10^{-2.5}$. However, the ionization parameter is 10 - 30 times higher than for B and C, suggesting that this component has some combination of lower density and\/or proximity to the ionizing source. We therefore conclude that this photoionizing model is plausible for all the species except the \\ion{O}{6}. \n\nThe \\ion{O}{6} detection suggests that some of this gas may reside in a hotter, collisionally ionized component. We pursued this hypothesis by adjusting the temperature and metallicity of a purely collisionally ionized model cloud to achieve matches to the observed ratios and column densities. No single temperature is clearly indicated; the well-measured ratio of \\ion{S}{4} to \\ion{S}{3} is matched by a model with 80000 K, but this model greatly underproduces both \\ion{Mg}{2} and \\ion{O}{6}, which has its highest ionization fraction at $\\sim 300000$ K. Single-temperature models that give \\ion{O}{6} column densities above $10^{14}$ cm$^{-2}$ typically give \\ion{Mg}{2} column densities 100$\\times$ lower than we measure here. A model with $\\log T = 5.3$ gives a ratio of \\ion{O}{6} to \\ion{S}{4} that is near the observed value, but overpredicts the observed ratio of \\ion{S}{4} to \\ion{S}{3}. Where therefore conclude that no single-temperature collisional ionization model can explain the absorption by component A; distinct models are needed for the low ionization gas up to \\ion{C}{3}, \\ion{Si}{3}, and \\ion{S}{3}, while a hotter, collisionally ionized phase could account for the \\ion{O}{6} and possibly the \\ion{S}{4}. In a combined model, the neutral H fraction associated with the collisional component is likely quite low with respect to the measured $\\log N$(\\ion{H}{1}) = 16.5, which would trace predominantly the photoionized component.  We thus have no constraint on the metallicity of the hotter component. \n\n\n\n\\begin{figure}[!t]\n\\epsscale{1.2}\n\\plotone{f10}\n\\caption{Parameter space for photoionization models corresponding to Component D. The strong detections of \\ion{C}{3} and \\ion{Si}{3} exclude models to the left of the green and blue lines, while the upper limit on \\ion{Mg}{2} requires models below the orange line and the upper limit on \\ion{S}{4} requires models to the left of the red line.  \\label{compDfig}}\n\\end{figure}\n\n\n\nThus we find that Component A can also be accommodated into a simple scenario in which low-metallicity gas in the halo(s) of the nearby galaxies is ionized by their star formation. Such a simple scenario can account for the observed column densities with the significant exception of the \\ion{O}{6}, for which a collisionally ionized phase may be preferred. \n\n\\subsection{Component D}\n\nComponent D has a column density $\\log N$(\\ion{H}{1}) = 14.8, and exhibits strong absorption in \\ion{O}{6}, saturated \\ion{C}{3} and \\ion{Si}{3}, and the problematic transition of \\ion{N}{3}. Significantly, no \\ion{Fe}{3} is seen here as in B and C. With fewer detected ions, models are more loosely constrained. In this case, a single temperature CIE model can account for the column densities of \\ion{C}{3}, \\ion{Si}{3}, \\ion{N}{3}, and \\ion{O}{6} in a narrow temperature range around $\\log T \\simeq 5.25$, provided the total H column density is $\\log N$(H) $\\sim 20$ and the metallicity is solar. However, this model conflicts with the observed doppler parameter $b = 16$ km s$^{-1}$\\ obtained by consistent fits to the profiles of Ly$\\beta$ and Ly$\\gamma$. At $\\log T = 5.3$, the doppler parameter for H should be no less than $b = \\sqrt{2kT\/m_H} = 57$ km s$^{-1}$, but the line profiles prefer a value $T \\lesssim 20000$ K, which corresponds to $b = 16$ km s$^{-1}$. We therefore rule out a single CIE model for all the detected absorption in this component. \n\nIf we consider a photoionization model for this component, we find that the number of detected ions is generally insufficient to constrain the ionization parameter to the degree that was possible for A, B, and C. Any model with $\\log U \\gtrsim -2.5$ gives the correct column densities of \\ion{C}{3} and \\ion{Si}{3}, but tends to produce too much \\ion{Mg}{2}, which is undetected, unless the metallicity is less than $0.3$ solar. This region of parameter space is shown in Figure~\\ref{compDfig}. The allowed area with $\\log U = 2.5$ to $-1$ and below $\\sim 25 - 40$\\% solar produces enough \\ion{C}{4} and \\ion{Si}{3} without producing too much \\ion{Mg}{2} or \\ion{S}{4}. The total gas column densities N(H) range from $10^{18} - 10^{19.5}$ cm$^{-2}$ as $\\log U$ increases from $-2.5$ to $-1$. This component would appear to have a very large ionization correction, with only $10^{-3}$ or $10^{-4}$ of the hydrogen in \\ion{H}{1}. As for the other components analyzed above, we find that the combined ionizing radiation can explain the detected absorption, but only if the gas is highly ionized and has a subsolar metallicity. \n\n\\subsection{What Creates the \\ion{O}{6}?}\n\n\\label{o6section}\n\nNone of the analysis just presented answers convincingly the question of what creates the highly ionized gas traced by the \\ion{O}{6}. Models with purely photoionized gas do not produce enough \\ion{O}{6}, or if they do they do not also explain the lower-ionization gas. It is the coexistence of this strong \\ion{O}{6}, together with the low ions, that make this absorber ``multiphase''. \n\nWe must consider the possibility that the highly ionized gas in this system -- the \\ion{O}{6} and perhaps some or all of the \\ion{S}{4}, \\ion{S}{3}, and \\ion{N}{3} -- arise in non-equilibrium situations such as shocks, conductive interfaces between hot and cold material, or simply cooling gas which is out of thermal and\/or ionization equilibrium. \\cite{Fox:04:738} and \\cite{Indebetouw:04:205, Indebetouw:04:309} have compiled models for many of these phenomena, and mapped out their predicted column-density ratios in \\ion{Si}{4}, \\ion{C}{4}, \\ion{N}{5}, and \\ion{O}{6}. Unfortunately our COS spectra have coverage of only the latter two ions, and \\ion{N}{5} is not detected. The ratio N(\\ion{N}{5}) \/ N(\\ion{O}{6}) provides only an upper limit for the four component groups, which ranges from $< -0.6$ to $< -1.0$. These limits provide no constraint, since they are consistent with shock ionization, radiative cooling, conductive interfaces, and turbulent mixing layers, as compiled by \\cite{Fox:04:738}. The uncertainty of comparing models to data in this fashion is complicated by the possibility that the gas under study has non-solar relative abundances of the diagnostic elements C, Si, N, and O, which allows the regions occupied by these various models to move within the column density ratio space. While \\cite{Fox:04:738} are able to correct these abundance ratios based on independent measurements of relative abundances, we do not have that much constraint here. So we are unable to produce quantitative constraints on ionization mechanisms using this technique. \n\nThis absorber is notable not only for its very high total column density on \\ion{O}{6}, but also for how it produces this high value. We do not see a monolithic, saturated component in \\ion{O}{6}, thermally broadened into a single profile in hot gas, nor do we see \\ion{O}{6} associated only with the strongest or weakest \\ion{H}{1}. Rather, we see a complex profile in which the \\ion{O}{6} absorption is spread over nearly 400 km s$^{-1}$\\ and breaks into at least two and possibly more distinct components, which line up with components at similar velocities in lower ions. With a sightline that extends over $> 100$ kpc through the halos of two galaxies, it seems natural to guess that the \\ion{O}{6} arises in multiple distinct objects within the halo which may or may not be associated with the individual galaxies. Strong shocks ($v \\sim 600-2000$ km s$^{-1}$) can exhibit column densities in the range of our detections \\citep{Gnat:09:1514}, but imply post-shock temperatures of $T \\sim 5 \\times 10^6$ K and \\ion{H}{1} thermal linewidths $b \\gtrsim 70$ km s$^{-1}$. The strength and complex component structure of the \\ion{H}{1} lines precludes a search for such a broad component, which would only be detectable in the Ly$\\alpha$ profile in this Lyman-limit system.  Thus while we cannot rule out shocks for some of this \\ion{O}{6}, we have no strong indication for them either.  We note that some \\ion{O}{6} absorbers reported in the literature have corresponding broad \\ion{H}{1} components that could indicate the expected temperature in shocked gas \\citep{Tripp:01:724, Richter:04:165, Narayanan:10:1443, 2011arXiv1102.2850S}.\n\nOne possible explanation for the observed kinematic and ionization pattern is that the \\ion{O}{6} arises in transition layers between hot and cold gas, such as might occur for neutral or photoionized clumps falling through a Galactic halo and interacting with the hot halo gas left there by the formation of the galaxy.  \\ion{O}{6} is believed to trace interfaces between stable cold ($T \\sim 10^4$ K) and hot ($T > 10^6$ K) phases of the ISM and circumgalactic medium because it achieves its maximum ionization fraction ($T = $100-300,000 K) near the peak of the radiative cooling curve and so is short-lived. In this simple scenario, interfaces arise where cold clouds interact with the hot medium and are generally unrelated to, or weakly dependent upon, the size and\/or quantity of gas in the cold clumps. Most simply there is just one interface per cloud, however large the cloud. \n\nQuantitative models of conductive interfaces and turbulent mixing layers typically produce column den fsities of \\ion{O}{6} in the range $10^{12-13}$ cm$^{-2}$, lower than we detect here \\citep[cf.][]{Borkowski:90:501, Slavin:93:83, Indebetouw:04:205, Indebetouw:04:309, Gnat:10:1315}, and so multiple interfaces must be invoked. The models of \\cite{Gnat:10:1315} find that \\ion{O}{6} column densities typically have N(\\ion{O}{6}) $< 10^{13}$ cm$^{-2}$ for cold clouds bounded by conductive interfaces within a hot corona, so that $\\sim 100$ interfaces are implied by the total column density of the J1009+0713 absorption-line system. A very large number of low-column density \\ion{O}{6} components probably could be accommodated by our data, since blending and thermal broadening would likely make them difficult to distinguish from a smaller number of higher column density components, which is which is similar to what we observe. \n\nHowever, the interface scenario implies a simple observational pattern: that \\ion{O}{6} should appear whenever low ions trace colder clumps in the halo, and that the strength of \\ion{O}{6} absorption near galaxies should increase with the number of detected components in the cooler material, and correlate with them in velocity space. The overall pattern observed so far is that is that the column density of \\ion{O}{6} is relatively insensitive to the total $N$(\\ion{H}{1}) in surveys of IGM absorbers,  though the patterns depend sensitively on the number of defined components, which vary from study to study especially for the strongest absorbers.  The column density of \\ion{O}{6} varies in our case by only about 0.2 dex across the four detected component groups, while the \\ion{H}{1} column density varies by at least three and possibly four orders of magnitude. This striking lack of correlation suggests that the presence and total quantity of \\ion{O}{6} is not related directly to the total quantity of gas. Indeed, this is a generic pattern of \\ion{O}{6} absorbers that has been noticed in systematic surveys \\citep{Danforth:08:194, Tripp:08:39, Thom:08:22}. In the compilation of \\ion{O}{6} absorbers by \\cite{Thom:08:22}, $N$(\\ion{O}{6}) varies by 1.5 dex over 5 decades of variation in $N$(\\ion{H}{1}). This is partly due to the dramatically multiphase character of some of the absorbers in the Thom \\& Chen sample, but even the ``simple'' \\ion{O}{6} absorbers (systems with well-aligned \\ion{O}{6} and \\ion{H}{1} components and simple component structure with no clear evidence of multiple phases) identified by \\cite{Tripp:08:39} show this effect, with N(\\ion{H}{1}) varying by 3 orders of magnitude while N(\\ion{O}{6}) only changes by 1 dex. Taken as a system, the J1009+0713 LLS occupies the extreme upper right of this diagram, with $\\log N$(\\ion{O}{6}) = 15.0 at $\\log N$(\\ion{H}{1}) $\\sim 18.5$, but taken by components the four groupings scatter over the full range found by \\cite{Thom:08:22}. Thus the comparison of \\ion{O}{6} to \\ion{H}{1} is roughly consistent with the interface scenario. \n\n\n\n\\begin{deluxetable*}{ccccccc}[!t]\n\\tablenum{3} \n\\tablecaption{Literature on OVI-bearing LLSs} \n\\tablehead{\n\\colhead{\\#}&\n\\colhead{Sightline}&\n\\colhead{$z_{abs}$}&\n\\colhead{$\\log N$(\\ion{H}{1})}&\n\\colhead{$\\log N$(\\ion{O}{6})}& \n\\colhead{$N_{comp}$}& \n\\colhead{Reference} }\n\\startdata\n1 & HE0153-4520\t& 0.022601 \t&\t$16.61^{+0.12}_{-0.17}$  & $14.21 \\pm 0.01$\t\t\t& 1 & 1   \\\\\n2 & PKS0312-77\t& 0.2026\t\t&     \t$18.22^{+0.19}_{-0.25}$\t& $14.95 \\pm 0.05$\t\t\t& 6 & 2 \\\\\n3 & PKS0405-123\t& 0.16692\t\t& \t$16.45\\pm 0.05$\t\t& $14.72\\pm0.02$\t\t\t& 5 & 3-7 \\\\\n4 & PG1116+215\t& 0.13847\t\t& \t$16.20 \\pm 0.04$\t\t& $13.68^{+0.10}_{-0.08}$\t& 2 & 8 \\\\\n5 & PG1216+069\t& 0.00632\t\t& \t$19.32 \\pm 0.03$\t\t& $< 14.26$\t\t\t\t& 2 & 9  \\\\\n6 &\t\"\t   \t& 0.12360         &     \t$> 15.95$               \t\t&  $14.83 \\pm 0.10$\t\t\t& 9 & 10  \\\\\n7 &   \"                  & 0.28189         &       $16.70 \\pm 0.04$\t\t& $14.02 \\pm 0.02$\t\t\t& 3 & 10 \\\\\n8& 3C 351.0\t\t& 0.22111\t\t& \t$> 17.0$ \t\t\t\t&  $14.27 \\pm 0.04$\t\t\t& 3 & 10 \\\\\n9 &PHL1811\t\t& 0.07765\t\t& \t$16.03 \\pm 0.07$\t\t& $13.56 \\pm 0.10$ \t\t\t& 2 & 10 \\\\\n10 &\t\"\t\t\t& 0.08093\t\t& \t$17.98 \\pm 0.05$\t\t& $< 13.59$\t\t\t\t& 2 & 11  \\\\\n11 & J1009+0713 \t& 0.3558   \t&    $ 18.0 - 18.8$\t\t\t& $15.0\\pm0.2$\t\t\t& 9 & 12 \n\\enddata\n\\tablerefs{\n(1) \\cite{2011arXiv1102.2850S}; \n(2) \\cite{Lehner:09:734};\n(3) \\cite{Chen:00:L9};\n(4) \\cite{Prochaska:04:718}; \n(5) \\cite{Williger:06:631};\n(6) \\cite{Lehner:07:680}; \n(7) \\cite{Savage:10:1526}; \n(8) \\cite{Sembach:04:351}; \n(9) \\cite{Tripp:05:714}; \n(10) \\cite{Tripp:08:39};\n(11) \\cite{Jenkins:05:767}.\n(12) this paper.}\n\\label{llstable}\n\\end{deluxetable*}\n\nThe low ion \\ion{Mg}{2} is another cold gas tracer that could provide information on the interface scenario. We have successfully built a photoionization scenario that works well for the detected \\ion{Mg}{2} and other low ions. The simplest interface scenario suggests that each ``cold cloud'' could contribute two interfaces to the sightline. We have grouped the major kinematic components in this absorber into four ranges, but if components are generously defined to maximize their number for the \\ion{Mg}{2} there could be approximately nine components over groups A - C, but none for group D. Thus at most 20 interfaces are expected in the \\ion{O}{6}, not the $\\gtrsim 100 $ that are implied by the observed column density and the interface models. While there could be numerous cold clouds and twice as many interfaces that fall below even the stringent detection limits of our HIRES data, there would need to be a large number of undetected \\ion{Mg}{2} to provide enough interfaces to reproduce the \\ion{O}{6}. \n\n\nThe features of this system recall the properties of some other low-redshift, \\ion{O}{6}-bearing LLS that have been intensively studied to which we can compare and contrast our results. These systems are tabulated in Table 3, where we list all the systems in the literature with well-studied UV spectra covering \\ion{O}{6} from {\\it HST} or {\\it FUSE} and having $\\log N$(\\ion{H}{1}) $> 16$ to within observational errors. The listed number of components is the number of \\ion{H}{1} and\/or metal-line components given in the references cited. Here we have assigned 9 components to the J1009+0713 LLS based on the number of observed \\ion{Mg}{2} components in the HIRES data, which has comparable resolution to the STIS\/E140M data used in most of the other cases from the literature.This compilation of results suggests two possible patterns of relevance to the J1009+0713 LLS and the origins of strong \\ion{O}{6} absorbers. First, the total quantity of \\ion{O}{6} is not well correlated with the column density of \\ion{H}{1}. The former spans less than 1 order of magnitude while the latter covers more than three dex. This poor correlation is one of the central puzzles in the origins of the \\ion{O}{6} absorbers (cf. Figure 6 of \\citet{Thom:08:22}). Second, the total {\\it system} column density of \\ion{O}{6} appears to correlate with the number of detected components in the system, such that $> 4$ components gives $\\log N$(\\ion{O}{6}) $>14.5$, while systems with a smaller number of components have lower column density. This effect is clearly seen in Figure~\\ref{o6compfig}, where we plot the \\ion{O}{6} column density over the number of components from Table 3. \n\nWhile the sample is still small, there is a marked correlation that is well outside the typical error on individual measurements (conservatively assigned as $\\pm 0.1$ dex and shown at lower right).  We apply a Spearman's rank correlation test and find that we can reject at 99.7\\% confidence the null hypothesis that there is no correlation between the \\ion{O}{6} column density for these systems and the number of kinematic components they exhibit\\footnote{The correlation is still strong if we assign only the 4 components detected at COS resolution to the J1009+0713 LLS; the data then reject the null hypothesis at 99.3\\% confidence.