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+{"text":"\\section{\\bf Introduction}\n\nIn the work, we prove some results for the viscosity solutions to some doubly nonlinear parabolic equations. The main focus of this work is the Trudinger's equation but we will also state some results for a parabolic equation involving the infinity-Laplacian. This is a follow-up of the works in \\cite{BM1, BM2}.\n\nTo describe our results more precisely, we introduce definitions and notations. We take $n\\ge 2$ in this work. \nLetters like $x,\\;y,\\;z$ etc, denote the spatial variable, $s,\\;t$ the time variable and $o$ stands for the origin in $\\mathbb{R}^n$. Let $\\overline{A}$ denote the closure of a set $A$. The ball of radius $R>0$ and center $x\\in \\mathbb{R}^n$ is denoted by $B_R(x)$. \nLet $\\Omega\\subset \\mathbb{R}^n,$ be a bounded domain and $0<T<\\infty$. We define\n$\\Omega_T=\\Omega\\times (0,T)$ and its parabolic boundary as $P_T=(\\overline{\\Omega}\\times\\{0\\})\\cup (\\partial \\Omega \\times (0,T)).$ Also set $H=\\mathbb{R}^n$, $H_T=H\\times(0,\\infty)$.\n\nFor $1<p<\\infty$, define the $p$-Laplacian $\\D_p$ and the infinity-Laplacian $\\D_{\\infty}$ as \n\\eqRef{1.1}  \n\\D_p u=\\mbox{div}(|Du|^{p-2}Du)\\;\\;\\;\\mbox{and}\\;\\;\\D_{\\infty} u=\\sum_{i,j=1}D_i uD_j u D_{ij}u,\n\\end{equation}\nwhere $u=u(x)$. We now define the parabolic operators of interest to us. Call\n\\eqRef{1.2}\n\\Gamma_p u=\\D_p u-(p-1) u^{p-2} u_t,\\;\\;2\\le p<\\infty,\\;\\;\\;\\mbox{and}\\;\\;\\;\\Gamma_{\\infty} u=\\D_{\\infty} u-3 u^2 u_t,\n\\end{equation}\nwhere $u=u(x,t)$. The equation $\\Gamma_p=0,\\;2\\le p<\\infty,$ is the well-known Trudinger's equation \\cite{TR}. See also\n\\cite{BM1,BM2} and the references therein. The operators $\\Gamma_p,\\;2< p\\le \\infty$ are doubly nonlinear and degenerate and, in this work, solutions will be understood to be in the viscosity\nsense. Note that in our work, we use $p=\\infty$ as a label. It is not clear to us what the limit of $\\Gamma_p$ (and $G_p$, see below) is for $p\\rightarrow \\infty$. For a detailed discussion about nonlinear parabolic equations, see \\cite{ED}.\n\nSuppose that $0<T<\\infty$. Let $f\\in C(\\overline{\\Omega})$ and $g(x,t)\\in C( \\partial\\Omega\\times[0,T))$. For ease of notation, we define \n\\eqRef{1.30}\nh(x,t)=\\left\\{ \\begin{array}{ccc} f(x), & \\forall x\\in \\overline{\\Omega},\\\\ g(x,t), & \\forall (x,t)\\in  \\partial\\Omega\\times[0,T) \\end{array}\\right.\n\\end{equation}\nIn most of this work, we take \n\\eqRef{1.3} \n0<\\inf_{P_T} h(x,t)\\le \\sup_{P_T}h(x,t)<\\infty.\n\\end{equation}\n\nFor $2\\le p\\le \\infty$, we consider positive viscosity solutions $u\\in C(\\Omega_T\\cup P_T)$ of \n\\eqRef{1.4} \n\\Gamma_p u=0,\\;\\;\\mbox{in $\\Omega_T$ and $u=h$ on $P_T$.}\n\\end{equation}\nIn \\cite{BM1} (see Theorem 5.2), we showed the existence of positive viscosity solutions of (\\ref{1.4}) for $p=\\infty$. The work \\cite{BM2} showed the existence of positive viscosity solutions\nfor $2\\le p<\\infty$, see Theorems 1.1 and 1.2 therein. For the case $2\\le p\\le n$, this result is proven for domains $\\Omega$ that satisfy a uniform outer ball condition. For $n<p<\\infty$, the result is shown for any $\\Omega$.\n\nWe will also have occasion to work with equations related to $\\Gamma_p$. As observed in Lemma 2.2 in \\cite{BM1} and Lemma 2.1 in \\cite {BM2}, \nif $u>0$ solves the doubly nonlinear equation $\\Gamma_p u=0,\\;2\\le p\\le \\infty,$ (see (\\ref{1.2})), then $v=\\log u$ solves\n$$\\Delta_pv+(p-1)|Dv|^p-(p-1)v_t=0,\\;\\;2\\le p<\\infty,\\;\\;\\mbox{and}\\;\\;\\;\\D_{\\infty} v+|Dv|^4-3v_t=0,$$\nFor convenience of presentation, call\n\\eqRef{1.5}\nG_p w=\\Delta_pw+(p-1)|Dw|^p-(p-1)w_t,\\;\\;2\\le p<\\infty,\\;\\;\\mbox{and}\\;\\;G_\\infty w=\\D_{\\infty} w+|Dw|^4-3w_t.\n\\end{equation}\n\nWe now introduce the notion of a viscosity solution. The set $usc(A)$ denotes the\nset of all upper semi-continuous functions on a set $A$ and $lsc(A)$ the set of all lower semi-continuous functions on $A$. We say $u\\in usc(\\Omega_T),\\;u>0,$ is a sub-solution of $\\Gamma_p w=0$, in $\\Omega_T$, or $\\Gamma_p u\\ge 0$ (see (\\ref{1.2}))\nif for any function $\\psi(x,t)$, $C^2$ in $x$ and $C^1$ in $t$, such that $u-\\psi$ has a local maximum\nat some $(y,s)\\in \\Omega_T$, we have\n$$\\D_p \\psi(y,s)-(p-1)u(y,s)^{p-2} \\psi_t(y,s)\\ge 0.$$\nSimilarly, $u\\in lsc(\\Omega_T),\\;u>0,$ is a super-solution of $\\Gamma_p w=0$ in $\\Omega_T$ or $\\Gamma_pu\\le 0$ (see (\\ref{1.2})) \nif for any function $\\psi(x,t)$, $C^2$ in $x$ and $C^1$ in $t$, such that $u-\\psi$ has a local minimum\nat some $(y,s)\\in \\Omega_T$, we have $\\D_p \\psi(y,s)-(p-1)u(y,s)^{p-2} \\psi_t(y,s)\\le 0.$ A function $u\\in C(\\Omega_T)$ is a solution of $\\Gamma_p w=0$, in $\\Omega_T$, or $\\Gamma_p u=0$, if $u$ is both a sub-solution and a super-solution. Analogous definitions can be provided for the equation $G_p w=0$, see (\\ref{1.5}).\n\nNext, we say $u\\in usc(\\Omega_T\\cup P_T),\\;u>0,$ is a viscosity sub-solution of (\\ref{1.4}) if $\\Gamma_p u\\ge 0$, in $\\Omega_T,$ and $u\\le h$ on $P_T$. Similarly, $u\\in lsc(\\Omega_T\\cup P_T),\\;u>0,$ is a viscosity super-solution of (\\ref{1.4}) if\n$\\Gamma_p u\\le 0$ in $\\Omega_T$, and $u\\ge h$ on $P_T$. A function $u\\in C(\\Omega_T\\cup P_T),\\;u>0,$ is a solution of (\\ref{1.4}) if $\\Gamma_p u=0$ in $\\Omega_T$ and $u=h$ on $P_T$.\n\nWe now state the main results of this work. Let $\\lambda_{\\Om}$ be the first eigenvalue of $\\Delta_p$ on $\\Omega$.\n\n\\begin{thm}\\label{1.6} Let $2\\le p<\\infty$ and $\\Omega\\subset \\mathbb{R}^n,\\;n\\ge 2,$ be a bounded domain. Call\n$\\Omega_\\infty= \\Omega\\times(0, \\infty)$ and \n$P_\\infty$ its parabolic boundary. Suppose that $h\\in C(P_\\infty)$ is as defined in (\\ref{1.30}) with\n$h\\ge 0$ and $\\sup_{P_\\infty} h<\\infty$.\nLet \n$u\\in usc(\\Omega_\\infty\\cup P_\\infty)),\\;u\\ge 0,$ solve\n$$\\Gamma_p u\\ge 0,\\;\\;\\mbox{in $\\Omega_\\infty$ and $u\\le h$ on $P_\\infty$}.$$\n\n(i) If $\\lim_{t\\rightarrow \\infty} (\\sup_{\\partial\\Omega}g(x,t) )=0$ then $\\lim_{t\\rightarrow \\infty} (\\sup_{\\Omega\\times [t, \\infty)}u)=0.$ \n\n(ii) Moreover, if $g(x,t)=0,\\forall(x,t)\\in \\partial\\Omega\\times[T_0,\\infty)$, for some $T_0\\ge 0$, then\n$$\\lim_{t\\rightarrow \\infty} \\frac{\\log(\\sup_\\Omega u(x,t) )}{t} \\le -\\frac{\\lambda_\\Omega}{p-1}. $$\n\\end{thm}\nThe above result is an analogue of the asymptotic result proven in Theorem 4.4 and Lemma 4.7 in \\cite{BM1} for \n$\\Gamma_\\infty u\\ge 0$. We provide an example where the rate $\\exp(-\\lambda_{\\Om} t\/(p-1))$ is attained, see Remark \n\\ref{3.15}. Note that we do not address existence for $h\\ge 0$. We also show\n\n\\begin{thm}\\label{1.61} Let $2\\le p\\le \\infty$, $\\Omega\\subset \\mathbb{R}^n,\\;n\\ge 2,$ be a bounded domain, \n$\\Omega_\\infty= \\Omega\\times(0, \\infty)$ and \n$P_\\infty$ be its parabolic boundary. Suppose that $h\\in C(P_\\infty)$ is as defined in (\\ref{1.30}). \nAssume that $0<\\inf_\\Omega f\\le 1\\le \\sup_\\Omega f<\\infty$ and $g(x,t)=1,\\;\\forall(x,t)\\in \\partial\\Omega\\times[0,\\infty)$.\n\nIf $u\\in C(\\Omega_\\infty\\cup P_\\infty)),\\;u>0,$ solves\n$$\\Gamma_p u= 0,\\;\\;\\mbox{in $\\Omega_\\infty$, $u(x,0)=f(x),\\;\\forall x\\in \\overline{\\Omega}$ and $u(x,t)=1,\\;\\forall(x,t)\\in \\partial\\Omega\\times[0,\\infty)$},$$\nthen for every $x\\in \\Omega$, $\\lim_{t\\rightarrow \\infty} u(x,t)=1.$ \n\\end{thm}\nFrom the proof, it follows that (i) $u(x,t)=\\exp(O(t^{-s})),\\;p=2$ and any $s>0$, (ii) $u(x,t)=\\exp(O(t^{-1\/(p-2)})),\\;2<p<\\infty,$ and (iii)\n$u(x,t)=\\exp(O(t^{-1\/2})),\\;p=\\infty$, as $t\\rightarrow \\infty$. \nFrom the works in \\cite{AJK, JL} one sees that (i) for $\\Delta_\\infty u=u_t$, the asymptotic decay \nis $t^{-1\/2}$ and (ii) for $\\D_p u=u_t$, the rate is $t^{-1\/(p-2)}.$ They do appear to agree if we consider $G_p.$\nHowever, at this time, it is not clear if the asymptotic rates in Theorem \\ref{1.61} are optimal and also if $u$ tends to a $p$-harmonic function when $g(x,t)=g(x)$, for all $t>0$.\n\nWe now state a Phragm\\'en-Lindel\\\"of type result for the unbounded domain $H_T$. A version was shown in Theorem 4.1 in \\cite{BM1} for $\\Gamma_\\infty$. We show an analogue for $\\Gamma_p$, $2\\le p < \\infty$, and include an improvement for $\\Gamma_\\infty$.\n\n\\begin{thm}\\label{1.7} Let $2\\le p\\le \\infty$, $T>0$, $H=\\mathbb{R}^n$ and $H_T=\\mathbb{R}^n\\times (0,T)$. \nAssume that $0<\\inf_{H}f(x)\\le \\sup_H f(x)<\\infty$.\nSuppose that $u\\in C(H\\times\\{0\\} \\cup H_T )$, $2\\le p\\le \\infty,$ solves\n$$\\Gamma_p u=0,\\;\\;\\mbox{in $H_T$},$$ \nand as $R\\rightarrow \\infty$, \n\\begin{eqnarray*}\n\\sup_{0\\le |x|\\le R,\\;0\\le t\\le T}u(x,t)\\le \\left\\{ \\begin{array}{llr}  \\exp\\left( o( R^{p\/(p-1)}\\right), &\\mbox{for $2\\le p<\\infty$, and}\\\\\n \\exp\\left( o( R^{4\/3}\\right),& \\mbox{for $p=\\infty$}.\\end{array}\\right.\n\\end{eqnarray*}\nIt follows that $\\inf_{H}f(x)\\le u(x,t) \\le\\sup_H f(x),\\;\\forall(x,t)\\in H_T$.\n\\end{thm}\nIt is not clear if the result in the theorem is optimal.\n\nThe proofs of Theorems \\ref{1.6}, \\ref{1.61} and \\ref{1.7} employ appropriate auxiliary functions and the comparison principle. We have divided our work as follows. Section 2 contains some previously proven results and includes some useful calculations. Proofs of Theorems \\ref{1.1} and {1.2} are in Section 3. Theorem \\ref{1.3} is proven in Section 4. Section 5 contains a discussion of the eigenvalue problem for $\\Delta_p$ in the viscosity setting and has relevance for Theorem \\ref{1.1}.\n\n\\section{Preliminaries and some observations}\n\nWe recall some previously proven results and include some useful calculations. \nSee Section 3 in \\cite{BM1, BM2} for proofs of Lemmas \\ref{2.1}, \\ref{2.3} and \\ref{2.5}, and Theorems \\ref{2.4} and \\ref{2.2}. Lemma \\ref{2.1} and Theorem \\ref{2.4} hold regardless of sign of $u$.\n\n\\begin{lem}\\label{2.1}{(Maximum principle)} Let $\\Omega\\subset \\mathbb{R}^n,\\;n\\ge 2$ be a bounded domain and $T>0$.\\\\\n(a) If $u\\in usc(\\Omega_T\\cup P_T)$ solves \n$$\\Delta_p u- (p-1)|u|^{p-2} \\phi_t\\ge 0,\\;\\;2\\le p<\\infty,\\;\\;\\mbox{or}\\;\\;\\D_{\\infty} u-3u^2 u_t\\ge 0,\\;\\; \\mbox{in $\\Omega_T$},$$\nthen $\\sup_{\\Omega_{T}}u\\le\\sup_{P_T} \\phi=\\sup_{\\Omega_T\\cup P_T}u.$\\\\\n(b) If $\\phi\\in lsc(\\Omega_T\\cup P_T)$ and \n$$\\Delta_p \\phi- (p-1)|\\phi|^{p-2}\\phi_t\\le 0,\\;\\;2\\le p<\\infty,\\;\\;\\D_{\\infty} u-3u^2u_t\\le 0,\\;\\;\\mbox{in $\\Omega_T$},$$ \nthen \n$\\inf_{\\Omega_T} \\phi\\ge \\inf_{P_T}\\phi=\\inf_{\\Omega_T\\cup P_T}\\phi.$\n\\end{lem}\nWe present a comparison principle for $G_p$ (see (\\ref{1.5})) that leads to Theorem \\ref{2.2}. \n\n\\begin{thm}\\label{2.4}{(Comparison Principle)} \nLet $2\\le p\\le \\infty$. Suppose that $\\Omega\\subset \\mathbb{R}^n,\\;n\\ge 2,$ is a bounded domain and $T>0$. Let $u\\in usc(\\Omega_T\\cup P_T)$ and $v\\in lsc(\\Omega_T\\cup P_T)$ satisfy\n$$G_p u\\ge 0,\\;\\;\\mbox{and}\\;\\;G_p v\\le 0,\\;\\;\\mbox{in $\\Omega_T$}.$$\nIf $u,\\;v$ are bounded and $u\\le v$ on $P_T$, then $u\\le v$ in $\\Omega_T$. \n\\end{thm}\n\nThe next is a comparison principle for $\\Gamma_p$ (see (\\ref{1.2})) that applies to positive solutions. \n\n\\begin{thm}\\label{2.2}{(Comparison Principle}) \nSuppose that $\\Omega\\subset \\mathbb{R}^n,\\;n\\ge 2,$ is a bounded domain and $T>0$. Let $u\\in usc(\\Omega_T\\cup P_T)$ and $v\\in lsc(\\Omega_T\\cup P_T)$ satisfy\n$$\\Gamma_p u\\ge 0,\\;\\;\\mbox{and}\\;\\;\\Gamma_p v\\le 0,\\;\\;\\mbox{in $\\Omega_T$},\\;\\;\\;\\;2\\le p\\le \\infty.$$\nAssume that $\\min(\\inf_{\\Omega_T\\cup P_T} u, \\inf_{\\Omega_T\\cup P_T} v)>0$. \nIf $\\sup_{P_T}v<\\infty$ then\n$$\\sup_{\\Omega_T} u\/v=\\sup_{P_T}u\/v.$$\nIn particular, if $u\\le v$ on $P_T$, then $u\\le v$ in $\\Omega_T$. Clearly, solutions to (\\ref{1.4}) are unique.\n\\end{thm}\n\n\\begin{rem}\\label{2.20}  We extend Theorem \\ref{2.2} to the case $u\\ge 0$ on $P_T$. Let $v$ be as in Theorem \\ref{2.2}.\n\n(i) If $u=0$ on $P_T$, then by Lemma \\ref{2.1}, $u=0$, in $\\Omega_T$, and the conclusion holds.\n\n(ii) Let $u\\ge 0$ be a sub-solution (see Theorem \\ref{2.2}) and $\\sup_{\\Omega_T}u>0$; clearly, \n$\\sup_{P_T}u>0$, by Lemma \\ref{2.1}. \nLet $\\varepsilon>0$, be small. Define $u_\\varepsilon(x,t)=\\max(u(x,t),\\;\\varepsilon),\\;\\forall(x,t)\\in \\Omega_t\\cup P_T.$ We show that $u_\\varepsilon\\in usc(\\Omega_T\\cup P_T)$ and $\\Gamma_p u_\\varepsilon\\ge 0,$ in $\\Omega_T$.\n\nLet $(y,s)\\in \\Omega_T\\cup P_T$. Since $\\limsup_{(x,t)\\rightarrow (y,s)}u(x,t)\\le u(y,s)$, we have\n$\\limsup_{(x,t)\\rightarrow (y,s)}u_\\varepsilon(x,t)\\le u_\\varepsilon(y,s)$ and $u_\\varepsilon\\in usc(\\Omega_T\\cup P_T)$. Next, let $\\psi$, $C^2$ in $x$ and $C^1$ in $t$, and $(y,s)\\in \\Omega_T$ be such that $u_\\varepsilon-\\psi$ has a maximum at $(y,s)$. If $u_\\varepsilon(y,s)=u(y,s)(\\ge \\varepsilon)$ then $u-\\psi$ has a maximum at $(y,s)$ (since $u\\le u_\\varepsilon$).\nSince $u$ is sub-solution, \nwe get \n$$\\Delta_p\\psi(y,s)-(p-1)u(y,s)^{p-2} \\psi_t(y,s)=\\Delta_p\\psi(y,s)-(p-1)u_\\varepsilon(y,s)^{p-2} \\psi_t(y,s)\\ge 0.$$\nNext, assume that $u_\\varepsilon(y,s)=\\varepsilon$. Rewriting $(u_\\varepsilon-\\psi)(x,t)\\le \\varepsilon-\\psi(y,s)$,\n$$0\\le u_\\varepsilon(x,t)-\\varepsilon\\le \\langle D\\psi(y,s), x-y\\rangle+\\psi_t(y,s)(t-s)+o(|x-y|+|t-s|),$$\nas $(x,t)\\rightarrow (y,s)$, where $(x,t)\\in \\Omega_T$. Clearly, $D\\psi(y,s)=0$ and $\\psi_t(y,s)=0$. Thus,\n$\\Delta_p\\psi(y,s)-(p-1)u_\\varepsilon(y,s)^{p-2} \\psi_t(y,s)=0$, if $p>2$. For $p=2$, we write the above Taylor expansion as $0\\le \\langle D^2\\psi(y,s)(x-y),x-y\\rangle\/2 +o(|x-y|^2+|t-s|)$, as $(x,t)\\rightarrow (y,s)$. It is clear that $\\Delta\\psi(y,s)\\ge 0$. Thus, $u_\\varepsilon$ solves $\\Gamma_pu_\\varepsilon \\ge 0$, in $\\Omega_T$, for $2\\le p<\\infty$. \n\nWe now apply Theorem \\ref{2.2} to obtain that $\\sup_{\\Omega_T}u_\\varepsilon\/v\\le \\sup_{P_T}u_\\varepsilon\/v$, for every small $\\varepsilon>0$. Since $u_\\varepsilon\\le u+\\varepsilon$ (note that $u\\ge 0$), we have\n$\\sup_{\\Omega_T}u\/v\\le \\sup_{\\Omega_T}u_\\varepsilon\/v\\le \\sup_{P_T}(u+\\varepsilon)\/v\\le \\sup_{P_T} u\/v+\\sup_{P_T} \\varepsilon\/v.$\nThe conclusion of Theorem \\ref{2.2} holds by letting $\\varepsilon\\rightarrow 0.$ $\\Box$\n\\end{rem}\n\nNext we state a change of variables result which relates $\\Gamma_p$ to $G_p$, see (\\ref{1.2}) and (\\ref{1.5}).\n\n\\begin{lem}\\label{2.3} Let $\\Omega\\subset \\mathbb{R}^n,\\;n\\ge 2,$ be a domain and $T>0$, and $2\\le p\\le \\infty$. Suppose $u:\\Omega_T\\rightarrow \\mathbb{R}^+$ and $v:\\Omega_T\\rightarrow \\mathbb{R}$ such that\n$u=e^v$. The following hold.\\\\\n (a) $u\\in usc(\\Omega_t\\cup P_T)$ and $\\Gamma_p u\\ge 0$ if and only if $v\\in usc(\\Omega_T\\cup P_T)$ and $G_p v\\ge 0$.\\\\\n (b) $u\\in lsc(\\Omega_t\\cup P_T)$ and $\\Gamma_p u\\le 0$ if and only if $v\\in lsc(\\Omega_T\\cup P_T)$ and $G_p v\\le 0$.\n\\end{lem}\nWe now present a separation of variable result that will be used for proving Theorem \\ref{1.6}.\nSee Lemma 2.14 in \\cite{BM1} and Lemma 2.3 in \\cite{BM2}.\n\n\\begin{lem}\\label{2.5} Let $\\lambda\\in \\mathbb{R}$, $\\mu \\in \\mathbb{R}$, $T>0$, and $\\psi:\\Omega\\rightarrow \\mathbb{R}^+$.  \n\n(a) Suppose that for some $2\\le p<\\infty$, $\\psi\\in usc(lsc)(\\Omega)$ solves\n$\\Delta_p\\psi+\\lambda \\psi^{p-1}\\ge (\\le) 0$ in $\\Omega$. If $u(x,t)=\\psi(x,t) e^{-\\mu t\/(p-1) }$ then \n$\\Gamma_p u\\ge (\\le) 0,\\;\\mbox{where $\\mu\\ge(\\le) \\lambda.$}$\n\n(b) Suppose that $\\psi\\in usc(lsc)(\\Omega)$ solves\n$\\D_{\\infty} \\psi+\\lambda \\psi^{3}\\ge (\\le) 0$ in $\\Omega$. If $u(x,t)=\\psi(x,t) e^{-\\mu t\/3 }$ then \n$\\Gamma_\\infty u\\ge (\\le) 0,\\;\\mbox{where $\\mu\\ge(\\le) \\lambda.$} $\n\\end{lem}\n\nWe include two results that will be used in Theorem \\ref{1.7}. We recall the radial form for the $\\D_p$, that is, if $r=|x|$, then, for $2\\le p\\le \\infty$, \n\\eqRef{2.50}\n\\D_p v(r)=|v^{\\prime}(r)|^{p-2} \\left( (p-1) v^{\\prime\\prime}(r)+\\frac{n-1}{r} v^{\\prime}(r) \\right)\\;\\;\\mbox{and}\\;\\;\\D_{\\infty} u= \\left(u^{\\prime}(r)\\right)^2 u^{\\prime\\prime}(r). \n\\end{equation}\n\n\\begin{lem}\\label{2.6} Let $R>0$; set\n$r=|x|,\\;\\forall x\\in \\mathbb{R}^n$ and $h(x)=1-(r\/R)^2,\\; \\forall 0\\le r\\le R.$\n\n(i) For $2\\le p<\\infty$, take\n$$k=p+n-2,\\;\\;\\;\\alpha=\\frac{2p+k-1}{2(p-1)},\\;\\;\\;\\theta^2=\\frac{k}{k+1}\\;\\;\\;\\mbox{and}\n\\;\\;\\;\\lambda_p= \\frac{k\\theta^{p-2}}{R^p} \\left( \\frac{2\\alpha}{1-\\theta^2}\\right)^{p-1}.$$\nCall $\\eta(x)=h(x)^\\alpha$ and $\\phi_p(x,t)=\\eta(x) e^{-\\lambda_p t\/(p-1)},\\;\\;\\forall 0\\le r\\le R.$\n\n(ii) For $p=\\infty$, define\n$$\\theta=1\/\\sqrt{2},\\;\\;\\;\\mbox{and}\\;\\;\\;\\lambda_\\infty=2^8\/R^4. $$\nSet $\\eta=h(x)^2$ and $\\phi_\\infty(x,t)=\\eta(x)e^{-\\lambda_\\infty t\/3},\\;\\forall 0\\le r\\le R.$\n\nThen, for $2\\le p\\le \\infty$, \n$\\Gamma_p \\phi_p\\ge 0,$ in $B_R(o)\\times(0,\\infty)$, $\\phi(0,0)=1$ and $\\phi(x, t)=0$, on $|x|=R$ and $t\\ge 0.$\n\\end{lem}\n\\begin{proof} Our goal is to show that for $2\\le p\\le \\infty$, $\\D_p \\eta+\\lambda_p \\eta^{p-1}\\ge 0$, in $0\\le r\\le R$. \n\n{\\bf Part (i): $2\\le p<\\infty$.} Observe that $\\alpha>1$. Differentiating $\\eta$, by using (\\ref{2.50}), and\nsetting $H=h^{(\\alpha-1)(p-1)-1}$, \n\\begin{eqnarray}\\label{2.61}\n\\D_p \\eta&+&\\lambda_p \\eta^{p-1}=\\D_p h^\\alpha+\\lambda_p h^{\\alpha(p-1)}\\nonumber\\\\\n&=&\\alpha^{p-1}\\left( h^{(\\alpha-1)(p-1)} \\D_p h+(\\alpha-1)(p-1)h^{(\\alpha-1)(p-1)-1}|Dh|^p\\right)\n+\\lambda_p h^{\\alpha(p-1)} \\nonumber\\\\\n&=&h^{(\\alpha-1)(p-1)-1} \\left[ \\lambda_p h^p+\\alpha^{p-1} \\left\\{ (\\alpha-1)(p-1)|Dh|^p+h\\D_p h\\right\\}\\right]\\nonumber\\\\\n&=&H\\left[ \\lambda_p h^p+\\alpha^{p-1} \\left\\{ (\\alpha-1)(p-1) \\left(\\frac{2r}{R^{2}}\\right)^p - \\frac{2^{p-1}r^{p-2}( p+n-2)h}{R^{2(p-1)}}  \\right\\} \\right] \\nonumber \\\\\n&=&H \\left[\\lambda_p h^{p} + \\alpha^{p-1} \\left\\{ (\\alpha-1)(p-1) \\left(\\frac{2r}{R^{2}}\\right)^p- \\frac{2^{p-1}r^{p-2}kh}{R^{2(p-1)}}\n \\right\\} \\right].\n\\end{eqnarray}\nWe now estimate the right hand side of (\\ref{2.61}) in $0\\le r\\le \\theta R$ and in $\\theta R\\le r\\le R$ separately.\n\nIn $0\\le r\\le \\theta R$, disregard the middle term in (\\ref{2.61}) and take $r=\\theta R$, to see\n\\begin{eqnarray*}\n\\D_p \\eta+\\lambda_p \\eta^{p-1}&\\ge& H \\left( \\lambda_p h^{p}- \\frac{(2\\alpha)^{p-1}r^{p-2}kh}{R^{2(p-1)}}\\right)=hH \\left( \\lambda_p h^{p-1}-\\frac{(2\\alpha)^{p-1}r^{p-2}k}{R^{2(p-1)}}\\right)\\\\\n&\\ge &hH \\left( \\lambda_p\\left(1-\\theta^2\\right)^{p-1} -\\frac{(2\\alpha)^{p-1}\\theta^{p-2}k}{R^p}\\right)= 0.\n\\end{eqnarray*}\n\nIn $\\theta R\\le r\\le R$, disregard the $\\lambda_p h^p$ term in (\\ref{2.61}), set $r=\\theta R$ in the second term and $h=1$ in the third term to obtain\n\\begin{eqnarray*}\n\\D_p \\eta+\\lambda_p \\eta^{p-1}&\\ge& \\alpha^{p-1}H  \\left\\{ (p-1)(\\alpha-1) \\frac{2^{p}r^p}{R^{2p}}-\\frac{2^{p-1}r^{p-2}k}{R^{2(p-1)}} h\n \\right\\}\\\\\n&\\ge & \\frac{(2\\alpha)^{p-1}Hr^{p-2}}{R^{2(p-1)}}  \\left\\{ \\frac{2(\\alpha-1)(p-1)r^2}{R^{2}}-k \\right\\}\\\\\n&=&\\frac{(2\\alpha)^{p-1}Hr^{p-2}}{R^{2(p-1)}} \\left\\{ 2(\\alpha-1)(p-1)\\theta^2-k \\right\\}=0,\n\\end{eqnarray*}\nsince $2(p-1)(\\alpha-1)\\theta^2=k$.\n\n{\\bf Part(ii): $p=\\infty$.} The work is similar to Part (i).\n\\begin{eqnarray}\\label{2.62}\n\\D_{\\infty} \\eta+\\lambda_\\infty \\eta^3&=&\\D_{\\infty} h^2+\\lambda_\\infty h^6=8h^3\\D_{\\infty} h+ 8h^2|Dh|^4+\\lambda_\\infty h^6\\nonumber\\\\\n&=&h^2\\left[ \\lambda_\\infty h^4+8\\left( |Dh|^4+h\\D_{\\infty} h\\right) \\right] \n=h^2\\left[ \\lambda_\\infty h^4+8\\left\\{ \\left(\\frac{2r}{R^2}\\right)^4-\\frac{8r^2}{R^6}h\\right\\} \\right]\n\\end{eqnarray}\nWe estimate (\\ref{2.62}) in $0\\le r\\le \\theta R$,\n\\begin{eqnarray*}\n\\D_{\\infty} \\eta+\\lambda_\\infty \\eta^3\\ge h^2\\left( \\lambda_\\infty h^4-\\frac{64r^2}{R^6}h \\right)=h^3\\left( \\lambda_\\infty h^3-\\frac{64r^2}{R^6} \\right)\\ge h^3\\left( \\lambda_\\infty (1-\\theta^2)^3-\\frac{64\\theta^2}{R^4} \\right)=0.\n\\end{eqnarray*}\nFrom (\\ref{2.62}), if $\\theta R\\le r\\le R$ then\n\\begin{eqnarray*}\n\\D_{\\infty} \\eta+\\lambda_\\infty \\eta^3\\ge 8h^2\\left\\{ \\left(\\frac{2r}{R^2}\\right)^4-\\frac{8r^2}{R^6}h\\right\\}\n\\ge \\frac{64h^2r^2}{R^6} \\left(\\frac{2r^2}{R^2}-1\\right)\\ge \\frac{64h^2r^2}{R^6} \n\\left(2\\theta^2-1\\right)=0.\n\\end{eqnarray*}\nThe claim holds by an application of Lemma \\ref{2.5}.\n\\end{proof}\n\nWe record a calculation we use in the various auxiliary functions we employ in our work.\n\n\\begin{rem}\\label{2.70} Let $f(t)\\in C^1,$ in $t\\ge 0,$ and $f(t)\\ge 0$. Set $r=|x|$ and \n$$u(x,t)=\\pm f(t) r^{p\/(p-1)},\\;\\;2\\le p<\\infty,\\;\\;\\mbox{and}\\;\\;u(x,t)=f(t) r^{4\/3},\\;\\;p=\\infty.$$\nCall $A=n(p\/(p-1))^{p-1}$ and $B=(p\/(p-1))^p.$\nWe show that in $r\\ge 0$, \n$$G_pu=\\left\\{ \\begin{array}{lcr} \\pm \\left\\{Af^{p-1}+(p-1)Bf^pr^{p\/(p-1)}-(p-1)r^{p\/(p-1)}f^{\\prime}\\right\\},& 2\\le p<\\infty,\\\\ \\pm\\left\\{(4^3\/3^4)f^3+f^4 r^{4\/3}-3f^{\\prime}r^{4\/3}\\right\\},& p=\\infty.\n\\end{array}\\right. $$\nWe prove the above for the $+$ case. The $-$ case can be shown similarly. Using (\\ref{2.50}) the above holds in $r>0$ and $u\\in C^2$ for $p=2$. We check at $r=0$ and for $2<p\\le \\infty$.\n\nSuppose that $\\psi$, $C^1$ in $t$ and $C^2$ in $x$, is such that $u-\\psi\\le u(o,s)-\\psi(o,s)$, for $(x,t)$ near $(o,s)$. Thus,\n$u(x,t)\\le \\langle D\\psi(o,s), x\\rangle+\\psi_t(o,s)(t-s)+o(|x|+|t-s|),$ as $(x,t)\\rightarrow (o,s)$. Clearly, $\\psi_t(o,s)=0$ and $D\\psi(o,s)=0$. Using the expansion\n$$u(x,t)=f(t)|x|^{p\/(p-1)}\\le \\langle D^2\\psi(o,s)x, x\\rangle\/2+\\psi_t(o,s)(t-s)+o(|x|^2+|t-s|),$$\nas $(x,t)\\rightarrow (o,s)$, we see that $D^2\\psi(o,s)$ does not exist. Hence, $u$ is a sub-solution. \n\nNext, let $\\psi$, $C^1$ in $t$ and $C^2$ in $x$, is such that $u-\\psi\\ge u(o,s)-\\psi(o,s)$, for $(x,t)$ near $(o,s)$. Thus,\n$u(x,t)\\ge \\langle D\\psi(o,s), x\\rangle+\\psi_t(o,s)(t-s)+o(|x|+|t-s|),$ as $(x,t)\\rightarrow (o,s)$. Clearly, $Du(o,s)=0$, $\\psi_t(o,s)=0$ and $G_p\\psi(o,s)=0$. \nHence, $u$ is a super-solution. A similar argument works for $G_\\infty$. $\\Box$\n\\end{rem}\n \nThe next is an auxiliary function which is employed in the proof of Theorem \\ref{1.7}.\n\n\\begin{lem}\\label{2.7} Let $T>0$ and $2\\le p\\le \\infty$. Set $r=|x|$ and for any fixed $\\alpha>0$, define $\\forall x\\in \\mathbb{R}^n$ and any $0\\le t\\le T$, \n\\begin{eqnarray*}\n&&\\phi_p(x,t)=\\exp\\left( a\\left[ (t+1)^{\\alpha (p-1)+1}-1\\right]+ b (t+1)^\\alpha r^{p\/(p-1)} \\right),\\;\\;2\\le p<\\infty,\\\\\n&&\\phi_\\infty(x,t)=\\exp\\left( a [ (t+1)^{3\\alpha+1}-1]+ b (t+1)^{\\alpha}r^{4\/3} \\right),\\;\\;\\;\\;p=\\infty.\n\\end{eqnarray*}\n\n(i) For $2\\le p<\\infty$, take $a$ and $b$ such that\n$$a= \\frac{n p^{p-1}b^{p-1}}{(p-1)^p\\{1+ \\alpha (p-1)\\}}\\;\\;\\mbox{and}\\;\\;0<b^{p-1} \\left( \\frac{p}{p-1}\\right)^p(T+1)^{\\alpha (p-1)+1}<\\alpha.$$\n\n(ii) For $p=\\infty$, take $a$ and $b$ such that\n$$a=\\frac{4^3b^3}{3^5(3\\alpha+1)}\\;\\;\\mbox{and}\\;\\; 0<b^3\\left(\\frac{4^4}{3^5}\\right)(T+1)^{3\\alpha+1}<\\alpha.$$\n\nThen for $2\\le p\\le \\infty$, $\\phi_p(0,0)=1$ and $\\Gamma_p \\phi_p\\le 0$, in $\\mathbb{R}^n\\times (0,T)$.\n\\end{lem}\n\\begin{proof} We set $v=\\log \\phi_p$ and use Lemma \\ref{2.3} and Remark \\ref{2.70} to show that\n$G_pv\\le 0$, in $\\mathbb{R}^n\\times(0,T).$ \n\n(i) Let $2\\le p<\\infty$. Then\n$$v=\\log\\phi_p= a\\left[ (t+1)^{\\alpha (p-1)+1}-1\\right]+ b (t+1)^\\alpha r^{p\/(p-1)}.$$\nA simple calculation shows that $\\D_p r^{p\/(p-1)}= n[ p\/(p-1)]^{p-1},\\;0\\le r<\\infty,\\;2<p<\\infty$ and  $\\Delta r^2=2n,\\;0\\le r<\\infty$, see Remark \\ref{2.70}.\n\nUsing the above, calculating in $0< r<\\infty$ and $0< t< T$, and using the definitions of $a$ and $b$, we see that\n\\begin{eqnarray*}\n&&\\Delta_pv+(p-1)|Dv|^p-(p-1)v_t\\\\\n&&=n\\left(\\frac{p}{p-1}\\right)^{p-1}b^{p-1}(t+1)^{\\alpha(p-1)} +(p-1) \\left(\\frac{p}{p-1}\\right)^p b^p (t+1)^{\\alpha p} r^{p\/(p-1)}\\\\\n&&\\qquad\\qquad\\qquad\\qquad\\qquad -(p-1) \\left\\{ a (\\alpha(p-1)+1) (t+1)^{\\alpha(p-1)}+ \\alpha b(t+1)^{\\alpha-1}  r^{p\/(p-1)} \\right\\}\\\\\n&&=\\left[ n\\left( \\frac{p }{p-1}\\right)^{p-1}b^{p-1} -a(p-1)(\\alpha(p-1)+1)\\right] (t+1)^{\\alpha(p-1)}\\\\\n&&\\qquad\\qquad\\qquad\\qquad\\qquad+ (p-1) r^{p\/(p-1)}\\left[ \\left(\\frac{p}{p-1}\\right)^pb^p(t+1)^{\\alpha p}-\\alpha b (t+1)^{\\alpha-1} \\right]\\\\\n&&=b(p-1)r^{p\/(p-1)}(t+1)^{\\alpha-1} \\left(\\left(\\frac{p}{p-1}\\right)^p b^{p-1}(t+1)^{\\alpha (p-1)+1}-\\alpha\\right)\\le 0.\n\\end{eqnarray*}\nThus, $\\phi$ is a super-solution in $0\\le r<\\infty$ and $0<t<T$. \n\nWe now show part (ii). Set\n$$v=\\log\\phi_\\infty= a [ (t+1)^{3\\alpha+1}-1]+ b (t+1)^{\\alpha}r^{4\/3}.$$\nNoting that $\\D_{\\infty} r^{4\/3}=4^3\/3^4$, in $0\\le r<\\infty$, and calculating,\n\\begin{eqnarray*}\n\\D_{\\infty} v+|Dv|^4-3v_t&=&\\frac{4^3}{3^4}b^3(t+1)^{3\\alpha} +b^4 \\left(\\frac{4}{3}\\right)^4(t+1)^{4\\alpha}r^{4\/3}\\\\\n&&\\qquad\\qquad\\qquad\\quad\\qquad\\qquad -3\\left\\{a(3\\alpha+1) (t+1)^{3\\alpha}+b \\alpha r^{4\/3}(t+1)^{\\alpha-1} \\right\\}\\\\\n&&\\le (t+1)^{3\\alpha} \\left( \\frac{4^3b^3}{3^4}-3a(3\\alpha+1)  \\right)\\\\\n&&\\qquad\\qquad\\qquad\\qquad+b r^{4\/3}(t+1)^{\\alpha-1}\\left( b^3\\left( \\frac{4}{3}\\right)^4(T+1)^{3\\alpha+1}-3\\alpha\\right)\\le 0,\n\\end{eqnarray*}\nwhere we have used the definitions of $a$ and $b$. Rest of the proof is similar to part (i).\n\\end{proof}\n\nWe now extend existence results in \\cite{BM1, BM2} to cylindrical domains $\\Omega\\times (0,\\infty)$. Set $\\Omega_\\infty=\\Omega\\times(0,\\infty)$ and $P_\\infty$ its parabolic boundary.\n \n\\begin{lem}\\label{2.8} Let $\\Omega\\subset \\mathbb{R}^n$ be bounded and $h\\in C(P_\\infty)$ with \n$0<\\inf_{P_\\infty}h\\le \\sup_{P_\\infty}h<\\infty$. Suppose that, for any $T>0$, \n$\\Gamma_p v=0,\\;\\mbox{in $\\Omega_T$ and $v=h$ on $P_T$,}\\;2\\le p\\le \\infty,$\nhas a unique positive solution. Then the problem\n\\eqRef{2.80}\n\\Gamma_p v=0,\\;\\;\\mbox{in $\\Omega{ _\\infty}$ and $v=h$ on $P_\\infty$,} \n\\end{equation}\nhas a unique positive solution $u\\in C(\\Omega_\\infty\\times P_\\infty).$ Moreover, \n$\\inf_{P_\\infty} h\\le u \\le \\sup_{P_\\infty}h.$\n\nIn particular, existence holds for any $\\Omega$, if $p>n$, and for any $\\Omega$ satisfying a uniform outer ball condition, if $2\\le p\\le n.$\n\\end{lem}\n\\begin{proof}\nFor any $T>0$, call $u_T$ to be the unique positive solution of $\\Delta_p u_T-(p-1) u_T^{p-2} (u_T)_t=0,\\;\\mbox{in $\\Omega_T$, $u_T=h$ on $P_T$.}$\n\nBy Theorem \\ref{2.2}, $u_{T_1}=u_{T}$ in $\\Omega_{T_1}$, for any $T>T_1>0$. Define\n$u=\\lim_{T\\rightarrow \\infty} u_T.$\nHence, $u$ solves the problem in $\\Omega_\\infty$. To show uniqueness, if $v$ is any other positive solution, then\n$v=u_T=u$ in $\\Omega_T$, by using Theorem \\ref{2.2}. The maximum principle in Lemma \\ref{2.1} shows that $\\inf_{P_\\infty}h\\le \\inf_{P_T}h\\le u_T\\le \\sup_{P_T}\\le \\sup_{P_\\infty} h.$\n\\end{proof}\n\n\\begin{rem}\\label{2.9} We record the following kernel functions of $\\Gamma_p$, for $2\\le p\\le \\infty$. Define the functions $K_p$, in $\\mathbb{R}^n\\times(0,\\infty)$, as follows.\n\\begin{eqnarray*}\n&&K_p(x,t)=t^{-n\/p(p-1)}\\exp\\left(- \\left(\\frac{p-1}{p^{p\/(p-1)}}\\right)\\left(\\frac{|x|^{p}}{t}\\right)^{1\/(p-1)} \\right),\\;\\;\\;2\\le p<\\infty,\\\\\n&&K_\\infty(x,t)=t^{-1\/12} \\exp\\left( - \\left(\\frac{3}{4} \\right)^{4\/3}\\left( \\frac{r^4}{t} \\right)^{1\/3} \\right),\\;\\;\\;\\;\\;p=\\infty.\n\\end{eqnarray*}\nFor $p=2$, $K_2(x,t)=t^{-n\/2}\\exp( -|x|^2\/(4t))$ is the well-known heat kernel for the heat equation.\nAlso,\n$$K_p(x,0^+)=0,\\;x\\ne o,\\;\\;\\lim_{t\\rightarrow 0^+}K_p(0, t)=\\infty,\\;\\;\\mbox{and}\\;\\;\\lim_{|x|+t\\rightarrow \\infty} K_p(x,t)=0.$$ \nWe omit the proof that $\\Gamma_p K_p=0$. $\\Box$\n\\end{rem}\n\n\\section{Proofs of Theorems \\ref{1.6} and \\ref{1.61}}\n\n{\\bf Proof of Theorem \\ref{1.6}.}  The proof is a some what simplified version of the one in \\cite{BM1} and we use Remark \\ref{2.20}. Let $\\lambda_{\\Om}$ be the first eigenvalue of\n$\\D_p$ on $\\Omega$, see Section 5.\n\nDefine $M=\\sup_{P_\\infty} h$. By Lemma \\ref{2.8}, we obtain $u\\ge 0$ and $\\sup_{\\Omega_\\infty}u\\le M.$\nFor every $t>0$, set  \n\\eqRef{3.1}\n\\mu(t)=\\sup_{\\overline{\\Omega}} u(x,t)\\;\\;\\;\\mbox{and}\\;\\;\\;\\nu(t)=\\sup_{\\partial\\Omega} g(x,t).\n\\end{equation}\n\n{\\bf Part (i).}  We observe by Remarks \\ref{6.13} and {\\ref{6.15} that \nfor a fixed $0<\\lambda<\\lambda_\\Omega$, one can find a solution $\\psi_\\lambda\\in C(\\overline{\\Omega}),\\;\\psi_\\lambda>0,$ such that \n\\eqRef{3.3}\n\\D_p \\psi_\\lambda+\\lambda \\psi_\\lambda^{p-1}=0,\\;\\;\\mbox{in $\\Omega$, and $\\psi_\\lambda=M$, on $\\partial\\Omega$.}\n\\end{equation}\nBy Remark \\ref{6.10},\n\\eqRef{3.4}\n\\psi_\\lambda>M,\\;\\;\\mbox{in $\\Omega$, and}\\;\\;\\psi_\\lambda(x)\\ge u(x,T_1),\\;\\forall x\\in \\overline{\\Omega}.\n\\end{equation} \n\nWe now construct an auxiliary function for the proof. Let $0<S<T<\\infty$; define\n\\begin{eqnarray}\\label{3.40}\n\\beta(t,T)=\\exp\\left( \\frac{\\lambda (T-t)}{p-1} \\right),\\;\\;\\forall\\;S\\le t\\le T.\n\\end{eqnarray}\nIn the rest of Part (i), we always choose $S$ and $T$ such that $\\beta(S,T)\\ge 2.$ Next, define the function\n\\eqRef{3.5}\nF(t; S, T)=\\frac{1}{2}\\left[1+ \\frac{\\beta(t,T)-1 }{ \\beta(S,T)-1 }\\right]=\\frac{1}{2}\\left[\\frac{\\beta(S,T)-2}{\\beta(S,T)-1}+ \\frac{\\beta(t,T)}{ \\beta(S,T)-1}\\right],\\;\\;\\forall t\\in [S, T].\n\\end{equation}\nUsing (\\ref{3.40}) and (\\ref{3.5}), we get $\\forall t\\in [S,T]$,\n\\eqRef{3.6}\nF_t=-\\frac{\\lambda \\beta(t,T)}{2(p-1) (\\beta(S,T)-1)},\\;\\;F(S; S,T)=1,\\;\\;F(T;S,T)=\\frac{1}{2},\\;\\;\\mbox{and}\\;\\;\\frac{1}{2}\\le F(t; S,T)\\le 1.\n\\end{equation}\nLet $\\phi=\\psi_\\lambda(x) F(t; S, T),$ $\\forall(x,t)\\in \\Omega\\times(S,T)$, where $\\psi_\\lambda$ is as in (\\ref{3.3}). Using Lemma \\ref{2.5}, (\\ref{3.5}) and (\\ref{3.6}), we get\n\\begin{eqnarray}\\label{3.7}\n\\Gamma_p\\phi&=&F^{p-1}\\D_p \\psi_\\lambda-(p-1)\\psi_\\lambda^{p-1}F^{p-2} F_t=-\\lambda F^{p-2} \\psi_\\lambda^{p-1}\n\\left(F{+}\\frac{p-1}{\\lambda}F_t\\right)\\nonumber\\\\\n&=&-\\lambda F^{p-2} \\psi_\\lambda^{p-1}\\left\\{ F-  \\frac{1}{2} \n\\left(  \\frac{\\beta(t,T) }{ \\beta(S,T)-1  } \\right) \\right\\}\\nonumber\\\\\n&=&-\\frac{\\lambda \\psi_\\lambda^{p-1} F(t)^{p-2}}{2}  \\left( \\frac{\\beta(S,T)-2 }{ \\beta(S,T)-1  } \\right)\\le 0, \\;\\;\\;\\;S<t< T,\n\\end{eqnarray}\nwhere we have used that $\\beta(S,T)\\ge 2.$\n\nRecalling (\\ref{3.1}) and the hypothesis of the theorem, there are $1<T_1<\\cdots<T_m<\\cdots<\\infty$ such that for $m=1,2,\\cdots,$ \n\\begin{eqnarray}\\label{3.8}\n(a)\\;\\;\\beta(T_m, T_{m+1})\\ge 2,\\;\\;\\mbox{and}\\;\\;(b)\\;\\;0\\le \\nu(t)\\le \\frac{M}{2^{m}},\\;\\forall t\\ge T_m,\n\\end{eqnarray}\nsee (\\ref{3.40}). Note that (\\ref{3.8}) (a) implies that $\\lim_{m\\rightarrow \\infty}T_m =\\infty$.\nFor each $m=1,2,\\cdots,$ set \n\\begin{eqnarray}\\label{3.9}\n&&\\mbox{$I_m=\\Omega\\times(T_m, T_{m+1})$, $J_m$ the parabolic boundary of $I_m$, }  \\\\\n&&\\eta_m(t)= F(t; T_m, T_{m+1}) ,\\;\\forall t\\in [T_m,\\;T_{m+1}], \\;\\;\\mbox{and}\\;\\;\\phi_m(x,t)=\\frac{\\psi_\\lambda(x) \\eta_m(t)}{2^{m-1}},\\;\\forall(x,t)\\in \\overline{I}_m,\\nonumber\n\\end{eqnarray}\nsee (\\ref{3.5}). \n\nTaking $m=1$, $\\phi_1(x,t)=\\psi_\\lambda \\eta_1(t),$ using (\\ref{3.3}), (\\ref{3.4}), (\\ref{3.6}) and (\\ref{3.9}),  \n\\eqRef{3.10}\n\\phi_1(x, T_1)\\ge M,\\;\\forall x\\in \\overline{\\Omega}, \\;\\;\\mbox{and}\\;\\;\\;\\frac{M}{2}\\le \\phi_1(x,t)\\le M,\\;\\;\\forall(x,t)\\in \\partial\\Omega\\times [T_1,T_2].\n\\end{equation}\nAlso, by (\\ref{3.4}) and (\\ref{3.8})(b),\n\\eqRef{3.101}\nu(x,T_1)\\le \\phi_1(x,T_1),\\;\\forall x\\in \\overline{\\Omega},\\;\\;\\mbox{and}\\;\\;u(x,t)\\le \\nu(t)\\le \\frac{M}{2},\\;\\forall(x,t)\\in \\partial\\Omega\\times[T_1, \\infty).\n\\end{equation}\nThus, $u\\le \\phi_1$, on $J_1$, and as $\\Gamma_p \\phi_1\\le 0$, in $I_1$, (see (\\ref{3.7})), Theorem \\ref{2.2} and Remark \\ref{2.20} imply that $u\\le \\phi_1(x,t)$ in $I_1.$ \nWe claim that $u\\le \\phi_1(x,t)$, in $\\overline{I}_1$ ($u$ is upper semi-continuous). Take $\\hat{T}_2>T_2$ and near $T_2$. The function\n$\\hat{\\phi}_1(x,t)=\\psi_\\lambda(x) F(t; T_1, \\hat{T}_2)$ (see (\\ref{3.6}) and (\\ref{3.9})) satisfies the conclusions in (\\ref{3.10}) and (\\ref{3.101}) if we replace $\\phi_1$ by $\\hat{\\phi}_1$.\nThus, $u\\le \\hat{\\phi}_1$, in $\\Omega\\times(T_1,\\hat{T}_2)$ and the conclusion that $u\\le \\phi_1$, in $\\overline{I}_1$, now follows by letting $\\hat{T}_2\\rightarrow T_2$. \nClearly,\n\\eqRef{3.102}\nu(x,T_2)\\le \\phi_1(x,T_2)=\\psi_\\lambda(x)\\eta_1(T_2)=\\frac{\\psi_\\lambda(x)}{2},\\;\\forall x\\in \\overline{\\Omega},\n\\end{equation}\nwhere we have used (\\ref{3.6}). Moreover, since $F$ is deceasing in $t$ (see (\\ref{3.6})), recalling (\\ref{3.1}), we have\n$$\\mu(t)\\le \\sup_{\\Omega}\\psi_\\lambda,\\;\\forall t\\in [T_1,\\;T_2],\\;\\;\\mbox{and}\\;\\;\\mu(T_2)\\le \\frac{\\sup_{\\Omega} \\psi_\\lambda}{2}.$$ \n\nWe now use induction and suppose that for some $m=1,2\\cdots$, \n\\eqRef{3.11}\nu(x,T_m)\\le \\frac{\\psi_\\lambda(x)}{2^{m-1}},\\;\\;\\forall x\\in \\overline{\\Omega},\n\\end{equation}\n(note that (\\ref{3.11}) holds for $m=1,2$, see (\\ref{3.4}) and (\\ref{3.102})). We will prove that\n\\eqRef{3.12}\nu(x,t)\\le \\frac{\\psi_\\lambda(x)}{2^{m-1}},\\;\\forall(x,t)\\in \\overline{I}_m, \\;\\;\\;\\mbox{and}\\;\\;\\;\\mu(T_{m+1})\\le \\frac{\\psi_\\lambda(x) } {2^m}.\n\\end{equation}\nthus proving part (i) of the theorem. \n\nBy (\\ref{3.7}) and (\\ref{3.9}), \n$\\Gamma_p\\phi_m\\le 0, \\;\\mbox{in $I_m$}.$ By (\\ref{3.6}), (\\ref{3.9}) and (\\ref{3.11}), \n\\begin{eqnarray*}\n&&u(x,T_m)\\le \\frac{\\psi_\\lambda(x)\\eta_m(T_m)}{2^{m-1}}=\\phi_m(x,T_m),\\;\\forall x\\in \\overline{\\Omega},\\;\\;\\mbox{and}\\\\\n&&0\\le g(x,t)\\le \\frac{M}{2^{m}}\\le \\phi_m(x,t),\\;\\forall(x,t)\\in \\partial\\Omega\\times[T_m, T_{m+1}).\n\\end{eqnarray*}\nThus, $\\phi_m\\ge u$ on $J_m$. Using Theorem \\ref{2.2} and Remark \\ref{2.20}, $u\\le \\phi_m$ in \n$\\overline{I}_m$, and using (\\ref{3.6}) \n$$u(x,t)\\le \\frac{\\psi_\\lambda(x) \\eta_m(t)}{2^{m-1}}\\le \\frac{\\psi_\\lambda(x)}{2^{m-1}},\\;\\forall(x,t)\\in \\overline{I}_m,\\;\\;\\mbox{and}\\;\\;u(x,T_{m+1})\\le \\frac{\\psi_\\lambda(x)}{2^m},\\;\\forall x\\in \\overline{\\Omega}.$$\nThus, (\\ref{3.12}) holds and part (i) is proven.\n\n{\\bf Part (ii).} Let $g(x,t)=0$, $\\forall(x,t)\\in \\partial\\Omega\\times[T_0,\\infty)$, for some $T_0>0$. \nWe make some elementary observations. From (\\ref{3.1}) one sees that \n$$M=\\sup_{P_\\infty} h=\\max( \\sup_{\\overline{\\Omega}} f, \\; \\sup_{\\partial\\Omega\\times[0,T_0]} g(x,t)).$$\nLemma \\ref{2.1} implies that $0\\le u(x,t)\\le M,\\;\\forall (x,t)\\in \\Omega_{T},$ for any $T>0$. \n\nWe claim that\n$\\mu(t)$ is decreasing in $[T_0,\\infty)$. Let $T_0\\le T<\\hat{T}<\\infty$. Since $g=0$ on\n$\\partial\\Omega\\times[T,\\hat{T})$, by Lemma \\ref{2.1}, $\\sup_{\\Omega\\times(T,\\hat{T})}u\\le \\mu(T)$. Since $u\\in usc(\\Omega_\\infty\\cup P_\\infty),\\;u\\ge 0$, it follows\nthat $\\mu(t)\\le \\mu(T),\\;T<t<\\hat{T}$. Combining this with Part (i), we obtain\n\\eqRef{3.13}\n\\mbox{$\\mu(t)$ is decreasing, in $t\\ge T_0$, and $\\lim_{t\\rightarrow \\infty} \\left(\\sup_{\\overline{\\Omega}}u(x,t)\\right)=0.$}\n\\end{equation}\n\nNext, let $T>T_0$ be large enough so that $\\mu(T)>0$ and small (if $\\mu(T)=0$ Part (ii) holds by Lemma \\ref{2.1} and (\\ref{3.13})).\nBy Remarks \\ref{6.13} and \\ref{6.15}, for any $0<\\lambda<\\lambda_\\Omega$, there is a $\\psi_\\lambda \\in C(\\overline{\\Omega}),\\;\\psi_\\lambda>0,$ that solves\n$$\\D_p \\psi_\\lambda+\\lambda |\\psi_\\lambda|^{p-2}\\psi_\\lambda=0,\\;\\;\\mbox{in $\\Omega$, with $\\psi_\\lambda=\\mu(T)$ on $\\partial\\Omega$.}\n$$\nBy Remark \\ref{6.10}, $\\psi_\\lambda\\ge \\mu(T)$, in $\\overline{\\Omega},$ and $\\psi_\\lambda(x)\\ge u(x,T),\\;\\forall x\\in \\overline{\\Omega}.$\n\nCall $D_T=\\Omega\\times(T,\\infty)$ and $Q_T$ its parabolic boundary. We fix $\\lambda<\\lambda_{\\Om}$, close to $\\lambda_{\\Om}$, in what  follows. Define\n$$L(x,t)=\\psi_\\lambda(x) \\exp\\left(-\\lambda (t-T)\/(p-1)\\right),\\;\\;\\mbox{in $\\overline{D}_T$,}$$\nand note that\n$$L(x,T)\\ge u(x,T),\\;\\forall x\\in \\overline{\\Omega},\\;\\; L(x,t)>0,\\;\\forall(x,t)\\in \\partial\\Omega\\times[T,\\infty).$$ \nBy Lemma \\ref{2.5}, $\\Gamma_p L=0$, in $D_T$. Since $L\\ge u$ on $Q_T$, Theorem \\ref{2.2} and Remark \\ref{2.20}\nimplies that $L\\ge u,$ $\\overline{D}_T,$ and \n$$\\lim_{t\\rightarrow \\infty} \\frac{\\log(\\sup_\\Omega u(x,t)  )}{t}\\le  \\lim_{t\\rightarrow \\infty} \\frac{\\log(\\sup_\\Omega L(x,t)  )}{t}=  -\\frac{\\lambda}{p-1}.$$\nChoosing $\\lambda$ arbitrarily close to $\\lambda_\\Omega$, we conclude\n$$\\lim_{t\\rightarrow \\infty} \\frac{\\log(\\sup_\\Omega u(x,t)  ) }{t}\\le -\\frac{\\lambda_{\\Om}}{p-1}.\\;\\;\\;\\;\\;\\;\\;\\Box$$\n\n\\begin{rem}\\label{3.14} Let $2\\le p\\le \\infty$.\nThe requirement that $u$ be a sub-solution is necessary in Theorem \\ref{1.6}. To see this, we take\n$\\Omega=B_R(o)$, and\n$$\\psi(x,t)=R^2-r^2,\\;\\;(x,t)\\in \\overline{B}_R(o)\\times(0,\\infty).$$\nOne sees that $\\Gamma_p \\psi<0$, except perhaps at $r=0$. Clearly, $\\psi$ does not decay in $t$. $\\Box$\n\\end{rem}\n\n\\begin{rem}\\label{3.15}  The decay rate in Part (i) of Theorem \\ref{1.6} may depend on the decay rate along $\\partial\\Omega\\times(0,\\infty)$. Take $\\ell(t):\\mathbb{R}^+\\rightarrow \\mathbb{R}^+\\cup\\{0\\}$, \n $C^2$ in $t$ and decreasing to $0$, as $t\\rightarrow \\infty$. \n \nPart (ii) shows that if $u=0$ on $\\partial\\Omega\\times(0,\\infty)$ then\nthe slowest rate of decay is $e^{-\\lambda_{\\Om} t\/(p-1)}$. \nLet $\\Omega=B_R(o)$; set $u(x,t)=\\psi(x)\\exp(-\\lambda_{\\Om} t\/(p-1))$, where $\\psi>0$ is a first eigenfunction of $\\D_p$ on $B_R(o)$, see Remark \\ref{6.19}. By Lemma \\ref{2.5}, $\\Gamma_pu=0$, in $B_R(o)\\times (0,\\infty)$. The decay rate in Theorem \\ref{1.6} is attained.\n\\quad $\\Box$\n\\end{rem}\n\nBefore presenting the proof of Theorem \\ref{1.61} we make a remark. \n\n\\begin{rem}\\label{3.17}  Let $0\\le S<T$ and $O=\\Omega\\times(S,T)$.\nWe look at three possibilities. Let $u\\in C(\\Omega_\\infty),\\;u>0$ solves\n$\\Gamma_pu=0,\\;u(x,0)=f(x),\\;\\forall x\\in \\overline{\\Omega},$ and $g(x,t)=1,\\;\\forall(x,t)\\in \\partial\\Omega\\times[0,\\infty).$ Set $\\mu(t)=\\sup_{\\overline{\\Omega}} u(x,t)$ and $m(t)=\\inf_{\\overline{\\Omega}} u(x,t)$. We apply Lemma \\ref{2.1}\n\n{\\bf (a) $\\inf_\\Omega f=1$:} For every $t>0$, $m(t)=1$ and $1\\le \\mu(t)\\le \\sup_\\Omega f$. Then $u(x,t)\\le \\mu(S)$, in $O$, and $\\mu(T)\\le \\mu(S)$. Hence, $\\mu(t)$ is decreasing in $t$.\n\n{\\bf (b) $\\sup_\\Omega f= 1$:} Clearly, $\\mu(t)=1$ and $m(t)\\le 1$, for every $t>0$. \nClearly, $m(t)$ is increasing in $t$, since $u(x,t)\\ge m(S)$, in $O$, and $m(T)\\ge m(S)$. \n\n{\\bf (c) $\\inf_\\Omega f\\le 1\\le \\sup_\\Omega f$:} Then $m(t)\\le 1\\le \\mu(t),\\;\\forall t>0$. Arguing as in (a) and (b) we see that $m(t)$ is increasing and $\\mu(t)$ is decreasing in $t$. \\qquad $\\Box$\n\\end{rem}\n\n{\\bf Proof of Theorem \\ref{1.61}.} Let $u>0$ be a solution, as stated in Theorem \\ref{1.61}. We assume that\n$0<\\inf_\\Omega f< 1< \\sup_\\Omega f,$ and set\n\\eqRef{3.18} \nm=\\inf_\\Omega f\\;\\;\\mbox{and}\\;\\;M=\\sup_\\Omega f.\n\\end{equation}\nLet $B_R(z)$ be the out-ball of $\\Omega$, where $z\\in \\mathbb{R}^n$; define $r=|x-z|$. Part (i) addresses the case $2\\le p<\\infty$, and Part (ii) discusses $p=\\infty$. Recall\nRemark \\ref{2.70}.\n\n{\\bf Part (i): $2\\le p<\\infty$.} Call\n\\eqRef{3.19}\nA=A(p,n)=n\\left(p\/(p-1)\\right)^{p-1},\\;\\; B=(p-1)\\left(p\/(p-1)\\right)^p R^{p\/(p-1)}.\n\\end{equation}\n\n{\\bf Upper Bound.} Let $T_0>0$, to be determined later. Recalling (\\ref{3.19}), take \n\\eqRef{3.20}\n\\phi(x,t)=\\exp\\left[ a \\left( \\frac{R^{p\/(p-1)}-r^{p\/(p-1)}+b}{(1+t)^{\\alpha}} \\right) \\right],\\;\\;\\;0\\le r\\le R,\n\\end{equation}\nwhere \n\\begin{eqnarray}\\label{3.201}\n&& (i)\\;\\;0<\\alpha\\le \\frac{1}{p-2},\\;\\;\\mbox{for $2<p<\\infty$, and}\\;\\;(ii)\\;\\;0<\\alpha<\\infty,\\;\\;\\mbox{for $p=2$,}\\nonumber\\\\\n&& ab=(1+T_0)^{\\alpha}\\log M\\;\\;\\;\\mbox{and}\\;\\;\\;a=\\frac{A(p-1) (1+T_0)^{\\alpha}}{pB}.\n\\end{eqnarray}\nTo make the calculations easier, we use Lemma \\ref{2.3} and work with $v=\\log \\phi$. \nRecalling that $G_p v=\\Delta_p v+(p-1) |Dv|^p-(p-1) v_t ,$ using (\\ref{3.19}), the value of $ab$ (see (\\ref{3.201})) and setting $C=\\alpha(p-1)$, we get in $0\\le r\\le R,$ and $t>0$,\n\\begin{eqnarray}\\label{3.21}\nG_p v&\\le&-\\frac{A a^{p-1}}{(1+t)^{\\alpha(p-1)}}+\\frac{B  a^p}{(1+t)^{\\alpha p }} + \\frac{\\alpha(p-1)a(R^{p\/(p-1)}-r^{p\/(p-1)}+b)}{(1+t)^{\\alpha+1}}  \\nonumber\\\\\n&\\le &  C\\left(\\frac{aR^{p\/(p-1)}+(1+T_0)^{\\alpha}\\log M}{(1+t)^{\\alpha+1}}\\right) +\\frac{B a^p}{(1+t)^{\\alpha p}}\n-\\frac{A a^{p-1}}{(1+t)^{\\alpha(p-1)}}\\nonumber\\\\\n&=&\\frac{1}{(1+t)^{\\alpha(p-1)}}\\left[ C\\left(\\frac{aR^{p\/(p-1)}+(1+T_0)^{\\alpha}\\log M}{(1+t)^{1-\\alpha(p-2)}} \\right)+a^{p-1}\\left( \\frac{B  a}{(1+t)^{\\alpha}}\n-A \\right) \\right].\n\\end{eqnarray}\nUsing (\\ref{3.201}) and calling $K=K(\\alpha,p,n,R)>0$ (see below), we calculate in $t\\ge T_0$,\n\\begin{eqnarray*}\na^{p-1}\\left( \\frac{B a}{(1+t)^{\\alpha}}-A \\right)&=&a^{p-1}A \\left( \\frac{p-1}{p} \\left(\\frac{1+T_0}{1+t}\\right)^{\\alpha}\n-1\\right)\n\\le a^{p-1} A\\left( \\frac{p-1}{p}-1\\right)\\\\\n&=&-\\frac{a^{p-1}A}{p}=-K(1+T_0)^{\\alpha(p-1)}.\n\\end{eqnarray*}\nUsing the above in (\\ref{3.21}) together with the value of $a$ in (\\ref{3.201}), we obtain in $t\\ge T_0$, \n\\begin{eqnarray}\\label{3.22}\nG_pv&\\le& \\frac{1}{(1+t)^{\\alpha(p-1)}}\\left[C\\left( \\frac{aR^{p\/(p-1)}+(1+T_0)^\\alpha\\log M  }{(1+t)^{1-\\alpha(p-2)}}\\right) -K(1+T_0)^{\\alpha(p-1)} \\right] \\nonumber\\\\\n&\\le&\\frac{1}{(1+t)^{\\alpha(p-1)}}\\left( \\frac{\\bar{K}(1+T_0)^{\\alpha}}{(1+T_0)^{1-\\alpha(p-2)}} -K(1+T_0)^{\\alpha(p-1)} \\right)\\nonumber\\\\\n&\\le& \\left(\\frac{1+T_0}{1+t}\\right)^{\\alpha(p-1)}\\left(\\frac{ \\bar{K}}{1+T_0}-K\\right),\n\\end{eqnarray}\nwhere $\\bar{K}=\\bar{K}(\\alpha,p,n,R, M)>0.$ Choose $T_0$, large enough, so that $G_p v\\le 0$, and \n$\\Gamma_p\\phi\\le 0$ in $\\Omega\\times(T_0,\\infty)$. Using (\\ref{3.20}) and (\\ref{3.201}), we see that\n$$\\inf_{\\Omega}\\phi(x,T_0)\\ge \\exp\\left( \\frac{ab}{(1+T_0)^\\alpha} \\right)=M,\\;\\;\\;\\mbox{and}\\;\\;\\;\\inf_{\\partial\\Omega}\\phi(x,t)\\ge 1,\\;\\;\\forall t>0.$$\nBy Theorem \\ref{2.2} (or Theorem \\ref{2.4}) we see that $u(x,t)\\le \\phi(x,t),\\;\\forall(x,t)\\in \\Omega\\times (T_0,\\infty)$, and \n\\eqRef{3.23}\n\\limsup_{t\\rightarrow \\infty} u(x,t)\\le \\lim_{t\\rightarrow \\infty} \\phi(x,t)=1,\\;\\;\\mbox{for any $x\\in \\Omega$.}\n\\end{equation}\n\n{\\bf Lower Bound.} Set in $B_R(z)\\times(0,\\infty)$,\n\\eqRef{3.24}\n\\varphi(x,t)=\\exp\\left[ -(1+T_1)^\\alpha \\left( \\frac{R^{p\/(p-1)}-r^{p\/(p-1)}-\\log m}{(1+t)^{\\alpha}} \\right) \\right],\n\\end{equation}\nwhere \n$$(i) \\;0<\\alpha\\le \\frac{1}{p-2},\\;\\;\\mbox{if $2<p<\\infty,$}\\;\\;\\;(ii)\\;0<\\alpha<\\infty,\\;\\;\\mbox{if $p=2$,}$$\nwhere $T_1>0$ is to be determined later. \nSet $w=\\log \\varphi$, we get in $0\\le r\\le R$ and $t\\ge T_1,$\n\\begin{eqnarray*}\nG_p w&\\ge& A\\left(\\frac{1+T_1}{1+t} \\right)^{\\alpha(p-1)} -\\frac{ \\alpha(p-1)(1+T_1)^\\alpha\\left( R^{p\/(p-1)}-\\log m\\right)}{(1+t)^{\\alpha+1}}\\\\\n&=&\\left(\\frac{1+T_1}{(1+t)}\\right)^{\\alpha(p-1)}\\left(A-\\frac{\\alpha(p-1) \\left( R^{p\/(p-1)}-\\log m\\right)}{ \n(1+T_1)^{\\alpha(p-2)} (1+t)^{1-\\alpha(p-2)}}\\right)\\\\\n&\\ge & \\left(\\frac{1+T_1}{1+t}\\right)^{\\alpha(p-1)}\\left( A-\\frac{K}{1+T_1}\\right),\n\\end{eqnarray*}\nwhere $K=K(\\alpha, p, R, m)$.\nThus, there is a $T_1=T_1(a,\\alpha, p, m, R)$ such that $\\Gamma_p \\varphi\\ge 0$, in $\\Omega\\times(T_1,\\infty)$. Next, we observe that\n$$0<\\varphi(x,T_1)\\le m,\\;\\forall x\\in \\Omega,\\;\\;\\;\\mbox{and}\\;\\;\\varphi(x,t)\\le 1,\\;\\forall(x,t)\\in \\partial\\Omega\\times[T_1,\\infty).$$\nClearly, $\\varphi\\le u$, in $P_\\infty$, and Theorem \\ref{2.2} implies that $\\varphi\\le u$, in $\\Omega \\times(T_1,\\infty)$, and\n\\eqRef{3.25}\n\\liminf_{t\\rightarrow \\infty}u(x,t)\\ge \\lim_{t\\rightarrow \\infty}\\varphi(x,t)=1, \\;\\forall x\\in \\Omega.\n\\end{equation}\nThus, (\\ref{3.23}) and (\\ref{3.25}) imply the claim.\n\n{\\bf Part(ii): $p=\\infty$.} The proof is similar to that in Part (i) and we provide the construction of a super-solution and a sub-solution. Set\n$$A= 4^3\/3^4\\;\\;\\;\\;\\mbox{and}\\;\\;\\;B=\\left(4\/3\\right)^4 R^{4\/3}.$$\n\n{\\bf Upper Bound:} Take \n$$\\phi(x,t)=\\exp\\left[ a \\left( \\frac{R^{4\/3}-r^{4\/3}+b}{(1+t)^{\\alpha}} \\right) \\right],\\;\\;\\;0\\le r\\le R,\\;\\forall t>0,$$\nwhere \n$$0<\\alpha\\le \\frac{1}{2},\\;\\;\\;ab=(1+T_0)^\\alpha \\log M,\\;\\;\\;\\mbox{and}\\;\\;\\;a=\\frac{3A(1+T_0)^\\alpha}{4B}.$$\nThe quantity $T_0>0$ is to be chosen later. \n\nAs done in Part(i), write $v=\\log \\phi$ and recall that $G_\\infty v=\\D_{\\infty} v+|Dv|^4-3v_t$. Using the values of $a$, $ab$ and calculating in $0\\le r\\le R,\\;t\\ge T_0$,\n\\begin{eqnarray*}\nG_\\infty v&=&-\\frac{a^3 A}{(1+t)^{3\\alpha}}+  \\left( \\frac{4}{3}\\right)^4  \\frac{a^4 r^{4\/3}}{(1+t)^{4\\alpha}}\n+\\frac{3\\alpha a(R^{4\/3}-r^{4\/3}+b)}{(1+t)^{\\alpha+1}}\\\\\n&\\le&\\frac{1}{(1+t)^{3\\alpha}}\\left(\\frac{3\\alpha a(R^{4\/3}+b)}{(1+t)^{1-2\\alpha}}+ \\frac{a^4 B}{(1+t)^{\\alpha}}-a^3 A\\right)\\\\\n&\\le&\\frac{1}{(1+t)^{3\\alpha}}\\left[ 3\\alpha\\left( \\frac{aR^{4\/3}+(1+T_0)^\\alpha\\log M}{(1+T_0)^{1-2\\alpha}}\\right)+ a^3\\left(\\frac{a B}{(1+T_0)^{\\alpha}}- A\\right)\\right]\\\\\n&=&\\frac{1}{(1+t)^{3\\alpha}}\\left[ \\frac{C(1+T_0)^\\alpha}{(1+T_0)^{1-2\\alpha}}-D(1+T_0)^{3\\alpha}\\right]=\\left(\\frac{1+T_0}{1+t}\\right)^{3\\alpha}\\left[ \\frac{C}{(1+T_0)}-D\\right].\n\\end{eqnarray*}\nHere the constants $C=C(\\alpha, R)>0$ and $D=D(\\alpha, R)>0$. We now choose $T_0$ large enough so that $\\Gamma_\\infty \\phi\\le 0$ in $\\Omega\\times[T_0,\\infty)$. Rest of the proof is similar to that in Part (i).\n\n{\\bf Lower Bound.} Set\n$$\n\\varphi(x,t)=\\exp\\left[ -a \\left( \\frac{R^{4\/3}-r^{4\/3}+b}{(1+t)^{\\alpha}} \\right) \\right],\\;\\;\\;0\\le r\\le R,\\;\\forall t>0,\n$$\nwhere $a>0$, $b>0$,\n$$ab=-\\log m,\\;\\;\\;\\mbox{and}\\;\\;\\;0<\\alpha\\le \\frac{1}{2},$$\nand $a$ is to be chosen later. Defining $w=\\log \\varphi$ and differentiating, in $t>0$,\n\\begin{eqnarray*}\nG_\\infty w&\\ge& \\frac{a^3 A}{(1+t)^{3\\alpha}}-\\frac{ 3\\alpha a\\left( R^{4\/3}+b\\right)}{(1+t)^{\\alpha+1}}=\\frac{1}{(1+t)^{3\\alpha}}\\left( a^3A- \\frac{3\\alpha\\left(aR^{4\/3}-\\log m\\right)}{(1+t)^{1-2\\alpha}} \\right)\\ge 0,\n\\end{eqnarray*}\nif $a>0$ is chosen large enough. Rest of the proof is similar to that in Part (i).\\qquad\\qquad $\\Box$\n\n\\section{Proof of Theorem \\ref{1.7}}\n\nWe make use of Lemmas \\ref{2.6}, \\ref{2.7} and Remark \\ref{2.70} to prove the theorem. \nLet $T>0;$ set $H=\\mathbb{R}^n,\\;n\\ge 2,$ and $H_T=\\mathbb{R}^n\\times(0,T)$. For some $2\\le p\\le \\infty$, let $u>0$ solve\n$\\Gamma_p u=0$, in $H_T$, and $u(x,0)=f(x),\\;\\forall x\\in H$. Set\n\\eqRef{4.1}\nm=\\inf_H f\\;\\;\\;\\mbox{and}\\;\\;\\;M=\\sup_H f.\n\\end{equation}\nWe assume that $0<m\\le M<\\infty$.\n\n{\\bf Proof of Theorem \\ref{1.7}.} \nFor any $\\rho>0$, $s>0$, and $x\\in H$, set \n$B_{\\rho,s}(x)=B_\\rho(x)\\times(0,s)$ and $P_{\\rho,s}(x)$ its parabolic boundary. Let $R>0$ be large.\n\n{\\bf (i) Lower bound.} Fix $y\\in H$ and set $r=|x-y|,\\;\\forall x\\in H$. Recall Lemma \\ref{2.6} and in $0\\le r\\le R$, take\n\\begin{eqnarray*}\n(i)\\; \\phi_p(x)=\\left(1- \\frac{r^2}{R^2}\\right)^\\alpha \\exp\\left(-\\frac{\\lambda_p t}{p-1}\\right),\\;\\;2\\le p<\\infty,\\;\\;(ii)\\; \\phi_\\infty(x,t)=\\left(1- \\frac{r^2}{R^2}\\right)^2 \\exp\\left(-\\frac{\\lambda_\\infty t}{3}\\right),\n\\end{eqnarray*}\nwhere $\\alpha=(3\/2)+n\/(2(p-1))$. Also, from Lemma \\ref{2.6}, one can write values of\n\\eqRef{4.2}\n\\lambda_p=\\frac{K_1}{R^p},\\;\\;2\\le p<\\infty,\\;\\;\\mbox{and}\\;\\;\\lambda_\\infty=\\frac{K_2}{R^4},\n\\end{equation}\nwhere $K_1=K_1(p,n)$ and $K_2$ is a universal constant. \n\nCall $\\phi_p(x,t)=\\phi_p(r,t),\\;\\overline{B}_R(y)\\times(0,T)$. By Lemma \\ref{2.6}, we see that the function $m\\phi_p$ is a sub-solution in\n$B_R(y)\\times (0,T)$, $\\phi_p(x,0)\\le m$, and $\\phi_p(x,t)=0$, in $|x-y|=R.$ Using Theorem \\ref{2.2} and Remark \\ref{2.20}, $\\phi_p(x,t)\\le u(x,t),\\;\\forall(x,t)\\in B_R(y)\\times(0,T)$. Writing $\\mu_p=\\lambda_p\/(p-1)$, for $2\\le p<\\infty$, and $\\mu_\\infty=\\lambda_\\infty\/3$, get\n$$\\phi_p(y,t)=m\\exp\\left(-\\mu_p t\\right)\\le u(y,t),\\;\\;0\\le t<T.$$\nUsing (\\ref{4.2}) and letting $R\\rightarrow \\infty$, we get $u(y,t)\\ge m.$ This shows the lower bound in the theorem.\n\n{\\bf (ii) Upper bound.} We use Lemma \\ref{2.7} and recall Remark \\ref{2.70}. Recall the expressions for $\\phi_p$ and take\n$\\alpha=1$ to obtain $\\phi_p(x,t)=\\phi_p(r,t)$ as \n\\begin{eqnarray}\\label{4.30}\n&&\\phi_p(x,t)=\\exp\\left( a\\left[ (t+1)^{p}-1\\right]+ b (t+1) r^{p\/(p-1)} \\right),\\;\\;2\\le p<\\infty,\\nonumber\\\\\n&&\\phi_\\infty(x,t)=\\exp\\left( a [ (t+1)^{4}-1]+ b (t+1)r^{4\/3} \\right),\\;\\;\\;\\;p=\\infty,\n\\end{eqnarray}\nwhere $r=|x|$.\nAlso, recall there are constants $K_1=K_1(p,n)$ and $K_2=K_2(p)$, and absolute constants\n$K_3$ and $K_4$  such that\n\\begin{eqnarray}\\label{4.3}\n&&(i)\\;\\;a= K_1 b^{p-1},\\;\\;\\mbox{and}\\;\\;0<b^{p-1}<\\frac{K_2}{(1+T)^{p}},\n\\;\\;\\mbox{for $2\\le p<\\infty$ and}\\nonumber\\\\\n&&(ii)\\;\\;a=K_3 b^3,\\;\\;\\mbox{and}\\;\\;0<b^3<\\frac{K_4}{(1+T)^4},\\;\\;\\mbox{for $p=\\infty$}.\n\\end{eqnarray}\nThen $\\phi_p(0,0)=1$ and $\\Gamma_p \\phi_p\\le 0$ in $\\mathbb{R}^n\\times(0,T).$\n\nLet $b=3\\varepsilon$, where $\\varepsilon>0$ is small so that the conditions in (\\ref{4.3}) are satisfied. We get from (\\ref{4.3}),\n\\eqRef{4.4}\na=(3\\varepsilon)^{p-1}K_1,\\;\\;\\mbox{for $2\\le p<\\infty$, and}\\;\\;a=27\\varepsilon^3K_3,\\;\\;\\mbox{for $p=\\infty.$} \n\\end{equation}\nSet \n$$\\beta=\\frac{p}{p-1},\\;\\;\\mbox{for $2\\le p<\\infty,$ and}\\;\\;\\beta=\\frac{4}{3},\\;\\;\\mbox{for $p=\\infty$.}$$\nFix $y\\in H$; set $r=|x-y|$ and $R>0$, so large that \n\\begin{eqnarray*}\n\\sup_{0\\le t\\le T} u(x,t)\\le \\exp( \\varepsilon r^\\beta), \\;\\;\\mbox{for $r\\ge R.$}\n\\end{eqnarray*}\n\nCall $C_{R,T}=B_R(y)\\times(0,T)$. \nDefine $v_p(x,t)=M\\phi_p(x,t),\\;\\forall(x,t)\\in C_{R,T}$. Then\nin $0\\le r\\le R$, for large enough $R$,\n$$v_p(y,0)=M,\\;\\;v_p(x,0)\\ge M,\\;\\;\\mbox{and}\\;\\;v_p(x,t)\\ge \\exp\\left(2\\varepsilon R^\\beta\\right),\\;\\mbox{on $|x-y|=R$ and $0\\le t\\le T$.}$$\nBy Theorem \\ref{2.2}, $u(x,t)\\le v_p(x,t),\\;\\forall(x,t)\\in C_{R,T}$. Using (\\ref{4.30}), (\\ref{4.4}) and $v_p(y,t)=M\\phi_p(0,t)$, we get\n$$u(y,t)\\le v_p(y,t)=M \\exp((3\\varepsilon)^{p-1}K),\\;2\\le p<\\infty, \\;\\mbox{and}\\;u(y,t)\\le v_\\infty(y,t)=M \\exp(27\\varepsilon^{3}K),$$\nwhere $K=K(p,T).$ Clearly, the above estimate holds for any $\\varepsilon>0$ and $u(y,t)\\le M$. The upper bound in the theorem holds and we obtain the statement of the theorem.\\qquad$\\Box$\n\\begin{rem}\\label{4.5} It is clear that an analogous version of the Phragm\\'en-Lindel\\\"of property also holds for the operator $G_p$. However, the optimality of Theorem \\ref{1.7} is not clear to us.\n\\quad$\\Box$\n\\end{rem}\n \n\\section{Positive solutions of $\\D_p u+\\lambda u^{p-1}=0,\\;\\;2\\le p<\\infty.$}\n\nIn the proofs of Parts (i) and (ii) of Theorem \\ref{1.6}, we used the existence of a function $\\psi_\\lambda$ for the problem in (\\ref{3.3}). In order to make our work self-contained, we now address the question of existence of $\\psi_\\lambda$ in the viscosity setting. We use ideas similar to those in \\cite{BM0} which addresses the case of the infinity-Laplacian. We refer to \\cite{CIL} for definitions.\n\nThe sets $usc(\\Omega)$ and $lsc(\\Omega)$ stand for the set of upper semi-continuous functions\nand the set of lower semi-continuous functions in $\\Omega$, respectively. We assume $2\\le p<\\infty$ and $\\Omega\\subset \\mathbb{R}^n$ is a bounded domain. To keep our work as brief as possible, we state our results as remarks. \n\n\n\\begin{rem}{(Maximum Principle)}\\label{6.1}\nSuppose that $u\\in usc(lsc)(\\overline{\\Omega})$ and $f:\\Omega\\times\\mathbb{R}\\rightarrow \\mathbb{R}$ is continuous. Assume that $f(x,t)<(>)0,\\;\\forall(x,t)\\in \\Omega\\times \\mathbb{R}$. \n$$\\mbox{If $\\D_p u+f(x,u)\\ge(\\le)0$, in $\\Omega$, then $\\sup_\\Omega u\\le \\sup_{\\partial\\Omega} u\\;(\\inf_\\Omega u\\ge \\inf_{\\partial\\Omega} u)$}.$$\n\\end{rem}\n\\begin{proof} We prove the maximum principle. Set $\\ell=\\sup_\\Omega u$ and $m=\\sup_{\\partial\\Omega} u$, and assume that $\\ell>m$. Let $\\varepsilon>0$ and $q\\in \\Omega$ be such that $2\\varepsilon=\\ell-m$ and $u(q)\\ge\\ell-\\varepsilon\/2$. Call $\\rho=\n\\sup_{x\\in \\partial\\Omega}|x-q|$ and set\n$$\\psi(x)=\\ell-\\varepsilon-\\varepsilon\\left(|x-q|\/\\rho\\right)^2,\\;\\;\\forall x\\in \\overline{\\Omega}.$$ \nThus, $(u-\\psi)(q)>0$ and $(u-\\psi)(x)\\le m-(\\ell-2\\varepsilon)=0,\\;\\forall x\\in \\partial\\Omega.$ Noting that $u-\\psi\\in usc(\\overline{\\Omega})$, let $z\\in \\Omega$ be such that $u-\\psi$ has a maximum. Using (\\ref{2.50}),\n$\\D_p \\psi(z)+f(z,u(z))\\le f(z, u(z))<0.$\nThis is a contradiction and the assertion holds. The proof of the minimum principle follows similarly.\n\\end{proof}\n\nWe prove a version of the strong maximum principle which is used in this work.\n\n\\begin{rem}{(Strong Maximum Principle)}\\label{6.10} Let $f\\in C(\\Omega\\times\\mathbb{R}, \\mathbb{R})$; assume that \n$\\inf_\\Omega |f(x,t)|=0$ if and only if $t=0$. \n\n(a) Suppose that $f\\le 0$ and $u\\in usc(\\overline{\\Omega})$ solves $\\D_p u+f(x,u)\\ge 0$, in $\\Omega$. If \n$\\sup_{\\partial\\Omega} u>0$ or $\\sup_{\\overline{\\Omega}}u<0$ then $u(x)<\\sup_{\\partial\\Omega} u,\\;\\forall x\\in \\Omega$.\n\n(b) Suppose that $f\\ge 0$ and $u\\in lsc(\\overline{\\Omega})$ solves $\\D_p u+f(x,u)\\le 0$, in $\\Omega$. If \n$\\inf_{\\partial\\Omega} u<0$ or $\\inf_{\\overline{\\Omega}}u>0$ then $u(x)>\\inf_{\\partial\\Omega} u,\\;\\forall x\\in \\Omega$.\n\\begin{proof} We show (a). Suppose that there is a point $z\\in \\Omega$ such that \n$u(z)=\\sup_{\\overline{\\Omega}}u\\ge \\sup_{\\partial\\Omega}u.$ Clearly, $u(z)\\ne 0.$ For $\\varepsilon>0$, small, define\n$v_\\varepsilon(x)=u(z)+\\varepsilon |x-z|^2$, in $\\Omega$. Then $v_\\varepsilon \\in C^2(\\Omega)$, $(u-v_\\varepsilon)(z)=0$ and\n$$u-v_\\varepsilon =u(x)-u(z)-\\varepsilon |x-z|^2=u(x)-\\sup_{\\overline{\\Omega}}u-\\varepsilon|x-z|^2<0, \\;\\forall x\\in \\overline{\\Omega},\\;x\\ne z.$$\nThus, $z$ is the only point of maximum of $u-v_\\varepsilon$. Noting that $\\Delta_p r^2=2^{p-1}r^{p-2}(p+n-2),\\;r=|x-z|,$ and using the definition of a viscosity sub-solution, we get for $2<p<\\infty$, \n$$(\\D_p v_\\varepsilon)(z)+f(z,u(z))=f(z,u(z))\\ge 0,\\;\\mbox{and}\\;(\\Delta v_\\varepsilon)(z)+f(z,u(z))=-(2\\varepsilon)^2n+f(z,u(z))\\ge 0.$$ \nLetting $\\varepsilon\\rightarrow 0$, \n$f(z,u(z))\\ge 0$, and $u(z))=0$. This is a contradiction and the claim holds. Proof of (b) is similar.\n \\end{proof}\n\\end{rem}\n\nLet $S^{n\\times n}$ be the set of symmetric $n\\times n$ matrices and $Tr$ denote the trace of a matrix.\nDefine for $(q,X)\\in \\mathbb{R}^n\\times S^{n\\times n}$ ($L_p$ is the differentiated version of $\\D_p$),\n\\eqRef{2.00}\nL_p(q, X)=\\left\\{ \\begin{array}{lcr} |q|^{p-2}Tr(X)+(p-2)|q|^{p-4} q_iq_jX_{ij},& p>2,\\;q\\ne 0,\\\\ Tr(X),& p=2,\\\\ 0& p>2, \\;q=0. \\end{array}\\right.\n\\end{equation}\n\n\\begin{rem}{(Comparison Principle)}\\label{6.2}\nLet $2\\le p<\\infty$, $f$ and $g\\in C(\\Omega, \\mathbb{R})$. Suppose that $u\\in usc(\\overline{\\Omega})$ and $v\\in lsc(\\overline{\\Omega})$ solve\n$\\Delta_p u+ f(x,u(x))\\ge 0,$ and $\\Delta_p v+g(x, v(x))\\le 0,$ in $\\Omega$.\nIf $sup_{\\Omega} (u-v)>\\sup_{\\partial\\Omega} (u-v)$ then there is a point $z\\in \\Omega$ such that\n$$(u-v)(z)=\\sup_\\Omega(u-v)\\;\\;\\;\\mbox{and}\\;\\;\\;g(z, v(z))\\le f(z, u(z)).$$\n\\end{rem}\n\\begin{proof} We adapt the proof in \\cite{CIL} (also see \\cite{BM0}) and provide a\nbrief outline. \n\nSet $M=\\sup _{\\Omega}(u-v)$. Then one may find a point $z\\in \\overline{\\Omega}$ and sequences $x_{\\varepsilon}$ and $y_{\\varepsilon}$ such that (i) $M=(u-v)(z)$, and (ii) \n$x_\\varepsilon,\\;y_\\varepsilon\\rightarrow z$ as $\\varepsilon\\rightarrow 0$. Moreover, since $M>\\sup_{\\partial \\Omega}(u-v)$, there is an open set $O$ such that $z,\\; x_{\\varepsilon}$ and $y_{\\varepsilon}\\in O\\subset\\subset \\Omega$. \nAlso, there exist (see \\cite{CIL})\n  $X_{\\varepsilon},\\;Y_{\\varepsilon}\\in S^{n\\times n}$ such that $((x_{\\varepsilon}-y_{\\varepsilon})\/\\varepsilon, X_{\\varepsilon})\\in \\bar{J}^{2,+}u(x_{\\varepsilon})$ and $((x_{\\varepsilon}-y_{\\varepsilon})\/\\varepsilon, Y_{\\varepsilon})\\in \\bar{J}^{2,-}v(y_{\\varepsilon})$. Moreover, $X_{\\varepsilon}\\le Y_{\\varepsilon}$. Using the definitions of $\\bar{J}^{2,+}$ and $\\bar{J}^{2,-}$, we see that \n$$-f(x_{\\varepsilon},u(x_{\\varepsilon}))\\le L_p((x_\\varepsilon-y_\\varepsilon)\/\\varepsilon, X_\\varepsilon)  \\le L_p((x_\\varepsilon-y_\\varepsilon)\/\\varepsilon, Y_\\varepsilon) \\le -g(y_{\\varepsilon}, v(y_{\\varepsilon})).$$\nNow let $\\varepsilon\\rightarrow 0$ to conclude that $g(z,v(z))\\le f(z,u(z)).$  \\end{proof}\n\nRemark \\ref{6.2} leads to the following comparison principle, see \\cite{BM0}.\n\n\\begin{rem}{(Quotient Comparison)}\\label{6.3} Let $u\\in usc(\\overline{\\Omega})$ and $v\\in lsc(\\overline{\\Omega})\\cap L^{\\infty}(\\Omega),\\;v>0$. Suppose that $\\lambda$ and $\\bar{\\lambda}$ are both positive. \n\n\\noindent (a) Let $\\lambda<\\bar{\\lambda}$, $u$ and $v$ solve\n$\\D_p u +\\lambda |u|^{p-2}u\\ge 0,\\;\\mbox{and}\\;\\D_p v+\\bar{\\lambda} v^{p-1}\\le 0,\\;\\mbox{in $\\Omega$},$\nthen \n$$\n\\mbox{either $u\\le 0$, in $\\Omega$,\\quad or }\\;\\;(u\/v)(x)\\le \\sup_{\\partial\\Omega}(u\/v),\\;\\;\\forall x\\in \\Omega.$$\n(b) Similarly, if, $\\bar{\\lambda}\\le \\lambda$, $u>0$ and $v>0$ solve $\\D_p u-\\lambda u^{p-1}\\ge 0,\\;\\mbox{and}\\;\\D_p v-\\bar{\\lambda} v^{p-1}\\le 0,\\;\\mbox{in $\\Omega$},$\nthen $\\sup_{\\Omega} (u\/v)\\le \\sup_{\\partial\\Omega}(u\/v).$\n\\end{rem}\n\\begin{proof} Set $\\mu=\\sup_{\\partial\\Omega} (u\/v)$ and $\\nu=\\sup_\\Omega (u\/v)$. We observe that \n\\eqRef{6.4}\nu-\\mu v\\le 0,\\;\\;\\mbox{in $\\partial\\Omega$, and}\\;\\;u-\\nu v\\le 0,\\;\\;\\mbox{in $\\Omega$}.\n\\end{equation}\nWe prove (a). Assume that $\\nu>0$ and $\\mu<\\nu$. Using (\\ref{6.4}), \n$\\sup_{\\partial\\Omega}(u-\\nu v)<0$ and $\\sup_\\Omega (u-\\nu v)=0$.\nSince $\\D_p (\\nu v)+\\bar{\\lambda}(\\nu v)^{p-1}\\le 0$, by Remark \\ref{6.2}, we conclude that there is a point $y\\in \\Omega$ such that $(u-\\nu v)(y)=\\sup_\\Omega(u-\\nu v)=0$, implying that $u(y)>0$, and\n$$\\bar{\\lambda} (\\nu v(y))^{p-1}\\le \\lambda u(y)^{p-1}= \\lambda (\\nu v(y))^{p-1}.$$\nWe have a contradiction and the assertion holds. To show (b), use (\\ref{6.4}), $\\mu<\\nu$ to conclude that $\\sup_\\Omega(u-\\mu v)>0$. Since $\\sup_{\\partial\\Omega}(u-\\mu v)=0$, in $\\partial\\Omega$, Remark \\ref{6.2} implies that there is a point $z\\in \\Omega$ such that $(u-\\mu v)(z)=\\sup_\\Omega (u-\\mu v)>0$ and\n$$\\lambda u(z)^{p-1}\\le \\bar{\\lambda} (\\mu v(z))^{p-1}<\\bar{\\lambda} u(z)^{p-1}.$$\nWe have a contradiction and the assertion holds. \\end{proof}\n\n\\begin{rem}\\label{6.5} We extend the result in Remark \\ref{6.3}(a) to include the case $\\lambda=\\bar{\\lambda}$, that is, $\\D_p v+\\lambda v^{p-1}\\le 0$, in $\\Omega$. \n\nSet $m=\\inf_{\\partial\\Omega} v$, $M=\\sup_\\Omega v$ and $v_t=v-t m$, where $0<t<1$. By Remark \\ref{6.10}, $v> m$ and $v_t> (1-t)m>0$, in $\\Omega$. \nSince $\\D_p v\\le -\\lambda v^{p-1}$, choose $\\varepsilon>0$, small (depending on $t$), so that\n\\begin{eqnarray}\\label{6.50}\n\\Delta_p v_t+(\\lambda+\\varepsilon) v_t^{p-1}&\\le& v^{p-1} \\left[-\\lambda+ (\\lambda+\\varepsilon) \\left(\\frac{v-t m}{v}\\right)^{p-1} \\right]\\nonumber\\\\\n&\\le& v^{p-1} \\left[ \\frac{\\lambda+\\varepsilon}{\\lambda} \\left(1-\\frac{t m}{M}\\right)^{p-1}-1 \\right]\\le 0.\n\\end{eqnarray}\nRemark \\ref{6.3}(a) holds for $\\lambda=\\bar{\\lambda}$, since $u\/v_t\\le \\sup_{\\partial\\Omega} u\/v_t$ for any $0<t<1$. Moreover, (\\ref{6.50}) leads to the estimates\n\\begin{eqnarray*}\n(i)\\quad 0<\\frac{\\varepsilon}{\\lambda}\\le\\left( \\frac{1}{1-t(m\/M)}\\right)^{p-1}-1,  \\;\\;\\;(ii)\\quad 1<\\frac{M}{m}\\le t \\left( \\frac{ (\\lambda+\\varepsilon)^{1\/(p-1)}} { (\\lambda+\\varepsilon)^{1\/(p-1)}-\\lambda^{1\/(p-1)} }\\right).\\qquad \\Box\n\\end{eqnarray*}\n\\end{rem} \n\n\\begin{rem}\\label{6.6} Remark \\ref{6.5} implies the following. Suppose that $\\delta>0$ and $u\\in C(\\overline{\\Omega})$ solves\n\\eqRef{6.7}\n\\Delta_p u+\\lambda u^{p-1}=0,\\;\\;\\mbox{in $\\Omega$, $u>0$, and $u=\\delta$ on $\\partial \\Omega$.}\n\\end{equation}\nFor each $0<t<1$, there is an $\\varepsilon>0$, depending on $\\sup_\\Omega u$, $\\delta$ and $t$ such that $u_t=u-t\\delta$ and \n$\\Delta_p u_t+(\\lambda+\\varepsilon)u_t^{p-1}\\le 0$, in $\\Omega$. Next, $\\bar{u}_t=u_t\/(1-t)$ is a super-solution of (\\ref{6.7}) with $\\lambda$ replaced by $\\lambda+\\varepsilon$, and \n$\\bar{u}_t=\\delta$, on $\\partial \\Omega$. Also, $v(x)=\\delta$ is a sub-solution. Both $v$ and $\\bar{u}_t$ attain the boundary data in (\\ref{6.7}) and $v\\le \\bar{u}_t$.\nRemark \\ref{6.5} and Perron's method (see \\cite{CIL}) imply there is a $\\psi\\in C(\\overline{\\Omega}),\\;\\psi>0,$ with\n$\\Delta_p \\psi+(\\lambda+\\varepsilon)\\psi^{p-1}=0,$ in $\\Omega$, and $\\psi=\\delta$ on $\\partial \\Omega$. \\quad $\\Box$\n\\end{rem}\n\nLet $\\delta>0$. We now discuss existence of positive solutions $u\\in C(\\overline{\\Omega})$ to the problem\n\\eqRef{6.8}\n\\D_p u+\\lambda u^{p-1}=0,\\;\\mbox{in $\\Omega,$ and $u=\\delta,$ in $\\partial\\Omega$.}\n\\end{equation}\nWe define\n\\eqRef{6.9}\nE_\\Omega=\\{\\lambda\\ge 0:\\;\\mbox{problem (\\ref{6.8}) has a positive solution $u$}\\},\\;\\mbox{and}\\;\\;\\lambda_{\\Om}=\\sup E_\\Omega.\n\\end{equation}\nWe show in Remark \\ref{6.11} below that $(0,\\lambda_{\\Om})\\subset E_\\Omega.