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+{"text":"\\section{Introduction}\\label{s-1}\nLet $G=(V(G), E(G))$ be a simple  undirected graph  on $n$ vertices with adjacency matrix $A=A(G)$. Denote by $\\lambda_1,\\lambda_2,\\ldots,\\lambda_t$  all the distinct eigenvalues of $A$ with multiplicities $m_1,m_2,\\ldots,m_t$ ($\\sum_{i=1}^tm_i=n$), respectively. These eigenvalues are also called the \\textit{eigenvalues} of $G$. All the eigenvalues together with their multiplicities are called the \\emph{spectrum} of $G$ denoted by $\\mathrm{Spec}(G)=\\big\\{[\\lambda_1]^{m_1},[\\lambda_2]^{m_2},\\ldots,[\\lambda_t]^{m_t}\\big\\}$.  If $G$ is a connected $k$-regular graph, then $\\lambda_1$ denotes $k$, and has multiplicity $m_1=1$.\n\n\nA graph $G$ is said to be \\emph{determined by its  spectrum} (DS for short) if  $G\\cong H$ whenever $\\mathrm{Spec}_A(G)=\\mathrm{Spec}_A(H)$ for any graph $H$. Also, a graph $G$ is called \\emph{walk-regular} if for which the number of walks of length $r$ from a given vertex $x$ to itself (closed walks) is independent of the choice of $x$ for all $r$ (see \\cite{Godsil}). Note that a walk-regular graph is always regular, but in general the converse is not true.\n\nThroughout this paper,  we denote  the \\emph{neighbourhood} of a vertex $v\\in V(G)$ by $N_{G}(v)$,  the complete graph on  $n$ vertices by $K_n$,  the complete multipartite graph with $s$ parts of sizes $n_1,\\ldots,n_s$  by $K_{n_1,\\ldots,n_s}$, and the graph obtained by removing a perfect matching from $K_{n,n}$ by $K_{n,n}^-$. Also,  the  $n\\times n$ identity matrix,  the $n\\times 1$ all-ones  vector and  the $n\\times n$ all-ones matrix will\nbe denoted by  $I_n$, $\\mathbf{e}_n$  and $J_n$, respectively.\n\nConnected graphs with a few  eigenvalues have aroused  a lot of  interest in the past several decades.  This problem was perhaps first raised by Doob \\cite{Doob}. It is well known that connected regular graphs having three distinct eigenvalues are  strongly regular graphs \\cite{Shrikhande}, and connected regular bipartite graphs having four distinct eigenvalues are the incidence graphs of  symmetric balanced incomplete block designs \\cite{Brouwer,Cvetkovic}. Furthermore, connected non-regular graphs with three distinct eigenvalues and least eigenvalue $-2$ were determined by Van Dam \\cite{Dam3}.  Very recently, Cioab\\u{a} et al.  in  \\cite{Cioaba1} (resp. \\cite{Cioaba})  determined all graphs with at most two eigenvalues (multiplicities included) not equal to $\\pm1$ (resp. $-2$ or $0$). De Lima et al. in \\cite{Lima} determined all connected non-bipartite graphs with all but two eigenvalues in the interval $[-1, 1]$. For more results on graphs with few eigenvalues, we refer the reader to \\cite{Bridges,Caen,Cheng,Dam1,Dam2,Dam4,Dam5,Dam7,Doob,Muzychuk,Rowlinson}.\n\n\nVan Dam in \\cite{Dam1,Dam2} investigated the connected regular graphs with four distinct eigenvalues. He classified such graphs into three classes according to the number of integral eigenvalues  (see Lemma \\ref{lem-1} below). Based on Van Dam's classification and the number of simple eigenvalues, we can classify such graphs more precisely, that is, if $G$ is a connected $k$-regular graphs with four distinct eigenvalues, then\n\\begin{enumerate}[(1)]\n\\vspace{-0.2cm}\n\\item $G$ has at least three  simple eigenvalues, or\n\\vspace{-0.25cm}\n\\item $G$ has two simple eigenvalues:\n\\vspace{-0.25cm}\n\\begin{enumerate}[(2a)]\n\\item $G$ has four integral eigenvalues in which two eigenvalues are simple;\n\\vspace{-0.2cm}\n\\item $G$ has two integral eigenvalues, which are simple, and two eigenvalues of the form $\\frac{1}{2}(a\\pm\\sqrt{b})$, with $a,b\\in \\mathbb{Z}$, $b>0$, with the same multiplicity, or\n\\end{enumerate}\n\\vspace{-0.35cm}\n\\item $G$ has one simple eigenvalue, i.e., its degree $k$:\n\\vspace{-0.25cm}\n\\begin{enumerate}[(3a)]\n\\item $G$ has four integral eigenvalues;\n\\vspace{-0.2cm}\n\\item $G$ has two integral eigenvalues, and two eigenvalues of the form $\\frac{1}{2}(a\\pm\\sqrt{b})$, with $a,b\\in \\mathbb{Z}$, $b>0$, with the same multiplicity;\n    \\vspace{-0.2cm}\n\\item $G$ has one integral eigenvalue, its degree $k$, and the other three have the same multiplicity $m=\\frac{1}{3}(n-1)$, and $k=m$ or $k=2m$.