}. We emphasize that this is not a homogeneously selected sample, though we have tried to gather all reported systems that meet the selection criteria. Nevertheless we find at least a strong suggestion that a large number of components leads to a strong  \\ion{O}{6} absorber. This correlation could indicate that the \\ion{O}{6} is generally associated with the low-ionization material even if it does not reside in the same physical layers of the gas. This general behavior is expected in the interface scenario discussed above, though the problem of how just a few interfaces give $\\log N$(\\ion{O}{6}) $>14.5$ remains. \n\n\n\n\nThe J1009+0713 system and other systems like it thus offer mild but not conclusive evidence in favor of the interface scenario. The broadening of strong O VI into many detectable components and a correlation with a large number of velocity components in the \\ion{H}{1} and low ions are consistent with the basic expectations of the interface scenario. But the total quantity of \\ion{O}{6} substantially exceeds the expectations of the interface models, given the number of apparent cold gas clouds along the sightline. Three possible solutions to this puzzle are that a large number of cold clouds go undetected but their interfaces are seen, or the \\ion{O}{6} from a single interface is well above that calculated in models, or the interface model does not apply. It is still possible, perhaps even likely, that some fraction of the \\ion{O}{6} we observe is produced in interfaces, along with additional contributions from another mechanism. This system shows that further measurements of \\ion{O}{6} and \\ion{Mg}{2} can provide more information about the correlation of hot gas with low-ionization gas for the same sample of absorbers. Such a study is planned as part of our larger survey. \n\n\\section{Summary and Discussion}\n\\label{interp-section}\n\nWe have examined the ionization, metallicity, and association with galaxies of a newly-discovered strong intervening \\ion{O}{6} absorption-line system discovered in our COS data on J1009+0713. This system exhibits $\\log N$(\\ion{O}{6}) = 15.0 spread kinematically over 400 km s$^{-1}$. It appears to be associated with at least two galaxies at projected separations of 14 and 46 kpc from the sightline that have redshifts coincident with the detected gas. The system includes two similar LLS-strength components separated by 60-80 km s$^{-1}$\\ with two weaker associated components 60 - 100 km s$^{-1}$\\ away from these. The direct line measurements only constrain the gas metallicity in the two strongest components to $Z = 0.1 - 1 Z_{\\odot}$ owing to saturation of the \\ion{H}{1}. However,  the detected ion ratios are well-matched by a photoionization model in which simple clouds of total column density $\\log N$(H) $\\simeq 20$ and $5 - 50$\\% solar metallicity are ionized to a 1 - 10\\% neutral fraction by a radiation field from star forming regions at a distance of 20 - 100 kpc. In short, it appears that the bulk of this absorber traces gas that resides in the common halo environment of these galaxies and is ionized by their ongoing star formation. Similar modeling applied to the two outlying components, with $\\log N$(\\ion{H}{1}) = 14.8 and 16.5, are also consistent with their detected absorption for models with $\\log U \\sim -2$ and $\\sim 25$\\% solar metallicity. These outlying clouds also appear to be relatively highly ionized, with neutral fractions of 0.1 - 1\\% even in the gas traced by low ions, but all with an additional, difficult-to-constrain, highly ionized component traced by \\ion{O}{6}.\n\n\\begin{figure}[!t]\n\\epsscale{1.2}\n\\plotone{f11}\n\\caption{ A correlation of the total system column density of \\ion{O}{6} versus the number of reported kinematic components for the literature compilation reported in Table 3. The typical error bar is generously assigned to be $\\pm0.1$ dex and is displayed at lower right; the apparent trend is well outside this margin of error. \\label{o6compfig}}\n\\end{figure}\n\nThis system exhibits two galaxies ranging over two magnitudes in the rest-frame optical, the brightest (170\\_9) has one with $M_* = 6 \\times 10^9 M_{\\odot}$ in stars\\footnote{The galaxy 92\\_7 appears in the SDSS photometric database with $r = 22$, and appears distinctly in the WFC3 image, but we have not yet obtained its redshift. It may also be associated with this absorption-line system.}. Galaxy 86\\_4 is 2 mag fainter with a total star formation rate of only about 1\/10 that of 170\\_9. Based on emission-line measurements and its inferred stellar mass, the brighter galaxy 170\\_9 appears to be consistent with solar metallicity, while 86\\_4 is either consistent with solar metallicity or with $\\sim 25$\\% solar (depending on which branch of the R23 indicator is adopted). Since none of the detected gas components appear to {\\it require} a solar metallicity (though B and C are consistent with solar if the photoionization modeling is discounted), we do not have positive evidence that any of the detected gas was brought into the sightline by outflows from the galaxies. \n\n\n\n   \n\n\n\n\n\nSeveral features of specific systems are worth comparing to the J1009+0713 LLS. The system at $z = 0.167$ toward PKS0405-1219 \\citep{Chen:00:L9}  lies $\\simeq 70$ kpc from two massive galaxies, is at least 0.1 solar and likely higher metallicity, and exhibits strong absorption from high ions up to \\ion{N}{5} and \\ion{O}{6} at $\\log N$(\\ion{O}{6}) = 14.6. The $z = 0.0809$ LLS toward PHL 1811 \\citep{Jenkins:05:767} lies 34 and 87 kpc from two nearby L* galaxies, and has nearly a solar metallicity, but is not associated with highly ionized gas. Both of these absorbers show evidence of a predominantly ionized gas, though the PHL 1811 system also exhibits absorption by low ions. While our simple photoionization models shown above indicate a sub-solar metallicity for the J1009+0713 system, this model-dependent value may not reflect the true gas metallicity in the LLS components: the \\ion{O}{1} measurements are consistent with solar metallicity.  The PKS0405-1219 system is a particularly interesting comparison to the J1009+0713 system, since the \\ion{O}{6} in both cases is stronger and more kinematically complex than would be expected from the lower ionization species. While is it difficult to draw general conclusions from only three cases, it appears that galaxy halos may commonly host LLS absorption, that it can be quite enriched in metals to nearly solar abundances, and that it can be associated with large quantities of highly ionized gas. Based on the PKS0405-1219 and J1009+0713 systems, we speculate that LLS-strength absorption may be necessary to produce the strongest intervening \\ion{O}{6} absorbers, $\\log N$(\\ion{O}{6}) $\\gtrsim 14.5$. The general incidence of LLS in galaxy halos and its relationship to hot gas will be addressed by our larger survey, which should also be able to assess the gas metallicity in many cases. High metallicity in dense, cool gas located at $> 50$ kpc from the nearest galaxies may indicate a major role for tidal stripping of dwarf satellites, or robust outflows from large galaxies or their dwarf satellites. \n\nThis sightline passes by a galaxy with nearly $L*$ and a dwarf galaxy possibly in its halo or interacting with it, which may have low metallicity. Their total velocity separation is only $25$ km s$^{-1}$, which suggests they are gravitationally associated. It is tempting to draw comparisons of this system to the Milky Way and its Magellanic Clouds. If the smaller galaxy has the metallicity of $25$\\% indicated by their lower-branch R23 measurements, it could be associated with the A and\/or D components in an outflow or tidal-stripping scenario. If so, some of the detected absorption components may arise in gas stripped from this dwarf galaxies, like the Magellanic Stream.\n\nAlternatively, all the detected absorption components could have sub-solar metallicity and could trace low-metallicity, infalling material not unlike the large HVC complexes of the Milky Way. This inference and the high ionization state of the detected gas would imply that infalling, metal-poor gas enters halos and\/or galaxies in ionized form, and is possibly associated with still more highly ionized material traced by \\ion{O}{6} that arises either in the hot diffuse galaxy halo itself or in interface zones of intermediate temperature between that hot ambient medium and the cooler infalling clouds. This scenario has been well developed for the Milky Way - further tests of its validity for external galaxies requires better knowledge of gas metallicities, kinematics in association to galaxies, and thus more well-studied cases.  \n\nIn our combined COS and HIRES data we have used two important diagnostic lines for gas in galaxy halos -- both \\ion{O}{6} and \\ion{Mg}{2} -- that are typically not available in combination for the practical reason that they require both UV and optical data and galaxies at $z \\gtrsim 0.1$. Recently, \\cite{Barton:09:1817}  and \\cite{Chen:10:1521} have systematically addressed the number density of \\ion{Mg}{2} absorbers in the $\\sim 100$ kpc regions surrounding galaxies using galaxies and QSO sightlines selected from SDSS. They find that the equivalent width of \\ion{Mg}{2} increases at low impact parameter in a well-defined trend that has a detectable dependence on galaxy luminosity for those galaxies that exhibit \\ion{Mg}{2} absorption. However, both \\cite{Bowen:11:47} and \\cite{Gauthier:10:1263} have found lower covering fractions of strong \\ion{Mg}{2} surrounding more massive galaxies (LRGs) that indicate the cool gas covering fraction depends, perhaps strongly, on galaxy type and\/or environment. For comparison with these samples, we obtained a total $W_r^{2796} = 1250 \\pm 15$ m\\AA\\ for the full LLS analyzed here, including 188 m\\AA\\ in component group A, 604 m\\AA\\ in component group B, and 472 m\\AA\\ in component group C. These findings are all within the range of the distributions found by these other studies. However, since we also detect \\ion{H}{1}, \\ion{O}{6} and other multiphase UV ions in these component groups, we can draw additional implications that are not possible based on \\ion{Mg}{2} alone. \n\nFirst, we have used the detections of \\ion{Mg}{2} and the ionization stages of C, N, O, Mg, S, Si, and Fe to infer that the absorbing clouds have only a small portion, 0.1 - 1\\% of their gas in the neutral phase. This finding implies that the gas mass traced by these absorbers could be significant. Indeed, \\cite{Chen:10:1521} inferred \\ion{Mg}{2} cloud masses of $\\sim 2 \\times 10^4$ M$_{\\odot}$ and a total baryonic mass in halos of $3 \\times 10^9$ M$_{\\odot}$, using the important assumptions of 10\\% solar metallicity and a 10\\% ionization fraction for \\ion{Mg}{2}. These values are within the range permitted for the J1009+0713 clouds, where the ionization corrections and metallicities are better constrained by multiple ionization stages. Thus our findings indicate that the assumption of a significant mass correction to the observed \\ion{Mg}{2} is reasonable, and such absorbers could harbor significant mass. Moreover, the \\ion{O}{6} suggests a possibly significant contribution of mass over and above that provided by \\ion{Mg}{2} systems. Finally, it may prove important to measure halo absorbers in {\\em both} \\ion{O}{6} and \\ion{Mg}{2}; components A, B, and C here hint that \\ion{O}{6} and \\ion{Mg}{2} can have some relationship but may vary in their ratio by over an order of magnitude. Our larger survey was designed with the goal of detecting both lines in a significant sample of galaxies over a range of mass. We will thus be able to test the findings of  \\cite{Barton:09:1817}  and \\cite{Chen:10:1521} that \\ion{Mg}{2} correlates with galaxy luminosity, and also the finding by \\cite{Bowen:11:47} and \\cite{Gauthier:10:1263} that massive galaxies have less \\ion{Mg}{2}, while also exploring these relations in \\ion{O}{6} for the same galaxies. These previous results, considered together with hints of a \\ion{Mg}{2} \/ \\ion{O}{6} relationship for the J1009+0713 system, suggests this will be a fruitful line of research. \n\nWhile it is among the strongest known \\ion{O}{6}\\ absorbers, this system at least superficially resembles the highly-ionized, multiphase, \\ion{O}{6}-bearing HVCs surrounding the Millky Way. For example, \\cite{Fox:05:332} report conditions very similar to those we find here for two HVC complexes toward HE0226-4110 and PG0953+414 in the Milky Way halo. Specifically, their ionization modeling finds $\\log U \\simeq -3.5$, subsolar metallicity, and $\\log n_H \\simeq -2$, but for clouds with $\\log N$(\\ion{H}{1}) $\\sim 16-16.5$. Like them, we have found consistent photoionization models for the present system that yet do not explain the observed abundance of \\ion{O}{6}. Our findings also resonate with the recent survey of ionized silicon in the Milky Way HVC population by \\cite{Shull:09:754} and \\cite{Collins:09:962}, who find typical values of $\\log U \\sim -3$ and 10 - 30\\% solar metallicity or lower in these clouds with neutral fractions of $\\simeq 1$\\%. These favorable comparisons suggests that highly-ionized, possibly metal-poor gas resides in the halos of star forming galaxies as it does in the Milky Way -- a key motivation for the survey that discovered this absorber. A more speculative but interesting inference can be drawn from the column density and ionization of the main components B and C. Were they exposed to a lower radiation field, because of a lower galactic $f_{esc}$ or SFR, they could instead appear as a sub-DLA or DLA with $N$(\\ion{H}{1}) $\\sim 10^{20}$ cm$^{-2}$ located at least 20 kpc from the nearest galaxy, and might have total column densities similar to the classical MW HVC complexes. We note that recent surveys for 21-cm absorption affiliated with nearby galaxies \\citep{Borthakur:10:131, Gupta:10:849} have found 21-cm absorbers to be quite rare in galaxy halos.  While this could be an indication that sub-DLAs and DLAs are rarely found at such large impact parameters \\citep[cf.][]{Meiring:11:2516, Peroux:11:2251}, this remains an open question since 21-cm absorption depends on spin temperature, and sub-DLAs and DLAs might be missed in 21-cm absorption surveys if the gas is too warm.  Our larger survey will address this problem by assessing the incidence of LLS, sub-DLAs, and DLAs in galaxy halos.\n\nWhile the similarity of this absorber to some well-studied Milky Way HVCs provides an encouraging link between the HVCs and galaxy halo gas at large, this system also illustrates some important limitations that must be overcome to fully map out diffuse gas in halos. First, \\ion{H}{1} column densities in the range that gives an LLS but not a DLA, as here, are extremely difficult to constrain to a narrow range. This is especially true if the gas obviously occupies several clearly separated kinematic components that are blended in the Lyman series, as shown by our HIRES data. This unfortunate circumstance makes metallicities hard to estimate, even in cases where conditions are otherwise favorable for obtaining metal-line column densities, as they are here. While LLS are rare in the IGM at large, they may be quite common close to galaxies where total gas densities are high, even if the gas is typically highly ionized. This system also illustrates vividly that relating halo gas to its host galaxies is complicated by projection effects and by multiple galaxies lying along the sightline. We clearly cannot associate the absorption, especially in the \\ion{O}{6}, with any of the three galaxy candidates in particular. \n\nFinally, this system illustrates that detection of many ionization stages can constrain the physical parameters of the gas, but only for well-defined hypothetical physical scenarios, e.g. pure photoionization or pure collisional ionization. Combinations of such models, or non-standard scenarios such as interfaces and\/or non-equilibrium calculations rapidly become too complex to test easily. Even for simple scenarios it is sometimes difficult to identify unique explanations for detected absorption. These considerations suggest that a large, systematic survey to assess possible variations could help  by smoothing out variations along individual lines of sight. Our systematic survey of gas in galaxy halos, which serendipitously discovered this absorber, aims to establish a set of empirical facts about the content of gaseous halos. Yet multiphase, kinematically complex absorbers like this one pose a stiff challenge to our ability to build theoretical models that are up to the task of relating complex physical processes in galaxy halos to observable quantities. Progress in this area could significantly advance the cause of understanding galaxy formation at large. \n\n\\acknowledgments\n\nWe are happy to acknowledge a very constructive referee's report from Mike Shull, and helpful comments from Andrew Fox. Support for program GO11598 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. TMT appreciates support for this work from NASA ADP grant NNX08AJ44G. Some of the data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community.  We are most fortunate to have the opportunity to conduct observations from this mountain.\n\n{\\it Facilities:} \\facility{HST (COS, WFC3)}, \\facility{Keck (LRIS, HIRES)}.\n\n\n\n\\bibliographystyle{\/Users\/tumlinson\/astronat\/apj\/apj}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe problem of Pillai states that for each fixed integer $ c\\ge 1 $, the Diophantine equation\n\\begin{eqnarray}\\label{Pillai}\na^x-b^y = c, ~~~~~\\min\\{x,y\\}\\ge 2,\n\\end{eqnarray}\nhas only a finite number of positive solutions $\\{a,b,x,y\\}$. This problem is still open; however, the case $ c=1 $, is the conjecture of Catalan and was proved by Mih\\u ailescu \\cite{Mihailescu}. In 1936 (see \\cite{Pillai:1936, Pillai:1937}), in the special case $ (a,b) =(3,2) $ which is a continutation of the work of Herschfeld \\cite{Herschfeld:1935, Herschfeld:1936} in 1935, Pillai conjectured that the only integers $ c $ admitting at least two representations of the form $ 2^{x}-3^{y} $ are given by\n\\begin{eqnarray}\\label{Pillaisolns}\n2^{3}-3^{2}=2^{1}-3^{1}=-1, ~~~ 2^{5}-3^{3}=2^{3}-3^{1}=5, ~~~ 2^{8}-3^{5}=2^{4}-3^{1}=13.