$ Let $M_\\lambda=\\sup_\\Omega u$, where $u$ solves (\\ref{6.8}). Note that $u\\ge \\delta.$ We observe that if $0<\\lambda_{\\Om}<\\infty$ (see Remark \\ref{6.15}) and $0<\\lambda<\\lambda_{\\Om}$ then by Remark \\ref{6.5}(i), for any $0<t<1$,\n\\eqRef{6.90}\n0<\\varepsilon\\le \\lambda \\left[\\left( \\frac{1}{1-t(\\delta\/M_\\lambda)}\\right)^{p-1}-1\\right]\\le \\lambda_{\\Om} -\\lambda,\\;\\;\\mbox{and}\\;\\;\nM_\\lambda\\ge \\delta  \\left( \\frac{\\lambda_{\\Om}^{1\/(p-1)}}{\\lambda_{\\Om}^{1\/(p-1)}-\\lambda^{1\/(p-1)} }\\right).\n\\end{equation}\nThus, $\\lim_{\\lambda\\rightarrow \\lambda_{\\Om}} M_\\lambda=\\infty.$\n\nOur goal is to show existence for small $\\lambda>0$ and to prove that $\\lambda_{\\Om}<\\infty$. This would provide the information necessary for Theorem \\ref{1.6}. Next, we show that \n(i) if $\\lambda\\in E_\\Omega,\\;\\lambda>0,$ then $[\\lambda,\\lambda_{\\Om})\\subset F_\\Omega$, and (ii) the domain monotonicity property of $\\lambda_{\\Om}$.\n\n\\begin{rem}\\label{6.11} Let $E_\\Omega$ be as in (\\ref{6.9}). Then $\\lambda_{\\Om}\\not\\in E_\\Omega$ and the following hold.\\\\\n(i) If $\\lambda\\in E_\\Omega$ then $(0,\\lambda^{\\prime})\\subset E_\\Omega$, for some $\\lambda^{\\prime}>\\lambda.$ \n Thus, $(0,\\lambda_{\\Om})\\subset E_\\Omega$.\\\\\n(ii) If $O\\subset\\Omega$ is a sub-domain then $\\lambda_{\\Om}\\le \\lambda_O$.\n\\begin{proof} \nIf $\\lambda_{\\Om}<\\infty$ and $\\lambda_{\\Om}\\in E_\\Omega$ then, by Remark \\ref{6.7}, $\\lambda_{\\Om}$ will not be the supremum.\n\nPart (i). Let $u_\\lambda\\in C(\\overline{\\Omega})$ solve\n$\\D_p u_\\lambda+\\lambda u_\\lambda^{p-1}=0,\\;\\mbox{in $\\Omega,\\;u_\\lambda>0,$ and $u_\\lambda=\\delta$, in $\\partial\\Omega.$}$\nClearly, $v=\\delta$, in $\\overline{\\Omega}$, is a sub-solution and $u_\\lambda$ is a super-solution of\n\\eqRef{6.12}\n\\D_p w+\\mu w^{p-1}=0,\\;\\;\\mbox{in $\\Omega$, and $w=\\delta$, in $\\partial\\Omega$,}\n\\end{equation}\nfor any $0<\\mu\\le \\lambda$. for some $\\lambda^{\\prime}>\\lambda.$ Since both $v$ and $u_\\lambda$\nattain the boundary data and $v\\le u_\\lambda$,\nrecalling Remarks \\ref{6.3}, \\ref{6.5} and applying Perron's method, we obtain a positive solution of (\\ref{6.12}), for each $0<\\mu\\le \\lambda.$ Combining this with Remark \\ref{6.6} we see that $(0,\\lambda^{\\prime})\\subset E_\\Omega$, for some $\\lambda^{\\prime}>\\lambda.$ Clearly, $(0,\\lambda_{\\Om})\\subset E_\\Omega.$\n\nPart (ii). Assume that $\\lambda_O<\\infty$ (otherwise we are done) and \n$\\lambda_{\\Om}>\\lambda_O$. By the definition of $E_\\Omega$ and Part (i), there is a $u\\in C(\\overline{\\Omega}),\\;u>0,$ solves \n$$\\D_p u+\\lambda_O u^{p-1}=0,\\;\\;\\mbox{in $\\Omega$, and $u=\\delta$, in $\\partial\\Omega.$}$$\nIf $\\lambda<\\lambda_O$ then there is a unique positive solution $v_\\lambda$ to\n$$\\D_p v_\\lambda +\\lambda v_\\lambda^{p-1}=0,\\;\\;\\mbox{in $O$, and $v_\\lambda=\\delta$, in $\\partial O$.}$$\nBy Remark \\ref{6.1}, $u\\ge v_\\lambda$, on $\\partial O$, and by Remark \\ref{6.3}, $u\\ge v_\\lambda$, in $O$. Since this holds for any $\\lambda<\\lambda_O$, we apply (\\ref{6.90})(on $O$) and let $\\lambda\\rightarrow \\lambda_O$ to conclude that $u$ is unbounded. This is a contradiction and the claim holds. \n\\end{proof} \n\\end{rem}\n\nWe record a consequence of (\\ref{6.90}) and Remark \\ref{6.11}. \n\\begin{rem}\\label{6.13} Let $h\\in C(\\partial\\Omega)$ with $\\inf_{\\partial\\Omega} h>0$. Suppose $\\lambda>0$ is such that the problem\n$$\\D_p u+\\lambda u^{p-1}=0,\\;\\;\\mbox{in $\\Omega$, $u=h$, in $\\partial\\Omega$,}$$\nhas a positive solution $u\\in C(\\overline{\\Omega})$. Call $E_{\\Omega,h}$ as the set of all $\\lambda$'s for which the above has a positive solution. Set $\\lambda_{\\Omega,h}=\\sup E_{\\Omega, h}$. We claim that\n$$\\lambda_{\\Omega, h}\\le \\lambda_{\\Om},$$\nwhere $\\lambda_{\\Om}$ is as in (\\ref{6.9}). We comment that the two are equal and since the proof of equality requires existence we will not address it here, see Theorem \\ref{1.6}.\n\n\\begin{proof} Assume that $\\lambda_{\\Om}<\\infty$ and $\\lambda_{\\Om}<\\lambda_{\\Omega, h}$ (otherwise we are done). Thus, there  is a $\\lambda_1$ with \n$\\lambda_{\\Om}<\\lambda_1\\le \\lambda_{\\Omega, h}$ and a function $u\\in C(\\overline{\\Omega}),\\;u>0,$ so that \n$$\\D_p u+\\lambda_1 u^{p-1}=0,\\;\\;\\mbox{in $\\Omega$, and $u=h$, in $\\partial\\Omega$.}$$\nBy Remark \\ref{6.11}(i), for any $0<\\lambda<\\lambda_{\\Om}$, there is a function $v_\\lambda$ so that $\\D_p v_\\lambda +\\lambda v_\\lambda^{p-1}=0,$ in $\\Omega$, and $v_\\lambda=\\delta$, in $\\partial\\Omega$,\nwhere $0<\\delta\\le \\inf_{\\partial\\Omega}h$. By Remark \\ref{6.3}, $v_\\lambda\\le u$, in $\\Omega$. Letting $\\lambda\\rightarrow \\lambda_{\\Om}$ and applying (\\ref{6.90}), we arrive at a contradiction. The claim holds.\n\\end{proof}\n\\end{rem}\n\nWe show existence for the problem in (\\ref{6.8}). We assume that (i) for $2\\le p\\le n$, $\\Omega$ satisfies a uniform outer ball condition, and (ii) for $n<p<\\infty$, $\\Omega$ is any domain.\n\n\\begin{rem}{(Existence:)}\\label{6.13} Consider the problem of finding a positive solution\n$u\\in C(\\overline{\\Omega})$ to \n\\eqRef{6.14} \\D_p u+\\lambda u^{p-1}=0,\\;\\;\\mbox{in $\\Omega$, and $u=\\delta$, on $\\partial\\Omega$.}\n\\end{equation}\n(i) Let $n<p<\\infty$ and $\\Omega$ be any bounded domain. Then there is $\\lambda_0=\\lambda_0(p,n,\\Omega)>0$ \nsuch that (\\ref{6.14}) has a solution $u$ for any $0<\\lambda<\\lambda_0$.\\\\\n(ii) The same holds for $2\\le p\\le n$, if $\\Omega$ satisfies a uniform outer ball condition.\n\n\\begin{proof} The function $v=\\delta$ is a sub-solution of (\\ref{6.14}) for any \n$\\lambda>0$ and any $2\\le p<\\infty$. We construct super-solutions to (\\ref{6.14}). Define\n$R=\\sup_{x,\\;y\\in \\partial\\Omega}|x-y|=\\mbox{diam}(\\Omega).$\n\n{\\bf (i) $n<p<\\infty$:} Fix $y\\in \\partial\\Omega$, $0<\\theta<1$ and set $r=|x-y|$. For $c>0$, (to be determined)\n$$\\alpha=\\theta(p-n)\/(p-1),\\;\\;\\mbox{and}\\;\\;w_y(x)=\\delta+c r^\\alpha,\\;\\;\\forall x\\in \\overline{\\Omega}.$$\nUsing (\\ref{2.50}), calculating in $0<r\\le R$, \n\\begin{eqnarray*}\n\\D_p w_y=(c \\alpha)^{p-1}r^{(\\alpha-1)(p-2)+\\alpha-2} \\left( (p-1)(\\alpha-1)+n-1 \\right)=\n-\\frac{(c\\alpha)^{p-1}(1-\\theta)(p-n)}{r^{p-\\alpha(p-1)}}\\nonumber\n\\end{eqnarray*}\nUsing the above, we obtain, in $\\Omega$,\n\\begin{eqnarray*}\n\\D_p w_y+\\lambda w_y^{p-1}&\\le& -\\frac{(c\\alpha)^{p-1}(1-\\theta)(p-n)R^{\\alpha(p-1)}}{R^{p}}+\\lambda (\\delta+c R^\\alpha)^{p-1}\\\\\n&=&(\\delta+cR^\\alpha)^{p-1}\\left[\\lambda-\\left(\\frac{cR^\\alpha}{\\delta+cR^\\alpha}\\right)^{p-1}\\left( \\frac{(1-\\theta)(p-n)\\alpha^{p-1}}{R^{p}}\\right) \\right].\n\\end{eqnarray*}\nIt is clear that if $0<\\lambda<(1-\\theta)(p-n)\\alpha^{p-1}R^{-p}$ then \none can find a value of $c>0$ such that $w_y$ is a super-solution. Since $w_y(y)=\\delta$ \nand $w_y\\ge \\delta$, on $\\overline{\\Omega}$, using Remarks \\ref{6.3}, \\ref{6.5}, and applying Perron's method, the problem (\\ref{6.14}) has a positive solution for $\\lambda>0$, small, and $E_\\Omega$ is non-empty.\n\n{\\bf (ii) $2\\le p\\le n$:} Let $\\rho>0$ be the optimal radius of the outer ball. Fix $y\\in \\partial\\Omega$ and let $z\\in \\mathbb{R}^n\\setminus \\Omega$ such that $B_\\rho(z)\\subset \\mathbb{R}^n\\setminus \\Omega$ and $y\\in \\overline{B}_\\rho(z)\\cap \\partial\\Omega$. Set $r=|x-z|$ and take, for $c>0$,\n$$\\alpha>\\max\\left\\{0,\\; (n-p)\/(p-1)\\right\\}\\;\\;\\;\\mbox{and}\\;\\;\\;w_y(x)=\\delta+c\\left(\\rho^{-\\alpha}-r^{-\\alpha}\\right),\\;\\;\\rho\\le r\\le R+\\rho.$$\nUsing (\\ref{2.50}), \n\\begin{eqnarray*}\n\\D_p w_y&=&(c\\alpha)^{p-1}r^{-(\\alpha+1)(p-2)-(\\alpha+2)}\\left( n-1-(\\alpha+1)(p-1)\\right)\\\\\n&=&\\frac{(c\\alpha)^{p-1} (n-p-\\alpha(p-1))}{r^{\\alpha(p-1)+p}}\n=-\\frac{c^{p-1}k}{r^{\\alpha(p-1)+p}},\n\\end{eqnarray*}\nwhere $k=k(n,p,\\alpha)>0$. Setting $J=\\rho^{-\\alpha}-(R+\\rho)^{-\\alpha}$ and using the above,\n\\begin{eqnarray*}\n\\D_p w_y+\\lambda w_y^{p-1}&\\le& -\\frac{c^{p-1}k}{(\\rho+R)^{\\alpha(p-1)+p}}+\\lambda\\left( \\delta+cJ\\right)^{p-1}\\\\\n&=&\\left( \\delta+cJ \\right)^{p-1}\n\\left[\\lambda-\\frac{k}{(\\rho+R)^{\\alpha(p-1)+p}} \\left( \\frac{c}{ \\delta+ cJ}\\right)^{p-1}\\right].\n\\end{eqnarray*}\nSince $c\/(\\delta+cJ)<1\/J$, one can find a $c>0$ such that $w_y$ is super-solution if \n$$0<\\lambda<\\frac{k}{(R+\\rho)^p}  \\left( \\frac{\\rho^{\\alpha}}{(R+\\rho)^\\alpha-\\rho^\\alpha }\\right)^{p-1}.$$\nRest of the proof is as in Part (i). \n\\end{proof}\n\\end{rem}\n\n\\begin{rem}{(Boundedness of $\\lambda_{\\Om}$)}\\label{6.15} Remark 6.13 shows that $\\lambda_{\\Om}>0$. We claim that $\\lambda_{\\Om}<\\infty$. By Remark \\ref{6.11}, this will follow if we show that $\\lambda_{B}<\\infty$, \nfor any ball $B$ in $\\Omega$. \n\\begin{proof} For ease of presentation, we take the origin $o\\in \\Omega$ and a ball $B_R(o)\\subset \\Omega$.\nSet $r=|x|$ and $\\lambda_R=\\lambda_{B_R}(o).$\n\nSuppose that $\\lambda_R=\\infty$. By (\\ref{6.9}) and Remark \\ref{6.11},\n$(0,\\infty)\\subset E_{B_R(o)}.$ Let $\\lambda>0$ and $\\lambda_m=m^p\\lambda,\\;m=0,1,\\cdots,$. \nFor each $m$, call $\\phi_m>0$ the solution of \n\\eqRef{6.16}\n\\Delta_p\\phi_m+\\lambda_m\\phi_m^{p-1}=0,\\;\\;\\mbox{in $B_R(o)$, and $\\phi_m=\\delta_m$, in $\\partial B_R(o)$}.\n\\end{equation}\nHere $\\delta_m>0$ is so chosen that $\\phi_m(o)=1$. Since $\\D_p$ is rotation and reflection invariant, \napplying Remark \\ref{6.5} to reflections about $n-1$ planes through $o$, it follows that $\\phi_m$ is radial. Next, using Remark \\ref{6.1} in concentric balls, it is clear that $\\phi_m$ is decreasing in $r$.\n\nBy Remarks \\ref{6.10} and \\ref{6.3}, if $\\ell<m$ then $1=\\phi_\\ell(o)\/\\phi_m(o)\\le \\delta_\\ell\/\\delta_m,\\;\\mbox{and}\\;\\;0<\\delta_m\\le \\delta_{m-1}\\le \\cdots\\le \\delta_1<1.$\n\nNext, we scale as follows. For $m=1,2,\\cdots,$ set\n$\\psi_m(y)=\\phi_m(x),$ where $y=mx,\\;x\\in B_R(o)$. Thus, $\\Delta_p\\phi_m=m^p\\Delta_p \\psi_m$ and recalling $\\lambda_m=m^p \\lambda$, we get\n$$\\Delta_p\\psi_m+\\lambda \\psi_m^p=0,\\;\\;\\mbox{in $B_{mR}(o)$, $\\psi_m(o)=1$ and $\\psi_m(mR)=\\delta_m$.}$$\nApplying Remark \\ref{6.5} in $B_{\\ell R}(o)$, $\\ell=1,2,\\cdots, m-1$,\n$$1=\\frac{\\psi_\\ell(o)}{\\psi_m(o)}\\le \\frac{\\psi_\\ell(r)}{\\psi_m(r)}\\le \\frac{\\delta_\\ell}{\\psi_m(\\ell R)}\\;\\;\\mbox{and}\\;\\;1\\le  \\frac{\\psi_m(r)}{\\psi_\\ell(r)}\\le \\frac{\\psi_m(\\ell R)}{\\delta_\\ell}.$$\nIt is clear that $\\psi_\\ell(r)=\\psi_m(r),\\;0\\le r\\le \\ell R,$ and $\\delta_\\ell=\\psi_m(\\ell R).$ For each $\\ell=1,2,\\cdots, m-1$, $\\psi_m$ extends $\\psi_\\ell$ to $B_{mR}(o)$, and in particular, extends $\\psi_1$ (defined on $B_R(o)$) to all \n$B_{mR}(o)$. \n\nThus, for any $r>0$, we can define $\\psi_1(r)=\\psi_m(r),\\;\\mbox{for any $m$ such that $mR>r$},$\nthus extending $\\psi_1$ to $\\mathbb{R}^n$. Also, $\\psi_1$ is decreasing and $\\psi_1(mR)=\\delta_m,\\;\\;\\forall m=1,2,\\cdots.$\n\nWe claim that $\\lim_{m\\rightarrow \\infty}\\delta_m=0$. For $1\\le \\ell\\le m$ and $0<\\alpha<1$, we calculate (see (\\ref{6.16})),\n\\begin{eqnarray*}\n\\D_p \\phi_m^\\alpha&+&\\lambda_\\ell \\phi_m^{\\alpha(p-1)}=\\alpha^{p-1}\\mbox{div}\\left(\\phi_m^{(\\alpha-1)(p-1)}|D\\phi_m|^{p-2}\\phi_m\\right)+\\lambda_\\ell \\phi_m^{\\alpha(p-1)}\\\\\n&=&\\alpha^{p-1}\\left( \\phi^{(\\alpha-1)(p-1)}_m\\D_p \\phi_m+ (\\alpha-1)(p-1)\\phi_m^{\\alpha (p-1)-p}|D\\phi_m|^p\\right)+  \\lambda_\\ell\\phi^{\\alpha(p-1)}_m\\\\\n&\\le & -\\alpha^{p-1}\\lambda_m \\phi_m^{\\alpha(p-1)}+\\lambda_\\ell \\phi_m^{\\alpha(p-1)}=\\left(\\lambda_\\ell- \\alpha^{p-1}\\lambda_m\\right)\\phi^{\\alpha(p-1)}_m=0,\n\\end{eqnarray*}\nif $\\alpha=(\\lambda_\\ell\/\\lambda_m)^{1\/(p-1)}=(\\ell\/m)^{p\/(p-1)}$. \nThus, $\\phi_m^\\alpha$ is a super-solution and Remark \\ref{6.5} shows that\n$1=\\phi_\\ell(o)\/\\phi^\\alpha_m(0)\\le \\delta_\\ell\/\\delta^\\alpha_m.$ Using the value of $\\alpha$, we have\n$(\\delta_m)^{(1\/m)^{p\/(p-1)}}\\le (\\delta_\\ell)^{(1\/\\ell)^{p\/(p-1)}},\\;\\forall \\ell=1,2,\\cdots,m-1.$\nHence, \n\\eqRef{6.17}\n\\delta_m\\le \\delta_1^{m^{p\/(p-1)}},\n\\end{equation}\nand since $\\delta_1<1$ (see Remark \\ref{6.10} and (\\ref{6.15})), \n$\\lim_{m\\rightarrow \\infty}\\psi_1(mR)=\\lim_{m\\rightarrow \\infty}\\delta_m=0$ and $\\lim_{r\\rightarrow \\infty}\\psi_1(r)=0.$\n\nWe now obtain lower bounds for $\\psi_1$. Note that $\\psi_1(o)=1$, $\\psi_1$ is decreasing and \n$\\psi_1(\\ell R)=\\delta_\\ell,\\;\\forall\\ell=1,2,\\cdots$ (see above). For any $m=2,3,\\cdots$, define in $R\\le r\\le 2mR$,\n\\begin{eqnarray*}\n&&f_m(r)=\\delta_1-\\left(\\delta_1-\\delta_m\\right) (r^{\\beta}-R^\\beta)\/((2mR)^\\beta-R^\\beta),\\;\\;\\mbox{where}\\;\\;\\beta=\\frac{p-n}{p-1},\\;p\\ne n,\\\\\n&&f_m(r)=\\delta_1- (\\delta_1-\\delta_m)\\log (r\/R)\/\\log (2m) ,\\;\\;p=n.\n\\end{eqnarray*}\nThen $f_m(R)=\\delta_1$, $f_m(2mR)=\\delta_{2m}$, $f_m> 0$ and $\\Delta_p f_m=0$, in $B_{2mR}(o)\\setminus B_R(o)$. Thus, $\\Delta_p f_m+\\lambda f_m^{p-1}\\ge 0$, in $B_{2mR}(o)\\setminus B_R(o)$. By Remark \\ref{6.5},\n$f_m\\le \\psi_1$, in $B_{2mR}(o)\\setminus B_R(o)$. Taking $r=mR$, we get, for large $m$, \n\\begin{eqnarray*}\n&&(i)\\;f_m(mR)\\ge \\delta_1(1-2^{-\\beta}), \\;\\;p>n,\\;\\;\\;(ii)\\;f_m(mR)\\ge \\delta_1(1-2^\\beta)\/m^{-\\beta}, \\;\\;2\\le p<n,\\\\\n&&(iii)\\;f_m(mR)\\ge \\delta_1\\log 2\/\\log (2m), \\;\\;p=n.\n\\end{eqnarray*}\nSince $\\delta_m=\\psi_1(mR)\\ge f_m(mR)$, the above and (\\ref{6.17}) lead to a contradiction. Thus, the claim holds and $\\lambda_R<\\infty$ and $0<\\lambda_{\\Om}<\\infty.$ \n\\end{proof}\n\\end{rem}\n\n\\begin{rem}{(Scaling property)}\\label{6.18} Let $\\lambda_R=\\lambda_{B_R(o)}.$ We claim that $\\lambda_R R^p=k$, for any $R>0$, where $k=k(p,n)>0.$ \nLet $R_1>0$, $R_2>0$ and $0<\\lambda<\\lambda_{R_1}$.  Suppose that $\\phi>0$ solves $\\Delta_p \\phi+\\lambda \\phi^{p-1}=0,$ in $B_{R_1}(o)$, with\n$\\phi_1=\\delta$ on $\\partial B_{R_1}(o)$. Set $\\psi(y)=\\phi(x)$ where $y=R_2 x\/R_1$. Then \n$\\Delta_p\\psi+\\lambda(R_1\/R_2)^p \\psi^{p-1}=0$, in $B_{R_2}(o)$, and $\\psi=\\delta$ on $\\partial B_{R_2}(o)$.\nClearly, $\\lambda_{R_1}R_1^p\\le \\lambda_{R_2}R_2^p$. Replacing $R_1$ by $R_2$ shows equality.\n$\\Box$\n\\end{rem}\n\n\\begin{rem}\\label{6.19}{(Eigenfunction)} The problem\n$\\Delta_p u+\\lambda_R u^{p-1}=0,\\;\\mbox{in $B_R(o)$ and $u=0$ on $\\partial B_R(o)$,}$\nhas a positive solution $u$, a first eigenfunction, that is radial and decreasing.\n\\end{rem}\n\\begin{proof} Fix $0<\\lambda<\\lambda_R$. By Remark \\ref{6.18}, let $\\bar{R}$ be such that $\\bar{R}^p\\lambda=\\lambda_R R^p$. Then $\\bar{R}>R$. \n\nFor each $k=1,2\\cdots$, let (i) $0<\\lambda<\\lambda_k<\\lambda_R$ be such that $\\lambda_k\\downarrow \\lambda$, (ii) \n$R_k=(\\lambda\/\\lambda_k)^{1\/p}\\bar{R}$, and (iii) a unique function $u_k>0$ and $\\delta_k>0$ such that (see Remark \\ref{6.11})\n\\eqRef{6.20}\n\\Delta_p u_k+\\lambda u_k^{p-1}=0,\\;\\;\\mbox{in $B_{R_k}(o)$, $u_k(o)=1$ and $u_k=\\delta_k$ on $\\partial B_{R_k}(o)$.}\n\\end{equation}\nAs seen in Remark \\ref{6.15}, $u_k$ is radial and decreasing. Also, \nClearly, $R_k<\\bar{R}$, for each $k$, and $R_k\\uparrow \\bar{R}.$ Let $1\\le \\ell<k$. Applying \nRemark \\ref{6.5} in $B_{R_\\ell}(o)$, we obtain \n$1\\le v_\\ell(r)\/v_k(r)\\le \\delta_\\ell\/v_k(R_\\ell)$\nand $1\\le v_k(r)\/v_\\ell(r)\\le v_k(R_\\ell)\/\\delta_\\ell$, for $0\\le r\\le R_\\ell.$ Thus, $v_k(R_\\ell)=\\delta_\\ell$, $v_k(r)=v_\\ell(r)$ and $v_k$ extends $v_\\ell$ to $B_{R_k}(o)$. \n\nFor any $x\\in B_{\\bar{R}}(o)$, define $v(x)=v(r)=\\lim_{k\\rightarrow\\infty} v_k(r)$. It clear that $v(x)=v_k(x)$ for any $k$ such that $|x|<R_k$. Also, $v(R_k)=\\delta_k$. Since every $v_k$ is  decreasing in $r$, $v(r)$ is decreasing in $r$ and solves\n$$\\Delta_p v+\\lambda v^{p-1}=0,\\;\\;\\mbox{in $B_{\\bar{R}}(o)$, $v>0$ and $v(o)=1$.}$$\nDefine $v(\\bar{R})=\\lim_{r\\rightarrow \\bar{R}}v(r).$ Thus, $v\\in C(\\overline{B}_{\\bar{R}}(o)).$ Since \n$\\lambda=\\lambda_{\\bar{R}}$, $v(\\bar{R})=0$, otherwise, by Remark \\ref{6.6}, $\\lambda_{\\bar{R}}>\\lambda.$\nUsing scaling, we get existence of a radial eigenfunction on $B_R(o)$.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSuperstring Theory is expected to be a consistent theory of Quantum\nGravity. Therefore, one would like to use it to study gravitational systems in\nwhich quantum-mechanical effects are believed to play an important role, such\nas black holes. In particular, one of the results that we expect from\nSuperstring Theory is a microscopical accounting of the entropy attributed to\nthem by macroscopic (thermodynamic) laws and calculations.\n\nAchieving this result demands, first of all, black-hole solutions of\nSuperstring Theory whose macroscopic entropy can be computed. These are\nclassical solutions of the Superstring effective action. Then, if one manages\nto associate the black-hole solution to a good Superstring Theory background\non which the theory can be quantized, the microscopic entropy can be\nassociated to the density of string states in that background.\n\nIn a seminal paper, \\cite{Strominger:1996sh} Strominger and Vafa completed the\nabove program for a extremal, static, 3-charge 5-dimensional black-hole\nsolution of the type~IIB Superstring Theory at lowest order in the Regge slope\nparameter $\\alpha'$, identifying the associated type~IIB string background as\none with intersecting D1- and D5-branes with momentum flowing along the\nintersection.  Strominger and Vafa argued that, although the black hole only\nsolved the zeroth-order in $\\alpha'$ equations of motion, the higher-order\ncorrections could be made small enough by imposing conditions on the charges\ncarried by the black hole. Under those conditions, the microscopic and\nmacroscopic entropies (the later given simply by the one fourth of the area of\nthe event horizon) matched to lowest order in $\\alpha'$.\n\nSince $\\alpha'$ is the square of the string length, the higher-order in\n$\\alpha'$ corrections to the string effective action, its solutions and the\nproperties of the solutions describe characteristic ``stringy'' deviations and\nthis makes their study most interesting. This study requires:\n\n\\begin{enumerate}\n\\item The knowledge of the higher-order terms in the string effective\n  field-theory actions.\n\\item The construction of solutions of those effective actions with\n  higher-order terms. These solutions can often be viewed as\n  $\\alpha'$-corrected zeroth-order solutions (recovered by setting\n  $\\alpha'=0$).\n\\item The computation of the physical properties of the $\\alpha'$-corrected\n  solutions.\n\\end{enumerate}\n\nTerms of higher-order in $\\alpha'$ are terms of higher order in curvatures and\ntheir complexity grows rapidly with the power of $\\alpha'$. This makes them\nvery difficult to compute and, consequently, our knowledge of the\n$\\alpha'$ corrections to the effective field theory actions of different\nSuperstring Theories is very limited. The $\\alpha'$ corrections to the\nHeterotic Superstring effective action are probably the best known, and they\nhave only been computed to cubic order (quartic in curvatures) in\nRef.~\\cite{Bergshoeff:1989de}, using supersymmetry completion of the Lorentz\nChern-Simons terms \\cite{Bergshoeff:1988nn}.\\footnote{The equivalence of this\n  effective action with previous results obtained in\n  Refs.~\\cite{Callan:1985ia,Gross:1986mw,Metsaev:1987zx,Hull:1987pc} was\n  established in Ref.~\\cite{Chemissany:2007he}.}\n\nWe can use, then, the Heterotic Superstring effective action given in\nRef.~\\cite{Bergshoeff:1989de} for the next step: computing $\\alpha'$\ncorrections to black-hole solutions.  As a matter of fact, the black-hole\nsolution studied by Strominger and Vafa in Ref.~\\cite{Strominger:1996sh} can\nalso be considered as a zeroth-order solution of the Heterotic Superstring\neffective action and it would certainly be interesting to compute its\n$\\alpha'$ corrections, at least to first order. Finding these corrections,\nthough, is a complicated problem. One of the problems is that the complete\nHeterotic Superstring effective action with higher-order corrections has not\nbeen compactified down to the 5 dimensions in which the black hole\nlives.\\footnote{A toroidal compactification to first order in $\\alpha'$ but\n  with no Yang-Mills fields has been recently constructed in\n  Ref.~\\cite{Eloy:2020dko}. The toroidal compactification with only Abelian\n  Yang-Mills fields (which occur at first order in $\\alpha'$) and no terms\n  involving the torsionful spin connection (so the 10-dimensional action is\n  that of $\\mathcal{N}=1,d=10$ supergravity coupled to Abelian vector\n  supermultiplets) was carried out in \\cite{Maharana:1992my}. An earlier\n  compactification of the Heterotic Superstring effective action to just $d=4$\n  at zeroth-order in $\\alpha'$ (so the 10-dimensional action is that of pure\n  $\\mathcal{N}=1,d=10$ supergravity) was carried out in\n  \\cite{Chamseddine:1980cp}.} Effective actions which would capture what are\nbelieved to be the most relevant $\\alpha'$ corrections in lower dimensions\nhave been proposed and used to compute corrections to black-hole solutions\n(see, \\textit{e.g.}~Ref.~\\cite{Mohaupt:2000mj} and references\ntherein). Alternatively, in order to simplify the problem, it has been\nproposed to work only with the near-horizon solution (see\n\\textit{e.g.}~Refs.~\\cite{Sen:2007qy,Prester:2010cw} and references therein\nand more recent work in the Type~IIA compactified on K3 setup\n\\cite{Pang:2019qwq,Chow:2019win}). It is fair to say that each of these\nsimplified approaches has problems of its own and that they do not offer a\ncomplete picture of what the $\\alpha'$-corrected black-hole solutions are\nlike.\n\nRecently, a different approach for computing $\\alpha'$ corrections without\nmaking assumptions about the lower-dimensional effective actions or\nconsidering only near-horizon limits has been proposed in\nRef.~\\cite{Cano:2018qev}: since the 10-dimensional first-order in $\\alpha'$\nHeterotic Superstring effective action is known without any ambiguities (beyond\npossible field redefinitions), first-order in $\\alpha'$ corrections to\nsolutions should be directly computed in 10 dimensions using the uplift of 4-\nor 5-dimensional solutions. Then, the $\\alpha'$-corrected solutions can be\ncompactified back to 4- or 5-dimensions. This approach has been successfully\nused to compute the first-order in $\\alpha'$ corrections to 5- and\n4-dimensional extremal black holes in Refs.~\\cite{Cano:2018qev} and\n\\cite{Chimento:2018kop,Cano:2018brq,Cano:2019oma}, respectively and, more\nrecently, to 4-dimensional non-extremal Reissner-Nordstr\\\"om black holes in\nRef.~\\cite{Cano:2019ycn}. The question of the regularity of the so-called\n\\textit{small black holes} has also been reviewed in\nRef.~\\cite{Cano:2018hut,Ruiperez:2020qda} in the light of those results.\n\nHaving the $\\alpha'$-corrected solutions we can compute their physical\nproperties. For black holes, these are their conserved charges and their\nthermodynamical properties: entropy and temperature. The Hawking temperature is\nalways determined by the value of the surface gravity of the metric. While the\nmetric can receive $\\alpha'$ corrections, the relation between Hawking\ntemperature and surface gravity does not change. This is not the case for the\nBekenstein-Hawking entropy, which, in presence of $\\alpha'$ corrections\n(higher-order in curvature corrections in general) is no longer determined by\nthe area of the horizon which also receives $\\alpha'$ corrections coming from\nthose of the metric. Based on previous work \\cite{Lee:1990nz,Wald:1993nt}, in\nRef.~\\cite{Iyer:1994ys} Iyer and Wald gave a prescription to derive an entropy\nformula in diffeomorphism-invariant theories. The main fact that characterizes\nthis prescription is that the entropy computed using it satisfies the first\nlaw of black-hole mechanics \\cite{Bardeen:1973gs}.\n\nIyer and Wald's prescription is based on a series of assumptions about the\nfield content, which has to consist of tensor fields only. The only tensor\nfield in our current understanding of Nature is the metric,  the rest\nbeing connections and sections of different gauge bundles or, in other words,\nfield with some kind of gauge freedom. The validity of Iyer and Wald's\nprescription has subsequently extended to theories that include fields with\ngauge freedoms in Refs.~\\cite{Jacobson:2015uqa,Prabhu:2015vua,Aneesh:2020fcr},\nbut the Heterotic Superstring effective action (and many other string\neffective actions) include a field which is not a connection or a section of\nsome gauge bundle: the Kalb-Ramond field. This complication has been ignored\nin most of the string literature\\footnote{An independent derivation of an\n  entropy formula using Wald's formalism and dealing with some of the problems\n  that the presence of the Kalb-Ramond field raises has been made in\n  Ref.~\\cite{Edelstein:2019wzg}. The final entropy formula derived there\n  depends on a compensating gauge parameter which was left undetermined. This\n  makes a comparison with the entropy formula we will derive impossible. For\n  instance, it is not possible to compute the entropy of the Strominger-Vafa\n  black hole using this formula, unless one can prove that the unknown term\n  does not contribute to it. Although in that reference it is argued that, at\n  least in certain relevant cases, this is indeed the case. In the same\n  reference it is also shown that the invariance of their entropy formula\n  under local Lorentz transformations depends on it, which seems\n  contradictory.}  and the Iyer-Wald prescription has been naively applied\nwith results that seem to be compatible with the microscopic calculations of\nthe entropy.\\footnote{Wald's formalism's first step consists in the proof of a\n  first law of black-hole mechanics for the theory under consideration. A\n  first law for the Heterotic Superstring effective action to first order in\n  $\\alpha'$ has not yet been proven, although it is widely assumed to exist\n  (for instance, in the derivation of the entropy formula of\n  Ref.~\\cite{Edelstein:2019wzg}).}\n\nFor instance, in Ref.~\\cite{Cano:2018qev}, the entropy of the (heterotic\nversion of the) $\\alpha'$-corrected Strominger-Vafa black hole was computed\nusing the Iyer-Wald prescription directly in the 10-dimensional action. The\nresult obtained was compatible with that of the microscopic calculation\ncarried out in Ref.~\\cite{Kraus:2005zm} to first-order in $\\alpha'$, with an\nappropriate identification between the charges carried by the black hole and\nassociated string background \\cite{Faedo:2019xii}. More precisely, the entropy\nobtained was interpreted in Ref.~\\cite{Cano:2018qev} as the\n$\\mathcal{O}(\\alpha')$ truncation of the expansion in powers of $\\alpha'$ of\nthe exact result found in Ref.~\\cite{Kraus:2005zm}.\n\nThis interpretation, however, was a bit puzzling, because in\nRef.~\\cite{Cano:2018qev}, it was argued that the near-horizon region of the\nblack-hole solution, which determines the entropy, should not receive further\n$\\alpha'$ corrections.\\footnote{The complete black-hole solution may receive\n  further corrections.} Furthermore, an explicit calculation shows that at\nleast the $\\mathcal{O}(\\alpha^{\\prime\\, 2})$ corrections to the entropy vanish\nidentically \\cite{kn:TOM}. All this suggests that the result obtained for the\nentropy in Ref.~\\cite{Cano:2018qev} should be exact to all orders in $\\alpha'$\nand, therefore, it should be identical to the result of the microscopic\ncalculation of Ref.~\\cite{Kraus:2005zm}.\n\nThis puzzle was solved in Ref.~\\cite{Faedo:2019xii}, where it was observed\nthat the dependence of the action on the Riemann curvature\\footnote{According\n  to the Iyer-Wald prescription, the entropy formula only depends on the\n  occurrences of the Riemann tensor in the action.} in the Lorentz\nChern-Simons term of the Kalb-Ramond field strength is changed by dimensional\nreduction. Taking into account this change, which amounts to a factor of 2\nwith respect to the result of Ref.~\\cite{Cano:2018qev}, the macroscopic\nentropy computed at first order in $\\alpha'$ using naively the Iyer-Wald\nformula matches the exact microscopic result. This gives further support to\nthe conjecture that the black-hole solution does not receive further $\\alpha'$\ncorrections and may be considered an exact Heterotic Superstring solution.\n\nThe results of Ref.~\\cite{Faedo:2019xii} made clear that, in the case of the\nHeterotic Superstring effective action, the entropy formula has to be derived\nfrom the dimensionally-reduced action in order to determine correctly the\ndependence of the action of the lower-dimensional Riemann tensor. One of our\ngoals in this paper is to perform the dimensional reduction of the Heterotic\nSuperstring effective action to first order in $\\alpha'$ over a circle to\napply to it the Iyer-Wald prescription and obtain an entropy formula. This\nentropy formula can only be applied to $d$-dimensional black holes that can be\nobtained by trivial compactification on T$^{9-d}$ and a non-trivial\ncompactification on a circle. For instance, it can be applied to the heterotic\nversion of the Strominger-Vafa black hole because it can be obtained from a\n10-dimensional solution by trivial compactification on T$^{4}$, to 6\ndimensions and a non-trivial compactification on a circle from 6 to 5\ndimensions. It can also be applied to the non-supersymmetric 4-dimensional\nReissner-Nordstr\\\"om black hole of Ref.~\\cite{Khuri:1995xq}, which can be\nobtained from pure 5-dimensional gravity and, therefore, can be obtained from\na purely gravitational 10-dimensional solution by trivial compactification on\nT$^{5}$ to 5 dimensions and, then, by a non-trivial compactification on a\ncircle from 5 to 4 dimensions. Actually, the entropy formula\nEq.~(\\ref{eq:entropyformulastringframe}) that we are going to derive in\nSection~\\ref{sec-entropyformula} has been applied to a non-extremal version of\nthe 4-dimensional Reissner-Nordstr\\\"om black hole we just discussed, in\nRef.~\\cite{Cano:2019ycn}. While the microscopic interpretation of the entropy\nof this black hole is unknown, being a black hole with finite temperature, one\ncan check that the first law of thermodynamics is indeed satisfied because the\ntemperature computed from the $\\alpha'$-corrected metric and the entropy\ncomputed from the $\\alpha'$-corrected metric with the $\\alpha'$-corrected\nentropy formula are related by the thermodynamic relation\n\n\\begin{equation}\n\\frac{\\partial S}{\\partial M}=\\frac{1}{T}\\, .\n\\end{equation}\n\nThis paper's second goal has to do with one of the most interesting and\ncharacteristic properties of String Theory: T~duality.\\footnote{For a review\n  with many early references see Ref.~\\cite{Alvarez:1994dn}.} T~duality\nrelates two string theories compactified in circles of dual radii. The spectra\nof the two theories can be put into one-to-one correspondence and, from the\nlower dimensional point of view, they are essentially identical, up to charge\nidentifications.\\footnote{Charges related to Kaluza-Klein momentum and charges\n  related to the winding number along the compact direction should be\n  interchanged.} More generally, Buscher \\cite{Buscher:1987sk,Buscher:1987qj}\nshowed that two string backgrounds with one isometry whose background fields\nare related by the so-called \\textit{Buscher T~duality rules} are equivalent.\n\nString backgrounds related by T~duality may have very different geometries and\nproperties in spite of their stringy equivalence. On the other hand, T~duality\n(via the Buscher rules) can be used actively to generate new (dual) string\nbackgrounds from known backgrounds. This is what makes this duality so\ninteresting.\n\nPerhaps not surprisingly, the Buscher rules can be derived from the string\neffective action: the dual\\footnote{That is, with fields related by the\n  Buscher rules.} Kaluza-Klein compactifications of two effective actions on a\ncircle give the same $(d-1)$-dimensional action and the same equations of\nmotion.  In practice, one can perform identical Kaluza-Klein\ncompactifications, determine the relation between the $(d-1)$-dimensional\nfields of the two actions (which is usually very simple because it does not\ninvolve the $(d-1)$-dimensional string metric or Kalb-Ramond field) and\nrewrite this relation in terms of the components of the original\n$d$-dimensional fields \\cite{Bergshoeff:1994dg}. This relation is just the\nBuscher T~duality rules. This strategy has been successfully used to find the\nextension of the Buscher T~duality rules that relates equivalent type~IIA and\ntype~IIB superstring backgrounds \\cite{Bergshoeff:1995as} and higher-rank\nRamond-Ramond potentials \\cite{Meessen:1998qm}.\n\nIn the context of the Heterotic Superstring, this strategy was used in\n\\cite{Bergshoeff:1995cg} to find the first-order in $\\alpha'$ corrections to\nthe Buscher rules.\\footnote{At zeroth-order in $\\alpha'$, the Heterotic\n  Superstring effective action only describes the so-called \\textit{common\n    sector} of Neveu-Schwarz-Neveu-Schwarz fields, so the Buscher rules are\n  just those found by Buscher.} Only the Yang-Mills fields were included at\norder $\\alpha'$, but, taking into account that the torsionful spin connection\nenters the action in exactly the same way as the Yang-Mills fields\n\\cite{Bergshoeff:1988nn}, it was possible to find the $\\alpha'$ corrections to\nthe Buscher rules.\n\nThe $\\alpha'$-corrected Buscher rules are of no use if there are no\n$\\alpha'$-corrected solutions at one's disposal to generate new solutions or to check their equivalence. For this reason,\nthe results of Ref.~\\cite{Bergshoeff:1995cg} were sleeping the ``sleep of the\njust''\\footnote{As a matter of fact, they have partially re-derived several\n  times \\cite{Serone:2005ge,Bedoya:2014pma}. Other studies of the effect of\n  $\\alpha'$ corrections on T~duality and O$(d,d)$ transformations in toroidal\n  compactifications, sometimes in extended set-ups (such as Double Field\n  Theory) can be found\n  \\cite{Meissner:1996sa,Kaloper:1997ux,Hohm:2014eba,Marques:2015vua,Baron:2017dvb,Eloy:2020dko}.}\nuntil quite recently, when they were first applied to $\\alpha'$-corrected\nself-T-dual solutions, providing a highly non-trivial test of both the\n$\\alpha'$ corrections of the solutions and of the T~duality rules.\n\nOur second goal will be to study the T~duality invariance of the complete\ndimensionally-reduced Heterotic Superstring effective action and of the\nentropy formula that follows from it. While the $\\alpha'$-corrected Buscher\nrules will be those of Ref.~\\cite{Bergshoeff:1995cg}, the complete reduced\naction will have many more $\\mathcal{O}(\\alpha')$ terms  than the action\nobtained there. The invariance of the action under T~duality suggests that\nthey will contribute to the entropy in a T~duality-invariant form, and we will\nprove that this is the case.\\footnote{\\label{foot:esa} It follows trivially\n  from the invariance of the lower-dimensional string metric and dilaton under\n  T~duality that the zeroth-order in $\\alpha'$ temperature and entropy (the\n  area) are also T~duality invariant. This property was proven by Horowitz and\n  Welch in Ref.~\\cite{Horowitz:1993wt} before the relation between the Buscher\n  rules and dimensional reduction was established in\n  Ref.~\\cite{Bergshoeff:1994dg}. Recently, it has been investigated again from\n  the same point of view in Refs.~\\cite{Edelstein:2018ewc,Edelstein:2019wzg}\n  to first order in $\\alpha'$, but, again, the relation between dimensional\n  reduction and T~duality and the invariance of the lower-dimensional string\n  metric and dilaton field lead, trivially, to the invariance of the\n  $\\alpha'$-corrected temperature. The invariance of the action under\n  T~duality at this order implies that of the entropy formula using the\n  Iyer-Wald prescription because the Riemann curvature is T~duality\n  invariant.}\n\nThis paper is organized as follows: we introduce the Heterotic Superstring\neffective action to first order in $\\alpha'$ following\nRef.~\\cite{Bergshoeff:1989de} in Section~\\ref{sec-heteroticalpha}. In\nSection~\\ref{sec-dimredO1}, we revisit the dimensional reduction on a circle of\nthe action at zeroth order in $\\alpha'$ as a warm-up exercise and also because\nwe will need some of the results when we consider the higher-order terms in\nSection~\\ref{sec-dimredOalpha}. In that section we will obtain the complete\ndimensionally-reduced action to first order in $\\alpha'$, we will find the\nT~duality rules and we will prove the invariance of the action under those\nT~duality rules.  In Section~\\ref{sec-entropyformula}, we will use the\ndimensionally-reduced T~duality-invariant action to derive an entropy formula\nusing the Iyer-Wald prescription and we will apply it to the heterotic version\nof the $\\alpha'$-corrected Strominger-Vafa black hole of\nRef.~\\cite{Cano:2018qev}.  We will end by discussing our results and future\nwork on these topics in Section~\\ref{sec-discussion}.\n\n\\section{The Heterotic Superstring effective action to\n  \\texorpdfstring{$\\mathcal{O}(\\alpha')$}{O(\u03b1')}}\n\\label{sec-heteroticalpha}\n\nLet us start by reviewing the Heterotic Superstring effective action to\n$\\mathcal{O}(\\alpha')$. We will use the formulation given in\nRef.~\\cite{Bergshoeff:1989de}, but written in the conventions of\nRef.~\\cite{Ortin:2015hya}.\\footnote{The relation with the fields in\n  Ref.~\\cite{Bergshoeff:1989de} can be found in\n  Ref.~\\cite{Fontanella:2019avn}.} In this formulation, the action is\nconstructed recursively order by order in $\\alpha'$.\n\nThe zeroth-order 3-form field strength of the Kalb-Ramond 2-form $B$\nis defined as\n\n\\begin{equation}\nH^{(0)}{}_{\\mu\\nu\\rho} \\equiv 3\\partial_{[\\mu}B_{\\nu\\rho]}\\, ,\n\\end{equation}\n\n\\noindent\nand it contributes as torsion to the zeroth-order torsionful spin\nconnections\n\n\\begin{equation}\n{\\Omega}^{(0)}_{(\\pm)\\, \\mu}{}^{{a}}{}_{{b}} \n=\n{\\omega}_{\\mu}{}^{{a}}{}_{{b}}\n\\pm\n\\tfrac{1}{2}{H}^{(0)}{}_{{\\mu}}{}^{{a}}{}_{{b}}\\, ,\n\\end{equation}\n\n\\noindent\nwhere ${\\omega}_{\\mu}{}^{{a}}{}_{{b}}$ is the (torsionless, metric-compatible)\nLevi-Civita spin connection 1-form.\n\nThe corresponding zeroth-order Lorentz curvature 2-forms and\nChern-Simons 3-forms are defined as\n\n\\begin{eqnarray}\n  \\label{eq:R0def}\n{R}^{(0)}_{(\\pm)\\, \\mu\\nu}{}^{{a}}{}_{{b}}\n& = & \n2\\partial_{[\\mu|} {\\Omega}^{(0)}_{(\\pm)\\, |\\nu]}{}^{{a}}{}_{{b}}\n-2 {\\Omega}^{(0)}_{(\\pm)\\, [\\mu|}{}^{{a}}{}_{{c}}\\,\n{\\Omega}^{(0)}_{(\\pm)\\, |\\nu]}{}^{{c}}{}_{{b}}\\, ,\n\\\\\n  & & \\nonumber \\\\\n  \\label{eq:oL0def}\n{\\omega}^{{\\rm L}\\, (0)}_{(\\pm)}\n& = &  \n3{R}^{ (0)}_{(\\pm)\\, [\\mu\\nu|}{}^{{a}}{}_{{b}} \n{\\Omega}^{ (0)}_{(\\pm)\\, |\\rho]}{}^{{b}}{}_{{a}} \n+2\n{\\Omega}^{ (0)}_{(\\pm)\\, [\\mu|}{}^{{a}}{}_{{b}} \\,\n{\\Omega}^{ (0)}_{(\\pm)\\, |\\nu|}{}^{{b}}{}_{{c}} \\,\n{\\Omega}^{ (0)}_{(\\pm)\\, |\\rho]}{}^{{c}}{}_{{a}}\\, .  \n\\end{eqnarray}\n\nThe gauge field 1-form is $A^{A}{}_{\\mu}$, where $A,B,C,\\ldots$ are the\nadjoint gauge indices of some group that we will not specify. The gauge field\nstrength and the Chern-Simons 3-forms are defined by\n\n\\begin{eqnarray}\n{F}^{A}{}_{\\mu\\nu}\n& = & \n2\\partial_{[\\mu}{A}^{A}{}_{\\nu]}+f_{BC}{}^{A}{A}^{B}{}_{[\\mu}{A}^{C}{}_{\\nu]}\\, , \n\\\\\n& & \\nonumber \\\\\n{\\omega}^{\\rm YM}\n& = & \n3F_{A\\, [\\mu\\nu}{A}^{A}{}_{\\rho]}\n-f_{ABC}{A}^{A}{}_{[\\mu}{A}^{B}{}_{\\nu}{A}^{C}{}_{\\rho]}\\, ,\n\\end{eqnarray}\n\n\\noindent\nwhere we have lowered the adjoint group indices using the Killing metric of\n$K_{AB}$: $f_{ABC}\\equiv f_{AB}{}^{D}K_{DC}$ and of the gauge fields\n$F_{A\\, \\mu\\nu}\\equiv K_{AB}F^{B}{}_{\\mu\\nu}$.\n\nThen, at first order\n\n\\begin{eqnarray}\nH^{(1)}{}_{\\mu\\nu\\rho}\n& = & \n3\\partial_{[\\mu}B_{\\nu\\rho]}\n+\\frac{\\alpha'}{4}\\left({\\omega}^{\\rm YM}{}_{\\mu\\nu\\rho}\n+{\\omega}^{{\\rm L}\\, (0)}_{(-)\\, \\mu\\nu\\rho}\\right)\\, ,  \n\\\\\n& & \\nonumber \\\\\n{\\Omega}^{(1)}_{(\\pm)\\, \\mu}{}^{{a}}{}_{{b}} \n& = & \n{\\omega}_{\\mu}{}^{{a}}{}_{{b}}\n\\pm\n\\tfrac{1}{2}{H}^{(1)}_{{\\mu}}{}^{{a}}{}_{{b}}\\, ,\n\\\\\n& & \\nonumber \\\\\n{R}^{(1)}_{(\\pm)\\, \\mu\\nu}{}^{{a}}{}_{{b}}\n& = & \n2\\partial_{[\\mu|} {\\Omega}^{(1)}_{(\\pm)\\, |\\nu]}{}^{{a}}{}_{{b}}\n-2 {\\Omega}^{(1)}_{(\\pm)\\, [\\mu|}{}^{{a}}{}_{{c}}\\,\n{\\Omega}^{(1)}_{(\\pm)\\, |\\nu]}{}^{{c}}{}_{{b}}\\, ,\n\\\\\n& & \\nonumber \\\\\n{\\omega}^{{\\rm L}\\, (1)}_{(\\pm)\\, \\mu\\nu\\rho}\n& = &  \n3{R}^{ (1)}_{(\\pm)\\, [\\mu\\nu|}{}^{{a}}{}_{{b}} \n{\\Omega}^{ (1)}_{(\\pm)\\, |\\rho]}{}^{{b}}{}_{{a}} \n+2\n{\\Omega}^{(1)}_{(\\pm)\\, [\\mu|}{}^{{a}}{}_{{b}}\\, \n{\\Omega}^{(1)}_{(\\pm)\\, |\\nu|}{}^{{b}}{}_{{c}}\\, \n{\\Omega}^{(1)}_{(\\pm)\\, |\\rho]}{}^{{c}}{}_{{a}}\\, .  \n \\\\\n& & \\nonumber \\\\\nH^{(2)}{}_{\\mu\\nu\\rho}\n& = & \n3\\partial_{[\\mu}B_{\\nu\\rho]}\n+\\frac{\\alpha'}{4}\\left({\\omega}^{\\rm YM}{}_{\\mu\\nu\\rho}\n+{\\omega}^{{\\rm L}\\, (1)}_{(-)\\, \\mu\\nu\\rho}\\right)\\, ,  \n\\end{eqnarray}\n\n\\noindent\netc.\n\nOnly\n$\\Omega^{(0)}_{(\\pm)\\, \\mu},{R}^{(0)}_{(\\pm)\\, \\mu\\nu}{}^{a}{}_{b},\n\\omega^{{\\rm L}\\, (0)}_{(\\pm)\\, \\mu\\nu\\rho}$ and $ H^{(1)}{}_{\\mu\\nu\\rho}$\n(plus the Yang-Mills fields) occur in the action.  In practice, though, it is\nmore convenient to work with the higher-order objects, neglecting the terms of\nhigher order in $\\alpha'$ when necessary. Thus, from now on we will suppress\nthe $(n)$ upper indices when they do not play a relevant role.\n\nIn terms of all these objects, the Heterotic Superstring effective action in\nthe string frame and to first-order in $\\alpha'$ can be written as\n\n\\begin{equation}\n\\label{heterotic}\n{S}\n=\n\\frac{g_{s}^{2}}{16\\pi G_{N}^{(10)}}\n\\int d^{10}x\\sqrt{|{g}|}\\, \ne^{-2{\\phi}}\\, \n\\left\\{\n{R} \n-4(\\partial{\\phi})^{2}\n+\\tfrac{1}{12}{H}^{2}\n-\\dfrac{\\alpha'}{8}\\left[\n{F}_{A}\\cdot {F}^{A}\n+\n{R}_{(-)}{}^{{a}}{}_{{b}}\\cdot\n{R}_{(-)}{}^{{b}}{}_{{a}}\n\\right]\n\\right\\}\\, ,\n\\end{equation}\n\n\\noindent\nwhere $G_{N}^{(10)}$ is the 10-dimensional Newton constant, $\\phi$ is the\ndilaton field, the vacuum expectation value of $e^{\\phi}$ is the Heterotic\nSuperstring coupling constant $g_{s}$, $R$ is the Ricci scalar of the\nstring-frame metric $g_{\\mu\\nu}$ and the dot indicates the contraction of the\nindices of 2-forms:\n${F}_{A}\\cdot {F}^{A}\\equiv {F}_{A\\, \\mu\\nu}{F}^{A\\, \\mu\\nu}$.\n\n\n\\section{Dimensional reduction on S$^{1}$ at zeroth order in $\\alpha'$}\n\\label{sec-dimredO1}\n\nAs a warm-up exercise (and also because of the recursive definition of the\naction that will make necessary the zeroth-order fields in the first-order\naction), we review the well-known dimensional reduction of the action at\nzeroth order in $\\alpha'$ using the Scherk-Schwarz formalism\n\\cite{Scherk:1979zr}. We add hats to all the 10-dimensional objects (fields,\nindices, coordinates) and split the 10-dimensional world indices as\n$(\\hat{\\mu})=(\\mu,\\underline{z})$ and the 10-dimensional indices as\n$(\\hat{a})=(a,z)$.\n\nThe Zehnbein and inverse-Zehnbein components $\\hat{e}_{\\hat{\\mu}}{}^{\\hat{a}}$\nand $\\hat{e}_{\\hat{a}}{}^{\\hat{\\mu}}$ can be put in an upper-triangular form\nby a local Lorentz transformation and, then, they can be decomposed in terms\nof the 9-dimensional Vielbein and inverse Vielbein components\n$e_{\\mu}{}^{a},e_{a}{}^{\\mu}$, Kaluza-Klein (KK) vector $A_{\\mu}$ and KK\nscalar $k$ as\n\n\\begin{equation}\n\\label{eq:standardVielbeinAnsatz}\n\\left( \\hat{e}_{\\hat{\\mu}}{}^{\\hat{a}} \\right) = \n\\left(\n\\begin{array}{c@{\\quad}c}\ne_{\\mu}{}^{a} & kA_{\\mu} \\\\\n&\\\\[-3pt]\n0       & k    \\\\\n\\end{array}\n\\right)\\!, \n\\hspace{1cm}\n\\left(\\hat{e}_{\\hat{a}}{}^{\\hat{\\mu}} \\right) =\n\\left(\n\\begin{array}{c@{\\quad}c}\ne_{a}{}^{\\mu} & -A_{a}  \\\\\n& \\\\[-3pt]\n0       & k^{-1} \\\\\n\\end{array}\n\\right)\\!,\n\\end{equation}\n\n\\noindent\nwhere $A_{a}= e_{a}{}^{\\mu} A_{\\mu}$. We will always assume that all the\n9-dimensional fields with Lorentz indices are 9-dimensional world tensors\ncontracted with the 9-dimensional Vielbeins. For instance, the KK fields\nstrength $F_{ab}$ is\n\n\\begin{equation}\n\\label{eq:KKfieldstrength}\nF_{ab} = e_{a}{}^{\\mu} e_{b}{}^{\\nu} F_{\\mu\\nu},\n\\hspace{1cm}\nF_{\\mu\\nu} \\equiv 2\\partial_{[\\mu}A_{\\nu]},\n\\end{equation}\n\n\nThe components of the 10-dimensional spin connection\n$\\hat{\\omega}_{\\hat{a}\\hat{b}\\hat{c}}$ decompose into those of the\n9-dimensional one $\\omega_{abc}$ and $F_{ab}$ as\n\n\\begin{equation}\n\\label{eq:standardspinconnectionreduction}\n\\begin{array}{rclrcl}\n\\hat{\\omega}_{abc} & = & \\omega_{abc}, &\n\\hat{\\omega}_{abz} & = & \\frac{1}{2} k F_{ab},\n\\\\\n& & & & & \\\\\n\\hat{\\omega}_{zbc} & = & -\\frac{1}{2} k F_{bc},\n\\hspace{1.5cm}& \n\\hat{\\omega}_{zbz} & = & -\\partial_{b} \\ln{k}.\\\\\n\\end{array}\n\\end{equation}\n\nThen, using the Palatini identity, it is not difficult to see that the first\ntwo terms in the action Eq.~(\\ref{heterotic}) take the following 9-dimensional\nform (up to a total derivative):\n\n\\begin{equation}\n  \\begin{aligned}\n    \\int d^{10}\\hat{x}\\sqrt{|\\hat{g}|}\\, e^{-2\\hat{\\phi}}\\, \\left\\{ \\hat{R}\n      -4(\\partial\\hat{\\phi})^{2} \\right\\}\n        & = \\\\\n        & \\\\\n        & \\hspace{-3cm}\n        \\int dz\\int d^{9}x\\sqrt{|g|}\\,\n        e^{-2\\phi}\\, \\left\\{ R -4(\\partial\\phi)^{2} +(\\partial \\log{k})^{2}\n          -\\tfrac{1}{4}k^{2}F^{2} \\right\\}\\, ,\n          \\end{aligned}\n\\end{equation}\n\n\\noindent\nwhere the 9-dimensional dilaton field is related to the 10-dimensional one by\n\n\\begin{equation}\n\\phi \\equiv \\hat{\\phi} -\\tfrac{1}{2}\\log{k}\\, .  \n\\end{equation}\n\nAt zeroth order in $\\alpha'$, the last term that we have to reduce is the\nkinetic term of the Kalb-Ramond 2-form $\\sim \\hat{H}^{(0)\\,2}$. Following\nScherk and Schwarz, we consider the Lorentz components of the 3-form field\nstrength, because they are automatically gauge-invariant combinations. The\n$\\hat{H}^{(0)}{}_{abz}$ components give\n\n\\begin{equation}\n  \\hat{H}^{(0)}{}_{abz}\n  =\n  k^{-1}e_{a}{}^{\\mu}e_{b}{}^{\\nu} \\hat{H}^{(0)}{}_{\\mu\\nu\\underline{z}} \n =\n k^{-1}e_{a}{}^{\\mu}e_{b}{}^{\\nu} 2\\partial_{[\\mu} \\hat{B}_{\\nu]\\underline{z}}\\, .\n\\end{equation}\n\nIt is, then, appropriate to define the zeroth-order ``winding''\\footnote{This\n  vector couples electrically to the string modes with non-vanishing winding\n  numbers, just as the KK vector field couples to those with non-vanishing\n  momentum in the internal direction.} vector field $B^{(0)}{}_{\\mu}$ and its\nfield strength $G^{(0)}{}_{\\mu\\nu}$ by\n\n\\begin{equation}\n  B^{(0)}{}_{\\mu} \\equiv \\hat{B}_{\\mu\\underline{z}}\\, ,\n  \\hspace{1cm}\n  G^{(0)}{}_{\\mu\\nu} \\equiv 2\\partial_{[\\mu}B^{(0)}{}_{\\nu]}\\, ,\n\\end{equation}\n\n\\noindent\nso that \n\n\\begin{equation}\n  \\hat{H}^{(0)}{}_{abz}\n  =\n  k^{-1}G^{(0)}{}_{ab}\\, .\n\\end{equation}\n\nThe second gauge-invariant combination is\n\n\\begin{equation}\n  \\hat{H}^{(0)}{}_{abc}\n  =\n  e_{a}{}^{\\mu}e_{b}{}^{\\nu}e_{c}{}^{\\rho}\n  \\left(\n    \\hat{H}^{(0)}{}_{\\mu\\nu\\rho} -3 A_{[\\mu}\\hat{H}^{(0)}{}_{\\nu\\rho]\\underline{z}}\n  \\right)\\, ,\n\\end{equation}\n\n\\noindent\nwhich suggests the definition\n\n\\begin{equation}\n  H^{(0)}{}_{\\mu\\nu\\rho}\n  \\equiv\n  \\hat{H}^{(0)}{}_{\\mu\\nu\\rho} -3A_{[\\mu}\\hat{H}^{(0)}{}_{\\nu\\rho]\\underline{z}}\n= 3\\partial_{[\\mu}\\hat{B}_{\\nu\\rho]} -6A_{[\\mu}\\partial_{\\nu}\nB^{(0)}{}_{\\rho]}\\, .\n\\end{equation}\n\nWe could simply identify $\\hat{B}_{\\nu\\rho}$ with the 9-dimensional\nKalb-Ramond field, but it is customary (and convenient) to use a\nT~duality-invariant definition. T~duality will interchange KK momentum and\nwinding, and therefore, will interchange $A_{\\mu}$ with $B^{(0)}{}_{\\mu}$,\nmodifying the Chern-Simons term in the above form of $H_{\\mu\\nu\\rho}$. We can,\nhowever, rewrite it in the form\n\n\\begin{equation}\n  H^{(0)}{}_{\\mu\\nu\\rho}\n  = 3\\partial_{[\\mu}\\left(\\hat{B}_{\\nu\\rho]} +A_{|\\nu}B^{(0)}{}_{\\rho]}\\right)\n  -\\tfrac{3}{2}A_{[\\mu}G^{(0)}{}_{\\nu\\rho]}-\\tfrac{3}{2}B^{(0)}{}_{[\\mu}F_{\\nu\\rho]} \\, ,\n\\end{equation}\n\n\\noindent\nand identify the T~duality-invariant 9-dimensional Kalb-Ramond 2-form\n\n\\begin{equation}\nB^{(0)}{}_{\\mu\\nu} \\equiv \\hat{B}_{\\mu\\nu} +A_{[\\mu}B^{(0)}{}_{\\nu]}\\, ,  \n\\end{equation}\n\n\\noindent\nwith the final result\n\n\\begin{equation}\n  H^{(0)}{}_{\\mu\\nu\\rho}\n  = 3\\partial_{[\\mu}\\hat{B}^{(0)}{}_{\\nu\\rho]}\n  -\\tfrac{3}{2}A_{[\\mu}G^{(0)}{}_{\\nu\\rho]}-\\tfrac{3}{2}B^{(0)}{}_{[\\mu}F_{\\nu\\rho]}\\, .\n\\end{equation}\n\nThen, after integrating over the length of the compact coordinate $z$\n($2\\pi\\ell_{s}$ by convention) the 9-dimensional action to zeroth order in\n$\\alpha'$ takes the form\n\n\\begin{equation}\n  \\label{eq:heterotic9order0}\n  S\n  =\n  \\frac{g_{s}^{2}(2\\pi\\ell_{s})}{16\\pi G_{N}^{(10)}}\n  \\int d^{9}x\\sqrt{|g|}\\,\n  e^{-2\\phi}\\, \\left\\{ R -4(\\partial\\phi)^{2} +(\\partial \\log{k})^{2}\n    -\\tfrac{1}{4}k^{2}F^{2}  -\\tfrac{1}{4}k^{-2}G^{(0)\\, 2}\n    +\\tfrac{1}{12}H^{(0)\\, 2}\\right\\}\\, .\n\\end{equation}\n\nThis action is invariant under the T~duality transformations\n\n\\begin{equation}\n  \\label{eq:0thorderTduality}\n  A_{\\mu}'\n  = B^{(0)}{}_{\\mu}\\, , \n  \\hspace{1cm}\n  B^{(0)}{}_{\\mu}'\n  = \n  A_{\\mu}\\, ,\n  \\hspace{1cm}\n  k'\n   = \n  1\/k\\, .\n\\end{equation}\n\nTaking into account the relations between the 10- and 9-dimensional fields,\ncollected in Appendix~\\ref{sec-10versus9atorder0}, it is easy to see that the\nabove T~duality transformations correspond to the following transformations of\nthe 10-dimensional fields known as \\textit{Buscher rules}\n\\cite{Buscher:1987sk,Buscher:1987qj}:\n\n\\begin{equation}\n\\label{eq:Buscherrules}\n\\begin{array}{rclrcl}\n\\hat{g}^{\\prime}_{\\underline{z}\\underline{z}} & = &\n1\/\\hat{g}_{\\underline{z}\\underline{z}}\\, , &\n\\hat{B}^{\\prime}_{\\mu \\underline{z}} & = &\n\\hat{g}_{\\mu \\underline{z}}\/\\hat{g}_{\\underline{z}\\underline{z}}\\, ,\n\\\\\n& & & & &\n\\\\\n\\hat{g}^{\\prime}_{\\mu \\underline{z}} & = &\n\\hat{B}_{\\mu \\underline{z}}\/\\hat{g}_{\\underline{z}\\underline{z}}\\, , &\n\\hat{B}^{\\prime}_{\\mu\\nu} & = &\n\\hat{B}_{\\mu\\nu}+2\\hat{g}_{[\\mu|\\underline{z}|}\n\\hat{B}_{\\nu] \\underline{z}}\/\\hat{g}_{\\underline{z}\\underline{z}}\\, ,\n\\\\\n& & & & &\n\\\\\n\\hat{g}^{\\prime}_{\\mu\\nu} & = &\n\\hat{g}_{\\mu\\nu}-(\\hat{g}_{\\mu \\underline{z}}\\hat{g}_{\\nu \\underline{z}}-\n\\hat{B}_{\\mu \\underline{z}}\\hat{B}_{\\nu \\underline{z}})\n\/\\hat{g}_{\\underline{z}\\underline{z}}\\, , &\n\\hat{\\phi}^{\\prime} & = & \\hat{\\phi} -\\frac{1}{2}\\ln\n|\\hat{g}_{\\underline{z}\\underline{z}}|\\, .\n\\end{array}\n\\end{equation}\n\n\n\\section{Dimensional reduction on S$^{1}$ at $\\mathcal{O}(\\alpha'$)}\n\\label{sec-dimredOalpha}\n\nThe reduction of the first two terms in the effective action is not modified\nby the inclusion of $\\alpha'$ corrections. The definitions of 9-dimensional\nmetric, dilaton and KK vector and scalar in terms of the 10-dimensional fields\nare not modified by them either.  We expect modifications in the definitions\nof the 9-dimensional Kalb-Ramond 2-form and of the winding vector, though,\nbecause of the presence of the Lorentz and Yang-Mills Chern-Simons 3-forms in\n$\\hat{H}^{(1)}$.\n\nIt is convenient to start by studying the dimensional reduction of the\nYang-Mills fields. The Lorentz-indices decomposition of the gauge field is\n\n\\begin{subequations}\n  \\begin{align}\n    \\hat{A}^{A}{}_{z}\n    & =\n      k^{-1}\\hat{A}^{A}{}_{\\underline{z}}\\, ,\n      \\\\\n    & \\nonumber \\\\\n    \\hat{A}^{A}{}_{a}\n    & =\n      e_{a}{}^{\\mu}( \\hat{A}^{A}{}_{\\mu}- \\hat{A}^{A}{}_{\\underline{z}}A_{\\mu})\\, ,\n  \\end{align}\n\\end{subequations}\n\n\\noindent\nwhich leads to the definition of the 9-dimensional adjoint scalars $\\phi^{A}$\nand gauge vectors\n\n\\begin{subequations}\n  \\begin{align}\n    \\varphi^{A}\n    & \\equiv\n      k^{-1}\\hat{A}^{A}{}_{\\underline{z}}\\, ,\n    \\\\\n    & \\nonumber \\\\\n    A^{A}{}_{\\mu}\n    & \\equiv\n      \\hat{A}^{A}{}_{\\mu}- \\hat{A}^{A}{}_{\\underline{z}}A_{\\mu}\\, .\n  \\end{align}\n\\end{subequations}\n\nIn terms of these variables, it is not difficult to see that the\ncomponents of 10-dimensional gauge field strength are given by\n\n\\begin{subequations}\n  \\label{eq:YMreduction}\n  \\begin{align}\n    \\hat{F}^{A}{}_{az}\n    & =\n      \\mathfrak{D}_{a}\\varphi^{A} +\\varphi^{A}\\partial_{a}\\log{k}\\, ,\n    \\\\\n    & \\nonumber \\\\\n    \\hat{F}^{A}{}_{ab}\n    & =\n      F^{A}{}_{ab} +k\\varphi^{A}F_{ab}\\, ,\n  \\end{align}\n\\end{subequations}\n\n\\noindent\nwhere $ F^{A}{}_{\\mu\\nu}$ is the standard Yang-Mills gauge field strength for\nthe 9-dimensional gauge fields $A^{A}{}_{\\mu}$.\n\nThe reduction of the first, second and fourth terms in the action\nEq.~(\\ref{heterotic}) gives (up to a total derivative)\n\n\\begin{equation}\n  \\begin{aligned}\n    \\int dz\\int d^{9}x \\sqrt{|g|}e^{-2\\phi} \\left\\{ R -4(\\partial\\phi)^{2}\n      +\\left(1+\\frac{\\alpha'}{4}\\varphi^{2}\\right)(\\partial\n      \\log{k})^{2}\n      +\\frac{\\alpha'}{4}(\\mathfrak{D}\\varphi)^{2}\n      +\\frac{\\alpha'}{4}\\partial_{a}\\log{k}\\partial^{a}\\varphi^{2}\n      \\right.\n      & \\\\\n      & \\\\\n      \\left.\n      -\\tfrac{1}{4}\\left(1+\\frac{\\alpha'}{2}\\varphi^{2}\\right)k^{2}F^{2}\n      -\\frac{\\alpha'}{8}\\left(F_{A}\\cdot F^{A}\n      +2\\varphi_{A}F^{A}\\cdot  kF\\right) \\right\\}\\, ,\n  &\n  \\end{aligned}\n\\end{equation}\n\n\\noindent\nwhere $\\varphi^{2}\\equiv \\varphi_{A}\\varphi^{A}$,\n$\\mathfrak{D}_{\\mu}\\varphi^{A}= \\partial_{\\mu}\\varphi^{A}\n+f_{BC}{}^{A}A^{B}{}_{\\mu}\\varphi^{C}$ etc.\n\nLet us now consider the reduction of the Kalb-Ramond 3-form field strength\n$\\hat{H}^{(1)}$, starting with the gauge-invariant combination\n\n\\begin{equation}\n  \\hat{H}^{(1)}{}_{abz}\n  =\n  k^{-1}e_{a}{}^{\\mu}e_{b}{}^{\\nu} \\hat{H}^{(1)}{}_{\\mu\\nu\\underline{z}} \n =\n k^{-1}e_{a}{}^{\\mu}e_{b}{}^{\\nu}\n \\left\\{ 2\\partial_{[\\mu} \\hat{B}_{\\nu]\\underline{z}}\n   +\\frac{\\alpha'}{4}\\left(\\hat{\\omega}^{\\rm YM}{}_{\\mu\\nu\\underline{z}}\n     +\\hat{\\omega}^{\\rm L (0)}_{(-)\\, \\mu\\nu\\underline{z}}\\right)\n   \\right\\}\\, .\n\\end{equation}\n\n\\noindent\nUsing the above results for the Yang-Mills fields we find that\n\n\\begin{equation}\n  \\label{eq:oYMmnz}\n  \\hat{\\omega}^{\\rm YM}{}_{\\mu\\nu\\underline{z}}\n  =\n  k\\varphi_{A}\\left(2F^{A}{}_{\\mu\\nu} +\\varphi^{A}kF_{\\mu\\nu} \\right)\n  -2\\partial_{[\\mu|}\\left(k\\varphi_{A}A^{A}_{|\\nu]} \\right)\\, .\n\\end{equation}\n\n\\noindent\nThe last term is a total derivative that can be absorbed into the definition\nof the 9-dimensional vector field $B^{(1)}{}_{\\mu}$ and the remaining terms\nare manifestly gauge-invariant 2-forms.\n\nWe can use this result in the reduction of the Lorentz Chern-Simons 3-form;\nafter all, the only difference with the Yang-Mills Chern-Simons 3-form is the\ngauge group, which now is the 10-dimensional Lorentz group. This is,\nnevertheless, an important difference because this group is broken down to the\n9-dimensional Lorentz group times U$(1)$ and we will have to take this fact\ninto account in a second step.\n\nIn order to profit from the previous result, we introduce the following\nnotation\n\n\\begin{subequations}\n  \\begin{align}\n    \\hat{A}^{\\hat{a}\\hat{b}}{}_{\\hat{\\mu}}\n    & \\equiv\n      \\hat{\\Omega}^{(0)}_{(-)\\, \\hat{\\mu}}{}^{\\hat{a}\\hat{b}}\\, ,\n    \\\\\n    & \\nonumber \\\\\n    \\hat{F}^{\\hat{a}\\hat{b}}{}_{\\hat{\\mu}\\hat{\\nu}}\n    & \\equiv\n      \\hat{R}^{(0)}_{(-)\\, \\hat{\\mu}\\hat{\\nu}}{}^{\\hat{a}\\hat{b}}\\, .\n  \\end{align}\n\\end{subequations}\n\n\\noindent\nThen, a straightforward application of Eq.~(\\ref{eq:oYMmnz}) gives\n\n\\begin{equation}\n  \\hat{\\omega}^{\\rm L (0)}_{(-)\\, \\mu\\nu\\underline{z}}\n  =\n  k\\varphi^{\\hat{a}}{}_{\\hat{b}}\\left(2F^{\\hat{b}}{}_{\\hat{a}\\, \\mu\\nu}\n    +\\varphi^{\\hat{b}}{}_{\\hat{a}}kF_{\\mu\\nu} \\right)\n  -2\\partial_{[\\mu|}\\left(k\\varphi^{\\hat{a}}{}_{\\hat{b}}A^{\\hat{b}}{}_{\\hat{a}\\,\n      |\\nu]} \\right)\\, ,\n\\end{equation}\n\n\\noindent\nwhere\n\n\\begin{subequations}\n  \\label{eq:lorentzgaugefields}\n  \\begin{align}\n    \\varphi^{\\hat{a}\\hat{b}}\n    & =\n      k^{-1}\\hat{\\Omega}^{(0)}_{(-)\\, \\underline{z}}{}^{\\hat{a}\\hat{b}}\\, ,\n    \\\\\n    & \\nonumber \\\\\n    A^{\\hat{a}\\hat{b}}{}_{\\mu}\n    & =\n      \\hat{\\Omega}^{(0)}_{(-)\\, \\mu}{}^{\\hat{a}\\hat{b}}\n      -A_{\\mu}\\hat{\\Omega}^{(0)}_{(-)\\, \\underline{z}}{}^{\\hat{a}\\hat{b}}\\, ,\n  \\end{align}\n\\end{subequations}\n\n\\noindent\nand where $F^{\\hat{a}\\hat{b}}{}_{\\mu\\nu}$ is the standard field strength of\nthe gauge field $A^{\\hat{a}\\hat{b}}{}_{\\mu}$ defined above. Decomposing now\nthe Lorentz indices, we obtain\n\n\\begin{equation}\n  \\label{eq:oL0decomposed}\n  \\begin{aligned}\n    \\hat{\\omega}^{\\rm L (0)}_{(-)\\, \\mu\\nu\\underline{z}}\n    & =\n    k\\varphi^{a}{}_{b}\\left(2F^{b}{}_{a\\, \\mu\\nu}\n      +\\varphi^{b}{}_{a}kF_{\\mu\\nu} \\right)\n    +2k\\varphi^{az}\\left(2F_{a}{}^{z}{}_{\\mu\\nu}\n      +\\varphi_{a}{}^{z}kF_{\\mu\\nu} \\right)\n    \\\\\n    & \\\\\n    &\n    -2\\partial_{[\\mu|}\\left[k\\left(\\varphi^{\\hat{a}}{}_{\\hat{b}}\n        A^{\\hat{b}}{}_{\\hat{a}\\,|\\nu]}\\right) \\right]\\, . \n  \\end{aligned}\n\\end{equation}\n\nThe components of these fields are\n\n\\begin{subequations}\n  \\begin{align}\n    \\varphi^{ab}\n    & =\n      -\\tfrac{1}{2}\\left(kF^{ab}+k^{-1}G^{(0)\\, ab}\\right)\\, ,\n    \\\\\n    & \\nonumber \\\\\n    \\varphi^{az}\n    & =\n      \\partial^{a}\\log{k}\\, ,\n    \\\\\n    & \\nonumber \\\\\n    A^{ab}{}_{\\mu}\n    & =\n      \\omega_{\\mu}{}^{ab}-\\tfrac{1}{2}H^{(0)}{}_{\\mu}{}^{ab}\n      \\equiv\n      \\Omega^{(0)}_{(-)\\, \\mu}{}^{ab}\\, ,\n    \\\\\n    & \\nonumber \\\\\n    A^{az}{}_{\\mu}\n    & =\n      -\\tfrac{1}{2}\\left(kF_{\\mu}{}^{a}-k^{-1}G^{(0)}{}_{\\mu}{}^{a}\\right)\\, ,\n    \\\\\n    & \\nonumber \\\\\n    F^{ab}{}_{\\mu\\nu}\n    & =\n    R^{(0)}_{(-)\\, \\mu\\nu}{}^{ab}\n      -\\tfrac{1}{2}\\left(kF_{[\\mu}{}^{a}-k^{-1}G^{(0)}{}_{[\\mu}{}^{a}\\right)\n      \\left(kF_{\\nu]}{}^{b}-k^{-1}G^{(0)}{}_{\\nu]}{}^{b}\\right)\\, ,  \n    \\\\\n    & \\nonumber \\\\\n    F^{az}{}_{\\mu\\nu}\n    & =\n      -\\mathcal{D}^{(0)}_{(-)\\, [\\mu}\\left(kF_{\\nu]}{}^{a}\n      -k^{-1}G^{(0)}{}_{\\nu]}{}^{a}\\right)\\, ,\n  \\end{align}\n\\end{subequations}\n\n\\noindent\nwhere $ R^{(0)}_{(-)\\, \\mu\\nu}{}^{ab}$ is the standard Lorentz curvature and\n$\\mathcal{D}^{(0)}_{(-)\\, \\mu}$ is the standard Lorentz-covariant derivative\nwith respect to the 9-dimensional torsionful spin connection\n$\\Omega^{(0)}_{(-)\\, \\mu}{}^{ab}$.\n\nReplacing the above expressions in Eq.~(\\ref{eq:oL0decomposed}) we obtain\n\n\\begin{equation}\n  \\begin{aligned}\n    \\hat{\\omega}^{\\rm L (0)}_{(-)\\, \\mu\\nu\\underline{z}}\n    & =\n-\\tfrac{1}{2}k\\left(kF^{a}{}_{b}+k^{-1}G^{(0)\\, a}{}_{b}\\right)\n\\left\\{\n 2 R^{(0)}_{(-)\\, \\mu\\nu}{}^{b}{}_{a}\n      -\\left(kF_{[\\mu|}{}^{b}-k^{-1}G^{(0)}{}_{[\\mu}{}^{b}\\right)\n      \\left(kF_{\\nu]\\, a}-k^{-1}G^{(0)}{}_{|\\nu]\\, a}\\right)\n    \\right.\n    \\\\\n    & \\\\\n    & \\hspace{.5cm}\n    \\left.\n-\\tfrac{1}{2}\\left(kF^{b}{}_{a}+k^{-1}G^{(0)\\, b}{}_{a}\\right)kF_{\\mu\\nu} \\right\\}\n    -2\\partial^{a}k\\left[2\\mathcal{D}^{(0)}_{(-)\\, [\\mu}\\left(kF_{\\nu]\\, a}\n      -k^{-1}G^{(0)}{}_{|\\nu]\\, a}\\right)\n      -\\partial_{a}k F_{\\mu\\nu} \\right]\n    \\\\\n    & \\\\\n    & \\hspace{.5cm}\n    -2\\partial_{[\\mu|}\\left[k\\left(\\varphi^{\\hat{a}}{}_{\\hat{b}}\n        A^{\\hat{b}}{}_{\\hat{a}\\,|\\nu]}\\right) \\right]\\, , \n  \\end{aligned}\n\\end{equation}\n\n\\noindent\nand\n\n\\begin{equation}\n  \\begin{aligned}\n  \\hat{H}^{(1)}{}_{cdz}\n  & =\n  k^{-1}e_{c}{}^{\\mu}e_{d}{}^{\\nu}\n\\left\\{ 2\\partial_{[\\mu}\\left[ \\hat{B}_{\\nu]\\underline{z}}\n    -\\frac{\\alpha'}{4}k\\left(\\varphi_{A}A^{A}{}_{|\\nu]}+\\varphi^{\\hat{a}}{}_{\\hat{b}}\n        A^{\\hat{b}}{}_{\\hat{a}\\,|\\nu]}\\right)\n    \\right]\n    \\right.\n    \\\\\n    & \\\\\n    & \\hspace{.5cm}\n  +\\frac{\\alpha'}{4}\n  k\\varphi_{A}\\left(2F^{A}{}_{\\mu\\nu} +\\varphi^{A}kF_{\\mu\\nu} \\right)\n\\\\\n& \\\\\n& \\hspace{.5cm}\n-\\tfrac{1}{2}k\\left(kF^{a}{}_{b}+k^{-1}G^{(0)\\, a}{}_{b}\\right)\n\\left[\n  2R^{(0)}_{(-)\\, \\mu\\nu}{}^{b}{}_{a}\n      -\\left(kF_{[\\mu|}{}^{b}-k^{-1}G^{(0)}{}_{[\\mu}{}^{b}\\right)\n      \\left(kF_{\\nu]\\, a}-k^{-1}G^{(0)}{}_{|\\nu]\\, a}\\right)\n    \\right.\n    \\\\\n    & \\\\\n    & \\hspace{.5cm}\n    \\left.\n      \\left.\n-\\tfrac{1}{2}\\left(kF^{b}{}_{a}+k^{-1}G^{(0)\\, b}{}_{a}\\right)kF_{\\mu\\nu} \\right]\n    -2\\partial^{a}k\\left[2\\mathcal{D}^{(0)}_{(-)\\, [\\mu}\\left(kF_{\\nu]\\, a}\n      -k^{-1}G^{(0)}{}_{|\\nu]\\, a}\\right)\n      -\\partial_{a}k F_{\\mu\\nu} \\right]\n\\right\\}\n   \\, .\n      \\end{aligned}\n\\end{equation}\n\n\\noindent\nSince the right-hand side has to be a gauge-invariant combination, it is\nnatural to define the first-order in $\\alpha'$ winding vector and its field\nstrength by\n\n\\begin{subequations}\n  \\begin{align}\n    B^{(1)}{}_{\\mu}\n    & \\equiv\n      \\hat{B}_{\\mu\\, \\underline{z}}\n      -\\frac{\\alpha'}{4}k\\left(\\varphi_{A}A^{A}{}_{\\mu}\n      +\\varphi^{\\hat{a}}{}_{\\hat{b}}A^{\\hat{b}}{}_{\\hat{a}\\, \\mu}\\right)\n      \\nonumber \\\\\n    & \\nonumber \\\\\n    & =\n      \\hat{B}_{\\mu\\, \\underline{z}}\n      -\\frac{\\alpha'}{4}\\left[\\hat{A}^{A}{}_{\\mu}\\hat{A}_{A\\, \\underline{z}}\n      +\\hat{\\Omega}^{(0)}_{(-)\\, \\mu}{}^{\\hat{a}}{}_{\\hat{b}}\n      \\hat{\\Omega}^{(0)}_{(-)\\, \\underline{z}}{}^{\\hat{b}}{}_{\\hat{a}}\n      \\right.\n      \\nonumber \\\\\n    & \\nonumber \\\\\n    & \\hspace{.5cm}\n      \\left.\n      -A_{\\mu}\\left(\\hat{A}^{A}{}_{\\underline{z}}\\hat{A}_{A\\, \\underline{z}}\n      +\\hat{\\Omega}^{(0)}_{(-)\\, \\underline{z}}{}^{\\hat{a}}{}_{\\hat{b}}\n      \\hat{\\Omega}^{(0)}_{(-)\\, \\underline{z}}{}^{\\hat{b}}{}_{\\hat{a}}\\right)\\right]\\, ,\n    \\\\\n    & \\nonumber \\\\\n    G^{(1)}{}_{\\mu\\nu}\n    & \\equiv\n      2\\partial_{[\\mu}B^{(1)}{}_{\\nu]}\\, .\n  \\end{align}\n\\end{subequations}\n\n\\noindent\nFurthermore, it is also natural to define the combinations\n\n\\begin{equation}\n    K^{(\\pm)}{}_{\\mu\\nu}\n    \\equiv\n    kF_{\\mu\\nu}\\pm k^{-1}G^{(0)}{}_{\\mu\\nu}\\, .\n\\end{equation}\n\n\\noindent\n$K^{(+)}{}_{\\mu\\nu}$ is invariant under the zeroth-order T~duality\ntransformations Eq.~(\\ref{eq:0thorderTduality}) while $K^{(-)}{}_{\\mu\\nu}$\ngets a minus sign under the same transformations. With this notation, we can\nfinally write\n\n\\begin{equation}\n  \\begin{aligned}\n  \\hat{H}^{(1)}{}_{abz}\n  & =\n  k^{-1}G^{(1)}{}_{ab} \n  +\\frac{\\alpha'}{4}\n\\left\\{\n  2\\varphi_{A}F^{A}{}_{ab}\n  +\\left[\\varphi^{2}-\\tfrac{1}{4}K^{(+)\\, 2}\n    +2(\\partial\\log{k})^{2}\\right]kF_{ab} \n\\right.\n\\\\\n& \\\\\n& \\hspace{.5cm}\n \\left.\n+R^{(0)}_{(-)\\, ab}{}^{cd}K^{(+)}{}_{cd}\n      -\\tfrac{1}{2}K^{(-)}{}_{a}{}^{c}K^{(-)}{}_{b}{}^{d}K^{(+)}{}_{cd}\n    -4\\mathcal{D}^{(0)}_{(-)\\, [a}K^{(-)}{}_{b]\\, c}\\partial^{c}\\log{k}\n   \\right\\}\\, .\n      \\end{aligned}\n\\end{equation}\n\nThis term contributes as\n$-\\tfrac{1}{4}\\hat{H}^{(1)}{}_{abz} \\hat{H}^{(1)\\, ab}{}_{z}$, which, at first\norder in $\\alpha'$ gives\n\n\\begin{equation}\n  \\begin{aligned}\n    -\\tfrac{1}{4}\\hat{H}^{(1)}{}_{abz} \\hat{H}^{(1)\\,  ab}{}_{z}\n    & =\n    -\\tfrac{1}{4}  k^{-2}G^{(1)\\, 2}\n    \\\\\n    & \\\\\n    & \\hspace{.5cm}\n  -\\frac{\\alpha'}{8}\n\\left\\{\n2  \\varphi_{A}F^{A}\\cdot k^{-1}G^{(0)}\n  +\\left[\\varphi^{2}-\\tfrac{1}{4}K^{(+)\\,\n      2}+2(\\partial\\log{k})^{2}\\right]F\\cdot G^{(0)} \n\\right.\n\\\\\n& \\\\\n& \\hspace{.5cm}\n+ k^{-1} R^{(0)}_{(-)\\, ab}{}^{cd}K^{(+)}{}_{cd}G^{(0)\\, ab}\n      -\\tfrac{1}{2}k^{-1}G^{(0)\\, ab}K^{(-)}{}_{a}{}^{c}K^{(-)}{}_{b}{}^{d}K^{(+)}{}_{cd}\n      \\\\\n      & \\\\\n      & \\hspace{.5cm}\n      \\left.\n        -4k^{-1}G^{(0)\\, ab}\\mathcal{D}^{(0)}_{(-)\\, [a}K^{(-)}{}_{b]\\, c}\n        \\partial^{c}\\log{k}\n   \\right\\}\\, .\n      \\end{aligned}\n\\end{equation}\n\nLet us now move to the gauge-invariant combination $\\hat{H}^{(1)}{}_{abc}$,\nthat we will identify with the 9-dimensional Kalb-Ramond 3-form field\nstrength. Using the zeroth-order result, we get\n\n\\begin{equation}\n  \\hat{H}^{(1)}{}_{abc}\n  =\n  H^{(0)}{}_{abc}\n  +\\frac{\\alpha'}{4}\\left(\\hat{\\omega}^{\\rm YM}{}_{abc}\n    +\\hat{\\omega}^{\\rm L\\, (0)}{}_{abc}\\right)\\, .\n\\end{equation}\n\n\\noindent\nUsing Eqs.~(\\ref{eq:YMreduction}) it is almost immediately seen that \n\n\\begin{equation}\n  \\hat{\\omega}^{\\rm YM}{}_{abc}\n  =\n  \\omega^{\\rm YM}{}_{abc} +3 k\\varphi_{A}F_{[ab}A^{A}{}_{c]}\\, .\n\\end{equation}\n\n\\noindent\nHalf of the last term has to be integrated by parts, and the final result is\n\n\\begin{equation}\n  \\hat{\\omega}^{\\rm YM}{}_{abc}\n  =\n  \\omega^{\\rm YM}{}_{abc}\n  +3e_{a}{}^{\\mu}e_{b}{}^{\\nu}e_{c}{}^{\\rho}\n  \\left[  \\partial_{[\\mu}\\left(kA_{\\nu|}\\varphi_{A}A^{A}{}_{|\\rho]}\\right)\n    +A_{[\\mu}\\partial_{\\nu|}\\left(k\\varphi_{A}A^{A}{}_{|\\rho]}\\right)\n  +\\tfrac{3}{2}k\\varphi_{A}A^{A}{}_{[\\mu}F_{\\nu\\rho]}\\right]\\, .\n\\end{equation}\n\n\\noindent\nThe second term in the above expression, a total derivative, will combine with\n$\\hat{B}_{\\mu\\nu}$ (and terms coming from\n$\\hat{\\omega}^{\\rm L\\, (0)}{}_{abc}$) to give $B^{(1)}{}_{\\mu\\nu}$ and the\nthird term, as we know, combines with $\\hat{B}_{\\mu\\underline{z}}$ (and terms\ncoming from $\\hat{\\omega}^{\\rm L\\, (0)}{}_{abc}$) to give $B^{(1)}{}_{\\mu}$.\n\nThe above result can be applied to $\\hat{\\omega}^{\\rm L\\, (0)}{}_{abc}$, using\nthe definitions Eq.~(\\ref{eq:lorentzgaugefields}). We get\n\n\\begin{equation}\n  \\begin{aligned}\n    \\hat{\\omega}^{\\rm L\\, (0)}{}_{abc}\n    & =\n    \\omega^{\\rm L\\, (0)}{}_{abc}     +3e_{a}{}^{\\mu}e_{b}{}^{\\nu}e_{c}{}^{\\rho} \\left[\n      \\mathcal{D}^{(0)}_{(-)\\, [\\mu}K^{(-)}{}_{\\nu}{}^{e}K^{(-)}{}_{\\rho]\\, e}\n      +\\partial_{[\\mu} \\left(\n        kA_{\\nu|}\\varphi^{\\hat{e}}{}_{\\hat{f}}A^{\\hat{f}}{}_{\\hat{e}\\, |\\rho]}\n      \\right)\n\\right.\n    \\\\\n    & \\\\\n    & \\hspace{.5cm}\n\\left.\n    +A_{[\\mu}\\partial_{\\nu|}\n      \\left(k\\varphi^{\\hat{e}}{}_{\\hat{f}}A^{\\hat{f}}{}_{\\hat{e}\\,\n          |\\rho]}\\right)\n      +\\tfrac{1}{2}k\\varphi^{\\hat{e}}{}_{\\hat{f}}A^{\\hat{f}}{}_{\\hat{e}\\,[\\mu}F_{\\nu\\rho]}\\right]\\, .\n  \\end{aligned}\n\\end{equation}\n\nDefining\n\n\\begin{equation}\n  B^{(1)}{}_{\\mu\\nu}\n  \\equiv\n  \\hat{B}_{\\mu\\nu}\n  +A_{[\\mu}\n  \\left[\n    \\hat{B}_{\\nu]\\, \\underline{z}}\n  +\\frac{\\alpha'}{4}\n k \\left(\n    \\hat{A}^{A}{}_{|\\nu]}\\hat{A}_{A\\, \\underline{z}}\n      +\\hat{\\Omega}^{(0)}_{(-)\\, |\\nu]}{}^{\\hat{a}}{}_{\\hat{b}}\n      \\hat{\\Omega}^{(0)}_{(-)\\, \\underline{z}}{}^{\\hat{b}}{}_{\\hat{a}}\n    \\right)\n    \\right]\\, ,\n\\end{equation}\n\n\\noindent\nwe find\n\n\\begin{subequations}\n  \\begin{align}\n  \\hat{H}^{(1)}{}_{abc}\n  & =\n    H^{(1)}{}_{abc}\\, ,\n    \\\\\n    & \\nonumber \\\\\n    \\label{eq:H1}\n    H^{(1)}{}_{\\mu\\nu\\rho}\n    & \\equiv\n      3\\partial_{[\\mu}B^{(1)}{}_{\\nu\\rho]}\n      -\\tfrac{3}{2}A_{[\\mu}G^{(1)}{}_{\\nu\\rho]}\n      -\\tfrac{3}{2}B^{(1)}{}_{[\\mu}F_{\\nu\\rho]}\n      \\nonumber \\\\\n    & \\nonumber \\\\\n    & \\hspace{.5cm}\n      +\\frac{\\alpha'}{4}\\left(\\omega^{\\rm YM}{}_{\\mu\\nu\\rho}+\\omega^{\\rm\n      L\\,(0)}_{(-)\\, \\mu\\nu\\rho}\n      +3\\mathcal{D}^{(0)}_{(-)\\, [\\mu}K^{(-)}{}_{\\nu}{}^{e}K^{(-)}{}_{\\rho]\\, e} \\right)\\, .\n  \\end{align}\n\\end{subequations}\n\nSummarizing, the reduction of all the terms in the action but the last one \ngives, to $\\mathcal{O}(\\alpha')$,\n\n\\begin{equation}\n  \\begin{aligned}\n    & \\int dz\\int d^{9}x \\sqrt{|g|}e^{-2\\phi} \\left\\{ R -4(\\partial\\phi)^{2}\n      +\\left(1+\\frac{\\alpha'}{4}\\varphi^{2}\\right)(\\partial \\log{k})^{2}\n      +\\frac{\\alpha'}{4}(\\mathfrak{D}\\varphi)^{2} \\right.\n          +\\frac{\\alpha'}{4}\\partial_{a}\\log{k}\\partial^{a}\\varphi^{2}\n    \\\\\n    & \\\\\n    & -\\tfrac{1}{4}\\left(1+\\frac{\\alpha'}{2}\\varphi^{2}\\right)k^{2}F^{2}\n    +\\tfrac{1}{12}H^{(1)\\, 2}\n    -\\tfrac{1}{4} k^{-2}G^{(1)\\, 2} -\\frac{\\alpha'}{8}\\left[F_{A}\\cdot F^{A}\n      +2\\varphi_{A}F^{A}\\cdot K^{(+)} \\right.\n    \\\\\n    & \\\\\n    & +\\left[\\varphi^{2}-\\tfrac{1}{4}K^{(+)\\, 2}+2(\\partial\\log{k})^{2}\\right]\n    F\\cdot G^{(0)}\n    +R^{(0)}_{(-)\\, ab}{}^{cd}K^{(+)}{}_{cd}k^{-1}G^{(0)\\, ab}\n    \\\\\n    & \\\\\n    & \\left.  \\left. -\\tfrac{1}{2}k^{-1}G^{(0)\\,\n          ab}K^{(-)}{}_{a}{}^{c}K^{(-)}{}_{b}{}^{d}K^{(+)}{}_{cd}\n        -4k^{-1}G^{(0)\\, ab}\\mathcal{D}^{(0)}_{(-)\\, [a}K^{(-)}{}_{b]\\,\n          c}\\partial^{c}\\log{k} \\right] \\right\\}\\, .\n  \\end{aligned}\n\\end{equation}\n\nNow, we must deal with the last term. We deal with it in the same way as we dealt\nwith the Yang-Mills kinetic term:\n\n\\begin{equation}\n  \\begin{aligned}\n    \\hat{R}^{(0)}_{(-)\\, \\hat{c}\\hat{d}}{}^{\\hat{a}}{}_{\\hat{b}}\\,\n    \\hat{R}^{(0)}_{(-)}{}^{\\hat{c}\\hat{d}\\, \\hat{b}}{}_{\\hat{a}} & =\n    \\hat{F}^{\\hat{a}}{}_{\\hat{b}\\, cd} \\hat{F}^{\\hat{b}}{}_{\\hat{a}}{}^{cd}\n    -2\\hat{F}^{\\hat{a}}{}_{\\hat{b}\\, cz}\n    \\hat{F}^{\\hat{b}}{}_{\\hat{a}}{}^{c}{}_{z}\n    \\\\\n    & \\\\\n    & = \\left(F^{\\hat{a}}{}_{\\hat{b}\\,\n        cd}+k\\varphi^{\\hat{a}}{}_{\\hat{b}}F_{cd} \\right) \\left(\n      F^{\\hat{b}}{}_{\\hat{a}}{}^{cd}+k\\varphi^{\\hat{b}}{}_{\\hat{a}}\n      F^{cd}\\right)\n    \\\\\n    & \\\\\n    & \\hspace{.5cm}\n    -2\\left( \\mathcal{D}_{c}\\varphi^{\\hat{a}}{}_{\\hat{b}}\n      +\\varphi^{\\hat{a}}{}_{\\hat{b}}\\partial_{c}\\log{k} \\right) \\left(\n      \\mathcal{D}^{c}\\varphi^{\\hat{b}}{}_{\\hat{a}}\n      +\\varphi^{\\hat{b}}{}_{\\hat{a}}\\partial^{c}\\log{k} \\right)\\, .\n\\end{aligned}\n\\end{equation}\n\n\\noindent\nThe Lorentz-covariant derivatives in the last line must be taken with respect\nto the connection $A^{\\hat{a}\\hat{b}}{}_{\\mu}$, which means that the $ab$\ncomponents contain contributions from $A^{az}{}_{\\mu}$ etc.  Taking this fact\ninto account, if we split the hatted indices into unhatted indices and $z$\ncomponents, we get\n\n\\begin{equation}\n\\begin{aligned}\n  & \\left(F^{a}{}_{b\\, \\mu\\nu}+k\\varphi^{a}{}_{b}F_{\\mu\\nu} \\right) \\left(\n    F^{b}{}_{a}{}^{\\mu\\nu}+k\\varphi^{b}{}_{a} F^{\\mu\\nu}\\right) +2\n  \\left(F^{az}{}_{\\mu\\nu}+k\\varphi^{az}F_{\\mu\\nu}\\right)\n  \\left(F_{a}{}^{z}{}^{\\mu\\nu}+k\\varphi_{a}{}^{z}F^{\\mu\\nu}\\right)\n  \\\\\n  & \\\\\n  & -2\\left( \\mathcal{D}_{c}\\varphi^{a}{}_{b} -A^{az}{}_{c}\\varphi_{b}{}^{z}\n    +A_{b}{}^{z}{}_{c}\\varphi^{az} +\\varphi^{a}{}_{b}\\partial_{c}\\log{k}\n  \\right) \\left( \\mathcal{D}^{c}\\varphi^{b}{}_{a}\n    -A^{bz}{}_{c}\\varphi_{a}{}^{z} +A_{a}{}^{z}{}_{c}\\varphi^{bz}\n    +\\varphi^{b}{}_{a}\\partial^{c}\\log{k} \\right)\n  \\\\\n  & \\\\\n  & -4\\left( \\mathcal{D}_{c}\\varphi^{az} +A^{bz}{}_{c}\\varphi^{a}{}_{b}\n    +\\varphi^{az}\\partial_{c}\\log{k} \\right) \\left(\n    \\mathcal{D}^{c}\\varphi_{a}{}^{z} +A^{bz\\, c}\\varphi_{ab}\n    +\\varphi_{a}{}^{z}\\partial^{c}\\log{k} \\right)\\, ,\n\\end{aligned}\n\\end{equation}\n\n\\noindent\nwhere, now $\\mathcal{D}_{c}$ is the Lorentz-covariant derivative with respect\nto the connection $A^{ab}{}_{\\mu}$.  \n\nSubstituting the components\n$A^{ab}{}_{\\mu},A^{az}{}_{\\mu},\\varphi^{ab},\\varphi^{az}$ by their values, we get\n\n\n\\begin{equation}\n  \\begin{aligned}\n    & \\left( R^{(0)}_{(-)\\, \\mu\\nu}{}^{a}{}_{b}\n      -\\tfrac{1}{2}K^{(-)}{}_{[\\mu}{}^{a}K^{(-)}{}_{\\nu]\\, b} -\\tfrac{1}{2}\n      K^{(+)\\, a}{}_{b}kF_{\\mu\\nu} \\right) \\left( R^{(0)}_{(-)}{}^{\\mu\\nu\\,\n        b}{}_{a} -\\tfrac{1}{2}K^{(-)\\, \\mu\\, b}K^{(-)\\, \\nu}{}_{a}\n      -\\tfrac{1}{2} K^{(+)\\, b}{}_{a}kF^{\\mu\\nu} \\right)\n    \\\\\n    & \\\\\n    & +2 \\left( \\mathcal{D}^{(0)}_{(-)\\, [\\mu}K^{(-)}{}_{\\nu]\\, a}\n      -\\partial_{a}\\log{k}\\, kF_{\\mu\\nu}\\right) \\left(\n      \\mathcal{D}^{(0)}_{(-)}{}^{[\\mu|}K^{(-)\\, |\\nu]\\, a}\n      -\\partial^{a}\\log{k}\\, kF^{\\mu\\nu} \\right)\n    \\\\\n    & \\\\\n    & +\\tfrac{1}{2}\\left( \\mathcal{D}^{(0)}_{(-)}{}^{c}K^{(+)\\, ab} -2K^{(-)\\,\n        c\\, [a}\\partial^{b]}\\log{k} +K^{(+)\\, ab}\\partial^{c}\\log{k} \\right)\n    \\\\\n    & \\\\\n    & \\left( \\mathcal{D}^{(0)}_{(-)\\, c}K^{(+)}{}_{ab} -2K^{(-)}{}_{c\\,\n        [a}\\partial_{b]}\\log{k} +K^{(+)}{}_{ab}\\partial_{c}\\log{k} \\right)\n    \\\\\n    & \\\\\n    & -4\\left( \\mathcal{D}^{(0)}_{(-)}{}^{c}\\partial^{a}\\log{k}\n      -\\tfrac{1}{4}K^{(-)\\, cb}K^{(+)}{}_{b}{}^{a}\n      +\\partial^{a}\\log{k}\\partial^{c}\\log{k} \\right)\n    \\\\\n    & \\\\\n    &\n    \\left(\n      \\mathcal{D}^{(0)}_{(-)\\, c}\\partial_{a}\\log{k}\n      -\\tfrac{1}{4}K^{(-)}{}_{c}{}^{b}K^{(+)}{}_{ba}\n      +\\partial_{a}\\log{k}\\partial_{c}\\log{k} \\right)\\, .\n  \\end{aligned}\n\\end{equation}\n\n\\noindent\nOperating, we finally get\n\n\\begin{equation}\n  \\begin{aligned}\n\\hat{R}^{(0)}_{(-)\\, \\hat{c}\\hat{d}}{}^{\\hat{a}}{}_{\\hat{b}}\\,\n\\hat{R}^{(0)}_{(-)}{}^{\\hat{c}\\hat{d}\\, \\hat{b}}{}_{\\hat{a}}\n& =\nR^{(0)}_{(-)\\,  \\mu\\nu}{}^{a}{}_{b} R^{(0)}_{(-)}{}^{\\mu\\nu\\, b}{}_{a}\n+R^{(0)}_{(-)\\,  \\mu\\nu}{}^{ab}K^{(-)\\, \\mu}{}_{a}K^{(-)\\, \\nu}{}_{b}\n+R^{(0)}_{(-)\\,  \\mu\\nu}{}^{ab}K^{(+)}{}_{ab}kF^{\\mu\\nu}\n\\\\\n& \\\\\n& \\hspace{.5cm}\n+\\tfrac{1}{4}K^{(-)}{}_{[\\mu|}{}^{a}K^{(-)}{}_{a}{}^{\\nu}K^{(-)}{}_{|\\nu]}{}^{b}\nK^{(-)}{}_{b}{}^{\\mu}\n-\\tfrac{1}{2}K^{(-)}{}_{\\mu a}K^{(-)}{}_{\\nu b}K^{(+)\\, ab}kF^{\\mu\\nu}\n\\\\\n& \\\\\n& \\hspace{.5cm}\n-\\tfrac{1}{4}(K^{(+)})^{2}k^{2}F^{2}\n+2\\mathcal{D}^{(0)}_{(-)}{}^{[\\mu|}K^{(-)\\, |\\nu]\\, a}\\mathcal{D}^{(0)}_{(-)\\,\n  [\\mu|}K^{(-)}{}_{|\\nu]\\, a}\n\\\\\n& \\\\\n& \\hspace{.5cm}\n-4\\mathcal{D}^{(0)}_{(-)}{}^{\\mu}K^{(-)\\, \\nu a}\\partial_{a}\\log{k} kF_{\\mu\\nu}\n+2(\\partial\\log{k})^{2}k^{2}F^{2}\n\\\\\n& \\\\\n& \\hspace{.5cm}\n+\\tfrac{1}{2}\n\\mathcal{D}^{(0)}_{(-)}{}^{c}K^{(+)\\, ab}\\mathcal{D}^{(0)}_{(-)\\, c}K^{(+)}{}_{ab}\n-2 \\mathcal{D}^{(0)}_{(-)}{}^{c}K^{(+)\\, ab}K^{(-)}{}_{c\\,\n  a}\\partial_{b}\\log{k}\n\\\\\n& \\\\\n& \\hspace{.5cm}\n+\\mathcal{D}^{(0)}_{(-)}{}^{c}K^{(+)\\, ab}K^{(+)}{}_{ab}\\partial_{c}\\log{k}\n+2K^{(-)\\, c\\, [a}\\partial^{b]}\\log{k}K^{(-)}{}_{ca}\\partial_{b}\\log{k}\n\\\\\n& \\\\\n& \\hspace{.5cm}\n+\\tfrac{1}{2}(K^{(+)})^{2}(\\partial\\log{k})^{2}\n\\\\\n& \\\\\n& \\hspace{.5cm}\n-4\\mathcal{D}^{(0)}_{(-)}{}^{c}\\partial^{a}\\log{k}\n\\mathcal{D}^{(0)}_{(-)\\, c}\\partial_{a}\\log{k}\n+2\\mathcal{D}^{(0)}_{(-)}{}^{c}\\partial^{a}\\log{k}K^{(-)}{}_{c}{}^{b}K^{(+)}{}_{ba}\n\\\\\n& \\\\\n& \\hspace{.5cm}\n-\\tfrac{1}{4}K^{(-)\\, a}{}_{b}K^{(+)\\, b}{}_{c}K^{(-)\\, c}{}_{d}K^{(+)\\, d}{}_{a}\n\\\\\n& \\\\\n& \\hspace{.5cm}\n-8\\mathcal{D}^{(0)}_{(-)}{}^{c}\\partial^{a}\\log{k}\n\\partial_{a}\\log{k}\\partial_{c}\\log{k}\n-4((\\partial\\log{k})^{2})^{2}\\, .\n\\end{aligned}\n\\end{equation}\n\nWith all these terms, the action takes the form\n\n\\begin{equation}\n  \\label{eq:9dimensionalheterotic}\n  \\begin{aligned}\n    S    & =\n      \\frac{g_{s}^{2}(2\\pi\\ell_{s})}{16\\pi G_{N}^{(10)}}\n\\int d^{9}x \\sqrt{|g|}e^{-2\\phi} \\left\\{ R -4(\\partial\\phi)^{2}\n      +\\frac{\\alpha'}{4}(\\mathfrak{D}\\varphi)^{2}\n      -\\partial_{a}k^{-1}\\partial^{a}k^{(1)} \\right.\n    \\\\\n    & \\\\\n    & \\hspace{.5cm}\n    -\\tfrac{1}{4}k^{(1)\\, 2}F^{2} -\\tfrac{1}{4} k^{-2}G^{(1)\\, 2}\n+\\tfrac{1}{2}(1-k^{(1)}k^{-1})F\\cdot G^{(1)}\n    +\\tfrac{1}{12}H^{(1)\\,  2}\n    \\\\\n    & \\\\\n    & \\hspace{.5cm}\n    -\\frac{\\alpha'}{8}\\left[ F_{A}\\cdot\n      F^{A}+R^{(0)}_{(-)}{}^{a}{}_{b}\\cdot R^{(0)}_{(-)}{}^{b}{}_{a}\n      +R^{(0)}_{(-)\\, ab}{}^{cd}\\left(K^{(-)\\, a}{}_{c}K^{(-)\\, b}{}_{d}\n        +K^{(+)\\, ab}K^{(+)}{}_{cd}\\right) \\right.\n    \\\\\n    & \\\\\n    & \\hspace{.5cm}\n   \n    +2\\varphi_{A}F^{A}\\cdot\n    K^{(+)}\n    \\\\\n    & \\\\\n    & \\hspace{.5cm}\n    -\\tfrac{3}{4}K^{(+)\\, a}{}_{b}K^{(-)\\, b}{}_{c}K^{(+)\\, c}{}_{d}K^{(-)\\,d}{}_{a}\n    +\\tfrac{1}{8}K^{(-)\\, a}{}_{b}K^{(-)\\, b}{}_{c}K^{(-)\\,c}{}_{d}K^{(-)\\,d}{}_{a}\n    -\\tfrac{1}{8}\\left(K^{(-)}\\cdot K^{(-)}\\right)^{2}\n    \\\\\n    & \\\\\n    & \\hspace{.5cm}\n    -4K^{(+)\\, ab}\\mathcal{D}^{(0)}_{(-)\\,\n      a}K^{(-)}{}_{bc}\\partial^{c}\\log{k} -2K^{(-)\\, ab}\n    \\mathcal{D}^{(0)}_{(-)\\, a}K^{(+)}{}_{bc}\\partial^{c}\\log{k}\n    \\\\\n    & \\\\\n    & \\hspace{.5cm}\n    +2\\mathcal{D}^{(0)}_{(-)}{}^{[a|}K^{(-)\\, |b]\\, c}\n    \\mathcal{D}^{(0)}_{(-)\\, [a|}K^{(-)}{}_{|b]\\, c}\n    +\\tfrac{1}{2}\\mathcal{D}^{(0)}_{(-)}{}^{c}K^{(+)\\, ab}\n    \\mathcal{D}^{(0)}_{(-)\\, c}K^{(+)}{}_{ab}\n    \\\\\n    & \\\\\n    & \\hspace{.5cm}\n    -4\\mathcal{D}^{(0)}_{(-)}{}^{c}\\partial^{a}\\log{k}\n    \\mathcal{D}^{(0)}_{(-)\\, c}\\partial_{a}\\log{k}\n    +2 K^{(-)\\, ac}K^{(+)}{}_{c}{}^{b}\n    \\mathcal{D}^{(0)}_{(-)\\, a}\\partial_{b}\\log{k}\n    \\\\\n    & \\\\\n    & \\hspace{.5cm}\n    \\left.  \\left.  +2K^{(-)\\, c\\,\n          [a}\\partial^{b]}\\log{k}K^{(-)}{}_{ca}\\partial_{b}\\log{k}  \\right]\n    \\right\\}\\, ,\n  \\end{aligned}\n\\end{equation}\n\n\\noindent\\\nwhere we have defined\n\n\\begin{equation}\nk^{(1)} \\equiv k\\left[1+\\frac{\\alpha'}{4}\\left(\\varphi^{2}-\\tfrac{1}{4}K^{(+)\\, 2}\n      +2(\\partial\\log{k})^{2}\\right)\\right]\\, , \n\\end{equation}\n\n\\noindent\nand we have added some $\\mathcal{O}(\\alpha'{}^{2})$ terms in order to obtain\nnicer or simpler expressions.\n\n\n\\subsection{T~duality}\n\\label{sec-Tduality}\n\nAll the $\\mathcal{O}(\\alpha')$ terms of the reduced action\nEq.~(\\ref{eq:9dimensionalheterotic}) are invariant under the zeroth-order\nT~duality transformations Eqs.~(\\ref{eq:0thorderTduality}), and the whole\naction is invariant to $\\mathcal{O}(\\alpha')$ under the transformations\n\n\\begin{equation}\n  \\label{eq:1storderTduality}\n  A_{\\mu}'\n  = B^{(1)}{}_{\\mu}\\, , \n  \\hspace{1cm}\n  B^{(1)}{}_{\\mu}'\n  = \n  A_{\\mu}\\, ,\n  \\hspace{1cm}\n  k'\n   = \n  1\/k^{(1)}\\, ,\n\\end{equation}\n\n\\noindent\nwhich reduce to the zeroth-order ones in Eqs.~(\\ref{eq:0thorderTduality}) when\nwe set $\\alpha'=0$. Furthermore, observe that\n\n\\begin{equation}\n  \\begin{aligned}\n    k^{(1)\\,\\prime}\n    & =\n    k' \\left[1+\\frac{\\alpha'}{4}\\left(\\varphi^{2}-\\tfrac{1}{4}K^{(+)\\, 2}\n        +2(\\partial\\log{k})^{2}\\right)\\right]\n    \\\\\n    & \\\\\n    & =\n    k^{(1)\\,-1}\\left[1+\\frac{\\alpha'}{4}\\left(\\varphi^{2}-\\tfrac{1}{4}K^{(+)\\,\n          2} +2(\\partial\\log{k})^{2}\\right)\\right]\n    \\\\\n    & \\\\\n    & =\n    k^{-1}\\left[1 +\\mathcal{O}(\\alpha^{\\prime\\, 2}) \\right]\\, .\n  \\end{aligned}\n\\end{equation}\n\n\nUsing the relation between the higher- and lower-dimensional fields, these\ntransformations can be expressed in terms of the higher-dimensional ones in\nthe form\n\n\\begin{equation}\n\\label{eq:buscheralphaprime}\n\\begin{aligned}\n  \\hat{g}'_{\\mu\\nu}\n  & = \n        \\hat{g}_{\\mu\\nu}+\\frac{\\hat{g}_{\\underline{z}\\underline{z}} \\hat{\\mathfrak{G}}^{(1)}{}_{\\underline{z}\\mu}\\hat{\\mathfrak{G}}^{(1)}{}_{\\underline{z}\\nu}}{\\hat{\\mathfrak{G}}^{(1)}{}_{\\underline{z}\\underline{z}}^{2}}\n-\\frac{2\\hat{\\mathfrak{G}}^{(1)}{}_{\\underline{z}(\\mu}\n\\hat{g}_{\\nu)\\underline{z}}}{\\hat{\\mathfrak{G}}^{(1)}{}_{\\underline{z}\\underline{z}}}\\, ,\n        \\hspace{-2cm} & & \\\\\n & \\\\\n  \\hat{B}'_{\\mu\\nu}\n  & = \n  \\hat{B}_{\\mu\\nu}\n  -\\frac{\\hat{\\mathfrak{G}}^{(1)}{}_{\\underline{z}[\\mu} \\hat{\\mathfrak{G}}^{(1)}{}_{\\nu]\\underline{z}}}{\\hat{\\mathfrak{G}}^{(1)}{}_{\\underline{z}\\underline{z}}}\\, ,\n  \\\\\n  & \\\\\n  \\hat{g}'_{\\underline{z}\\mu}\n  & = \n-\\frac{\\hat{g}_{\\underline{z}\\mu}}{\\hat{\\mathfrak{G}}^{(1)}{}_{\\underline{z}\\underline{z}}}\n+\\frac{\\hat{g}_{\\underline{z}\\underline{z}}\\hat{\\mathfrak{G}}^{(1)}{}_{\\underline{z}\\mu}}{\\hat{\\mathfrak{G}}^{(1)}{}_{\\underline{z}\\underline{z}}^{2}}\\, ,\n        \\hspace{1cm}\n  &\n    \\hat{B}'_{\\underline{z}\\mu}\n    & = \n          -\\frac{\\hat{B}_{\\underline{z}\\mu}}{\\hat{\\mathfrak{G}}^{(1)}{}_{\\underline{z}\\underline{z}}}\n          -\\frac{\\hat{\\mathfrak{G}}^{(1)}{}_{\\underline{z}\\mu}}{\\hat{\\mathfrak{G}}^{(1)}{}_{\\underline{z}\\underline{z}}}\\, ,\n  \\\\\n  &  \\\\\n  \\hat{g}'_{\\underline{z}\\underline{z}}\n  & = \n\\frac{\\hat{g}_{\\underline{z}\\underline{z}}}{\\hat{\\mathfrak{G}}^{(1)}{}_{\\underline{z}\\underline{z}}^{2}}\\, ,\n  &\n    e^{-2\\hat{\\phi}'}\n    & = \n          e^{-2\\hat{\\phi}}|\\hat{\\mathfrak{G}}^{(1)}{}_{\\underline{z}\\underline{z}}|\\, ,\n  \\\\\n  & \\\\\n  \\hat{A}'^{A}{}_{\\underline{z}}\n  & = \n        -\\frac{\\hat{A}^{A}{}_{\\underline{z}}}{\\hat{\\mathfrak{G}}^{(1)}{}_{\\underline{z}\\underline{z}}}\\, ,\n  &\n    \\hat{A}'^{A}{}_{\\mu}\n    & = \n          \\hat{A}^{A}{}_{\\mu}\n          -\\frac{\\hat{A}^{A}{}_{\\underline{z}}\\hat{\\mathfrak{G}}^{(1)}{}_{\\underline{z}\\mu}}{\\hat{\\mathfrak{G}}^{(1)}{}_{\\underline{z}\\underline{z}}}\\, ,\n\\end{aligned}\n\\end{equation}\n\n\\noindent\nwhere the  tensor $\\hat{\\mathfrak{G}}^{(1)}{}_{\\hat{\\mu}\\hat{\\nu}}$ is defined by\n\n\\begin{equation}\n  \\hat{\\mathfrak{G}}^{(1)}{}_{\\hat{\\mu}\\hat{\\nu}}\n  \\equiv\n  \\hat{g}_{\\hat{\\mu}\\hat{\\nu}} -\\hat{B}_{\\hat{\\mu}\\hat{\\nu}}\n  -\\frac{\\alpha'}{4} \\left\\{\n    \\hat{A}^{A}{}_{\\hat{\\mu}}^{A}\\hat{A}_{A\\, \\hat{\\nu}}\n    +\\hat{\\Omega}^{(0)}_{(-)\\, \\hat{\\mu}}{}^{\\hat{a}}{}_{\\hat{b}}\n    \\hat{\\Omega}^{(0)}_{(-)\\, \\hat{\\nu}}{}^{\\hat{b}}{}_{\\hat{a}} \\right\\}\\, .\n\\end{equation}\n\nThese are the $\\alpha'$-corrected Buscher rules first found in\nRef.~\\cite{Bergshoeff:1995cg} and later rediscovered elsewhere\n\\cite{Serone:2005ge,Bedoya:2014pma}. \n\nIt is well known that $\\mathcal{N}=1,d=10$ supergravity\n\\cite{Bergshoeff:1981um,Chapline:1982ww} coupled to $n_{V}$ Abelian vector\nmultiplets \\cite{Bergshoeff:1981um,Chapline:1982ww} and dimensionally reduced\non a T$^{n}$ has a global O$(n,n+n_{V})$ symmetry which was shown in\nRef.~\\cite{Maharana:1992my} to be related to string T~duality. In the case at\nhand, the YM vectors are, generically, non-Abelian, which reduces the symmetry\nto just O$(n,n)$ \\cite{Hohm:2014eba} or just O$(1,1)$ here. This group\nconsists of the discrete transformation that give rise to the Buscher rules\nEq.~(\\ref{eq:1storderTduality}) and rescalings of just certain\nlower-dimensional fields:\n\n\\begin{equation}\n  A_{\\mu}' = \\lambda^{-1}A_{\\mu}\\, ,\n  \\hspace{1cm}\n  B^{(1)\\,\\prime}{}_{\\mu} = \\lambda B^{(1)\\,\\prime}{}_{\\mu}\\, ,\n  \\hspace{1cm}\n  k' = \\lambda k\\, .\n\\end{equation}\n\nUnder these rescalings $K^{\\pm},H^{(1)}$ and the Lorentz curvature terms\nremain invariant while\n\n\\begin{equation}\nk^{(1)\\,\\prime} = \\lambda k^{(1)}\\, .\n\\end{equation}\n\nIt can be checked that the dimensionally-reduced action\nEq.~(\\ref{eq:9dimensionalheterotic}) is invariant under these transformations\nand, therefore, under the whole O$(1,1)$ group.\n\nWe observe that the kinetic term of the KK and winding vectors is the sum of\ntwo separately O$(1,1)$-invariant terms\n\n\\begin{equation}\n  -\\tfrac{1}{4}\n  (F_{\\mu\\nu}\\,\\,\\,,\\,\\,\\,G^{(1)}{}_{\\mu\\nu})\n  \\left(\n    \\begin{array}{cc}\n      k^{(1)\\,2} & 0 \\\\\n      & \\\\\n      0 &  1\/k^{2} \\\\\n    \\end{array}\n  \\right)\n  \\left(\n    \\begin{array}{c}\n      F^{\\mu\\nu} \\\\ \\\\ G^{(1)\\, \\mu\\nu} \\\\\n    \\end{array}\n  \\right)\n  + \\tfrac{1}{2}(1-k^{(1)}\/k)F\\cdot G^{(1)}\\, ,\n\\end{equation}\n\n\\noindent\nand that the diagonal kinetic matrix transforms consistently under O$(1,1)$\ntransformations even though, as different to the zeroth-order case, the\nkinetic matrix is not an O$(1,1)$ matrix itself. The consistency is related to\nthe fact that it is part of a O$(1,1+n_{V})$ matrix.\n\n\\section{Entropy formula}\n\\label{sec-entropyformula}\n\nWe can use the dimensionally reduced action we have obtained to calculate the\nentropy of some $d$-dimensional heterotic string black holes using the\nIyer-Wald prescription \\cite{Wald:1993nt,Iyer:1994ys}. These black holes must\nbe solutions of the theory defined by the action\nEq.~(\\ref{eq:9dimensionalheterotic}) understood as a $d$-dimensional\naction. Therefore, they must be solutions of the theory defined by the action\nEq.~(\\ref{heterotic}) understood as a $(d+1)$-dimensional action\\footnote{The\n  constant in front of the action should now contain the volume of a\n  $(10-d)$-dimensional torus instead of that of circle, that is\n  \\begin{equation}\n    \\frac{g_{s}^{2}(2\\pi\\ell_{s})^{10-d}}{16\\pi G_{N}^{(10)}}\n    =\n     \\frac{(g^{(d)}_{s})^{2}}{16\\pi G_{N}^{(d)}}\\, , \n   \\end{equation}\n   %\n   where $g^{(d)}_{s}$ is the $d$-dimensional string coupling constant or the\n   vacuum expected value of the $d$-dimensional dilaton\n   $<e^{\\phi}>=e^{\\phi_{\\infty}}$ and $G_{N}^{(d)}$ the $d$-dimensional Newton\n   constant. The relations of the 10-dimensional and $d$-dimensional ones with\n   the volume of the $(10-d)$-dimensional compact space, $V_{10-d}$  is\n   %\n      \\begin{subequations}\n     \\begin{align}\n       g_{s}^{2} & = V_{10-d}\/(2\\pi\\ell_{s})^{10-d}g_{s}^{(d)\\, 2}\\, , \\\\\n                 & \\nonumber \\\\\n     \\label{eq:relationsbetweenconstants}\n        G_{N}^{(10)} & = G_{N}^{(d)} V_{10-d}\\, .\n     \\end{align}\n   \\end{subequations}\n } admitting an isometry. Since this $(d+1)$-dimensional action\n can be obtained from the 10-dimensional one by a trivial compactification on\n a $10-(d+1)$-dimensional torus, the metrics of the 10-dimensional solutions\n corresponding to the $d$-dimensional black holes are the direct products of\n non-trivial $(d+1)$-dimensional metrics and the metric of a\n $10-(d+1)$-dimensional torus. The non-extremal 4-dimensional\n Reissner-Nordstr\\\"om black hole of Ref.~\\cite{Cano:2019ycn} or the heterotic\n version of the 5-dimensional Strominger-Vafa black hole of\n Ref.~\\cite{Cano:2018qev} are two interesting examples of this kind of\n solution.\n\n Applying directly the Iyer-Wald prescription to the $d$-dimensional action\n Eq.~(\\ref{eq:9dimensionalheterotic}) we obtain the following entropy formula\n expressed in string-frame variables:\n\n\\begin{subequations}\n  \\begin{align}\n  \\label{eq:entropyd4}\nS\n  & =\n  -2\\pi\\int_\\Sigma d^{d-2}x\\sqrt{|h|}\n  \\frac{\\partial \\mathcal{L}}{\\partial R_{abcd}}\n    \\epsilon_{ab}\\epsilon_{cd}\\, ,\n    \\\\\n    & \\nonumber \\\\\n\\label{eq:entropyformulastringframe}\n      \\frac{\\partial\\mathcal{L}}{\\partial R_{abcd}}\n    & =\n      \\frac{e^{-2(\\phi-\\phi_{\\infty})}}{16\\pi G_{N}^{(d)}}\n      \\left\\{ g^{ab,\\, cd}\n      -\\frac{\\alpha'}{8} \\left[H^{(0)\\, abg}\n        \\left(\\omega_{g}{}^{cd}-H^{(0)}_{g}{}^{cd}\\right)\n      \\right.\\right.\n\\nonumber  \\\\\n    & \\nonumber \\\\\n    & \\hspace{.5cm}\n      \\left. \\left.\n        -2R_{(-)}^{(0)\\, abcd}\n      +K^{(-)\\, [a|c}K^{(-)\\, |b]d} +K^{(+)\\, ab}K^{(+)\\, cd}\n      \\right]\n      \\right\\}\\, ,\n  \\end{align}\n\\end{subequations}\n\n\\noindent\nwhere $|h|$ is the absolute value of the determinant of the metric induced\nover the event horizon, $g^{ab,cd}= \\tfrac{1}{2}(g^{ac}g^{bd}-g^{ad}g^{bc})$,\n$\\epsilon^{ab}$ is the event horizon's binormal normalized so that\n$\\epsilon_{ab}\\epsilon^{ab}=-2$ and $R_{abcd}$ is the Riemann tensor.\n\n\\subsection{The Wald entropy of the $\\alpha'$-corrected Strominger-Vafa black hole}\n\\label{sec-entropystromingervafa}\n\n\nThe entropy formula Eq.~(\\ref{eq:entropyformulastringframe}) has been shown in\nRef.~\\cite{Cano:2019ycn} to give an entropy which is related to the Hawking\ntemperature by the thermodynamic relation\n\n\\begin{equation}\n\\frac{\\partial S}{\\partial M}=\\frac{1}{T}\\, ,\n\\end{equation}\n\n\\noindent\nfor the particular case of $\\alpha'$-corrected, 4-dimensional, non-extremal\nReissner-Nordstr\\\"om black holes. In this section we want to recalculate the\nWald entropy of the $\\alpha'$-corrected Strominger-Vafa black hole. Being an\nextremal black hole, we will not be able to check that the entropy obtained is\nrelated to the temperature as above, but, instead, we will be able to compare\nwith other results obtained in the literature and with the microscopic\ncalculations.\n\nThe 5-dimensional $\\alpha'$-corrected Strominger-Vafa black hole corresponds\nto the 10-dimensional solution of the Heterotic Superstring effective action\n\\cite{Cano:2018qev,Chimento:2018kop}\n\n\\begin{subequations}\n\\label{eq:10dsolution}\n\\begin{align}\nd\\hat{s}^{2}\n& = \n\\frac{2}{\\mathcal{Z}_{-}}du\\left(dv-\\tfrac{1}{2}\\mathcal{Z}_{+}du\\right)\n-\\mathcal{Z}_{0}(d\\rho^{2}+\\rho^{2}d\\Omega_{(3)}^{2})-dy^{i}dy^{i}\\, ,\n\\hspace{.5cm}\ni=1,\\ldots,4\\, ,\n\\\\\n& \\nonumber \\\\\n\\hat{H}^{(1)}\n& = \nd\\mathcal{Z}_{-}^{-1}\\wedge du\\wedge dv +\\star_{4}d\\mathcal{Z}_{0}\\, ,\n\\\\\n& \\nonumber \\\\\ne^{-2\\hat{\\phi}}\n& = \ne^{-2\\hat{\\phi}_{\\infty}}\n\\mathcal{Z}_{-}\/\\mathcal{Z}_{0}\\, ,\n\\end{align}\n\\end{subequations}\n\n\\noindent\nwhere $\\star_{4}$ stands for the Hodge dual in the 4-dimensional Euclidean\nspace with metric $d\\rho^{2}+\\rho^{2}d\\Omega_{(3)}^{2}$, and where the\n$\\mathcal{Z}$ functions take the values\\footnote{The Regge slope parameter\n  $\\alpha'$ in Refs.~\\cite{Cano:2018qev,Chimento:2018kop} has been replaced by\n  $\\alpha'\/8$ here to obtain the correct form of the action and solutions.}\n\n\\begin{subequations}\n\\label{eq:Zs}\n\\begin{align}\n\\mathcal{Z}_{0}\n& = \n1+\\frac{\\tilde{q}_{0}}{\\rho^{2}}\n-\\alpha' \n\\frac{\\rho^{2}+2\\tilde{q}_{0}}{(\\rho^{2}+\\tilde{q}_{0})^{2}}\n+\\mathcal{O}(\\alpha'^{2})\\, , \n\\\\\n& \\nonumber \\\\\n\\mathcal{Z}_{-}\n& =\n1+\\frac{\\tilde{q}_{-}}{\\rho^{2}}\n+\n\\mathcal{O}(\\alpha'^{2})\\, ,\n\\\\\n&  \\nonumber \\\\\n\\mathcal{Z}_{+}\n& =\n1+\\frac{\\tilde{q}_{+}}{\\rho^{2}}\n+2\\alpha'\\frac{\\tilde{q}_{+}(\\rho^{2}+\\tilde{q}_{0}+\\tilde{q}_{-})}\n{\\tilde{q}_{0}(\\rho^{2}+\\tilde{q}_{0})(\\rho^{2}+\\tilde{q}_{-})}\n+\\mathcal{O}(\\alpha'^{2})\\, .\n\\end{align}\n\\end{subequations}\n\nCompactifying this solution in a T$^{4}$ parameterized by the coordinates\n$y_{i}$ is trivial. Then, we just have to compactify the resulting\n6-dimensional solution to $d=5$ using the results obtained here along the\ncoordinate $z\\equiv u\/k_{\\infty}$, where $k_{\\infty}$ is the asymptotic value\nof the KK scalar $k$. It is helpful to rewrite the 6-dimensional solution in the form\n\n\n\\begin{subequations}\n\\label{eq:6dsolution}\n\\begin{align}\nd\\hat{s}^{2}\n  & =\n    \\frac{1}{\\mathcal{Z}_{+}\\mathcal{Z}_{-}}dt^{2}\n    -\\mathcal{Z}_{0}(d\\rho^{2}+\\rho^{2}d\\Omega_{(3)}^{2})\n    -\\frac{k_{\\infty}^{2}\\mathcal{Z}_{+}}{\\mathcal{Z}_{-}}\n    \\left(dz-    \\frac{1}{k_{\\infty}\\mathcal{Z}_{+}}dt\\right)^{2}\\, ,\n\\\\\n& \\nonumber \\\\\n\\hat{H}^{(1)}\n& = \n   d\\left( -\\frac{k_{\\infty}}{\\mathcal{Z}_{-}}dt \\wedge dz\\right)\n                +\\star_{4}d\\mathcal{Z}_{0}\\, ,\n\\\\\n& \\nonumber \\\\\ne^{-2\\hat{\\phi}}\n& = \ne^{-2\\hat{\\phi}_{\\infty}}\n\\mathcal{Z}_{-}\/\\mathcal{Z}_{0}\\, ,\n\\end{align}\n\\end{subequations}\n\n\n\\noindent\nwhere we have set $v=t$, to identify immediately the following 5-dimensional\nfields:\\footnote{We have only computed $G^{(0)}$ and not $G^{(1)}$ because of\n  its complication and because it is unnecessary to do it for the calculation\n  of the entropy. On the other hand, the Kalb-Ramond field is customarily\n  dualized into another vector field to which the third charge $\\tilde{q}_{0}$\n  is associated.}\n\n\\begin{subequations}\n\\label{eq:5dsolution}\n\\begin{align}\nds^{2}\n  & =\n    \\frac{1}{\\mathcal{Z}_{+}\\mathcal{Z}_{-}}dt^{2}\n    -\\mathcal{Z}_{0}(d\\rho^{2}+\\rho^{2}d\\Omega_{(3)}^{2})\\, ,\n\\\\\n& \\nonumber \\\\\nH^{(1)}\n& = \n\\star_{4}d\\mathcal{Z}_{0}\\, ,\n\\\\\n& \\nonumber \\\\\n  F\n  & =\n    d\\left(-\\frac{1}{k_{\\infty}\\mathcal{Z}_{+}}dt\\right)\\, ,\n  \\\\\n& \\nonumber \\\\\n  G^{(0)}\n  & =\n    d\\left(-\\frac{k_{\\infty}}{\\mathcal{Z}_{-}}dt\\right)\\, ,\n  \\\\\n& \\nonumber \\\\\ne^{-2(\\phi-\\phi_{\\infty})}\n& = \n\\sqrt{\\mathcal{Z}_{+}\\mathcal{Z}_{-}}\/\\mathcal{Z}_{0}\\, ,\n\\\\\n& \\nonumber \\\\\n  k\/k_{\\infty}\n  & =\n    \\sqrt{\\mathcal{Z}_{+}\/\\mathcal{Z}_{-}}\\, ,\n\\end{align}\n\\end{subequations}\n\n\\noindent\nand the T~duality even and odd 2-forms\n\n\\begin{equation}\n  K^{\\pm} = -\\frac{1}{\\sqrt{\\mathcal{Z}_{+}\\mathcal{Z}_{-}}}\n  \\left(\n    \\frac{\\mathcal{Z}_{+}'}{\\mathcal{Z}_{+}}\n  \\pm  \\frac{\\mathcal{Z}_{-}'}{\\mathcal{Z}_{-}}\n  \\right) d\\rho \\wedge dt\\, ,  \n\\end{equation}\n\n\\noindent\nwhere a prime indicates derivative with respect to $\\rho$.\n\nIn the Vielbein basis\n\n\\begin{equation}\n  e^{0} = \\frac{1}{\\sqrt{\\mathcal{Z}_{+}\\mathcal{Z}_{-}}}dt\\, ,\n  \\hspace{.5cm}\n  e^{1} = \\sqrt{\\mathcal{Z}_{0}}d\\rho\\, ,\n  \\hspace{.5cm}\n  e^{i} = \\tfrac{1}{2}\\sqrt{\\mathcal{Z}_{0}}\\rho \\theta^{i}\\, ,\n\\end{equation}\n\n\\noindent\nwhere the $\\theta^{i}$ are the left-invariant SU$(2)$ Maurer-Cartan 1-forms\nthat satisfy $d\\Omega^{2}{}_{(3)}=\\tfrac{1}{4}\\theta^{i}\\theta^{i}$, the\nbinormal is given by just $\\epsilon^{01}=+1$ and the entropy formula in\nEqs.~(\\ref{eq:entropyd4}) and (\\ref{eq:entropyformulastringframe}) becomes\n\n\\begin{equation}\nS\n   =\n        \\frac{1}{4G_{N}^{(5)}}\\int_{\\Sigma} d^{3}xe^{-2(\\phi-\\phi_{\\infty})}\\sqrt{|h|}\n    \\left\\{1\n    +\\frac{\\alpha'}{4} \\left[\n        -2R^{0101}\n      +(K^{(-)\\, 01})^{2}+(K^{(+)\\, 01})^{2}\n      \\right]\n      \\right\\}\\, .\n\\end{equation}\n\nThe fields in the integrand are only functions of $\\rho$ and we can perform\nthe integral over S$^{3}$. Evaluating the zeroth-order term at $\\rho=0$, where\nthe horizon is located, we get\n\n\n\\begin{equation}\n\\begin{aligned}\n  \\label{eq:entropydSVbh1}\nS\n    &   =\n      \\frac{1}{4G_{N}^{(5)}}\n    \\left\\{A_{\\mathcal{H}}\n    +\\alpha' \\pi^{2}\\lim_{\\rho\\rightarrow 0}\n      \\rho^{3}\\sqrt{\\mathcal{Z}_{0}\\mathcal{Z}_{+}\\mathcal{Z}_{-}}\\left[\n      -\\sqrt{\\frac{\\mathcal{Z}_{+}\\mathcal{Z}_{-}}{\\mathcal{Z}_{0}}}\n      \\left[\\frac{1}{\\sqrt{\\mathcal{Z}_{0}}}\n          \\left(\\frac{1}{\\sqrt{\\mathcal{Z}_{+}\\mathcal{Z}_{-}}}\\right)'\n      \\right]'\n    \\right.    \\right.\n    \\\\\n    & \\\\\n    & \\hspace{.5cm}\n    \\left.    \\left.\n      +\\frac{1}{\\mathcal{Z}_{0}}\n        \\left(\\frac{\\mathcal{Z}_{+}'}{\\mathcal{Z}_{+}}\\right)^{2}\n        +\\frac{1}{\\mathcal{Z}_{0}}\n        \\left(\\frac{\\mathcal{Z}_{-}'}{\\mathcal{Z}_{-}}\\right)^{2}\n      \\right]\n      \\right\\}\\, ,\n    \\end{aligned}\n\\end{equation}\n\n\\noindent\nwhere $A_{\\mathcal{H}}$, the area of the horizon, is given by\n\n\\begin{equation}\n  A_{\\mathcal{H}}\n  =\n  2\\pi^{2}\\lim_{\\rho\\rightarrow 0}\n  \\rho^{3}\\sqrt{\\mathcal{Z}_{0}\\mathcal{Z}_{+}\\mathcal{Z}_{-}}\n  = 2\\pi^{2}\\sqrt{\\tilde{q}_{0}\\tilde{q}_{+}\\tilde{q}_{-}}\\, .\n\\end{equation}\n\n\\noindent\nFinally, we arrive at\n\n\\begin{equation}\n  \\label{eq:entropydSVbh2}\n  S\n  =\n    \\frac{A_{\\mathcal{H}}}{4G_{N}^{(5)}}\n    \\left\\{ 1 +\\frac{2\\alpha'}{\\tilde{q}_{0}}\\right\\}\\, .\n\\end{equation}\n\nIn order to compare this result with the microscopic entropy in\nRef.~\\cite{Kraus:2005zm}, we have to express the charges\n$\\tilde{q}_{+},\\tilde{q}_{-},\\tilde{q}_{0}$ in terms of the asymptotic\ncharges\\footnote{See Refs.~\\cite{Cano:2018qev,Faedo:2019xii}, specially\n  Eqs.~(2.18),(2.20),(2.21) of the later.}. First, we have to take into\naccount the relation between $\\tilde{q}_{+},\\tilde{q}_{-},\\tilde{q}_{0}$ and\nthe numbers of fundamental strings $n$, momentum $w$ and S5-branes $N$\n\n\\begin{equation}\n  \\tilde{q}_{+}\n  =\n  \\frac{\\alpha^{\\prime\\, 2}g_{s}^{2}n}{R_{z}^{2}}\\, ,\n  \\hspace{.5cm}\n  \\tilde{q}_{-}\n  =\n  \\alpha' g_{s}^{2}w\\, ,\n  \\hspace{.5cm}\n  \\tilde{q}_{0}\n  =\n  \\alpha' N\\, .\n\\end{equation}\n\n\\noindent\nSecond, 10-dimensional Newton constant $G_{N}^{(10)}$ is given in terms of the\nRegge slope parameter $\\alpha'=\\ell_{s}^{2}$ and the 10-dimensional string\ncoupling constant $g_{s}$ by\n\n\\begin{equation}\n  \\label{eq:d10newtonconstant}\nG_{N}^{(10)}=8\\pi^{6}g_{s}^{2} \\alpha^{\\prime\\, 4}\\, .\n\\end{equation}\n\nThis and Eq.~(\\ref{eq:relationsbetweenconstants}) allow us to rewrite the \nentropy Eq.~(\\ref{eq:entropydSVbh2}) in the form\n\n\\begin{equation}\n  \\label{eq:entropydSVbh3}\n  S\n  =\n  2\\pi \\sqrt{nwN}\\left(1+\\frac{2}{N}\\right)\\, .\n\\end{equation}\n\nFinally, in terms of the asymptotic charges $Q_{+},Q_{-},Q_{0}$, which are\nrelated to the numbers of branes by\n\n\\begin{equation}\n  Q_{+} = n\\left(1+\\frac{2}{N}\\right)\\,\n  \\hspace{.5cm}\n  Q_{-}\n  = w\\, ,\n  \\hspace{.5cm}\n  Q_{0}\n  =\n  N-1\\, ,\n\\end{equation}\n\n\\noindent\nthe entropy takes the final form that can be compared with the microscopic\nformula\n\n\\begin{equation}\n  \\label{eq:entropydSVbh4}\n  S\n  =\n  2\\pi \\sqrt{Q_{+}Q_{-}(Q_{0}+3)}\\, .\n\\end{equation}\n\n\\section{Discussion}\n\\label{sec-discussion}\n\nIn this paper we have performed the complete dimensional reduction of the\nHeterotic Superstring effective action to first order in $\\alpha'$ using the\nformulation based on the supersymmetry completion of the Lorentz Chern-Simons\nterms that occur in the Kalb-Ramond field strength\n\\cite{Bergshoeff:1988nn,Bergshoeff:1989de}. We have found a $\\mathbb{Z}_{2}$\ntransformation of the dimensionally-reduced action that leaves it invariant\nand that is an $\\mathcal{O}(\\alpha')$ generalization of the standard\ntransformations that interchange KK and winding vectors and invert the KK\nscalar. In 10-dimensional variables (the components of the 10-dimensional\nfields) these transformations are nothing but the $\\alpha'$-corrected Buscher\nrules of the Heterotic Superstring theory, first found in\n\\cite{Bergshoeff:1995cg}.\n\nThen, we used the dimensionally-reduced action to find, following the\nIyer-Wald prescription \\cite{Wald:1993nt,Iyer:1994ys} an entropy formula for\nstringy black holes that can be obtained from a 10-dimensional solution by a\nsingle non-trivial compactification on a circle, supplemented by a trivial\ncompactification on a torus. This formula was successfully applied to a\nnon-extremal 4-dimensional Reissner-Nordstr\\\"om black hole in\nRef.~\\cite{Cano:2019ycn} and, in this paper, we have applied it to the\n$\\alpha'$-corrected heterotic version of the Strominger-Vafa black hole of\nRef.~\\cite{Cano:2018qev} obtaining an entropy formula that matches the\nmicroscopic result obtained in \\cite{Kraus:2005zm} once the relations between\nintegration constants and asymptotic brane charges have been correctly taken\ninto account. As explained in Ref.~\\cite{Faedo:2019xii}, the result obtained\nin Ref.~\\cite{Cano:2018qev} misses a factor of 2 that we recover here.\n\nIt would be desirable to obtain an entropy formula that could be applied to\nmore general black holes, in particular, to the 4-dimensional 4-charge\nones. This requires a more general toroidal compactification of the action\nalong the lines of the one recently obtained in Ref.~\\cite{Eloy:2020dko},\nincluding YM fields and closer to the Bergshoeff-de Roo formulation\n\\cite{kn:TOM2}. It is also necessary to prove the first law of black hole\nmechanics for the Heterotic Superstring effective action in order to make sure\nthat the Iyer-Wald prescription can be unambiguously applied as we have done\nhere or that the entropy is just given by the integral of the Noether charge\nas assumed in Ref.~\\cite{Edelstein:2019wzg}, since, in presence of matter\nfields, some terms of the total Noether charge can be related to other\nterms in the first law \\cite{Compere:2007vx}.\n\nFinally, let us comment on the invariance of the black-hole temperature and\nentropy under $\\alpha'$-corrected T~duality, which we already mentioned in\nfootnote~\\ref{foot:esa} in the Introduction, which was discussed in\nRefs.~\\cite{Horowitz:1993wt,Edelstein:2018ewc,Edelstein:2019wzg}. T~duality\nmanifests itself in the non-compact dimensions in which black-hole solutions\nappear as such as a symmetry that only acts on some lower-dimensional vector\nfields (KK and winding vector fields) and on some scalars (KK scalars and\nscalars originating in the Kalb-Ramond 2-form, if one compactifies more than\none dimension). In particular, the lower-dimensional dilaton, the string\nmetric and the lower-dimensional Kalb-Ramond field are\nT~duality-invariant.\\footnote{More precisely, in toroidal compactifications,\n  the lower-dimensional Kalb-Ramond 2-form is only invariant up to a\n  compensating O$(n)$ gauge transformation \\cite{kn:TOM2}, but its field\n  strength is exactly invariant.} Since the surface gravity is computed\ndirectly on the dimensionally-reduced metric, its invariance implies\nimmediately the invariance of the Hawking temperature. The invariance of the\nlower-dimensional action and the invariance of the Riemann tensor (which\nfollows that of the metric) automatically imply the invariance of the Wald\nentropy formula. In our case, the invariance of the entropy formula in\nEqs.~(\\ref{eq:entropyd4}),(\\ref{eq:entropyformulastringframe}) is manifest.\n\n\\section*{Acknowledgments}\n\nTO would like to thank Pedro F.~Ram\\'{\\i}rez and JJ.~Fern\\'andez-Melgarejo for\nmany useful conversations and Pablo A.~Cano for useful comments on the\nmanuscript.  This work has been supported in part by the MCIU, AEI, FEDER (UE)\ngrant PGC2018-095205-B-I00 and by the Spanish Research Agency (Agencia Estatal\nde Investigaci\\'on) through the grant IFT Centro de Excelencia Severo Ochoa\nSEV-2016-0597. The work of ZE has also received funding from ``la Caixa''\nFoundation (ID 100010434), under the agreement LCF\/BQ\/DI18\/11660042.  TO\nwishes to thank M.M.~Fern\\'andez for her permanent support.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nDigital holography is a fast-growing research field\nthat has drawn increasing attention. The main advantage\nof digital holography is that, contrary to holography\nwith photographic plates, the holograms\nare recorded by a CCD, and the image of the object is digitally reconstructed\nby a computer, avoiding photographic\nprocessing \\cite{schnars1994direct}.  To extract two quadratures of the holographic signal (i.e. to get the amplitude and the phase of the optical signal) two main optical configurations have been developed: off-axis and phase-shifting.\n\n\n\nOff-axis holography is the oldest and\nthe simplest configuration adapted to digital\nholography.  In that configuration, the reference or local oscillator\n(LO) beam is angularly tilted with respect to the\nobject observation axis. It is then possible to record,\nwith a single hologram, the two quadratures of the\nobject complex field \\cite{schnars1994direct}. However, the object field of view\nis reduced, since one must avoid the overlapping of\nthe image with the conjugate image alias.\nPhase-shifting digital holography \\cite{yamaguchi1997phase} makes possible to get phase information on the whole camera area by recording several\nimages with a different phase for the reference (called here local oscillator or LO) beam.\nIt is then possible to obtain the two quadratures of\nthe field in an on-axis configuration even though the\nconjugate image alias and the true image overlap, because\naliases can be removed by taking image differences.  In a typical phase-shifting holographic setup, the phase of the reference is shifted by moving a mirror with a PZT.\n\nOn the other hand, there is a big demand for full field vibration measurements, in particular in industry. Different\nholographic techniques are able to image and analyze such vibrations. Double pulse holography\n\\cite{pedrini1995digital,pedrini1997digital} records a double-exposure hologram with time separation in the 1...1000 $\\mu$s range, and\nmeasures the instantaneous velocity of a vibrating object from the phase difference. The method\nrequires a quite costly double pulse ruby laser system, whose repetition rate is low. Multi pulse\nholography \\cite{pedrini1998transient} is able to analyse transient motions, but the setup is still heavier (4 pulses laser,\nthree cameras).\n\nThe development of fast CMOS camera makes possible to analyze vibration efficiently by\ntriggering the camera on the motion in order to record a sequence of holograms that allows to\ntrack the vibration of the object as a function of the time \\cite{pedrini2006high,fu2007vibration}. The  analysis of the\nmotion can be done by phase difference or by Fourier analysis in the time domain. The method\nrequires a CMOS camera, which can be costly. It is also limited to low frequency vibrations,\nsince a complete analysis of the motion requires a camera frame rate higher than the vibration\nfrequency, because the bandwidth $\\textrm{BW}$ of the holographic signal, which is sampled at the camera at angular frequency $\\omega_{CCD}$ must be lower than corresponding Nyquist-Shannon limit: $BW < \\frac{1}{2}~\\omega_{CCD}$.\n\nFor a periodic vibration motion, the bandwidth $\\textrm{BW}$ is close to zero.  Measurements can thus be done with much slower cameras.\nPowell and Stetson \\cite{powell1965interferometric} have shown for example that an harmonically vibrating\nobject yields  alternate dark and bright fringes, whose analysis yields informations on the vibration motion.\nPicard et al. \\cite{picart2003time} has simplified the processing of\nthe data by performing time averaged holography with a digital CCD camera.\nTime averaged\nholography has no limit in vibration frequency and do not involve costly laser system, nor an\nexpensive CMOS fast camera. Although the time-averaging method gives a way to determine the amplitude\nof vibration \\cite{picart2005some} quantitative measurement remain quite difficult for low and high vibration\namplitudes.\n\nHeterodyne holography \\cite{le2000numerical,le2001synthetic} is a variant of phase shifting holography, in which the frequency, phase and amplitude of both reference and signal signal beam are  controlled by acousto optic modulators (AOM).  Heterodyne holography is thus extremely versatile.\nBy shifting the frequency  $\\omega_{LO}$ of the local oscillator beam with respect to the  frequency  $\\omega_0$ of illumination, it is for example possible  to detect the holographic signal at a frequency $\\omega$ different  than illumination $\\omega_0$. This ability will be extremely useful to analyze vibration, since  heterodyne holography can detect selectively the signal that is scattered by the object on a vibration sideband of frequency   $\\omega_m=\\omega_0+ m \\omega_A$, where $\\omega_A$ is the vibration frequency and  $m$ and integer index.\n\n\nIn this chapter we will first present in section  \\ref{section_Heterodyne holography}\nheterodyne holography and its advantages in section \\ref{section_accurate_phase} and \\ref{section_shot_noise}. Then in section  \\ref{section_vibrometry}, we will apply  heterodyne holography to vibration analysis. We will show in section \\ref{section_Selective detection of the sideband components}, how heterodyne holography  can be used to detect the vibration sidebands, in section \\ref{section_strobe_detection} how this sideband holography can be combined with stroboscopic techniques to record instantaneous velocity maps of motion, and in sections  \\ref{Section_large_vibration} and \\ref{section_small_vibration} how it can retrieve both small and large vibration amplitudes.\n\n\n\n\\section{Heterodyne holography}\\label{section_Heterodyne holography}\n\n\n\n\n\\begin{figure}\n\\begin{center}\n   \n  \\includegraphics[width=9 cm]{fig_heterodyne_setup}\\\\\n \\caption{Typical heterodyne holography setup. L: main\nlaser; BS: beam splitter; AOM1, AOM2: acousto-optic\nmodulators; BE: beam expander; M, mirror; A1, A2: light\nattenuators; USAF, transmission USAF 1951 test pattern;\nCCD:  camera; $E_L$, $E_I$, $E_{LO}$, and $E$: laser, illumination,\nreference (i.e. local oscillator LO) and object fields; $\\omega_{AOM1\/2}$: driving frequencies of the\nacousto optics modulators AOM1 and AOM2; $\\theta$: off-axis angular tilt.}\\label{Fig_fig_heterodyne_setup}\n\\end{center}\n\\end{figure}\n\n\nLet us first describe heterodyne holography. A example of setup is shown on Fig. \\ref{Fig_fig_heterodyne_setup}. The object is an U.S. Air Force (USAF) resolution target whose hologram is recorded in transmission geometry (the target is back illuminated). The camera records the interference of the signal field ${\\cal E}(t)$ of optical angular frequency $\\omega_0$, with the reference (or local oscillator field) ${\\cal E}_{LO}(t)$ of optical frequency\n\\begin{eqnarray}\n \\omega_{LO}&=&\\omega_0- \\Delta \\omega\n\\end{eqnarray}\n where $\\Delta \\omega$ is the frequency shift. Let us introduce the slowly varying complex fields $E$ and $E_{LO}$.\n\n\\begin{eqnarray}\n {\\cal{E}} (t)& = &E e^{j\\omega_0 t} + E^* e^{-j\\omega_0 t}\\\\\n\\nonumber {\\cal{E}}_{LO} (t) &= &E_{LO}  e^{j\\omega_{LO}  t} + E_{LO} ^* e^{-j\\omega_{LO}  t}\n\\end{eqnarray}\nwhere $E^*$ and $E_{LO} ^*$ are the complex conjugates of $E$ and $E_{LO}$, and $j$ is the imaginary unit. The camera signal $I$ is proportional to the intensity intensity of the total field $ |{\\cal{E}} (t)+{\\cal{E}}_{LO} (t)|^2$. We have thus:\n\\begin{eqnarray}\\label{Eq_I}\n   I&=&\\left|E_{LO}~ e^{j\\omega_{LO}t}~+ ~E~ e^{j\\omega_0 t}\\right|^2 \\\\\n \\nonumber  &=& \\left|E_{LO} \\right|^2 + \\left|E \\right|^2 + E_{LO}^* E~e^{+j\\Delta \\omega t} +  E_{LO} E^*~e^{-j\\Delta \\omega t}\n\\end{eqnarray}\nSince the camera signal is slowly varying, we have neglected in Eq. \\ref{Eq_I} the fast varying terms  (which evolve at frequency $\\simeq 2\\omega_0$). Moreover, to simplify the present discussion, we  have not considered  the spatial variations of $I$, $E$ and $E_{LO}$ with $x$ and $y$, in particular the spatial variations that are related to the off axis tilt angle $\\theta$ of Fig. \\ref{Fig_fig_heterodyne_setup}.\nIn  equation \\ref{Eq_I},   $E_{LO}^* E~e^{-j\\Delta \\omega t}$ is the +1 grating order term, that contains the useful information (since this term  is proportional to $E$). The others terms: $\\left|E_{LO} \\right|^2 + \\left|E \\right|^2$ (zero grating order term),    and  $E_{LO} E^*~e^{+j\\Delta \\omega t}$ ( -1 grating order term or twin image term) are unwanted terms that must be cancelled. To filter off these terms, and to select the wanted +1 grating order signal,   4-phase detection is made. The AOMs driving angular frequencies  $\\omega_{AOM1}$ and  $\\omega_{AOM2}$ are tuned to have:\n\\begin{eqnarray}\\label{Eq_4_phases}\n  \\Delta \\omega =  \\omega_{AOM2}-  \\omega_{AOM1} =\\omega_{CCD}\/4\n\\end{eqnarray}\nwhere $\\omega_{CCD}$ is is the angular frequency of the camera frame rate. Lets us consider 4 successive camera frames: $I_0$, $I_1$ ...$I_3$  that are recorded at times $t=0$, $T$ ... $3T$ with $T = 2 \\pi\/( \\omega_{CCD})$.\nFor these  frames, the  phase factor  $e^{+j\\Delta \\omega t}$ is equal to $1$, $j$, -1 and $-j$. We thus get:\n\\begin{eqnarray}\\label{Eq_4_phases_I}\n  I_0&=&    \\left|E_{LO} \\right|^2 + \\left|E \\right|^2 + ~E_{LO}^* E + ~ E_{LO} E^*\\\\\n\\nonumber  I_1&=&    \\left|E_{LO} \\right|^2 + \\left|E \\right|^2 +j~ E_{LO}^* E - j ~E_{LO} E^*\\\\\n\\nonumber  I_2&=&    \\left|E_{LO} \\right|^2 + \\left|E \\right|^2 -~E_{LO}^* E  - ~E_{LO} E^*\\\\\n\\nonumber  I_3&=&    \\left|E_{LO} \\right|^2 + \\left|E \\right|^2 - j ~E_{LO}^* E  + j~E_{LO} E^*\n\\end{eqnarray}\nBy linear combination of $4$ frames, we get the 4-phase hologram $H$ that  obeys the demodulation equation:\n\\begin{eqnarray}\\label{Eq_4_phase_Hs}\n H&=& (I_0-I_2)+j(I_1-I_3)\\\\\n \\nonumber &=& 4~E_{LO}^* E\n\\end{eqnarray}\nAs wanted, the 4-phase demodulation equation yields a quantity $H$ that proportional to the signal complex field $E$. Note that the coefficient $4 E_{LO}$ is supposed to be known and do not depends on the object.\n\nHeterodyne holography exhibits several advantages with respect to other holographic\ntechniques:\n\\begin{enumerate}\n  \\item The phase shift is very accurate;\n  \\item The holographic detection is shot noise limited;\n  \\item Since the holographic  detection is  made somewhere near $\\omega_{LO}$ (depending on the demodulation equation that is chosen), it is possible to perform the holographic detection  with any frequency shift with respect to the object illumination angular frequency $\\omega_0$.\n\\end{enumerate}\nThe first advantage will be discussed in section \\ref{section_accurate_phase},\nthe second one in section \\ref{section_shot_noise}, while the third one, which  is the heart of the sideband holographic technique used to analyse vibration, will be discussed in\nsection \\ref{section_Selective detection of the sideband components}.\n\n\n\n\n\n\\subsection{Accurate phase shift  and holographic detection bandwidth}\\label{section_accurate_phase}\n\n\n\n\\begin{figure}[h]\n\\begin{center}\n   \n \n    \\includegraphics[width=8 cm]{fig_usaf_accurate_phase_ab}\n  \\includegraphics[width= 8 cm]{fig_usaf_accurate_phase_cd}\n \\caption{ USAF target reconstructed images (a,b) displayed with linear grey scale for the reconstructed  field intensity  $|E|^2$ with    $ \\omega_0- \\omega_{LO} = \\omega_{CCD}\/4=+2.5$ Hz (a) and   $ \\omega_0- \\omega_{LO} = - \\omega_{CCD}\/4= -2.5$ Hz (b).  Zooms  (c,d)  of the bright zones of the (a) and (b) images. Images (a,b) and zoom (c,d)  correspond to the $m=+1$ (a,c) and $m=-1$ (b,d) grating orders. }\\label{Fig_fig_usaf_accurate_phase_abcd}\n\\end{center}\n\\end{figure}\n\nIn the  typical heterodyne holography setup of Fig.\\ref{Fig_fig_heterodyne_setup}, the signals that drive the  AOMs  are generated by frequency synthesizers, phase-locked with a common 10 MHz clock.. The phase shift  $\\Delta \\varphi= \\Delta \\omega T$ (that is equal to  $\\pi\/2$ in the 4 phase demodulation case) can thus be adjusted with quartz accuracy.\n\nTo illustrate this accurate phase shift,  holograms of a USAF target have been recorded  with the Fig.\\ref{Fig_fig_heterodyne_setup} setup, while sweeping the LO frequency $\\omega_{LO}$. Figure \\ref{Fig_fig_usaf_accurate_phase_abcd} shows  USAF reconstructed images that are obtained for different values of the frequency shift $ \\omega_0-\\omega_{LO}$  \\cite{atlan2007accurate}.\n\\begin{enumerate}\n  \\item For $\\omega_0- \\omega_{LO} = \\omega_{CCD}\/4=+2.5$ Hz, the image that is reconstructed is sharp and  correspond to the +1 grating order.  On the other hand, the -1 grating order signal is very low; the magnitude of the -1 parasitic twin image image is negligible in front of the +1 contribution. Image and zoom of image are seen on Fig.\\ref{Fig_fig_usaf_accurate_phase_abcd} (a) and (c).\n  \\item For $\\omega_0- \\omega_{LO}  = -\\omega_{CCD}\/4= -2.5$ Hz, the image that is reconstructed is blurred and  corresponds to the -1 grating order twin image. The magnitude of the +1 grating order image is negligible in front of its -1 counterpart. Image and zoom of image are seen on Fig.\\ref{Fig_fig_usaf_accurate_phase_abcd} (b) and (d).\n\\end{enumerate}\n\n\n\n\\begin{figure}\n\\begin{center}\n   \n \n  \\includegraphics[width=12 cm]{fig_accurate_phase_curves}\n \\caption{ Field energy  of the +1 (a) and -1 (b) grating order. The vertical axis is in logarithmic scale arbitrary units. The points correspond to the experimental data obtained by sweeping the LO frequency  with   $(\\omega_0-\\omega_{LO}) =-5$ to +5 Hz with 0.1 Hz increments. The solid gray curves are  $+U_{+}(\\omega-\\omega_{LO})$ (a) and $+U_{-}(\\omega-\\omega_{LO})$ (b) given by  Eq.\\ref{Eq_total_versus_delta_omega} in the same frequency range:   $(\\omega-\\omega_{LO}) =-5$ to +5 Hz. }\\label{Fig_fig_accurate_phase_curves}\n\\end{center}\n\\end{figure}\n\nWe have measured the total field energy $U_{\\pm}$ in both +1 and -1 grating order reconstructed images:\n\\begin{eqnarray}\\label{Eq_total_energy_definition}\n  U_{\\pm} &=&   \\sum_{x,y}\\left|E(x,y) \\right|^2\n\\end{eqnarray}\nwhere $\\sum_{x,y}$ is the sum over  the pixels of the reconstructed field image $E(x,y)$ corresponding to the  $\\pm 1$ grating order zones.\nAs seen on Fig. \\ref{Fig_fig_accurate_phase_curves}, the energy of the signal that is measured in the +1 grating order (i.e. $U_+$)    is maximum for  $\\omega_0-\\omega_{LO}=\\omega_{CCD}\/4 = +2.5$ Hz, and null  $\\omega_0 -\\omega_{LO}=-2.5$ Hz. Similarly,  $U_-$ is maximum for  $ \\omega_0 -\\omega_{LO}=-2.5$ Hz, and null for $\\omega_0 -\\omega_{LO}=+2.5$ Hz.  By adjusting $\\omega_0 -\\omega_{LO}$, it is then possible to select the grating order that is detected.\n\nFor 4 phase detection with a local oscillator of frequency $\\omega_{LO}$, the total energy  $U_{\\pm}=|E|^2$ detected at frequency $\\omega$ in the $\\pm 1$ grating order can be easily calculated   \\cite{verpillat2010digital}:\n\\begin{eqnarray}\\label{Eq_total_versus_delta_omega}\n   U_{\\pm}(\\omega-\\omega_{LO}) &=& \\left| ~\\frac{1}{4T'}\\sum_{n=0}^{n=3} (\\pm j)^n \\int_{t=nT}^{nT+T'}~dt~ e^{ \\pm j ( \\omega -\\omega_{LO}) t} \\right|^2\\\\\n \\nonumber  &=& \\left|\\frac{ \\textrm{sinc}(\\pi (\\omega-\\omega_{LO}) T')}{4} \\sum_{n=0}^{n=3} j^n\n    e^{j  n (\\omega-\\omega_{LO})T} \\right|^2\n\\end{eqnarray}\nwhere $ \\textrm{sinc} (x) = \\sin x\/x$. In equation \\ref{Eq_total_versus_delta_omega}, $T$ is the frame period, and  $T'$ the exposure time. The coefficient   $1\/4T'$ is thus a normalization factor. On the other hand,\nthe factors $(\\pm j)^n$ corresponds to the coefficients of  the   demodulation equation (see Eq. \\ref{Eq_4_phase_Hs}), and   $ e^{\\pm j (\\omega-\\omega_{LO})  t}$ is the interference instantaneous phase factor, which must be integrated  over the exposure time from  $t=nT$ to $t=nT+T'$.\n\nIn the  experiment of Fig.\\ref{Fig_fig_accurate_phase_curves}, we have $T=T'=2\\pi\/\\omega_{CCD}=100 $ ms. We have plotted the field energy  $U_{\\pm}$ given by Eq.\\ref{Eq_total_versus_delta_omega} as a function of $ \\omega- \\omega_{LO}$ on Fig. \\ref{Fig_fig_accurate_phase_curves} (solid grey lines). As seen, experiment agrees  with the theoretical curve of Eq. \\ref{Eq_total_versus_delta_omega}. The shape of the curves  represents here the frequency response spectrum of the holographic device considered as a detector. For the +1 grating order, detection is centered at frequency $ \\omega = \\omega_{LO} + \\omega_{CCD} \/ 4$. The measurement bandwidth  BW is  2.5 Hz. It is equal to the inverse of the measurement time of 4 frames i.e. $\\textrm{BW}= 1 \/ 4T$.  It illustrates the coherent character (in time) of holographic detection.\n\n\n\n\n\\begin{figure}\n\\begin{center}\n   \n \n  \\includegraphics[width=5.5 cm]{fig_Ua}\n  \\includegraphics[width=5.5 cm]{fig_Ub}\n \\caption{ Theoretical field energy  $U_{+}(\\omega-\\omega_{LO})$ given by Eq.\\ref{Eq_total_versus_delta_omega}    plotted in linear (a) and  logarithmic scale (b) as a function of $( \\omega-\\omega_{LO})$ in Hz Units for the +1  grating order. The number of frame is $n_{max}=4$ (solid grey line), $n_{max}=8$ (solid black line) and $n_{max}=16$ (dashed black line).\n }\\label{Fig_fig_U_curves}\n\\end{center}\n\\end{figure}\n\nIt is possible to increase the selectivity of the holographic coherent detection by increasing the measurement time, i.e. by increasing the number of frames $n_{max}$ used for demodulation. In that case, Eq. \\ref{Eq_4_phase_Hs} and Eq.\\ref{Eq_total_versus_delta_omega}  must be replaced by similar equations involving   $n_{max}$ frames in place of 4. We have:\n\\begin{equation}\\label{EQ_H_n_max}\n    H= \\sum_{n=0}^{n_{max}-1} j^n ~I_n\n\\end{equation}\nand\n\\begin{eqnarray}\\label{Eq_total_versus_delta_omega_nmax}\n   U_{\\pm}(\\omega-\\omega_{LO}) &=& \\left| ~\\frac{1}{n_{max}T}\\sum_{n=0}^{n_{max}-1} (\\pm j)^n \\int_{t=nT}^{nT+T'}~dt~ e^{\\pm j (\\omega-\\omega_{LO}) t} \\right|^2\\\\\n    \\nonumber  &=& \\left|\\frac{ \\textrm{sinc}(\\pi (\\omega-\\omega_{LO}) T')}{n_{max}} \\sum_{n=0}^{n_{max}-1} j^n\n    e^{j n (\\omega-\\omega_{LO})T} \\right|^2\n\\end{eqnarray}\nFigure  \\ref{Fig_fig_U_curves} plots  the  detection frequency spectrum $U_{+}$   for $n_{max}=4$, 8 and 16.    As seen, the detection bandwidth BW  decreases with $n_{max}$. It is equal to:\n\\begin{equation}\\label{Eq_BW}\n    \\textrm{BW} = \\frac{1}{n_{max} T}\n\\end{equation}\nThe noise, uniformly distributed in frequency, is expected to decrease accordingly.  We will show in section \\ref{section_shot_noise}, that very high detection sensitivity can  be obtained by holographic detection with large number of frames $n_{max}$.\n\n\n\\subsection{Shot noise holographic detection}\\label{section_shot_noise}\n\n\nBecause the holographic signal results in the interference of the object\nsignal complex field $E$ with a reference (LO) complex\nfield $E_{LO}$ whose amplitude can be much larger (i.e.,\n$E_{LO} \\gg E$),  the holographic detection benefits from ''heterodyne'' or ''holographic'' gain (i.e.,  $|E E^*_{LO}| \\gg |E|^2$), and is thus well suited for  detection  of  weak signal fields $E$.  Holographic detection can reach the theoretical limit of noise which corresponds to a noise equivalent signal of 1 photo electron per pixel during the total measurement time~\\cite{gross2007digital,gross2008noise,verpillat2010digital,lesaffre2012noise}.\n\n\n\n\n\\begin{figure}\n\\begin{center}\n   \n \n  \\includegraphics[width=11 cm]{fig2_shot_noise}\n\n \\caption{Heterodyne detection of 1 photon per pixel of signal with N photon\nlocal oscillator.\n }\\label{Fig_fig2_shot_noise}\n\\end{center}\n\\end{figure}\n\n\n\nTo illustrate this point, let us consider the interference of a weak signal  of about 1 photo-electron per pixel and per camera frame, with a large local oscillator reference signal  of $N=10^4$ photo-electrons: see Fig. \\ref{Fig_fig2_shot_noise}. In this example, the signal intensity is $|E|^2=1$, while the holographic interference term $|E E^*_{LO}|=100$ is much larger. The detected signal is  $|E_{LO}+E|^2 \\simeq 10^4$ photo-electrons.\n\n\n\n\nBecause of the quantum nature of the process involved in digital holography (laser emission, photodetection...), the detected signal in photo electron Units is random Gaussian integer number, whose average value is $ N=10^4$, and whose standard deviation is $ \\sqrt{N}=100$. These  fluctuations of the number of photo electrons are called shot noise. Here, with 1 photo electron of signal, the heterodyne signal is  $100$ photo electrons and the noise  100 too. The noise equivalent signal (for the energy $|E|^2$) is thus 1 photo electron per pixel and per frame.\n\nLet us now study how shot noise varies with the number of frames  $n_{max}$ used for  detection.  As in any detection process, the noise in energy is proportional to the measurement time and to the detection bandwidth BW. Since  the measurement time is $n_{max} T$, and since  the detection bandwidth is $\\textrm{BW}=1\/n_{max} T$ (see Eq. \\ref{Eq_BW} ),  the shot noise in energy do not depends on $n_{max}$. The shot noise equivalent signal is thus the same than for one frames. It remains equal to   1 photo electron per pixel whatever the number  of frames $n_{max}$ and measurement time $n_{max}T$ are.\n\nLet us now discuss the ability to reach this  shot noise optimal sensitivity  in\nreal life holographic experiments. Since we consider implicitly\na very weak signal, the noises that must be\nconsidered are\n\\begin{enumerate}\n  \\item the read noise and dark current of the  camera,\n  \\item the quantization noise of the camera A\/D converter,\n  \\item the technical noise of the LO beam,\n  \\item and the LO beam shot noise, which yield the theoretical\nnoise limit.\n\\end{enumerate}\nFor a typical camera, the full well capacity is $2 ~10^4$ photo electrons, and a good practice is to work with $N=10^4$ photo electrons for the local oscillator. Shot noise on the camera signal (100 photo electrons) is thus much larger than the camera read noise (1 to 20 photo electrons) and the camera dark current (a few photo electrons per second). If the camera is 12 bit, the  full well capacity   corresponds $\\sim 2^{12}$ digital count (DC). As a results, the quantization noise ($\\sim 7 $ photo electrons) can be neglected too.\n\n\n\\begin{figure}\n\\begin{center}\n   \n \n  \\includegraphics[width=8 cm]{fig_usaf_1_4phases}\n\n \\caption{Fourier space hologram intensity (i.e. $|\\tilde H(k_x,k_y)^2$) displayed in arbitrary log scale for one phase ($H=I_0$) and 4 phases ($H=(I_0-I_2)+j(I_1-I_3)$ detection.\n }\\label{Fig_fig_usaf_1_4phases}\n\\end{center}\n\\end{figure}\n\n\n\nLet us discuss on the technical noise of the LO beam, and on the way to filter it off.