\n\\end{eqnarray}\nThis  was confirmed by Stroeker and Tijdeman \\cite{StroekerTijdeman} in 1982. The general problem of Pillai is difficult to solve and this has motivated the consideration of special cases of this problem. In the past years, several special cases of the problem of Pillai have been studied. See, for example, \\cite{Luca16, Chim1, Chim2, Ddamulira11, Ddamulira12, Ddamulira1, Hernane, Hernandez}.\n\nLet $k \\geqslant 2$ be an integer. We consider a generalization of Fibonacci sequence called the $k$--generalized Fibonacci sequence $\\lbrace F_n^{(k)} \\rbrace_{n\\geqslant 2-k}$ defined as\n\\begin{equation}\nF_n^{(k)} = F_{n-1}^{(k)} + F_{n-2}^{(k)} + \\cdots + F_{n-k}^{(k)},\\label{fibb1}\n\\end{equation}\nwith the initial conditions\n\\[F_{-(k-2)}^{(k)} = F_{-(k-3)}^{(k)} = \\cdots = F_{0}^{(k)} = 0 \\quad {\\text {\\rm  and  }}\\quad F_{1}^{(k)} = 1.\\]\nWe call $F_{n}^{(k)}$ the $n$th $k$--generalized Fibonacci number. Note that when $k=2$, it  coincides with the Fibonacci numbers and when $k=3$ it is the Tribonacci number. The first $k+1$ nonzero terms in $F_{n}^{(k)}$ are powers of $2$, namely\n\\begin{equation*}\nF_{1}^{(k)}=1,\\quad F_{2}^{(k)}=1,\\quad F_{3}^{(k)}=2, \\quad F_{4}^{(k)}=4,\\ldots, F_{k+1}^{(k)}=2^{k-1}.\n\\end{equation*}\nFurthermore, the next term is $F_{k+2}^{(k)}=2^{k}-1$.\nThus, we have that\n\\begin{equation}\\label{Fibbo111}\nF_{n}^{(k)} = 2^{n-2} \\quad {\\text{\\rm holds for all}}\\quad 2\\leq n\\leq k+1.\n\\end{equation}\nWe also observe that the recursion \\eqref{fibb1} implies the three--term recursion\n$$\nF_{n}^{(k)} =2F_{n-1}^{(k)} - F_{n-k-1}^{(k)} \\quad {\\text{\\rm for all}}\\quad n\\geq 3,\\label{fibb2}\n$$\nwhich can be used to prove by induction on $m$ that $F_n^{(k)}<2^{n-2}$ for all $n\\geq k+2$ (see also \\cite{BBL17}, Lemma $2$).\n\nThe generalised Fibonacci analogue of the problem of Pillai under the same conditions as in \\eqref{Pillai}, concerns studying for fixed $(k,\\ell)$ all values of the integer $c$ such that the equation\n\\begin{eqnarray}\\label{Pillai2}\nF_n^{(k)} - F_m^{(\\ell)} = c\n\\end{eqnarray}\nhas at least two solutions $(n,m)$. We are not aware of a general treatment of equation \\eqref{Pillai2} (namely, considering $k$ and $\\ell$ parameters), although the particular case when $\\{k,\\ell\\}=\\{2,3\\}$ was treated in \\cite{Chim1}.\n\nDdamulira, G\\'omez and Luca \\cite{Ddamulira2}, studied the Diophantine equation\n\\begin{eqnarray} \\label{Pillai3}\nF_{n}^{(k)}-2^{m}=c,\n\\end{eqnarray}\nwhere $k$ is also a parameter, which is a variation of equation \\eqref{Pillai2}. They determined all integers $c$ such that equation \\eqref{Pillai3} has at least two solutions $(n,m)$. \nThese $c$ together with their multiple representations as in \\eqref{Pillai3} turned out to be grouped into four parametric families. \n\nIn this paper, we study a related problem and we find all integers $c$ admitting at least two representations of the form $ F_n^{(k)} - 3^m $ for some positive integers $k$, $n$ and $m$. This can be interpreted as solving the equation\n\\begin{align}\n\\label{Problem}\n F_n^{(k)} - 3^m = F_{n_1}^{(k)} - 3^{m_1}~~~~(=c)\n\\end{align}\nwith $(n, m) \\neq (n_1, m_1)$. The cases $k=2$ and $ k=3 $ have been solved completely by the first author in \\cite{Ddamulira11} and \\cite{Ddamulira12}, respectively. So, we focus on the case  $k \\geqslant 4$.\\\\\n\n\n\\begin{theorem}\\label{Main}\nFor fixed integer $ k\\ge 4 $, the Diophantine equation \\eqref{Problem} with $ n>n_1\\geq 2 $ and $ m>m_1\\ge 1 $ has:\n\\begin{itemize}\n\\item[(i)] solutions with $c\\in \\{-1, 5, 13\\}$ and $ 2\\le n \\le k+1 $, which arise from the classical Pillai problem for $(a,b)=(2,3)$, namely: \n\\begin{eqnarray*}\nF_{5}^{(k)}-3^{2}&=&F_{3}^{(k)}-3^{1} ~~=~ -1, ~~~k\\ge 4, \\\\\nF_{7}^{(k)}-3^{3}&=&F_{5}^{(k)}-3^{1} ~~=~~ 5, ~~~k\\ge 6,\\\\\nF_{10}^{(k)}-3^{5}&=&F_{6}^{(k)}-3^{1} ~~=~~ 13, ~~~k\\ge 9;\n\\end{eqnarray*}\n\\item[(ii)] solutions with $c\\in \\{-25, -7,5\\} $ and $ n\\geq k+2 $ and $ k\\in \\{4,5,6\\} $. Futhermore, all the representations of $ c $ in this case are given by\n\\begin{eqnarray*}\nF_{8}^{(4)}-3^{4}&=&F_{3}^{(4)}-3^{3} ~~=~ -25,\\\\\nF_{10}^{(5)}-3^{5}&=&F_{3}^{(5)}-3^{2} ~~=~ -7,\\\\\nF_{10}^{(6)}-3^5&=&F_6^{(6)}-3^1~~=~~5.\n\\end{eqnarray*}\nfor $ k=4,5 $ and $6$, respectively.\n\\end{itemize}\n\\end{theorem}\n\n\\section{Preliminary Results}\n\nIn this section, we recall some general results from algebra number theory and diophantine approximations \nand properties of the $k$-generalized Fibonacci sequence.\n\n\\subsection{Notations and terminology from algebraic number theory} \n\nWe begin by recalling some basic notions from algebraic number theory.\n\nLet $\\eta$ be an algebraic number of degree $d$ with minimal primitive polynomial over the integers\n$$\na_0x^{d}+ a_1x^{d-1}+\\cdots+a_d = a_0\\prod_{i=1}^{d}(x-\\eta^{(i)}),\n$$\nwhere the leading coefficient $a_0$ is positive and the $\\eta^{(i)}$'s are the conjugates of $\\eta$. Then the \\textit{logarithmic height} of $\\eta$ is given by\n$$ \nh(\\eta):=\\dfrac{1}{d}\\left( \\log a_0 + \\sum_{i=1}^{d}\\log\\left(\\max\\{|\\eta^{(i)}|, 1\\}\\right)\\right).\n$$\nIn particular, if $\\eta=p\/q$ is a rational number with $\\gcd (p,q)=1$ and $q>0$, then $h(\\eta)=\\log\\max\\{|p|, q\\}$. The following are some of the properties of the logarithmic height function $h(\\cdot)$, which will be used in the next sections of this paper without reference:\n\\begin{eqnarray}\nh(\\eta\\pm \\gamma) &\\leq& h(\\eta) +h(\\gamma) +\\log 2,\\nonumber\\\\\nh(\\eta\\gamma^{\\pm 1})&\\leq & h(\\eta) + h(\\gamma),\\\\\nh(\\eta^{s}) &=& |s|h(\\eta) \\qquad (s\\in\\mathbb{Z}). \\nonumber\n\\end{eqnarray}\n\n\n\\subsection{$k$-generalized Fibonacci numbers}\n\nIt is known that the characteristic polynomial of the $k$--generalized Fibonacci numbers $F^{(k)}:=(F_m^{(k)})_{m\\geq 2-k}$, namely\n$$\n\\Psi_k(x) := x^k - x^{k-1} - \\cdots - x - 1,\n$$\nis irreducible over $\\mathbb{Q}[x]$ and has just one root outside the unit circle. Let $\\alpha := \\alpha(k)$ denote that single root, which is located between $2\\left(1-2^{-k} \\right)$ and $2$ (see \\cite{Dresden}). \nThis is called the dominant root of $F^{(k)}$. To simplify notation, in our application we shall omit the dependence on $k$ of $\\alpha$. We shall use $\\alpha^{(1)}, \\dotso, \\alpha^{(k)}$ for all roots of $\\Psi_k(x)$ with the convention that $\\alpha^{(1)} := \\alpha$.\n\nWe now consider for an integer $ k\\geq 2 $, the function\n\\begin{eqnarray}\\label{fun12}\nf_{k}(z) = \\dfrac{z-1}{2+(k+1)(z-2)} \\qquad {\\text{for}}\\qquad z \\in \\mathbb{C}.\n\\end{eqnarray}\nWith this notation, Dresden and Du presented in  \\cite{Dresden} the following ``Binet--like\" formula for the terms of $F^{(k)}$:\n\\begin{eqnarray} \\label{Binet}\nF_m^{(k)} = \\sum_{i=1}^{k} f_{k}(\\alpha^{(i)}) {\\alpha^{(i)}}^{m-1}.\n\\end{eqnarray}\nIt was proved in \\cite{Dresden} that the contribution of the roots which are inside the unit circle to the formula (\\ref{Binet}) is very small, namely that the approximation\n\\begin{equation} \\label{approxgap}\n\\left| F_m^{(k)} - f_{k}(\\alpha)\\alpha^{m-1} \\right| < \\dfrac{1 }{2} \\quad \\mbox{holds~ for~ all~ } m \\geqslant 2 - k.\n\\end{equation}\nIt was proved by Bravo and Luca in \\cite{BBL17} that\n\\begin{eqnarray}\\label{Fib12}\n\\alpha^{m-2} \\leq F_{m}^{(k)} \\leq \\alpha^{m-1}\\qquad {\\text{\\rm holds for all}}\\qquad m\\geq 1\\quad {\\text{\\rm and}}\\quad k\\geq 2.\n\\end{eqnarray}\n\nBefore we conclude this section, we present some useful lemma that will be used in the next sections on this paper. The following lemma was proved by Bravo and Luca in \\cite{BBL17}.\n\\begin{lemma}[Bravo, Luca]\\label{fala5}\nLet $k\\geq 2$, $\\alpha$ be the dominant root of $\\{F^{(k)}_m\\}_{m\\ge 2-k}$, and consider the function $f_{k}(z)$ defined in \\eqref{fun12}. \n\\begin{itemize}\n\\item[(i)]\\label{kat1} The inequalities\n$$\n\\dfrac{1}{2}< f_{k}(\\alpha)< \\dfrac{3}{4}\\qquad \\text{and}\\qquad |f_{k}(\\alpha^{(i)})|<1, \\qquad  2\\leq i\\leq k\n$$\nhold. In particular, the number $f_{k}(\\alpha)$ is not an algebraic integer.\n\\item[(ii)]\\label{kat2}The logarithmic height of $f_k(\\alpha)$ satisfies $h(f_{k}(\\alpha))< 3\\log k$.\n\\end{itemize}\n\\end{lemma}\n\n\n\\subsection{Linear forms in logarithms and continued fractions}\n\n\nIn order to prove our main result Theorem \\ref{Main}, we need to use several times a Baker--type lower bound for a nonzero linear form in logarithms of algebraic numbers. There are many such in the literature  like that of Baker and W{\\\"u}stholz from \\cite{bawu07}. We use the following result by Matveev \\cite{MatveevII}, which is one of our main tools in this paper.\n\n\\begin{theorem}[Matveev]\\label{Matveev11} Let $\\gamma_1,\\ldots,\\gamma_t$ be positive real algebraic numbers in a real algebraic number field \n$\\mathbb{K}$ of degree $D$, $b_1,\\ldots,b_t$ be nonzero integers, and assume that\n\\begin{equation}\n\\label{eq:Lambda}\n\\Lambda:=\\gamma_1^{b_1}\\cdots\\gamma_t^{b_t} - 1\n\\end{equation}\nis nonzero. Then\n$$\n\\log |\\Lambda| > -1.4\\times 30^{t+3}\\times t^{4.5}\\times D^{2}(1+\\log D)(1+\\log B)A_1\\cdots A_t,\n$$\nwhere\n$$\nB\\geq\\max\\{|b_1|, \\ldots, |b_t|\\},\n$$\nand\n$$A\n_i \\geq \\max\\{Dh(\\gamma_i), |\\log\\gamma_i|, 0.16\\},\\qquad {\\text{for all}}\\qquad i=1,\\ldots,t.\n$$\n\\end{theorem} \n\nDuring the course of our calculations, we get some upper bounds on our variables which are too large, thus we need to reduce them. To do so, we use some results from the theory of continued fractions. Specifically, for a nonhomogeneous linear form in two integer variables, we use a slight variation of a result due to Dujella and Peth{\\H o} (see \\cite{dujella98}, Lemma 5a), which  itself is a generalization of a result of Baker and Davenport \\cite{BD69}.\n\nFor a real number $X$, we write  $||X||:= \\min\\{|X-n|: n\\in\\mathbb{Z}\\}$ for the distance from $X$ to the nearest integer.\n\\begin{lemma}[Dujella, Peth\\H o]\\label{Dujjella}\nLet $M$ be a positive integer, $p\/q$ be a convergent of the continued fraction of the irrational number $\\tau$ such that $q>6M$, and  $A,B,\\mu$ be some real numbers with $A>0$ and $B>1$. Let further \n$\\varepsilon: = ||\\mu q||-M||\\tau q||$. If $ \\varepsilon > 0 $, then there is no solution to the inequality\n$$\n0<|u\\tau-v+\\mu|<AB^{-w},\n$$\nin positive integers $u,v$ and $w$ with\n$$ \nu\\le M \\quad {\\text{and}}\\quad w\\ge \\dfrac{\\log(Aq\/\\varepsilon)}{\\log B}.\n$$\n\\end{lemma}\n\nThe above lemma cannot be applied when $\\mu=0$ (since then $\\varepsilon<0$). In this case, we use the following criterion of Legendre.\n\n\\begin{lemma}[Legendre]\n\\label{lem:legendre}\nLet $\\tau$ be real number and $x,y$ integers such that\n\\begin{equation}\n\\label{eq:continuedfraction}\n\\left|\\tau-\\frac{x}{y}\\right|<\\frac{1}{2y^2}.\n\\end{equation}\nThen $x\/y=p_k\/q_k$ is a convergent of $\\tau$. Furthermore, \n\\begin{equation}\n\\label{eq:continuedfraction1}\n\\left|\\tau-\\frac{x}{y}\\right|\\ge \\frac{1}{(a_{k+1}+2)y^2}.\n\\end{equation}\n\\end{lemma}\n\n\nFinally, the following lemma is also useful. It is Lemma 7 in \\cite{guzmanluca}. \n\n\\begin{lemma}[G\\'uzman, Luca]\n\\label{gl}\nIf $m\\geqslant 1$, $T>(4m^2)^m$  and $T>x\/(\\log x)^m$, then\n$$\nx<2^mT(\\log T)^m.\n$$\n\\end{lemma}\n\n\\section{The connection with the classical Pillai problem}\n\nAssume that $ (n,m)\\neq (n_{1}, m_{1}) $ are such that\n$$F_{n}^{(k)} - 3^{m} = F_{n_{1}}^{(k)}-3^{m_{1}}.$$\nIf $ m=m_{1} $, then $ F_{n}^{(k)} = F_{n_{1}}^{(k)} $ and since $ \\min\\{n,n_{1}\\}\\geq 2$, we get that $ n=n_{1} $. Thus, $ (n,m) = (n_{1}, m_{1}) $, contradicting our assumption. Hence, $ m\\neq m_{1} $, and we may assume without loss of generality that $ m>m_{1}\\geq 1 $. Since\n\\begin{eqnarray}\n\\label{fala1}\nF_{n}^{(k)} - F_{n_{1}}^{(k)} &=& 3^{m}-3^{m_{1}},\n\\end{eqnarray}\nand the right--hand side of \\eqref{fala1} is positive, we get that the left--hand side of\n\\eqref{fala1} is also positive and so $ n>n_{1} $. Furthermore, since $ F_{1}^{(k)}=F_{2}^{(k)}=1 $,   we may assume that $ n> n_{1}\\geq 2 $. \n\n\\medskip\nWe analyse the possible situations.\n\\medskip\n\n\\noindent {\\bf Case 1.}\nAssume that $ 2\\leq n_{1}<n\\leq k+1 $. Then, by \\eqref{Fibbo111}, we have\n\\begin{eqnarray*}\nF_{n_{1}}^{(k)}=2^{n_{1}-2}\\qquad {\\rm and} \\qquad F_{n}^{(k)}=2^{n-2}\n\\end{eqnarray*}\nso, by substituting them in \\eqref{fala1}, we get\n\\begin{eqnarray*}\n2^{n-2}-3^{m} = 2^{n_1-2}-3^{m_{1}}.\n\\end{eqnarray*}\nBy comparing with the classical solutions in \\eqref{Pillaisolns}, and by using the fact that $F_n^{(k)}$ is a power of $2$ if and only if $n\\le k+1$ (see \\cite{BravoLucapowersof2}), we get the solutions\n\\begin{eqnarray}\nF_{5}^{(k)}-3^{2}&=&F_{3}^{(k)}-3^{1} ~~=~ -1, ~~~k\\ge 4,\\nonumber \\\\\nF_{7}^{(k)}-3^{3}&=&F_{5}^{(k)}-3^{2} ~~=~~ 5, ~~~k\\ge 6,\\\\\nF_{10}^{(k)}-3^{5}&=&F_{6}^{(k)}-3^{1} ~~=~~ 13, ~~~k\\ge 9.\\nonumber\n\\end{eqnarray}\n\n\n\\medskip\n\n\\noindent {\\bf Case 2.}\nAssume $ n \\ge k+2 $. The following lemma is useful.\n\\begin{lemma}\\label{classical1}\nFor $n\\ge k+2$, the conditions \n$$\nF_n^{(k)}-3^m=F_{n_1}^{(k)}-3^{m_1}\\quad {\\text{and}}\\quad 2^{n-2}-3^m=2^{n_1-2}-3^{m_1}\n$$\ncannot simultaneously hold.\n\\end{lemma}\n\n\\begin{proof} If they do, then\n$$\n2^{n-2}-F_{n}^{(k)}=2^{n_1-2}-F_{n_1}^{(k)}.\n$$\nThe sequence $\\{2^{n-2}-F_{n}^{(k)}\\}_{n\\ge 2}$ is $0$ at $n=2,3,\\ldots,k+1$ and is $1$ at $n=k+2$. We show that from here on it is increasing. That is \n$$\n2^{n-1}-F_{n+1}^{(k)}>2^{n-2}-F_n^{(k)}\\quad {\\text{\\rm holds~for}}\\quad n\\ge k+2.\n$$\nThis is equivalent to\n$$\n2^{n-2}>F_{n+1}^{(k)}-F_n^{(k)}=F_{n-1}^{(k)}+\\cdots+F_{n+1-k}^{(k)},\n$$\nand this last inequality holds true because in the right--hand side we have $F_i\\le 2^{i-2}$ for $i=n+1-k,n+2-k,\\ldots,n-1$ and then \n$$\n\\sum_{i=n-k+1}^{n-1} F_i\\le \\sum_{i=n-k+1}^{n-1} 2^{i-2}<1+2+\\cdots+2^{n-3}<2^{n-2}.\n$$\n\\end{proof}\n\n\n\n\n\\section{Bounding $n$ in terms of $m$ and $k$}\n\nBy the results of the previous section, we assume that $n\\ge k+2$. Thus, $ 2^{n-2}-3^{m}\\neq 2^{n_1-2}-3^{m_1} $. Since $ n>n_{1}\\geq 2 $, we have that $ F_{n_{1}}^{(k)}  \\leq F_{n-1}^{(k)}$ and therefore\n$$F_{n}^{(k)} = F_{n-1}^{(k)} + \\cdots + F_{n-k}^{(k)} \\geq F_{n-1}^{(k)} + \\cdots + F_{n-k-1}^{(k)} \\geq F_{n_{1}}^{(k)} + \\cdots + F_{n-k-1}^{(k)}.$$\nSo, from the above, \\eqref{Fib12} and \\eqref{fala1}, we have\n\\begin{eqnarray}\\label{fala4}\n\\alpha^{n-4}&\\leq& F_{n-2}^{(k)}\\leq F_{n}^{(k)}-F_{n_{1}}^{(k)} = 3^{m}-3^{m_{1}} < 3^{m}, \\text{  and  }\\\\\n\\alpha^{n-1}&\\geq& F_{n}^{(k)} > F_{n}^{(k)}-F_{n_{1}}^{(k)} = 3^{m}-3^{m_{1}}\\geq 3^{m-1},\\nonumber\n\\end{eqnarray}\nleading to\n\\begin{eqnarray}\\label{fala2}\n1+\\left(\\dfrac{\\log 3}{\\log\\alpha}\\right) (m - 1) < n< \\left(\\dfrac{\\log 3}{\\log \\alpha}\\right)m + 4.\n\\end{eqnarray}\n\nWe note that the above inequality \\eqref{fala2} in particular implies that $ m < n< 1.6m+4 $. \nWe assume for technical reasons that $ n>600 $. By \\eqref{approxgap} and \\eqref{fala1}, we get\n\\begin{eqnarray*}\n\\left|f_{k}(\\alpha)\\alpha^{n-1} - 3^{m}\\right|&=&\\left|(f_{k}(\\alpha)\\alpha^{n-1}-F_{n}^{(k)})+(F_{n_{1}}^{(k)}-3^{m_{1}})\\right|\\\\\n&=&\\left|(f_{k}(\\alpha)\\alpha^{n-1}-F_{n}^{(k)})+(F_{n_{1}}^{(k)}-f_{k}(\\alpha)\\alpha^{n_{1}-1})+(f_{k}(\\alpha)\\alpha^{n_{1}-1}-3^{m_{1}})\\right|\\\\\n&<&\\dfrac{1}{2}+\\dfrac{1}{2}+\\alpha^{n_{1}-1}+3^{m_{1}}\\\\\n&<& \\alpha^{n_{1}}+3^{m_{1}}\\\\\n&<& 2\\max\\{\\alpha^{n_{1}}, 3^{m_{1}}\\}.\n\\end{eqnarray*}\nIn the above, we have also used the fact that $ |f_{k}(\\alpha)| < 1 $ (see Lemma \\ref{fala5}). Dividing through by $ 3^{m} $, we get\n\\begin{eqnarray}\\label{fala3}\n&&\\left|f_{k}(\\alpha)\\alpha^{n-1}3^{-m} -1\\right|< 2\\max\\left\\{\\dfrac{\\alpha^{n_{1}}}{3^{m}}, 3^{m_{1}-m}\\right\\} < \\max\\{\\alpha^{n_{1}-n+6}, 3^{m_{1}-m+1}\\},\n\\end{eqnarray}\nwhere for the right--most inequality in \\eqref{fala3} we used \\eqref{fala4} and the fact that $ \\alpha^{2}> 2 $.\n\nFor the left-hand side of \\eqref{fala3} above, we apply Theorem \\ref{Matveev11} with the data\n$$\nt:=3, \\quad \\gamma_{1}:=f_{k}(\\alpha), \\quad \\gamma_{2}: = \\alpha,\\quad \\gamma_{3}:=3, \\quad b_{1}:=1, \\quad b_{2}:=n-1, \\quad b_{3}:=-m .\n$$\nWe begin by noticing that the three numbers $ \\gamma_{1}, \\gamma_{2}, \\gamma_{3} $ are positive real numbers and belong to the field $ \\mathbb{K}: = \\mathbb{Q}(\\alpha)$, so we can take $ D:= [\\mathbb{K}:\\mathbb{Q}]= k$. Put\n$$\\Lambda :=f_{k}(\\alpha)\\alpha^{n-1}3^{-m} -1. $$\nTo see why $ \\Lambda \\neq 0 $, note that otherwise, we would then have that $ f_{k}(\\alpha) = 3^{m}\\alpha^{-(n-1)} $ and so $ f_{k}(\\alpha) $ would be an algebraic integer, which contradicts Lemma \\ref{fala5} (i).\n\nSince $ h(\\gamma_{2})= (\\log \\alpha)\/k <(\\log 2)\/k $ and $ h(\\gamma_{3})= \\log 3$, it follows that we can take $ A_{2}:= \\log 2$ and $ A_{3}:= k\\log 3 $. Further, in view of Lemma \\ref{fala5} (ii), we have that $ h(\\gamma_{1})<3\\log k$, so we can take $ A_{1}:=3k\\log k $. Finally, since $ \\max\\{1, n-1, m\\} = n-1$, we take $ B:=n $.\n\nThen, the left--hand side of \\eqref{fala3} is bounded below, by Theorem \\ref{Matveev11}, as\n$$\\log |\\Lambda| >-1.4\\times 30^6 \\times 3^{4.5} \\times k^4 (1+\\log k)(1+\\log n)(3\\log k)(\\log 2)(\\log 3).$$\nComparing with \\eqref{fala3}, we get\n$$\\min\\{(n-n_{1}-6)\\log\\alpha , (m-m_{1}-1)\\log 3\\} < 6.54\\times 10^{11} k^4 \\log^{2}k(1+\\log n),$$\nwhich gives\n$$\\min\\{(n-n_{1})\\log\\alpha , (m-m_{1})\\log 3\\} < 6.60\\times 10^{11} k^4 \\log^{2}k(1+\\log n).