\n\\begin{enumerate}\n  \\item\nWe have displayed,   on Fig. \\ref{Fig_fig_usaf_1_4phases}, examples of Fourier space hologram $\\tilde H(k_x,k_y) =\\textrm{FFT}( ~H(x,y))$ (where FFT is the Fourier transform operator). The signal fields yields the interference term $E E^*_{LO}$ that is located in the left hand side of the Fourier space images of Fig. \\ref{Fig_fig_usaf_1_4phases} (a) and (b). It corresponds to the grating order +1.  The local oscillator signal  $|E_{LO}|^2$ is located in the center of the Fourier space. It is visible on Fig. \\ref{Fig_fig_usaf_1_4phases} (a) that is obtained with 1 phase hologram. As seen, the LO signal are  separated in the Fourier space  because of the off axis configuration.\nIt is then possible to filter off the parasitic LO signal by a proper spatial filtering in the Fourier space as shown by  \\cite{cuche2000spatial}.\nAs noticed, this operation filters-off both  the LO signal and the LO technical noise.\n  \\item On the other hand, because of phase shifting, the  signal and LO   fields  $E$ and $E_{LO}$  evolve at two different time frequencies  $\\omega$ and $\\omega_{LO}$ with $\\omega=\\omega_{LO} + \\omega_{CCD}\/4$. It is then possible to filter off the LO signal in time. This is done by the 4 phases demodulation process since the LO term $|E_{LO}|^2$ vanish in Eq. \\ref{Eq_4_phase_Hs} and \\ref{Eq_total_versus_delta_omega_nmax}. This filtering in time is illustrated by Fig. \\ref{Fig_fig_usaf_1_4phases}, since the 0 grating order signal that is large for single phase off-axis holography  (Fig. \\ref{Fig_fig_usaf_1_4phases} (a)), roughly vanishes in the four phase case (Fig. \\ref{Fig_fig_usaf_1_4phases} (b)).\nTime filtering is also illustrated by  Fig.\\ref{Fig_fig_accurate_phase_curves} and Fig.\\ref{Fig_fig_U_curves} and Eq. \\ref{Eq_4_phase_Hs} and \\ref{Eq_total_versus_delta_omega_nmax}, since the detected energy  $U_{+\/-}$ vanishes for  $\\omega- \\omega_{LO} =0 $ i.e. for detection at local oscillator frequency $\\omega_{LO}$. Here again, one filters off both  the LO signal and the LO technical noise.\n\\end{enumerate}\nBy combining off axis and phase shifting, it is then possible apply to a double filtering (in space and time) that filter off the LO technical noise very effectively. One can gets then the shot noise ultimate sensitivity that corresponds to a noise equivalent signal of  1 photon per pixel whatever the measurement times is.\n\n\n\\begin{figure}\n\\begin{center}\n   \n\\includegraphics[width=12 cm]{fig_usaf_10mn}\n \n\n \\caption{Reconstructed image of a USAF target in dim light. Hologram are recorded by 4 phases heterodyne holography with $T=100$ ms, and $n_{max}=600$ (a,b) and $n_{max}=6000$  (c). The coherent measurement time $n_{max} T $ is thus 1 minute (a,b) and 10 minutes (c). Illumination is adjusted so that the USAF signal integrated is   1 photon per pixel over the whole measurement time $n_{max}T$  in (a). Holograms (b) and (c) are recorded by adding a neutral density filter D=1.0 on illumination. The USAF signal is thus 0.1 photon per pixel in (b) and  1 photon per pixel in (c). Display is made with arbitrary linear scale for intensity.}\\label{Fig_fig_usaf_10mn}\n\\end{center}\n\\end{figure}\n\n\nFigure \\ref{Fig_fig_usaf_10mn} illustrates  heterodyne holography ability of imaging an object\n(here an USAF target) in dim light with shot noise limited sensitivity. In figure \\ref{Fig_fig_usaf_10mn} (a), the measurement time is 1 minute and the illumination power is adjusted such a way the USAF signal is about 1 photon per pixel in 1 minute. The visual quality of USAF  image is quite good. We can say that  SNR is about 1.  In figure \\ref{Fig_fig_usaf_10mn} (b), the  measurement time is the same, but illumination is divided by 10 by using a neutral  filter of density $D=1.0$. SNR is low, and one cannot see the grooves of the USAF target. Figure \\ref{Fig_fig_usaf_10mn} (c) is obtained with the same illumination level that Fig. \\ref{Fig_fig_usaf_10mn} (b) (i.e. with neutral density filter) but the measurement time is multiplied by 10 (i.e. 10 minutes in place of 1 minute). The visual quality of reconstructed image is the same than for Fig. \\ref{Fig_fig_usaf_10mn} (a) with  SNR about 1. This experiment shows that the reconstructed image quality depends on the total amount of signal, and do not depends on the  time needed to get that amount of signal: to get $\\textrm{SNR}\\sim 1$ one needs about 1 photon per pixel, for any measurement time.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Holographic vibrometry}\\label{section_vibrometry}\n\nLets us now  apply heterodyne holography\nto vibration analysis.\n\n\\subsection{Optical signal scattered by a vibrating object}\n\n\n\\begin{figure}\n\\begin{center}\n   \n\\includegraphics[width=9 cm]{fig_setup-reed}\n \n\n \\caption{Heterodyne holography setup applied to analyse vibration of a clarinet reed. L: main laser; AOM1, AOM2: acousto-optic modulators; M: mirror; BS:\nbeam splitter; BE: beam expander; CCD:  camera; LS: loud-speaker exiting the vibrating  clarinet\nreed at frequency $\\omega_A\/2\\pi$.}\\label{Fig_fig_setup_reed}\n\\end{center}\n\\end{figure}\n\nConsider a point of the  objet  (for example a clarinet reed) that is studied by heterodyne holography (see Fig. \\ref{Fig_fig_setup_reed}), vibrating  at frequency\n$\\omega_A$  with amplitude $z_{max}$. The displacement $z$ along the out of plane direction is\n\\begin{equation}\\label{Eq_z_t}\n  z(t) = z_{max} \\sin{\\omega_A t}\n\\end{equation}\nIn backscattering geometry, this corresponds to a\nphase modulation $\\varphi(t)$ of the signal:\n\\begin{eqnarray}\\label{Eq_varphi_t}\n \\varphi(t) &=& 4\\pi z(t)\/\\lambda\\\\\n  \\nonumber  &=&\\Phi \\sin{\\omega_A t}\n\\end{eqnarray}\nwhere $\\lambda$ is the optical wavelength and  $\\Phi$ the amplitude of the phase modulation of the signal  at angular frequency $\\omega_A$:\n\\begin{eqnarray}\\label{Eq_Phi}\n\\Phi&=&4\\pi z_{max}\/\\lambda\n\\end{eqnarray}\nLet us define the slowly varying complex  amplitude $E(t)$ of the field ${\\cal E}(t)$ scattered by the vibrating object. We have:\n\\begin{eqnarray}\n{\\cal E}(t)&=& E(t) e^{j\\omega_0 t} + \\textrm{c.c.}\n\\end{eqnarray}\nBecause of the Jacobi-Anger expansion, we get:\n\\begin{eqnarray}\\label{Eq_cal_E_sum_over_harmonic}\n\\nonumber E(t)&=&E ~e^{j\\varphi(t)}=E ~e^{j \\Phi \\sin{\\omega_A t}  }\\\\\n\\nonumber &=& E ~\\sum_m J_m(\\Phi)~e^{j n\\omega_A t}\n\\end{eqnarray}\nwhere $E$ is the complex amplitude without vibration, and $J_m$\n the mth-order Bessel function of the first kind, with\n$J_{-m}(z)=-1^m J_m(z)$ for integer $ m$ and real $z$.\nThe scattered\nfield ${\\cal E}(t)$ is then the sum of the carrier and sideband field components ${\\cal E}_m(t)$ of frequency $\\omega_m$, where $m$ is the sideband index with:\n\\begin{eqnarray}\\label{Eq_cal_E_m}\n {\\cal E}(t) &=& \\sum_{m=-\\infty}^{+\\infty}{\\cal E}_m(t)\\\\\n\\nonumber   {\\cal E}_m(t) &=& E_m e^{j\\omega_m t} +  E_m^* e^{-j\\omega_m t}\\\\\n\\nonumber   \\omega_m &=& \\omega_0 + m \\omega_A\n\\end{eqnarray}\nwhere $ E_m$ is the complex amplitude of the field component ${\\cal E}_m(t)$. Equation \\ref{Eq_cal_E_sum_over_harmonic} yields:\n\\begin{equation}\\label{Eq_E_m}\n    E_m = J_m(\\Phi)~ E\n\\end{equation}\n\n\\begin{figure}\n\\begin{center}\n   \n\\includegraphics[width=5.8 cm]{fig_Jn_a}\n\\includegraphics[width=5.8 cm]{fig_Jn_b}\n \n\n \\caption{Relative amplitude $|E_m|^2\/|E|2=|J_m(\\Phi)|^2$ of the sideband component $m$ for an  amplitude modulation of the phase equal to  $\\Phi=0.3$  (a) and $\\Phi=3.0$ (b)  rad.}\\label{Fig_fig_sideband _energy}\n\\end{center}\n\\end{figure}\nFigure \\ref{Fig_fig_sideband _energy} presents the distribution of the field energy   on the sidebands components $|E_m|^2$. If the amplitude of modulation $\\Phi$ is low, most of the energy is on the carrier $|E_0|\/|E|^2\\simeq 1$, and energy $|E_m|^2$  decreases rapidly with the sideband index $m$. If the amplitude $\\Phi$ is large, the energy of the carrier is low $|E_0|\/|E|^2 \\ll 1$, while  energy is distributed over many sidebands  $|E_m|^2$ .\n\n\n\n\\subsection{Selective detection of the sideband components $E_m$: sideband holography}\n\\label{section_Selective detection of the sideband components}\n\n\n\\begin{figure}\n\\begin{center}\n   \n\\includegraphics[width=8 cm]{fig_clarinet_reed}\n \n\n \\caption{Reconstructed holographic images of a clarinet reed vibrating at frequency $\\omega_A\/2\\pi =\n2143$ Hz perpendicularly to the plane of the figure. Figure (a) shows the carrier image\nobtained for $m = 0$. Fig. (b)-(d) show the frequency sideband images respectively for $m = 1$,\n$m = 10$, and $m = 100$. A logarithmic grey scale has been used}\\label{Fig_fig_clarinet_reed}\n\\end{center}\n\\end{figure}\nHeterodyne holography that is able to perform\nthe holographic detection with any frequency shift $\\omega-\\omega_0 $ with respect to the\nobject illumination angular frequency $\\omega_0$ is well suited to detect the vibration sideband components $E_m$.\nTo selectively detect  by  4 phase demodulation the sideband $m$ of frequency $\\omega_m$,\nthe frequency $\\omega_{LO}$ must be adjusted to fulfil the condition :\n\\begin{eqnarray}\\label{Eq_Delata omega 4phase_sideband}\n     \\omega_{LO}&=& \\omega_m-\\omega_{CCD}\/4\\\\\n   \\nonumber    &=& \\omega_0 + m \\omega_A -\\omega_{CCD}\/4\n\\end{eqnarray}\n\n\n\nFigure \\ref{Fig_fig_clarinet_reed} shows  images obtained by detecting different sideband $m$ of a clarinet reed \\cite{joud2009imaging}.\nThe clarinet reed is\nattached to a clarinet mouthpiece and its vibration is driven by a sound wave propagating inside\nthe mouthpiece, as in playing conditions, but the sound wave is created by a\nloudspeaker excited at frequency $\\omega_A$ and has a lower intensity than inside a clarinet. The excitation frequency\nis adjusted to be resonant with the first flexion mode (2143 Hz) of the reed.\n\n\nFigure \\ref{Fig_fig_clarinet_reed} (a) is obtained at the unshifted carrier frequency $\\omega_0$. It corresponds to an image obtained by time averaging holography \\cite{picart2003time}.\nThe left side of the reed\nis attached to the mouthpiece, and the amplitude of vibration is larger at the tip of the reed\non the right side; in this region the fringes become closer and closer and difficult to count.\nThe mouthpiece is also visible, but without fringes since it does not vibrate. Similar images\nof clarinet reeds have been obtained in \\cite{demoli2004detection,picart20052d}, with more conventional techniques and lower\nimage quality. Figures \\ref{Fig_fig_clarinet_reed} (b) to (d) show images obtained for the sidebands $m=1$, 10 and 100. As\nexpected, the non-vibrating mouthpiece is no longer visible. Figure \\ref{Fig_fig_clarinet_reed} (b) shows the $m = 1$\nsideband image, with $J_1$ fringes that are slightly shifted with respect to those of $J_0$. Figure\n\\ref{Fig_fig_clarinet_reed} (c) shows the image of sideband  $m = 10$ and Fig.\n\\ref{Fig_fig_clarinet_reed} (d) the  $m = 100$ one. The left side region of the image\nremains dark because, in that region, the vibration amplitude is not sufficient to generate these\nsidebands, $J_m(z)$ being evanescent for $z < m$.\n\n\n\\begin{figure}\n\\begin{center}\n   \n\\includegraphics[width=10 cm]{fig_opex_curve}\n \n\n \\caption{A slice of the data along the $y = 174$ line is used in this figure; the $x$ horizontal\naxis gives the pixel index (100 pixels correspond to 3.68 mm.), the vertical axis the vibration\namplitude $\\Phi$. The lower part of the figure shows the normalized signal\n$|E_m(x)|^2\/|E(x)|^2$  (where $E(x)$ is obtained loud-speaker off) for  $m = 5$, with a downwards axis. The left part\nshows the corresponding square of the Bessel function $|J_5(\\Phi)|^2 $ with a leftward axis. The\nzeroes of the two curves are put in correspondence, which provides the points in the central\nfigure.   Similar correspondences are made harmonic order $m = 0, 1, 5, 10...100$.\nDifferent gray densities are used for  different $m$.    The crosses\ncorrespond to $m = 5$. The juxtaposition of the points for all values of $m$ gives an accurate\nrepresentation of the amplitude of vibration $\\Phi$ as a function of $x$.}\\label{Fig_fig_opex_curve}\n\\end{center}\n\\end{figure}\n\nTo quantitatively visualize the vibration amplitude $\\Phi$, cuts of the reconstructed images signal $|E_m(x,y)|^2$ are made  for different\nsideband orders $m$ along the horizontal line $y = 174$. This value has been chosen because it\ncorresponds to a region where the fringes are orthogonal to the $y$ axis.\nTo build the central part of Fig. \\ref{Fig_fig_opex_curve},  the position of the antinodes of $|E_m(x,174)|^2$ are put in correspondence  with the antinodes of  $|J_m(\\Phi)|^2 $. Correspondence is made  for  $m =$ 0, 1, 5, 10, . . . 100. Note that this  method is insensitive to inhomogeneities in the illumination zone. Therefore, no normalization is required. The curve seen in the central zone of Fig. \\ref{Fig_fig_opex_curve} represents the amplitude of phase oscillation  $\\Phi$ in radian  as a function of pixel index along $x$ for $y=174$.\n\n\n\n\\begin{figure}\n\\begin{center}\n   \n\\includegraphics[width=10 cm]{fig_taillard}\n \n\n \\caption{Clarinet reed  reconstructed images  obtained on sideband $m=1$.  Frequency $\\omega_A$ is swept  from 1.4 KHz up to 20 KHz, and images are displayed from left to right and   top to bottom ($26 \\times 7$ images). The display is made with arbitrary grey scale for the intensity $|E_0(x,y)|^2$. }\\label{Fig_fig_taillard}\n\\end{center}\n\\end{figure}\n\n\n\\textbf{Remark:} In a typical heterodyne holography setup, digital synthesizers drive the acousto optic modulator at $\\omega_{AOM1}$ and  $\\omega_{AOM2}$,\nthe camera at  $\\omega_{CCD}$, and the vibration frequency at $\\omega_{A}$. These synthesizers use a common 10 MHz reference frequency, and  are driven by the computer.   It is then possible to automatically  sweep $\\omega_{A}$, and $\\omega_{AOM1}$  (or $\\omega_{AOM2}$) in order to fulfil Eq. \\ref{Eq_Delata omega 4phase_sideband} so that detection remains ever tuned on a given sideband.  Figure \\ref{Fig_fig_taillard} shows an example \\cite{taillard2014statistical}. A series of $26 \\times 7$ images of a clarinet reed  are obtained on sideband $m=1$  by sweeping the frequency $\\omega_A$ from 1.4 kHz up to 20 kHz by steps of 25 cents. The amplitude of the excitation signal is  exponentially increased in the range 1.4 to 4 kHz, from 0.5 to 16 V, then kept constant at 16 V up to 20 kHz. This crescendo limits the amplitude of vibration of the first two resonances of the reed.  The different vibration  modes  of the reed can be easily recognized on the reconstructed reed images of Fig. \\ref{Fig_fig_taillard}.\n\n\n\\subsection{Sideband holography for large amplitude of vibration}\\label{Section_large_vibration}\n\nIn the previous section (section \\ref{section_Selective detection of the sideband components} and Fig. \\ref{Fig_fig_opex_curve})\nwe have shown how the comparison of dark fringes for different sideband leads to a determination of the vibration\namplitude $\\Phi(x, y)$ at each point of the object. This determination is non-local, since it involves counting fringes\nfrom one reference point of the image to the point of interest, so that large amplitudes are not accessible.\nThe vibration amplitude $\\Phi(x, y)$ can be determined by another approach that\ncompletely eliminates the necessity of counting fringes,\ngiving a local measurement of the amplitude of vibration $\\Phi$,\neven for large $\\Phi$ \\cite{joud2009fringe}.\n\n\n\\begin{figure}\n\\begin{center}\n   \n\\includegraphics[width=7 cm]{fig_Jm_303}\n \n\n \\caption{Relative intensity $|E_m|^2\/|E|2=|J_m(\\Phi)|^2$ of the sideband component $m$ for an  amplitude modulation of the phase equal to  $\\Phi=30.3$  radiant. The light grey shade shows the Doppler spectrum obtained\nfrom the vibration velocity distribution, with a continuous\nvariable $m$. Both spectra\nfall abruptly beyond $m=30.3$, which corresponds to the\nDoppler shift $\\pm \\omega_A \\Phi$ associated with the maximum velocity.}\\label{Fig_fig_Jm_303}\n\\end{center}\n\\end{figure}\n\n\nFor large amplitude of vibration ( $\\Phi\\gg 1$), the  distribution of the sideband energy $|E_m|^2$  over $m$    exhibits a sharp variation from maximum to zero near $m \\simeq \\Phi$, as seen on  Fig. \\ref{Fig_fig_Jm_303} that plot $|E_m|^2\/|E|^2$ for  $\\Phi=30.3$. This property can be understood if one consider the limits $z_{max}\\gg \\lambda$. In that case, one can define an instantaneous  Doppler angular frequency shift $\\omega_D(t)$, and an instantaneous sideband index $m(t)$ that are continuous variables :\n\\begin{eqnarray}\n \\omega_D(t)&=&\\omega_A \\Phi \\cos( \\omega_A t) \\\\\n \\nonumber m(t)&=& \\frac{\\omega_D(t)}{ \\omega_A}=  \\Phi \\cos( \\omega_A t)\n\\end{eqnarray}\nBecause of its sinusoidal variation, $m(t)$ spend more time near the extreme points $m= \\pm \\Phi$. The Doppler continuous distribution spectrum of $m(t)$ that is displayed in  light grey shade on Fig. \\ref{Fig_fig_Jm_303} are thus maximum near the extreme  points $m= \\pm \\Phi$.\n\nThe vibration amplitude $\\Phi(x,y)$ can be determined for each location $x,y$,  by measuring $|E_m(x,y)|^2$  for all sidebands $m$, and by determining for each location $x,y$ the sideband index $m$  that correspond to a fast drop of the signal  $|E_m(x,y)|^2$.    The method is robust and can easily be used even when the fringes become so narrow that they cannot be resolved, which gives immediate\naccess to large amplitudes of vibration.\n\n\n\\begin{figure}\n\\begin{center}\n   \n\\includegraphics[width=10 cm]{Fig_cube_of_data}\n \n\n \\caption{Cube of data obtained from the reconstructed holographic images of a vibrating clarinet reed. The sideband images\nwith $m = 0$, 20, 40 ...120 are shown in arbitrary linear scale.\n The white dashed lines correspond to $x = 249$\nand $y = 750$, i.e. to the point chosen for Fig. \\ref{Fig_fig3__OL}.}\\label{Fig_fig_cube_of_data}\n\\end{center}\n\\end{figure}\n\n\nBy successively adjusting the frequency $\\omega_{LO}$ of\nthe local oscillator to appropriate values, one records\nthe intensity images $|E_m(x,y)|^2$ of the sidebands as a\nfunction of $x$ $y$ and $m$. One then obtains a cube of data\nwith three axes $x$, $y$ and $m$. Figure \\ref{Fig_fig_cube_of_data} shows the images obtained for for $m = 0$, 20 ...120\nthat correspond to cuts of the cube along $x,y$ planes.\n The images illustrate how, when $m$ increases, the fringes move towards regions with larger amplitudes of vibrations. Since the right part of the reed ($x > 800$) is clamped on\nthe mouthpiece,  no signal is obtained in regions near the clamp\nwhere $\\Phi =4\\pi z_{max}\/\\lambda \\leq m$ (right part of the images).\n\n\n\n\\begin{figure}\n\\begin{center}\n   \n  \\includegraphics[width=8 cm]{Fig_3OL}\\\\\n  \\caption{Images corresponding to cuts of 3D data of the reconstructed\n images along the planes $y=750$ (a) and $x=249$\n(b). Fig~(a) shows the deformation of the object along its axis and\nFig~(b) a transverse cut with  a slight vibration asymmetry. A\nlogarithmic intensity scale is used.} \\label{Fig_fig3__OL}\n\\end{center}\n\\end{figure}\n\n\nFigure \\ref{Fig_fig3__OL} (a) displays a 2D cut (coordinates $x,m$) of the cube of\ndata along the  horizontal plane $y = 750$ (horizontal\nwhite dashed line in Fig. \\ref{Fig_fig_cube_of_data}). The envelope of the non-zero\n(non black) part of the image provides a measurement of\nthe amplitude of vibration in units of $\\lambda\/4\\pi$. One actually\nobtains a direct visualization of the shape of the reed at\nmaximal elongation, from the right part clamped on the\nmouthpiece to the tip on the left. The maximum amplitude correspond to $\\Phi=120$ rad.\n\n\\begin{figure}\n\\begin{center}\n   \n  \\includegraphics[width=6 cm]{fig4_OL}\\\\\n \\caption{(a): image reconstructed with sideband $n=330$, with a large amplitude of vibration. (b) is the equivalent of\nFig.~\\ref{Fig_fig3__OL}(a), but with positive  and negative $m$\nvalues. One measures a maximum vibration amplitude of\n$z_{ max} \\simeq 60$ $\\mu$m. A logarithmic intensity\nscale is used.} \\label{Fig_fig4__OL}\n\\end{center}\n\\end{figure}\n\nFigure \\ref{Fig_fig4__OL} shows  images\nobtained at higher excitation amplitudes, about\n10 times larger that for Fig.\\ref{Fig_fig3__OL}. Figure \\ref{Fig_fig4__OL} (a) shows the images\nobtained for $m = 330$: the fringes are now completely\nunresolved, but the transition from zero to non-zero intensity remains very clear. With a single hologram, and\nwithout fringe counting, one obtains a clear marker of the line where $\\Phi(x, y) = 330$ rad.\nFigure \\ref{Fig_fig4__OL}(b) shows the equivalent of Fig. \\ref{Fig_fig3__OL}(a), but with\na higher excitation level, and this time for positive and\nnegative values of $m$.  Data range up to about $|m|\\simeq 1140$,\ncorresponding to $z_{max}\\simeq 58.4$ $\\mu$m. Since the vibration\namplitude is much larger than $\\lambda$, the continuous approximation for $m$ is valid, and the images of Fig. \\ref{Fig_fig4__OL} can\nbe reinterpreted in term of classical Doppler effet.\n\nTaking advantage of\nthe sideband order $m$  of the light scattered by a vibrating object adds a new dimension to digital holography.\nEach pixel $x,y$ of the image then provides an information\nthat is completely independent from others, which results in redundancy and robustness of the measurements.\nLooking at the edges of the spectrum provides an accurate determination of the vibration amplitude and avoids\na cumbersome analysis of the whole cube of data cube,\ngiving easy access to a measurements of large amplitudes\nof vibration.\n\n\n\\subsection{Sideband holography with strobe illumination}\\label{section_strobe_detection}\n\n\n\n\n\\begin{figure}[h]\n\\begin{center}\n   \n  \\includegraphics[width=7 cm]{Fig_chronogramme}\\\\\n \\caption{Chronogram of the signals.  Sinusoidal signal of period $T_A = 2\\pi\/\\omega_A$  exciting\nthe reed (a). Rectangular gate that is applied to both illumination and reference beams (b) .Gate delay is $t_d$ with respect to the origin of reed sinusoidal motion. Gate duration is $\\delta t$.  Gated AOM1 and AOM2 signals at $\\simeq 80 $ MHz (c). These signals drive the acousto optic modulator  AOM1 and AOM2 and switch on and off the illumination and reference beams.} \\label{Fig_fig_chronogramme}\n\\end{center}\n\\end{figure}\n\n\nBoth time averaged \\cite{picart2003time} and sideband digital holography \\cite{joud2009imaging} record the holographic signal over a large number a vibration periods. These two techniques are not sensitive to the phase of the vibration,\nand are thus unable to measure the instantaneous velocities of the object. To respond this problem,\nLeval et al. \\cite{leval2005full} combine time averaged holography with stroboscopic illumination, but,\nsince Leval uses a mechanical stroboscopic device, the Leval technique suffer of a quite low\nduty cycle (1\/144), and is limited in low vibration frequencies ($\\omega_A\/2\\pi < 5$ kHz).\n\n\n\\begin{figure}[h]\n\\begin{center}\n   \n  \\includegraphics[width=5.0 cm]{Fig_Am_a}\n  \\includegraphics[width=5.0 cm]{Fig_Am_b}\n  \\includegraphics[width=5.0 cm]{Fig_Am_c}\n  \\includegraphics[width=5.0 cm]{Fig_Am_d}\n \\caption{Doppler spectrum $|E_m|^2\/|E|^2=|A(m,t_d,\\delta t)|^2$ calculated for  a vibration amplitude  $\\Phi=0.50$, a gate width $\\delta t=0.1~ T_A$ and a gate time $t_c=0.25$ (a), 0.35 (b) , 0.45 (c) and 0.50 $T_A$ (d).} \\label{Fig_fig_Am}\n\\end{center}\n\\end{figure}\n\n\nTo overcome these two limitations, it is possible to combine sideband digital holography  with stroboscopic illumination and detection\nsynchronized with the vibration \\cite{verpillat2012imaging}. This can be achieved without changing the experimental sideband holography setup of Fig.\\ref{Fig_fig_setup_reed} by switching electronically on and off the Radio Frequency signals that drive the AOMs at $\\omega_{AOM1}$ and $\\omega_{AOM2} \\simeq 80 $ MHz. Figure \\ref{Fig_fig_chronogramme} shows a typical chronogram of stroboscopic illumination and detection.\n\nWithout stroboscopic illumination (and detection), the Doppler  velocity spectrum is similar to the ones observed in Fig.\\ref{Fig_fig_Jm_303} section \\ref{Section_large_vibration}. It covers the entire range of speeds that can be reached during the sinusoidal motion, that is $\\Phi<m<+\\phi$, with two maxima near  $m \\simeq \\pm \\Phi$. With stroboscopic illumination, the Doppler spectrum is modified, since it  corresponds to the narrower velocities range that is reached during gated illumination.\nOn the other hand, the finite gate duration   $\\delta t$ (that is much shorter than the vibration period $T_A= 2\\pi\/\\omega_A$) yields a broadening of  the spectrum of about  $T_A\/\\delta t$ for $m$.\nIn order to take quantitatively these effects into account, we have calculated the spectrum. The slowly varying complex field $E(t)$ must be multiplied by a gate\nfunction $ H(t,t_d,\\delta t)$ of period $T_A$. Within the interval $[0,T_A]$ we have:\n\\begin{eqnarray}\\label{Eq_gate}\n  H(t,t_d,\\delta t) &=& 0 ~~~~\\textrm{for}~~~~0<t<t_d-\\delta t\/2\\\\\n\\nonumber   &=&  1~~~~\\textrm{for}~~~~t_d-\\delta t\/2<t<t_d+\\delta t\/2\\\\\n \\nonumber  &=&0~~~~\\textrm{for}~~~~t_d+\\delta t\/2<t<T_A\n\\end{eqnarray}\nThe slowly varying complex field signal $E(t)$ becomes thus:\n\\begin{eqnarray}\\label{Eq_gated_signal}\nE(t)&=& E~ H(t,t_d,\\delta t)~ e^{j(\\Phi \\sin \\omega_A t) }\\\\\n\\nonumber &=&E~\\sum_m ~A(m,t_d,\\delta t,\\Phi)~e^{j m \\omega_A t}\\\\\n\\nonumber E_m&=& E ~A(m,t_d,\\delta t,\\Phi)\n\\end{eqnarray}\nwhere $A(m,t_d,\\delta t)$ is the $m^{\\textrm{th}}$ Fourier component of the periodic function $E(t)\/E$.\nWe have calculated $A(m,t_d,\\delta t,\\Phi)$ by Fourier transformation of $E(t)\/E$, and we have plotted on Fig. \\ref{Fig_fig_Am} the Doppler spectrum  $|E_m|^2\/|E|^2=|A(m,t_d,\\delta t)|^2$ for  different gate  times $t_c$ and for $\\delta t\/T_A=0.1$. In Fig. \\ref{Fig_fig_Am} (a), the gate  coincides with the maximum velocity ($t_d=0.5~T_A$). The Doppler spectrum is narrow and centered near $m\\simeq  - \\Phi = -50$ rad. In figure \\ref{Fig_fig_Am} (b) and (c) i.e. for gate time $t_c=0.25$ (a), 0.35 (b), the absolute velocity decreases, and the center of  the Doppler spectrum moves towards $m=0$. On the other hand,  the Doppler spectrum broadens. In figure \\ref{Fig_fig_Am} (c) the gate  coincides with the minimum  velocity ($t_d=0.5~T_A$). The center of the spectrum is $m=0$, and  the   broadening is maximum.\n\n\n\n\n\n\\begin{figure}\n\\begin{center}\n   \n  \\includegraphics[width=11 cm]{anche___xz}\n \\caption{Successive velocities of the reed on a period $T_A$. These images are obtained by\nmaking a cut  in the $y = 256$  plane  of the  3D stack of reconstructed images, whose coordinates are $x,y$ and $m$ with $m =$ -100 to +100. Gate duration is $\\delta t=0.1 ~T_A$. From (a) to (j), the gate time $t_d$ is swept from $t_d=0.25 ~T_A$ to  $t_d=1.15~T_A$ by 0.1 $T_A$ step.   The images are displayed in logarithmic scale for the sideband complex field intensity: $|E_m(x,y=256)|^2$. } \\label{Fig_anche___xz}\n\\end{center}\n\\end{figure}\n\n\nTo measure the instantaneous velocities of the object at time $t=t_d$, the gate is adjusted at $t_d$, and the holograms of all sidebands $m$ are recorded.  This can be done automatically  by the computer, by adjusting the frequency $\\omega_{LO}$  to fulfil  Eq.\\ref{Eq_Delata omega 4phase_sideband}. The sideband images  $E_m(x,y)$ are then reconstructed for all $m$, and  a 3D  cube of data (coordinate $x$, $y$ and $m$) is  obtained as illustrated by Fig. \\ref{Fig_fig_cube_of_data}.\nOne can get velocity images by making cuts along $x$ or $y$. Figure \\ref{Fig_anche___xz} shows an examples of  velocity images obtained at different times $t_d$ of the vibration period. Here, the reed is oriented in the same direction than in Fig.\\ref{Fig_fig_cube_of_data}, and  the velocity images of coordinates $x,m$ are obtained  with horizontal cuts in plane $y=256$.\n\n\n\n\\subsection{Sideband holography for small amplitude of vibration}\\label{section_small_vibration}\n\n\nWhen the vibration amplitude $\\Phi$ becomes small, the energy  within sidebands  decrease very rapidly with the sideband indexes $m$, and  one has only to consider the carrier $m=0$, and the two first sideband $m=\\pm1$. Time averaged holography \\cite{picart2003time} that detects the carrier field $E_0$  is  not efficient in detecting small amplitude vibration $\\Phi$, because  $E_0$  varies quadratically  with $\\Phi$:\n\\begin{eqnarray}\\label{Eq_E_0_lim0}\nE_0(\\Phi) &=&  E~J_0(\\Phi)  \\\\\n \\nonumber  & \\simeq & E~(1- \\Phi^2)\/6\n\\end{eqnarray}\nOn the other hand,  sideband holography that is able to detect the two  first sidebands fields $E_{\\pm 1}$ is  much more sensitive, because  $E_{\\pm 1}$  varies   linearly with $\\Phi$:\n\\begin{eqnarray}\\label{Eq_E_1_lim0}\n  E_{\\pm 1}(\\Phi) &=& E~ J_{\\pm 1}(\\Phi)  \\\\\n \\nonumber   &\\simeq &\\pm E ~\\Phi\/2\n\\end{eqnarray}\nThis point has been noticed about 40 years ago by Ueda et al. \\cite{ueda1976signal}, who has made a  comprehensive study of the signal-to-noise\nratio (SNR) observed for classical (photographic film) sideband\nholography. The authors managed to observe\nvibration amplitudes of a few Angtroms, and linked\nthe smallest detectable amplitude to the SNR in the\nabsence of spurious effects. Later on, sub nanometric\nvibration amplitude measurements were achieved with\nsideband digital holography \\cite{psota2012measurement,verrier2013absolute,bruno2014phase,bruno2014holographic,bruno2014non}, and comparison with  single point  laser interferometry has been  made \\cite{psota2012comparison,bruno2014holographic}.\n\n\n\nOne must notice that the complex field amplitude $E$ scattered by the sample without vibration  depends strongly on the $x$ $y$ position. In a typical experiment, the sample rugosity is such that  $E$ is a speckle that is fully developed. The field $E(x,y)$ is thus random in amplitude and phase from one speckle to the next. One cannot thus  extract the phase of the vibration motion from a measurement made on single a sideband, for instance the sideband $m=1$. To get the phase, one needs to measure the field components on several sidebands, for example  the carrier field $E_0$  and the first sideband $E_1$. For small amplitude $\\Phi$, the carrier field $E_0(\\Phi)$ is very close to the field $E=E_0(\\Phi=0)$ measured without vibration.   By measuring both $E_0(\\Phi)$ and $E_1(\\Phi)$ by sideband digital holography, one can then eliminate $E$ in Eq. \\ref{Eq_E_0_lim0} and Eq. \\ref{Eq_E_1_lim0}\ngetting by the way $J_0(\\Phi)$, $J_1(\\Phi)$  and $\\Phi$. This can be made by calculating  either the ratio $E_1\/E_0$ \\cite{verrier2013absolute}:\n\\begin{eqnarray}\\label{Eq_E1_E0_ratio}\n  \\left|\\frac{E_1}{E_0}\\right| &=& \\frac{J_1(\\Phi)}{J_0(\\Phi)}\\\\\n\\nonumber  &\\simeq& \\Phi\/2\n\\end{eqnarray}\nor the correlation $E_1~E^*_0$:\n\\begin{eqnarray}\\label{Eq_E1_E0_correlation}\n\\left| E_1~E^*_0 \\right| &=& |E|^2~ J_1(\\Phi) ~J_0(\\Phi)\\\\\n\\nonumber &\\simeq & |E|^2 \\Phi\/2\n\\end{eqnarray}\nBoth methods yield a signal ($\\Phi\/2$ or $|E|^2 \\Phi\/2 $) that is proportional to $\\Phi$.\n\n\n\\begin{figure}\n\\begin{center}\n   \n  \\includegraphics[width=11 cm]{fig_2_sideband_setup}\n \\caption{Sideband holography setup able to acquire the carrier and sideband signal simultaneously} \\label{Fig_fig_2_sideband_setup}\n\\end{center}\n\\end{figure}\n\n\nFrom a practical point of view,   $E_0$ and $E_1$ are measured successively in a standard sideband holography setup like the one of  Fig.\\ref{Fig_fig_setup_reed} \\cite{joud2009imaging,psota2012measurement}.  A sequence of $n_{max}$ frames with $\\omega_{LO}=\\omega_{0}- \\omega_{CCD}\/4$ is first recorded yielding  $E_0$. A second sequence  with $\\omega_{LO}=\\omega_{0}+\\omega_A -\\omega_{CCD}\/4$ is then recorded yielding  $E_1$.\n\nIt is also possible to record $E_0$ and $E_1$ simultaneously \\cite{verrier2013absolute,bruno2014phase,bruno2014holographic,bruno2014non}. In that case, the RF signal that drives the second acousto optic modulator AOM2  is made of two  frequency components (Fig. \\ref{Fig_fig_2_sideband_setup}) at $\\omega_{AOM2}^a$ and  $ \\omega_{AOM2}^b$:\n\\begin{eqnarray}\n  \\omega_{AOM2}^a &=& \\omega_{AOM1}+\\omega_{CCD}\/4 \\\\\n\\nonumber \\omega_{AOM2}^b &=& \\omega_{AOM1}+\\omega_A-\\omega_{CCD}\/4\n\\end{eqnarray}\n\\begin{itemize}\n   \\item The first component at $\\omega_{AOM2}^a$, whose weight is  $\\alpha$,  yields the carrier signal $E_0$, if the hologram $H$ is calculated with demodulation on the -1 grating order i.e. with $ H=\\sum_{n=0}^{n_{max}} (-j)^n I_n $.\n   \\item The second component at $\\omega_{AOM2}^a$, whose weight is  $\\beta$, yields the sideband $E_1$ with demodulation on the +1 grating order i.e. with $ H=\\sum_{n=0}^{n_{max}} (+j)^n I_n $.\n \\end{itemize}\n\n\n\\begin{figure}\n\\begin{center}\n   \n  \\includegraphics[width=11 cm]{fig_canteviller_mike}\n \\caption{Vibration amplitude $z_{max}$ (a) and phase $\\varphi$ (b) of a lamellophone, averaged over the first upper cantilever,  versus excitation frequency $\\omega_A\/(2\\pi)$ (Hz). Insets: retrieved vibration amplitude and phase maps in the neighborhood of the resonance. The theoretical resonance line is in gray.} \\label{Fig_fig_canteviller_mike}\n\\end{center}\n\\end{figure}\n\n\n\nSimultaneous detection of $E_0$ and $E_1$, and calculation of $\\Phi$ by the ratio method of Eq. \\ref{Eq_E1_E0_ratio} has been used to study various vibrating samples. Figure \\ref{Fig_fig_canteviller_mike} shows example of signals obtained with a lamellophone of a musical box \\cite{bruno2014phase}. In that experiment, all the frequencies $\\omega_{A}$, $\\omega_{CCD}$, $\\omega_{AOM1}$, $\\omega_{AOM2}^a$ and  $\\omega_{AOM2}^b$  are driven by numerical synthesizers. In this particular case, harmonics are extracted by a temporal Fourier transform of a sequence of 8 raw interferograms. Here, optical sidebands of interest are downconverted to demodulated frequencies in the camera bandwidth. In order to avoid signal phase drifts from the measurement of one sequence to the next, AOM2 is driven using a phase ramp. As a consequence, the phases of $E_0$ and $E_1$ are well defined, and  yield the phase $\\varphi$ of the mechanical motion. We have:\n\\begin{eqnarray}\\label{equ_phase_mechnical}\n  \\varphi &=&\\textrm{Arg} (E_1\/E_0) +C\n\\end{eqnarray}\nwhere $C$ is a additive constant that depends of the relative phase of the synthesizers. Figure \\ref{Fig_fig_canteviller_mike} shows  the averaged  amplitude $z_{max}$ and phase $\\varphi$ of the upper cantilever of the lamellophone, which is driven though its resonance frequency $\\simeq 541.5$ Hz. As expected, the phase $\\varphi$ makes a jump of about $\\pi$ when excitation frequency $\\omega_A$ crosses the resonance frequency.\n\n\n\n\n\n\\begin{figure}\n\\begin{center}\n   \n  \\includegraphics[width=11 cm]{fig_acoustic_wave} \\caption{Amplitude maps of the out-of-plane vibration of a thin metal plate versus excitation frequency $\\omega_A\/(2\\pi)$. Holographic\nimages at 40.1 kHz (a), 61.7 kHz (b). Scalebar: 5 mm.} \\label{Fig_fig_acoustic_wave}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n   \n  \\includegraphics[width=11 cm]{curve_holo_versus_laser_doppler}\n \\caption{ Comparison of quantitative out-of-plane vibration\namplitudes $z_{max} $ retrieved with the single-point laser vibrometer ($ \\square$\nsymbols) and holographic vibrometer with $\\beta=\\alpha$ ($\\circ$ symbols)\nand $\\beta\\simeq 50 \\alpha$ ($+$ symbols).} \\label{Fig_curve_holo_versus_laser_doppler}\n\\end{center}\n\\end{figure}\n\n\nThe method has been also applied to get full field  images of  surface acoustic  waves as shown on Fig. \\ref{Fig_fig_acoustic_wave} \\cite{bruno2014holographic}. Comparison with single point Doppler vibrometry is reported in Fig.~\\ref{Fig_curve_holo_versus_laser_doppler}. If the carrier and sideband components of the RF signal at $\\omega_{AOM2}^a$ and  $\\omega_{AOM2}^b$ have the same amplitude (i.e. if $\\alpha=\\beta$), some spurious detection of the  carrier field $E_0$ (which is much bigger) is observed  when the detection is tuned on the grating order -1 (used to detect  $E_0$). This spurious effect limits the detection sensitivity to $z_{max}\\sim 1 $ nm, as seen on Fig. \\ref{Fig_fig_acoustic_wave} ($\\circ$ symbols). Much better results are obtained by reducing the weight of the carrier component (i.e. with $\\beta\\gg \\alpha$). The detection sensitivity becomes  $z_{max}\\sim 0.01 $ nm with $\\beta\\simeq 50 \\alpha$  as seen on Fig. \\ref{Fig_fig_acoustic_wave} ($+$ symbols).\n\nThe ratio method of Eq. \\ref{Eq_E1_E0_ratio} used above is simple, since it gives directly  $\\Phi$. Nevertheless, the method is unstable for the points $x,y$ of the object where $E$ is close to zero.  Since $\\Phi$ varies slowly with the location $x,y$, one can improve the accuracy by averaging  $\\Phi$ over neighbor points. In that case, the ratio method is not optimal for noise, since all points  $x,y$ are weighted equally in the average, while the accuracy on the ratio depends strongly on the location  $x,y$, since the accuracy is low when  $|E (x,y)|$ is low.\n\nFor this issue, the correlation method of Eq.\\ref{Eq_E1_E0_correlation} is less simple, since it gives $|E|^2\\Phi$, and not $\\Phi$. It is nevertheless better for noise, since spatial averaging over neighbor points $x,y$ is weighted by the scattered energy $|E(x,y)|^2$.\n\n\\begin{figure}\n\\begin{center}\n   \n  \\includegraphics[width=11 cm]{fig_vibrating_cube}\n \\caption{(a,b) Reconstructed images of the correlation $E_1 E^*_0$ obtained with a cube of wood; (a) amplitude: $|E_1 E^*_0|$ ,  and (b) phase: $\\arg E_1 E^*_0$. The correlation $E_1(x,y) E^*_0(x,y)$ is averaged with a 2D gaussian blur filter of radius 4 pixels. The display is made in arbitrary linear scale. (c) 3D display of  the phase: $x,y,\\arg E_1 E^*_0$} \\label{Fig_fig_vibrating_cube}\n\\end{center}\n\\end{figure}\n\nFigure \\ref{Fig_fig_vibrating_cube} shows reconstructed images of the correlation $E_1 E^*_0$ obtained with a cube of wood (2 cm $\\times$ 2 cm) vibrating at its resonance frequency $\\omega_A=21.43 $ kHz. Here, the fields $E_0$ and $E_1$ are measured sucessively.  In order to get better SNR, the complex correlation signal $E_1(x,y) E^*_0(x,y)$ is averaged over neighbor $x,y$ points by using a 2D gaussian blur filter of radius 4 pixels. Figure \\ref{Fig_fig_vibrating_cube}  shows the amplitude (a) and phase (b) of the the filtered correlation $E_1(x,y) E^*_0(x,y)$. Figure \\ref{Fig_fig_vibrating_cube} (c) displays the phase of  the correlation   in 3D. As seen,  the opposite corners (upper left and bottom right for example) vibrate in phase, while the neighbor corners (upper left and upper right for example) vibrate in phase opposition. Note that the cube is excited in one of its corner by a needle. This may explain why the opposite corners are not perfectly in phase in Fig. \\ref{Fig_fig_vibrating_cube} (b,c).\n\n\n\\begin{figure}\n\\begin{center}\n   \n  \\includegraphics[width=9 cm]{fig_reed_1pt}\n \\caption{ (a) Clarinet reed with illumination beam focused in $x_0,y_0$. (b) Sideband $m=1$ reconstructed image of the vibrating reed. The display is made in arbitrary log scale for the field intensity $|E_1(x,y)|^2$.   } \\label{Fig_fig_reed_1pt}\n\\end{center}\n\\end{figure}\n\n\n\nIn order to increase the ability to analyze vibration of small amplitude, one can  increase the illumination on the zone where the correlation averaging is performed. This is  done by using a collimated illumination beam. Figure \\ref{Fig_fig_reed_1pt} shows an example. The illumination laser has been focused on point $x_0, y_0$ of  the clarinet read to be studied (see Fig. \\ref{Fig_fig_reed_1pt} (a) ). Most of the  energy in the reconstructed image is thus located near $x_0,y_0$ (see Fig. \\ref{Fig_fig_reed_1pt} (b) ). To measure the vibration amplitude at location $x_0,y_0$, we have calculated the averaged correlation  $\\langle E_1 E^*_0 \\rangle $ and the averaged energy $\\langle |E_0|^2 \\rangle $:\n\\begin{eqnarray} \\label{Eq_averaged E0E1}\n  \\langle E_1 E^*_0 \\rangle &=& (1\/N_{pix}) \\sum_{x,y} E_1(x,y) E^*_0(x,y) \\\\\n  \\nonumber  &=&  J_1(\\Phi)  J_0(\\Phi)~ (1\/N_{pix})\\sum_{x,y} |E(x,y)|^2\n\\end{eqnarray}\n\\begin{eqnarray} \\label{Eq_averaged E0E0}\n\\langle |E_0|^2 \\rangle &=&(1\/N_{pix}) \\sum_{x,y} |E_0(x,y)|^2 \\\\\n  \\nonumber  &=&    J^2_0(\\Phi)~ (1\/N_{pix})\\sum_{x,y} |E(x,y)|^2\n\\end{eqnarray}\nwhere $\\sum_{x,y}$ is the summation over the $N_{pix}$ pixels of the illuminated region  located near $x_0,y_0$. In that region, the vibration amplitude $\\Phi$ is supposed to  not depend on $x$ and $y$. We get then:\n\\begin{eqnarray} \\label{Eq_Phi_E0EO_E1E0}\n   \\frac{ \\langle E_1 E^*_0 \\rangle   }{ \\langle |E_0|^2 \\rangle }=\\frac{J_1(\\Phi)}{J_0(\\Phi)} \\simeq \\Phi\/2\n\\end{eqnarray}\n\n\\begin{figure}\n\\begin{center}\n   \n  \\includegraphics[width=9 cm]{fig_lobna}\n \\caption{ Ratio $ \\langle E_1 E^*_0 \\rangle\/ \\langle |E_0|^2 \\rangle$  (vertical axis) as function of the reed excitation voltage $V_{pp}$ in volt Units.  The light grey curve  is $J_1(\\Phi)\/J_0(\\Phi)$ with $\\Phi = z_{max}\/(4 \\pi \\lambda)$.} \\label{Fig_fig_lobna}\n\\end{center}\n\\end{figure}\n\n\nTo verify the ability of measuring small vibration, the reed has been excited with lower and lower loudspeaker levels, the illumination laser being focused on the point $x_0,y_0$ of Fig. \\ref{Fig_fig_reed_1pt} (b), we want to study. A Sequence of $n_{max}=128$ frames have been  recorded for both carrier ($E_0$) and sideband ($E_1$). Four phase detection has been then  made by calculating $H$  with $n_{nmax}=128$ in Eq. \\ref{EQ_H_n_max}. We have then  calculated the reconstructed fields $E_0(x,y)$ and $E_1(x,y)$, the averaged correlation $\\langle E_1 E^*_0 \\rangle$ and averaged carrier intensity $\\langle |E_0|^2 \\rangle$ and the ratio $\\langle E_1 E^*_0 \\rangle  \/\\langle |E_0|^2 \\rangle$. This ratio is plotted on Fig.\\ref{Fig_fig_lobna} ($y$ axis)  as a function of the peak to peak voltage $V_{pp}$ of the sinusoidal signal that excites the loudspeaker  ($x$ axis). The measured ratio varies linearly with $V_{pp}$. For low voltage $V_{pp}\\leq 5$ mV, the ratio reached a noise floor corresponding to $\\langle E_1 E^*_0 \\rangle  \/\\langle |E_0|^2 \\rangle \\simeq 10^{-4}$. Since this voltage $V_{pp}$ is proportional to the vibration amplitude $z_{max}$ and thus to $\\Phi$, we have convert the  voltage $V_{pp}$ in vibration amplitude  $z_{max}$. The conversion factor ($z_{max}=0.02$ nm for $V_{pp}=10$ mV) is obtained by assuming that  $J_1(\\Phi)\/J_0(\\Phi)$ fits the ratio $\\langle E_1 E^*_0 \\rangle  \/\\langle |E_0|^2 \\rangle$ that is  measured, as predicted by Eq. \\ref{Eq_Phi_E0EO_E1E0} (see solid grey curve of Fig.\\ref{Fig_fig_lobna}). We get here a noise floor noise corresponding to $z_{max}\\simeq 0.01 $ nm. It   is about $\\times 10$ lower than the limit $\\lambda\/3500 =0.22$ nm predicted by Ueda \\cite{ueda1976signal}, and  $\\times 10$ lower than in  previous experiments at similar  frequency \\cite{psota2012measurement,verrier2013absolute} in the kHz range. Similar noise floor $0.01$ nm has been  obtained  by the ratio method on Fig. \\ref{Fig_curve_holo_versus_laser_doppler}, but at higher vibration frequency $\\omega \\simeq 40 $ kHz  \\cite{bruno2014holographic}.\n\n\nWe have tried to make the experiment with more frames i.e. with $n_{max}>128$, but this does not lower the noise floor. The remaining noise floor can  be related  to a spurious detection of the carrier field $E_0$ when detection is tuned to detect $E_1$, or to some technical noise on the carrier signal that is not totally filter off.  Since the carrier and sideband fields $E_0$ and $E_1$ are within the same spatial mode, no space filtering can be applied to filter off $E_0$.  Here, the spurious carrier field $E_0$ is filtered off in the time domain, while the local oscillator field $E_{LO}$ is filtered off both in time and frequency domain.\n\n\\section{Conclusion}\n\n\nIn this chapter we have presented the digital heterodyne holography technique that is able to fully control the amplitude, phase and frequency of both illumination and reference beams. Full automatic data acquisition of the holographic signal can be made, and   ultimate shot noise sensitivity can be reached. Heterodyne holography is an extremely versatile and powerful tool, in particular when applied to vibration analysis. In that case, heterodyne holography is able to detect in wide field (i.e. in all points of the object 2D surface at the same time) the vibrating object signal at any optical sideband of rank $m$. Since the control of the intensity of the illumination and reference beams is fast, instantaneous measurements of the vibration signal, sensitive to the mechanical phase, can be made. For the measurement of large vibration amplitudes, the possible ambiguity of measurements  can be removed by making measurements at different sideband indexes $m$. For small  vibration amplitudes, the mechanical phase can be obtained from measurements made on the carrier ($m=0$) and on the first sideband ($m=1$). Moreover, extremely low vibration amplitudes, below 10 picometers, can be measured.\n\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nThere is an increasing interest in strongly correlated systems containing \nonedimensional (1D) structures, both for their novel properties as well\nas for their possible applications. For example, chains and two-leg \nladders are present in transition metal oxides and in organic conductors. \nOther systems of increasing interest involving strongly correlated\nfeatures are nanoscopic devices which have been recently fabricated\nand experimentally studied.\\cite{goldhaber} In many of these devices,\na droplet of electrons confined in the three spatial directions,\nthe quantum dot (QD), is connected to metallic leads.\nIn this case the leads are described by tight-binding models, or in \ngeneral by Fermi liquids. The transport through such devices involve\nthe formation of a Kondo resonance in the QD. Kondo physics is \nthen relevant to understand this kind of devices. \\cite{hewson}\n\nIt is also possible to connect the QD to conducting leads with \nelectron correlations. This is the case of spin valves, one of the\nsimplest devices used in spintronics,\\cite{wolf,pasupathy} where a\nQD is attached to ferromagnetic leads.\nThese leads can be built from materials such as manganese oxides\n\\cite{manganato_tube} or other compounds containing \ntransition metal oxides such as vanadates.\\cite{vanadato_tube}\nThere have also been recent developments in producing devices where\nthe leads are formed by carbon nanotubes (although ultimately these\nleads are connected to metallic terminals) which can be considered\nquasi-onedimensional and where electron correlations are presumably\nimportant.\\cite{tsukagoshi,biercuk,nygard} In fact, a Luttinger\nliquid behavior, typical of correlated 1D systems, has been found in\ncarbon nanotubes.\\cite{bockrath}\n\nThe problem of a magnetic impurity or QD in a correlated \nonedimensional system is then an interesting topic both for fundamental\nand for technological reasons. From the theoretical point of view, the\nmost important results for this kind of systems were\nobtained by considering a single impurity attached to leads described\nby Luttinger liquids. Much less is known when the leads are described\nby a Hubbard model. In this case, this model has been treated by\nlinearizing the dispersion relation and, using bosonization\ntechniques, the long wavelength behavior is captured. Within this\napproach, it was shown that in a system of a magnetic impurity \nconnected to Luttinger chains by a Kondo exchange interaction $J$,\nthe Kondo temperature acquires the power-law expression\\cite{leetoner}\n$T_K \\approx (J )^{2\/(1-K_\\rho)}$ instead of the conventional exponential\nlaw for noninteracting chains. The Luttinger exponent $K_\\rho$ also\ndetermines the conductance through an interacting wire, which is \ngiven by $G=2 K_\\rho e^2\/h$.\\cite{kanefisher}\n\nIn this article we consider the even less studied problem of a \nsingle-impurity Anderson-Hubbard Hamiltonian on finite-size systems.\nOur aim is to provide a detailed microscopic picture of the\ncompetition between the electron correlations on the impurity and the\nones on the leads, and to understand the role of finite length of\nthe chains.  Although the lengths considered in this study are\nstill smaller than the physical length of current devices, the study \non finite-size systems are interesting per se. In fact, it has been\nsuggested that the Kondo temperature can be very different in finite\nchains with respect to the one in infinite leads.\\cite{simon-affleck}\nIt is also expected that the size of nanoelectronic devices will be\nfurther reduced in the near future.\nThe fundamental problem of the competition between the Kondo effect\nand the Luttinger liquid starting from a microscopic Hamiltonian \nwas recently studied on infinite systems using an approximated \nanalytical renormalization group technique.\\cite{andergas05} The\nproblem of an Anderson impurity in a $t-J$ model has been also\naddressed within the Bethe {\\it Ansatz} formalism.\\cite{schlott03}\n\nAlthough manganese oxides should be described by a generalized \nferromagnetic Kondo-lattice model\\cite{elbio_manganitas}, and \nvanadates or carbon nanotubes by some variants of the Hubbard or $t-J$\nmodels on ladders\\cite{nanotube_model,vanadato_model}, in order to\nget an insight on the physical behavior, it is necessary to consider\nfirst the simpler case of leads described by a 1D Hubbard model.\nOf course, some of the results obtained are relevant to magnetic \nimpurities in strongly correlated quasi-1D compounds such as those\nmentioned at the beginning. Results for LDOS for example, could\nbe compared with those obtained by scanning tunneling microscopy\n(STM) experiments.\\cite{derro}\n\nNumerical techniques are ideally suited to cope with finite systems.\nIn particular we mainly employ the DMRG technique,\\cite{white} which\nprovides essentially exact results for various real-space properties\non finite chains. These real-space properties of real-space models\ncan shed light on the functioning of QD devices.\\cite{gazza} In this\nsense, in this work we will determine the way in which the Kondo\neffect is affected by varying the electron density or the Hubbard\nrepulsion on the leads by computing the electron occupation and the\nsquare of the $z$-component of the spin at the impurity site. More\nimportantly, from the spin-spin correlation functions we will study\nthe ``spin compensation cloud\", a possible measure of the more\nelusive Kondo screening cloud.\\cite{gubernatis} Using this technique,\nit has been shown that the screening cloud is reduced by electron\ncorrelations on the leads in a system with two Kondo\nimpurities.\\cite{hallberg-egger} We will extend this approach to\nthe Anderson impurity where we will make a careful study of\nfinite-size effects. DMRG results for the spin compensation cloud\nhave been reproduced to a good approximation by using a recently\ndeveloped quantum Monte Carlo technique\\cite{sandvik}.\nThis technique also allows us to study this quantity at finite\ntemperature.\n\nIn order to study transport properties, we will study the LDOS at\nthe impurity, which is related to the conductance in an\nessential way. By implementing some recent developments in\nDMRG\\cite{feiguin,schollwock}, we will compute the conductance\nas the response of the system to a finite but small voltage bias\napplied to the leads. We will examine this property as the \nHubbard repulsion $U$ is varied. \nWe will also show that the effect of this variation on both the\nLDOS at the impurity and the conductance,\ndepend in turn on the values of the electron interactions on the\nimpurity. We will explore various types of\nimpurities, from the mixed valence to Kondo regimes.\n\nThe paper is organized as follows. In Section \\ref{model} we\ndescribe the model studied and we provide details of the numerical\ntechniques employed. In Section \\ref{varying_density} we study \nthe effect of reducing the electron density from half-filling\nto quarter-filling for the case of noninteracting (tight-binding)\nleads. In Section \\ref{varying_U}, we analyze how the Kondo effect \nvaries with the Hubbard repulsion $U$ on the leads at quarter-filling.\nFinally, the relation between the results obtained and those\nin the previous literature, and the relevance of the present study\nto real systems or nanoscopic devices, are discussed in Section\n\\ref{conclusions}.\n\n\\section{Model and methods}\n\\label{model}\n\nIn this paper we use the terms ``impurity\" and QD indistinctly, and\nwe refer to the remainder sites of the chain as the ``leads\".\nWe consider a one-dimensional single-impurity Anderson-Hubbard model \ndefined by the Hamiltonian:\n\\begin{eqnarray}\n{\\cal H} = &-& t \\sum_{i=\\leq -2,\\sigma} (c^{\\dagger}_{i \\sigma}\nc_{i+1 \\sigma} + H.c. ) + U \\sum_{i=\\leq -1} n_{i, \\uparrow}\nn_{i, \\downarrow} \\nonumber \\\\\n&-& t \\sum_{i=\\geq 1,\\sigma} (c^{\\dagger}_{i \\sigma} c_{i+1\n\\sigma}+ H.c. ) + U \\sum_{i=\\geq 1} n_{i, \\uparrow}\nn_{i, \\downarrow} \\nonumber \\\\\n&-& t' \\sum_{\\sigma} (c^{\\dagger}_{-1 \\sigma} c_{0 \\sigma} +\nc^{\\dagger}_{0 \\sigma} c_{1 \\sigma} + H.c. ) \\nonumber\\\\\n&+& \\epsilon' ~ n_{0} + U'  n_{0, \\uparrow} n_{0, \\downarrow}\n\\label{hamilt}\n\\end{eqnarray}\n\\noindent where conventional notation was used. The first two terms\ncorrespond to the left lead and the following two terms to the right\nlead. The next term is the impurity-lead interaction, and the last\ntwo terms are the on-site energy and Hubbard repulsion at the\nimpurity site, located at site $0$, respectively.  We adopt $t$ as\nthe scale of energy. In the case of $U=0$, the Hamiltonian \n(\\ref{hamilt}) reduces to the single-impurity Anderson model, and\nfor $\\epsilon'=0$, $U'=U$, $t'=t$, to the Hubbard model.\n\nHamiltonian (\\ref{hamilt}) will be studied mostly using the numerical\ntechnique DMRG\\cite{white,review} on finite-size clusters of length\n$L$ with open boundary conditions (OBC). This algorithm provides\nnumerically exact results for static properties at zero temperature\nwith a precision which depends on the number $M$ of states retained.\nMost of the results here reported were obtained for $M=600$, except\notherwise stated, asserting us that the integrated weight of discarded\nstates are of order $10^{-6}$ in the worst case.  On the other hand, \nresults for dynamical properties, such as LDOS, should be taken\nqualitatively, as discussed below. The same apply to the results\nobtained for the time evolution of the current on the links\nconnecting the impurity with the leads. In order to assess even-odd\neffects most of the results reported below were obtained by using \neven and odd chain lengths. Throughout all this paper,\nDMRG calculations will be done in the subspace of total $S_z=0$\n($S_z=1\/2$) for $L$ even (odd). The QD is located in one of two\ncentral sites of the chain when $L$ is even (in the central site when\n$L$ is odd).\n\nWe have computed static properties such as the electron occupancy \non each site, $\\langle n_{i,\\sigma}\\rangle$\n($\\sigma=\\uparrow, \\downarrow$) and spin-spin correlations from the QD,\n$S(j)=\\langle S_0^z~S_j^z\\rangle-\\langle S_0^z\\rangle\\langle S_j^z\\rangle$.\n\nA very important quantity related to the Kondo effect is the Kondo\nscreening length, which is somewhat elusive to \ncompute.\\cite{simon-affleck} A possible measure of this Kondo length\nis the length of the ``spin compensation cloud\"\\cite{gubernatis}\ndefined as the length $\\xi$ such that\n\\begin{eqnarray}\n \\sum_{ j=-\\xi \/2, j \\neq 0}^ {j=\\xi \/2} S(j) = x S(0),\n\\label{sccdef}\n\\end{eqnarray}\n\\noindent\nwhere $x$ is an arbitrary parameter. In the following we adopt a fixed\nvalue $x=0.9$ in order to compare  $\\xi$ for the different cases.\n\nOne of the main quantities we have studied is the local density of\nstates, $\\rho(\\omega)$, at the QD. In the first place, from this\nquantity it would be possible to evaluate the conductance in the\nlinear response regime (see below). In the second place, recent\nadvances in STM have made it\npossible to directly measure this quantity. Since in our DMRG\ncalculation the measurements are performed when the two added sites\nare at the center of the chain (symmetrical configuration), the QD\nthen is one of these two sites which are exactly treated. Then, we\nadopt the approximation of applying the creation and annihilation\noperators at the QD on the ground state vector and then determine\n$\\rho(\\omega)$ following the well-known continued fraction\nformalism. A more accurate approach would be, after the\napplication of each of those creation and annihilation operators,\nto run additional sweeps for an enlarged density\nmatrix.\\cite{hallberg} In any case, the truncation of the Hilbert \nspace is the essential source of error in DMRG, and to estimate the\nprecision of our approach we have compared results for various\nnumbers of retained states $M$, from $M=300$ to 600. We would\nalso like to stress the fact that the conductance in linear\nresponse is related to the LDOS near $\\omega=0$ where the\napproximation is more precise.  Our calculation starts to be less\nprecise as we move away from the chemical potential, since then\nhigher excited states are involved.\n\nFor this kind of systems, specially in connection to nanoscopic devices,\nthe most interesting properties are those related to transport, in\nparticular, the conductance $G$. When the chains or leads are described\nby a tight-binding model, or in general by a Fermi liquid, various\nexpressions relating it to the LDOS at the impurity are available in\nlinear response,\nas mentioned before. In addition, the Friedel's sum rule (see \nEq. (\\ref{Gfriedel}) below) allows a simple and precise calculation\nof $G$, particularly for numerical techniques. In the case of Hubbard\nchains, or in general Luttinger liquid leads, one must resort to the\nexpression, in linear response,\\cite{kanefisher}\n\\begin{eqnarray}\nG=\\lim_{\\omega \\rightarrow 0}\n\\frac{1}{\\hbar L\\omega}\n\\int_{0}^{L} \\int dx~dt~e^{i\\omega t} \\langle J(x,t) J(0,0) \\rangle\n\\label{Gdef}\n\\end{eqnarray}\nwhere the current is defined as\n\\begin{eqnarray}\nJ(x,t) = i \\frac{e}{\\hbar} t_{x} \\sum_{\\sigma} \\langle \\Psi(t)|\n(c^{\\dagger}_{x+1\\sigma} c_{x\\sigma} - H. c.)| \\Psi(t)\\rangle,\n\\label{current_op}\n\\end{eqnarray}\nand $| \\Psi(t)\\rangle$ is the ground state of the system at time\n$t$.\n\nAlthough such a calculation can be carried out within DMRG, some\nrecent developments collectively known as time-dependent\nDMRG\\cite{feiguin,schollwock} allow to compute directly \n$\\langle J(x,t) \\rangle$ as\n$\\langle \\Psi(t)| J(x) |\\Psi(t)\\rangle$. The time-evolution is\ntriggered by the application of a bias voltage $\\Delta V$ between the\nleft and right leads at $t=0$.\\cite{schmitteckert,alhassanieh}\nThat is, at time $t=0$, a potential $-\\Delta V\/2 n_i$\n($\\Delta V\/2 n_i$) on the sites on the left (right) leads is switched\non. Then, charge moves from one lead to the other in a non-equilibrium\nprocess. This non-equilibrium process was studied analytically in\nRef. \\onlinecite{wingreen}. In finite size systems,\n$\\langle J(x,t) \\rangle$ follows an oscillatory evolution as charge\nis bounced between the left and right leads. As the length of the\nleads are extended to infinity, the period of this oscillation\nbecomes infinitely long (the leads become infinite charge reservoirs)\nand the usual concept of conductance is recovered. It was shown in\nRefs. \\onlinecite{schmitteckert,alhassanieh} that even for moderate\nsize clusters, on each period, $\\langle J(x,t) \\rangle$ reaches a\nplateau that corresponds to the value of the bulk limit.\nHence a measure of the conductance can be obtained as the value of\nthis plateau or the amplitude of this oscillation, that is:\n\\begin{eqnarray}\nG(\\Delta V) = \\max_t \\frac{\\langle J(t)\\rangle}{\\Delta V}\n\\label{condJt}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nJ(t) = (J_L(t) + J_R(t))\/2\n\\end{eqnarray}\nand $J_L(t)$ ($J_R(t)$) is the current on the link connecting the\nimpurity site with the left (right) lead. Notice that\nEq. (\\ref{condJt}) is valid for an\narbitrary finite bias $\\Delta V$. The evolution of the ground state\n$|\\Psi(t)\\rangle$ from $t$ to $t+\\Delta t$ is computed with a\n{\\it static} Runge-Kutta treatment of the time-dependent\nSchr\\\"odinger equation.\\cite{cazalilla}\nAgain in this case, a much precise calculation could be performed by\nincluding $|\\Psi(t)\\rangle$ at intermediate times $\\tau$ as targeted\nstates in the density matrix and by running additional\nsweeps.\\cite{feiguin} This approach is of course much more expensive\ncomputationally.\n\nIn the calculations reported below we adopted a bias $\\Delta V=0.01$\nand a time step $\\Delta t=0.1$.\n\nFinally, we report finite temperature results obtained with quantum\nMonte Carlo simulations using the stochastic series expansion (SSE)\nalgorithm.\\cite{sandvik} In this case, we consider chains with \nperiodic boundary conditions (PBC). The use of PBC helps to assess\nthe effects of the Friedel oscillations which appear with OBC, used\nwith DMRG, but are absent with PBC. SSE with the loop upgrade is a\nvery efficient QMC algorithm which allows to reach much lower\ntemperatures and larger sizes than other QMC algorithms. It also\navoids the problem of extrapolation of the Trotter dimension\ncommon in ``world-line\" algorithms. It is also\nworth to note that SSE works in the grand canonical ensemble, i.e.,\nthe chemical potential has to be tuned to give a desired density\n{\\it on average}. Notice that the ``minus sign problem\" which affects\nquantum Monte Carlo simulations for fermionic models is absent in\n1D with hopping between nearest neighbor sites only since fermion\npermutations appear just as a boundary effect.\n\n\\section{Noninteracting leads, electron densities away from half-filling}\n\\label{varying_density}\n\nThe main purpose of this work is to study the effects of correlations\non the leads. This study will be performed in the next Section. \nMost theoretical studies on Kondo effect have considered the half-filled\ncase ($n=1$), where the system is particle-hole symmetric.\nNow, since an infinitesimal value of $U$ drives the system into an\ninsulating state, in order to keep the leads metallic in the presence\nof correlations it is necessary to study the system\naway from half-filling. In Section \\ref{varying_U} we\nwill work at quarter-filling ($n=0.5$).