$$\nNow the argument is split into two cases.\\\\\n\n\\textbf{Case 1.} $\\min \\lbrace (n-n_1) \\log \\alpha , (m-m_1) \\log 2 \\rbrace  = (n - n_{1}) \\log \\alpha$.\n\n\\medskip\n\nIn this case, we rewrite \\eqref{fala1} as\n\\begin{eqnarray*}\n\\left| f_{k}(\\alpha)\\alpha^{n-1} - f_{k}(\\alpha)\\alpha^{n_{1}-1} - 3^{m}\\right| &=& \\left|(f_{k}(\\alpha)\\alpha^{n-1}- F_{n}^{(k)}) + (F_{n_{1}}^{(k)} - f_{k}(\\alpha)\\alpha^{n_{1}-1}) - 3^{m_{1}}\\right|\\\\\n&<&\\dfrac{1}{2}+\\dfrac{1}{2}+3^{m_{1}} \\leq 3^{m_{1}+1}.\n\\end{eqnarray*}\nDividing through by $ 3^{m} $ gives\n\\begin{eqnarray}\\label{fala6}\n\\left| f_{k}(\\alpha)(\\alpha^{n-n_{1}}-1)\\alpha^{n_{1}-1}3^{-m} - 1\\right|&<&3^{m_{1}-m+1}.\n\\end{eqnarray}\nNow we put\n$$\n\\Lambda_{1} := f_{k}(\\alpha)(\\alpha^{n-n_{1}}-1)\\alpha^{n_{1}-1}3^{-m} - 1.\n$$\nWe apply again Theorem \\ref{Matveev11} with the following data\n$$\nt:=3,\\quad \\gamma_{1} :=f_{k}(\\alpha)(\\alpha^{n-n_{1}}-1), \\quad \\gamma_{2}:=\\alpha, \\quad \\gamma_{3}:=3,  \\quad b_{1}:=1, \\quad b_{2}:=n_{1}-1, \\quad b_{3}:=-m.\n$$\nAs before, we begin by noticing that the three numbers $ \\gamma_{1}, \\gamma_{2}, \\gamma_{3} $ belong to the field $ \\mathbb{K} := \\mathbb{Q}(\\alpha) $, so we can take $ D:= [\\mathbb{K}: \\mathbb{Q}] = k$. To see why $ \\Lambda_{1} \\neq 0$, note that otherwise, we would get the relation $ f_{k}(\\alpha)(\\alpha^{n-n_{1}}-1) = 3^{m}\\alpha^{1-n_{1}} $. Conjugating this last equation with any automorphism $ \\sigma$ of the Galois group of $ \\Psi_{k}(x) $ over $ \\mathbb{Q} $ such that $ \\sigma(\\alpha) = \\alpha^{(i)} $ for some $ i\\geq 2 $, and then taking absolute values, we arrive at the equality $ |f_{k}(\\alpha^{(i)})((\\alpha^{(i)})^{n-n_1}-1)| = |3^{m}(\\alpha^{(i)})^{1-n_1}| $. But this cannot hold because, $ |f_{k}(\\alpha^{(i)})||(\\alpha^{(i)})^{n-n_1}-1|<2 $ since $ |f_{k}(\\alpha^{(i)})|<1 $ by Lemma \\ref{fala5} (i), and $ |(\\alpha^{(i)})^{n-n_1}|<1 $, since $ n>n_1$, while $ |3^{m}(\\alpha^{(i)})^{1-n_1}|\\geq 3$.\n\nSince\n$$\nh(\\gamma_{1})\\leq h(f_{k}(\\alpha)) +h(\\alpha^{n-n_{1}}-1) \n< 3\\log k +(n-n_{1})\\dfrac{\\log\\alpha}{k}+\\log 2,\n$$\nit follows that\n$$\nkh(\\gamma_{1}) < 6k\\log k + (n - n_1)\\log\\alpha < 6k\\log k + 6.60 \\times 10^{11} k^4 \\log^{2}k(1+\\log n).\n$$\nSo, we can take $ A_{1}:= 6.80\\times 10^{11} k^4 \\log^{2}k(1+\\log n) $. Further, as before, we take $ A_{2} :=\\log 2 $ and $ A_{3}: = k\\log3 $. Finally, by recalling that $ m<n $, we can take $ B:=n $.\n\n\nWe then get that\n$$\\log|\\Lambda_{1}|>-1.4\\times 30^6 \\times 3^{4.5}\\times k^{3}(1+\\log k)(1+\\log n)(6.80\\times 10^{11} k^4 \\log^{2}k(1+\\log n))(\\log 2)(\\log 3),$$\nwhich yields\n$$ \\log |\\Lambda_{1}|>-7.41 \\times 10^{22} k^7\\log^3 k(1+\\log n)^2.$$\nComparing this with \\eqref{fala6}, we get that\n$$(m-m_{1})\\log 3 < 7.50\\times 10^{22} k^7\\log^3 k(1+\\log n)^{2}.$$\n\n\\medskip\n\n\\textbf{Case 2.} $\\min \\lbrace (n-n_1) \\log \\alpha , (m-m_1) \\log 3 \\rbrace  = (m - m_{1} ) \\log 3$.\n\n\\medskip\n\nIn this case, we write \\eqref{fala1} as\n\\begin{eqnarray*}\n\\left|f_{k}(\\alpha)\\alpha^{n-1} - 3^{m} +3^{m_{1}}\\right| &=& \\left|(f_{k}(\\alpha)\\alpha^{n-1} -F_{n}^{(k)}) + (F_{n_{1}}^{(k)} - f_{k}(\\alpha)\\alpha^{n_{1}-1})+ f_{k}(\\alpha)\\alpha^{n_{1}-1} \\right| \\\\\n&<&\\dfrac{1}{2}+\\dfrac{1}{2}+\\alpha^{n_{1}-1} ~~<~~\\alpha^{n_{1}},\n\\end{eqnarray*}\nso that\n\\begin{eqnarray}\\label{fala7}\n&&\\left|f_{k}(\\alpha)(3^{m-m_{1}}-1)^{-1}\\alpha^{n-1}3^{-m_{1}} - 1\\right|<\\dfrac{\\alpha^{n_{1}}}{3^{m}-3^{m_{1}}}\\leq \\dfrac{2\\alpha^{n_{1}}}{3^{m}}<\\alpha^{n_{1}-n+6}.\n\\end{eqnarray}\nThe above inequality \\eqref{fala7} suggests once again studying a lower bound for the absolute value of\n$$\n\\Lambda_{2} := f_{k}(\\alpha)(3^{m-m_{1}}-1)^{-1}\\alpha^{n-1}3^{-m_{1}} - 1.\n$$\nWe again apply Matveev's theorem with the following data\n$$\nt: =3,\\quad \\gamma_{1}: =f_{k}(\\alpha)(3^{m-m_{1}}-1)^{-1},\\quad \\gamma_{2}: = \\alpha, \\quad \\gamma_{3}: = 3, \\quad b_{1}:=1,\\quad b_{2}:=n-1, \\quad b_{3}:=-m_{1}.\n$$\nWe can again take $ B:=n $ and  $ \\mathbb{K} := \\mathbb{Q}(\\alpha) $, so that $ D:=k $. We also note that, if $ \\Lambda_{2} =0 $, then $ f_{k}(\\alpha) = \\alpha^{-(n-n_{1})} 3^{m_{1}} (3^{m-m_{1}}-1) $ implying that $ f_{k}(\\alpha) $ is an algebraic integer, which is not the case. Thus, $ \\Lambda_{2} \\neq 0 $.\n\nNow, we note that\n$$\nh(\\gamma_{1})\\leq  h(f_{k}(\\alpha))+h(3^{m-m_{1}}-1)\n<3\\log k +(m-m_{1}+k)\\dfrac{\\log 3}{k}.\n$$\nThus, $ kh(\\gamma_{1})< 4k\\log k + (m-m_{1})\\log 3  < 6.80 \\times 10^{11} k^4\\log^2k(1+\\log n)$, and so we can take $ A_{1} := 6.80 \\times 10^{11} k^4\\log^2k(1+\\log n) $. As before, we take $ A_{2}: = \\log 2 $ and $ A_{3} := k\\log  3 $.\nIt then follows from Matveev's theorem, after some calculations, that\n$$\n\\log |\\Lambda_{2}| > -7.41\\times 10^{22}k^7\\log^3 k(1+\\log n)^2.\n$$\nFrom this and \\eqref{fala7}, we obtain that\n$$(n-n_{1})\\log\\alpha < 7.50\\times 10^{22}k^7\\log^3 k(1+\\log n)^2.$$\nThus, in both Case $ 1 $ and Case $ 2 $, we have\n\\begin{eqnarray}\\label{fala8}\n\\min\\{(n-n_{1})\\log\\alpha , (m-m_{1})\\log 2\\} & < & 6.6\\times 10^{11} k^4 \\log^{2}k(1+\\log n),\\\\\n\\max\\{(n-n_{1})\\log\\alpha , (m-m_{1})\\log 2\\} & < & 7.5\\times 10^{22}k^7\\log^3 k(1+\\log n)^2.\\nonumber\n\\end{eqnarray}\nWe now finally rewrite equation \\eqref{fala1} as\n$$\n\\left|f_{k}(\\alpha)\\alpha^{n-1} -f_{k}(\\alpha)\\alpha^{n_{1}-1}-3^{m}+3^{m_{1}}\\right| = \\left|(f_{k}(\\alpha)\\alpha^{n-1} - F_{n}^{(k)})+(F_{n_{1}}^{(k)} - f_{k}(\\alpha)\\alpha^{n_{1}-1})\\right| < 1.\n$$\nWe divide through both sides by $ 3^{m}-3^{m_{1}} $ getting\n\\begin{eqnarray}\\label{fala9}\n&& \\left|\\dfrac{f_{k}(\\alpha)(\\alpha^{n-n_{1}}-1)}{3^{m-m_{1}}-1}\\alpha^{n_{1}-1}3^{-m_{1}} - 1\\right|<\\dfrac{1}{3^{m}-3^{m_{1}}} \\leq \\dfrac{2}{3^{m}} <3^{5 - 0.8n},\n\\end{eqnarray}\nsince $n < 1.6m+4$. To find a lower--bound on the left--hand side of \\eqref{fala9} above, we again apply Theorem \\ref{Matveev11} with the data\n$$\nt:=3,\\quad \\gamma_{1}: =\\dfrac{f_{k}(\\alpha)(\\alpha^{n-n_{1}}-1)}{3^{m-m_{1}}-1},\\quad \\gamma_{2} := \\alpha, \\quad \\gamma_{3} := 3, \\quad b_{1}:=1,\\quad b_{2}:=n_{1}-1, \\quad b_{3}:=-m_{1}.\n$$\nWe also take $ B:=n $ and we take  $ \\mathbb{K} := \\mathbb{Q}(\\alpha) $ with $ D := k $. From the properties of the logarithmic height function, we have that\n\\begin{eqnarray*}\nkh(\\gamma_{1})&\\leq& k\\left(h(f_{k}(\\alpha))+h(\\alpha^{n-n_1}-1)+h(3^{m-m_{1}}-1)\\right)\\\\\n&<&3k\\log k +(n-n_{1})\\log\\alpha +k(m-m_{1})\\log3 + 2k\\log2\\\\\n&<&8.3\\times 10^{22}k^8\\log^3 k(1+\\log n)^2,\n\\end{eqnarray*}\nwhere in the above chain of inequalities we used the bounds \\eqref{fala8}. So we can take $ A_{1} :=8.3\\times 10^{22}k^8\\log^3 k(1+\\log n)^2  $, and certainly as before we take $ A_{2} := \\log 2 $ and $ A_{3}: = k\\log  3 $. We need to show that if we put\n$$\n\\Lambda_{3}:=\\dfrac{f_{k}(\\alpha)(\\alpha^{n-n_{1}}-1)}{3^{m-m_{1}}-1}\\alpha^{n_{1}-1}3^{-m_{1}} - 1,\n$$\nthen $ \\Lambda_{3} \\neq 0 $. To see why $ \\Lambda_{3} \\neq 0$, note that otherwise, we would get the relation \n$$ \nf_{k}(\\alpha)(\\alpha^{n-n_{1}}-1) = 3^{m_{1}}\\alpha^{1-n_{1}}(3^{m-m_{1}}-1).\n$$ \nAgain, as for the case of $ \\Lambda_{1} $, conjugating the above relation with an automorphism $ \\sigma $ of the Galois group of $ \\Psi_{k}(x) $ over $ \\mathbb{Q} $ such that $ \\sigma(\\alpha) = \\alpha^{(i)} $ for some $ i\\geq 2 $, and then taking absolute values, we get that $ |f_{k}(\\alpha^{(i)})((\\alpha^{(i)})^{n-n_1}-1)| = |3^{m_1}(\\alpha^{(i)})^{1-n_1}(3^{m-m_{1}}-1)| $. This cannot hold true because in the left--hand side we have $ |f_{k}(\\alpha^{(i)})||(\\alpha^{(i)})^{n-n_1}-1|<2 $, while in the right--hand side we have $ |3^{m_{1}}||(\\alpha^{(i)})^{1-n_1}||3^{m-m_1}-1|\\geq 4 $. Thus,\n$ \\Lambda_{3} \\neq 0 $. Then Theorem \\ref{Matveev11} gives\n$$\\log |\\Lambda_{3}|>-1.4\\times 30^{6}\\times 3^{4.5}k^{11}(1+\\log k)(1+\\log n)\\left(8.3\\times 10^{22}\\log^3 k(1+\\log n)^2\\right)(\\log 2)(\\log 3),$$\nwhich together with \\eqref{fala9} gives\n$$(0.8n - 5)\\log 3 < 9.05\\times 10^{33} k^{11}\\log^{4}k(1+\\log n)^{3}.$$\nThe above inequality leads to\n\\begin{eqnarray*}\nn < 6.2\\times 10^{34} k^{11}\\log^{4}k\\log^{3}n,\n\\end{eqnarray*}\nwhich can be equivalently written as\n\\begin{eqnarray}\\label{fala10}\n\\dfrac{n}{(\\log n)^{3}} & < & 6.2\\times 10^{34} k^{11}\\log^{4}k.\n\\end{eqnarray}\nWe apply Lemma \\ref{gl} with the data $ m=3, ~~ x=n, ~~ T=6.2\\times 10^{34} k^{11}\\log^{4}k  $. Inequality \\eqref{fala10} yields\n\\begin{eqnarray}\nn & < & 8\\times(6.2\\times 10^{34} k^{11}\\log^{4}k) \\log (6.2\\times 10^{34} k^{11}\\log^{4}k)^{3}\\nonumber\\\\\n&<&4\\times 10^{42}k^{11}(\\log k)^{7}.\n\\end{eqnarray}\nWe then record what we have proved so far as a lemma.\n\\begin{lemma}\\label{lemmaBD}\nIf $ (n,m,n_{1},m_{1}, k) $ is a solution in positive integers to equation \\eqref{Problem}  with $ (n,m)\\neq (n_{1},m_{1}) $, $ n> \\min\\{k+2, n_1+1\\}$, $n_{1}\\geq 2 $, $ m>m_{1}\\geq 1 $ and $ k\\geq 4 $, we then have that $ n < 4\\times 10^{42}k^{11}(\\log k)^{7}$.\n\\end{lemma}\n\n\\section{Reduction of the bounds on $ n $}\n\n\\subsection{The cutoff $k$}\n\nWe have from the above lemma that Baker's method gives\n$$\nn < 4\\times 10^{42}k^{11}(\\log k)^7.\n$$\nBy imposing that the above amount is at most $ 2^{k\/2} $, we get\n\\begin{eqnarray*}\n4\\times 10^{42}k^{11}(\\log k)^7 &<& 2^{k\/2}. \n\\end{eqnarray*}\nThe inequality above holds for $k>600$. \n\n\nWe now reduce the bounds and to do so we make use of Lemma \\ref{Dujjella} several times.\n\n\\subsection{The Case of small $ k $}\nWe now treat the cases when $ k\\in [4, 600] $. First, we consider equation \\eqref{fala1} which is equivalent to \\eqref{Problem}. For $ k\\in[4, 600] $ and $ n\\in[3, 600] $, consider the sets\n\\begin{eqnarray*}\nF_{n,k}:=\\left\\{F_{n}^{(k)}-F_{n_1}^{(k)} (\\text{mod} ~10^{20}):~n\\in [3, 600], ~ n_1 \\in [2, n-1]\\right\\}\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\nD_{n,k}:=\\left\\{3^{m}-3^{m_1}(\\text{mod}~ 10^{20}): ~m\\in [2, 600], ~m_1 \\in [1, m-1]\\right\\}.\n\\end{eqnarray*}\nWith the help of \\textit{Mathematica}, we intersected these two sets and found the only solutions listed in Theorem \\ref{Main}.\n\n\n\n\n\n Next, we note that for these values of $ k $, Lemma \\ref{lemmaBD} gives us absolute upper bounds for $ n $. However, these upper bounds are so large that we wish to reduce them to a range where the solutions can be easily identified by a computer. To do this, we return to \\eqref{fala3} and put\n\\begin{eqnarray}\n\\Gamma:=(n-1)\\log\\alpha-m\\log 3+\\log(f_k(\\alpha)).\n\\end{eqnarray}\nFor technical reasons we assume that $ \\min\\{n-n_1, m-m_1\\} \\ge 20 $. In the case that this condition fails, we consider one of the following inequalities instead:\n\\begin{itemize}\n\\item [(i)] if $ n-n_1<20 $ but $ m-m_1\\ge 20 $, we consider \\eqref{fala6};\n\\item[(ii)] if $ n-n_1\\ge 20 $ but $ m-m_1< 20 $, we consider \\eqref{fala7};\n\\item[(iii)] if $ n-n_1<20 $ but $ m-m_1< 20 $, we consider \\eqref{fala9}.\n\\end{itemize}\nWe start by considering \\eqref{fala3}. Note that $ \\Gamma\\neq 0 $; thus we distinguish the following two cases. If $ \\Gamma>0 $, then $ e^{\\Gamma}-1>0 $, then from \\eqref{fala3} we get\n\\begin{eqnarray*}\n0<\\Gamma<e^{\\Gamma}-1< \\max\\left\\{\\alpha^{n_1-n+6}, 3^{m_1-m+1}\\right\\}.\n\\end{eqnarray*}\nNext we suppose that $ \\Gamma<0 $. Since $ \\Lambda = |e^{\\Gamma}-1|<\\frac{1}{2} $, we get that $ e^{|\\Gamma|}<2 $. Therefore,\n\\begin{eqnarray*}\n0<|\\Gamma|\\le e^{|\\Gamma|}-1 = e^{|\\Gamma|}|e^{\\Gamma}-1|<2\\max\\left\\{\\alpha^{n_1-n+6}, 3^{m_1-m+1}\\right\\}.\n\\end{eqnarray*}\nTherefeore, in any case, the following inequality holds\n\\begin{eqnarray}\\label{BD111}\n0<|\\Gamma|<2\\max\\left\\{\\alpha^{n_1-n+6}, 3^{m_1-m+1}\\right\\}.\n\\end{eqnarray}\nBy replacing $ \\Gamma $ in the above inequality by its formula and dividing through by $ \\log 3 $, we then conclude that\n\\begin{eqnarray*}\n0<\\left|(n-1)\\left(\\dfrac{\\log\\alpha}{\\log 3}\\right)-m+\\dfrac{\\log(f_k(\\alpha))}{\\log 3}\\right|<\\max\\left\\{(2\\alpha^{6})\\cdot\\alpha^{-(n-n_1)}, \\dfrac{6}{\\log 3}\\cdot 3^{-(m-m_1)}\\right\\}\n\\end{eqnarray*}\nThen, we apply Lemma \\ref{Dujjella} with the following data\n\\begin{eqnarray*}\nk\\in[4, 600], ~~~~ \\tau_{k}:=\\dfrac{\\log\\alpha}{\\log 3}, ~~~~ \\mu_{k}:=\\dfrac{\\log(f_k(\\alpha))}{\\log 3}, ~~~~(A_{k}, B_{k}):= (2\\alpha^{6}, \\alpha) \\text{  or  } \\left(\\dfrac{6}{\\log 3},3\\right).\n\\end{eqnarray*}\nNext, we put $ M_{k}:=\\lfloor 4\\times 10^{42}k^{11}(\\log k)^{7} \\rfloor $, which is the absolute upper bound on $ n $ by Lemma \\ref{lemmaBD}.  An intensive computer search in \\textit{Mathematica} revealed that the maximum value of $ \\lfloor \\log (2\\alpha^6q\/\\varepsilon)\/\\log\\alpha \\rfloor $ is $ < 600$ and the maximum value of $ \\lfloor \\log ((6\/\\log3 )q\/\\varepsilon)\/\\log 3 \\rfloor $ is $ <375 $. Thus, either\n\\begin{eqnarray*}\nn-n_1 < \\dfrac{\\log(2\\alpha^6q\/\\varepsilon)}{\\log\\alpha}< 600, ~~~\\text{  or  } m-m_1 < \\dfrac{\\log((6\/\\log 3)q\/\\varepsilon)}{\\log 3}< 375.\n\\end{eqnarray*}\nTherefore, we have that either $ n-n_1 \\leq 600 $ or $ m-m_1 \\le  375 $.\n\n\nNow, let us assume that $ n-n_1\\leq 600 $. In this case, we consider the inequality \\eqref{fala6} and assume that $ m-m_1 \\geq 20 $. Then we put\n\\begin{eqnarray*}\n\\Gamma_{1}:=(n_1-1)\\log\\alpha -m\\log 3+\\log ((f_k(\\alpha)(\\alpha^{n-n_1}-1)).\n\\end{eqnarray*}\nBy similar arguments as in the previous step for proving \\eqref{BD111}, from \\eqref{fala6} we get\n\\begin{eqnarray*}\n0<|\\Gamma_1|<\\dfrac{6}{3^{m-m_1}},\n\\end{eqnarray*}\nand replacing $ \\Gamma_1 $ with its formula and dividing through by $ \\log 3 $ gives\n\\begin{eqnarray}\\label{BD112}\n0<\\left|(n_1-1)\\left(\\dfrac{\\log\\alpha}{\\log 3}\\right)-m+\\dfrac{\\log(f_k(\\alpha)(\\alpha^{n-n_1}-1))}{\\log 3}\\right|<\\dfrac{6}{\\log 3}\\cdot3^{-(m-m_1)}.\n\\end{eqnarray}\nAs before, we keep the same $ \\tau_k, ~~M_k,~~ (A_k, B_k):=((6\/\\log 3),3) $ and put\n\\begin{eqnarray*}\n\\mu_{k,l}:=\\dfrac{\\log(f_k(\\alpha)(\\alpha^{l}-1))}{\\log 3}, ~~~~k\\in[4, 600], ~~~~l:=n-n_1 \\in[1, 600].\n\\end{eqnarray*}\nWe apply Lemma \\ref{Dujjella} to the inequality \\eqref{BD112} with the above data. A computer search in \\textit{Mathematica} revealed that the maximum value of $ \\lfloor \\log(Aq\/\\varepsilon)\/\\log B \\rfloor $ over the values of $ k\\in [4, 600] $ and $ l\\in [1, 600] $ is $ < 377 $. Hence, $ m-m_1 \\le 377 $.\n\n\nNext, we assume that $ m-m_1 \\le 375 $. Here, we consider the inequality \\eqref{fala7} and also assume that $ n-n_1 \\geq 20$. We put\n\\begin{eqnarray*}\n\\Gamma_{2}:=(n-1)\\log\\alpha-m_1\\log 3+\\log\\left(f_k(\\alpha)\/(3^{m-m_1}-1)\\right).\n\\end{eqnarray*}\nThus, by the same arguments as before, we get\n\\begin{eqnarray*}\n0<|\\Gamma_{2}|<\\dfrac{2\\alpha^{6}}{\\alpha^{n-n_1}}.\n\\end{eqnarray*}\nBy substituting for $ \\Gamma_{2} $ with its formula and dividing through by $ \\log 3 $ in the above inequality, we get\n\\begin{eqnarray*}\n0<\\left|(n-1)\\left(\\dfrac{\\log\\alpha}{\\log 3}\\right)-m_1+\\dfrac{\\log\\left(f_k(\\alpha)\/(3^{m-m_1}-1)\\right)}{\\log 3}\\right|<\\dfrac{2\\alpha^6}{\\log 3}\\cdot\\alpha^{-(n-n_1)}.\n\\end{eqnarray*}\nAs before, we apply Lemma \\ref{Dujjella} with the same $ \\tau_k,~~M_k, ~~(A_k, B_k):=(2\\alpha^{6}\/\\log 3, \\alpha) $ and put\n\\begin{eqnarray*}\n\\mu_{k,j}:=\\dfrac{\\log\\left(f_k(\\alpha)\/(3^{m-m_1}-1)\\right)}{\\log 3}, ~~~~ k\\in[4, 600], ~~~j:=m-m_1\\in [1, 375].\n\\end{eqnarray*}\nA computer search with \\textit{Mathematica} revealed that the maximum value of $ \\lfloor \\log(Aq\/\\varepsilon)\/\\log B \\rfloor$, for $ k\\in[4, 600] $ and $ j\\in [1, 375] $ is $ < 603 $. Hence, $ n-n_1 \\leq 603 $.\n\nTo conclude the above computations, first we got that either $ n-n_1\\leq 600 $ or $ m-m_1\\leq 375 $. If $ n-n_1 \\leq 600 $, then $ m-m_1 \\leq  377 $, and if $ m-m_1 \\leq 375 $, then $ n-n_1 \\leq 603 $. Therefore, we can conclude that we always have\n\\begin{eqnarray*}\nn-n_1\\leq 603 ~~~\\text{  and  } m-m_1 \\leq 377.\n\\end{eqnarray*}\n\nFinally, we go to \\eqref{fala9} and put\n\\begin{eqnarray*}\n\\Gamma_{3}:=(n_1-1)\\log\\alpha - m_1\\log 3 + \\log\\left(\\dfrac{f_k(\\alpha)(\\alpha^{n-n_1}-1)}{3^{m-m_1}-1}\\right).\n\\end{eqnarray*}\nSince $ n>600 $, from \\eqref{fala9} we can conclude that\n\\begin{eqnarray*}\n0<|\\Gamma_3|<\\dfrac{2\\cdot 3^{5}}{3^{0.8n}}.\n\\end{eqnarray*}\nHence, by substituting for $ \\Gamma_{3} $ by its formula and dividing through by $ \\log 3 $, we get\n\\begin{eqnarray*}\n0<\\left|(n_1-1)\\left(\\dfrac{\\log\\alpha}{\\log 3}\\right) - m_1 + \\dfrac{\\log\\left(f_k(\\alpha)(\\alpha^{n-n_1}-1)\/(3^{m-m_1}-1)\\right)}{\\log 3}\\right|< 1328\\cdot 3^{-0.8n}.\n\\end{eqnarray*}\nWe apply Lemma \\ref{Dujjella} with the same $ \\tau_k, ~~M_k, ~~(A_k, B_k):=(1328, 3), ~~k\\in [4, 600], $\nand put\n\\begin{eqnarray*}\n\\mu_{k, l, j}:=\\dfrac{\\log\\left(f_k(\\alpha)(\\alpha^l-1)\/(3^{j}-1)\\right)}{\\log 3},  ~~l:=n-n_1\\in[1, 603], ~~j:=m-m_1\\in [1, 377].\n\\end{eqnarray*}\nA computer search in \\textit{Mathematica} revealed that the maximum value of $ \\lfloor \\log(1328q\/\\varepsilon)\/\\log 3\\rfloor $, for $ k\\in[4,600], ~~l\\in[1, 603]  $ and $ j\\in [1, 377] $ is $ < 378 $. Hence, $ n< 473 $, which contradicts the assumption that $ n> 500 $ in the previous section.\n\n\n\\subsection{The case of large $k$}\nWe now assume that $ k>600 $. Note that for these values of $ k $ we have\n\\begin{eqnarray*}\nn<4\\times 10^{42}k^{11}(\\log k)^{7}.\n\\end{eqnarray*}\nSince, $ n\\ge k+2 $, we have that $ n\\ge 602 $.\nThe following lemma is useful.