\nIn this Section, as a preliminary step, we study the evolution of the \nKondo effect as the filling is reduced from $n=1$ to $n=0.5$\nin the absence of interactions on the lead ($U=0$). In particular we\nare interested in following the evolution of what at half-filling is\nthe Kondo resonance, as the electron density decreases.\n\nModel Eq.(\\ref{hamilt}), which as stated above reduces to the \nsingle-impurity Anderson model, was studied by DMRG for several $L$ \nup to 96 sites, and for fillings $n=1$, 0.875, 0.75, 0.625 and 0.5.\nSince, as we said in the Introduction, there is a competition between\nthe correlations on the leads and the correlations at the impurity\nor QD, we consider four sets of interactions on the impurity:\n$\\{U'=8, t'=0.5\\}$, $\\{U'=8, t'=1.0\\}$, $\\{U'=4, t'=0.5\\}$, and \n$\\{U'=2, t'=0.5\\}$. In all cases we consider the symmetric case,\n$\\epsilon' = -U'\/2$. For these sets of parameters, the effective\nKondo coupling $J_{eff}=4t'^2\/U'$ takes the values 0.125, 0.5, 0.25,\nand 0.5 respectively, although strictly this relationship\nis only valid for $U' >> t'$, and hence not much applicable to the\nlast case. In spite of sharing the same $J_{eff}$, the second and\nfourth sets behave quite differently as we will show below.\n\n\\begin{figure}\n\\includegraphics[width=0.42\\textwidth]{fig1.eps}\n\\caption{(Color online) (a) Electron occupancy and (b)\n$\\langle S_z^2\\rangle$ at the impurity site as a function of the\nfilling for $U'=8$, $t'=0.5$ (circles), $U'=8$, $t'=1.0$ (squares),\n$U'=4$, $t'=0.5$ (diamonds) and $U'=2$, $t'=0.5$ (triangles).\n$L=96$}\n\\label{fig1}\n\\end{figure}\n\nLet us start by examining the dependence of the electron occupancy\n$\\langle n_0\\rangle$ and $\\langle S_{z,0}^2\\rangle$ at the impurity site \nwith the electron density. This variation depends in turn strongly\non the interactions at the impurity as it can be seen in\nFig.~\\ref{fig1}.\nIf $\\langle n_0\\rangle=1$, $\\langle S_{z_0}^2\\rangle$ takes its\nmaximum value (1\/4) when the impurity\nis a perfect spin-1\/2, corresponding to the full Kondo regime. In\nthe mixed valence regime, it takes the value 1\/12.\nFor the parameter set $\\{U'=8, t'=0.5\\}$, the\nsystem is in a well defined Kondo regime, and the dependence with\n$n$ is relatively weak. On the opposite case, for\n$\\{U'=2, t'=0.5\\}$ the system is not in a well-defined Kondo regime\nat half-filling, and the dependence with the electron filling is\nstronger. The reduction of electron density drives the system into\nthe mixed valence regime. Notice that in this second case, the\noccupancy of the impurity is smaller than $n$. Besides, it is well\nknown that there are Friedel oscillations in $\\langle n_i\\rangle$\nbeginning at the ends of the chains with OBC.\\cite{whitaffscal} It is\nthen valid to ask if these oscillations could have some effect on\nthe electron occupancy at the impurity. In the present case, any\nimportant influence of the Friedel oscillations on the impurity\nproperties can be ruled out because the results for $L=96$ are virtually \nidentical within error bars to those obtained with SSE and using\nPBC where these oscillations are absent, at the lowest temperatures\nwe considered. Of course, the presence of the impurity induces other\ndensity oscillations which overimpose to the Friedel oscillations\ninduced by the open ends. In fact we have observed in some cases that \nthe presence of the impurity changes the period of the open end \noscillations from $2k_F$ to $4k_F$. This issue is worth to be\nexamined more thoroughly but it is somewhat outside the scope of the\npresent work.\n\n\\begin{figure}\n\\includegraphics[width=0.43\\textwidth]{fig2.eps}\n\\caption{(Color online) (a) Conductance obtained from the Friedel's sum\nrule, and (b) length of the compensation cloud $\\xi$, as a function of\nthe filling for $U'=8$, $t'=0.5$ (circles), $U'=8$, $t'=1.0$ (squares),\n$U'=4$, $t'=0.5$ (diamonds) and $U'=2$, $t'=0.5$ (triangles). $L=64$ \n(full symbols, dot lines), $L=96$ (open symbols, full lines).}\n\\label{fig2}\n\\end{figure}\n\nFrom the values of the electron occupancy it is possible to obtain\nthe conductance by using the Friedel's sum rule:\\cite{hewson,langreth}\n\\begin{eqnarray}\nG_\\sigma=sin(\\pi n_\\sigma)^2\n\\label{Gfriedel}\n\\end{eqnarray}\n($\\sigma=\\uparrow, \\downarrow$) which is valid when the bandwidth\nis much larger than the Kondo coupling. Results for\n$G=G_\\uparrow +G_\\downarrow$ are shown in \nFig.~\\ref{fig2}(a), for the four types of impurities considered.\nAs expected from the results in Fig.~\\ref{fig1}(a), the\nconductance decreases by reducing electron density from half-filling.\n\nThe length of the spin compensation cloud $\\xi$ is depicted in\nFig.~\\ref{fig2}(b) for the same four types of impurities. $\\xi$\ndepends on two factors: (i) the effective exchange coupling between\nthe magnetic impurity and the spins of the conduction electrons, \nand (ii), to a lesser extent, how much defined is the magnetic\ncharacter of the impurity or the Kondo regime. The\nfinite length of the cluster imposes an upper bound to the\ncompensation length. It has been pointed out that the inability of\nthe finite system to accommodate the Kondo cloud would modify in turn\nthe Kondo effect, in particular the Kondo\ntemperature.\\cite{simon-affleck} For the parameter set $U'=8$, \n$t'=0.5$ ($J_{eff}=0.125$) the compensation cloud is large, quite\nlikely exceeding the system size since $\\xi \\approx L\/2$, and\nscaling linearly with the system size. For $U'=4$, $t'=0.5$\n($J_{eff}=0.25$), $\\xi$ is somewhat smaller but still scales linearly\nwith $L$. On the other hand, for the $U'=8$, $t'=1$ ($J_{eff}=0.5$)\nimpurity, where $\\langle S_{z,0}^2\\rangle \\approx 0.21$, the length of\nthe compensation cloud is much smaller and it has a weak dependence\nwith $L$. An intermediate situation occurs for the fourth type of\nimpurity, $U'=2$, $t'=0.5$ which has the same $J_{eff}=0.5$ but with\na poorly defined magnetic character ($\\langle S_{z,0}^2\\rangle < 0.2$),\nleading to a more extended compensation cloud. Notice an overall weak\ndependence of $\\xi$ with the electron density.\n\n\\begin{figure}\n\\includegraphics[width=0.42\\textwidth]{fig3ab.eps}\n\\vspace{0.6cm}\n\n\\includegraphics[width=0.42\\textwidth]{fig3cd.eps}\n\\caption{(Color online) Evolution of the LDOS at the impurity site as the\nelectron density is reduced from half-filling to quarter-filling. Titles\non each panel show the corresponding impurity parameters. For each panel,\nthe fillings are $n=1.0$, 0.875, 0.75, 0.625 and 0.5 from top to bottom,\n$L=95$, $M=500$. Dashed lines in the top left panel correspond to \n$L=96$, $M=300$}\n\\label{fig3}\n\\end{figure}\n\nIt is important to understand the behavior of the conductance shown\nin Fig.~\\ref{fig2}(a) by studying the LDOS at the impurity site,\n$\\rho(\\omega)$. In Fig.~\\ref{fig3} we show $\\rho(\\omega)$ close to\nthe Fermi level, for the four types of impurities and for several\nelectron densities from half-filling to quarter-filling.\n$\\rho(\\omega)$ has been shifted to the chemical potential $\\mu$\nfor the sake of comparison, since starting from half-filling, $\\mu$\nshifts to $\\omega < 0$. In this figure and all the following\nsimilar ones, we adopted a Lorentzian broadening of the peaks of\n$\\delta=0.1$. A measure of the precision of the results presented\ncan be inferred from the comparison between $L=95$, $M=500$, and\n$L=96$, $M=300$. Clearly, the main features are present in both\ncases but it can be seen that the ones for $L=95$, $M=500$ are\nmore consistent as $n$ is reduced.\nFor the impurity  $U'=8$, $t'=0.5$, the Kondo peak at half-filling\nremains at the Fermi level upon reducing the density down to $n=0.5$.\nThis is consistent with the relative large conductance of this case.\nFor the impurity $U'=8$, $t'=1$, the peak shifts slightly away from\nresonance as the filling is reduced, going into the electron part of\nthe spectrum ($\\omega -\\mu < 0$). Also in this case, the behavior\nof $\\rho(0)$ is correlated with the conductance weakly varying with\nthe density. For these two impurities, the Coulomb or holon peaks\nfall outside of the frequency range adopted in the plot. In the\ncase of $U'=4$, $t'=0.5$, the Coulomb peak at $\\omega \\approx -U\/2$\nshows up as $n$ decreases. The overall behavior of the Kondo peak\nis similar to that of the first impurity and this is related to the \nsmall $J_{eff}$ of these cases. On the other\nhand, for the case $U'=2$, $t'=0.5$, the chemical potential shifts\n{\\it below} the Coulomb peak at $\\omega \\approx -U\/2$ (the Coulomb\npeaks are those located at $\\omega -\\mu \\sim \\pm 1$ at half-filling).\nThis behavior is clearly correlated with the important suppression\nof the conductance for this impurity observed in Fig.~\\ref{fig2}(a).\nFor the $U'=8$, $t'=1$, and $U'=2$, $t'=0.5$ impurities, the observed\nsplitting of the central peak at $n=1$ is just a finite size effect;\nthe split\nbetween the two central peaks, which correspond to electron creation\n($\\omega -\\mu > 0$) and annihilation ($\\omega -\\mu < 0$), close\nwhen $L\\rightarrow \\infty$. Notice also the different amplitudes of\nthe spectral weight at the Fermi level, roughly proportional to\n$J_{eff}$.\n\n\\section{Interacting leads, quarter-filling}\n\\label{varying_U}\n\nAfter examining the effect of varying the filling in a system\nwith an Anderson impurity in a tight-binding chain, we now turn\nto the central issue of this paper, that is the effect of the\npresence of a Hubbard on-site potential on the chain. Then,\nwe consider the full Hamiltonian defined in (\\ref{hamilt}), at\nquarter-filling, and taking $U=1$, 2, 4 and 8. In the pure\nHubbard chain, the system behaves as a Luttinger liquid, and \nhence, from the point of view of transport properties is a metal.\n\n\\begin{figure}\n\\includegraphics[width=0.42\\textwidth]{fig4.eps}\n\\caption{(Color online) (a),(b) Electron occupancy and (c),(d)\n$\\langle S_z^2\\rangle$ at the\nimpurity site as a function of the inverse of the chain length $L$\nfor $U=0$, 1, 2, 4, and 8, from bottom to top. $U'=8$, $t'=0.5$\n(left panel), $U'=2$, $t'=0.5$ (right panel). $n=0.5$.}\n\\label{fig4}\n\\vspace{0.6cm}\n\\end{figure}\n\nWe start by looking at finite size\nbehavior of the electron occupancy and $\\langle S_z^2\\rangle$ at the\nimpurity.  Results for the impurities  $U'=8$, $t'=0.5$ and \n$U'=2$, $t'=0.5$ as a function of $1\/L$ are shown in Fig.~\\ref{fig4}.\nFor each value of $U$ a weak variation can be observed in\n$\\langle n_0\\rangle$. For the case of $U'=8$, $t'=0.5$ there is a\nsomewhat larger variation of  $\\langle S_{z,0}^2\\rangle$ which becomes \nweaker as $U$ is increased.\n\n\\begin{figure}\n\\includegraphics[width=0.43\\textwidth]{fig5.eps}\n\\caption{(Color online) (a) Electron occupancy and (b)\n$\\langle S_z^2\\rangle$ at the\nimpurity site as a function of the interaction on the leads $U$ for\n$U'=8$, $t'=0.5$ (circles), $U'=8$, $t'=1.0$ (squares),\n$U'=4$, $t'=0.5$ (diamonds), and $U'=2$, $t'=0.5$ (triangles). \n$L=96$, $n=0.5$.}\n\\label{fig5}\n\\end{figure}\n\nThe fact that as $U$ increases, both $\\langle n_0\\rangle$ and\n$\\langle S_{z,0}^2\\rangle$\nincrease, is a general behavior observed for all the values of $U'$\nand $t'$ that we examined, as can be seen in Fig.~\\ref{fig5}. Note\nthat for $U'=8, t'=1$ in Fig.~\\ref{fig5}(a) the system reaches the\nlimit of the pure Hubbard model at $U=8$, with $\\langle n_0\\rangle=1$,\nwhile the other cases represent different kind of impurities, with\ndensities not necessarily equal to 1. This forces the crossing of the\ntwo $U'=8$ curves found near $U=5$. For the cases $U'=2$ and $U'=4$,\nfor $U>U'$ we can observe a change in the growing behavior of\nboth $\\langle n_0\\rangle$ and $\\langle S_{z,0}^2\\rangle$ curves.\nIn other words, a Hubbard repulsion on the chains favors a more defined\nmagnetic character of the impurity. This is particularly apparent for\nthe impurity with parameters $U'=2$, $t'=0.5$, which most likely is in\na mixed valence state at $U=0$, as discussed in the previous Section.\nAgain these results are consistent with those obtained with SSE within\nerror bars. One may conjecture that as $U$ increases, the $4k_F$\nelectron correlations are enhanced forcing an increased occupation\nat the impurity. The enhancement of $\\langle n_0\\rangle$ and\n$\\langle S_{z,0}^2\\rangle$ with  $U$ occurs also at density\n$n=0.75$ at least for this impurity parameters.\n\n\\begin{figure}\n\\includegraphics[width=0.43\\textwidth]{fig6.eps}\n\\caption{(Color online) (a) Integral of the spin-spin correlations as a\nfunction of the\ndistance from the impurity site (Eq.(\\ref{sccdef})), $U'=8$, $t'=0.5$,\nand $U=0$, 1, 2, 4, and 8, from top to bottom; $L=96$, $n=0.5$.\n(b) Length of the compensation cloud as a function of $U$ for\nthe four set of interactions at the impurity, $U'=8$, $t'=0.5$,\n(circles), $U'=8$, $t'=1.0$ (squares), $U'=4$, $t'=0.5$ (diamonds),\nand $U'=2$, $t'=0.5$ (triangles). $L=64$ (open symbols), $L=96$\n(filled symbols), $n=0.5$.}\n\\label{fig6}\n\\end{figure}\n\nLet us examine now the behavior of the spin compensation cloud when\n$U$ is increased. In Fig.~\\ref{fig6}(a), the integral of the\nspin-spin correlations is plotted as a function of the distance\nfrom the impurity. The typical $2 k_F$ oscillations which at $n=0.5$\nhas a period equal to 4, are clearly seen. $S(r)$ increases (in \nabsolute value) as $U$ increases, implying that the impurity spin\nbecomes screened at shorter distances. This is precisely the \nbehavior of $\\xi$, as it can be seen in Fig.~\\ref{fig6}(b) for the\nfour types of impurities considered. For $U'=8$, $t'=0.5$ and $U'=4$,\n$t'=0.5$ the reduction of\n$\\xi$ with $U$ is clear in spite of the fact that the dependence\nof $\\xi$ with the system size suggests that the bulk limit has not\nbeen reached. On the other hand, for $U'=8$, $t'=1.0$ and for\n$U'=2$, $t'=0.5$, results for $L=64$ and 96 are very similar suggesting\nthat the compensation clouds observed are those of the bulk limit.\nThis behavior of the spin compensation cloud is correlated with \nthe increase of $\\langle S_{z,0}^2\\rangle$ shown in\nFig.~\\ref{fig5}(b). The relative values of $\\xi$ for the various\nimpurities follows in first approximation the values of the Kondo\ncouplings, as was discussed with respect to Fig.~\\ref{fig2}(b).\nThe overall reduction of $\\xi$ with $U$ is consistent with previous\ncalculations for the two-impurity Kondo coupling\\cite{hallberg-egger}\nand expected on general grounds due to the increase of the Kondo\ntemperature with correlations.\\cite{leetoner}\n\n\\begin{figure}\n\\includegraphics[width=0.42\\textwidth]{fig7ab.eps}\n\\vspace{0.8cm}\n\n\\includegraphics[width=0.42\\textwidth]{fig7cd.eps}\n\\caption{Evolution of the LDOS at the impurity site as $U$ in the\nleads is increased. Titles on each panel show the corresponding\nimpurity parameters. For each panel, $U=0.0$, 1.0, 2.0, 4.0 and\n8.0 from top to bottom. $L=95$, $n=0.5$.}\n\\label{fig7}\n\\end{figure}\n\nThe LDOS at the impurity presents also interesting features as $U$\nis increased. Let us first consider the impurity $U'=8$, $t'=0.5$\n(top left panel Fig.~\\ref{fig7}). In this case the peak retains its\nresonant character as $U$ increases (only for $U=8$ it appears at a\nsmall negative frequency), and its amplitude is slightly reduced.\nFor the second type of impurity, $U'=8$, $t'=1.0$ (top right panel\nFig.~\\ref{fig7}), there is a pronounced transfer of spectral weight\nto the occupied part of the spectrum, and also \n$\\rho(\\omega -\\mu = 0)$ is much reduced. For the impurity $U'=4$, \n$t'=0.5$ (bottom left panel), the remains of the Kondo peak stays\nat the chemical potential, which moves towards positive $\\omega$.\nThe same effect of $U$ is observed for $U'=2$, $t'=0.5$ (bottom right\npanel Fig.~\\ref{fig7}). In this case, the peak located at\n$\\omega -\\mu > 0$ for $U=0$, crosses the Fermi level at $U\\sim 4$, \nand finally appears at $\\omega -\\mu < 0$ for $U=8$, as the spectral\nweight is transfered from the unoccupied to the occupied part of\nthe spectrum. Taking into account the strong enhancement of \n$\\langle S_{z,0}^2\\rangle$ for these impurity parameters shown \nin Fig.\\ref{fig5}(b), one may conjecture that the states close\nto the Fermi level that are shifted by increasing $U$ correspond\nto magnetic excitations becoming increasingly occupied.\n\n\\begin{figure}\n\\includegraphics[width=0.39\\textwidth]{fig8new.eps}\n\\caption{(Color online) $J(t)\/\\Delta_V$ for different values of $U$,\n(a) $U'=8$, $t'=1.0$,\n(b) $U'=4$, $t'=0.5$, $L=96$, $n=0.5$, $\\Delta V=0.01$, $\\Delta t=0.1$.\nThe horizontal lines show the time interval over which the average\nof the current is taken to compute the conductance.\n(c) Relative conductance (see text) as a function of $U$ for \ndifferent values of the interactions $(U',t')$. The values of \n$K_{\\rho}$ at $n=0.5$, extracted from Ref.\\onlinecite{schulz},\nare shown with stars.}\n\\label{fig8}\n\\end{figure}\n\nOur most relevant results regarding the transport properties in a\nHubbard chain in the presence of an Anderson impurity are shown in\nFig.~\\ref{fig8}. In Fig.~\\ref{fig8}(a,b), we show the average current\n$J(t)$ on the links connected to the impurity divided by $\\Delta V$\non a $L=96$ chain (results for $L=95$ are virtually identical)\nfor two sets of impurity interactions. \nAs discussed in Section \\ref{model},  $J(t)$\nhas an oscillatory behavior which is typical of these time-evolving\nsystems.\\cite{alhassanieh,meir-wingreen} The period of these \noscillations for the cases shown in Fig.~\\ref{fig8} is approximately\n$800 \\Delta t$. In this figure, we only show $J(t)$ up to \n$t =250  \\Delta t$ because after that the time-evolution becomes \nunstable. In some cases,\n(Fig.~\\ref{fig8}(a)), these wiggles make the conductance to exceed\nthe value $2 e^2\/h$ indicating that the time-evolution has already\nbecome unstable.\n\nFrom $J(t)\/\\Delta V$, we have computed the \nconductance according to Eq.~(\\ref{condJt}). Typical time intervals\nor ``plateaus\" over which the time average was taken are shown in\nFig.~\\ref{fig8}(a,b). The values of the conductance $G=G(U)$, for\na given impurity, relative to the value for noninteracting leads,\n$G_0=G(U=0)$, are shown in Fig.~\\ref{fig8}(c). Although there is\nsome arbitrariness in defining these plateaus, a qualitatively\nclear trend emerges. For the impurities ($U'=8$, $t'=1.0$) and\n($U'=4$, $t'=0.5$) $G\/G_0$ decreases as U increases. In these\ncases, the suppression is consistent with the reduction of\nspectral weight at the Fermi level in the impurity LDOS, shown in\nFig.~\\ref{fig7}. In these cases, where a Kondo resonance subsists\nfor finite $U$, and particularly for ($U'=8$, $t'=1.0$), the\nsuppression of the conductance is consistent with the predicted\nrelation\\cite{kanefisher} $G\\sim K_\\rho$, as shown in\nFig.~\\ref{fig8}(c). On the other hand, for the case of the impurity\nparameters  $U'=2$, $t'=0.5$, $G\/G_0$ follows a more\ncomplex behavior. From the non-interacting limit\nto approximately $U=1$, $G\/G_0$ increases and then as $U$ \nincreases further, it starts to decrease. This non-monotonous \nbehavior again follows the one of the spectral weight at the \nFermi level in the impurity LDOS (Fig.~\\ref{fig7}, bottom right \npanel).\n\n\\begin{figure}\n\\includegraphics[width=0.43\\textwidth]{fig9.eps}\n\\caption{(Color online) (a) Evolution of the spin compensation cloud as\na function of distance from the impurity at various temperatures\nobtained with SSE, $U'=2$, $t'=0.5$, $L=96$.\n(b) Length of the spin compensation cloud as a function of temperature\n$U'=2$, $t'=0.5$, $L=96$, $n=0.5$, for $U=0$, and 1. The $T=0$ values\nwere obtained with DMRG. Error bars are shown for $T=0.0125$. Temperature\nin units of $t$.}\n\\label{fig9}\n\\end{figure}\n\nFinally, we report a finite temperature study of the spin compensation\ncloud. We chose the impurity defined by the couplings $U'=2$, $t'=0.5$.\nIn this case, since $J_{eff}$ is large (as long as the relation\n$J_{eff}= 4t'^2\/U'$  is applicable to these parameters), we expect a\nrelatively small \nlength $\\xi$ of the spin compensation cloud. In addition, the \nexpression for the Kondo temperature for non-interacting leads ($U=0$),\n$T_K=t' \\sqrt{U'\/(2 t)} \\exp{(-\\pi t U'\/({8 t'}^2))}$, although not\nmuch precise for these impurity parameters, gives $T_K \\approx 0.02$\nwhich is reasonably accessible by SSE simulations. In\nFig.~\\ref{fig9}(a) the spin compensation cloud $S(r)$ is shown at\nseveral temperatures for the case of noninteracting leads, $U=0$. The\ntemperature is in units of $t$, $k_B=1$.\nIt is remarkable the convergence of finite temperature results\nobtained with SSE to the zero temperature ones obtained with DMRG.\nNotice that both techniques are in principle exact but SSE results\ncorrespond to PBC systems while DMRG ones to OBC chains.\n\nThe length $\\xi$ is shown in Fig.~\\ref{fig9}(b). Errors were computed\nfrom the deviation of results obtained in four independent runs.\nFor $T > T^*$, $T^* \\approx 0.375$,  $\\xi$ is equal to half\nthe chain length, implying that the spin compensation cloud exceeds\nthe limits of the chain. For $T < T^*$, $\\xi$ starts to be smaller\nthan half the chain length indicating that now the cloud is fitting\nin the chain. That is to say, for $T < T^*$ the system manages to\ncompensate the spin of the impurity but this is not possible for\n$T > T^*$. If the Kondo temperature is thought as the crossover\ntemperature around which the impurity spin becomes compensated then\n$T^*$ would be a measure of the Kondo temperature in finite systems,\nthat is, $T^*=T^*(L)$. As mentioned in Section \\ref{model}, there is\na certain degree of arbitrariness in the estimation of $\\xi$ because\nof the cutoff $x$ but what is relevant is to determine how $T^*$\nbehaves when a parameter of the system is varied. For the case of\n$U=1$, $T^*$ reduces its value to $\\approx 0.25$. This seems a trend\nas the correlations on the leads are increased but results for larger\nvalues of $U$ have larger error bars and so they are not precise\nenough to confirm this trend.\n\n\\section{Conclusions}\n\\label{conclusions}\n\nIn summary, we have studied the single-impurity Anderson-Hubbard \nmodel on finite chains with numerical techniques. In the first place,\nwe analyzed how the reduction of the electron density from \nhalf-filling to quarter-filling affects the Kondo resonance for\nnon-interacting leads. Physical quantities defining the magnetic\ncharacter of the impurity show in general a weak behavior with the\nelectron density except when the impurity is, at half-filling,\nclose to a mixed valence regime. In this case there is a steep\nlost of magnetic character by reducing the density. This behavior is\nreflected also in the impurity LDOS and in the conductance in linear\nresponse. Next, we turn to the main topic of this \nwork which is the study of the effects of a Hubbard interaction\non the leads at quarter-filling. A very clear and interesting result\nis that $U$ drives the impurity into a more defined Kondo regime\n(Fig.~\\ref{fig5}) although accompanied in most cases by a reduction\nof the spectral weight of the impurity LDOS. This reduction is\nhowever not general since for an impurity  close to\nthe mixed valence regime we\nobserved a nonmonotonic behavior. As we mentioned in the\nIntroduction, there is a number of strongly correlated materials\nwhich can be or have already been implemented\\cite{biercuk} in\nnanoscopic devices and where some of the present results for the\nimpurity LDOS could be contrasted.\nThe conductance, computed for a small finite bias applied to the\nleads, follows the behavior of the impurity LDOS. For the impurities\nin a well-defined Kondo regime, the conductance is suppressed by\n$U$ in finite chains, in agreement with what is expected for an \nimpurity in a Luttinger liquid. Again, for the impurity close to\na mixed valence regime, $G$ increases for small $U$ but eventually\nstarts to decrease for large values of $U$. This nonmonotonic\nbehavior indicates a complex interplay between the interactions\non the impurity and the ones on the leads (in addition, of course,\nto the impurity-leads interactions).\n\nFinally, we exploit one of the important advantages of the real space\nmethods we used, that is, the possibility of getting a measure of the\nspin screening or compensation of the impurity. The ``spin\ncompensation cloud\" gives an insight on the internal working of the\nKondo effect and it does not depend just on properties at the\nimpurity site but also on $U$ on the leads. As shown in\nFig.~\\ref{fig6},\nfinite-size effects are more important for small values if $U$.\nThe proposed determination of the Kondo\ntemperature in finite systems through the spin compensation cloud, \nhas to be thoroughly explored as a function of density and coupling\nparameters. We emphasize that this numerical estimation of $T^*$ is\ncompletely general in the sense that it is not limited to 1D\nAnderson-Hubbard model but could be used for any Hamiltonian for the\nleads, for example the Kondo lattice model, and also not limited to\n1D leads, although beyond 1D the ``minus sign problem\" makes\ndifficult the application of QMC methods.  Moreover,  we believe\nthat the dependence of $T^*$ with the finite size of the system\ncould be experimentally measured and compared with theoretical\nresults.\n\n\n\\acknowledgments\nThis work was supported in part by grant \nPICT 03-12409 (ANPCYT).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Technical Appendix}\\label{sec:append}\n\n\nIn this section we provide technical definitions and  details of proofs of the results.\n\n\\input{window-movie-lemma}\n\n\n\\subsection{Proof of Lemma \\ref{lem:depend-path}}\\label{sec:depend-path-proof}\n\n\n\nRestatement of Lemma \\ref{lem:depend-path} (Dependence paths):\nGiven a singly-seeded aTAM system $\\calT$, producible assembly $\\alpha \\in \\prodasm{\\calT}$, and sets of locations $L_1,L_2 \\subset \\dom{\\alpha}$, if $L_2$ strictly depends upon $L_1$, then in each valid assembly sequence of $\\calT$ there must be a dependence path from some  $l_1 \\in L_1$ to some $l_2 \\in L_2$.\n\n\n\\begin{proof}\nWe prove Lemma \\ref{lem:depend-path} by contradiction. Therefore, assume that, given singly-seeded aTAM system $\\calT = (T,\\sigma, \\tau)$, producible assembly $\\alpha \\in \\prodasm{\\calT}$, and sets of locations $L_1,L_2 \\subset \\dom{\\alpha}$, that $L_2$ strictly depends upon $L_1$. However, for the sake of contradiction assume that there exists some valid assembly sequence in $\\calT$ in which there is not a dependence path from any point $l_1 \\in L_1$ to any of the points $l_2 \\in L_2$.\n\nBy the definition of strict dependence, we know that a tile is placed in $L_1$ before a tile is placed in $L_2$. Therefore, we have two cases to consider: (1) $\\dom{\\sigma} \\in L_1$, i.e. $L_1$ contains the location of the seed of $\\calT$, or (2) $L_1$ does not contain the seed location. We'll show that case (1) can't hold by induction. The induction hypothesis is that, given a producible assembly $\\alpha$ in which every tile has a dependency path from the seed (which is in $L_1$) to it, then any tile which binds has a dependency path to the seed. The base case is the first tile attachment, which must be directly to the seed and therefore that tile has a dependency path to the seed. We prove the induction hypothesis simply by noting that if every tile of $\\alpha$ has a dependency path to the seed and a new tile attaches to $\\alpha$, the dependency path of any tile to which it binds is simply extended by 1 to be a dependency path from the location of the seed to the new tile. Therefore, there is a dependency path from the seed to any location in a producible assembly, and thus case (1) cannot hold.\n\nNow, consider case (2). Assume that $L_2$ strictly depends upon $L_1$, but that there is no dependency path from any location in $L_1$ to any of the locations in $L_2$. Let $\\vec{\\alpha}$ be any assembly sequence which places tiles in $L_1$ and $L_2$. Since $L_2$ strictly depends upon $L_1$, $\\vec{\\alpha}$ must place a tile in $L_1$ before $L_2$ (by definition of strict dependence). However, we can now use $\\vec{\\alpha}$ to make a new assembly sequence $\\vec{\\alpha}'$ as follows. Step through $\\vec{\\alpha}$ one tile placement at a time. Add all tile placements from $\\vec{\\alpha}$ to $\\vec{\\alpha}'$ until any tile placement in $L_1$. Do not add that tile placement to $\\vec{\\alpha}'$, and from that point add all tile placements from $\\vec{\\alpha}$ which are not in $L_1$ and the tiles are able to attach to the assembly growing from $\\vec{\\alpha}'$ to that sequence as well, until and including the first tile placement in $L_2$. By the assumption that there is no dependency path from $L_1$ to $L_2$, it must be possible for $\\vec{\\alpha}'$ to also place the tile in $L_2$. Then, in $\\vec{\\alpha}'$, place the same first tile in $L_1$ which $\\vec{\\alpha}$ places there. This must be possible because no tile has been placed in that location in $\\vec{\\alpha}'$ and whichever tiles bordered that location in $\\vec{\\alpha}$ to allow a tile to attach there also border it in $\\vec{\\alpha}'$. However, this would mean that in $\\vec{\\alpha}'$, a tile is placed in $L_2$ before $L_1$, which is a contradiction. Therefore, there must be a dependence path from $L_1$ to $L_2$ in any valid assembly sequence in $\\calT$ and Lemma \\ref{lem:depend-path} is proven.\n\n\\end{proof}\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:multi-imposs}}\\label{sec:multi-imposs-proof}\n\n\nRestatement of Theorem \\ref{thm:multi-imposs}:\nThere exist an infinite number of aTAM systems with multi-tile seeds that cannot be shape-simulated by any aTAM system with a single-tile seed at (1) scale factors 1 or 2, or (2) scale factor 3 without using cheating fuzz.\n\n\\begin{proof}\n\nTo prove Theorem \\ref{thm:multi-imposs}, we present one system which can't be shape-simulated by any aTAM system with a single-tile seed at (1) scale factors 1 or 2, or (2) scale factor 3 without using cheating fuzz, and note how an infinite set of such systems can easily be derived from it. Let $\\calT = (T,\\sigma,1)$, whose seed is depicted in Figure \\ref{fig:multi-scale-system}, be such a system. (An infinite set of similar systems can be derived by increasing the lengths of the arms of the seed to arbitrary values and adjusting the tile set as necessary to accommodate the growth that will be described.) In $\\calT$, there are five locations on the perimeter of the seed in which glues are exposed and to which tiles can attach. They are depicted in green. Each arm can grow a uniquely colored subassembly, each of which uses tile and glue types unique to that subassembly. An example assembly is depicted in Figure \\ref{fig:pumped-paths-terminal}.\nAn infinite number of unique terminal assemblies, and assembly shapes, are possible in $\\calT$, since, for instance, each path labeled 1 can be of arbitrary length.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=5.0in]{images\/multi-scale-pumped-paths.png}\n    \\caption{Depiction of an example assembly in $\\calT$ (from the proof of Theorem \\ref{thm:multi-imposs}) in which paths have grown from each arm of the seed. In order to show all paths, their lengths are greatly reduced, but all path segments marked with an arrow are assumed to be of length greater than the pumping length $p$. The blue, yellow, pink, and red sets of paths, which grow from seed arms A, B, C, and E, respectively, are similar modulo rotation\/reflection and the lengths of the repeating periods of some of the paths labeled 4 and 5 (i.e. some of those paths travel distance 5 horizontally and 5 vertically before repeating, and some travel 6 horizontally and 6 vertically before repeating). Path segments marked with capped lines (i.e. those marked 2 and 3 in the blue yellow, pink, and red, and that marked 6 in the gold) are fixed lengths. Note that the slopes of the lines marked 7, 8, and 9 in the gold section are $\\pm 6\/4$, and those of the pink are red sections are all $\\pm 1\/1$, so all pumpable paths can be greater than the pumping length without colliding with each other and still be aimed at the appropriate seed locations.}\n    \\label{fig:pumped-paths-terminal}\n\\end{figure}\n\nOur proof will be by contradiction, so assume that there exists singly-seeded aTAM system $\\mathcal{S} = (S, \\sigma_\\mathcal{S}, \\tau)$ such that $\\mathcal{S}$ shape-simulates $\\calT$ at either (1) scale factor 1 or 2, or (2) scale factor 3 but does not use cheating fuzz. Let $m$ be the scale factor used by $\\mathcal{S}$ to shape-simulate $\\calT$, and let $g$ be the number of unique glues types on the tile types of $\\mathcal{S}$. We define $p$ to be the ``pumping length'' of a simulated one-tile-wide path in $\\mathcal{S}$ and set $p = ((g+1)^{6m}\\cdot (6m)!+1)\\cdot 6 + 1$. This value is derived such that we can apply the Window Movie Lemma of \\cite{temp1notIU}. Let $R: B^T_m \\dashrightarrow S$ be the representation function that maps $m$-block supertiles over $S$ to tiles of $T$.\n\n\n\n\n\n\n\n\n\nFigure \\ref{fig:pumped-paths-terminal} shows an example of a producible, terminal assembly $\\alpha \\in \\termasm{\\calT}$ (with the pumpable path segments shortened to fit on the page). From each green location of the seed a path grows outward from the seed (these paths are labeled 1 in the figure). These paths can be arbitrarily long, and we will only consider lengths significantly longer than the pumping length of $\\mathcal{S}$, $p$. Each path labeled 1 then splits, ultimately resulting in 3 paths which grow back toward the seed. Since the paths labeled 1 are significantly longer than $p$, so can all of these others be, and we consider an $\\alpha$ in which this is the case.\n    \nWe'll now refer to the blue paths originating from the seed point labeled A, and note that the same arguments hold for the yellow, pink, and red sections up to rotation. The gold section has a different geometry, but the same general principles hold.\nIn this assembly $\\alpha$, all paths terminate, but only after growing longer than length $p$. Each has a periodic section which can repeat an arbitrary number of times consecutively, but for the paths labeled 4, 5, and 7, nondeterministically there is also a chance, after each repetition, for a ``capping'' tile to attach causing the path to terminate. However, note that the shape and slope of path 4 would allow it, if it were not capped, to extend via continued repetitions until it crashed into the green location of arm B (i.e. it would be able to place a tile in that location if there wasn't already a tile there). Similarly, the path labeled 5 could crash into the green location of arm E. Each colored section has the potential to grow paths that can crash into the green locations of the two neighboring arms.\n\n    \n    \n\\begin{figure}\n    \\centering\n    \\includegraphics[width=4.0in]{images\/multi-scale-crashed-paths.png}\n    \\caption{Depiction of an assembly (derived from $\\alpha$ in the proof of Theorem \\ref{thm:multi-imposs}) in which blue paths from arm A have pumped and crashed into the green locations of arms B and E, and gold paths from arm D have pumped and crashed into the green locations of arms C and E. Note that such an assembly is not actually possible in $\\calT$ since all tiles of the seed, including the green tiles, are present before any paths begin growth. Additionally, the figure depicts the blue and gold paths growing through each other in arm E, which is also impossible but depicted to show the trajectory of each path.}\n    \\label{fig:multi-scale-crashed-paths}\n\\end{figure}\n\nSince $\\mathcal{S}$ is assumed to shape-simulate $\\calT$, there must be a terminal assembly $\\beta \\in \\termasm{\\mathcal{S}}$ that maps, via $R$, to an assembly with the same shape as $\\alpha$. Thus, there must be an assembly sequence in $\\mathcal{S}$ that creates $\\beta$, which we'll call $\\vec{\\beta}$. By the Window Movie Lemma, since each of paths 4 and 5 are of lengths greater than $p$, there must also be valid assembly sequences in $\\mathcal{S}$ that produce extended versions of each path (by pumping an earlier segment before the terminating ``capping'' tile attaches). We examine a few such possible extensions and potential collision now (and a depiction of an assembly impossible to form in $\\calT$, but to which the assembly maps can be seen in Figure \\ref{fig:multi-scale-crashed-paths}).\n    \nLet $l_B$ refer to the location of the blue highlighted seed tile in arm B of $\\alpha$, $l_D$ to the blue highlighted location in arm $D$, $l_{p4}$ to a location on path 4 which is greater than distance $p$ from the beginning of path 4, and $l_{p5}$ a location on path 5 which is greater than distance $p$ from the beginning of path 5. Given some location $l \\in \\mathbb{Z}^2$, let $R^{-1}(l)$ refer to the set of locations in $\\mathcal{S}$ which map to location $l$ in $\\calT$ under $R$.\n    \n\\begin{lemma}\\label{lem:dep}\n$R^{-1}(l_{p4})$ strictly depends upon $R^{-1}(l_B)$.\n\\end{lemma}\n\n\\begin{proof}\nLemma \\ref{lem:dep} says that all valid assembly sequences in $\\mathcal{S}$ place a tile in the macrotile location mapping to $l_B$ before placing any tile in the macrotile location mapping to $l_{p4}$. We prove this by contradiction, so therefore assume that there exists an assembly sequence which places a tile in the location mapping to $l_{p4}$ before placing any tiles in the macrotile location mapping to $l_B$. Because the length of path 4 is greater than $p$, so is the portion of $\\beta$ which maps to path 2. This means that, by the Window Movie Lemma, there must be a repeated window movie along cuts of the subassembly mapping to path 2, and therefore valid producible assemblies can also be made by pumping the repeated section of that subpath arbitrarily many times. (A schematic depiction of the pumping of a path can be seen in Figure \\ref{fig:pumped-example}. Let $\\beta'$ be an assembly, which must therefore be producible in $\\mathcal{S}$, in which the subassembly mapping to path 2 is pumped so that it passes through the green location of arm B and places tiles which resolve to at least one more period of the path (i.e. it places the 5 tiles of each of a vertical and horizontal section of the path). This must be possible by the Window Movie Lemma (which allows the pumping) and the fact that no tiles can previously have been in locations occupied by the newly extended path since, if no tile has been placed in $l_B$, then there also cannot be tiles in the fuzz locations to its left and right (since those are not legal fuzz locations before $l_B$ has resolved to a tile in $\\calT$, which it cannot since it's empty). Thus, the tiles mapping to the blue path as well as any allowable fuzz positions around it can be freely placed (in exact duplication of the pumpable ordering that previous copies must have been placed in for an earlier segment of the path). Since the macrotiles of the previous section of the path map to tiles of path 4, so must the newly pumped macrotiles since the representation function remains constant.\n\nAt this point we can note that, regardless of whatever additional tile additions may be made to $\\beta'$, it must map to an assembly whose shape cannot be that of a terminal assembly of $\\calT$. This is because there is no possible assembly sequence in $\\calT$ in which a path with the repeating period (5 horizontal and 5 vertical positions) can extend from the right side of the green location of arm B. This is because the green tile of arm B can only initiate paths that would not be able to return to the location to its immediate east, and any path that could extend into that location from the pink growth would have a different period (i.e. 6 horizontal and 6 vertical positions). Therefore, $\\mathcal{S}$ would fail to shape-simulate $\\calT$ in this case. This is a contradiction to the assumption that $R^{-1}(l_{p4})$ does not strictly depend upon $R^{-1}(l_B)$, and thus Lemma \\ref{lem:dep} is proven.\n\n\\end{proof}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=6.0in]{images\/multi-pumped-example.png}\n    \\caption{Example of a pumpable segment.}\n    \\label{fig:pumped-example}\n\\end{figure}\n\nThe same logic of Lemma \\ref{lem:dep} can be applied to show that strict dependence occurs between two of the pumpable paths growing from each arm of the seed and the locations marked in blue in Figure \\ref{fig:multi-scale-crashed-paths} of each of the two neighboring arms. By Lemma \\ref{lem:depend-path}, this means that there is a dependence path going from a point in each blue-marked location, through each of the two neighboring arms, to the locations on those pumpable paths. In Figure \\ref{fig:multi-scale-system}, the blue line across each arm shows a minimal cut across each. In $\\alpha$ that cut crosses two locations (one tiled and one that could validly include the growth of fuzz). In $\\beta$, the scaled version of each cut crosses two macrotile locations (one that must resolve to a tile under $R$, and one that is potentially allowed to contain growth of fuzz and also potentially able to eventually resolve into a tile). The scale factor $m$ of the simulation determines the size of each of these macrotiles.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=3.5in]{images\/multi-scale-4-paths.png}\n    \\caption{Schematic depiction of the dependence paths between the arms of $\\beta$ from the proof of Lemma \\ref{lem:4-paths}. Arrows above a label indicate which of the two neighbors has the required dependence path leading to that arm. (Top) Every dependence path is separate (i.e. formed by a disjoint set of tiles). This leads to all arms having minimal cuts across 4 tiles. (Middle) A single path originates in arm A, visits every other arm, then returns to arm B. That path satisfies the dependencies of arms A, B, and E. A separate path grows from E to D, satisfying the dependencies of D. At this point, the only ways to satisfy the missing dependency for arm C (i.e. a path from D), are to (1) continue the path from B (since it has passed through arm D) or to grow a new path directly from arm D. The first results in a minimal cut across 4 tiles in arm B, and the second results in a minimal cut across 4 tiles in arm D. (Bottom) Besides the circular path which begins and ends at arm A (satisfying dependencies for A and E), paths grow from arm C to arm B, and from arm E to arm D (satisfying the dependencies for B and D, respectively). At this point, the most succinct way to satisfy the missing dependency for arm C requires a path from D to C, which causes a minimal cut across 4 tiles in arm D. All other options either require increasing additional cut sizes and\/or are identical modulo rotation.}\n    \\label{fig:4-paths}\n\\end{figure}\n\n\\begin{lemma}\\label{lem:4-paths}\nIn assembly $\\beta$ of $\\mathcal{S}$ (as depicted in Figure \\ref{fig:pumped-paths-terminal}), a minimal cut across the macrotiles of at least one arm (and its associated fuzz) must cross at least 4 tiles.\n\\end{lemma}\n\n\\begin{proof}\nWe prove Lemma \\ref{lem:4-paths} by contradiction, so assume that there is a cut across at least one of the arms of $\\beta$ that crosses fewer than 4 tiles. We first note that two consecutive tiles on a dependence path cannot also both be part of a dependence path traveling in the opposite direction. This is by the definition of a dependence path, since the $i$th tile of a dependence path must be placed before the $(i+1)$th tile, which only allows one direction for a dependence path across a pair of tiles.\n\nWe now refer to Figure \\ref{fig:4-paths}, which gives a schematic depiction of the directed paths that must exist between each neighboring pairs of arms. The topmost figure depicts each path growing separately, and shows that in this case, there would need to be four paths traveling through each arm, resulting in minimal cuts across 4 tiles for every arm, so this cannot be the case. Since all dependencies must be resolved by dependence paths, the only alternative is to concatenate paths (which can allow dependencies to be resolved in a cyclic manner). Concatenating paths only serves to reduce the need for some separate path if a single path is created which visits all arms.  This is depicted in the middle and bottom figures of Figure \\ref{fig:4-paths}, which show that it is impossible for all necessary dependence paths to exist without requiring at least 4 paths passing through each arm. Even though, in some cases, at different points the same path may pass into or out of an arm (e.g. when the same path travels through all arms and extends to revisit one or more arms), in each such case an additional path of tiles is required to maintain the dependencies of the full path and\/or the correct directionality. Therefore, in all cases, at least one arm must have a minimal cut crossing at least 4 tiles. This is a contradiction to the assumption that some cut crosses less than 4 tiles, and proves Lemma \\ref{lem:4-paths}.\n\\end{proof}\n\nWe now know by Lemma \\ref{lem:4-paths} that the minimal cut across some arm must cross at least 4 tiles. This immediately means that the simulation scale factor of $\\mathcal{S}$, $m$, must be greater than 1 since the shortest cut across each arm at scale factor one (depicted as blue lines in Figure \\ref{fig:multi-scale-system}) crosses only one tile and one valid fuzz location. Additionally, for the case of $m = 3$ where cheating fuzz is not allowed, we note that the cuts represented by the blue lines in Figure \\ref{fig:multi-scale-system} each cross only one macrotile location which can resolve to a tile, providing a maximum of three tiles to be crossed by the cut, and the fuzz macrotile location (outlined in red) is not allowed to receive any tiles. This is because no glues are exposed on the exterior edges of the seed tiles adjacent to those locations and since $\\beta$ is terminal, no paths can crash into those locations. This means that those fuzz locations could never resolve to tiles or be in locations that map to locations adjacent to exposed glues, and thus represent cheating fuzz. Therefore, $\\mathcal{S}$ must not be simulating $\\calT$ at scale factor 3 without using cheating fuzz, since at least one of the dependence paths would have to place tiles in locations of cheating fuzz.\n\nThus, the only remaining case to consider is that in which the scale of the simulation is $m = 2$. In this case, the cuts marked in blue cross exactly 4 locations which can receive tiles. This provides enough locations for the dependence paths to grow, but requires that\nif there are any unfilled locations within any of the fuzz location outlined in red, they must be contained between a pair of dependence paths and unreachable from the exterior, which is a fact we will utilize shortly.\n\nWe now have two cases to consider: (1) there exists even one assembly sequence that results in $\\beta$ and which fills one of those locations in a way that the representation function $R$ maps to a tile in $\\calT$, or (2) in all assembly sequences, such locations are filled so that $R$ maps them to empty space in $\\calT$. In case (1), we follow such an assembly sequence and since $\\beta$ is terminal, the domain of the assembly that it maps to in $\\calT$ via $R$ is a domain that no terminal assembly in $\\calT$ matches (since there is no chance for a tile to grow from the seed at that location and no path crashes into it) and thus $\\mathcal{S}$ fails to shape-simulate $\\calT$.\nTherefore, case (1) cannot hold.\nThis leaves us with the final case to consider, that in which there is always at least one red outlined fuzz position that maps to empty space. Since any unfilled locations in such macrotiles are sealed off from the exterior, and we've already determined that no tiles can be placed within those cavities, there is no way to add tiles to those macrotile locations and thus they must always map to empty space. We can now simply note that the paths labeled 7 have the potential to grow so that they crash into each of those locations (examples seen as the two filled locations outlined in red in Figure \\ref{fig:empty-crash}). This means that there are producible assemblies in $\\calT$ that have tiles in those locations, but there are no producible assemblies in $\\mathcal{S}$ that can map to assemblies that have tiles in those locations. Therefore, $\\mathcal{S}$ also fails in the case, meaning that it fails in all cases. This is a contradiction that $\\mathcal{S}$ shape-simulates $\\calT$ at either (1) scale factor 1 or 2, or (2) scale factor 3 without using cheating fuzz. Since the only assumption made about $\\mathcal{S}$ is that it is singly-seeded, it holds that no singly-seeded aTAM system can shape-simulate $\\calT$ at (1) scale factors 1 or 2, or (2) scale factor 3 without using cheating fuzz. As previously mentioned, $\\calT$ can be easily modified by changing the lengths of the seed arms (and appropriately modifying the path-growing tiles) through an infinite range, yielding an infinite set of aTAM systems with multi-tile seeds that cannot be shape-simulated by singly-seeded systems at these scale factors, and thus Theorem \\ref{thm:multi-imposs} is proven.\n\n\n\\end{proof}\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:scale4-sim}}\\label{sec:scale4-sim-proof}\n\n\nRestatement of Theorem \\ref{thm:scale4-sim}:\nGiven an arbitrary aTAM system $\\mathcal{T} = (T,\\sigma,\\tau)$, there exists an aTAM system $\\mathcal{T}_4 = (T_4, \\sigma_0, \\tau_4=\\max(2, \\tau))$ which seed-first-simulates $\\mathcal{T}$ at scale factor 4 and does not use cheating fuzz, where $|\\sigma_0| = 1$ and $|T_4| \\leq 28s + 16g + 6t$ given that $s = |\\sigma|$, $t = |T|$, and $g$ is the total number of unique glue\/strength combinations in $T$ and $\\sigma$.\n\n\\begin{proof}\nWe prove Theorem~\\ref{thm:scale4-sim} by construction. Let $\\mathcal{T} = (T,\\sigma,\\tau)$ be an arbitrary aTAM system, and let $s = |\\dom \\sigma|$ be the number of tiles in the seed $\\sigma$.\n\nWe define the system $\\calT_4$ which seed-first-simulates $\\calT$ by $\\calT_4 = (T_4, \\sigma_0, \\tau_4)$. The temperature is defined by the function $\\tau_4 = \\max(2, \\tau)$.\nTemperature 1 systems can be trivially simulated by temperature 2 systems, and cooperative growth is utilized in the perimeter path to allow for correct seed-first-simulation. \nWe define $T_4$ as the combination of two sets of tiles, $T_4 = T_\\sigma \\cup T_{IO}$.\n$T_\\sigma$ are the $16s$ tiles which will self-assemble the scaled version of $\\sigma$.\n$T_{IO}$ is the scaled expansion of the tileset $T$.\n\n\nWe begin by the generation of $T_\\sigma$.\n\n\\begin{observation}\\label{obs:scale4-contains-scale2}\nEvery scale 4 supertile contains a scale 2 square.\n\\end{observation}\n\nObservation~\\ref{obs:scale4-contains-scale2} is demonstrated in Figure~\\ref{fig:S4-7_scale_4_contains_scale_2}, and from this we define this scale 2 square as the \\emph{core tiles} of a scale 4 supertile.\nFigure~\\ref{fig:S4-7_scale_4_contains_scale_2} also demonstrates that the core tiles of adjacent supertiles can be connected via 2 additional tiles in each supertile, demonstrated by the red lines.\nThe presence of a Hamiltonian cycle is proven to exist for shapes of scale factor 2, as shown in \\cite{SummersTemp}.\nThis allows us to guarantee that a core path can be created which visits each supertile contained in the seed.\nThe generation of this Hamiltonian cycle follows the procedure developed in \\cite{griffith2004growing} which consists of the following two steps.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.25\\textwidth]{images\/S4-7_scale_4_contains_scale_2.png}\n    \\caption{The scale 2 square contained inside the scale 4 supertiles is defined by the green tiles. Note that scale 2 squares of adjacent supertiles can be connected by 2 tiles, indicated by the tiles under red lines.}\n    \\label{fig:S4-7_scale_4_contains_scale_2}\n\\end{figure}\n\nFirst, a spanning tree must be generated; this can be found utilizing breadth-first or depth-first search algorithms.\nFor convenience, we define the \\emph{origin} supertile as the westernmost tile in the southernmost row of the seed.\nThe root of the spanning tree is set as the origin from which either breadth-first or depth-first search is carried out.\n\nSecond, we are able to replace each tile in the spanning tree with an associated supertile based upon its neighbors - we begin with the origin supertile.\nIf $s=1$, the system $\\calT$ already contains a single-tile seed, and we utilize a specific origin tile; this is case 0).\nDue to the location of the origin supertile, only 3 additional cases exist for how the origin tile is connected to the remainder of the seed: 1) the origin tile is connected to the remainder of the seed by its north edge only, 2) the origin tile is connected to the remainder of the seed by its east edge only, 3) the origin tile is connected along both its north and east edges.\nCases 0), 1), 2) and 3) are illustrated as $\\sigma$, N, E and N+E in Figure~\\ref{fig:S4-1_origin_macrotile}, respectively.\nThe new seed tile of our single-tile seed ($\\sigma_0$) is defined by the teal tiles in Figure~\\ref{fig:S4-1_origin_macrotile}; these are the tiles which encode the start of core path.\nThe remaining tiles of the spanning tree are represented by a set of five supertiles (adapted from \\cite{griffith2004growing}), shown in Figure~\\ref{fig:S4-2_seed_path_macrotiles}.\nThese five supertiles (and their rotations) allow for the core path to be connected to any supertile in the representation of a seed.\nThe tiles of the core path are assigned once all supertiles are assigned to the spanning tree.\nThe first tile of the core path is $\\sigma_0$, and to ensure that the core path is a dependence path tiles are added with unique strength $\\tau_4$ glues along adjacent edges.\nAdditionally, each tile in the core path assigns a unique strength $\\tau_4 - 1$ glue on the edge facing the exterior of the supertile; this glue is utilized in the creation of the precedence path.\n\n\n    \\begin{figure}\n        \\centering\n        \\includegraphics[width=0.65\\textwidth]{images\/S4-1_origin_macrotile.png}\n        \\caption{The four possible supertiles which allow for the new single-tile seed system to begin growth. The green arrows indicate the order of tile connections of the core path, and the blue arrows indicate the order of the perimeter path. The teal tile indicates the location of the new seed tile $\\sigma_0$, the yellow tile indicates the first tile placed by the perimeter path. The yellow tile is connected by a strength 2 glue to the final tile of the core path. The rightmost origin tile is the case of system being simulated already containing a single-tile seed.}\n        \\label{fig:S4-1_origin_macrotile}\n    \\end{figure}\n    \n    \\begin{figure}\n        \\centering\n        \\includegraphics[width=0.5\\textwidth]{images\/S4-2_seed_path_macrotiles.png}\n        \\caption{The five possible supertiles which allow for the core and perimeter path to be connected to each tile in the spanning tree of the seed. Green arrows indicate order in which the tiles of the core path are connected to each other. Blue arrows indicate the order in which the tiles of the perimeter path follow the core path. Note that these supertile templates may be rotated to connect adjacent supertiles as necessary.}\n        \\label{fig:S4-2_seed_path_macrotiles}\n    \\end{figure} %\n\nThe final process in the creation of the seed is creating the tiles of the \\emph{perimeter path}.\nThis is the set of tiles that allow for the encoding of the glues which are present on the exterior of the seed.\nThe first tile of the perimeter path is the yellow tile in the origin supertiles of Figure~\\ref{fig:S4-1_origin_macrotile}, and is connected via a unique strength $\\tau$ glue to the last tile of the core path (indicated by the vertical bar).\nThe remaining tiles of the perimeter path then `follow' the direction of the core path.\nFor perimeter path tiles which share an edge with tiles of the core path, a unique strength $\\tau_4 - 1$ glue is present along that edge for each supertile.\nA strength 1 glue of type $p$ is present along the the edge the perimeter tile shared with its predecessor in the perimeter path.\nCertain perimeter path tiles may not share an edge with a tile of the core path - this occurs in tiles of types d and e (on the corners) in Figure~\\ref{fig:S4-2_seed_path_macrotiles} when the perimeter path takes a 90 degree turn.\nIn this case, the dependence path tiles which share two edges with other dependence path tiles have one edge with a unique strength $\\tau$ glue connecting to the preceding tile in the dependence path.\nThe second edge contains a strength 1 glue of type $p$, allowing for the next tile in the dependence path to attach to the seed assembly using cooperative growth.\nOn the exterior-facing edges of all perimeter path tiles, the glues which represent the glues of the simulation of the tile being simulated by the supertile are added to the tiles of the perimeter path following the encoding and point of competition demonstrated in Figure~\\ref{fig:S4-8_scale_4_template}.\nThe glues exposed by the perimeter path function identically to that of the remaining scale 4 supertile representations, allowing for growth of fuzz into points of cooperation (the process by which will be outlined).\n\nA visualization of the process of seed tile generation is presented in Figure~\\ref{fig:S4-4_spanning_tree}, from the initial scale 1 seed to the creation of a perimeter path.\n\n    \nTo complete the creation of the full tile set, we must create a supertile representation of each tile type in $T$ that can attach outside of the assembly representing the seed $\\sigma$.\nAn interesting item to note here is that, for each tile type $t \\in T$ for which one or more tiles of type $t$ appear in the seed assembly $\\sigma$, there will be two logically different types of supertiles that represent $t$ in $\\calT_4$.\nOne will be that which is grown via the tiles of $T_\\sigma$ (and which contains the core and precedence paths), and one which is a ``standalone'' supertile that represents a copy of a tile of type $t$ outside of the seed.\nThe result of the transformation to make the $T_{IO}$ (a.k.a. ``inward\/outward'') tile set is that it is never possible to regrow any portion of a version of a supertile that is specific to the seed in any location outside of that representing the seed, and to always instead allow supertiles of the standalone version for type $t$ to grow in those locations.\n\n\nWe generate this tile set in the manner similar to that outlined in \\cite{Versus} in the sections ``minimal glue sets'' and ``inward-outward glues'' (and note that this technique is now relatively common in tile assembly results).\n$T_{IO}$ is an expansion of the tiles of $T$ so that for each $t \\in T$, we make a new tile for every subset of glues that share glue labels with the seed whose strengths sum to ``just barely'' $\\tau_4$.\nEach glue in $T$ is expanded to have N, E, S, W variants corresponding to the the direction they are pointing, also represented by the unit vectors $\\{(0,1),\\: (1,0),\\:(0,-1),\\:(-1,0)\\}$.\nFor example, a glue of type $x$ would be represented by types $x_N,\\: x_E,\\: x_S,\\: x_W$.\nWe say a glue is pointing \\emph{inwards} if the the glue type has a direction opposite of the edge which it resides (e.g. $x_N$ on a south edge of a tile or $x_E$ on a west edge).\nAlternatively, a glue is pointing \\emph{outwards} if the glue type matches the edge which it resides (e.g., $x_N$ on a north edge of a tile or $x_E$ on an east edge).\nFor each tile $t \\in T$, we consider every combination of that tile's glues.\nWe define a \\emph{minimal glue set S} as a set of glues such that, if any glue is removed from S, the set falls below combined strength $\\tau_4$.\nFor each minimal glue set, we generate a tile where the glue labels of the minimal set are such that they are pointing inwards (i.e. they act as ``input'' glues), and the remaining glue labels are pointing outwards (i.e. they act as ``output'' glues).\nFinally, the only tiles of $T_\\sigma$ which are modified are perimeter path tiles which represent exterior-facing glues.\nThese perimeter path tiles are modified such that their glues labels are modified to be outward facing.\nThis transformation ensures the invariant that any tile which is designed to be part of a supertile that represents a tile in $\\sigma$ appears only in the assembly mapping to $\\sigma$, and prevents incorrect regrowth of any portion of the seed outside of that, since the seed supertiles are designed to only grow a full copy of the seed, rather than allowing individual supertiles to grow independently.\n\n\n\n\n\n\n\n\nNext, we create the tiles to form the scale four supertiles representing each tile generated by $T_{IO}$.\nWe begin by defining a point of cooperation and\/or competition in each supertile which allows for supertiles of $T_{IO}$ to be mapped from assemblies of $\\calT_4$ to $\\calT$ by $R^*$.\nFigure~\\ref{fig:S4-8_scale_4_template} shows this cooperation point with the tile labeled `C'.\nGiven this point of cooperation, for each $t \\in T_{IO}$ we provide a single tile with the minimum glue set of inward glue labels at the prescribed strength.\nWhichever tile type attaches in location C of a supertile fully determines the tile type in $T$ that the supertile maps to under $R$.\nEach neighboring supertile that has grown to represent a tile in $T$ grows a path toward the C locations in the neighboring supertiles that haven't yet filled in.\nFor those whose edges represent adjacent glues of strength $\\tau$, this path, if unblocked, can grow into the neighboring supertile's C location without cooperation to determine the type of tile that that supertile will represent all by itself, which is by design since the $\\tau$-strength glue being represented would be able to cause the neighboring tile to bind all by itself.\nIn this case, the location C acts as a point of ``competition'' where various growth paths may be competing to be the first to arrive and place a tile there.\nFor those that edges that represent glues of strength less than $\\tau$, upon reaching the C location in a neighboring supertile the path stops and exposes a glue of strength less than $\\tau_4$, which forces a tile placed in the C location to use cooperation if it is going to bind using that path.\nThis is analogous to the cooperation that would be required with the glue exposed by the tile which is represented in $\\calT$, and in this case location C acts as a point of ``cooperation''.