\n\\begin{lemma}\\label{Kala224}\nFor $1\\le n<2^{k\/2}$ and $k\\ge 10$, we have \n$$\nF_n^{(k)}=2^{n-2}\\left(1+\\zeta\\right)\\quad {\\text{where}}\\quad |\\zeta|<\\frac{5}{2^{k\/2}}.\n$$\n\\end{lemma}\n\n\\begin{proof}\nWhen $n\\le k+1$, we have $F_{n}^{(k)}=2^{n-2}$ so we can take $\\zeta:=0$. So, assume $k+2\\le n<2^{k\/2}$. It follows from (1.8) in \\cite{BGL} that\n$$\n|f_k(\\alpha)\\alpha^{n-1}-2^{n-2}|<\\frac{2^n}{2^{k\/2}}.\n$$\nBy \\eqref{approxgap},  we also have $\\left|F_{n}^{(k)}-f_k(\\alpha)\\alpha^{n-1}\\right|<1\/2$. Thus,\n\\begin{eqnarray*}\n|F_n^{(k)}-2^{n-2}| & \\le & |f_k(\\alpha)\\alpha^{n-1}-2^{n-2}|+|F_n^{(k)}-f_k(\\alpha)\\alpha^{n-1}|\\\\\n& < & \\frac{2^n}{2^{k\/2}}+\\frac{1}{2}=\\frac{2^n}{2^{k\/2}}\\left(1+\\frac{1}{2^{n-k\/2+1}}\\right)\\le \n\\frac{2^n}{2^{k\/2}}\\left(1+\\frac{1}{2^{k\/2+3}}\\right)\\\\\n& < & \\frac{2^n\\cdot 1.25}{2^{k\/2}}=\\left(\\frac{5}{2^{k\/2}}\\right) 2^{n-2}.\n\\end{eqnarray*}\n\\end{proof}\nBy the above lemma, we can rewrite \\eqref{fala1} as\n$$\n2^{n-2}(1+\\zeta)-2^{n_1-2}(1+\\zeta_1)=3^m-3^{m_1},\\qquad \\max\\{|\\zeta|,|\\zeta_1|\\}<\\frac{5}{2^{k\/2}}.\n$$\nSo,\n\\begin{eqnarray}\n\\label{Kalai1}\n|2^{n-2}-3^m| & = & |-\\zeta\\cdot 2^{n-2}+2^{n_1-2}(1+\\zeta_1)-3^{m_1}|\\nonumber \\\\\n& \\le & 2^{n-2}\\left(\\frac{5}{2^{k\/2}}\\right)+2^{n_1-2}\\left(1+\\frac{5}{2^{k\/2}}\\right)+3^{m_1}.\n\\end{eqnarray}\nNext, we have \n$$\n2^{n-2} > F_n^{(k)}-F_{n_1}^{(k)}=3^m-3^{m_1}\\ge 2\\cdot 3^{m-1},\\quad {\\text{\\rm so}} \\quad 2^{n-2}\/3^m>2\/3.\n$$\nFurther,\n\\begin{eqnarray*}\n3^m>3^m-3^{m_1} & = & F_n^{(k)}-F_{n_1}^{(k)} \\ge  F_{n}^{(k)}-F_{n-1}^{(k)}\\\\\n& \\ge & F_{n-2}^{(k)}>2^{n-4}\\left(1-\\frac{5}{2^{k\/2}}\\right)\\\\\n& > & 2^{n-4}\\left(\\frac{27}{32}\\right)\\quad (k>10),\n\\end{eqnarray*}\nso \n\\begin{equation}\n\\frac{128}{27}>\\frac{2^{n-2}}{3^m}>\\frac{2}{3}.\n\\end{equation}\nGoing back to \\eqref{Kalai1}, we have\n$$\n|3^{m}2^{-(n-2)}-1|<\\frac{5}{2^{k\/2}}+\\frac{1.25}{2^{n-n_1}}+\\frac{3^{m_1}}{(2\/3) 3^m}=\\frac{5}{2^{k\/2}}+1.5\\left(\\frac{1}{2^{n-n_1}}+\\frac{1}{3^{m-m_1}}\\right).\n$$\nThus, \n\\begin{eqnarray}\\label{Kalai2}\n|3^{m}2^{-(n-2)}-1|<8\\max\\left\\{\\frac{1}{2^{n-n_1}},\\frac{1}{3^{m-m_1}}, \\frac{1}{2^{k\/2}}\\right\\}.\n\\end{eqnarray}\nWe now apply Theorem \\ref{Matveev11} on the left-hand side of \\eqref{Kalai2} with the data\n\\begin{eqnarray*}\n\\Gamma := 3^{m}2^{-(n-2)}-1,\\quad t:=2, \\quad \\gamma_{1}:=3, \\quad \\gamma_{2}:=2, \\quad b_{1}:=m, \\quad b_{2}:=-(n-2).\n\\end{eqnarray*}\nIt is clear that $ \\Gamma \\neq 0 $, otherwise we would get $ 3^{m}=2^{n-2} $ which is a contradiction since $ 3^{m} $ is odd  while $ 2^{n-2} $ is even. \nWe consider the field $ \\mathbb{K}=\\mathbb{Q} $, in this case $ D=1 $. Since $ h(\\gamma_1)=h(3)=\\log 3 $ and $ h(\\gamma_2)=h(2)=\\log 2 $, we can take $ A_{1}:=\\log 3 $ and $ A_{2}:=\\log 2 $. We also take $ B:=n $. Then, by Theorem \\ref{Matveev11}, the left-hand side of \\eqref{Kalai2} is bounded below as\n\\begin{eqnarray}\n\\log|\\Gamma|>-5.86\\times 10^{8}(1+\\log n).\n\\end{eqnarray}\nBy comparing with \\eqref{Kalai2}, we get\n\\begin{eqnarray*}\n\\min\\{(n-n_1-3)\\log 2, ~~(m-m_1-2)\\log 3, ~~(k\/2-3)\\log 2\\}<5.86\\times 10^{8}(1+\\log n),\n\\end{eqnarray*}\nwhich implies that\n\\begin{eqnarray}\n\\min\\{(n-n_1)\\log2, ~~(m-m_1)\\log 3, ~~(k\/2)\\log 2\\}<5.88\\times 10^{8}(1+\\log n).\n\\end{eqnarray}\nNow the argument is split into four cases.\n\n\\medskip\n\n\\textbf{Case 5.3.1.} $ \\min\\{(n-n_1)\\log2, ~~(m-m_1)\\log 3, ~~(k\/2)\\log 2\\}=(k\/2)\\log2 $.\n\n\\medskip \n\n\\noindent In this case, we have\n\\begin{eqnarray*}\n(k\/2)\\log 2<5.88\\times 10^{8}(1+\\log n),\n\\end{eqnarray*}\nwhich implies that\n\\begin{eqnarray*}\nk<1.70\\times 10^{9}(1+\\log n).\n\\end{eqnarray*}\n\n\\medskip\n\n\\textbf{Case 5.3.2.} $ \\min\\{(n-n_1)\\log2, ~~(m-m_1)\\log 3, ~~(k\/2)\\log 2\\}=(n-n_1)\\log2 $.\n\n\n\\medskip \n\\noindent We rewrite \\eqref{fala1} as\n\\begin{eqnarray*}\n|3^{m}-2^{n_1-2}(2^{n-n_1}-1)|&=&|3^{m_1}+2^{n-2}\\zeta-2^{n_1-2}\\zeta_1|\\\\\n&<&3^{m_1}+2^{n-2}\\left(\\dfrac{10}{2^{k\/2}}\\right),\n\\end{eqnarray*}\nwhich implies that\n\\begin{eqnarray}\\label{Kalai3}\n\\left|3^{m}2^{-n_1}(2^{n-n_1}-1)^{-1}-1\\right|<20\\max\\left\\{\\dfrac{1}{3^{m-m_1}}, \\dfrac{1}{2^{k\/2}}\\right\\}.\n\\end{eqnarray}\nWe now apply Matveev's theorem,  Theorem \\ref{Matveev11} on the left-hand side of \\eqref{Kalai3} to \n$$ \n\\Gamma_{1}=3^{m}2^{-(n_1-2)}(2^{n-n_1}-1)^{-1}-1,\n$$\n\\begin{eqnarray*}\n t:=3, \\quad \\gamma_{1}:=3, \\quad \\gamma_{2}:=2, \\quad \\gamma_{3}:=2^{n-n_1}-1,\\quad b_1:=m, \\quad b_2:=-(n_1-2), \\quad b_3:=-1.\n\\end{eqnarray*}\nNote that $ \\Gamma_{1} \\neq 0 $. Otherwise, $ 3^{m}=2^{n-2}-2^{n_1-2} $, so $n_1=2$, and $2^{n-2}-3^m=1$, so $n\\le 4$ by classical results on Catalan's equation,  which is a contradiction  \nbecause $ n\\geq k+2>602 $.  We use the same values, $ A_1:=\\log 3 $, $ A_2:=\\log 2 $, $ B:=n $ as in the previous step. In order to find $A_3$, note that\n\\begin{eqnarray*}\nh(\\gamma_3)=h(2^{n-n_1}-1)\\leq (n-n_1+1)\\log 2 < 5.90\\times 10^{8}(1+\\log n).\n\\end{eqnarray*}\nSo, we take $ A_3:=5.90\\times 10^{8}(1+\\log n) $. By Theorem \\ref{Matveev11}, we have\n\\begin{eqnarray*}\n\\log|\\Gamma_{1}|>-6.43\\times 10^{19}(1+\\log n)^{2}.\n\\end{eqnarray*}\nBy comparing with \\eqref{Kalai3}, we get\n\\begin{eqnarray*}\n\\min\\{(m-m_1-3)\\log3, ~(k\/2-5)\\log 2\\}<6.43\\times 10^{19}(1+\\log n)^{2},\n\\end{eqnarray*}\nwhich implies that\n\\begin{eqnarray*}\n\\min\\{(m-m_1)\\log 3, (k\/2)\\log 2\\}<6.44\\times 10^{19}(1+\\log n)^{2}.\n\\end{eqnarray*}\nAt this step, we have that either $$ (m-m_1)\\log 3 < 6.44\\times 10^{19}(1+\\log n)^{2} $$ or $$ k<1.86\\times 10^{20}(1+\\log n)^{2}. $$\n\n\\medskip\n\n\\textbf{Case 5.3.3.} $ \\min\\{(n-n_1)\\log2, ~~(m-m_1)\\log 3, ~~(k\/2)\\log 2\\}=(m-m_1)\\log3 $.\n\n\\medskip\n\n\\noindent We rewrite \\eqref{fala1} as\n\\begin{eqnarray*}\n|(3^{m_1}(3^{m-m_{1}}-1)-2^{n-2}|&=&|2^{n-2}\\zeta-2^{n_1-2}(1+\\zeta_1)|\\\\\n&<&2^{n-2}\\left(\\frac{5}{2^{k\/2}}\\right)+2^{n_1-2}\\left(1+\\frac{5}{2^{k\/2}}\\right),\n\\end{eqnarray*}\nwhich implies that\n\\begin{eqnarray}\\label{Kalai4}\n\\left|3^{m_1}(3^{m-m_1}-1)2^{-(n-2)}-1\\right|<20\\max\\left\\{\\dfrac{1}{2^{n-n_1}}, \\dfrac{1}{2^{k\/2}}\\right\\}.\n\\end{eqnarray}\nWe again apply Matveev's theorem,  Theorem \\ref{Matveev11} on the left-hand side of \\eqref{Kalai3} which is\n$$ \n\\Gamma_{2}=3^{m_1}2^{-(n-2)}(3^{m-m_1}-1)-1,\n$$\n\\begin{eqnarray*}\n t:=3, \\quad \\gamma_{1}:=3, \\quad \\gamma_{2}:=2, \\quad \\gamma_{3}:=(3^{m-m_1}-1), \\quad b_1:=m_1, \\quad b_2:=-(n-2), \\quad b_3:=1.\n\\end{eqnarray*}\nNote that $ \\Gamma_2 \\neq 0 $. Otherwise, $ 3^{m}-3^{m_1}=2^{n-2}$, which is impossible since the left--hand side is a multiple of $3$ and the right--hand side isn't.\n We use the same values, $ A_1:=\\log 3 $, $ A_2:=\\log 2 $, $ B:=n $ as in the previous steps. In order to determine $A_3$, note that\n\\begin{eqnarray*}\nh(\\gamma_3)=h(3^{m-m_1}-1)\\leq (m-m_1+1)\\log 3 < 5.90\\times 10^{8}(1+\\log n).\n\\end{eqnarray*}\nSo, we take $ A_3:=5.90\\times 10^{8}(1+\\log n) $. By Theorem \\ref{Matveev11}, we have the lower bound\n\\begin{eqnarray*}\n\\log|\\Gamma_{2}|>-6.43\\times 10^{19}(1+\\log n)^{2}.\n\\end{eqnarray*}\nBy comparing with \\eqref{Kalai4}, we get\n\\begin{eqnarray*}\n\\min\\{(n-n_1-5)\\log3, ~(k\/2-5)\\log 2\\}<6.43\\times 10^{19}(1+\\log n)^{2},\n\\end{eqnarray*}\nwhich implies that\n\\begin{eqnarray*}\n\\min\\{(n-n_1)\\log 3, (k\/2)\\log 2\\}<6.44\\times 10^{19}(1+\\log n)^{2}.\n\\end{eqnarray*}\nAs before, at this step we have that either $$ (n-n_1)\\log 3 < 6.44\\times 10^{19}(1+\\log n)^{2} $$ or $$ k< 1.86\\times 10^{20}(1+\\log n)^{2}.  $$\n\n\\medskip\n\nTherefore, in all the three cases above, we got\n\\begin{eqnarray}\n\\min\\{(n-n_1)\\log2, ~~(m-m_1)\\log 3, ~~(k\/2)\\log 2\\}&<&5.88\\times 10^{8}(1+\\log n)\\nonumber\\\\\n\\max\\{(n-n_1)\\log2, ~~(m-m_1)\\log 3, ~~(k\/2)\\log 2\\} &<&6.44\\times 10^{19}(1+\\log n)^{2}.\n\\end{eqnarray}\n\n\\medskip\n\n\\textbf{Case 5.3.4.} $(k\/2)\\log 2>6.44\\times 10^{19}(1+\\log n)^{2}$.\n\n\\medskip\n\nFrom the previous analysis, we conclude that one of $(n-n_1)\\log 2$ and $(m-m_1)\\log 3$ is bounded by $5.88\\times 10^{8}(1+\\log n)$ and the other one by \n$6.44\\times 10^{19}(1+\\log n)^{2}.$ We rewrite \\eqref{fala1} as\n\\begin{eqnarray*}\n\\left|3^{m_1}(3^{m-m_1}-1)-2^{n_1-2}(2^{n-n_1}-1)\\right|=|\\zeta|\\cdot 2^{n-2}+|\\zeta_1|\\cdot 2^{n_1-2}\\le 2^{n-2} \\left(\\frac{10}{2^{k\/2}}\\right),\n\\end{eqnarray*}\nwhich implies that\n\\begin{eqnarray}\\label{Kalai5}\n\\left|3^{m_{1}}2^{-(n_1-2)}\\left(\\dfrac{3^{m-m_1}-1}{2^{n-n_1}-1}\\right)-1\\right|<\\dfrac{20}{2^{k\/2}}.\n\\end{eqnarray}\nWe apply Matveev's Theorem to \n$$ \n\\Gamma_{3}=3^{m_{1}}2^{-(n_1-2)}\\left(\\dfrac{3^{m-m_1}-1}{2^{n-n_1}-1}\\right)-1,\n$$\nwith the data \n\\begin{eqnarray*}\nt=:3, \\quad \\gamma_{1}:=3, \\quad \\gamma_{2}:=2, \\quad \\gamma_{3}:=\\left(\\dfrac{3^{m-m_1}-1}{2^{n-n_1}-1}\\right), \\quad b_1:=m_1, \\quad b_2:=-(n_1-2),\\quad  b_3:=1.\n\\end{eqnarray*}\nNote that $ \\Gamma_{3}\\neq 0 $, otherwise, we get $ 2^{n}-3^{m}=2^{n_1}-3^{m_1}$ which is impossible by Lemma \\ref{classical1}. \n\n As before we take $ B:=n $, $ A_1:=\\log 3 $, $ A_2:=\\log 2 $. In oder to determine an acceptable value for $ A_3 $, note that\n\\begin{eqnarray*}\nh(\\gamma_{3})&\\leq& h(3^{m-m_1}-1)+h(2^{n-n_1}-1)<(m-m_1+1)\\log 3+(n-n_1+1)\\log 2\\\\\n&<&2\\times 6.46\\times 10^{19}(1+\\log n)^{2}<1.30\\times 10^{20}(1+\\log n)^{2}.\n\\end{eqnarray*}\nThus, we take $ A_3:=1.30\\times 10^{20}(1+\\log n)^{2} $. By Theorem \\ref{Matveev11}, we have\n\\begin{eqnarray*}\n\\log|\\Gamma_{3}|>-1.86\\times 10^{31}(1+\\log n)^{3}.\n\\end{eqnarray*}\nBy comparing with \\eqref{Kalai5}, we get\n\\begin{eqnarray*}\n(k\/2-5)\\log 2 < 1.86\\times 10^{31}(1+\\log n)^{3},\n\\end{eqnarray*}\nwhich implies that\n\\begin{equation}\n\\label{eq:finalk}\nk<5.42\\times 10^{31}(1+\\log n)^{3}.\n\\end{equation}\nThus, inequality \\eqref{eq:finalk} holds in all four cases. Since $ n<4\\times 10^{42}k^{11}(\\log k)^{7} $, then \n\\begin{eqnarray}\nk<5.42\\times 10^{31}\\left(1+\\log\\left(4\\times 10^{42}k^{11}(\\log k)^{7}\\right)\\right)^{3},\n\\end{eqnarray}\nwhich gives the absolute upper bounds $$ k<8.631\\times 10^{40}~<10^{41} $$ and $$ m<n<3.44\\times 10^{506}<10^{507}. $$\n\nWe record what we have proved.\n\n\\begin{lemma}\n\\label{lem:abs}\nWe have\n$$\nk<10^{41}\\qquad {\\text{and}}\\qquad m<10^{507}.\n$$\n\\end{lemma}\n\n\\subsection{The final reduction}\n\n\nThe previous bounds are too large, so we need to reduce them by applying a Baker-Davenport reduction procedure. First, we go to \\eqref{Kalai2} and let\n\\begin{eqnarray*}\nz:=m\\log 3 - (n-2)\\log 2.\n\\end{eqnarray*}\nAssume $m-m_1>1066$, $ n-n_1>1690 $  and $ k>600 $. \nThen, we note that \\eqref{Kalai3} can be rewritten as\n\\begin{eqnarray*}\n\\left|e^{z}-1\\right|<\\max\\{2^{n_1-n+3}, ~~3^{m_1-m+2}, ~~2^{-k\/2+3}\\}.\n\\end{eqnarray*}\nIf $ z>0 $, then $ e^{z}-1 >0$, so we obtain\n\\begin{eqnarray*}\n0<z<e^{z}-1<\\max\\{2^{n_1-n+3}, ~~3^{m_1-m+2}, ~~2^{-k\/2+3}\\}.\n\\end{eqnarray*}\nSuppose now that $ z<0 $. Since $ \\Gamma=|e^{z}-1|<1\/2 $, we get that $ e^{|z|}<2 $. Thus,\n\\begin{eqnarray*}\n0<|z|\\leq e^{|z|}-1=e^{|z|}|e^{z}-1|<2\\max\\{2^{n_1-n+3}, ~~3^{m_1-m+2}, ~~2^{-k\/2+3}\\}.\n\\end{eqnarray*}\nTherefore, in any case we have that the inequality\n\\begin{eqnarray}\n0<|z|<2\\max\\{2^{n_1-n+3}, ~~3^{m_1-m+2}, ~~2^{-k\/2+3}\\}\n\\end{eqnarray}\nalways holds. By replacing $ z $ in the above inequality by its formula and dividing through by $ m\\log 2 $, we get that\n\\begin{eqnarray}\n0<\\left|\\dfrac{\\log 3}{\\log 2}-\\dfrac{n}{m}\\right|<\\max\\left\\{\\dfrac{24}{2^{n-n_1}m},~~\\dfrac{26}{3^{m-m_1}m}, ~~\\dfrac{24}{2^{k\/2}m}\\right\\}.\n\\end{eqnarray}\nThen\n\\begin{eqnarray*}\n\\max\\left\\{\\dfrac{24}{2^{n-n_1}m},~~\\dfrac{26}{3^{m-m_1}m}, ~~\\dfrac{24}{2^{k\/2}m}\\right\\}<\\dfrac{1}{2m^{2}},\n\\end{eqnarray*}\nbecause $ m<10^{507} $.\nBy the Legendre criterion Lemma \\ref{lem:legendre}, it follows that $ n\/m $ is a convergent of $ \\log3\/\\log2 $. So $ n\/m $ is of the form $ p_l\/q_l $ for some $ l=0, 1, 2, \\ldots, 972 $. Then $n\/m=p_l\/q_l$ implies that $m=dq_l$ for some $d\\ge 1$. Thus,\n\\begin{eqnarray*}\n\\dfrac{1}{(a_{l+1}+2)q_{k}q_{l+1}}<\\left|\\dfrac{\\log 3}{\\log 2}-\\dfrac{p_l}{q_l}\\right|<\\max\\left\\{\\dfrac{24}{2^{n-n_1}dq_l},~~\\dfrac{26}{3^{m-m_1}dq_l}, ~~\\dfrac{24}{2^{k\/2}dq_l}\\right\\}.\n\\end{eqnarray*}\nSince $ \\max\\{a_{l+1}: l=0, 1, 2, \\ldots, 972]=3308 $, we get that\n\\begin{eqnarray*}\n\\min\\{2^{n-n_1}, 3^{m-m_1}, 2^{k\/2}\\}\\le 26\\cdot3310q_{973}.\n\\end{eqnarray*}\nWith the help of \\textit{Mathematica}, we have $ q_{973} \\approx 1.6834\\times 10^{507} $. We then conclude that one of the following inequalities holds:\n\\begin{eqnarray*}\nn-n_1<1690, \\qquad m-m_1<1066, \\qquad k<3380.\n\\end{eqnarray*}\n\n\nSuppose first that $ m-m_1>10 $ and $ k\\ge 20 $, we go back to \\eqref{Kalai3} and let\n\\begin{eqnarray}\nz_1:=m\\log 3-(n_1-2)\\log 2-\\log(2^{n-n_1}-1).\n\\end{eqnarray}\nThen we note that \\eqref{Kalai3} can be rewritten as\n\\begin{eqnarray*}\n\\left|e^{z_1}-1\\right|<\\max\\{3^{m_1-m+3}, ~~2^{-k\/2+5}\\}.\n\\end{eqnarray*}\nThis implies that\n\\begin{eqnarray*}\n0<|z_1|<2\\max\\{3^{m_1-m+3}, ~~2^{-k\/2+5}\\}.\n\\end{eqnarray*}\nThis also holds when $ m-m_1<10 $ and $k<20$. By substituting for $ z_1 $ and dividing through by $ \\log 2 $, we get\n\\begin{eqnarray*}\n0<\\left|m\\left(\\dfrac{\\log 3}{\\log 2}\\right)-(n_1-2)+\\dfrac{\\log(1\/(2^{n-n_1}-1))}{\\log 2}\\right|<\\max\\{98\\cdot3^{-(m-m_1)}, ~~94\\cdot2^{-k\/2}\\}.\n\\end{eqnarray*}\nWe put \n\\begin{eqnarray*}\n\\tau: = \\dfrac{\\log 3}{\\log 2}, \\qquad \\mu: = \\dfrac{\\log(1\/(2^{n-n_1}-1))}{\\log 2},\\qquad  (A,B):=(78,3) \\quad \\text{ or } \\quad (94,2),\n\\end{eqnarray*}\nwhere $n-n_1 \\in [1, 1690]$. We take $M:=10^{507}$. A computer search in \\textit{Mathematica} reveals that $ q=q_{977}\\approx 5.708\\times 10^{510}> 6M $ and the minimum positive value of $ \\varepsilon:=||\\mu q||-M||\\tau q||>0.0186 $. Thus, Lemma \\ref{Dujjella} tells us that either $ m-m_1\\leq 1078 $ or $ k\\leq 3418 $.\n\nNext, we suppose that $ n-n_1 > 10 $, $ k>20 $ and  go to \\eqref{Kalai4} and let\n\\begin{eqnarray}\nz_2:=m_1\\log 3-(n-2)\\log 2+\\log(3^{m-m_1}-1).\n\\end{eqnarray}\nThen we also  note that \\eqref{Kalai4} can be rewritten as\n\\begin{eqnarray*}\n\\left|e^{z_2}-1\\right|<\\max\\{2^{n_1-n+5}, ~~2^{-k\/2+5}\\}.\n\\end{eqnarray*}\nThis gives\n\\begin{eqnarray*}\n0<|z_2|<2\\max\\{2^{n_1-n+5}, ~~2^{-k\/2+5}\\}.\n\\end{eqnarray*}\nThis also holds for $ n-n_1< 10 $ and $ k<20 $ as well.\nBy substituting for $ z_2 $ and dividing through by $ \\log 2 $, we get\n\\begin{eqnarray*}\n0<\\left|m_1\\left(\\dfrac{\\log 3}{\\log 2}\\right)-(n-2)+\\dfrac{\\log(3^{m-m_1}-1)}{\\log 2}\\right|<\\max\\{94\\cdot2^{-(n-n_1)}, ~~94\\cdot2^{-k\/2}\\}.\n\\end{eqnarray*}\nWe put \n\\begin{eqnarray*}\n\\tau: = \\dfrac{\\log 3}{\\log 2}, \\qquad\\mu: = \\dfrac{\\log(3^{m-m_1}-1)}{\\log 2},\\qquad (A,B):=(94,2),\n\\end{eqnarray*}\nwhere $m-m_1 \\in [1, 1066]$. We keep the same $ M $ and $ q $ as in the previous step. A computer search in \\textit{Mathematica} reveals that the minimum positive value of $ \\varepsilon:=||\\mu q||-M||\\tau q||>0.0372 $. Thus, Lemma \\ref{Dujjella} tells us that either $ n-n_1\\leq 1708 $ or $ k\\leq 3416 $.\n\nLastly, we assume that $ k>20 $ and go to \\eqref{Kalai5} and let\n\\begin{eqnarray}\nz_3:=m_1\\log 3-(n_1-2)\\log 2-\\log((3^{m-m_1}-1)\/(2^{n-n_1}-1)).\n\\end{eqnarray}\nWe note that \\eqref{Kalai5} can be rewritten as\n\\begin{eqnarray*}\n\\left|e^{z_3}-1\\right|<2^{-k\/2+5}.\n\\end{eqnarray*}\nThis gives\n\\begin{eqnarray*}\n0<|z_3|<2^{-k\/2+6},\n\\end{eqnarray*}\nwhich also holds when $ k<20 $. By substituting for $ z_3 $ and dividing through by $ \\log 2 $, we get\n\\begin{eqnarray*}\n0<\\left|m_1\\left(\\dfrac{\\log 3}{\\log 2}\\right)-(n_1-2)+\\dfrac{\\log((3^{m-m_1}-1)\/(2^{n-n_1}-1))}{\\log 2}\\right|<94\\cdot2^{-k\/2}.\n\\end{eqnarray*}\nWe put \n\\begin{eqnarray*}\n\\tau: = \\dfrac{\\log 3}{\\log 2}, \\qquad \\mu: = \\dfrac{\\log((3^{m-m_1}-1)\/(2^{n-n_1}-1))}{\\log 2},\\qquad (A,B):=(94,2),\n\\end{eqnarray*}\nwhere $n-n_1 \\in [1, 1708]$ and $ m-m_1 \\in [1, 1074] $. We keep the same $ M $ and  $ q $ as before. A computer search in \\textit{Mathematica} reveals that the minimum positive value of $ \\varepsilon:=||\\mu q||-M||\\tau q||>0.00058 $. Thus, Lemma \\ref{Dujjella} tells us that $ k\\leq 3428 $. \n\nTherefore, in all cases we found out that $ k<3428 $ which gives that $ n<7.2741\\times10^{87}<10^{88} $. These bounds are still too large. We repeat the above procedure several times by adjusting the values of $ M $ with respect to the new bounds of $ n $. We summarise the data for the iterations performed in Table \\ref{tab1}\n\\begin{table}[H]\n\\caption{Computation results}\\label{tab1}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n & $M$& $n-n_1 \\le$ & $m-m_1 \\le$ & $k\\le$\\\\\n\\hline\n$ 1 $& $ ~10^{507} $& $ 1708 $& $ 1074 $& $ 3428 $\\\\\n$ 2 $& $ 10^{88} $& $ ~319 $& $ ~197 $& $ ~662 $\\\\\n$ 3 $& $ 10^{80} $& $ ~287 $& $ ~180 $& $ ~590 $\\\\\n$ 4 $& $ 10^{79} $& $ ~282 $& $ ~180 $& $ ~584 $\\\\\n$ 5 $& $ 10^{79} $& $ ~282 $& $ ~180 $& $ ~584 $\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\noindent\nFrom the data displayed in the above table, it is evident that  after four times of the iteration, the upper bound on $ k $ stabilizes at $ 584 $.  Hence, $ k<600 $ which contradicts our assumption that $ k>600 $. Therefore, we have no further solutions to the Diophantine equation \\eqref{Problem} with $ k>600 $.\n\n\\section*{Acknowledgements}\nM.~D. was supported by the Austrian Science Fund (FWF) grants: F5510-N26---Part of the special research program (SFB), ``Quasi-Monte Carlo Methods: Theory and Applications'' and W1230---``Doctoral Program Discrete Mathematics''. F.