\nAs soon as the C location of a supertile receives a tile, that supertile represents the corresponding tile in $T$, and the output sides of the tile in location C each have $\\tau_4$ strength glues which match with a set of 3 tiles that grow towards the cooperation point of an adjacent supertile.\nWe note that these paths are legal fuzz, since the supertile with the C location filled in now maps to a tile and those locations are thus (non-diagonally) adjacent to at least one mapping supertile.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.3\\textwidth]{images\/S4-6_perim_path_collision.png}\n    \\caption{A possible case of the perimeter path growth around the supertiles representing $\\sigma$ being blocked by growth of non-seed representing supertiles. The supertile represented by the fuchsia tiles, which itself is allowed to grow from the supertile represented by the yellow tiles, grows north to the point of cooperation. Further growth north of the fuchsia tiles is blocked by the presence of the core path, but the currently placed tiles of the perimeter path can no longer cooperate with the core path to allow for placement of additional perimeter path tiles. The next tile of the perimeter path is indicated by the grey tile with dashed edges - it cooperates with a strength 1 `p' glue provided the fuchsia tiles in the path along with the $\\tau_4 - 1$ strength glue provided by the core path tile to continue the growth of the perimeter path.\n   \n    }\n    \\label{fig:S4-6_perim_path_collision}\n\\end{figure}\n\nAt this point, we've described both portions of the tile set of $\\calT_4$: $T_\\sigma$ which contains the tiles that grow the supertiles representing the seed $\\sigma$, and $T_{IO}$ which contains the tiles that form the supertiles that represent all tiles not contained within the seed-representing assembly.\nWe've shown how the single seed tile in $\\calT_4$ initiates growth of first a core path that goes through the center of every supertile of the seed-representing assembly (providing the correctness of the ``seed-first'' portion of the simulation, since these supertiles can then map to the full seed assembly before allowing any growth outside of the seed).\nIt then initiates the growth of the perimeter path, which circles the entire assembly and effectively ``activates'' each supertile by placing glues on their perimeters that allow them to initiate growth away from the supertile.\nThe growth away from the supertile is handled by the tile types in $T_{IO}$, which are able to correctly grow the $4 \\times 4$ supertiles representing each tile type of $T$ by growing paths representing glues from neighboring supertiles to points of cooperation\/competition that correctly handle the selection of valid supertiles to grow into, which in turn then grow the necessary output glue paths to neighbors.\nOnly in cases where the tile represented by a supertile has a glue in direction $d$ that wasn't used in its initial binding (i.e. an ``output'' glue) does that supertile grow a path into the neighboring supertile.\nSince the supertile originating the output path maps to a tile in $T$ at this point, this represents legal fuzz, and since sides which do not have output glues do not grow such fuzz, cheating fuzz is never grown.\nFurthermore, the careful design of $T_{IO}$ and the exterior glues of the seed-representing structure using the ``inward\/outward'' glue conventions ensures that the tiles of $T_\\sigma$ never appear outside of the seed-representing supertiles.\n\nTo complete the proof of correctness of the seed-first-simulation by $\\calT_4$, the final case left to consider is that where a portion of the assembly representing the seed has completed its perimeter path, allowing growth to proceed away from the seed while the perimeter path hasn't yet completed for another portion of the seed-representing assembly.\nThis growth away from the seed could potentially eventually crash into a portion of the seed-representing assembly which hasn't yet completed growth of its perimeter path. Potentially this could obstruct the completion of the perimeter path and thus stall the completion of the seed structure.\nMore specifically, this may occur where the tiles attempting to grow an output path from a completed supertile into the location of a neighboring supertile (which happens to be one in the seed structure that hasn't yet completed its perimeter path), to the cooperation\/competition point of that supertile. This may `obstruct' the normal growth of the perimeter path (and is demonstrated in Figure~\\ref{fig:S4-6_perim_path_collision}).\nTo allow for the continued growth of the perimeter path, the tile of the output path growing into the seed supertile's location contains a strength 1 $p$ glue on exposed sides along the direction of the perimeter path. Thus in the case that growth of the perimeter path is `cut off'; east and west growing output tiles have a $p$ glue on their north and south edges, and north and south growing output tiles have a $p$ glue along their east and west edges.\nAdditionally, depending upon the direction from which the output path grows into the perimeter path, it may block the corners diagonal to the growth path.\nThe output path growth from each direction will place a tile in the diagonal corners of the supertiles, denoted by the locations highlighted in white in Figure~\\ref{fig:S4-8_scale_4_template}.\nIn this way, it is ensured that a perimeter path can always complete and that growth away from the seed cannot cause incorrect or incomplete growth.\nNote that in cases of these collisions, the tiles on both sides of the collision map to the correct tiles in $T$ under $R$, and these cases are representative of situations where there is blocking performed by a seed tile.\n\nWe investigate the overall tile complexity of $\\calT_4$.\nFor the tile complexity of $\\calT_4$, we define the function $C_4(s,t,g) : \\mathbb{N} \\rightarrow \\mathbb{N}$ where $s = |\\sigma|$, $t = |T|$, $g$ is the total number of unique glue\/strength combinations in $T$ and $\\sigma$.\nWe consider the worst possible upper bound for each of the items, and thus our tile complexity is a conservative estimate.\nThis function takes into account the number of tiles utilized by both $T_\\sigma$ and $T_{IO}$.\n\nFor $T_\\sigma$, each tile utilizes all $16s$ tiles to include the perimeter path and core path.\nWe then consider the fuzz required for each edge of a tile which contains an exterior facing glue.\nAt maximum, each tile in the seed can have 3 exterior facing glues; we take this as our upper bound.\nFor each edge with tiles will have up to 2 additional tiles required to grow to the point of cooperation from the perimeter.\nThis leads to an additional $3 \\times s \\times 2 = 6s$ tiles.\n\nLet us consider the tiles of $T_{IO}$.\nFor output glues, fuzz can be shared between different tiles; it is the tile at the point of cooperation\/competition which causes the output to be differentiated.\nGlue has an input and output variant, and we assume this is present for all 4 sides.\n4 tiles of fuzz are required per side to grow to adjacent points of cooperation\/competition, leading to $2 \\times 4 \\times 4 \\times g = 16g$.\nFinally, we consider the $T_{IO}$ expansion.\nFrom \\cite{Versus}, the size of the minimal glue set is at most 6 (4 choose 2) for a tile at maximum, thus we require at most 6 tiles for each tile in $t$ for our expansion.\nWe additionally include tiles of the seed in this count - leading to an additional $6t + 6s$ tiles.\n\nThus, the conservative estimate for the tile complexity is $C_4 = 16s + 6s + 16g + 6t +6s = 28s + 16g + 6t$; as such, $T_4 \\leq 28s + 16g + 6t$.\n\nThus, $\\calT_4$ correctly seed-first-simulates an arbitrary aTAM system $\\calT$ at scale factor 4 without making use of cheating fuzz using $T_4 \\leq 28s + 16g + 6t$ tiles, and so Theorem~\\ref{thm:scale4-sim} is proven.\n\\end{proof}\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:scale3-sim}}\\label{sec:scale3-sim-proof}\n\n\nRestatement of Theorem \\ref{thm:scale3-sim}:\nGiven an arbitrary aTAM system $\\mathcal{T} = (T,\\sigma,\\tau)$, there exists an aTAM system $\\mathcal{T}_3 = (T_3, \\sigma_0, \\tau_3= \\max(2, \\tau))$ which seed-first-simulates $\\mathcal{T}$ at scale factor 3 utilizing cheating fuzz, where $|\\sigma_0| = 1$ and $|T_3| \\leq 20s + 16g + 6t$ given that $s = |\\sigma|$, $t = |T|$, and $g$ is the total number of unique glue\/strength combinations in $T$ and $\\sigma$.\n\n\\begin{proof}\nWe prove Theorem~\\ref{thm:scale3-sim} by construction.\nLet $\\mathcal{T} = (T,\\sigma,\\tau)$ be an arbitrary aTAM system.  \nWe generate the tile system $\\mathcal{T}_3 = (T_3, \\sigma_0, \\tau_3= \\max(2, \\tau))$ taking as input tileset $T$, seed $\\sigma$ and temperature $\\tau$.\nThe first step is to generate the tileset $T_\\sigma \\subset T_3$ which allows for growth of the seed assembly.\nWe develop a template for all scale 3 supertiles which allows for the creation of tiles which allow for seed-first simulation, shown in Figure~\\ref{fig:S3-14_core_path}.\nWe show by induction that utilizing this template (without specifying exact tile locations at this point),  \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{images\/S3-14_core_path.png}\n    \\caption{The basic outline of a scale 3 supertile $t$. This general tile makes no assumption on the path which is utilized to represent the tiles of the dependence path within the supertile. The green edge represents the edges shared with the neighbor supertile $t_{in}$ which contains tiles of the dependence path with an index smaller than any tile in $t$ (i.e., assembled earlier than $t$). The red edges represent the edges shared with up to 3 neighbor supertiles $t_{out,1},\\:t_{out,2},\\:t_{out,3}$ such that there exists tiles of $t$ in the dependence path with indices smaller than any tile in $t_{out,1},\\:t_{out,2},\\:t_{out,3}$. Two edges are taken, as this allows for the dependence path to both extend and return through the edges shared with $t_{in}$. The three tiles on the right demonstrate the rotations of this tile.}\n    \\label{fig:S3-14_core_path}\n\\end{figure}\n\n\\begin{lemma}\\label{lem:S3-supertile-template-connectivity}\nGiven a finite scale 1 assembly, $\\alpha_1$, we can generate an arbitrary finite scale 3 assembly, $\\alpha_3$, such that a dependence path $l$ exists which visits each supertile and returns to the supertile which contains the tile.\n\\end{lemma}\n\\begin{proof}\nWe prove by induction.\nLet us consider the binding graph of $\\alpha_1$.\nThe base case is the westernmost tile of the southernmost row; we assign the supertile of Figure~\\ref{fig:S3-14_core_path} to this location.\nDue to the location of the tile, no other tile will be present along the location $t_{in}$; as such, this contains the minimal value of the dependence path.\nWe call this tile the `origin' tile.\nFor this tile, we investigate the tile locations adjacent to the origin tile in $\\alpha_1$.\nFor each location of $t_{out,i}$ which contains a tile where $i \\in \\{1,2,3\\}$, we rotate the supertile of Figure~\\ref{fig:S3-14_core_path} as necessary such that $t_{in}$ matches up with the edge of $t_{out,i}$\nOnce all supertiles adjacent to the origin (which is at most 2) have been added, we select the an arbitrary supertile which has neighbor tiles adjacent to it in $\\alpha_1$ which are not part of $\\alpha_3$. \nThis repeats until all locations of $\\alpha_1$ have an associated supertile in $\\alpha_3$.\nAt termination, the resulting $\\alpha_3$ then contains a supertile in each location of $\\alpha_1$ with a dependence path $l$ which visits each supertile and begins and ends the origin.  \n\\end{proof}\n\nWith Lemma~\\ref{lem:S3-supertile-template-connectivity} we have a template which to define our supertiles, which will guarantee a dependence path through an assembly $\\alpha_3$.\nIn addition, we define the single tiles which provide input and output to the system by specific edges of Figure~\\ref{fig:S3-14_core_path}.\nThe light green edge shares a glue from the precedent path of the prior supertile adjacent to the current supertile.\nLight red edges are those where the precedent path exits the current tile.\nDark red edges are those where the precedent path re-enters the current tile.\nFinally, the dark green edge is that which the precedent path returns to the prior supertile.\nAs such, we then are restricted for the precedent path traveling between specific edges; tiles adjacent to light green can only have a precedent path which leads to a tile adjacent to dark green or light red.\nTiles adjacent to light red can only have a precedent path which leads to a tile adjacent to dark red or dark green.\n\n\nUsing the prior restrictions, the process of seed tile creation begins by defining the origin supertile.\nAs in Lemma~\\ref{lem:S3-supertile-template-connectivity}, we define the origin supertile as the westernmost tile in the southernmost row of the seed.\nIf the system $\\calT$ already contains a single-tile seed, we utilize a specific origin tile; this is case 0).\nDue to the location of the origin supertile, only 3 possible cases exist for how the origin tile is connected to the remainder of the seed: 1) the origin tile is connected to the remainder of the seed by its north edge only, 2) the origin tile is connected to the remainder of the seed by its east edge only, 3) the origin tile is connected along both its north and east edges.\nCases 0), 1), 2) and 3) are illustrated as $\\sigma$, N, E and N+E in Figure~\\ref{fig:S3-8_origin_macrotile}, respectively.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{images\/S3-8_origin_macrotile.png}\n    \\caption{The 4 possible origin supertiles of westernmost tile of the southernmost row of the seed. `N+E' is the case where the origin supertile is connected to two neighbors along both its North and East edges, whereas `N' and `E' are connected to only one neighbor along their North or East edge, respectively. $\\sigma$ is utilized in the case that the system being simulated is already singly seeded. The teal tile represents the location of the new single tile seed. Green bars indicate location of the final tile of the precedent path. The yellow tile represents the location of the first tile of the precedent path.}\n    \\label{fig:S3-8_origin_macrotile}\n\\end{figure}\n\nAs compared to scale 4, the number of general supertiles is increased in scale 3.\nSimply applying rotation to a single case of a supertile with 3 neighbors (similar to supertile b in Figure~\\ref{fig:S4-2_seed_path_macrotiles}) would result in the dependence path unable to be connected between neighbor tiles.\nAn additional difference is that instead of the dependence path taking the the clockwise most neighbor, it utilizes the counter-clockwise most neighbor.\nThis results in a total of 8 supertiles, as shown in Figure~\\ref{fig:S3-2_seed_path_macrotiles}.\nWe note that these \nThese tiles have an additional invariant - any possible perimeter path which utilizes fuzz outside the borders of the scale 3 supertile does not utilize diagonal fuzz.\nWhile each of the tile types may need cheating fuzz, we must guarantee that fuzz is not diagonal.\nWe note that diagonal fuzz is only reachable from a subset of the supertiles in Figure~\\ref{fig:S3-2_seed_path_macrotiles} which may be present on convex corners of $\\sigma$: these are a, c and d.\nThe perimeter path for this scale 3 construction also follow the dependence path, similar to scale 4.\nBy inspection and comparing the locations of diagonal fuzz shown as red locations in Figure~\\ref{fig:S3-10_cheating_fuzz_locations}, no locations which contain cheating fuzz overlap with diagonal fuzz.\nAny fuzz which may be required by the perimeter path to be in a diagonal location is present on corners where two tiles are adjacent - as such, this is a location of acceptable cheating fuzz.\n\n\\begin{lemma\nThe supertiles of Figure~\\ref{fig:S3-2_seed_path_macrotiles} generate a dependence path which visits each supertile and returns to the supertile which contains the origin. \n\\end{lemma}\n\\begin{proof}\nThe tiles of Figure~\\ref{fig:S3-2_seed_path_macrotiles} obey the edge restrictions of the template scale 3 tile; as such, Lemma~\\ref{lem:S3-supertile-template-connectivity} follows for these tiles.\n\\end{proof}\n \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.85\\textwidth]{images\/S3-2_seed_path_macrotiles.png}\n    \\caption{The supertile types which allow for the dependence path to follow the depth-first spanning tree depending upon the side of the origin tile which the tile can be connected through. Note, we can rotate these to allow for all possible path through the seed assembly. Blue dashed boxes indicate the locations of cheating fuzz which may be needed for the perimeter path.}\n    \\label{fig:S3-2_seed_path_macrotiles}\n\\end{figure}\n\nThe application of general set of scale 3 supertiles to the spanning tree of an arbitrary seed has the potential to isolate cavities from the perimeter path.\nFor full seed-first-simulation (and thus, also shape-simulation) it may be required for tiles to attach to glues inside of cavities.\nFigure~\\ref{fig:S3-6_cavity_enclosure} demonstrates a case where a seed which is assigned supertiles from Figure~\\ref{fig:S3-2_seed_path_macrotiles} and a cavity is prevented from being accessed by the perimeter path.\nTo resolve this issue, we require two solutions: 1) developing a set of tiles which we guarantee allow for the perimeter path to access a cavity regardless of its location, and 2) correctly placing the tiles in the seed.\nAt a high level, we choose the northernmost corner of the westernmost edge of the cavity and draw a line in the $+y$ direction until it reaches either another cavity or the perimeter of the seed.\nAll the edges which this line passes through we remove.\nWe demonstrate his can be done safely in the case of \\emph{cyclic cavities} - cavities for which there exists a cycle between the vertices of the binding graph of the seed which contain only the vertices within a Chebyshev distance of 1 from any location within the cavity.\nIn the case of \\emph{non-cyclic cavities}, cavities for which there does not exist a cycle between the vertices of the binding graph of the seed which contain only the vertices within a Chebyshev distance of 1 from any location within the cavity, we demonstrate that they can be recursively combined until either they form a cyclic cavity or they are connected to the perimeter path.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{images\/S3-6_cavity_enclosure.png}\n    \\caption{A valid assignment of supertiles which leads to cavities which are not reachable by the perimeter path. The red line indicate the cavities which are connected together but unable to be accessed from the perimeter path on the exterior of the seed. Due to the orientations of tiles, the dependence paths are adjacent to one another and block any possible perimeter path from reaching the internal cavities.}\n    \\label{fig:S3-6_cavity_enclosure}\n\\end{figure}\n\nWe first provide a description of cyclic and non-cyclic cavities, with the smallest possible cyclic cavity (a single tile missing from the center of a $3 \\times 3$ square) demonstrated in Figure~\\ref{fig:S3-4_cavity_types}.\nWe define \\emph{regions} as (potentially infinite) connected subsets of $\\mathbb{Z}^2$.\nThe regions split into 3 categories - cavities, the assembly, and \\emph{free space} (the remaining plane).\nWhen taking the complement of the domain of an assembly, there exists the infinite connected subset of $\\mathbb{Z}^2$ which is the free space.\nAdditionally, for shapes which contain cavities (sometimes called holes in the literature), we will find finite connected subsets - these are the cavities within an assembly.\nGiven that a cycle is formed around cyclic cavities, we can safely remove a single edge from the cycle without causing a tile in that cycle to become disconnected.\nNon-cyclic corners either separate two cavities in scale 1, or separate a cavity from the exterior of the shape.\nFigure~\\ref{fig:S3-5_non_cyclic_corners} demonstrates that in the worst case, two tiles which are at the non-cyclic corners of a scale 3 supertile will have a 1 tile wide path.\nThis path allows for tiles to be generated which connect the two regions - as such, we then can assign a new cavity from the union of the two connected subsets of $\\mathbb{Z}^2$.\nEither the two regions are a both non-cyclic cavities and can be combined, or the non-cyclic cavity is connected to the perimeter (and thus, free space).\nIn the latter case that the non-cyclic cavity is connected to the perimeter, we are done; no further modifications need to be made.\nIn the former case, the two non-cyclic cavities are joined and can the resulting cavity may remain a non-cyclic cavity.\nWe continue the process of joining the non-cyclic cavity together with its neighbor region.\nOtherwise, the two non-cyclic regions may become a cyclic region themselves, at which point we halt.\nOnce all non-cyclic cavities have been combined and are either part of cyclic cavities or connected to the perimeter, the binding graph can then be modified.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{images\/S3-4_cavity_types.png}\n    \\caption{The two types of cavities possible in assemblies.}\n    \\label{fig:S3-4_cavity_types}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{images\/S3-5_non_cyclic_corners.png}\n    \\caption{For a non-cyclic cavity, we demonstrate how even in the worst case of corner tiles connected to two neighbors themselves in a non-cyclic corner that a 1 tile wide path exists between the non-cyclic cavity and the adjacent space. This applies to all possible rotations of non-cyclic cavities.}\n    \\label{fig:S3-5_non_cyclic_corners}\n\\end{figure}\n\nAfter the prior steps have finished, all cavities are now cyclic cavities.\nThis provides us with the property that a single edge can be removed from all cavities, and the vertices which are contained in the cycle remain connected.\nFor all remaining cyclic cavities, we remove the edge between the vertices surrounding the cyclic cavity adjacent to the northern most vertex of the western most column of the cavity.\nFrom this vertex, we can then define the two vertices which will have their edge removed; for a northwest vertex location of $(i,j)$ we add the vectors $(-1,1)$ and $(0,1)$ to identify the two vertices to separate.\nThis provides us with the tuple $((x_0, y_0), (x_1, y_1))$ which corresponds to the coordinates of the two vertices.\nOf the two vertices, we take $x_0$ as the $x$ coordinate of the two vertices with the smallest value (i.e., the western-most vertex).\nWe then increment $y_0$ and $y_1$ by 1 successively, removing all edges with the coordinates until either vertex is in the location of another cavity or free space.\nWhen edges are removed from the binding graph such that two cavities are connected, similar to the combination of non-cyclic cavities we combine the two cavities.\nWe track the values of the vertices defined by $(x_0,y_0)$ and store these values for modifying the supertiles in these locations, we additionally include the vertex at $(-1+i, j)$; this guarantees the 1-tile wide path will reach into the cavity.\nWe call this tracked set of vertices $M$.\nThis process is repeated for all cavities until the perimeter is connected to all cavities.\nUpon connecting all cavities to the perimeter, we then assign supertiles of Figure~\\ref{fig:S3-2_seed_path_macrotiles} to the modified binding graph demonstrated in Figure~\\ref{fig:S3-11_combining_cavities}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.8\\textwidth]{images\/S3-11_combining_cavities.png}\n    \\caption{(left) The initial binding graph of a seed which contain cavities. We note that cavities (B) and (C) are, by themselves, non-cyclic cavities. These can be combined to form a larger cyclic cavity. (middle) After being combined, from the northernmost corner of the westernmost edge of each cavity we remove edges of the connection graph along the yellow line, guaranteeing access of the perimeter path to each cavity. We note that both cyclic cavities (B)(C) and (D) are connected to the cavity of (A), which is then connected to the perimeter. (right) A spanning tree is formed from the assignment of the scale 3 supertiles in Figure~\\ref{fig:S3-2_seed_path_macrotiles}.}\n    \\label{fig:S3-11_combining_cavities}    \n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{images\/S3-12_scale_3_example.png}\n    \\caption{The seed resulting from the addition of supertile templates assigned to each vertex of the modified binding graph from Figure~\\ref{fig:S3-11_combining_cavities}. The teal square represents the new single-tile seed for $\\calT_3$. Green squares represent tiles of the dependence path, with the black arrows demonstrating the order of tile addition of the dependence path. Blue dashed squares represent locations of cheating fuzz available for the perimeter path. The blue arrow indicates the growth of the tiles of the perimeter path, barring a collision occurring.}\n    \\label{fig:S3-12_scale_3_example}\n   \n\\end{figure}\n\nWith the modifications of the binding graph, we additionally must assign new supertiles which allow for the perimeter path to take advantage of the new binding graph structure.\nThe deletion of the edges from each cyclic cavity provides opportunity for a 1 tile wide path to exist in two manners.\nEither the dependence path exists in a north-south direction, following the removed edges, or the dependence path enters from the west and terminates at a vertex which was adjacent to the deleted edges.\nIn the latter case, tile a from Figure~\\ref{fig:S3-2_seed_path_macrotiles} is utilized in the presented rotation.\nThis supertile in its presented state allows for a 1 tile wide path.\nIf the dependence path follows north-south growth, additional tiles must be utilized which allow for the 1 tile wide path.\nWe note that for proper dependence path alignment, five new variants of tiles are defined.\nThe three left supertiles in Figure~\\ref{fig:S3-13_n_to_e_conversion_tiles} allow for dependence path utilized by the supertiles in Figure~\\ref{fig:S3-2_seed_path_macrotiles} to be shifted; in turn, this facilitates the usage of the right two supertiles which shift the dependence path to the left 2\/3 tiles and leaving the necessary 1 tile wide gap for the perimeter path.\nTiles from each set of vertices which describe the 1 tile wide path, $M$, are replaced by the supertiles in Figure~\\ref{fig:S3-13_n_to_e_conversion_tiles}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{images\/S3-13_n_to_e_conversion_tiles.png}\n    \\caption{The five cavity connector supertiles which allow for the perimeter path to reach all cavities. The left three tile types allow for the dependence path of the cavity connector tiles to be joined to the dependence path of the remaining seed. The right two tiles shift the dependence path to the west side of the supertiles, guaranteeing that a 1 tile wide path exists to the cavity in question. We note that the tile `a' from Figure~\\ref{fig:S3-2_seed_path_macrotiles} may also be utilized in the cavity connector as shown; it also provides a 1 tile wide path for the perimeter path to utilize.}\n    \\label{fig:S3-13_n_to_e_conversion_tiles}\n\\end{figure}\n\nThe final remaining task for the creation of the seed tiles is assigning the tiles of the dependence path and perimeter path.\nAs with the scale 4 construction, the dependence path tiles are assigned according to the supertile templates and are assigned unique strength $\\tau_3$ glues shared between adjacent tiles of the dependence path.\nWhen the dependence path terminates at the origin supertile, another unique strength $\\tau_3$ glue is assigned to the edge between the final tile of the dependence path and the first tile of the perimeter path (yellow tile, Figure~\\ref{fig:S3-8_origin_macrotile})\nThe perimeter path tiles begin by ``following'' the outermost tiles of the dependence path in a clockwise fashion.\nThe core tiles contain a unique strength $\\tau_3 - 1$ glue which is shared with the perimeter path tile adjacent to it, and tiles in the perimeter path exposes a strength 1 glue of type $p$ along the the edge the perimeter tile shared with its predecessor in the perimeter path.\nAs in the scale 4 simulation, strength $\\tau_3$ glues are required for attachment of perimeter path tiles which carry out 90 degree turns.\nA key difference between the perimeter paths of scale 3 versus scale 4 is that scale 3 perimeter paths may be required to branch; in particular, this occurs when the perimeter path may need to access cavities which have only a width-1.\nThis requires that perimeter path tiles may require having two outwards facing $p$ glues; these are denoted by the circle with the surrounding.\nAdditionally, not all tiles of of the supertile may be immediately defined by either the perimeter path or the dependence path.\nThis may require small `offshoots' from the dependence path to provide a path to follow; we can see an example of this in Figure~\\ref{fig:S3-12_scale_3_example} in the tile type a which is rotated 90 degrees clockwise.\nWe note that in cases like these, they manifest when type a supertiles from Figure~\\ref{fig:S3-2_seed_path_macrotiles} are leafs of a spanning tree adjacent to the perimeter path.\n\n\nFor the remaining simulations of $3 \\times 3$ tiles, we define the point of cooperativity\/competition as the central tile in a $3 \\times 3$ square, as shown in Figure~\\ref{fig:S3-7_cooperativity_scale3}.\nThe process of developing $T_{IO}$ and the modification of $T_\\sigma$ to prevent regrowth of the dependence path follows directly from the scale 4 example.\nIn the case of tiles crashing into the perimeter path, the fuzz growing from the point of cooperation\/competition must be able take additional locations, as the perimeter path is guaranteed to be located in any particular location due to the use of cheating fuzz.\nThese locations have been highlighted in white - tiles which grow into these locations have $p$ glues facing towards the interior of the supertile which the fuzz is growing into.\n\nWe investigate the overall tile complexity of $\\calT_3$.\nFor the tile complexity of $\\calT_3$, we define the function $C_3(s,t,g) : \\mathbb{N} \\rightarrow \\mathbb{N}$ where $s = |\\sigma|$, $t = |T|$ and $g$ is the total number of unique glue\/strength combinations in $T$ and $\\sigma$.\nWe consider the worst possible upper bound for each of the items, and thus our tile complexity is a conservative estimate.\nThis function takes into account the number of tiles utilized by both $T_\\sigma$ and $T_{IO}$.\n\nFor $T_\\sigma$, there exists a variable number of tiles, depending upon the perimeter path taken.\nFigure~\\ref{fig:S3-2_seed_path_macrotiles} c requires at most 14 tiles for both perimeter path and dependence path - we take that as our high value; $14s$.\nWe then consider the fuzz required for each edge of a tile which contains an exterior facing glue.\nBased upon our assumption high value, the fuzz is already present and reaches the cooperation\/competition point.\nComparing with the all other tiles, the most possible fuzz required (Figure~\\ref{fig:S3-2_seed_path_macrotiles} a) also requires 14 tiles.\nThus, we have $14s$ tiles required at most for the seed.\n\nLet us consider the tiles of $T_{IO}$.\nFor output glues, fuzz can be shared between different tiles; it is the tile at the point of cooperation\/competition which causes the output to be differentiated.\nGlue has an input and output variant, and we assume this is present for all 4 sides.\n4 tiles of fuzz are required per side to grow to adjacent points of cooperation\/competition, leading to $2 \\times 4 \\times 4 \\times g = 16g$.\nFinally, we consider the $T_{IO}$ expansion.\nFrom \\cite{Versus}, the size of the minimal glue set is at most 6 (4 choose 2) for a tile at maximum, thus we require at most 6 tiles for each tile in $t$ for our expansion.\nWe additionally include tiles of the seed in this count - leading to an additional $6t + 6s$ tiles.\n\nThe conservative estimate for the tile complexity is $C_3 = 14s + 16g + 6t + 6s = 20s + 16g + 6t$, and it follows that $|T_3| \\leq 20s + 16g + 6t$.\n\nThus, $\\calT_3$ correctly seed-first-simulates an arbitrary aTAM system $\\calT$ at scale factor 3 requiring the use of cheating fuzz using at most $|T_3| \\leq 20s + 16g + 6t$ tile types, and so Theorem~\\ref{thm:scale3-sim} is proven.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\input{scaled-shape-simulation-appendix}\n\\section{Definitions}\\label{sec:definitions}\nIn this section we provide definitions of the aTAM and also related to the simulation of one tile assembly system by another.\n\n\n\\subsection{The abstract Tile Assembly Model}\n\\label{sec:tam-informal}\n\nThis section gives a brief informal sketch of the abstract Tile Assembly Model (aTAM) \\cite{Winf98} and uses notation from \\cite{RotWin00} and \\cite{jSSADST}. For more formal definitions and additional notation, see \\cite{RotWin00} and \\cite{jSSADST}.\n\nA \\emph{tile type} is a unit square with four sides, each consisting of a \\emph{glue label} which is often represented as a finite string.\nAn aTAM system has a finite set $T$ of tile types, but an infinite number of copies of each tile type, with each copy being referred to as a \\emph{tile}.\nA \\emph{glue function} is a symmetric mapping from pairs of glue labels to a non-negative integer value which represents the strength of binding between those glues.\nAn \\emph{assembly}\nis a positioning of tiles on the integer lattice $\\Z^2$, described  formally as a partial function $\\alpha:\\Z^2 \\dashrightarrow T$.\nLet $\\mathcal{A}^T$ denote the set of all assemblies of tiles from $T$, and let $\\mathcal{A}^T_{< \\infty}$ denote the set of finite assemblies of tiles from $T$.\nWe write $\\alpha \\sqsubseteq \\beta$ to denote that $\\alpha$ is a \\emph{subassembly} of $\\beta$, which means that $\\dom\\alpha \\subseteq \\dom\\beta$ and $\\alpha(p)=\\beta(p)$ for all points $p\\in\\dom\\alpha$.\nWe write $\\alpha \\setminus \\beta$ to denote the assembly $\\alpha$ without any of the tiles in locations in $\\beta$, i.e. the result of starting with $\\alpha$ and removing tiles from any locations which are in both $\\alpha$ and $\\beta$.\nTwo adjacent tiles in an assembly \\emph{interact}, or are \\emph{attached}, if the glue labels on their abutting sides match.\nEach assembly induces a \\emph{binding graph}, a grid graph whose vertices are tiles, with an edge between two tiles if they interact.\nThe assembly is \\emph{$\\tau$-stable} if every cut of its binding graph has strength at least~$\\tau$, where the strength   of a cut is the sum of all of the individual glue strengths in the cut.\n\n\nA \\emph{tile assembly system} (TAS) is a 3-tuple $\\calT = (T,\\sigma,\\tau)$, where $T$ is a finite set of tile types, $\\sigma:\\Z^2 \\dashrightarrow T$ is a finite $\\tau$-stable \\emph{seed assembly},\nand $\\tau$ is the \\emph{temperature} parameter (a.k.a. \\emph{binding threshold}).\nGiven an assembly $\\alpha$, the \\emph{frontier}, $\\frontiert{\\alpha}$, is the set of locations to which tiles can $\\tau$-stably attach.\nAn assembly $\\alpha$ is \\emph{producible} if either $\\alpha = \\sigma$ or if $\\beta$ is a producible assembly and $\\alpha$ can be obtained from $\\beta$ by the stable binding of a single tile to a location in $\\frontiert{\\beta}$. In this case we write $\\beta\\to_1^\\calT \\alpha$ (to mean $\\alpha$ is producible from $\\beta$ by the attachment of one tile), and we write $\\beta\\to^\\calT \\alpha$ if $\\beta \\to_1^{\\calT*} \\alpha$ (to mean $\\alpha$ is producible from $\\beta$ by the attachment of zero or more tiles).\nAn \\emph{assembly sequence} in a TAS $\\calT$ is a (finite or infinite) sequence $\\vec{\\alpha} = (\\alpha_0,\\alpha_1,...)$ of assemblies in which each $\\alpha_{i+1}$ is obtained from $\\alpha_i$ by the addition of one tile, i.e. $\\alpha_i \\to_1^{\\calT} \\alpha_{i+1}$.\n\nWe let $\\prodasm{\\calT}$ denote the set of producible assemblies of $\\calT$.\nAn assembly $\\alpha$ is \\emph{terminal} if no tile can be $\\tau$-stably attached to it, i.e. $|\\frontiert{\\alpha}| = 0$.\nWe let   $\\termasm{\\calT} \\subseteq \\prodasm{\\calT}$ denote  the set of producible, terminal assemblies of $\\calT$.\nA TAS $\\calT$ is \\emph{directed} if $|\\termasm{\\calT}| = 1$. Hence, although a directed system may be nondeterministic in terms of the order of tile placements, it is deterministic in the sense that exactly one terminal assembly can be produced\n\n\nWe define the \\emph{producible shapes} of an aTAM system $\\calT$, denoted $\\proddom{T}$, as the set of shapes (i.e. domains) of all producible assemblies of $\\calT$. More formally, $\\proddom{T} = \\{\\dom \\alpha \\mid \\alpha \\in \\prodasm{T}\\}$. Similarly, we define the \\emph{terminal shapes} of an aTAM system $\\calT$, denoted $\\termdom{T}$, as the set of shapes (i.e. domains) of all terminal assemblies of $\\calT$. More formally, $\\termdom{T} = \\{\\dom \\alpha \\mid \\alpha \\in \\termasm{T}\\}$. We say that aTAM system $\\calT$ \\emph{self-assembles shape $S$} if and only if $|\\termdom{T}| = 1$ and $S \\in \\termdom{T}$, that is, $\\calT$ produces terminal assemblies of only a single shape, which is $S$.\n\n\\begin{definition}[TAS equivalence]\\label{def:tas-equiv}\nGiven two aTAM systems $\\calT$ and $\\mathcal{S}$, and assembly $\\gamma$, we say that $\\calT$ and $\\mathcal{S}$ are \\emph{equivalent modulo $\\gamma$} if and only if for every producible assembly $\\alpha \\in \\prodasm{\\calT}$ there exists a producible assembly $\\beta \\in \\prodasm{\\mathcal{S}}$ such that $(\\alpha \\setminus \\gamma) = (\\beta \\setminus \\gamma)$, and vice versa (i.e. for every producible assembly $\\beta \\in \\prodasm{\\mathcal{S}}$ there exists a producible assembly $\\alpha \\in \\prodasm{\\mathcal{T}}$ such that $(\\beta \\setminus \\gamma) = (\\alpha \\setminus \\gamma)$).\n\\end{definition}\n\nNote that the notion of equivalence between aTAM systems is quite strict, requiring all of the same tile types to be used outside of the region of $\\gamma$, and not allowing for any scaling factor. For more general notions of equivalence between systems, see Section \\ref{sec:simulation_def}.\n\n\\begin{definition}[Strict dependence]\\label{def:strict-dep}\nGiven an aTAM system $\\calT$ and sets of locations $L_1,L_2 \\subset \\mathbb{Z}^2$, we say that $L_2$ \\emph{strictly depends upon} $L_1$ if, for every valid assembly sequence in $\\calT$, if a tile is placed in some location in $L_2$, a tile was previously placed in some location in $L_1$.\n\\end{definition}\n\nWe define a \\emph{path} $p$ as an ordered list of distinct locations in $\\mathbb{Z}^2$, and refer to the $i$th location in $p$ as $p[i]$, so that each location $p[i]$, for $0 \\le i < |p|-1$, is adjacent to $p[i+1]$.\n\n\\begin{definition}[Dependence path]\\label{def:dep-path}\nGiven an aTAM system $\\calT$, a producible assembly $\\alpha \\in \\prodasm{\\calT}$, assembly sequence $\\vec{\\alpha}$ which produces $\\alpha$, and locations $l_1,l_2 \\in \\dom{\\alpha}$, we say that there is a \\emph{dependence path from $l_1$ to $l_2$} if and only if there exists a path $p$ from $1_1$ to $l_2$ in $\\dom{\\alpha}$ such that, in $\\vec{\\alpha}$, for each location $p[i]$, for $0 < i < |p|-1$, (1) the tile at location $p[i]$ was placed before the tile in $p[i+1]$, and (2) the tile at location $p[i+1]$ was required to form a bond with the tile at location $p[i]$ in order to attach.\n\\end{definition}\n\nIntuitively, a dependence path from $l_1$ to $l_2$ is a path formed by sequential tile attachments that leads from $l_1$ to $l_2$. Note that, by definition, a dependence path is directional and therefore a dependence path from $l_1$ to $l_2$ cannot be the same path as a dependence path from $l_2$ to $l_1$.\n\n\\begin{lemma}[Dependence paths]\\label{lem:depend-path}\nGiven an aTAM system $\\calT$ whose seed consists of a single tile, a producible assembly $\\alpha \\in \\prodasm{\\calT}$, and sets of locations $L_1,L_2 \\subset \\dom{\\alpha}$, if $L_2$ strictly depends upon $L_1$, then in each valid assembly sequence of $\\calT$ there must be a dependence path from some  $l_1 \\in L_1$ to some $l_2 \\in L_2$.\n\\end{lemma}\n\nThe proof of Lemma \\ref{lem:depend-path} can be found in Section \\ref{sec:depend-path-proof}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=6.0in]{images\/crashing-example.png}\n    \\caption{A basic example of blocking by a seed location. (Left) The seed tile, (Middle) Growth from the seed with tiles specific to each location, (Right) Growth back toward the seed using a single, repeating tile type. This row ``crashes'' into the seed (i.e. it would place a tile in the location of the seed tile if that tile weren't there), so the seed location blocks placement of a tile (of the type with $x$ glues on its east and west).}\n    \\label{fig:blocking}\n\\end{figure}\n\n\\begin{definition}[Blocking]\\label{def:blocking}\nGiven a TAS $\\calT$ and producible assembly $\\alpha \\in \\prodasm{\\calT}$, a tile $t$ at location $\\vec{l}$ in $\\alpha$ \\emph{blocks} a tile if there exists a valid assembly sequence which begins with $\\alpha$ and results in one or more tiles adjacent to $t$ which did not require glues of $t$ to bind to the assembly and to which a tile of a type different than $t$ could bind with $\\tau$ strength in location $\\vec{l}$ if $t$ was removed.\n\\end{definition}\n\nSee Figure \\ref{fig:blocking} for an example of blocking. Note that we still say blocking occurs even if the tile adjacent to $t$ strictly depends upon $t$, meaning it would technically be impossible to remove $t$ and replace it with a tile of another type.\n\n\n\\subsection{Simulation of tile assembly systems}\n\\label{sec:simulation_def}\n\nFirst, we give a very brief intuitive definition of what it means for one tile assembly system to simulate another, and then provide more technically detailed definitions related to simulation, especially as it relates to scale factors greater than 1. We define new notions of simulation that are necessary to allow, and capture, the dynamics of simulating systems that don't begin from seed assemblies that represent the full seed assemblies of the systems they are simulating, and thus they have to grow those missing portions which the standard definition of simulation assumes exist at the beginning of the simulation. For the technical definitions related to the standard model of simulation, see \\cite{DirectedNotIU}.\n\nIntuitively, simulation of a system $\\calT$ by a system $\\mathcal{S}$ requires that there is some scale factor $m \\in \\Z^+$ such that $m \\times m$ squares of tiles in $\\mathcal{S}$ represent individual tiles in $\\calT$, and there is a ``representation function'' capable of inspecting assemblies in $\\mathcal{S}$ and mapping them to assemblies in $\\calT$. A representation function $R$ takes as input an assembly in $\\mathcal{S}$ and returns an assembly in $\\calT$ to which it maps. In order for $\\mathcal{S}$ to correctly simulate $\\calT$, it must be the case that for any producible assembly $\\alpha \\in \\prodasm{\\calT}$ that there is a corresponding assembly $\\beta \\in \\prodasm{\\mathcal{S}}$ such that $R(\\beta) = \\alpha$. (Note that there may be more than one such $\\beta$.) Furthermore, for any $\\alpha' \\in \\prodasm{\\calT}$ which can result from a tile addition to $\\alpha$, there exists $\\beta' \\in \\prodasm{\\mathcal{S}}$, where $R(\\beta') = \\alpha'$, which can result from the addition of one or more tiles to $\\beta$, and conversely, $\\beta$ can only grow into assemblies which can be mapped into valid assemblies of $\\calT$ into which $\\alpha$ can grow.\n\nWe now present a formal, rigorous definition of what it means for one tile assembly system to ``simulate'' another. Our definitions are based on those of \\cite{temp1notIU}, but are adapted to account for the simulating system to grow a representation of the original system's seed rather than beginning from a seed assembly which already represents it. Also, note that a great amount of the complexity required for the definitions arises due to the possible dynamics of simulations with scale factors $> 1$, and that otherwise the mapping of assemblies and equivalence of production and dynamics are much more straightforward.\n\n\nFrom this point on, let $T$ be a tile set, and let $m\\in\\Z^+$.\nAn \\emph{$m$-block supertile} over $T$ is a partial function $\\alpha : \\Z_m^2 \\dashrightarrow T$, where $\\Z_m = \\{0,1,\\ldots,m-1\\}$.\nLet $B^T_m$ be the set of all $m$-block supertiles over $T$.\nThe $m$-block with no domain is said to be $\\emph{empty}$.\nFor a general assembly $\\alpha:\\Z^2 \\dashrightarrow T$ and $(x,y)\\in\\Z^2$, define $\\alpha^m_{x,y}$ to be the $m$-block supertile defined by $\\alpha^m_{x,y}(i_x, i_y) = \\alpha(mx+i_x, my+i_y)$ for $0 \\leq i_x,i_y< m$.\nFor some tile set $S$, a partial function $R: B^{S}_m \\dashrightarrow T$ is said to be a \\emph{valid $m$-block supertile representation} from $S$ to $T$ if for any $\\alpha,\\beta \\in B^{S}_m$ such that $\\alpha \\sqsubseteq \\beta$ and $\\alpha \\in \\dom R$, then $R(\\alpha) = R(\\beta)$.\nNote that we use the term \\emph{macrotile} interchangeably with supertile, to mean the same thing.\n\nFor a given valid $m$-block supertile representation function $R$ from tile set~$S$ to tile set $T$, define the \\emph{assembly representation function}\\footnote{Note that $R^*$ is a total function since every assembly of $S$ represents \\emph{some} assembly of~$T$; the functions $R$ and $\\alpha$ are partial to allow undefined points to represent empty space.}  $R^*: \\mathcal{A}^{S} \\rightarrow \\mathcal{A}^T$ such that $R^*(\\alpha') = \\alpha$ if and only if $\\alpha(x,y) = R\\left(\\alpha'^m_{x,y}\\right)$ for all $(x,y) \\in \\Z^2$.\nFor an assembly $\\alpha' \\in \\mathcal{A}^{S}$ such that $R(\\alpha') = \\alpha$, $\\alpha'$ is said to map \\emph{cleanly} to $\\alpha \\in \\mathcal{A}^T$ under $R^*$ if for all non empty blocks $\\alpha'^m_{x,y}$, $(x,y)+(u_x,u_y) \\in \\dom \\alpha$ for some $u_x,u_y \\in \\{-1,0,1\\}$ such that $u_x^2 + u_y^2 \\leq 1$. In other words, $\\alpha'$ may have tiles on supertile blocks representing empty space in $\\alpha$, but only if that position is adjacent to a tile in $\\alpha$.  We call such growth ``around the edges'' of $\\alpha'$ \\emph{fuzz} and thus restrict it to be adjacent to only valid supertiles, but not diagonally adjacent (i.e.\\ we do not permit \\emph{diagonal fuzz}). Additionally, if tiles grow as fuzz into a supertile region which maps to a location in the simulated system in which there is never an incident glue (other than the null glue) from any adjacent tile, we call this \\emph{cheating fuzz}. The justification for this distinction is that the goal of an intrinsic simulation is for the simulator to only utilize the dedicated macrotile space to simulate a tile or to compute if a tile may grow into a location, but cheating fuzz would be tile growth into a macrotile location that never has any input glues and therefore no chance of ever becoming a tile. See Figure \\ref{fig:S3-10_cheating_fuzz_locations} for an example.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.3\\textwidth]{images\/S3-10_cheating_fuzz_locations.png}\n    \\caption{When simulating an aTAM tile `A' with glues on its north and east faces and null glues on south and west faces, although a simulation is allowed to grow fuzz into each of the adjacent north, east, south, and west marcortile locations, we further restrict the locations for ``legal fuzz'' to grow only in the adjacent macrotiles to its north and east, indicated by blue hashed tiles. Locations for placement of ``cheating fuzz'', in adjacent directions in which no glue is presented by A, are indicated by the green hashed tiles.}\n    \\label{fig:S3-10_cheating_fuzz_locations}\n\\end{figure}\n\n\n\nIn the following definitions, let $\\mathcal{T} = \\left(T,\\sigma_T,\\tau_T\\right)$ be a tile assembly system, let $\\mathcal{S} = \\left(S,\\sigma_S,\\tau_S\\right)$ be a tile assembly system, and let $R$ be a valid $m$-block representation function $R:B^S_m \\rightarrow T$.\n\n\n\\begin{definition}\\label{def-shape-sim}\nWe say that $\\mathcal{S}$ \\emph{shape-simulates} $\\calT$ if the following conditions hold:\n\\begin{enumerate}\n        \\item $\\left\\{ \\dom R^*(\\alpha') | \\alpha' \\in \\termasm{\\mathcal{S}}\\right\\} = \\termdom{T}$\n        \\item For all $\\alpha'\\in \\termasm{\\mathcal{S}}$, $\\alpha'$ maps cleanly to $R^*(\\alpha')$.\n\\end{enumerate}\n\\end{definition}\n\nEssentially, for $\\mathcal{S}$ to shape-simulate $\\calT$, its terminal assemblies must map to the exact same set of domains as those of the terminal assemblies of $\\calT$.\n\n\\begin{definition}\n\\label{def-equiv-prod-mod} We say that $\\mathcal{S}$ and $\\mathcal{T}$ have \\emph{equivalent productions modulo $\\sigma_\\calT$} (under $R$), and we write $\\mathcal{S} \\Leftrightarrow^{\\sigma_\\calT}_R \\mathcal{T}$ if the following conditions hold:\n\\begin{enumerate}\n        \\item $\\left\\{R^*(\\alpha') | \\alpha' \\in \\prodasm{\\mathcal{S}}\\right\\} = \\prodasm{\\mathcal{T}} \\cup \\{ \\alpha \\mid \\alpha \\sqsubseteq \\sigma_\\calT\\}$.\n        \\item $\\left\\{R^*(\\alpha') | \\alpha' \\in \\termasm{\\mathcal{S}}\\right\\} = \\termasm{\\mathcal{T}}$.\n        \\item For all $\\alpha'\\in \\prodasm{\\mathcal{S}}$, $\\alpha'$ maps cleanly to $R^*(\\alpha')$.\n\\end{enumerate}\n\\end{definition}\n\nNote that the definition of equivalent productions modulo the seed assembly differs slightly from the definition of equivalent productions in other papers (e.g. \\cite{IUSA,DirectedNotIU,temp1notIU}) since we must allow the producible assemblies of the simulating system to represent subsets of the seed of the simulated system. This is because we are interested in being able to begin with a singly-seeded system that \\emph{first} grows a complete representation of the seed of the simulated system, and then continues with its simulation. Alternatively, the next definition can be used to define a notion of simulation in which growth can extend beyond the simulated seed before the full representation of the seed has completed.\n\n\\begin{definition}\n\\label{def-equiv-prod-minus} We say that $\\mathcal{S}$ and $\\mathcal{T}$ have \\emph{equivalent productions minus $\\sigma_\\calT$} (under $R$), and we write $\\mathcal{S} \\Leftrightarrow^{-\\sigma_\\calT}_R \\mathcal{T}$ if the following conditions hold:\n\\begin{enumerate}\n        \\item For all $\\alpha' \\in \\prodasm{\\mathcal{S}}$, all locations of $\\alpha'$ that map to a location in $\\dom{\\sigma_\\calT}$ map, under $R$, to the tile of $\\calT$ in that location of $\\sigma_\\calT$.\n        \\item $\\left\\{R^*(\\alpha') \\cup \\sigma_\\calT | \\alpha' \\in \\prodasm{\\mathcal{S}}\\right\\} = \\prodasm{\\mathcal{T}}$.\n        \\item $\\left\\{R^*(\\alpha') | \\alpha' \\in \\termasm{\\mathcal{S}}\\right\\} = \\termasm{\\mathcal{T}}$.\n        \\item For all $\\alpha'\\in \\prodasm{\\mathcal{S}}$, $\\alpha'$ maps cleanly to $R^*(\\alpha')$.\n\\end{enumerate}\n\\end{definition}\n\n\n\\begin{definition}\n\\label{def-t-follows-s} We say that $\\mathcal{T}$ \\emph{follows $\\mathcal{S}$ modulo $\\sigma_\\calT$} (under $R$), and we write $\\mathcal{T} \\dashv^{\\sigma_\\calT}_R \\mathcal{S}$ if $\\alpha' \\rightarrow^\\mathcal{S} \\beta'$, for some $\\alpha',\\beta' \\in \\prodasm{\\mathcal{S}}$, implies that $R^*(\\alpha') \\cup \\sigma_\\calT \\to^\\mathcal{T} R^*(\\beta') \\cup \\sigma_\\calT$.\n\\end{definition}\n\nThe next definition essentially specifies that every time $\\mathcal{S}$ simulates an assembly $\\alpha \\in \\prodasm{\\mathcal{T}}$, there must be at least one valid growth path in $\\mathcal{S}$ for each of the possible next steps that $\\mathcal{T}$ could make from $\\alpha$ which results in an assembly in $\\mathcal{S}$ that maps to that next step.\n\n\\begin{definition}\n\\label{def-s-models-t} We say that $\\mathcal{S}$ \\emph{models} $\\mathcal{T}$ (under $R$), and we write $\\mathcal{S} \\models_R \\mathcal{T}$, if for every $\\alpha \\in \\prodasm{\\mathcal{T}}$, there exists $\\Pi \\subset \\prodasm{\\mathcal{S}}$ where $\\Pi \\neq$ \\O\nand $R^*(\\alpha') = \\alpha$ for all $\\alpha' \\in \\Pi$, such that, for every $\\beta \\in \\prodasm{\\mathcal{T}}$ where $\\alpha \\rightarrow^\\mathcal{T} \\beta$, (1) for every $\\alpha' \\in \\Pi$ there exists $\\beta' \\in \\prodasm{\\mathcal{S}}$ where $R^*(\\beta') = \\beta$ and $\\alpha' \\rightarrow^\\mathcal{S} \\beta'$, and (2) for every $\\alpha'' \\in \\prodasm{\\mathcal{S}}$ where $\\alpha'' \\rightarrow^\\mathcal{S} \\beta'$, $\\beta' \\in \\prodasm{\\mathcal{S}}$, $R^*(\\alpha'') = \\alpha$, and $R^*(\\beta') = \\beta$, there exists $\\alpha' \\in \\Pi$ such that $\\alpha' \\rightarrow^\\mathcal{S} \\alpha''$.\n\\end{definition}\n\nWe now present the two new definitions for types of simulations that are relevant when we want to use systems with single-tile seeds to simulate systems with multi-tile seeds. The first, seed-first-simulation, requires that a complete representation of the multi-tile-seed of the simulated system self-assembles before any growth may occur away from the seed, and once the seed is complete, simulation continues in the standard way. The second, seed-growth-simulation, simply requires that the entire seed is eventually grown\n, but doesn't restrict growth away from the seed from beginning before the representation of the multi-tile-seed is complete. However it must still correctly simulate the behavior of the system with a multi-tile seed.\n\n\\begin{definition}\n\\label{def-s-sf-simulates-t} We say that $\\mathcal{S}$ \\emph{seed-first-simulates} $\\mathcal{T}$ (under $R$) if $\\mathcal{S} \\Leftrightarrow^{\\sigma_\\calT}_R \\mathcal{T}$ (they have equivalent productions modulo the seed of $\\calT$), $\\mathcal{T} \\dashv^{\\sigma_\\calT}_R \\mathcal{S}$ and $\\mathcal{S} \\models_R \\mathcal{T}$ (they have equivalent dynamics).\n\\end{definition}\n\n\\begin{definition}\n\\label{def-s-sg-simulates-t} We say that $\\mathcal{S}$ \\emph{seed-growth-simulates} $\\mathcal{T}$ (under $R$) if $\\mathcal{S} \\Leftrightarrow^{-\\sigma_\\calT}_R \\mathcal{T}$ (they have equivalent productions minus the seed of $\\calT$), $\\mathcal{T} \\dashv^{\\sigma_\\calT}_R \\mathcal{S}$ and $\\mathcal{S} \\models_R \\mathcal{T}$ (they have equivalent dynamics).\n\\end{definition}\n\n\\begin{corollary}\\label{cor:seed-growth-implies-shape}\nIf system $\\mathcal{S}$ seed-growth-simulates $\\calT$ or seed-first-simulates $\\calT$, then $\\mathcal{S}$ shape-simulates $\\calT$.\n\\end{corollary}\n\nCorollary \\ref{cor:seed-growth-implies-shape} follows immediately from the fact that both seed-growth-simulation and seed-first-simulation require equivalent sets of terminal assemblies between the simulated system $\\calT$ and the simulating system $\\mathcal{S}$ (under mapping by $R$), which implies shape-simulation.\n\n\n\\begin{corollary}\\label{cor:seed-first-implies-seed-growth}\nIf system $\\mathcal{S}$ seed-first-simulates $\\calT = (T,\\sigma,\\tau)$, then $\\mathcal{S}$ seed-growth-simulates $\\calT$.\n\\end{corollary}\n\nCorollary \\ref{cor:seed-first-implies-seed-growth} follows immediately from the fact that the only difference between the two types of simulation is that seed-first-simulation requires equivalent productions modulo $\\sigma$, while seed-growth-simulation requires equivalent productions minus $\\sigma$. Seed-first-simulation means that assemblies that map strictly to subassemblies of the seed are first produced, and all of these are valid under equivalent productions minus $\\sigma$, and from there all producible assemblies contain the full seed and must map to producible assemblies in $\\calT$, and all such assemblies are also valid under equivalent productions minus $\\sigma$.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Limits of Single-Tile Seeds}\\label{sec:single-tile-limits}\n\n\nWhile we previously provided a relatively simple example to show that some systems with multi-tile seeds cannot be simulated at scale factor 1 by systems with single-tile seeds, in this section we maximize the scale factor for which that is true (since Theorems \\ref{thm:scale3-sim} and \\ref{thm:scale4-sim} prove that simulation is possible above the bounds shown). Namely, we present a result, Theorem \\ref{thm:multi-imposs}, showing that there are systems with multi-tile seeds which require at least scale factor 3 plus the use of cheating fuzz, or scale factor 4 (without cheating fuzz), to shape-simulate using systems with single-tile seeds. Note that since seed-first simulation implies shape-simulation, this also proves that seed-first-simulation requires such scaling for some shapes. This section contains a brief description of the proof, and the full details can be found in Section~\\ref{sec:multi-imposs-proof}.\n\n\\begin{theorem}\\label{thm:multi-imposs}\nThere exist an infinite number of aTAM systems with multi-tile seeds that cannot be shape-simulated by any aTAM system with a single-tile seed at (1) scale factors 1 or 2, or (2) scale factor 3 without using cheating fuzz.\n\\end{theorem}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=3.0in]{images\/multi-scale-seed.png}\n    \\caption{(left) The seed of aTAM system $\\calT$ from the proof of Theorem \\ref{thm:multi-imposs}. Grey and green locations represent tiles, white positions outlined in black represent potentially valid fuzz locations, tile positions outlined in blue and fuzz positions outlined in red are referenced in the proof. Green locations indicate the only seed tiles which have exterior-facing glues capable of initiating growth. The light blue lines show minimal cuts across each arm and its fuzz (each crossing two locations). The ``arms'' of the seed are labeled A-E for convenience.}\n    \\label{fig:multi-scale-system}\n\\end{figure}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=4.0in]{images\/multi-scale-crashed-paths-empty.png}\n    \\caption{Depiction of an assembly producible in $\\calT$ in the proof of Theorem \\ref{thm:multi-imposs}. (The seed alone is shown in Figure \\ref{fig:multi-scale-system}.) From each arm of the seed, paths can grow away from the seed, and two such (each labeled 1) are depicted here. The paths labeled 1 can grow arbitrarily far away from the seed and initiate the growth of paths back toward the seed. For example, the blue path 7 from arm A, and gold path 7 from arm D, have pumped and crashed into the red locations of those arms\n    }\n    \\label{fig:empty-crash}\n\\end{figure}\n\n\nTo prove Theorem \\ref{thm:multi-imposs}, we present one system which can't be shape-simulated by any aTAM system with a single-tile seed at (1) scale factors 1 or 2, or (2) scale factor 3 without using cheating fuzz, and note how an infinite set of such systems can easily be derived from it.\nLet $\\calT = (T,\\sigma,1)$, the seed of which is shown in Figure \\ref{fig:multi-scale-system} and a producible assembly of which is depicted in Figure \\ref{fig:empty-crash}, be such a system.\nIn $\\calT$, there are five locations on the perimeter of the seed in which glues are exposed and to which tiles can attach. They are depicted in green.\nEach arm can grow a uniquely colored subassembly, each of which uses tile and glue types unique to that subassembly.\nAn infinite number of unique terminal assemblies, and assembly shapes, are possible in $\\calT$, since, for instance, each path labeled 1 can be of arbitrary length. The intuition behind the proof is that it is possible for arbitrarily long paths that start at the ends of the seed's arms to grow away from the seed assembly, and then to grow paths that come back toward the seed, where they can crash into other arms. In order for a system with a single-tile seed to simulate this system, it would have to ensure that there are tiles in place in each arm to provide blocking before any arm is able to grow crashing paths. However, at the small scale factors allowed there is not enough room for all necessary dependence paths to grow between all arms, so simulation fails.\n\n\n\n\n\n\n\n\n\n\n\n\n\n    \n\n    \n    \n\n    \n    \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n    \n    \n    \n    \n    \n        \n    \n\n\n\n\\subsection{Scale factor 1}\n\n\nIn this section, we prove that there are infinitely many aTAM systems whose seeds consist of more than one tile, but for none of them does there exist a singly-seeded aTAM system which can produce the exact same set of terminal assembly shapes, which also implies that none of them can be seed-growth-simulated by any singly-seeded aTAM system.\n\n\\begin{theorem}\\label{thm:imposs-scale1}\nThere exists an infinite class of aTAM systems, $\\mathfrak{T}$, such that for every $\\mathcal{T} \\in \\mathfrak{T}$, there exists no singly-seeded aTAM system $\\mathcal{S}$ which shape-simulates $\\mathcal{T}$ at scale factor 1.\n\\end{theorem}\n\n\\begin{proof}\nWe prove Theorem~\\ref{thm:imposs-scale1} by first presenting an aTAM system $\\calT = (T,\\sigma,1)$ and then showing why it cannot be shape-simulated by any singly-seeded aTAM system. A high-level overview of $\\calT$ is shown in Figure~\\ref{fig:no-single-seed-at-scale-factor-1}. The seed $\\sigma$ consists of $5$ tiles (shown in grey and labeled $A$ through $E$). Growth can happen in two locations: from the north of tile $A$, and south from tile $D$. The green tile can attach to itself and grow an arbitrarily tall column of green tiles. However, nondeterministically and at any point either a red or blue tile can attach. The red tile has no glues on its north, east, or west sides, and thus terminates the growth of the column. The blue tile is among a set of tiles which are able to attach and grow four positions to the east, and then a single tile type which attaches to itself (on the north and south sides) and grows downward until it crashes into tile $E$ of the seed. Analogously, the yellow and pink portions may grow on the south.\n\n\\begin{figure}[t]\n    \\centering\n    \\subfigure{\n    \\includegraphics[height=50mm]{images\/no-single-seed-at-scale-factor-1-2.png}\n    \\label{fig:no-single-seed-at-scale-factor-1}\n    }\n    \\quad\n    \\subfigure{\\raisebox{19mm}{\n    \\includegraphics[height=31mm]{images\/SF1-blocking.png}}\n    }\n    \\caption{An example system from the class $\\mathfrak{T}$ which cannot be shape-simulated by any singly-seeded aTAM system at scale factor 1. The left and right show two possible terminal assemblies. The grey tiles (labeled $A-E$) are the seed. Green tiles can grow upward arbitrarily far, and at any nondeterministic location either a red tile can attach to terminate the growth of the column, or a blue tile can attach which causes growth of four additional tiles to the right and then growth downward of a column which crashes into the tile labeled $E$. Similar (but slightly narrower) growth can occur with the yellow and pink tiles, initiating from the tile labeled $D$.