~L. was also supported by grant CPRR160325161141 from the NRF of South Africa, grant RTNUM19 from CoEMaSS, Wits, South Africa. Part of the work in this paper was done when both authors visited the Max Planck Institute for Mathematics Bonn, in March 2018 and the Institut de Math\\'ematiques de Bordeaux, Universit\\'e de Bordeaux, in May 2019. They thank these institutions for hospitality and fruitful working environments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nHydrogen is a clean energy carrier and an alternative to carbon based fuels in the long run. \\cite{coontz2004nss} Mobile applications require a compact, dense and safe storage of hydrogen\nwith a high-rate loading and unloading capability. \\cite{zuttel2003mhs,zuttel2004hsm} Lightweight\nmetal hydrides could satisfy these requirements. \\cite{schlapbach2001hsm,bogdanovic2003ihs} Metal\nhydrides are formed by binding hydrogen atoms in the crystal lattice, resulting in very high\nvolumetric densities. Reasonable hydrogen gravimetric densities in metal hydrides can be achieved\nif lightweight metals are used.\n\n$\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$ has been studied intensively since it has a relatively high hydrogen gravimetric density of\n$7.7$ wt. $\\%$. Bottlenecks in the application of $\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$ are its thermodynamic stability and slow (de)hydrogenation kinetics. These lead to excessively high operating\ntemperatures ($573-673$ K) for hydrogen release. \\cite{stampferjr1960mhs,huot2001mam,grochala2004tdn}                 The hydrogen (de)sorption rates can be improved by decreasing the particle size down to nanoscales. \\cite{zaluska2001sca,dornheim2006ths,li2007mne} It is predicted that particles smaller than $1$ nm have a markedly decreased hydrogen desorption enthalpy, which would lower the operating temperature.\\cite{wagemans2005hsm} The production of such small particles is nontrivial, however, and the hydrogen (de)sorption rates of larger nanoparticles are still too low.\n\nAn additional way of improving the (de)hydrogenation kinetics of $\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$ is to add transition\nmetals (TMs).\\cite{zaluska2001sca,pelletier2001hdm,vonzeppelin2002hdk,yao2006:jpcb} Usually only a few wt. \\% is added, since TMs are thought to act as catalysts for the dissociation of hydrogen\nmolecules. Recently however, Notten and co-workers have shown that the (de)hydrogenation kinetics\nis markedly improved by adding more TM and making alloys $\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}$, TM$=$Sc, Ti, $x\\lesssim 0.8$.\n\\cite{notten2004hed,niessen2005ehs,niessen2005hst, kalisvaart2006ehs,\nniessen2006epc,vermeulen2006hsm,\nborsa2006mth,borsa2007soa,vermeulen2007ter,kalisvaart2007mtb,gremaud2007hoc} The basic ansatz is\nthat the rutile crystal structure of $\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$ enforces an unfavorably slow diffusion of\nhydrogen atoms. \\cite{buschow1982hfi} $\\textrm{S\\lowercase{c}}\\textrm{H}_{2}$ and $\\textrm{T\\lowercase{i}}\\textrm{H}_{2}$ have a fluorite structure, which would be more favorable for fast hydrogen kinetics. By adding a sufficiently large fraction of these TMs one could force the $\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}$ compound to adopt the fluorite structure.\n\nIn this paper we examine the structure and stability of $\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}$, TM = Sc, Ti, V, Cr, compounds by first-principles calculations. In particular, we study the relative stability of the rutile versus the fluorite structures. This paper is organized as follows. In Sec.~\\ref{sec:computational} we discuss the computational details. The calculations are benchmarked on the $\\textrm{TM}\\textrm{H}_{2}$ simple hydrides. The structure and formation enthalpies of the compounds $\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}$ are studied in Sec.~\\ref{sec:results} and an analysis of the electronic structure is given. We discuss the hydrogenation enthalpy of the compounds in Sec.~\\ref{sec:discussion} and summarize our main results in Sec.~\\ref{sec:summary}.\n\n\\section{Computational Methods and Test Calculations}\\label{sec:computational}\nWe perform first-principles calculations at the level of density functional theory (DFT) with the\nPW91 functional as the generalized gradient approximation (GGA) to exchange and correlation. \\cite{perdew1996gga} As transition metals have partially filled 3$d$ shells we include spin polarization and study ferromagnetic and simple antiferromagnetic orderings where appropriate.\nA plane wave basis set and the projector augmented wave (PAW) formalism are used, \\cite{blochl1994paw,kresse1999upp} as implemented in the VASP code. \\cite{kresse1993aim,kresse1996eis} The cutoff kinetic energy for the plane waves is set\nat $650$ eV. Standard frozen core potentials are applied for all the elements, except for Sc,\nwhere we include 3\\textit{s} and 3\\textit{p} as valence shells, in addition to the usual 4\\textit{s} and 3\\textit{d} shells. The Brillouin zone (BZ) is integrated using a regular\n$\\textbf{k}$-point mesh with a spacing $\\sim 0.02$ \\AA$^{-1}$ and the Methfessel-Paxton scheme\nwith a smearing parameter of $0.1$ eV. \\cite{methfessel1989hps}\nThe self-consistency convergence criterion for the energy difference between two consecutive\nelectronic steps is set to 10$^{-5}$ eV. Structural optimization is assumed to be complete when\nthe total force acting on each atom is smaller than $0.01$ eV\/\\upshape{\\AA}. The volumes of the unit cells are relaxed, and, where appropriate, also their shapes. Finally, we calculate\naccurate total energies for the optimized geometries using the linear tetrahedron method.\n\\cite{blochl1994itm}\n\nTo calculate the formation enthalpies of the metal hydrides we consider the following reaction.\n\\begin{equation}\n\\label{MHreaction}\n     x\\textrm{M\\lowercase{g}}  +  (1-x)\\textrm{TM}  +  \\textrm{H}_{2}(g) \\longrightarrow  \\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}.\n\\end{equation}\nThe formation enthalpies (at $T=0$) are then obtained by subtracting the total energies of the reactants from that of the product. A cubic box of size 10 \\AA\\ is applied for the $\\textrm{H}_{2}$ molecule. The calculated H$-$H bond length, binding energy, and vibrational frequency are $0.748$ \\upshape{\\AA}, $-4.56$ eV and $4351$ cm$^{-1}$, respectively, in good agreement with the experimental values of $0.741$ \\upshape{\\AA}, $-4.48$ eV and $4401$ cm$^{-1}$. \\cite{hubher1979cdm,codata1989kvt}\n\nSince hydrogen is a light element, the zero point energy (ZPE) due to its quantum motion, is not negligible. We find that the correction to the reaction enthalpies of Eq.~(\\ref{MHreaction}) resulting from the ZPEs, is  $0.15 \\pm 0.05$ eV\/H$_2$, as function of the composition $x$ and the transition metal TM. Since we are mainly interested in\nrelative formation enthalpies, we omit the ZPE energy correction in the following.\n\n\\begin{figure}[tb]\n    \\centering\n        \\includegraphics[width=4cm]{ready_MgH2_136.eps}\n        \\includegraphics[width=4cm]{ready_TiH2_225.eps}\n        \\caption{(Color online) (Left) rutile crystal structure of $\\alpha$-$\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$, and (right) fluorite crystal structure of $\\alpha$-$\\textrm{TM}\\textrm{H}_{2}$. The white spheres represent the hydrogen atoms.}\n    \\label{fig:binarydihydrides}\n\\end{figure}\n\n\nBefore discussing the $\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}$ compounds we benchmark our calculations on the simple compounds\n$\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$ and $\\textrm{TM}\\textrm{H}_{2}$. Under standard conditions magnesium-hydride has the rutile\nstructure, $\\alpha$-$\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$, see Fig. \\ref{fig:binarydihydrides}, with space group P4$_{2}$\/mnm (136) and Mg and H\natoms in the 2$\\textit{a}$ and 4$\\textit{f}$ ($x=0.304$) Wyckoff positions, respectively. Each Mg atom is\ncoordinated octahedrally by H atoms, with two Mg-H distances of $1.94$ \\upshape{\\AA} and four\ndistances of $1.95$ \\upshape{\\AA}. First row early transition metal hydrides\ncrystallize in the fluorite structure, $\\alpha$-$\\textrm{TM}\\textrm{H}_{2}$, see Fig. \\ref{fig:binarydihydrides}, with space group\nFm$\\overline{3}$m (225) and TM atoms in $\\textit{a}$ and H atoms in 8$\\textit{c}$ Wyckoff positions.\nEach TM has a cubic surrounding of H atoms with calculated TM-H bond lengths of $2.07, 1.92, 1.82$\nand $1.79$ \\upshape{\\AA} for Sc, Ti, V, and Cr, respectively. By breaking the cubic symmetry by\nhand and reoptimizing the geometry we have confirmed that the fluorite structure indeed represents a stable minimum.\n\nThe optimized cell parameters and the calculated formation enthalpies of the simple hydrides are\ngiven in Table \\ref{BHstr}. The structural parameters are in good agreement both with available\nexperimental data and with previous DFT calculations. The formation enthalpies of $\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$ and $\\vh$\nare somewhat underestimated by the calculations, whereas those of $\\textrm{S\\lowercase{c}}\\textrm{H}_{2}$ and $\\textrm{T\\lowercase{i}}\\textrm{H}_{2}$ are in\nexcellent agreement with experiment. $\\textrm{C\\lowercase{r}}\\textrm{H}_{2}$ is predicted to be unstable with respect to\ndecomposition.\n\n\\begin{table}[tb]\n\\centering\n\\caption{Optimized cell parameters $a$ $(c)$, and calculated formation enthalpies $E_f$, of elemental dihydrides in their most stable ($\\alpha$) forms. All $\\textrm{TM}\\textrm{H}_{2}$ have a fluorite structure, space group Fm$\\overline{3}$m (225), whereas $\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$ has a rutile structure, space group P4$_{2}$\/mnm (136). }\n\\label{BHstr}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccc}\nCompound        & \\multicolumn{2}{c}{$a$ $(c)$ \\AA} & \\multicolumn{2}{c}{$E_f$ (eV\/f.u.)} \\\\\n                &    Calc          & Exp                            &   Calc      &  Exp\\\\\n\\hline\n$\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$          &   4.494 (3.005)  &  4.501 (3.010)\\footnotemark[1] &   $-0.66$   & $-0.76$ \\\\\n$\\textrm{S\\lowercase{c}}\\textrm{H}_{2}$          &   4.775          &  4.78\\footnotemark[2]          &   $-2.09$   & $-2.08$ \\\\\n$\\textrm{T\\lowercase{i}}\\textrm{H}_{2}$          &   4.424          &  4.454\\footnotemark[3]         &   $-1.47$   & $-1.45$ \\\\\n$\\vh$           &   4.210          &  4.27\\footnotemark[4]          &   $-0.65$   & $-0.79$ \\\\\n$\\textrm{C\\lowercase{r}}\\textrm{H}_{2}$          &   4.140          &  3.861\\footnotemark[4]         &   $+0.13$\\footnotemark[5]   &   -     \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\footnotetext[1]{Ref. \\onlinecite{bortz1999shp}}\n\\footnotetext[2]{Ref. \\onlinecite{mueller1968mh}}\n\\footnotetext[3]{Ref. \\onlinecite{villars1991psh}}\n\\footnotetext[4]{Ref. \\onlinecite{snavely1949ucd}}\n\\footnotetext[5]{Antiferromagneticly ordered.}\n\\end{table}\n\n\\section{Results $\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}$}\\label{sec:results}\n\\subsection{Structures and formation enthalpies}\\label{sec:forment}\n$\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}$ has the fluorite structure for $x=0$, and the rutile structure for $x=1$. We want to\nestablish which of the two structures is most stable at intermediate compositions $x$. First we\nsummarize the current status of the experimental work on $\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}$ alloys.\n\nExperimentally it has been demonstrated that $\\textrm{M\\lowercase{g}}_{x}\\textrm{S\\lowercase{c}}_{(1-x)}$ alloys can be reversibly hydrogenated, both\nin thin films, as well as in bulk form. \\cite{notten2004hed,niessen2005ehs,niessen2006epc,kalisvaart2006ehs,latroche2006csm,magusin2008hsa} Mg and Ti do not form a stable bulk alloy, but thin films of $\\textrm{M\\lowercase{g}}_{x}\\textrm{T\\lowercase{i}}_{(1-x)}$ have been made, which are\nreadily and reversibly hydrogenated. \\cite{vermeulen2006hsm,vermeulen2006edt,borsa2006mth,borsa2007soa,vermeulen2007ter,kalisvaart2007mtb,gremaud2007hoc} Thin films of $\\mgva$ and $\\textrm{M\\lowercase{g}}_{x}\\textrm{C\\lowercase{r}}_{(1-x)}$ can also be easily hydrogenated. \\cite{niessen2005ehs} Attempts\nto produce non-equilibrium bulk $\\textrm{M\\lowercase{g}}_{x}\\textrm{T\\lowercase{i}}_{(1-x)}$ alloys by ball milling of Mg and Ti or their hydrides\nhave had a limited success so far. \\cite{liang1999cet,bobet2000sma,liang2003smt,choi2008hsp}\nHowever, Mg$_7$TiH$_{y}$ crystals have been made using a high pressure anvil technique. \\cite{kyoi2004ntm}            The same technique has been applied to produce the hydrides Mg$_6$VH$_{y}$\nand Mg$_3$CrH$_{y}$. \\cite{kyoi2003fmc,kyoi2004nmv,ronnebro2004scm}\n\nThe crystal structure of $\\textrm{M\\lowercase{g}}_{x}\\textrm{S\\lowercase{c}}_{(1-x)}\\textrm{H}_{y}$ and $\\textrm{M\\lowercase{g}}_{x}\\textrm{T\\lowercase{i}}_{(1-x)}\\textrm{H}_{y}$ in thin films, $x\\lesssim 0.8$, $y\\approx 1$-2, is\ncubic, with the Mg and TM atoms at fcc positions. No detectable regular ordering of Mg and TM\natoms at these positions has been found. \\cite{latroche2006csm,borsa2007soa,magusin2008hsa}\nIn contrast, the Mg and TM atoms form simple ordered structures in the high pressure phases. \\cite{kyoi2003fmc,kyoi2004ntm,kyoi2004nmv,ronnebro2004scm,ronnebro2005hsa} The hydrogen atoms in\nMg$_{0.65}$Sc$_{0.35}$H$_y$, $y\\approx 1$-2, assume tetrahedral interstitial positions, as is\nexpected for the fluorite structure. \\cite{latroche2006csm,magusin2008hsa} In the Mg$_7$TiH$_{16}$ high\npressure phase the metal atoms are in fcc positions and are ordered as in the\nCa$_7$Ge structure. \\cite{kyoi2004ntm} The H atoms are in interstitial sites, but displaced from\ntheir ideal tetrahedral positions. \\cite{ronnebro2005hsa}\n\nThe latter structure can be used as a starting point to construct simple, fluorite-type structures\nfor $\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}$, $0<x<1$. For $x=0.125, 0.875$ we use the Ca$_7$Ge structure to order the metal\natoms, for $x=0.25, 0.75$ the Cu$_3$Au ($L1_2$) structure, and for $x=0.5$ the CuAu ($L1_0$)\nstructure. The H atoms are placed at or close to tetrahedral interstitial positions. To model $\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}$ in\nrutile-type structures we use the $\\alpha$-MgH$_2$ structure as a starting point. For each\ncomposition $x$ we choose the smallest supercell of that structure corresponding to that\ncomposition. As the atomic volumes of the various TM atoms differ, in each of the structures and\ncompositions the cell parameters are optimized, as well as the positions of all atoms within the\ncell. Care is taken to allow for breaking the symmetry in the atomic positions. In particular the\nhydrogen atoms are often displaced from their ideal tetrahedral positions. Although these structures are then no longer ideal fluorite structures anymore, we still use the term fluorite in the following. Test calculations show that the total energies of the hydrides of (quasi-)random alloys are similar to those of the simple ordered structures. In particular, the relative stability of the fluorite vs. the rutile structures is not extremely sensitive to the relative ordering of the metal atoms.\n\n\\begin{figure}[tb]\n    \\centering\n        \\includegraphics[width=9cm]{Vol_data.eps}\n      \\caption{(Color online) The volumes per formula unit in \\AA$^3$ of $\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}$ in the fluorite structure, as a function of the composition $x$ for TM $=$ Sc, Ti, V, Cr (from top to bottom).}\n    \\label{fig:cellparameters}\n\\end{figure}\n\nThe calculated volumes $V({\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}})$ of the fluorite structures, normalized per formula unit\n(f.u.), are shown in Fig.~\\ref{fig:cellparameters}. The volumes of a few bulk compounds and\ncompositions can be extracted from experimental data, thus providing a check on the calculations.\nInterpolating the results in Ref.~\\onlinecite{latroche2006csm} gives\n$V(\\mathrm{Mg}_{0.65}\\mathrm{Sc}_{0.35}\\mathrm{H}_{2})=27.5$ \\AA$^3$, whereas the interpolated\ncalculated value from Fig.~\\ref{fig:cellparameters} is 26.9 \\AA$^3$.\nRef.~\\onlinecite{ronnebro2005hsa} gives\n$V(\\mathrm{Mg}_{0.875}\\mathrm{Ti}_{0.125}\\mathrm{H}_{2})=27.3$ \\AA$^3$, compared to the calculated\nvalue 26.6 \\AA$^3$. These differences between experimental and calculated volumes are consistent\nwith the differences between experimental and calculated lattice parameters of the simple\nhydrides, see Table~\\ref{BHstr}. Experiments on thin films give\n$V(\\mathrm{Mg}_{0.7}\\mathrm{Ti}_{0.3}\\mathrm{H}_{2})=26.4$ \\AA$^3$, if one assumes cubic symmetry. \\cite{borsa2007soa}   The interpolated computational value is 24.4 \\AA$^3$. The difference between\nthese values is not excessively large, but considering the smaller difference found for the bulk\ncomposition $\\mathrm{Mg}_{0.875}\\mathrm{Ti}_{0.125}\\mathrm{H}_{2}$, it may suggest that the\nstructure of thin films is slightly different from that of bulk.\n\nThe trends observed in Fig.~\\ref{fig:cellparameters} can be interpreted straightforwardly. The\ncell volumes of $\\alpha$-$\\textrm{S\\lowercase{c}}\\textrm{H}_{2}$ and the cubic $\\beta$-$\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$ structure \\cite{vajeeston2002pis} are within 1.4\\% of one another,\nwhich explains why the volumes calculated for $\\textrm{M\\lowercase{g}}_{x}\\textrm{S\\lowercase{c}}_{(1-x)}\\textrm{H}_{2}$ only weakly depend on the composition $x$.