}\n    \n\\end{figure}\n\nWe now prove that $\\calT$ cannot be shape-simulated by any singly-seeded aTAM system by contradiction. Therefore, assume that aTAM system $\\mathcal{S} = (S,\\sigma',\\tau)$, where $|\\sigma'|=1$, shape-simulates $\\calT$ at scale factor 1. We now do a case analysis to show that there is no valid location for the tile of $\\sigma'$ to be placed such that $\\calT$ can be correctly shape-simulated.\n\nEach of the following cases assumes a different location for the tile of $\\sigma'$ and shows why that leads to a failure to correctly shape-simulate $\\calT$.\n\n\\begin{enumerate}\n    \\item Location of $A$: Since there exist terminal assemblies of $\\calT$ which have green tiles north of $A$ capped by a red tile, and also tiles to the east of $A$, with no connected path between them other than through $A$, it must be the case that tiles can attach in $\\mathcal{S}$ to both the north and the east of the tile of $\\sigma'$.  Let $n = |S|$ be the number of tile types in $\\mathcal{S}$, and note that in $\\calT$ there exist terminal assemblies where the green columns of tiles grow to a height of $h > n$. In order for $\\mathcal{S}$ to shape-simulate $\\calT$, it must also be able to grow columns of tiles in the locations of green tiles to a height of $> h$. Note that this means that some tile type of $S$ must appear in such a column more than once, and since it is only a single-tile-wide column of tiles with a repeating tile type, it must be possible to ``pump'' it up arbitrarily far (i.e. there exists a valid assembly sequence in $\\mathcal{S}$ which can grow a column in the location of the green tiles of $\\calT$ arbitrarily tall).\n    \n    Furthermore, since $\\mathcal{S}$ is assumed to shape-simulate $\\calT$, it must be possible for tiles to grow so that they form a loop similar to that created by the green and blue tiles of $\\calT$ (which we'll call a ``north loop'').\n    \n    First, assume that the ``blue portion'' can grow from the ``green column''. [TODO: show this means there is a valid assembly sequence in which the blue passes the location of $E$, which is invalid.]\n    \n    [TODO: make images of invalid assemblies from the cases]\n    \n    Since that cannot be the case, the ``blue portion'' must grow upward from the $E$ location. [TODO: could pump green to a taller height and then crash blue into it, which is invalid.]\n    \n    \\item Locations of $B$, $C$, or $D$: The same argument holds as for the location of $A$. (With the slight caveat of a connected path which avoids $C$ connecting $A$ and $E$, which doesn't change the argument\n\n    \\item Location of $E$: Although $E$ is now in place to block the north loop before it begins, an analogous argument that was used for location $A$ can now be used to show that the south loop cannot be correctly grown. [TODO: formalize]\n    \n    \\item Location of a green tile: We can use the same argument as for the location of $A$.\n    \n    \\item Location of a blue tile: [TODO: similar to green]\n    \n    \\item Location of a yellow tile: [TODO: similar to green]\n    \n    \\item Location of a pink tile: [TODO: similar to blue]\n\\end{enumerate}\n\nThe above case analysis shows that there is no location, within the shape of a valid terminal assembly of $\\calT$, in which $\\sigma'$ can be placed and $\\mathcal{S}$ can then shape-simulate $\\calT$. Therefore, $\\mathcal{S}$ does not shape-simulate $\\calT$, and since the only assumption made about $\\mathcal{S}$ was that it is singly-seeded, no singly-seeded aTAM system can shape-simulate $\\calT$ at scale factor 1.\n\n\n[TODO: to finish the proof for the full class...]\nTo add infinitely many additional aTAM systems to $\\mathfrak{T}$, we could extend the seed arbitrarily far in either direction and could have arbitrarily more upward and\/or downward growing arms\/loops along those extensions. It is clear that the previous case analysis could be extended to any such system to show that it also cannot be shape-simulated by any singly-seeded aTAM system.\n\nTherefore, we have shown an infinite class of aTAM systems, $\\mathfrak{T}$, such that none can be shape-simulated by any singly-seeded aTAM system at scale factor 1 and thus Theorem~\\ref{thm:imposs-scale1} is proven.\n\n\\end{proof}\n\n\n\\begin{corollary}\\label{cor:imposs-scale1}\nThere exists an infinite class of aTAM systems, $\\mathfrak{T}$, such that for every $\\mathcal{T} \\in \\mathfrak{T}$, there exists no singly-seeded aTAM system $\\mathcal{S}$ which seed-growth-simulates $\\mathcal{T}$ at scale factor 1.\n\\end{corollary}\n\n\\begin{proof}\nCorollary \\ref{cor:imposs-scale1} follows immediately from Theorem \\ref{thm:imposs-scale1} since seed-growth-simulation implies shape-simulation.\n\\end{proof}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{images\/scale2-T.png}\n    \\caption{An example aTAM system $\\mathcal{T} = (T,\\sigma,1)$ which cannot be shape-simulated at scale factor 2. The grey and green tiles represent the seed tiles (which have unshown strength-1 glues between all adjacent sides). The blue tiles can attach to the green tiles of the seed and nondeterministically allow for growth from any green tile that could potentially crash into the seed at any location.}\n    \\label{fig:scale2-T}\n\\end{figure}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{images\/no-single-seed-at-scale-factor-2.png}\n    \\caption{The system shown in Figure~\\ref{fig:scale2-T} at scale factor 2. Each $2 \\times 2$ block represents one tile of the seed of $\\calT$. Green blocks represent locations from which growth can be initiated outward from the seed.}\n    \\label{fig:no-single-seed-at-scale-factor-2}\n\\end{figure}\n\n\n\\begin{theorem}\\label{thm:imposs-scale2}\nThere exists an infinite class of aTAM systems, $\\mathfrak{T}$, such that for every $\\mathcal{T} \\in \\mathfrak{T}$, there exists no singly-seeded aTAM system $\\mathcal{S}$ which shape-simulates $\\mathcal{T}$ at scale factor $\\le 2$.\n\\end{theorem}\n\n\\begin{proof}\nFor the proof of Theorem \\ref{thm:imposs-scale2}, we present an aTAM system $\\mathcal{T} = (T,\\sigma,1)$ and show that no singly-seeded aTAM system can shape-simulate it at scale factor $\\le 2$. Figure \\ref{fig:scale2-T} shows $\\mathcal{T}$ and Figure \\ref{fig:no-single-seed-at-scale-factor-2} depicts a version at scale factor 2.\n\nOur proof will be by contradiction, so assume that there exists an aTAM system $\\mathcal{S} = (S,\\sigma_\\mathcal{S},\\tau)$ which shape-simulates $\\calT$ at scale factor 2.\n\nWe first note that the green tiles of the seed of $\\calT$ allow for the attachment of blue tiles. From each green tile, it is possible for a path of blue tiles to attach which can (1) be arbitrarily long, and (2) grow to any location on the perimeter of the seed. The nondeterministic nature of the aTAM means that any such path has a possibility of growing from any one green tile before a single tile attaches to any other green lite. We will only focus on such paths which grow far enough away from the seed then turn and grow a straight line path into a location adjacent to the seed such that the path that returns to the seed ``pumps'' (TODO: refer to the Window Movie Lemma). We will call such a path a ``crashing path'' .\n\nSince every green location can initiate growth of a crashing path which can crash into any other green location, and since such a crashing path must be able to resolve a tile of a blue type in a location where a green tile should be if there is no tile already in the green macrotile location, it must be the case that before any crashing path can grow, at least one tile is present in each green macrotile location. [TODO: there are a few trivial details to walk through to make this completely clear.]\n\nSince $\\mathcal{S}$ is singly-seeded, $\\sigma_\\mathcal{S}$ can have at most a single tile in a single green macrotile location. Without loss of generality, assume that $\\sigma_\\mathcal{S}$ consists of a tile in the green macrotile location $A$. (If not, it must grow a path to a green macrotile location, w.l.o.g. assume $A$ is the first, and our argument continues from that point.) It must be the case that a non-branching path grows from $A$ to every other green macrotile location before any of them can initiate growth of a simulated crashing path (by our previous argument). We will call this path the ``blocking path'' and without loss of generality assume that the first green macrotile to have a tile is $A$ and the last is $E$. (Our arguments will apply for any other choices.)\n\nBecause it must not be possible to initiate growth of any crashing path before completion of the blocking path, i.e. the blocking path must first place at least $1$ tile in each green macrotile, after the blocking path places a tile in the last green macrotile location it must then initiate growth of paths that go to every other green macrotile location to allow them to initiate growth of crashing paths. We call these paths ``activation paths''.\n\nWe now analyze the paths that must grow. It $A$ is the first and $E$ the last macrotile on the path, then $B$ and $D$ must be visited in between. Since the path cannot branch, it must be the case that it grows upward to, then back down from, each of $B$ and $D$. This requires the use of two tile locations across the cuts marked as diagonal red lines. (The outlined locations show all valid locations for tile placements allowed strictly during growth of the seed, accounting for allowable fuzz. The missing $2\\times 2$ square would be the location of invalid diagonal fuzz.) Once the blocking path completes at $E$, it must then be the case that activation paths grow back to both $B$ and $D$, using the third of the four valid tile locations along the red cuts. However, at least one of either the path to $B$ or $D$ must then continue to activate $C$. (This is because the blocking path will have cut $C$ off from any paths except those that go around the blocking path at those locations.) In order to do this, that path would require use of the fourth of four locations along the associate red diagonal. At this point, one of the macrotile locations marked as yellow must be completely filled. By the definition of simulation, since no additional tiles can ever be added to that macrotile, whatever tile in $\\calT$ that it maps to at this point must be permanent. If it now maps to a tile (rather than empty space), we can grow crashing paths that miss that location, yielding an invalid assembly. If it maps to empty space, we can instead grow a crashing path that should terminate in that location but is unable to do so, also yielding an invalid assembly.\n\n[TODO: more details are needed. I'm trying to prove the stronger statement of shape self-assembly and need to make sure that there isn't some weird nondeterministic crap that could occur. I don't think so because of pumping...]\n\n\n\n\n\n\\end{proof}\n\n\n\n\\section{Seed-Growth-Simulation at Scale Factor 3}\n\n\\begin{theorem}\\label{thm:imposs-scale3}\nThere exists an infinite set of poly-seeded aTAM systems such that none of them can be seed-growth simulated by any singly-seeded aTAM system at scale factor $\\le 3$ without the use of cheating fuzz.\n\\end{theorem}\n\n[TODO: the proof of Theorem \\ref{thm:imposs-scale2} applies directly to this, so the following proof should be removable.]\n\nThroughout the following, a \\emph{single-tile-wide path} of tiles in an assembly $\\alpha$ is an ordered sequence of tiles $(t_0,t_1,...,t_{n-1})$ such that, for $0<i<n$, $t_i$ binds to $\\alpha$ via a single $\\tau$-strength glue on $t_{i-1}$.\n\nThe following lemma states that if arbitrarily long paths can grow and crash into tiles of a system's seed, then any system which seed-growth-simulates that system must ensure that all those locations which map (under the representation function) to those seed locations which can be crashed into must have tiles within them before those crashing paths can grow.\n\n\\begin{lemma}\\label{lem:seed-first}(Seed First Lemma)\nLet $\\calT = (T,\\sigma_\\calT,\\tau)$ be a poly-seeded aTAM system, and let $\\mathcal{S}$ be a singly-seeded aTAM system which seed-growth-simulates $\\calT$. Let $L \\subseteq \\dom \\sigma_\\calT$ be a set of locations in the seed of $\\calT$ such that for any value $d \\in \\mathbb{Z}^+$, for each $l \\in L$, there exists a valid assembly sequence in $\\calT$ where a single-tile-wide path of tiles, $p_l$, begins at distance $d$ away from $l$ and grows until the tile at location $l$ blocks further growth of $p_l$. For each $l$, let $C_l$ be the set of all such unique paths $p_l$ which can crash into $l$ (across all valid assembly sequences) such that, if the seed tile was not in location $l$, a tile of a different type would be able to attach in location $l$ by attaching to the last tile placed in $p_l$. In $\\mathcal{S}$, for each $l \\in L$ the macrotile location which maps to location $l$ must have at least one tile within it before it is possible to grow the path of macrotiles which represent any $p_l \\in C_l$.\n\\end{lemma}\n\n\\begin{proof}\nWe prove Lemma~\\ref{lem:seed-first} by contradiction. Therefore, assume that in $\\mathcal{S}$ there exists a macrotile which maps to a location $l \\in L$ which has no tiles in it, and macrotiles which represent some $p_l \\in C_l$ are able to assemble.\n\nFirst, we note that by the Window Movie Lemma of \\cite{temp1notIU}, that there is a constant $p$ such that, if $p_l$ grows beyond length $p$ (which must be possible by the definition of the paths in $C_l$), there must be valid assembly sequences which allow $p_l$ to become ``pumpable''. That is, there must exist at least two separate cuts in the binding graph across the macrotiles representing $p_l$ with identical window movies (i.e. the same glues attaching in the same order), which means there is a valid assembly sequence allowing the assembly between those cuts to be repeated arbitrarily many times (i.e. pumped) unless growth collides with previously existing tiles. Therefore, $p_l$ must be able to grow a macrotile which represents the next tile of $p_l$ into the empty seed tile location. This would be a failure of $\\mathcal{s}$ to correctly (seed-growth-)simulate $\\calT$. Therefore, every such $l \\in L$ must have at least one tile within in to prevent that growth.\n\\end{proof}\n\n\\begin{definition}\nA \\emph{precedence path} in an assembly $\\alpha$ is a path in the biding graph of $\\alpha$ following an ordered sequence of tiles in $\\alpha$, $(t_0,t_1,...t_{n-1})$, such that for each $0<i<n$, tile $t_{i-1}$ provides an output glue which is necessary for the biding of an input glue on $t_i$ to allow it to bind into $\\alpha$. \n\\end{definition}\n\n[TODO: rework the following lemma by defining locations that ``output'' (i.e. allow the growth away from the seed which eventually crashes) and then saying that no location can output until all locations into which its output can crash must have a precedence path connecting them before the output location grows.]\n\n\\begin{lemma}\\label{lem:single-path}(Single Path Lemma)\nGiven the same systems $\\calT$ and $\\mathcal{S}$ and sets $L$ and $C_l$ as defined for Lemma~\\ref{lem:seed-first}, for every producible assembly in $\\mathcal{S}$ in which macrotiles exist that represent any path $p_l$ in any $C_l$, there exists a precedence path which begins at the seed tile, includes a minimum of one tile in every macrotile representing a location in $L$, and ends with the macrotiles of $p_l$.\n\\end{lemma}\n\n\\begin{proof}\nWe prove Lemma~\\ref{lem:single-path} by contradiction. Therefore, assume that there exists some producible assembly $\\beta \\in \\prodasm{\\mathcal{S}}$ such that $\\beta$ contains the macrotiles of some $p_l$ but no precedence path leading from the seed tile to the macrotiles of $p_l$ contains at least one tile in every macrotile representing a location in $L$.\n\\end{proof}\n\n\n\\begin{proof}\nWe prove Theorem~\\ref{thm:imposs-scale3} by giving a concrete example of a poly-seeded aTAM system $\\mathcal{T}$ which cannot be simulated by any singly-seeded aTAM system at scale factor $3$ without the use of cheating fuzz. We will then show how an infinite set of poly-seeded aTAM systems can be derived from that example.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=1.0\\textwidth]{images\/no-single-seed-at-scale-factor-3-examples.png}\n    \\caption{This system should work for a proof that singly-seeding can't work at scale factor 3 without cheating fuzz.[TODO: note that without cheating fuzz, only depicted macrotiles can ever have tiles within them.]}\n    \\label{fig:no-single-seed-at-scale-factor-3}\n\\end{figure}\n\nLet $\\calT = (T,\\sigma,1)$ be the system represented in Figure~\\ref{fig:no-single-seed-at-scale-factor-3}, which shows examples of its growth. Note that, in the figure, each tile of $\\calT$ is represented as a $3 \\times 3$ block of squares surrounded by a darker line, thus showing the subdivision of the macrotiles of a simulator at scale 3 which is attempting to simulate $\\calT$. The grey blocks represent locations within the seed $\\sigma$.\n\nThe tiles at the locations labeled $X$ and $Y$ both initiate growth of northward and southward columns which nondeterministically grow to arbitrary lengths before either randomly being ``capped' by a red tile or growing right, or left (respectively) one of two fixed distances before then growing columns which eventually crash back into the seed locations $A$,$B$,$C$, or $D$.\n\nThe intuitive reason behind why $\\calT$ cannot be seed-growth-simulated by a singly-seeded system is that, in $\\calT$, all 4 locations $A$,$B$,$C$, and $D$ of the seed are tiled before any non-seed growth can occur, and therefore they are in place and able to block any growth which might eventually crash into them. Assume, for the sake of contradiction, that some singly-seeded aTAM system $\\mathcal{S}$ seed-growth-simulates $\\calT$. [TODO: switch to reference the Seed First Lemma]\n\nIf there is a valid assembly sequence in $\\mathcal{S}$ in which such a column grows through one of the locations $A$,$B$,$C$, or $D$, then $\\mathcal{S}$ fails to (seed-growth-)simulate $\\calT$. Therefore, it must be the case that before any such column can validly grow, starting from $X$ or $Y$, there exists at least one tile in each macrotile location corresponding to $A$,$B$,$C$, and $D$. However, since $\\mathcal{S}$ is singly-seeded, at most one location in one of the grey macrotiles can have $\\mathcal{S}$'s seed tile, and then one or more paths of tiles must grow which place tiles so that there is at least one tile in each of the macrotiles in locations $A$,$B$,$C$, and $D$.\n\nWe first show that it must be a single, unbranching path which grows into each of the macrotiles representing locations $A$,$B$,$C$, and $D$ and only after doing so can growth of the macrotiles at $X$ and $Y$ complete to the point that they can grow upward or downward columns (which we will refer to as ``outputting''). Therefore, assume otherwise, namely that growth of a path $p_{xy}$ can occur into at least one of the macrotiles of $X$ or $Y$ and allow it to output, and that $p_{xy}$ is independent of (i.e. a separate path or a branch off of) any path which places a tile in at least one of $A$,$B$,$C$, or $D$. (Without loss of generality, we will assume that $p_{xy}$ grows into $X$, and that $p_{xy}$ is independent of any path growing into $A$.) This means that there exists a valid assembly sequence in $\\mathcal{S}$ in which $p_{xy}$ grows into $X$, then a column grows upward out of $X$ to length $\\ge l$, then growth representing blue tiles occurs which grows rightward to a location over $A$, and then a column grows downward. If $\\mathcal{S}$ correctly simulates $\\calT$, that downward growing column must be able to grow to length $>l$. [TODO: make use of the Seed First Lemma]\n\nThus, by the following argument, it must be the case that a single, unbranching path (which we'll call $p$) grows from the single seed tile of $\\mathcal{S}$, places at least one tile in each of $A$,$B$,$C$, and $D$, and only then can growth continue (now via one or more paths) into locations $X$ and $Y$ and allow them to output.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{images\/no-single-seed-at-scale-factor-3-pathss.png}\n    \\caption{Possible paths through the macrotiles of $\\mathcal{S}$ which represent the seed of $\\calT$.}\n    \\label{fig:no-single-seed-at-scale-factor-3-paths}\n\\end{figure}\n\nWithout loss of generality, assume that either (1) the seed tile of $\\mathcal{S}$ is in the macrotile of $A$, or (2) path $p$ visits $A$ before visiting $B$,$C$, or $D$. Figure~\\ref{fig:no-single-seed-at-scale-factor-3-paths} shows possible routes for $p$. The left two show possible routes if $B$ is visited before $C$ and $D$, with the differences being whether $C$ or $D$ is visited next. In the leftmost, $D$ is visited last, so (as previously shown) only from that point can one or more paths grow to $X$ and $Y$ and allow them to output. However, since the macrotiles marked as $C'$ and $B'$ must necessarily each have two rows of tiles across them, it is impossible for any path to grow from $D$ to $X$, thus $X$ can never output and the simulation fails. In the middle image of Figure~\\ref{fig:no-single-seed-at-scale-factor-3-paths}, $C$ is visited last by $p$ but then it is impossible to grow a path from there to $Y$, so the simulation would fail. The final case to consider is depicted in the rightmost image of Figure~\\ref{fig:no-single-seed-at-scale-factor-3-paths}, in which $C$ and $D$ are visited by $p$ before $B$. However, in this case, depending on which two columns of the macrotiles in the vertical section of the path are occupied, either $X$ or $Y$ cannot be reached by a path. Therefore, in all cases the simulation fails, and therefore $\\mathcal{S}$ cannot correctly seed-growth-simulate $\\calT$ and there can be no such simulator.\n\nWe now show that there is an infinite class of systems of which $\\calT$ is a member. It is trivial to build them by making systems similar to $\\calT$ but with successively wider seeds, each with its sets of colored tiles modified by adding extra tiles so that the collisions with locations $A$,$B$,$C$, and $D$ can still occur.\n \nTo prove the final point of Theorem~\\ref{thm:imposs-scale3}, that it holds for scale factors 1 and 2 as well, we simply note that no simulator could succeed in growing the necessary paths with even less allowed space than $\\mathcal{S}$ was given at scale factor 3.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\n    \n   \n    \n   \n\n   \n   \n   \n            \n   \n            \n   \n   \n    \n   \n    \n   \n\n   \n    \n   \n    \n   \n\n   \n    \n\n   \n\n    \n    \n            \n            \n\n    \n    \n        \n\n\nMathematical models of self-assembly provide high-level abstractions of the self-assembling behaviors of systems that typically function on a molecular level. While many natural systems harness the power of self-assembly, scientists and engineers are rapidly increasing their abilities to design self-assembling systems. One of the most versatile molecules that we have discovered is DNA. While biology utilizes DNA primarily as a storage medium, it is now being recognized for its impressive capabilities for structure-building, augmented by its ease of synthesis, durability, and programmability. Engineers are designing impressive DNA-based self-assembling systems whose complexities are increasing at a nearly exponential rate (e.g.  \\cite{woods2019diverse,zhang2020programming,MonaLisa,AndersenBox2009}). Not to be outdone, theoreticians are rapidly expanding the mathematical toolkit for modeling, predicting, and analyzing such systems. The level of granularity in modeling can vary from highly realistic atomic-level modeling to purely mathematical abstractions, and computer science is impacting all levels. In this paper, we utilize computational theory, complexity theory, and algorithm design and analysis to explore a fundamental aspect of mathematical modeling: the ways in which so-called ``seed'' assemblies, i.e. preexisting assemblies that serve as nucleation points for growth, can influence and impact the behaviors of self-assembling systems.\n\nA popular and widely studied mathematical model of self-assembly called the abstract Tile Assembly Model (aTAM) was introduced by Winfree \\cite{Winf98} and immediately shown to be capable of complex algorithmic behavior and computationally universal. In the aTAM, the basic components are square tiles that have glues on their edges that allow them to bind together, and as a ``seeded'' model growth begins with tile attachments to an input seed assembly and proceeds by growing that assembly. This is in contrast to ``seedless'' models such as the 2-Handed Assembly Model (2HAM) \\cite{AGKS05g,2HAMIU}, a.k.a. Hierarchical Assembly Model, in which growth can begin with the combination of any pair of base components and proceeds by the combination of pairs of base or previously formed components. Prior work has directly compared these two models \\cite{Versus,jVersus1}, and although the power of hierarchical assembly grants the 2HAM many advantages, the few results showing advantages for the aTAM resulted from aspects related to the seed structures, particularly due to the geometric hindrance (a.k.a. blocking) they can provide.\n\nOne of the major metrics used to analyze self-assembling systems is called \\emph{tile complexity}, which is the number of unique types of tiles that are required to self-assemble a target structure. Various results have exemplified the power of algorithmic self-assembly by demonstrating theoretical constructions that meet information theoretic lower bounds on tile complexity. For instance, the self-assembly of $n \\times n$ squares in the aTAM has been shown to be possible with $\\Theta(\\frac{\\log n}{\\log \\log n})$ tile types\\cite{RotWin00,AdChGoHu01}, and general shapes (at large scale factors) with $\\Theta(\\frac{K(S)}{\\log K(S)})$ tile types (where $S$ is the definition of the target shape and $K(S)$ is the Kolmogorov complexity of $S$)\\cite{SolWin07}.\n\nThe aTAM allows seed assemblies to be arbitrarily complex (as long as they are finite), but if the goal is to measure the complexities of systems, the information contained in the seed assemblies should be factored in along with the tile complexity. However, it isn't immediately clear how to completely quantify the amount of information contained within a seed assembly. Clearly, the amount of information encoded in the glues exposed by an assembly is important. However, information can also be implicitly encoded in the shapes of tiles and assemblies (e.g. \\cite{GeoTiles,jDuples,Polygons,Polyominoes,OneTile}), and variations in shapes can allow systems to utilize another tool: geometric hindrance. Since quantifying the information embedded in a seed assembly can be difficult, a standard basis for these complexity results is the requirement that the system have a seed consisting of a single tile (i.e. a single-tile seed). In this way, all information is transferred into the definitions of the tiles, providing a more even foundation for comparison. Such single-tile seeds are the basis of results such as the self-assembly of $n \\times n$ squares and also of generals shapes previously mentioned. Of course, given the ability of seed assemblies to encode arbitrary amounts of information, it is also possible to use a constant tile set and vary the seed assemblies to instead shift all of the information to the seed. Several results have been based on this approach, including those related to \\emph{intrinsic universality} (IU), a notion of simulation in which one tile assembly system is used to simulate another in a way that attempts to capture the full dynamics of the simulated system, modulo only scaling in time and space. In \\cite{IUSA} it was shown that the aTAM is intrinsically universal, which means that there is a single universal aTAM tile set (and functions that specify how to arrange those tiles into seed assemblies and interpret blocks of them as individual tiles in the simulated systems) that can be used to simulate \\emph{any} aTAM system. This is a powerful closure property of the model, and the notion of intrinsic simulation has been utilized to compare and contrast the powers of various models and classes of systems (e.g. \\cite{DirectedNotIU,j2HAMIU,temp1notIU,WoodsMeunierSTOC}). When designing and utilizing IU tile sets, the tile set is constant across differing simulations, and therefore the information serving as input to the system comes completely from the structure and glues of the seed assembly.\nSlight variations to the aTAM have also resulted in models in which there are alternative methods for providing information to systems to seed their growth. These include \\emph{temperature programming} in which the temperature parameter of a system can be programmed to change following a prescribed, and arbitrarily complex, series of values that causes a growing structure to periodically grow and\/or partially break apart \\cite{AGKS05g,KS07,SummersTemp}, \\emph{concentration programming} in which a very precise concentration value can be specified for each tile type in order to influence the probabilities of attachment of specific tile types \\cite{KaoSchS08,Dot10}, and others.\n\nIn this paper, we present a wide array of results that serve to quantify the types and magnitudes of impacts that seed structures can have on aTAM systems. We first provide a few simple results and observations to show (1) systems that can encode arbitrary amounts of information in seeds and utilize ``Garden of Eden'' assemblies (which are assemblies that have no pathway for growth and can only stably exist if they completely form instantaneously) in Section \\ref{sec:seed-encoding-GoE}, (2) ``blocking'' (which occurs when potential placement of later tiles is prevented by the prior placement of earlier tiles) by tiles in a seed assembly can impact the dynamics of systems in important ways but that it is uncomputable, in general, to determine if one or any seed tile locations have the potential to block growth (Observation \\ref{obs:uncomp-blocking}), (3) the same sets of shapes of the assemblies produced by even some relatively simple multi-tile seed systems cannot be produced by systems with single-tile seeds if they are not allowed to be scaled up in size (i.e. if each single tile is replaced by an $n \\times n$ square of tiles, we say that the system is scaled by factor $n$) in Section \\ref{sec:imp-scale-1}, and (4) that, since the aTAM does not allow for seeds of infinite size, there is an infinite set of shapes that cannot self-assemble in any aTAM system (Observation \\ref{obs:infinite-seed}).\n\nGiven the utility of transferring information encoded in seed assemblies into tile complexity, by trading multi-tile seeds for single-tile seeds, we explore when and how it is possible to do so.  In order to fully explore the capabilities and limitations of converting systems with multi-tile seeds to those with single-tile seeds, we provide definitions for new notions of simulation (Section \\ref{sec:simulation_def}).\nIn standard definitions, both the simulated system and the simulating system start from the same assembly (modulo scale factor and interpreted through a mapping function). In these new definitions, the simulator is allowed to begin with seed assemblies as small as a single tile, and then are allowed to grow assemblies representing the seed structures. In the most permissive definition (which we call ``shape-simulation''), we only stipulate that both systems have to produce terminal assemblies of the same shapes (modulo scale factor and mapping). In the most restrictive definition (which we call ``seed-first-simulation''), we require that the simulating system completely grow an assembly mapping to the seed structure of the original system before allowing any of the rest of the growth permitted by that system to occur.\nUsing these notions of simulation, we prove the tightest results possible.\nWe prove that even with the most relaxed version, shape-simulation, there are aTAM systems with multi-tile seeds that no systems with single-tile seeds can correctly shape-simulate at scale factors 1 or 2, or also at scale factor 3 with a technical restriction applied (i.e. ``cheating fuzz'', as defined in Section \\ref{sec:simulation_def}, is not allowed) (Theorem \\ref{thm:multi-imposs}). However, using even the most restrictive notion of simulation, seed-first-simulation, we prove that any aTAM system with a multi-tile seed can be seed-first-simulated by a system with a single-tile seed at scale factor 3 with the technical restriction removed (i.e. cheating fuzz is allowed) (Theorem \\ref{thm:scale3-sim}) or at scale factor 4 with the technical restriction once again applied (i.e. cheating fuzz is not allowed) (Theorem \\ref{thm:scale4-sim}). Using the restrictive notion of seed-first-simulation makes it possible to guarantee the full seed-representing assembly is grown in the simulator before any outward growth can occur, making it behave identically to the simulated system after an initial seed-building phase\n\nWhile we prove that the scale 3 and 4 simulations achieve the lower bound for scale factor, they pay the price in terms of tile complexity, which is $O(|\\sigma|+|T|)$ for the simulation of a system with tile set $T$ and seed assembly $\\sigma$. In order to achieve optimality for the tile complexity metric, our next result (Theorem \\ref{thm:scaled-simulation}) proves that the tile complexity can be reduced to $O(\\frac{K(\\calT)}{\\log(K(\\calT)})$ (which, by an information theoretic argument, is the lower bound) for the seed-first-simulation of an arbitrary aTAM system $\\calT$ at the trade-off of a (massive) increase in scale factor (which is proportional to the running time of a relatively complex Turing machine).\n\nAlthough the tradeoff in scale factor is immense for the previous construction, beyond approaching optimal tile complexity, it provides a basis for additional theoretically interesting results. As a module of that construction, we use a (minimally modified) version of the intrinsically universal aTAM tile set from \\cite{IUSA}. This machinery allows us to extend the construction to first present a new construction that simultaneously and in parallel simulates all aTAM systems (Theorem \\ref{thm:simult-sim}). This, of course, requires a modified definition of simulation to take into account the fact that no fixed value for a scale factor could suffice to simulate arbitrary aTAM systems, so we define a type of ``mixed-scale-simulation''. Then, for our final result (Theorem \\ref{clm:cross-model-IU}), we show how we can make use of the ability of systems in a different model to simulate systems of the aTAM by providing a construction that utilizes ``nested simulations'', allowing us to make connections between models that can simulate arbitrary systems in the aTAM and intrinsic universality for the aTAM (notions we also show are not necessarily correlated).\n\nIn summary, we prove a wide assortment of results that expose the impacts of seed assemblies on the dynamics and complexities of aTAM systems and provide tight bounds that show how it is possible to replace multi-tile seeds with single-tile seeds and what the tradeoffs are.\n\nThe paper is laid out as follows. Section \\ref{sec:definitions} contains the definitions of, and related to, the aTAM, as well as definitions for the various types of simulations used throughout the paper. Section \\ref{sec:basic} contains a set of simple results and observations that lay the foundation for the more complex results to follow. Section \\ref{sec:single-tile-limits} sketches the result showing the lower bound for the scale factor required to simulate systems with multi-tile seeds by those with single-tile seeds.\nSection \\ref{sec:min-scaled-sims} gives overviews of the constructions that provide a tight upper bound for the scale factor of such simulations. In Section \\ref{sec:scaled-sim} we present the results showing simulation by systems with single-tile seeds using optimal tile complexity, as well as simultaneous simulation of all aTAM systems, and one relating to intrinsic universality and the aTAM. Then, the Technical Appendix, Section \\ref{sec:append}, contains proofs and technical details omitted from the other sections.\n\n\n\n   \n    \n   \n    \n   \n    \n   \n    \n   \n    \n   \n    \n    \n   \n    \n   \n    \n   \n    \n\n   \n    \n   \n    \n   \n    \n   \n    \n   \n    \n   \n    \n   \n    \n   \n    \n   \n    \n   \n    \n   \n\n\\section*{Acknowledgements}\nThe authors would like to thank Trent Rogers for helpful comments and creative brainstorming, and credit him for the idea of Claim \\ref{clm:cross-model-IU}. Additionally, we thank the anonymous reviewers who gave extremely helpful comments and suggestions that have greatly improved this version of the paper.\n\n\n\n\n\\bibliographystyle{plainurl\n\n\\section{Basic Results and Observations About Seeds}\\label{sec:basic}\n\nIn this section, we provide a set of relatively basic results and observations that display the amount of information that can be contained in a seed, that it is uncomputable to know whether or not blocking may be caused by the tiles of a seed, the fact that single-tile seeds cannot always be used to replace multi-tile seeds due to the potential for blocking, and that some shapes would require infinite-sized seeds to self-assemble in the aTAM.\n\n\\subsection{Encoding information in a seed structure}\\label{sec:seed-encoding-GoE}\n\nAn aTAM system is defined as a triple. For example, system $\\calT = (T,\\sigma,\\tau)$. $T$ is the tile set and can be encoded using an amount of information proportional to the number of tile types, $t = |T|$, multiplied by the amount of information required to represent each tile type, which given the number of glue types $g$ and the maximum glue strength $\\tau$, is $O(\\log(g)\\log(\\tau))$ bits. The seed structure $\\sigma$ can be encoded using an amount of information proportional to the number of locations it contains $|\\sigma|$, multiplied by the amount of information required to represent which tile type is at each location, which is $O(\\log(t))$, for a total of $O(|\\sigma|\\log(t))$ bits.\nWhile the number of tile types in $T$ impacts the number of bits required to represent each seed location, a constant number of tile types can clearly be used to define an infinite number of seeds.\nFigure \\ref{fig:garden-of-eden} gives an example of a tile set which can be used to form an infinite number of seeds that can encode any binary number and are stable at temperature $\\tau = 2$. It also gives an example of a seed structure which may encode information in its geometry.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=6.0in]{images\/garden-of-eden.png}\n    \\caption{(Left) An example of a constant-sized tile set which can be used to form an infinite number of stable (at temperature 2) assemblies in a ``Garden of Eden'' manner (i.e. they are not stable if even a single tile is missing, and thus there is no valid growth process in which they form). (Middle) An example assembly encoding a binary number via its north-facing glues. (Right) An example assembly potentially encoding information in its geometry (i.e. information that could be ``read'' by paths of tiles that may or may not crash into tiles in specific locations).}\n    \\label{fig:garden-of-eden}\n\\end{figure}\n\n\n\\input{uncomputable-blocking}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Impossible simulation at scale factor 1}\\label{sec:imp-scale-1}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=2.5in]{images\/simple-scale-1.png}\n    \\caption{A simple nondeterministic system with a multi-tile seed (green) that can be neither seed-growth-simulated, nor even shape-simulated, by any system with a single-tile seed at scale factor 1. This is because the center column grows upward to a nondeterministic height before possibly growing the pink row that initiates the downward growing blue columns. Each blue column may nondeterministically terminate with a red tile at any height, or continue until crashing into the seed. Any system attempting to simulate it at scale factor 1 must be able to grow a single side of the green row before growing rows that can crash into either side - or continue downward through and beyond the unfilled location of an incomplete side, which would lead to an invalid simulation and\/or shape. (Note that the spacing of two empty tile locations between pairs of columns prevents the use of fuzz from enabling a simulation at scale factor 1, and the nondeeterministic ability of blue columns to terminate with red tiles before crashing prevents shape simulation.)}\n    \\label{fig:simple-scale-1}\n\\end{figure}\n\nAlthough it is uncomputable to know if the leftmost seed tile will block growth in the system on the left in Figure \\ref{fig:TM-crash}, it is still possible to simulate that system at scale factor 1 with a system utilizing a single-tile seed because the leftmost tile of the multi-tile seed could be used as the single seed tile of the simulating system. The seed could then grow to the right, and if the simulated TM ever halts and a column grows downward, it is guaranteed that the blocking tile of the seed must already be there. (Note that this simulation is only seed-growth simulation if the first row of the zig-zag TM growth starts from the right side and grows right-to-left, since only then can the seed be guaranteed to be fully grown before other growth begins. Otherwise, it would be a seed-growth-simulation.)\n\nHowever, the system depicted on the right of Figure \\ref{fig:TM-crash} is different in this respect. Since there are two locations on opposite ends of the seed which have the potential to block growth, it turns out to be uncomputable to determine whether or not that system can be simulated at scale factor 1 by a system with a single-tile seed. This is because any green location which may be chosen as the location of the single seed tile in the simulator, which we'll call $\\mathcal{S}$, has the potential to result in an assembly in which a seed tile that would block growth in the original system, say $\\calT$, isn't yet placed when the growth to be blocked in $\\mathcal{S}$ arrives. Since the seed of $\\calT$ is a single-tile-wide row, the glues must be $\\tau$-strength between tiles in the green locations so growth can begin from the seed and directly proceed through the shortest sequence which completes placement of all tiles that serve as input to the TM simulation. At this point, it must be the case that one of the green end tiles (the leftmost or rightmost) has not been placed. From here, a valid assembly sequence is one which places no other tiles in the green locations but instead grows the TM simulations for an arbitrary amount of time. If both TMs halt, it is possible for the assembly sequence to proceed to grow both of the blue paths back down to the seed and for the one on the incomplete side to grow through the seed location.\nThus, it is only possible to always block one side, and it is uncomputable to know if both sides need to be blocked, and therefore uncomputable to know whether or not the simulation will fail\n\nWe now use Figure \\ref{fig:simple-scale-1} to present a much simpler system and give intuition as to why even the relaxed notion of shape-simulation (i.e. only needing to generate terminal assemblies with the same shapes as the original system, without needing to follow the same dynamics) using single-tile seeds is impossible for certain systems at scale factor 1. Note that properly accounting for all technicalities, such as growth in fuzz locations, requires more complexities and in-depth analysis such as that which is provided for the proof of Theorem \\ref{thm:multi-imposs} (which includes the impossibility of shape-simulation at scale factor 1, among others), and here we will ignore issues that arise with fuzz, etc. to present a high-level sketch that gives the general idea.\n\n\nWe will refer to the system depicted in Figure \\ref{fig:simple-scale-1} as $\\calT$. In  $\\calT$, the green row is the seed assembly. A column grows upward, nondeterministically attaching one of two tile types at each step. The first has the same glue on the north and south and thus allows continued growth, and the second has no glue on the north but has glues on the east and west. Thus, this column can grow arbitrarily high before stopping upward growth. Once a tile of the second type attaches, a row of three tiles grows to each side. Then, a tile of either of two tile types can attach to form a downward growing column on each side.\nOne of them has the same north and south glue and allows for the column to grow arbitrarily long, and the other (labeled `Y') has no southern glue so stops the growth of the column. The growth of each downward column is nondeterministically either stopped by the attachment of a Y tile, or continues until it is eventually blocked by a seed tile. System $\\calT$ makes an infinite number of terminal assembly shapes, which include assemblies with center columns of all heights, and for each of those, assemblies with all combinations of left and right columns of all lengths from 2 to that of the center column.\n\nTo show that $\\calT$ cannot be shape-simulated at scale factor 1, we'll assume that aTAM system $\\mathcal{S}$ is a system with a single-tile seed which attempts to do so.  First, we note that we will make (relatively trivial) use of the Window Movie Lemma from \\cite{temp1notIU} (which we include in Section \\ref{sec:wml} for completeness), that essentially says that as a constant-width path of tiles becomes arbitrarily long, it must eventually repeat the glues placed along multiple cuts (a.k.a. windows), as well as their ordering (a.k.a. window movies). This means that the segment of the path between those repeated cuts can be ``pumped'' (i.e. there is a valid assembly sequence in which that segment can be repeated an arbitrary number of times). The grey path growing upward and the blue paths which grow down to crash into the seed can be arbitrarily long. Therefore, no matter how many tile types are in $\\mathcal{S}$ it is possible for the columns to be tall enough that they must be pumpable.\n\nWe now consider all possible locations for the single seed tile of $\\mathcal{S}$, which we'll call $\\vec{s}$. We'll call the leftmost green location $\\vec{l}$, the middle $\\vec{m}$, and the rightmost $\\vec{r}$. If $\\vec{s}$ is located within a green location, it must be possible for the tile that fills location $\\vec{m}$ to initiate upward growth of the grey column since that is the only connection between the seed and an infinite number of assemblies whose blue columns stop before crashing into the seed. There must be a valid assembly sequence that grows a row directly from the seed location to place a tile at $\\vec{m}$ (noting that if $\\vec{m}$ is the seed tile location, no tiles need to be added), at which point an arbitrarily tall grey column must be able to grow upward. It must be possible for this growth, after reaching the distance necessary for pumping, to place the pink tile which initiates the growth to the sides and the downward growing blue columns. Since the blue columns must also be tall enough to pump, it must be possible to pump them until they reach the seed. Since the only portion of the green row to have grown was that directly from the seed location to the grey tile, there must be at least one of the locations $\\vec{l}$ or $\\vec{r}$ without a tile before the blue column arrives, allowing the blue column to continue pumping past the green row. This creates a shape which is invalid (i.e. it cannot be made in $\\calT$), so the simulator fails. Trivial case analysis shows that if the seed location is in any other location, rather than one of the green locations, it is still impossible to ensure that pumpable columns will be blocked, and thus $\\mathcal{S}$ must fail.\n\nThese scenarios display the difficulty associated with simulating systems with multi-tile seeds by those with single-tile seeds. Given an arbitrary system, it is uncomputable to know which seed locations, if any, can block growth. If multiple seed locations exist which can block growth, it is necessary that there be the ability to grow dependence paths from each to the portion of the assembly that could initiate the crashing growth. Without increasing the scale factor, it is not always possible to do so.\n\n\n    \n\n\n\n\n\\subsection{Shapes requiring infinite-sized seeds}\n\nTrivially, any finite shape (where we define a shape as a connected subset of $\\mathbb{Z}^2$) can self-assemble in the aTAM. For instance, given a target shape $S$, for every point in $S$ a unique tile type can be created which, for every adjacent tile when positioned correctly in $S$, has a strength-1 glue that is unique to that pair of adjacent tiles on their abutting sides. Thus, the size of the tile set is the number of points in $S$. The seed can be any single tile from the tile set, and the temperature $\\tau=1$. This system will be directed and self-assemble a single terminal assembly of shape $S$.\n\nTherefore, we consider infinite shapes. The aTAM requires that any seed assembly be finite, that is, a seed can contain only a finite number of tiles. Because of this, there are shapes that cannot self-assemble in any system in the aTAM. It has previously been shown that there exists an infinite class of shapes that cannot self-assemble in the aTAM \\cite{jSSADST,jSADSSF} (the Sierpinski triangle and many other discrete self-similar fractals), and it would be easy to show that these could self-assemble if allowed infinite seeds (e.g. the seeds could trivially be the entire shapes). However, for completeness in our discussion of how seeds impact self-assembly in the aTAM, here we present a simpler example of a class of shapes which would also require infinite seeds to self-assemble in the aTAM and provide a short proof. An example can be seen in Figure \\ref{fig:infinite-seed-shape}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=2.0in]{images\/infinite-seed-shape.png}\n    \\caption{The left side of an infinite shape which could only self-assemble in the aTAM if seeds of infinite size were allowed. The full shape consists of an infinite horizontal line (along the bottom), an infinite diagonal line up and to the right, and vertical lines between those two in every third column.}\n    \\label{fig:infinite-seed-shape}\n\\end{figure}\n\n\\begin{observation}\\label{obs:infinite-seed}\nThere exist shapes which could only self-assemble in the aTAM if seeds with an infinite number of tiles were allowed.\n\\end{observation}\n\n\\begin{proof}\nTo prove Observation \\ref{obs:infinite-seed}, we refer to the shape depicted in Figure \\ref{fig:infinite-seed-shape}, which we'll call $S$. $S$ consists of two infinite paths. One is horizontal and goes infinitely far to the right. The other begins at the left end of the first and goes diagonally upward and to the right infinitely far. At every third $x$-coordinate, a vertical column exists between the two lines. Our proof will be by contradiction, so assume that $\\mathcal{S} = (T,\\sigma,\\tau)$ is a system with a finite seed (i.e. $|\\sigma| < \\infty$) which self-assembles $S$. Let $x_{max}$ be the $x$-coordinate of the easternmost tile of $\\sigma$. Since $S$ is infinite to the right, each of the two main paths extends infinitely far to the right of $x_{max}$, and there are an infinite number of vertical columns to the right of $x_{max}$, each taller than the one to its left.  Clearly, regardless of the size of $\\mathcal{S}$'s tile set, each column eventually becomes tall enough to be pumpable. Let that height be denoted by $p$. Let $c_{2p}$ be the first column of height $2p$ which is to the east of any point in $\\sigma$. We note that, in order for $\\mathcal{S}$ to correctly create an assembly of shape $S$, it must be possible for growth to proceed from $\\sigma$ to at least one of the bottom or top tiles of $c_{2p}$, and for that tile to initiate growth of at least half of column $c_{2p}$. (If that is not possible, then either the assembly never gets to column $c_{2p}$, or neither the top nor the bottom can initiate growth that gets to the middle, so column $c_{2p}$ isn't completed and either way $\\mathcal{S}$ fails to make shape $S$.) Without loss of generality, assume that $\\sigma$ can grow until placing the bottom tile of $c_{2p}$, which is then able to initiate growth upward at least half the height of that column, namely distance $p$. (If instead it is only the top tile that can grow this far, simply flip the directions in the rest of the proof.) \nThere must exist some valid, minimal assembly sequence which grows from $\\sigma$ to place the bottom tile of $c_{2p}$. Let $c_{2p-1}$ denote the column which is the first to the left of $c_{2p}$. \nNote that it must be possible to grow the path from the bottom of $c_{2p-1}$ to the bottom of $c_{2p}$ without growing the path from the top of $c_{2p-1}$ to the top of $c_{2p}$, since they are disjoint and each one-tile-wide and thus there cannot be a dependency that prevents this.\nNow, in $\\mathcal{S}$ grow column $c_{2p}$ upward to the mid-point, which it is known to be able to do. However, since the length of this path is the pumping length, it must also be possible to increase the length of this path arbitrarily. Therefore, do so and pump the growth of the column beyond the height $2p$, which must be possible since the top tile of $c_{2p}$ is not in place to block. This places at least one tile outside of shape $S$ and this is a contradiction to the claim that $\\mathcal{S}$ self-assembles shape $S$. The only assumption made of $\\mathcal{S}$ was that it had a finite seed, thus $S$ cannot be self-assembled within any aTAM system with a finite seed. However, if the seed was allowed to be the infinite bottom row, for example, a system using that seed could easily self-assemble $S$ by growing the diagonal row upward and to the right, and initiating all column growth downward from that. Each column would be guaranteed to crash into the seed row, so $S$ would be correctly self-assembled and Observation \\ref{obs:infinite-seed} is proven.\n\\end{proof}\n\nIt is worth noting that this shape actually cannot self-assemble at any scale factor. As the columns get arbitrarily long, any fixed-width provided by a constant scale factor would eventually be unable to prevent pumping, and it must always be the case that either the top or the bottom could grow after the other, and thus the pumping arm could go past that boundary. There exist an infinite set of similar shapes, each differing by just the spacing between the vertical columns and each similarly impossible to self-assemble in the aTAM with a finite seed. Thus, this exhibits an infinite set of shapes, each of which cannot self-assemble in the aTAM at any scale factor.\n\n\n\n\\subsection{Seed-First-Simulation at Scale Factor 3}\\label{sec:scale3-sim-fuzz}\n\n\n\nWe demonstrate that an arbitrary aTAM system is able to be seed-first-simulated at scale factor 3 utilizing cheating fuzz.\n\n\\begin{theorem}\\label{thm:scale3-sim}\nGiven an arbitrary aTAM system $\\mathcal{T} = (T,\\sigma,\\tau)$, there exists an aTAM system $\\mathcal{T}_3 = (T_3, \\sigma_0, \\tau_3= \\max(2, \\tau))$ which seed-first-simulates $\\mathcal{T}$ at scale factor 3 utilizing cheating fuzz, where $|\\sigma_0| = 1$ and $|T_3| \\leq 20s + 16g + 6t$ given that $s = |\\sigma|$, $t = |T|$, and $g$ is the total number of unique glue\/strength combinations in $T$ and $\\sigma$.\n\\end{theorem}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.25\\textwidth]{images\/S3-7_cooperativity_scale3.png}\n    \\caption{For a scale 3 supertile, the point of cooperativity\/competition resides in the center of the $3 \\times 3$ square (denoted with `C').}\n    \\label{fig:S3-7_cooperativity_scale3}\n\\end{figure}\n\nThe full proof of Theorem~\\ref{thm:scale3-sim} is found in Section~\\ref{sec:scale3-sim-proof}.\nThis proof is by construction which, while similar to that presented in the construction of Theorem~\\ref{thm:scale4-sim}, requires extra care in the routing of the both the dependence path and perimeter path through the seed-representing assembly to prevent diagonal fuzz by the tiles in $T_\\sigma$.\nIn a scale 4 supertile, the perimeter path contains tiles in every supertile, regardless of its location in $\\sigma$.\nThis is not possible in scale 3 seed assemblies, specifically with regards to supertiles which are not on the perimeter of a structure.\nThis potentially inhibits the placement of perimeter tiles in supertiles adjacent to \\emph{cavities} (locations in $\\mathbb{Z}^2$ which do not contain a tile but are surrounded by seed tiles) which may be present in a multi-tile seed.\nThis is necessary for seed-first-simulation.\nTo ensure the perimeter path may reach these internal cavities, we provide a modification to the connectivity of the seed prior to assigning supertiles from a set of supertile templates  which leverages the changes in connectivity.\nFor the generation of the remaining tiles of $\\calT$, we utilize the the same techniques of Theorem~\\ref{thm:scale4-sim} to develop a $T_{IO}$, and the design for the point of competition\/cooperation in the $3 \\times 3$ supertiles is slightly modified, as shown in Figure \\ref{fig:S3-7_cooperativity_scale3}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Seed-First-Simulation at Scale Factor 4}\\label{sec:scale4-sim}\nWe begin by introducing the construction for seed-first-simulation at scale 4 without cheating fuzz.\nAlthough it is not the smallest possible scale factor for simulation, scale factor 4 has favorable characteristics that allow us to build an overall process for seed-first-simulation in a relatively straightforward manner that can be re-utilized in our scale factor 3 construction. \n\n\\begin{theorem}\\label{thm:scale4-sim}\nGiven an arbitrary aTAM system $\\mathcal{T} = (T,\\sigma,\\tau)$, there exists an aTAM system $\\mathcal{T}_4 = (T_4, \\sigma_0, \\tau_4= \\max(2, \\tau))$ which seed-first-simulates $\\mathcal{T}$ at scale factor 4 and does not use cheating fuzz, where $|\\sigma_0| = 1$ and $|T_4| \\leq 28s + 16g + 6t$ given that $s = |\\sigma|$, $t = |T|$, and $g$ is the total number of unique glue\/strength combinations in $T$ and $\\sigma$.\n\\end{theorem}\n\nThe full proof of Theorem~\\ref{thm:scale4-sim} is located in Section~\\ref{sec:scale4-sim-proof}, and here we provide a brief overview.\nThe construction of a simulating system $\\mathcal{T}_4$ consists of two main parts.\nFirst, we create a tileset $T_\\sigma$ which assembles a scale-4 version of the seed of $\\calT$, $\\sigma$.\nThe generation of $T_\\sigma$ comprises the brunt of the construction.\nThe new assembly that grows to represent the seed $\\sigma$ can be logically thought of as having 2 parts: the \\emph{core path} and the \\emph{perimeter path}. Each scale 4 supertile of $T_\\sigma$ is shown to contain an interior scale 2 square which can be easily connected to its neighboring scale 2 squares; these 4 tiles are called \\emph{core tiles}.\nThe core path is a dependence path which represents a Hamiltonian path through the core tiles, allowing for all supertiles in the seed to be connected to the remainder of the seed assembly, enabling seed-first simulation.\nBy focusing on the connectivity of the core tiles within the supertiles of the seed assembly, the core path is generated utilizing the fact that all shapes with scale factor $2$ contain a Hamiltonian cycle (see proof in \\cite{SummersTemp}).\nWe generate both the core path and the perimeter path from a set of scale factor 4 template tiles which take advantage of the presences of a Hamiltonian cycle in the core tiles.\nOne of the supertiles represented as a template is the \\emph{origin supertile} which contains the single-tile seed $\\sigma_0$, and the beginning of the perimeter path.\nWe connect the end of the core path to the beginning of the perimeter path; this is the key part of the construction which allows for seed-first simulation.\nOn the edges of tiles of the perimeter path, exterior glues allow for the attachment of tiles.\nA high level visualization of the steps of generating the new tileset $\\calT_4$ is found in Figure~\\ref{fig:S4-4_spanning_tree}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.6\\textwidth]{images\/S4-4_spanning_tree.png}\n    \\caption{The process of creating the tiles for a simulation at scale 4 with a single-tile seed. Beginning with the binding graph of a seed assembly $\\sigma$ (left), we generate a spanning tree from the westernmost tile of the southernmost row (middle). From this spanning tree, we replace each vertex with a tile from the template provided by Figure~\\ref{fig:S4-2_seed_path_macrotiles}, leading to the tileset which comprises the scale factor 4 simulation of our initial multi-tile seed (right). The new single-tile seed of $\\calT_4$ is shown as a teal tile.}\n    \\label{fig:S4-4_spanning_tree}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.25\\textwidth]{images\/S4-8_scale_4_template.png}\n    \\caption{High-level scheme for scale-4 supertiles, showing the point of competition\/cooperation. Consider the fully colored $4 \\times 4$ supertiles that are filled in to already be representing tiles of the simulated system, and the central $4 \\times 4$ supertile to not yet be filled in enough to represent a tile. Locations in the central supertile with the same colors as neighboring supertile locations depict the paths by which tiles from the filled-in supertiles grow into an empty neighboring supertile. The location labeled 'C' is the point of competition\/cooperation where tile placement decides the identify of the supertile.}\n    \\label{fig:S4-8_scale_4_template}\n\\end{figure}\n\nSecond, we create scale 4 versions of the tileset $T$.\nBefore creating the scaled versions of $T$, we additionally need to prevent the re-growth of portions of the seed which attempt to re-grow the perimeter path.\nRe-growth is prevented by generating $T_{IO}$, an expansion of $T$, which identifies all possible $\\tau$ strength combinations which can allow a tile to attach (serving as its ``input'' glues) and then generating new tiles with specific `inward' and `outward' glues. (This is a standard technique in tile assembly constructions, and can be found in the construction of \\cite{Versus} and others).\nFinally, we generate the scale 4 representations of $T_{IO}$ based upon a single point of cooperation and\/or competition between the supertiles.\n(See Figure \\ref{fig:S4-8_scale_4_template} for a high-level depiction.)\nIn addition to assigning tiles which grow into legal fuzz regions, we must take into account collisions of an arbitrary supertile with the seed assembly and add glues which allow for the continuation of the dependence path.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n    \n\n\n\n\n    \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Technical details of the optimal tile complexity for seed-first-simulation}\\label{sec:scaled-sim-append}\n\nIn this section we provide the proof and technical details for Theorem \\ref{thm:scaled-simulation}.\n\nFirst, we define a few structures and auxiliary tile sets that will be useful in our construction.\n\nIn \\cite{IUSA}, a tile set $U$ was given that is intrinsically universal for the aTAM. This means that, given an arbitrary aTAM system $\\calT = (T,\\sigma,\\tau)$, there is a function that specifies how the tiles of $U$ can be arranged to form a seed structure, say $\\sigma_\\calT$, and another function that returns a representation function $R$ mapping macrotiles of over $U$ to tiles of $T$, so that the resulting system correctly simulates $\\calT$ (at temperature $2$, regardless of the value of $\\tau$ in $\\calT$) using $R$. In that construction, the macrotiles (a.k.a. supertiles) use a well-defined format for their sides, called \\emph{supersides}. The format used in \\cite{IUSA} is shown on the top in Figure \\ref{fig:supersides}. For our construction, we will slightly modify the encoding of supersides by adding ``spacer'' sections, which are potentially very large sections that are simply passed over by the gadgets of the construction and used merely to allow for correct alignment and a larger scale factor that is driven by the space required to simulate a relatively complex Turing machine. In our construction, we will utilize the superside encoding on the bottom, and it will frequently be split in half into portions we'll refer to as $g_l$ and $g_r$. For a given $\\calT$, all segments will have constant encodings across all macrotiles except for the ``glue'' sections which will consist of the binary encoding of the specific glue that each superside is simulating. To correctly utilize the superside encoding to perform simulations following the procedure of \\cite{IUSA}, our construction will use a slightly modified version of their tile set $U$, which we will call $U'$. The only differences with $U'$ will be its ability to skip over the (arbitrarily long) spacer sections (effectively just absorbing the larger scale factor imparted by them, and a slight change to the tiles that form the left end of a so-called ``frame'' row (which will be discussed when the tiles for growing the ``activation row'' are explained\n\n\\begin{proof}\nWe prove Theorem~\\ref{thm:scaled-simulation} by construction. Therefore, let $P_\\calT$ be a program that outputs $\\calT = (T,\\sigma,\\tau)$, an arbitrary aTAM system.\nWe will define $\\mathcal{S}_\\calT = (T_\\calT,\\sigma_\\calT,2)$ such that $\\sigma_\\calT$ consists of a single tile and $\\mathcal{S}_\\calT$ seed-first-simulates $\\calT$ at some scale factor $c \\in \\mathbb{Z}^+$ while using $O(\\frac{|P_\\calT|}{\\log{|P_\\calT|}})$ tile types.\n\nFirst, we will use the technique of \\cite{AdChGoHu01} to compress $P_\\calT$, allowing for the use of an optimal number of tile types.\nWe note that the binary string $P_\\calT$ can be encoded in a higher base, which we'll refer to as $b$.\nThis allows for the length of the compressed encoding to be reduced to $\\frac{|P_\\calT|}{\\log(b)}$. Tiles can be designed so that from the seed tile a row of $\\frac{|P_\\calT|}{\\log(b)}$ tiles assemble with that compressed encoding on their north sides (requiring $\\frac{|P_\\calT|}{\\log(b)}$ unique tile types).\nFollowing the construction of \\cite{AdChGoHu01}, tiles of $O(b)$ types can then assemble to the north of that row to ``decompress'' the string, resulting in the encoding of $P_\\calT$ on the northern glues of the tiles of their northern row.\nNote that, in order to correctly integrate with the tiles to be described later, these decompression tiles also perform binary counting to count the number of rows that they assemble during the decompression. (This only increases the tile complexity by a constant multiplicative factor, thus the decompression still only requires $O(b)$ tile types.)\nThe subest of tile types described so far will be referred to as $T_\\sigma$ later in this proof.\nAt this point, the tile complexity is $\\frac{|P_\\calT|}{\\log(b)} + O(b)$.  As discussed in \\cite{AdChGoHu01}, the base $b$ can be selected such that this quantity is equal to $O(\\frac{|P_\\calT|}{\\log(|P_\\calT))})$. Finally, we note that the length of the shortest program $P_\\calT$ which outputs $\\calT$ is the Kolmogorov complexity of $\\calT$, denoted $K(\\calT)$. Thus, this portion of the construction requires $O(\\frac{K(\\calT)}{\\log(K(\\calT)})$ tile types. As will be shown, all other tile types used are from a constant tile set, independent of the system $\\calT$, and thus the overall tile complexity is $O(\\frac{K(\\calT)}{\\log(K(\\calT)})$. This is the information theoretic lower bound for an aTAM tile set representing $\\calT$.\n\nWe now define the rest of the construction.\nLet $\\sigma^2$ be a mapping of each point of $\\sigma$ to a $2 \\times 2$ square (i.e. the points of $\\sigma$ at scale factor 2).\nLet $H'$ be a Hamiltonian cycle through $\\sigma^2$ and consider it to begin and end in the leftmost of the bottom-most $2 \\times 2$ square of $\\sigma^2$, starting in its top-left and ending in its bottom-left. (See Figure \\ref{fig:ham-cycle-encoding} for an example.)\n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=1.0\\textwidth]{images\/supertile-sides.png}\n    \\caption{(Top) Encoding of the information presented on a superside in the intrinsically universal construction of \\cite{IUSA}. The length of each section is listed below it, with respect to the tile set $T$ that is being simulated, (Bottom) Modified superside encoding for the construction in Theorem \\ref{thm:scaled-simulation}. Note the additional ``spacer'' sections (whose sizes will be related to the running time of Turing machine $D$), and the division into halves called ``$g_l$'' and ``$g_r$''.}\n    \\label{fig:supersides}\n\\end{figure}\n\nDefine Turing machine $D$ that uses a one-way-infinite-to-the-right tape, takes as input program $P_\\calT$, and does the following:\n\\begin{enumerate}\n    \\item $D$ moves its head across the input from left-to-right then back from right-to-left. (This is merely a technicality to guarantee that the runtime is longer than the output, a fact which will be utilized later.\n    \n    \\item $D$ runs $P$ to obtain the string encoding the definition of $\\calT$, which we will refer to as $\\langle \\calT \\rangle$.\n    \n    \\item $D$ uses $\\langle \\calT \\rangle$ to run the algorithm of \\cite{IUSA} and compute the modified encoding of a superside for a macrotile simulating $\\calT$ (as shown in Figure \\ref{fig:supersides}). In the base encoding, the spacer sections will be a single tile wide and the glue encoding will be of the \\emph{null} glue. This encoding will be split into a left half and a right half, which we will call $g_l$ and $g_r$, respectively.\n\n    \\item $D$ uses the definition of the seed $\\sigma$ from $\\langle \\calT \\rangle$ to compute the set of points contained within $\\sigma^2$, which we will refer to as $\\langle \\sigma^2 \\rangle$. (See Figure \\ref{fig:ham-cycle-encoding} for an example.)\n    \n    \\item $D$ uses $\\langle \\sigma^2 \\rangle$ to compute the Hamiltonian cycle $H'$. (See Figure \\ref{fig:ham-cycle-encoding} for an example.)\n    \n    \\item $D$ creates a string, which we'll call $H$, that contains a section encoding each location on the cycle $H'$. The full string $H$ is a concatenation of one section for each point on the path. The encoding for each point is the following:\n    \n        \\begin{enumerate}\n            \\item Each contains 3 portions, labeled $F$, $L$, and $R$ for ``forward'', ``left'', and ``right'', respectively.\n            \n            \\item For all sections other than the one for the final point, exactly one of the portions will be marked with either a ``$^*$'' or a ``\\#'', indicating that that is the direction of travel to the next point. (For example ``$F^*$'' means the path continues forward in the same direction as from the previous point, ``$L^*$'' means it turns left, and ``$R^*$'' means it turns right.) The difference between the ``$^*$'' or a ``\\#'' markings are that ``$^*$'' indicates that the next location is within the same $2 \\times 2$ macrotile as the current location, and ``\\#'' indicates that the location is in a different, neighboring $2 \\times 2$ macrotile.\n            \n            \\item Each of the two remaining un-marked portions will either have (i) the symbol ``\\_'' to indicate that its side is an interior side of $\\sigma^2$ (i.e. it is adjacent to another block in $\\sigma^2$) and thus no information needs to be output to that side, or (ii) if the side is an exterior side of $\\sigma^2$ (i.e. it is on the perimeter of $\\sigma^2$, whether that is the external perimeter or the perimeter of a cavity enclosed by $\\sigma$) one of the strings $g_l$ or $g_r$, with either the \\emph{null} glue or the encoding of a glue type in the portion of $g_l$ or $g_r$ labeled ``glue'' in Figure \\ref{fig:supersides}. (Figure \\ref{fig:ham-cycle-encoding} shows a shorthand encoding of the first 7 locations of the path. Rather than giving a full encoding of each of $g_l$ or $g_r$, it simply notes which is used for that location and gives the name of the glue that is encoded within it, or ``\\_'' if it is the \\emph{null} glue.) Traveling clockwise around the exterior of a $2 \\times 2$ square of $\\sigma^2$, the counterclockwise most of the two edges making up each side receives the $g_l$ string, and the clockwise most receives the $g_r$ string. Additionally, there are special markings that may be added to the end of the $g_r$ encoding as follows:\n                \\begin{enumerate}\n                    \\item If the right side of the $g_r$ encoding will be at the end of a convex corner, the symbol `c' is added to the rightmost glue of $g_r$.\n                    \n                    \\item If the right side of the $g_r$ encoding will be at the boundary between two different $2 \\times 2$ blocks of macrotiles but not at a concave corner, the symbol '4' is added to the rightmost glue of $g_r$.\n                    \n                    \\item If the right side of the $g_r$ encoding will be at the boundary between two different $2 \\times 2$ blocks of macrotiles at a concave corner, the symbol `7' is added to the rightmost glue of $g_r$.\n                \\end{enumerate}\n        \\end{enumerate}\n        \n        Additionally, another type of special marker is added to $H$ as follows:\n        \n        \\begin{enumerate}\n            \\item In the encoding of the last location in $H$, for the portion labeled $L$, the leftmost symbol of the encoding of $g_r$ is given the additional special marker `A'. This will be used to eventually trigger the initiation of the pre-activation row which will begin its growth around the exterior of the seed-representing assembly after this final macrotile of $H$ has grown.\n            \n            \\item If the seed $\\sigma$ contains any internal cavities that are completely enclosed by $\\sigma$, special handling for those cavities is required. For each such cavity, $D$ determines which side of which macrotile will be the final to be grown along it. Let $d \\in \\{F,L,R\\}$ be the direction of that side relative to the growth of that macrotile. Let $g_d \\in \\{g_l,g_r\\}$ be the string that is to be placed on the $d$ side of the macrotile. If $g_d = g_l$, then the rightmost symbol is given the additional special marker `A'. Else, if $g_d = g_r$, then the leftmost symbol is given the additional special marker `A'. This will be used to eventually trigger the initiation of the pre-activation row which will begin its growth around the the interior cavity to which it is adjacent after this final macrotile of on the perimeter of that cavity has grown.\n        \\end{enumerate}\n        \n    \\item $D$ outputs $H$ and halts.\n\\end{enumerate}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=4.0in]{images\/optimal-macrotile-growth.png}\n    \\caption{Growth of the first seed macrotile, representing one location of a $2 \\times 2$ block in $\\sigma^2$. Note that this is a generic example that doesn't match with the first macrotile of the example from Figure \\ref{fig:ham-cycle-encoding}. For an example which does match, and depicts the first two macrotiles, see Figure \\ref{fig:two-optimal-macrotiles}.}\n    \\label{fig:optimal-macrotile-growth}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=4.0in]{images\/optimal-macrotiles-3.png}\n    \\caption{Example of the first two macrotiles from the example in Figure \\ref{fig:ham-cycle-encoding}.}\n    \\label{fig:two-optimal-macrotiles}\n\\end{figure}\n\nThe tile set $T_\\calT$ can now be constructed as follows. $T_\\calT$ is the union of the following tile sets:\n\\begin{enumerate}\n    \\item The previously defined tile set $U'$, which is a modification of the tile set $U$ from \\cite{IUSA}.\n    \n    \\item The tile set $T_\\sigma$, which was previously discussed, is hard-coded to grow from the seed tile a row whose northern glues encode the compressed version of $P_\\calT$, and then decompress the bits of $P_\\calT$ (the program which outputs $\\calT$) in binary. It will also display a binary number encoding the number of rows which have grown upward to that point. The rightmost tile of the initial row will be the seed tile, $\\sigma_\\calT$ of $\\mathcal{S}_\\calT$. The glues between the tiles of this initial row will be strength-2, allowing the full row to grow directly from the seed tile. Additionally, the northern glue of the leftmost tile of the northern row will not only encode the first bit of $P_\\calT$, but it will also encode the start state of Turing machine $D$ and be a strength-2 glue.\n    \n    \\item The tile set $T_D$, which is a tile set that simulates Turing machine $D$ in a standard zig-zag manner (see \\cite{DirectedNotIU}, among many others, for examples of zig-zag Turing machine simulations). Additionally, (1) each row of the simulation of $D$ grows one additional tile to each of the left and right sides, and (2) the bits of a binary counter are overlaid on the tiles of the simulation, with the least significant bit being on the leftmost tape cell (although each row extends by one additional tile to both the left and right ends, the Turing machine $D$'s tape has a fixed left end which never moves) and the bits extending to the right. This counter starts from the value encoded by the binary counter of $T_\\sigma$ and is incremented during the growth of every row so that it effectively keeps a count of the runtime of $D$.\n    \n    \\item The tile set $T_{splitter}$. (There are four rotated versions of these tiles, so we will explain the version that grows from north to south. The other three versions simply consist of rotated versions of the described tile types.) These tiles grow on top of row which encodes (from left to right) an arbitrary number of ``blank encoding'' tiles, followed by an encoding of $H$ overlaid with the bits of a binary counter, followed by an arbitrary number of ``blank encoding'' tiles. The grow in a zig-zag manner and cause portions of the information in the input row to be rotated to each of the left and right, as well as propagated northward. These tiles only copy a subset of the input row to be propagated in each direction. (See Figure \\ref{fig:optimal-macrotile-growth} for an example of the ``splitter'' in the middle square.) The subset for each direction is determined by the leftmost section of the encoding of $H$, which contains a portion for each direction ($F$, $L$, and $R$).\n    \n        \\begin{enumerate}\n            \\item For the direction marked with either a ``$^*$'' or a ``\\#'', the full encoding of $H$ is copied to the side minus the leftmost section (i.e. the leftmost $F$, $L$, and $R$ portions are not copied, but instead replaced by ``blank'' values). This serves to erase that portion from $H$ as it is passed to the next block, so that the leftmost remaining portion will encode the correct instructions for that block. If the erased direction was marked with ``\\#'', that symbol is still included on the left side of $H$ to signal to the next component (the copier) that the information is being copied across the boundary of one $2 \\times 2$ macrotile into another. Additionally, the overlaid bits of the binary counter are copied in this direction.\n            \n            \\item For any direction which consist of either $g_l$ or $g_r$ (with the appropriate glue encodings), that string is rotated to that side, along with the overlaid bits of the binary counter. The rest of the positions encode ``blank'' values.\n            \n            \\item For any direction consisting of just the blank, ``\\_'', nothing is rotated to that side, and that side of the macrotile is not grown (since it is interior to the $2 \\times 2$ block of $\\sigma^2$ and thus can't be used to initiate any additional growth).\n        \\end{enumerate}\n        \n\\begin{figure}\n    \\centering\n    \\includegraphics[width=5.0in]{images\/optimal-macrotile-dimensions.png}\n    \\caption{A schematic depiction of dimensions of portions of the macrotiles in $\\mathcal{S}$ and how the shifting of the information from the encoding of either $g_l$ or $g_r$ occurs.}\n    \\label{fig:optimal-macrotile-dimensions}\n\\end{figure}\n\n    \\item The tile set $T_{shifter}$. (There are four rotated versions of these tiles, so we will explain the version that grows from north to south. The other three versions simply consist of rotated versions of the described tile types.) These tiles grow on top of row which encodes (from left to right) an arbitrary number of ``blank encoding'' tiles, followed by an encoding of either $g_l$ or $g_r$, followed by an arbitrary number of ``blank encoding'' tiles, with a subset of the tiles overlaid with the bits of a binary counter. These tiles cause zig-zag rows to grow which decrement a copy of the binary counter value until it reaches the value 0\n    , extend each row by an additional tile on each of the left and right sides, and copy the encoding of $g_l$ or $g_r$ upward while shifting its two halves to either end of the final row. As mentioned previously, and shown in the bottom of Figure \\ref{fig:supersides}, each of $g_l$ and $g_r$ have a ``spacer'' section which is initially a side tile wide. As these zig-zag rows grow upward, they shift the information on the left of the spacer to the left, and the information to the right of the spacer to the right. Since rows alternate direction of growth, they alternate the direction in which they can shift information. Initially, the information which is being shifted in the same direction as the growth of a row is shifted by 6 positions. This is possible by having glues that encode up to the last 6 ``remembered'' values at a time. Once the information being shifted reaches the end of the row, the shifting switches to 2 positions per shift so that the information remains at the end of the row until arriving at the final row. (See Figure \\ref{fig:optimal-macrotile-dimensions} for a schematic depiction of the dimensions of a macortile and how the encodings of $g_l$ and $g_r$ are shifted.) The reason for initially shifting 6 spaces at a time is that, since shifting occurs every second row for each direction, this provides a slope for the information being shifted of $2\/6$, i.e. $1\/3$. The dimensions of the macrotiles are largely dictated by dictated by the runtime of $D(\\calT)$, which is denoted by the value $t$ in Figure \\ref{fig:optimal-macrotile-dimensions}. Since $D$ is designed so that its runtime must always be greater than the length of its output, the length of $H$ must always be less than $t$ (by - significantly - more than 2). Therefore, the maximum amount that any information could need to be shifted is bounded above by $3t$ (see Figure \\ref{fig:optimal-macrotile-dimensions}). Since the shifter grows upward a distance of $t-2$ rows, that means a slope of $1\/3$ must always suffice to shift the information to the correct information before reaching the top row. Once reaching the edge of the row, the information can then be shifted at a slope of $1\/1$ for the remaining rows to stay at the end of the rows. The tiles that perform the shifting can easily accommodate this by having a special marking on the tiles at the end of the rows to detect when the shifting information overlaps that mark.\n    \n    \\item The tile set $T_{copier}$. (There are four rotated versions of these tiles, so we will explain the version that grows from north to south. The other three versions simply consist of rotated versions of the described tile types.)  These tiles grow on top of row which encodes (from left to right) an arbitrary number of ``blank encoding'' tiles, followed by an encoding of $H$ overlaid with the bits of a binary counter, followed by an arbitrary number of ``blank encoding'' tiles. There may also be a ``\\#'' symbol on the left end of the encoding of $H$. These tiles cause zig-zag rows to grow which simply decrement a copy of the binary counter value until it reaches 0, while also copying the original value of the binary counter as well as the encoding of $H$ upward. If a ``\\#'' symbol is on the left end of $H$, an additional 4 rows are grown upward, copying all of the same information but erasing that ``\\#'' symbol. (This accounts for growth from one $2 \\times 2$ macrotile into another and the spacing required to account for the pre-activation and activation rows that are between each pair of macrotiles where at least one is a non-seed macrotile. These will be described later.) Since this copying behavior will always need to be performed twice in a row (except for the possible spacing of the extra 4 rows), a special marking is also copied upward through the first occurrence of the copier so that a second occurrence of the copier is initiated. (An example can be seen at the top of the first macrotile and bottom of the second macrotile in Figure \\ref{fig:two-optimal-macrotiles}.)\n    \n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{images\/scaled-simulation-seed-macrotile-frame.png}\n    \\caption{Depiction of the growth of the pre-activation (green) and activation (blue) rows around a simple assembly of seed macrotiles representing a single-tile seed $\\sigma$, and the locations of possible future surrounding macrotiles outside of the representation of $\\sigma$. Note that the rows showing the locations of pre-activation and activation rows, and between surrounding macrotile locations, are not remotely to scale but shown much larger for clarity, as the scale factor is very large and the size of rows is significantly smaller in the depicted macrotiles in the center.}\n    \\label{fig:scaled-simulation-seed-macrotile-frame}\n\\end{figure}\n\n    \\item The tile set $T_{act}$. This tile set creates two rows that grow around a perimeter of the assembly consisting of the seed-representing macrotiles. This includes the external perimeter around the entire seed-representing assembly, as well as the perimeters of any internal cavities (if there are any). The first row to grow is called the \\emph{pre-activation} row, and begins growth when the special marker `A' makes it to the exterior row of a macrotile tile.\n   \n   \n   \n    This special marker causes the tile that is placed there \n   \n    to include an exterior facing strength 2 glue that initiates the growth of the pre-activation row that grows in a clockwise direction around the perimeter of which that side is a part. (This is shown as the green row in Figure \\ref{fig:scaled-simulation-seed-macrotile-frame}.)  The tiles of this row (and the activation row which will piggyback on it) grow via cooperation between a glue on the side of the tile most recently placed in the path facing the direction of growth, and a glue on the exterior of the macrotile over which it is growing. However, there are a few special cases in which such cooperation is not possible during the growth of the pre-activation row. Each of these is detected by the growing row due to a special glue marking on the rightmost glue of a $g_r$ encoding. They are as follows:\n    \n        \\begin{enumerate}\n            \\item Convex corner:  Since such cooperation isn't possible when the row needs to grow around a convex corner, whenever the row reaches the end of the encoding of a copy of $g_r$ and the rightmost glue is marked with a `c', this signifies that it is at a convex corner, so a tile is placed with a strength 2 glue allowing for the placement of a tile that facilitates turning the corner. (Examples can be seen as purple tiles in the pre-activation row in Figure \\ref{fig:scaled-simulation-seed-macrotile-frame}.)\n            \n            \\item Straight gap between macrotiles of different $2 \\times 2$ blocks: Since there will be an additional spacing of 4 tiles between such macrotiles, the rightmost glue of $g_r$ will have been marked with a '4', and this will cause a pre-activation row tile to attach that is able to initiate the growth of a set of 4 tiles that can grow across the gap. They can be generic tiles, unspecific to the encodings of $g_l$ or $g_r$ because they will be in locations don't need to propagate any such information since they will be interfacing with the tiles from the IU construction of \\cite{IUSA} and in the corresponding locations there are generic `frame' tiles.\n            \n            \\item Concave corner between macrotiles of different $2 \\times 2$ blocks: Since there will be a spacing requiring 7 non-cooperative tiles between such macrotiles to go around the corner crossing the two 4-tile-wide gaps, the rightmost glue of $g_r$ will have been marked with a '7', and this will cause a pre-activation row tile to attach that is able to initiate the growth of a set of 7 tiles that can grow around that corner, across the gaps. As before, these can be generic tiles.\n        \\end{enumerate}\n    \n    This pre-activation row is guaranteed to be able to successfully complete for each perimeter on which it grows, even if it has to temporarily pause progress as some output sides generated by shifter modules may need extra time to complete. However, all macrotiles are guaranteed to already be in place and to have initiated their outputs to the point of supporting growth of the pre-activation row, but none of them can initiate any growth beyond that point, so there is no growth that can interfere with the completion of the pre-activation row. The tiles of the pre-activation row copy the information from the exterior of the macrotiles to their exterior sides, resulting in the perimeter of each pair of adjacent macrotiles in the same $2 \\times 2$ square containing a complete superside representation in the form shown in Figure ~\\ref{fig:supersides}, which is equivalent to that of the IU construction in \\cite{IUSA} with the addition of the two extra ``spacer'' segments. This is because the two adjacent macrotiles will expose the encodings of $g_l$ and $g_r$ so that they are concatenated. In this way, each $2 \\times 2$ block of seed-representing macrotiles in $\\mathcal{S}_\\calT$ will represent a single tile of the seed in $\\calT$.\n    \n    When a pre-activation row completes by placing the final tile, which is adjacent to the first tile,\n    (the final tile of the external perimeter is shown as yellow in Figure \\ref{fig:scaled-simulation-seed-macrotile-frame}) that final tile presents a strength 2 glue outward to initiate growth of the \\emph{activation} row. The activation row also grows in a clockwise manner, on the outside of the pre-activation row. The activation row is the final row to grow around the edges of the $2 \\times 2$ blocks of macrotiles that each represent single tiles of the seed of $\\calT$. This row is always able to cooperate using a tile from the pre-activation row except for the case when it goes around a convex corner. In this case, the glue of the tile in the pre-activation row at that corner signals the upcoming corner and a tile attaches with a strength 2 glue, allowing the activation row to grow around the corner.\n    \n    As each segment completes, it exposes the glues of a superside that, modulo the spacing regions that are ignored, are identical to those used in the construction of \\cite{IUSA}. What this means is that, at that point, each superside can behave identically to the simulated side of a tile in the intrinsically universal simulator. All of the information that needs to be propagated to newly growing adjacent supertiles is presented, and from the exterior (other than the larger scale factor accommodated by the spacer sections), the supertiles representing the seed structure are identical to those used by that simulator. Because of this, we call this the activation row since as soon as it grows across a supertile side, if that supertile simulates a tile with a glue exposed on the corresponding side, growth can now proceed outward, out of the representation of the seed into adjacent supertile locations. Thus, the supertile has been ``activated'' to begin simulation of growth beyond the seed assembly.\n    \n    Since no growth outside of the macrotiles representing the seed can occur before the activation row grows over them, and the activation row cannot even begin until the entire pre-activation row is complete, which can only occur once every macrotile representing the seed that is on the exterior of the seed assembly has completed, nothing can interfere with the growth of the assembly representing the seed until the activation row is growing. Therefore, we only need to consider interactions between growth outside of the seed and macrotiles that have already completed pre-activation rows but don't necessarily have completed activation rows. The only potential conflict then occurs if a supertile outside of the representation of the seed, but adjacent to the seed, attempts growth into the location that would have been occupied by the activation row, potentially blocking its further growth. To handle this situation, since we are leveraging the construction of \\cite{IUSA} to handle all growth beyond the seed, we make use of the growth pattern used by that construction. The supertiles of that construction have empty $2 \\times 2$ squares in the corners between them (which can be seen in Figures \\ref{fig:scaled-simulation-seed-macrotile-frame} and \\ref{fig:scaled-simulation-seed-macrotile-frame-blocking}). The row in which the activation row would have been attempting to grow but was blocked would instead be filled by a row of the `frame' of the IU construction. A simple modification to the the tiles of such a row from that tile set (specifically the tiles which are the leftmost tiles - from the perspective of the supertile growing that row - of those rows) adds a strength 2 glue that allows a newly designed tile to attach to the left of that tile into the otherwise empty $2 \\times 2$ square. In most circumstances, this tile will be completely unused but out of the way. However, in circumstances in which the row to which it is attached blocked formation of the activation row, then this tile will be able to cooperate with a tile from the next segment of the pre-activation row to place a tile of the activation row and begin a new segment of its growth. This can been seen occurring three times in Figure \\ref{fig:scaled-simulation-seed-macrotile-frame-blocking}, with correct growth of the activation row being restarted each time. Note that if such an encroaching supertile (as depicted in red in Figure \\ref{fig:scaled-simulation-seed-macrotile-frame-blocking}) were to beat the activation row to the right side of the macrotile edge instead of the left side, then all that does is pause growth until that supertile's growth also occupies the left side (which it must eventually do, by design of the frame tiles in \\cite{IUSA} which attempt to occupy both ends of such a row and then grow toward each other in the middle, since growth would pause in the middle until that supertile's growth occupies the left side). Therefore, growth of the activation row can be temporarily paused but never stopped.\n    \n    As each macrotile completes its activation row, any simulation growth which would be valid can commence. Collisions with any inactive portions of the seed do not cause errors, as don't previously mentioned collisions that pause the growth of the activation row, since the seed representation of the seed tile is sufficiently completed by that point to represent the appropriate tile, and there is no need for it to grow output from the colliding side since there is clearly already a supertile in that location.\n\\end{enumerate}\n\n\nAlthough the necessary glue markings aren't always explicitly described above, each module places specially augmented glues in one or more positions (as needed) of its output row that allow the correct next module to begin growth from that output. This is possible because whenever a module completes growth, it is known which module will use that output to begin the next phase of growth. \n\n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{images\/scaled-simulation-seed-macrotile-frame-blocking.png}\n    \\caption{Extending the example of Figure~\\ref{fig:scaled-simulation-seed-macrotile-frame}, a depiction of the growth protocol which ensures that, even if the growth of surrounding macrotiles completes and attempt to initiate growth into seed macrotiles locations, the activation of the seed remains correct and no erroneous growth can occur.}\n    \\label{fig:scaled-simulation-seed-macrotile-frame-blocking}\n\\end{figure}\n\n\n\nWith the seed $\\sigma_\\calT$ and tile set $T_\\calT$ defined as above, and $\\tau = 2$, the definition of $\\mathcal{S}$ is complete. We now provide a sketch of the growth and correctness of the construction.\n\nStarting from a single seed tile, a hard-coded row grows which encodes a program that outputs $\\calT$ (potentially the shortest such program). A Turing machine is then simulated by the growth of a set of tiles designed to both simulate that particular Turing machine and also to maintain a counter which keetps track of the runtime of that Turing machine. The Turing machine computes a Hamiltonian cycle through a $2 \\times 2$ scaled version of the seed, and then computes the necessary encodings to navigate the growth of the assembly along the cycle and also the strings to be encoded along the exposed sides of those microtiles. During the computation, the increase of width, on each side, of the subassembly performing the computation, is equal to the height it grows (which is equivalent to the runtime). Utilizing these dimensions, a few relatively simple sets of modular construction components are able to assemble the macrotiles along the Hamiltonian cycle and place the necessary encodings along the outer perimeter of the seed-representing assembly so that each $2 \\times 2$ block of macrotiles is able to map to a single tile in the seed of $\\calT$, with the appropriate information encoded on their perimeters for the assembly to eventually initiate growth that appears from the outside as though it is coming from supertiles of the original IS construction \\cite{IUSA} (apart for the extra spacing fields), and thus growth outside of the simulated seed structure follows the design of the IU construction, which has been proven to correctly simulate the system whose information is encoded into the supersides.\n\nThe representation function $R$ is able to map a $2 \\times 2$ square of macrotiles to a tile of $\\calT$ once the fourth macrotile begins forming. At that point, $R$ is able to inspect the encodings of $g_l$ and $g_r$ designated for every side, and within them read the glues encoded for those sides, and then map the $2 \\times 2$ block of macrotiles to a single tile in the seed of $\\calT$. This ensures that all supertiles simulating tiles in the seed of $\\calT$ resolve to those tiles before any growth can occur away from the seed (since all tiles will then be represented before the pre-activation and activation rows grow, which are required for growth away for the seed to begin). Therefore, $\\mathcal{S}_\\calT$ completely grows the seed first, following the requirement for seed-first-simulation. By connecting with the tiles of the IU construction \\cite{IUSA}, we can utilize the full construction and proof of it to assert that as growth now proceeds, it correctly simulates $\\calT$. Therefore, given an optimally short program that outputs an arbitrary aTAM system $\\calT$, $\\mathcal{S}_\\calT$ seed-first-simulates $\\calT$ using an optimal $O(\\frac{K(\\calT)}{\\log(K(\\calT))}$ tile types and a single-tile-seed, so Theorem \\ref{thm:scaled-simulation} is proven.\n\n\\end{proof}\n\\section{Optimal Tile Complexity for Seed-First-Simulation}\\label{sec:scaled-sim}\n\nIn this section, we give an overview of a universal construction that takes as input a program $P_\\calT$, where $P_\\calT$ outputs any arbitrary aTAM system, say $\\calT$, and the construction outputs an aTAM system $\\mathcal{S}$ with a single-tile seed that seed-first-simulates $\\calT$. Technical details can be found in Section \\ref{sec:scaled-sim-append}. The tile complexity of $\\mathcal{S}$ is asymptotically related to the length of the input program. Thus, the tile complexity is $O(\\frac{K(\\calT)}{\\log(K(\\calT)})$, where $K(\\calT)$ is the Kolmogorov complexity of $\\calT$, i.e. the length of the shortest program that outputs $\\calT$.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=4.0in]{images\/ham-cycle-encoding.png}\n    \\caption{(Left) The seed $\\sigma$ of $\\calT$, (Middle) $\\sigma^2$ with a Hamiltonian cycle through it (beginning in the green location and ending in the red location), (Right) A shorthand representation of the sections representing the first 7 locations along the Hamiltonian cycle in the string encoding it.}\n    \\label{fig:ham-cycle-encoding}\n\\end{figure}\n\n\n\\begin{theorem}\\label{thm:scaled-simulation}\nGiven an arbitrary aTAM system $\\calT$, there exists $c \\in \\mathbb{Z}^+$ and aTAM system $\\mathcal{S}_\\calT = (T_\\calT,\\sigma_\\calT,2)$ such that $|\\sigma_\\calT| = 1$, $|T_\\calT| = O(\\frac{K(\\calT)}{\\log(K(\\calT)})$, and $\\mathcal{S}_\\calT$ seed-first-simulates $\\calT$ at scale factor $c$.\n\\end{theorem}\n\nAt a high level, the construction works by taking as input the program $P_\\calT$ and creating a hard-coded set of tiles (the first of which will be the single seed tile) that self-assemble a row whose north glues encode a compact version of that program that is first ``unpacked'' by a set of tiles that present the full program $P_\\calT$ along the northern glues of a row of tiles. A tile set that simulates a Turing machine then uses that program as input, runs that program to obtain the description of aTAM system $\\calT$, then uses that description to run a variety of subprograms. One subprogram executes the algorithm from the intrinsic universality construction of \\cite{IUSA} which computes a string encoding information about $\\calT$ in a format that can be used by that construction to simulate the tiles of $\\calT$ as large supertiles with that information on their exteriors.\nAnother creates a $2 \\times 2$ scaling of the seed assembly of $\\calT$ and computes a Hamiltonian cycle through it. (An example can be seen in Figure \\ref{fig:ham-cycle-encoding}.) It then creates a string containing entries for every stop along the Hamiltonian cycle that contain the information to be put on each output side. During this computation, a binary counter tile set keeps track of the number of computational steps so that the scale factor is computed and utilized.\nThen, the information is moved along the cycle and the information copied to the exterior sides of the assembly representing the scaled version of the seed of $\\calT$.\nOnce that structure is complete, the IU construction of \\cite{IUSA} (only minimally changed) takes over and manages the simulation of the rest of steps in the simulation.\n\n\n\n\n\n\n\n    \n    \n    \n    \n    \n    \n    \n\n\n\n\n\n\n\n\n\\subsection{Simultaneous simulation}\\label{sec:simult-sim}\n\nIn this section, we describe a relatively simple modification to the construction of Theorem \\ref{thm:scaled-simulation} that results in a single aTAM system that simultaneously and in parallel simulates every aTAM system.\nAt a high level, this construction works by the (standard aTAM) assumption that there are an infinite number of copies of the seed, which in this case is a single tile. From each copy, an arbitrarily long row encoding an arbitrary binary number nondeterministically grows. Each row terminates at a random point when a special tile attaches to ``cap'' the growth of the row. At this point, the row encodes a random binary number $n$ in its northern glues (and every binary number has some chance of being represented). Each such assembly encoding some number $n$ then proceeds to grow into a simulation of the $n$th tile assembly system in the aTAM using the construction of Theorem \\ref{thm:scaled-simulation}. Thus, in parallel, the infinite set of seeds grows into an infinite set of assemblies, each simulating one aTAM system from the infinite set, and each aTAM system is simulated (with some non-zero probability)\\footnote{An alternative, but also common, interpretation of the aTAM model is that, similar to the behavior of nondeterministic versions of automata such as Turing machines, whenever a nondeterministic choice occurs the system splits into a separate instance to follow each option. In this interpretation, it is considered that there is only a single copy of the seed, and then for each assembly to which more than one tile can attach, a new instance of the assembly is created for each possible attachment. Thus (possibly in the limit) all assemblies which can form from a seed assembly do so. In this interpretation, this construction also validly simulates all systems in parallel.}\n\nIn order to discuss such a simulator, we must define a new type of representation function and slightly modify the definition of simulation which allows multiple scale factors to be utilized in parallel.\nThis is because there are a countably infinite number aTAM systems and therefore for any scale factor $m \\in \\mathbb{Z}^+$, there must exist an infinite number of aTAM systems, with increasingly large tile sets to represent, that require scale factor greater than $m$ to simulate.\n\nWe will call this new notion of simulation \\emph{mixed-scale-simulation}. For use with our result, we will base it off of the definition of seed-first-simulation with the following modification.\n\n    \nRather than the standard definition of a representation function, due to the fact that different scale factors must be allowed for different simulations occurring simultaneously in the same system, we define an \\emph{adaptive representation function} as one which is not confined to a grid of fixed-size squares, but which instead is allowed to inspect any locations of an assembly and only be restricted by the requirement that it is able to inspect only a finite portion of an assembly before reading enough information to (1) compute the scale factor of the simulation being carried out, and (2) compute the mapping from supertiles in that assembly to tiles from the simulated system, and then (3) to correctly identify supertiles at that scale.\n\n\\begin{theorem}\\label{thm:simult-sim}\nThere exists an aTAM system $U_\\infty$ that mixed-scale-simulates all aTAM systems, simultaneously and in parallel.\n\\end{theorem}\n\n\\begin{proof}\n\nWe prove Theorem \\ref{thm:simult-sim} by construction. Therefore, we present aTAM system $\\mathcal{U}_\\infty = (U,\\sigma,2)$ and discuss how it is constructed and how it behaves.\n\nLet $D$ be the Turing machine defined in the proof of Theorem \\ref{thm:scaled-simulation}. We now define a new Turing machine $D'$ which takes as input a binary string $b \\in 1(0 \\cup 1)^*$ (i.e. any binary string beginning with a 1) that is immediately followed by the letter `x', and does the following:\n\n\\begin{enumerate}\n    \\item $D'$ enumerates over the set of all aTAM systems following some enumeration\\footnote{The set of all aTAM systems is countably infinite since it is clearly infinite (e.g. for every $i \\in \\mathbb{N}$ there exists an aTAM system with $i$ tile types that has a single-tile seed and self-assembles an $i \\times 1$ line at temperature 1), but every component of an aTAM system must be finite by definition of the aTAM, and therefore the set of systems must be countably infinite. Since any countably infinite set can be enumerated, there exists some enumeration of all aTAM systems.}, and stops after printing the $b$th aTAM system (where the binary string $b$ is interpreted as a positive integer). Let $\\langle \\calT \\rangle$ be the string representing the $b$th aTAM system.\n    \n    \\item $D'$ creates a Turing machine, which we'll call $P_\\calT$, that takes no input and simply prints $\\langle \\calT \\rangle$ and halts (by having the characters of $\\langle \\calT \\rangle$ hard-coded into its transition rules).\n    \n    \\item $D'$ runs $D(P_\\calT)$ and outputs what $D(P_\\calT)$ outputs.\n\\end{enumerate}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=3.0in]{images\/binary-string-tiles.png}\n    \\caption{(Left) Assuming that $q_0$ is the start state of Turing machine $D'$, using the tile on the left (which we'll refer to as $b_{seed}$) as the seed tile, this set of tiles can self-assemble to represent any binary string of the form $1(0 \\cup 1)^*$ with a terminating $x$ and that assembly can be used as input to $D'$. (Right) An example assembly encoding the binary string ``10100''.}\n    \\label{fig:binary-string-tiles}\n\\end{figure}\n\nWe now define tile set $U_\\infty$ as the tile set $T_\\calT$ from the proof of Theorem \\ref{thm:scaled-simulation} minus the tile sets $T_\\sigma$ and $T_D$. Instead of the tiles of $T_\\sigma$ which encode some specific program, we add the set of tiles $T_b$ pictured on the left in Figure \\ref{fig:binary-string-tiles}. Then, instead of the tiles $T_D$ we add tiles that simulate $D'$ in the same way that the tiles of $T_D$ simulated $D$ (i.e. zig-zag growth, an embedded counter, etc.).\n\nWe can now fully define $\\mathcal{U}_\\infty = (U_\\infty, \\sigma, 2)$, where $\\sigma$ is one copy of a tile of type $b_{seed}$ located at $(0,0)$.\n$\\mathcal{U}_\\infty$ does the following. For every $i \\in \\mathbb{Z}^+$, there exists a producible assembly that self-assembles from the tiles of $T_b$ that represents $i$ in binary, followed by the character `x'. Each such assembly initiates the growth of an assembly that simulates $D'$ on that value of $i$. This causes $D'$ to print out the $i$th aTAM system and then input that to $D$, which will then cause the tiles from the construction of Theorem \\ref{thm:scaled-simulation} to use that assembly to grow in such a way that it correctly seed-first-simulates the $i$th aTAM system.\n\nThe adaptive representation function $R^*$ works as follows. Given an arbitrary assembly, it begins at location $(0,0)$, which is the location of the seed tile, $b_{seed}$.\nIt reads the number encoded by the tiles to the right of that tile until encountering a tile encoding an `x' and thus the end of the binary number. If no such tile is found, the assembly maps to empty space as it doesn't (yet) simulate any system. For those that do encode a complete number, $R^*$ runs Turing machine $D'$ with that number as input to get the full specification of the system being simulated by that assembly (which we'll call $\\calT$), and from that it is also able to compute the scale factor of the simulation, $m$. $R^*$ is then able to inspect the $m \\times m$ squares and match the information encoded in sufficiently completed supertiles to tiles in $\\calT$.\n\nSince, for every aTAM system, there exists an assembly which simulates it, and there exists an adaptive representation function capable of identifying the system being simulated by each assembly and correctly mapping the supertiles to tiles of that simulated system, $\\mathcal{U}_\\infty$ mixed-scale-simulates all aTAM systems, simultaneously and in parallel.\n\n\\end{proof}\n\n\n\\vspace{-20pt}\n\\subsection{Cross-model intrinsic universality}\n\nIn the following we claim that, given some model $M$, if for any arbitrary aTAM system $\\calT$, there exists a system in $M$ that simulates it, then there is a tile set in model $M$ that is intrinsically universal for the aTAM (using seed-growth-simulation). This means that a single tile set in $M$ is capable of being used to simulate any aTAM system. First, we note that our use of the term ``model'' in this case applies not only to the complete set of systems within a model, but also to what is commonly referred to as a ``class of systems within a model''. For instance, the aTAM is a model and so is the 2-Handed Assembly Model, so referring directly to one of those models would mean the full set of systems within those models. Alternatively, the subset of systems within the aTAM which are directed is often referred to as the class of directed aTAM systems. The following result applies to both models and classes of systems within models.\n\nWhile it may seem like the existence of a one-to-one relationship between systems in some model $A$ and systems in another model $B$ that can simulate them may imply the existence of a tile set in $B$ which is intrinsically universal for model $A$, this is not the case. A counterexample occurs with the class of systems in the aTAM which are directed. Trivially, using the identity function as the representation function and scale factor 1, for every system in the directed aTAM there exists a system within the directed aTAM which simulates it. However, as proven in \\cite{DirectedNotIU}, there exists no tile set which can be used within the directed class of aTAM systems to simulate all directed aTAM systems\n\n\n\\begin{claim}\\label{clm:cross-model-IU}\nLet $M$ be a model of tile-based self-assembly such that, for every system $\\calT$ in the aTAM, there exists a system $\\calT' \\in M$ that simulates system $\\calT$ of the aTAM. Then, there exists a tile set in $M$ which is intrinsically universal for the aTAM by performing seed-first-simulations.\n\\end{claim}\n\nTo support Claim \\ref{clm:cross-model-IU}, we now define aTAM system $\\mathcal{U}_b$ as follows. Use the tile set $U$ from $\\mathcal{U}_\\infty$ of the proof of Theorem \\ref{thm:simult-sim}, set the temperature to 2, select an arbitrary binary number $b$ and create a seed assembly, $\\sigma_b$, that is pre-built from the tiles of $T_b$ (starting with $b_{seed}$) to represent $b$ with an `x' after the last bit (i.e., rather than letting such assemblies nondeterministically form, the seed is now a single, pre-selected number encoded into a seed assembly).\n\nLet $M$ be a model of tile-based self-assembly such that, for every aTAM system $\\calT$, there exists a system $\\calT'$ in $M$ that simulates $\\calT$. If this is true, then there must be a system in $M$, which we'll call $\\mathcal{M}_b$, that simulates $\\mathcal{U}_b$. Let $T_M$ be the tile set used by $\\mathcal{M}_b$, $c$ be the scale factor of the simulation, and $R$ the representation function. \nWe now claim that, since a seed assembly using $T_M$, $c$, and $R$ was made for an arbitrary value of $b$, it should be possible to use the same $T_M$, $c$ and $R$ to make seed assemblies encoding all values $i \\in \\mathbb{Z}^+$. The intuition behind this claim is that it must be possible to reuse the macrotiles representing tiles of individual bit values, since that would have to be the case if $b$ was large enough, so it must be possible to combine them to form any desired value. Assuming that claim,\nit is possible to generate system $M_i$, using $T_M$, $R$, and $c$ to create an assembly over $T_M$ that maps to $\\sigma_i$ so that $M_i$ simulates $\\mathcal{U}_i = (U, \\sigma_i, 2)$ for arbitrary $i$, and note that not only does $M_i$ simulate $U_i$ in the standard way (i.e. with a pre-built seed structure for a multi-tile seed), using $R$ and $c$, but we can also determine the representation function $R''$ and scale factor $c''$ such that we can interpret the assemblies in $\\mathcal{M}_i$ as assemblies in $\\calT_i$, which $\\mathcal{M}_i$ is also seed-first-simulating via $R''$ at scale factor $c''$.\n\nLet Turing machine $M^*$ be a Turing machine that takes as input an assembly $\\sigma_i$ and runs an aTAM simulator on the system $(U,\\sigma_i,2)$ until the it completes the simulation of $D'(i)$ (where $D'$ is the TM defined in the proof of Theorem \\ref{thm:simult-sim}). \nAt that point, $M^*$ will be able to compute the scale factor $c'$ at which $(U,\\sigma_i,2)$ simulates aTAM system $\\calT_i$, as well as the representation function $R'$ for that simulation.\n\nDefine $R''$ as follows: On input $\\alpha'$ which is a producible assembly in $\\mathcal{M}_i$ (i.e. an assembly over the tiles of $T_M$),\n\\begin{enumerate}\n   \n    \\item Run $M^*$ on input assembly $\\sigma_i$ to obtain $c'$ and $R'$. (Note that the assembly $\\sigma_i$ is hard-coded into $R''$ so that it is specific for the simulation of aTAM system $i$.)\n    \\item Compute $c'' = c * c'$\n    \\item Return $R'(R(\\alpha')$ (i.e. the result is an assembly in $\\calT_i$)\n\\end{enumerate}\n\nSince we claim it's possible, for an arbitrary $i \\in \\mathbb{Z}^+$, to create an assembly from the tiles of $T_M$ that represents $\\sigma_i$ under $R$ at scale factor $c$, and that such an assembly seed-first-simulates the $i$th aTAM system, it would then be the case that $T_M$ is intrinsically universal for the aTAM by performing seed-first-simulations.\n\n\n\n\n\n\\section{Single-tile Seeds via Seed-First-Simulation at Small Scales}\\label{sec:min-scaled-sims}\n\nGiven that we have shown the impossibility of shape-simulation (the broadest form of simulation via singly seeding) at scale factors 1 and 2, a question that naturally follows is: do systems exist which are able to shape-simulate just above the proven lower scale bounds?\nThe answer is yes - when allowing cheating fuzz, this is possible at scale factor 3 (Section~\\ref{sec:scale3-sim-fuzz}), and restricting ourselves to no cheating fuzz allows for simulation at scale factor 4 (Section~\\ref{sec:scale4-sim}).\nBoth constructions utilize seed-first-simulation.\n\n\\input{scale4-simulation}\n\n\\input{scale3-simulation}\n\n\n\\subsection{Uncomputability of blocking}\n\n\\begin{observation}\\label{obs:uncomp-blocking}\nGiven an arbitrary aTAM system $\\calT = (T,\\sigma,\\tau)$, it is uncomputable to know if some locations in the seed assembly $\\sigma$ can block tiles (in one or more valid assembly sequences).\n\\end{observation}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=6.0in]{images\/scale-1-crashes.png}\n    \\caption{(Left) Schematic example of an assembly simulating a Turing machine and showing the uncomputability of blocking. The seed row (green) encodes the input to the Turing machine (TM) via exposed glues on the north and initiates growth of an assembly where the rows grow in a zig-zag pattern and simulate the TM in a standard way (i.e. each pair of rows represents the configuration of the TM at the time step following that of the pair below it). If and only if the TM halts on the input (the halt state is depicted by the red tile), a row of blue tiles grows to the left and then a column of tiles of the same type grows downward. This column will crash into the seed. Since the crashing column only grows if the TM halts, it is uncomputable to know if growth will crash into the seed.\n    (Right) Slight modification to the construction on the left in which, upon halting, the TM initiates a crashing column on the left and another on the right side. This makes it uncomputable to know if the leftmost and\/or rightmost seed locations will block and impossible to start from a single seed tile yet guarantee sufficient growth of the seed to provide whatever blocking may be needed.}\n    \\label{fig:TM-crash}\n\\end{figure}\n\n\\begin{proof}\nWe prove Observation \\ref{obs:uncomp-blocking} by providing a simple reduction to the Halting problem. We'll call the language that consists of binary strings representing aTAM systems that have some seed location that can block the growth of a tile the \\texttt{BLOCKING} language. Assume that \\texttt{BLOCKING} is computable. Then, there exists some Turing machine, say $M_{block}$, which decides the language. We now construct a new Turing machine, $M_{halt}$ as follows: On input $\\langle M,b \\rangle$, where $M$ is an encoding of an arbitrary Turing machine and $b$ is an arbitrary binary string, $M_{halt}$ first constructs aTAM system $\\calT_M = (T_M, \\sigma_b, 2)$ following the design of the system on the left of Figure \\ref{fig:TM-crash}. The seed $\\sigma_b$ is the green row which encodes $b$ in its northern glues with an additional 4 tiles to the left that have no northern glues. The tile set $T_M$ consists of the seed tile types, plus a set of tiles that simulate a Turing machine in a standard manner (e.g. see \\cite{DirectedNotIU} for a definition of zig-zag Turing machines and Figure \\ref{fig:TM-crash} for a schematic depiction), plus a small constant-sized set (depicted by the blue tiles) that grows to the left from the tile representing the halting state (if it appears in the assembly, as it does if and only if $M(b)$ halts), past the left edge by four tiles, and then a single tile type that can repeatedly attach to copies of itself in a downward growing column. Note that this column can only form if $M$ halts on $b$, and in this case the downward growing column crashes into the leftmost seed tile, which blocks its growth. With $\\calT_M$ thus constructed, $M_{halt}$ gives $\\calT_M$ to $M_{block}$ as input. $M_{block}(\\calT_M)$ accepts if some location of the seed blocks, which occurs if and only if $M$ halts on $b$, and rejects otherwise. $M_{halt}$ simply outputs the answer given by $M_{block}$. Thus, $M_{halt}$ solves the halting problem, which we know is uncomputable. Therefore, \\texttt{BLOCKING} must also be uncomputable.\n\\end{proof}\n\n\n\\subsection{Window Movie Lemma}\\label{sec:wml}\n\nHere we include definitions related to the Window Movie Lemma from \\cite{temp1notIU}, as well as restate that lemma, for completeness.\n\nA \\emph{window} $w$ is a set of edges forming a cut-set in the infinite grid graph over $\\mathbb{Z}^2$.\nGiven a window $w$ and an assembly $\\alpha$, a window that {\\em intersects} $\\alpha$ is a partioning of $\\alpha$ into two  configurations (i.e.\\ after being split into two parts, each part may or may not be  disconnected).\nIn this case we say that the window $w$ cuts the assembly $\\alpha$ into two configurations $\\alpha_L$ and~$\\alpha_R$, where $\\alpha = \\alpha_L \\cup \\alpha_R$.\nGiven a window $w$, its translation by a vector $\\vec{c}$,  written $ w + \\vec{c}$ is simply the translation of each of $w$'s elements (edges) by~$\\vec{c}$.\n\nGiven an assembly sequence $\\vec{\\alpha}$ and a window $w$, the associated {\\em window movie} is the maximal sequence $M_{\\vec{\\alpha},w} = (v_{0}, g_{0}) , (v_{1}, g_{1}), (v_{2}, g_{2}), \\ldots$ of pairs of grid graph vertices $v_i$ and glues $g_i$, given by the order of the appearance of the glues along window $w$ in the assembly sequence $\\vec{\\alpha}$.\nFurthermore, if $k$ glues appear along $w$ at the same instant (this happens upon placement of a tile which has multiple  sides  touching $w$) then these $k$ glues appear contiguously and are listed in lexicographical order of the unit vectors describing their orientation in $M_{\\vec{\\alpha},w}$.\n\n\\begin{lemma}[Window movie lemma]\n\\label{lem:windowmovie}\nLet $\\vec{\\alpha} = (\\alpha_i \\mid 0 \\leq i < l)$ and $\\vec{\\beta} = (\\beta_i \\mid 0 \\leq i < m)$, with\n$l,m\\in\\Z^+ \\cup \\{\\infty\\}$,\nbe assembly sequences in $\\mathcal{T}$ with results $\\alpha$ and $\\beta$, respectively.\nLet $w$ be a window that partitions~$\\alpha$ into two configurations~$\\alpha_L$ and $\\alpha_R$, and $w' = w + \\vec{c}$ be a translation of $w$ that partitions~$\\beta$ into two configurations $\\beta_L$ and $\\beta_R$.\nFurthermore, define $M_{\\vec{\\alpha},w}$, $M_{\\vec{\\beta},w'}$ to be the respective window movies for $\\vec{\\alpha},w$ and $\\vec{\\beta},w'$, and define $\\alpha_L$, $\\beta_L$ to be the subconfigurations of $\\alpha$ and $\\beta$ containing the seed tiles of $\\alpha$ and $\\beta$, respectively.\nThen if $M_{\\vec{\\alpha},w} = M_{\\vec{\\beta},w'}$, it is the case that  the following two assemblies are also producible:\n(1) the assembly $\\alpha_L \\beta'_R = \\alpha_L \\cup \\beta'_R$ and\n(2) the assembly $\\beta'_L \\alpha_R = \\beta'_L \\cup \\alpha_R$, where $\\beta'_L=\\beta_L-\\vec{c}$ and $\\beta'_R=\\beta_R-\\vec{c}$.\n\\end{lemma}\n\nEssentially, the Window Movie Lemma states that if the same window movie occurs in two different assembly sequences of some TAS $\\calT$, but in different locations, valid producible assemblies in $\\calT$ include (1) an assembly with the ``left'' half created by the first sequence and the ``right'' half created by the second sequence, and (2) an assembly with the ``left'' half created by the second sequence and the ``right'' half created by the first sequence. Typical use of this lemma includes showing that some portion of a sufficiently large growing assembly must ``pump'', i.e. have repetitive structure exhibited by repetition of identical window movies. When this occurs, there must exist valid assembly sequences in $\\calT$ in which the portions of the assembly that grow after each occurrence of that window movie can be swapped, and\/or the subassembly between the identical window movies can be repeated (a.k.a. \\emph{pumped}) an arbitrary number of times.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}