\nThe cell volumes of the other $\\textrm{TM}\\textrm{H}_{2}$ are smaller, hence one expects the volumes $V({\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}})$ to increase\nwith $x$. At fixed composition $x$, the volumes $V({\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}})$ decrease along the series Sc, Ti,\nV, Cr, as the atomic volumes of the TMs decrease correspondingly. According to Zen's law of additive\nvolumes one would expect \\cite{hafner1985nvs}\n\\begin{equation}\nV({\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}}) = xV({\\beta\\textrm{-}\\textrm{M\\lowercase{g}}\\textrm{H}_{2}}) + (1-x)V({\\textrm{TM}\\textrm{H}_{2}}).\n\\end{equation}\nThe curves shown in Fig.~\\ref{fig:cellparameters} deviate slightly, but distinctly, from straight\nlines, with a maximum deviation of $\\sim 5$~\\%. This deviation is consistent with the experimental\nobservations on $\\textrm{M\\lowercase{g}}_{x}\\textrm{T\\lowercase{i}}_{(1-x)}\\textrm{H}_{2}$. \\cite{vermeulen2006hsm,borsa2007soa} It is also observed in simple metal alloys. \\cite{hafner1985nvs}\n\nThe calculated formation enthalpies of $\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}$ are shown in Fig.~\\ref{fig:deltaH}.\nClearly for all TMs the fluorite structure is more stable than the rutile structure for all $x$\nsmaller than a critical value, $x_c$. The critical composition $x_c$ at which the rutile structure\nbecomes more stable can be guessed by interpolation and is in the range $x_c\\approx 0.8$-0.85 for\nall TMs. The fact that the critical composition is fairly high might be guessed from the energies of the fluorite and rutile structures of the simple hydrides. As a\nfirst estimate of the critical composition $x_c$ below which the fluorite structure is stable one\nmay try a linear interpolation between the pure $\\textrm{TM}\\textrm{H}_{2}$ compounds, $x=0$, and $\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$, $x=1$. The\nrutile structure of $\\textrm{TM}\\textrm{H}_{2}$ is more unstable than the fluorite structure by $\\Delta(\\textrm{TM}\\textrm{H}_{2})=$ 0.75,\n0.65, 0.68, 0.36 eV\/f.u. for TM $=$ Sc, Ti, V, Cr, respectively. The difference in formation\nenthalpy between the $\\alpha$ (rutile) and $\\beta$ (cubic) phases of $\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$ is $\\Delta(\\textrm{M\\lowercase{g}}\\textrm{H}_{2})=0.10$\neV\/f.u. Linear interpolation then gives $x_c=\\Delta(\\textrm{TM}\\textrm{H}_{2})\/(\\Delta(\\textrm{TM}\\textrm{H}_{2})+\\Delta(\\textrm{M\\lowercase{g}}\\textrm{H}_{2}))$, which\nresults in $x_c\\approx0.9$ for Sc, Ti and V, and $x_c=0.8$ for Cr. Whereas these values may seem a good first guess, they are somewhat too high as compared to the crossing points $x_c$ observed in\nFig.~\\ref{fig:deltaH}. Moreover, as this figure shows, in particular the curves for Ti and V are far from linear, so the results for the linear interpolation may be somewhat fortuitous.\n\n\\begin{figure}[tb]\n    \\centering\n        \\includegraphics[width=9cm]{Energy.eps}\n      \\caption{(Color online) The formation enthalpy (per formula unit) of the $\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}$ compounds as obtained from spin polarized calculations. The values for the fluorite and rutile structures are represented by squares (solid lines) and triangles (dashed lines), respectively.}\n    \\label{fig:deltaH}\n\\end{figure}\n\nExperimental results indicate that Mg$_{0.7}$Ti$_{0.3}$H$_2$ has the fluorite structure and Mg$_{0.9}$Ti$_{0.1}$H$_2$ has the rutile structure. \\cite{borsa2007soa} This agrees with the results shown in Fig.~\\ref{fig:binarydihydrides}, where a fluorite to rutile phase transition takes place at the composition $x=x_c\\approx0.83$. It has been suggested that the fluorite structure allows for a much faster kinetics of hydrogen loading and unloading. Experimentally it has been observed that the dehydrogenation kinetics of $\\textrm{M\\lowercase{g}}_{x}\\textrm{S\\lowercase{c}}_{(1-x)}$ and $\\textrm{M\\lowercase{g}}_{x}\\textrm{T\\lowercase{i}}_{(1-x)}$ becomes markedly slower if $x\\gtrsim x_0 = 0.8$ . The results shown in Fig.~\\ref{fig:binarydihydrides} suggest that $x_0 = x_c$, i.e. the composition at which the phase transition between fluorite and rutile structures takes place.\n\nThe dehydrogenation kinetics of $\\textrm{M\\lowercase{g}}_{x}\\textrm{T\\lowercase{i}}_{(1-x)}$ also becomes gradually slower with\ndecreasing $x$, for $x<x_c$, i.e. where the compound remains in the fluorite structure, although\nit is still faster than for $x>x_c$. \\cite{kalisvaart2006ehs,vermeulen2006hsm} Whereas kinetic studies\nare beyond the scope of the present paper, we speculate that a volume effect might play a role\nhere. In the fluorite structure the hydrogen atoms occupy interstitial positions close to the\ntetrahedral sites. Diffusion of hydrogen atoms is likely to take place via other interstitial\nsites such as the octahedral sites. The smaller the volume, the shorter the distance between such\nsites and the occupied positions, or in other words, the shorter the distance between a diffusing\nhydrogen atom and other hydrogen atoms in the lattice. This may increase the barrier for\ndiffusion. As the volume of $\\mgva$ and $\\textrm{M\\lowercase{g}}_{x}\\textrm{C\\lowercase{r}}_{(1-x)}$ is smaller than that of $\\textrm{M\\lowercase{g}}_{x}\\textrm{S\\lowercase{c}}_{(1-x)}$ and $\\textrm{M\\lowercase{g}}_{x}\\textrm{T\\lowercase{i}}_{(1-x)}$ (at\nthe same composition $x$), this might also explain why the dehydrogenation kinetics of the former\ncompounds is much slower. \\cite{niessen2005ehs} We note that the smaller volume of $\\mgva$ and\n$\\textrm{M\\lowercase{g}}_{x}\\textrm{C\\lowercase{r}}_{(1-x)}$ is accompanied by a distortion of the structures consistent with the limited space\navailable to accommodate the hydrogen atoms. For instance, in $\\mgvhb$ and $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{C\\lowercase{r}}_{0.25}\\textrm{H}_{2}$ the hydrogen\natoms are displaced considerably from the tetrahedral positions, and the coordination number of V\nand Cr (by hydrogen) is 7, instead of 8 as in case of a perfect fluorite structure.\n\n\\subsection{Electronic structure}\nTo analyze the electronic structure of the compounds $\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}$, we start with the density of states (DOS) of the pure hydrides $\\alpha$-$\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$ and $\\textrm{TM}\\textrm{H}_{2}$ as shown in Fig.~\\ref{fig:dostmh2}. The bonding in $\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$ is dominantly ionic; occupied hydrogen orbitals give the main\ncontribution to the valence states, whereas the conduction bands have a significant contribution\nfrom the Mg orbitals. \\cite{vansetten2007esa} As usual, ionic bonding between main group elements\nresults in an insulator with a large band gap. In contrast, the transition metal\ndihydrides are metallic, as demonstrated by Fig.~\\ref{fig:dostmh2}. The peak in the DOS at low\nenergy, i.e. between $-9$ and $-2$ eV in $\\textrm{S\\lowercase{c}}\\textrm{H}_{2}$ to between $-12$ and $-4$ eV in $\\textrm{C\\lowercase{r}}\\textrm{H}_{2}$, is dominated by hydrogen\nstates. The broad peak around the Fermi level consists of transition metal $d$-states. It suggests that bonding in $\\textrm{TM}\\textrm{H}_{2}$ is at least partially ionic. The TM $s$-electrons are transferred to the H atoms, whereas the $d$-electrons largely remain on the TM atoms. The DOSs of $\\textrm{TM}\\textrm{H}_{2}$, TM$=$Sc, Ti, V, Cr, are very similar in shape. As the number of $d$-electrons increases from one in Sc to four in Cr, the Fermi level moves up the $d$-band in this series. As the DOS at the Fermi level increases, it enhances the probability of a magnetic instability. Indeed we find $\\textrm{C\\lowercase{r}}\\textrm{H}_{2}$ to be antiferromagnetic with a magnetic moment of $1.5\\mu_B$ on the Cr atoms. The antiferromagnetic ordering is $51$ meV\/f.u. more stable than the ferromagnetic ordering, which is $7$ meV\/f.u. more stable than the non-polarized solution. In the other $\\textrm{TM}\\textrm{H}_{2}$ we do not find magnetic effects.\n\n\\begin{figure}[tb]\n    \\centering\n        \\includegraphics[width=9.5cm]{DOSall.eps}\n      \\caption{(Color online) Densities of states of $\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$ and $\\textrm{TM}\\textrm{H}_{2}$ (left column) and of $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{TM}_{0.25}\\textrm{H}_{2}$ (right column) for TM $=$ Sc, Ti , V, Cr. The shaded areas give the projected densities of states on the TM $d$ orbitals. For $\\textrm{C\\lowercase{r}}\\textrm{H}_{2}$ the nonmagnetic DOS is given for simplicity reasons; $\\textrm{C\\lowercase{r}}\\textrm{H}_{2}$ is antiferromagnetic (see text).}\n    \\label{fig:dostmh2}\n\\end{figure}\n\nAdditional information on the type of bonding can be obtained from a Bader charge analysis. \\cite{henkelman2006far}      In $\\alpha$-$\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$ the Bader charges are $Q_\\mathrm{Mg}=+1.59e$ (and\n$Q_\\mathrm{H}=-0.80e$, since the compound is neutral), which confirms that this compound is dominantly ionic. The results for $\\textrm{TM}\\textrm{H}_{2}$\nare shown in Table~\\ref{table:badertmh2}. They indicate that the ionicity in $\\textrm{S\\lowercase{c}}\\textrm{H}_{2}$ is\ncomparable to that in $\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$. Furthermore, the ionicity decreases along the series Sc, Ti, V and $\\textrm{C\\lowercase{r}}\\textrm{H}_{2}$.\nComparison to Table~\\ref{BHstr} shows that the decrease in ionicity correlates with a decrease in\nformation enthalpy.\n\nThese results on the simple hydrides help us to analyze the electronic structure and bonding in\n$\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}$. We show results for the fluorite structure only, since that is the more stable structure over most of the composition range. As an example, Fig.~\\ref{fig:dostmh2} shows the calculated DOSs of $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{TM}_{0.25}\\textrm{H}_{2}$. One can qualitatively interpret these DOSs as a superposition of\nthe DOSs of $\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$ and $\\textrm{TM}\\textrm{H}_{2}$. The bonding states at low energy, comprising the first broad peak in the DOS, consist mainly of filled hydrogen states. The peaks close to the Fermi level are\ndominated by TM $d$-states. The Fermi level moves up the $d$-band through the series Sc, Ti, V, Cr. At higher energy we find the (unoccupied) Mg $s$ states. The basic structure of the DOSs remains the same for all compositions $\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}$. As $x$ increases, the TM $d$ contribution of course decreases. In addition, the TM $d$-peak becomes narrower with increasing $x$, as the distance between the TM atoms increases.\n\nNarrowing of the $d$ peak can give rise to magnetic instabilities. The tendency to such instabilities increases along the series Sc, Ti, V and Cr. The development of nonzero magnetic moments of course strongly depends upon the structure. Nevertheless, for $\\textrm{M\\lowercase{g}}_{x}\\textrm{S\\lowercase{c}}_{(1-x)}\\textrm{H}_{2}$ and $\\textrm{M\\lowercase{g}}_{x}\\textrm{T\\lowercase{i}}_{(1-x)}\\textrm{H}_{2}$ we see a tendency to form magnetic moments on the TMs only if $x\\gtrsim0.8$. For $\\mgvha$ this occurs if $x\\gtrsim0.5$, and for $\\textrm{M\\lowercase{g}}_{x}\\textrm{C\\lowercase{r}}_{(1-x)}\\textrm{H}_{2}$ one can find magnetic instabilities over the whole composition range. Most of the structures have a finite DOS at the Fermi level, which, might indicate a metallic behavior. One cannot conclude this on the basis of a DOS alone, however, but should also critically evaluate possible localization and on-site correlation effects. There are a few exceptions. In particular cases low spin states can be more stable, such as for $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{C\\lowercase{r}}_{0.25}\\textrm{H}_{2}$ in the fluorite structure. Cubic crystal field splitting by the hydrogens surrounding the Cr atom results in a gap between $e_g$ and $t_{2g}$ states, the $e_g$ states being lowest in energy. The latter are filled by the four $d$ electrons of Cr, which makes this particular structure insulating, see Fig.~\\ref{fig:dostmh2}. The DOS of $\\mgvhb$ in the fluorite structure is explained by the same mechanism. However, as V only has three $d$ electrons, each V atom obtains a magnetic moment of 1~$\\mu_B$. The distance between the TM atoms is fairly large in most compositions that have nonzero magnetic moments, which suggests a small magnetic coupling between the TM atoms, a low N\\'{e}el or Curie temperature, and paramagnetic behavior at room temperature. Exceptions are the Cr compounds with a substantial amount of Cr, as discussed above.\n\nA Bader charge analysis of $\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}$ can be made, similar to the simple hydrides. For all\ncompositions $Q_\\mathrm{Mg}\\approx +1.6e$, i.e. close to the value found in $\\alpha$-$\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$. As an example the Bader charges on the TM and H atoms in $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{TM}_{0.25}\\textrm{H}_{2}$ are given in\nTable~\\ref{table:badertmh2}. The charge on the TM atoms decreases\nalong the series Sc, Ti, V, and Cr as in the simple hydrides, but compared to the latter, it is somewhat smaller on V and Cr. The charges on the H atoms in $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{TM}_{0.25}\\textrm{H}_{2}$ are roughly the\nproportional average of the charges on the H atoms in $\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$ and $\\textrm{TM}\\textrm{H}_{2}$. The charge analysis of the $\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}$ compounds is consistent with the bonding picture extracted from the DOSs.\n\n\\begin{table}[tb]\n\\centering\n\\caption{Bader charge analysis of $\\textrm{TM}\\textrm{H}_{2}$ and $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{TM}_{0.25}\\textrm{H}_{2}$. All charges $Q$ are given in units of $e$. }\n\\label{table:badertmh2}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccc}\n          & \\multicolumn{2}{c}{$\\textrm{TM}\\textrm{H}_{2}$}  &  \\multicolumn{2}{c}{$\\textrm{M\\lowercase{g}}_{0.75}\\textrm{TM}_{0.25}\\textrm{H}_{2}$}  \\\\\nTM        & $Q_\\mathrm{TM}$ ($e$)  & $Q_\\mathrm{H}$ ($e$)  & $Q_\\mathrm{TM}$ ($e$)  & $Q_\\mathrm{H}$ ($e$) \\\\\n\\hline\nSc        &   $+1.51$  &  $-0.75$  &   $+1.57$  &  $-0.80$  \\\\\nTi        &   $+1.17$  &  $-0.59$  &   $+1.18$  &  $-0.76$  \\\\\nV         &   $+1.09$  &  $-0.55$  &   $+0.98$  &  $-0.73$  \\\\\nCr        &   $+0.89$  &  $-0.45$  &   $+0.68$  &  $-0.69$  \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\section{Discussion}\\label{sec:discussion}\nWe discuss to what extend the Mg-TM alloys are suitable as hydrogen storage materials.\nThe formation enthalpies of $\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}$ are shown in Fig.~\\ref{fig:deltaH}.  Lightweight materials require a high content of magnesium, but to have a stable fluorite structure it should not exceed the critical composition $x_c$, as discussed in Sec.~\\ref{sec:forment}. We focus upon the composition $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{TM}_{0.25}\\textrm{H}_{2}$ in the following discussion. The calculated formation enthalpies are $-0.83,-0.59,-0.43$ and $-0.30$ eV\/f.u. for TM $=$ Sc, Ti, V, and Cr, respectively. For applications the binding enthalpy of hydrogen in the lattice should be $\\lesssim 0.4$ eV\/H$_2$,\\cite{zuttel2003mhs,zuttel2004hsm,schlapbach2001hsm} which indicates that the Sc and Ti compounds are too stable. The formation enthalpies of the V and Cr compounds could be in the right range. However, the parameter that is most relevant for hydrogen storage is the hydrogenation enthalpy. Assuming that the alloy does not dissociate upon dehydrogenation, the hydrogenation enthalpy corresponds the reaction\n\\begin{equation}\n\\label{hydrogenation}\n\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}  +  \\textrm{H}_{2}(g) \\longrightarrow  \\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}.\n\\end{equation}\n\nTo assess the hydrogenation enthalpy, one can break down the formation enthalpy associated with Eq.~(\\ref{MHreaction}) into components, similar to the decomposition used in Ref.~\\onlinecite{miwa2002fps}. We write the formation enthalpy as a sum of three terms. (i) The enthalpy required to make the Mg-TM alloy in the fcc structure from the elements in their most stable form. (ii) The energy required to expand the fcc lattice in order to incorporate the hydrogen atoms. (iii) The energy associated with inserting the hydrogen atoms. The results of this decomposition for $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{TM}_{0.25}\\textrm{H}_{2}$ are given in Fig.~\\ref{fig:edecomposed}. To facilitate the discussion, a similar decomposition is shown for the simple hydrides, where (i) only consists of transforming the pure metal into the fcc structure. In contrast to Ref.~\\onlinecite{miwa2002fps}, we use the spin-polarized fcc alloy for calculating the contributions (i) and (ii), as this will make the extraction of the hydrogenation enthalpy easier. In the cases where the magnetic moment is nonzero, we study both ferromagnetic and antiferromagnetic ordering. As for the simple hydrides, Cr compounds generally have an antiferromagnetic ordering.\n\nThe lattice expansion energy (ii) of the compounds $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{TM}_{0.25}\\textrm{H}_{2}$ is $\\leq 0.1$ eV for all TMs, see Fig.~\\ref{fig:edecomposed}(b). It is in fact comparable to that of pure Mg, see Fig.~\\ref{fig:edecomposed}(a). At the composition $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{TM}_{0.25}\\textrm{H}_{2}$, the effect on the energy of changing the unit cell volume is dominated by Mg. For these compounds the lattice expansion only plays a minor role in the formation energy, in contrast to the simple hydrides, where the lattice expansion gives a significant contribution. The hydrogen insertion energies (iii) are also remarkably similar for the Sc, Ti, and V compounds. Again this is in sharp contrast to the corresponding energies for the simple hydrides, which strongly depend on the TM. The hydrogen insertion energies for the compounds are in fact similar to that of pure Mg. At the composition $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{TM}_{0.25}\\textrm{H}_{2}$ also this energy is then dominated by Mg. Only the compound $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{C\\lowercase{r}}_{0.25}\\textrm{H}_{2}$ has a somewhat smaller hydrogen insertion energy. The reason for this is that the energy gained by magnetic ordering of the alloy $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{C\\lowercase{r}}_{0.25}$ is relatively high, as compared to the other compounds. This contribution stabilizes the alloy with respect to the hydride, which is nonmagnetic.\n\\begin{figure}[tb]\n    \\centering\n        \\includegraphics[width=8.5cm]{E_decomposed_TMH.eps}\n        \\includegraphics[width=8.5cm]{E_decomposed_MgTMH.eps}\n      \\caption{Decomposition of the formation energy into (i) the formation energy of the spin-polarized fcc metal $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{TM}_{0.25}$ (black), (ii) the lattice expansion energy (gray), and (iii) the hydrogen insertion energy (white); (a) the simple hydrides $\\textrm{M\\lowercase{g}}\\textrm{H}_{2}$, $\\textrm{TM}\\textrm{H}_{2}$; (b) $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{TM}_{0.25}\\textrm{H}_{2}$.}\n    \\label{fig:edecomposed}\n\\end{figure}\n\nThe formation enthalpy of the fcc alloys $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{TM}_{0.25}$ (i) shows the largest variation as a function of the TM, relative to the contributions (ii) and (iii). Whereas the alloy formation energy is negative for TM $=$ Sc, indicating that this alloy is stable, it is positive for Ti, V, and Cr, meaning that these alloys are unstable. This result agrees with the experimental finding that of the Mg-TM alloys considered here, only a stable Mg-Sc alloy exists in bulk form. The substantial increase of the alloy formation enthalpy in the series Sc, Ti, V, is largely responsible for the variation of the formation energy of the corresponding hydrides $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{TM}_{0.25}\\textrm{H}_{2}$. The alloy formation energy of $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{C\\lowercase{r}}_{0.25}$ is similar to that of $\\mgvb$, due to a relatively high spin-polarization energy, as discussed in the previous paragraph.\n\nThe hydrogenation enthalpy according to Eq.~(\\ref{hydrogenation}) can be determined by summing the contributions (ii) and (iii) of Fig.~\\ref{fig:edecomposed}. Since the most stable structure of the alloys is not always the fcc structure, one should however subtract the energy required to convert the alloys from their most stable structure to an fcc structure. We find, for instance, that for $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{T\\lowercase{i}}_{0.25}$ the fcc structure is 0.04 eV\/f.u. less stable than the hcp structure. Indeed thin film experiments on $\\textrm{M\\lowercase{g}}_{x}\\textrm{T\\lowercase{i}}_{(1-x)}$ yield yield an hcp structure.\\cite{kalisvaart2006ehs,borsa2007soa} For $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{S\\lowercase{c}}_{0.25}$ the fcc structure is more stable than the hcp structure by 0.05 eV\/f.u..\n\nThe calculated hydrogenation enthalpy of $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{S\\lowercase{c}}_{0.25}$ is $-0.79$ eV\/f.u., in good agreement with the experimental value of $-0.81$ eV\/f.u..\\cite{kalisvaart2006ehs} The calculated hydrogenation enthalpy of $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{T\\lowercase{i}}_{0.25}$ is $-0.76$ eV\/f.u., which is in good agreement with the experimental value of $-0.81$ eV\/f.u. of Ref. \\onlinecite{gremaud2007hoc}, obtained if the thin film correction suggested there is included. These hydrogenation enthalpies are remarkably similar to that of pure Mg, strongly suggesting that alloying Mg with these TMs does not improve this energy as compared to pure Mg. The most stable structures of $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{TM}_{0.25}$, TM $=$ V, Cr, are not known, but judging from the Sc and Ti compounds the energy difference between the fcc and the most stable structures will be small.\nNeglecting this energy difference upper bounds for the hydrogenation enthalpies of $\\mgvb$ and $\\textrm{M\\lowercase{g}}_{0.75}\\textrm{C\\lowercase{r}}_{0.25}$ are $-0.72$ and $-0.57$ eV\/f.u., respectively. Again this indicates that alloying Mg with these TMs does not improve the hydrogenation enthalpy substantially.\n\n\\section{Summary}\\label{sec:summary}\nIn summary, we have studied the structure and stability of $\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}\\textrm{H}_{2}$, TM = Sc, Ti, V, Cr, compounds by first-principles calculations. We find that for $x < x_c \\approx 0.8$ the fluorite structure is more stable than the rutile structure, whereas for $x > x_c$ the rutile structure is more stable. This phase transition correlates with the observed slowing down of the (de)hydrogenation kinetics in these compounds if $x$ exceeds the critical composition $x_c$. The density of states of these compounds is characterized by the valence bands being dominated by contributions from the hydrogen atoms, wherease the TMs have partially occupied $d$ states around the Fermi level. As $x$ increases and\/or one moves down the TM series, the tendency to magnetic instabilities increases. The formation enthalpy of Mg$_x$TM$_{1-x}$H$_2$ can be tuned over a substantial range, i.e. 0-2 eV\/f.u., by varying TM and $x$. To a large part this reflects the variation of the formation enthalpy of the alloy $\\textrm{M\\lowercase{g}}_{x}\\textrm{TM}_{(1-x)}$, however. Assuming that the alloys do not decompose upon dehydrogenation, the hydrogenation enthalpy then shows much less variation. For compounds with a high magnesium content ($x=0.75$) it is close to that of pure Mg.\n\n\\section*{ACKNOWLEDGMENTS}\nThis work is part of the research programs of ``Advanced Chemical Technologies for Sustainability (ACTS)'' and the ``Stichting voor Fundamenteel Onderzoek der Materie (FOM)''. The use of supercomputer facilities was sponsored by the ``Stichting Nationale Computerfaciliteiten (NCF)''. These institutions are financially supported by ``Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)''.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Description of UBAT}\nThe Ultra-Fast Flash Observatory pathfinder (UFFO-p) \\cite{Chen11,Nam13,Park13} is a new space mission dedicated to detect Gamma-Ray Bursts (GRBs) and rapidly follow their afterglows to provide early optical\/ultraviolet measurements. It consists of two scientific instruments. One is the UFFO Burst Alert \\& Trigger telescope (UBAT) \\cite{Jung11,Lee13,Chang14} which employs the coded mask technique providing X-ray\/gamma-ray imaging observations in the energy range of $10-150$\\,keV with half-coded field of view (FOV) $70.4^{\\circ}\\times70.4^{\\circ}$ and angular resolution $\\le10'$. The second instrument is the Slewing Mirror Telescope (SMT) \\cite{Jeong13,Kim13} with the field of view of $17'\\times17'$ and optical\/ultraviolet range of $200-650$\\,nm. It consists of a Ritchey-Chr\\'{e}tien telescope with an Intensified Charge-Coupled Device in its focal plane together with a plane slewing mirror. In this article the laboratory tests of UBAT flight module (FM) (Figure~\\ref{fig:uffo_ubat}) are briefly described. UBAT and SMT have been assembled and integrated with the control electronics on UFFO-p which is planned to be launched on the Lomonosov Moscow State University satellite \\cite{Panasyuk11}.\n\n\\begin{figure}\n\\centering\n    \\makebox[0.45\\linewidth][c]{\\includegraphics[width=0.35\\linewidth]{fig-uffo.eps}}   \n    \\makebox[0.45\\linewidth][c]{\\includegraphics[width=0.35\\linewidth]{fig-ubat.eps}}\n    \\caption{\n\t    \\textit{Left:} A photo shows the assembled UFFO-p with its two scientific instruments: UBAT and SMT.\n\t    \\textit{Right:} A schematic view of UBAT is drawn. It consists of a coded mask, hopper, detector and readout electronics.\n\t    }\n    \\label{fig:uffo_ubat}\n\\end{figure}\n\nUBAT is a wide-field X-ray imager which provides triggers of GRBs and determines their location via the coded mask technique. It consists of a coded mask, hopper, detector, and readout electronics. The coded mask is a 1\\,mm thick plate of tungsten with a random pattern of $68\\times68$ opened and closed tiles, having 44.5\\,\\% of open area. The size of each tile is 5.76\\,mm$\\times$5.76\\,mm, i.e. double the detector pixels size. It is placed 280\\,mm above the detector surface and it casts a mosaic shadow onto the detector. From the cross-correlation of the position of the projected shadow and the coded mask pattern one can determine the direction of the X-ray source \\cite{Connell13}. The hopper supports the coded mask and provides shielding against the diffuse cosmic X-ray background. It is made of 2\\,mm thick aluminium with 0.1\\,mm thick layer of tungsten.\nThe detector comprises a Cesium-doped Yttrium Oxyorthosilicate (YSO) (Y$_{2}$SiO$_{5}$ : 0.2\\,\\% Ce) scintillator crystal array and Hamamatsu \\textit{R11265-03-M64} Multi-Anode Photomultiplier Tubes (MAPMTs), together with the analogue and digital electronics. There are 36 MAPMTs and each of them has $8 \\times 8 = 64$ channels, thus providing 2304 imaging pixels in total. On the top of each MAPMT there is a 8$\\times$8 YSO crystal array. The volume of each single crystal is 2.68 (length) $\\times$ 2.68 (width) $\\times$ 3.0 (height)\\,mm$^{3}$.\n\n\n\\section{Pixel-to-Pixel Response}\nFor the best imaging performance a uniform response of the detector is required because the imaging algorithm effectively subtracts any uniform background. In order to test the pixel-to-pixel response, to set the thresholds for each MAPMT and to determine the noisy pixels an X-ray tube was placed on axis in front of the detector without the coded mask attached. A tube with adjustable current and voltage of the accelerated electrons was used. It produced a broad-band Bremsstrahlung X-ray radiation up to 50\\,keV. Figure~\\ref{fig:hitmap-exp-spec-50kV} shows results for the following setup: source-detector distance of 8088\\,mm, tube voltage (current) of 50\\,kV (5.3\\,$\\mu$A), total exposure time of 186\\,s and with a 1.6\\,mm thick Cu filter placed in front of the tube to attenuate the flux.\n\nThe recorded data were processed off-line with analysis software which used a geometrical center hit-finder to eliminate the optical crosstalk caused by total reflections at the glass-vacuum (photo-cathode) interface \\cite{Chang14}. There were 30\\,460 registered counts due to the background and 137\\,256 counts due to the source, and the area of the enabled detector pixels was 160.2\\,cm$^2$ thus giving a count rate due to the source of 4.6 cnt\\,s$^{-1}$\\,cm$^{-2}$. Figure~\\ref{fig:hitmap-exp-spec-50kV} shows the expected count spectrum given by the radiation produced by the X-ray tube after attenuation in the air and in the 1.6\\,mm thick Cu filter. It was calculated from the measured X-ray spectrum given by the tube's manufacturer and considering the attenuation law.\n\n\\begin{figure\n    \\makebox[0.45\\linewidth][c]{\\includegraphics[width=0.4\\linewidth]{hitmap_total_frames_50kV.eps}}   \n   \n    \\makebox[0.53\\linewidth][c]{\\includegraphics[trim=-50pt -50pt 0pt 0pt, width=0.49\\linewidth]{50kV_8088mm_Cu1_6mm_scaled.eps}}\n    \\caption{\n\t    \\textit{Left:} The UBAT FM detector response to X-rays of energies $\\leq 50$\\,keV is shown. The color scale indicates the total number of counts in each pixel. White color means zero counts at noisy detector pixels, which have been disabled noisy.\n\t    \\textit{Right:} The expected normalized spectrum given by the radiation produced by the X-ray tube illuminating the detector after attenuation in the air and in the Cu filter is displayed.}\n    \\label{fig:hitmap-exp-spec-50kV}\n\\end{figure}\n\n\n\\section{Test of Imaging}\n\nThe results of imaging with X-rays of energies $\\leq 50$\\,keV are shown in Figure~\\ref{fig:imaging_50kV_1.8mmCu}. The X-ray tube was placed on the axis of UBAT with voltage (current) set to 50\\,kV (5.3\\,$\\mu$A). The source-detector distance was 8\\,090\\,mm and a 1.8\\,mm thick Cu filter was used to additionally attenuate the flux. For the image reconstruction a conical beam algorithm was used because at this distance the beam illuminating the coded mask does not have exactly parallel rays, but they diverge at an angle of about $\\pm0.6^{\\circ}$.\n\nData were processed off-line with analysis software where an FM hit-finder (the same concept as implemented in the UFFO-p electronics) was applied on each 10\\,ms data frame in order to reduce the crosstalk and increase the peak signal-to-noise ratio (SNR) in the reconstructed image. The FM hit-finder utilizes a hit pattern recognition technique. The total exposure was 31\\,s. The coded mask was attached and it has an open area fraction of about 0.445. There were 4\\,026 registered counts due to the background and 4\\,781 counts due to the source, and the area of enabled detector pixels was 154.6\\,cm$^2$ thus giving a count rate due to the source under the open mask pixels of 2.2 cnt\\,s$^{-1}$\\,cm$^{-2}$.\n\n\\begin{figure\n     \\makebox[0.4\\linewidth][c]{\\includegraphics[width=0.35\\linewidth]{UBAT_GRB_imaging_location_zen.eps}}   \n   \n    \\makebox[0.6\\linewidth][c]{\\includegraphics[trim=-50pt -50pt 0pt 0pt, width=0.55\\linewidth]{UBAT_GRB_imaging_snr.eps}}\n    \\caption{\n\t    \\textit{Left:} Reconstructed X-ray image as an SNR map. Color scale corresponds to the SNR value. The X-ray tube was placed on axis with voltage 50\\,kV. The X-ray source is clearly visible in the middle of the FOV.\n\t    \\textit{Right:} The image peak SNR evolution during the exposure time.}\n    \\label{fig:imaging_50kV_1.8mmCu}\n\\end{figure}\n\n\n\\section{Angular Resolution}\n\nThe X-ray source was placed in a fixed direction, on axis, and the recorded data of one exposure totalling 30\\,s were split into 15 independent sub-samples each of duration 2\\,s. The imaging algorithm for each sub-sample was run separately. The variance of the calculated directions of the source indicates the angular resolution (Figure~\\ref{fig:angular_resolution_method1}). The image reconstruction algorithm was run for three different thresholds of SNR$_{\\textrm{thr}}=5, 7, 9$ at which the algorithm was forced to stop. This allows comparison of the dependence of the localization accuracy on SNR. The experimental setup was the same as in the previous section except that here a 1.6\\,mm thick Cu filter was used. The count rate due to the source beneath the open mask pixels was 4.1 cnt\\,s$^{-1}$\\,cm$^{-2}$.\n\nFor the detection of SNR\\,=\\,$5.0\\sim5.6$ the standard deviations of the calculated positions in the x-axis and y-axis were $\\sigma_{X}=8.2'$ and $\\sigma_{Y}=10.3'$.\nFor SNR\\,=\\,$7.0\\sim7.4$ they were $\\sigma_{X}=6.5'$ and $\\sigma_{Y}=6.4'$.\nFor SNR\\,=\\,$8.6\\sim9.4$ they were $\\sigma_{X}=5.1'$ and $\\sigma_{Y}=6.7'$.\nConsidering that the angular radius of the source was $1'$, the result is close to the designed resolution of $\\pm5\\,'$ for $>7\\sigma$ detection.\n\n\\begin{figure\n    \\mbox{\\includegraphics[width=0.33\\linewidth]{angular_resolution_thr5.eps}}   \n   \n    \\mbox{\\includegraphics[width=0.33\\linewidth]{angular_resolution_thr7.eps}}\n   \n    \\mbox{\\includegraphics[width=0.33\\linewidth]{angular_resolution_thr9.eps}}   \n    \\caption{The angular resolution of UBAT FM for different detection thresholds is shown. The variance of the measured source positions of 15 different exposures indicate the resolution. SMT has FOV$=17'\\times17'$.}\n    \\label{fig:angular_resolution_method1}\n\\end{figure}\n\n\n\\section{Combined Test with BDRG Detector}\nThe UBAT FM detector was also tested together with the laboratory version of the Block for X-ray and Gamma-Radiation Detection (BDRG) \\cite{Amelyushkin13} scintillator gamma-ray spectrometer. BDRG will be one of the instruments on the Lomonosov satellite. It consists of 75\\,mm (diameter) $\\times$ 3\\,mm (thickness) NaI(Tl) scintillator optically coupled with 80\\,mm (diameter) $\\times$ 17\\,mm (thickness) CsI(Tl). Both crystals are attached to the same photomultiplier tube. The efficiency in the range $30-100$\\,keV is close to 100\\,\\%.\n\nThe X-ray tube was set to 50\\,kV, 5.3\\,$\\mu$A, a 1.6\\,mm thick Cu filter was placed in front of the tube and the source-detector distance was 7760\\,mm. The measured background rate by BDRG in the range $10-100$\\,keV was 0.15\\,cnt\\,s$^{-1}$\\,cm$^{-2}$ and the rate due to the source at energy $>$30\\,keV was 5.18\\,cnt\\,s$^{-1}$\\,cm$^{-2}$. This count rate can be compared with the measurement done with UBAT FM for the same setup as described in the previous section. However, in that case the source-detector distance was 8090\\,mm. The count rate due to the source and under the open mask pixels was 4.10 cnt\\,s$^{-1}$\\,cm$^{-2}$. Taking into account this distance difference the expected count rate detected by UBAT at distance 7760\\,mm would be $\\approx 4.46$\\,cnt\\,s$^{-1}$\\,cm$^{-2}$. Comparing this with the count rate from BDRG one can infer the efficiency of UBAT FM at the range of $30-50$\\,keV to be $4.46\/5.18 = 86\\,\\%$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}