\n\\end{enumerate}\n   \\vspace{-0.3cm}\n\\end{enumerate}\n\nIn this paper, we continue to focus on connected regular  graphs with four distinct eigenvalues. Concretely, we  show that there are no graphs in (1), and   give a complete characterization of the graphs belonging to $\\mathcal{G}(4,2,-1)$: if $-1$ is a non-simple eigenvalue, we determine all such graphs; if $-1$ is a simple eigenvalue,  we prove that such graphs cannot belong to (2a) and (2b), respectively,  and so do not exist. In the process, we determine all the graphs  in $\\mathcal{G}(4,\\geq -1)$ and $\\mathcal{G}(4,2,0)$, respectively, and show that all these graphs  are DS.\n\n\\section{Main tools}\\label{s-2}\nIn this section, we recall some results from the literature  that will be useful in the next section.\n\n\\begin{lem}\\label{lem-1} (See \\cite{Dam1,Dam2}.)\nIf $G$ is a connected $k$-regular graph on $n$ vertices\nwith four distinct eigenvalues, then\n\\begin{enumerate}[(i)]\n\\vspace{-0.3cm}\n\\item $G$ has four integral eigenvalues, or\n\\vspace{-0.3cm}\n\\item $G$ has two integral eigenvalues, and two eigenvalues of the form $\\frac{1}{2}(a\\pm\\sqrt{b})$, with $a,b\\in \\mathbb{Z}$, $b>0$, with the same multiplicity, or\n\\vspace{-0.3cm}\n\\item $G$ has one integral eigenvalue, its degree $k$, and the other three have the same multiplicity $m=\\frac{1}{3}(n-1)$, and $k=m$ or $k=2m$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{lem}\\label{lem-5}(See \\cite{Brouwer}.)\nIf $G$ is connected and regular with four distinct eigenvalues,\nthen $G$ is walk-regular.\n\\end{lem}\n\nLet $G$ be a $k$-regular graph. We say that $G$ admits a \\emph{regular partition into halves} with degrees $(a,b)$ ($a+b=k$) if we can partition the vertices of $G$ into two parts of equal size such that every vertex has $a$ neighbors in its own part and $b$ neighbors in the other part \\cite{Dam1}.\n\\begin{lem}\\label{lem-2}(See \\cite{Dam1}.)\nLet $G$ be a connected walk-regular graph on $n$ vertices\nand degree $k$, having distinct eigenvalues $k,\\lambda_2,\\lambda_3,\\ldots,\\lambda_t$,\nof which an eigenvalue unequal to $k$, say $\\lambda_j$, has multiplicity $1$.\nThen $n$ is even and $G$ admits a regular partition into halves with degrees\n$(\\frac{1}{2}(k+\\lambda_j),\\frac{1}{2}(k-\\lambda_j))$. Moreover, $n$ is a divisor of\n$$\\prod_{i\\neq j}(k-\\lambda_i)+\\prod_{i\\neq j}(\\lambda_j-\\lambda_i)~~and~~\\prod_{i\\neq j}(k-\\lambda_i)-\\prod_{i\\neq j}(\\lambda_j-\\lambda_i).$$\n\\end{lem}\n\nFrom the proof of Lemma \\ref{lem-2}, we  obtain  the following corollary immediately.\n\n\\begin{cor}\\label{cor-1}\nUnder the assumption of Lemma \\ref{lem-2},  the eigenvector of $\\lambda_j$ can be written as $\\mathbf{x}_j=\\frac{1}{\\sqrt{n}}(\\mathbf{e}_{\\frac{n}{2}}^T,-\\mathbf{e}_{\\frac{n}{2}}^T)^T$, and the vertex partition $V(G)=V_1\\cup V_2$ with $V_1=\\{v\\in V\\mid \\mathbf{x}_{j}(v)=1\\}$ and $V_2=\\{v\\in V\\mid \\mathbf{x}_{j}(v)=-1\\}$ is just the regular partition of $G$ into halves with degrees $(\\frac{1}{2}(k+\\lambda_j),\\frac{1}{2}(k-\\lambda_j))$ described in Lemma \\ref{lem-2}.\n\\end{cor}\nA \\emph{balanced incomplete block design}, denoted by \\emph{BIBD}, consists of $v$ elements and $b$ subsets of these elements called \\emph{blocks} such that each element is contained\nin $t$ blocks,  each block contains $k$ elements, and  each pair of elements is simultaneously contained in $\\lambda$ blocks (see \\cite{Cvetkovic}). The integers $(v,b,r,k,\\lambda)$ are called the \\emph{parameters} of the design. In the case $r=k$ (and then $v=b$) the design is called \\emph{symmetric} with parameters $(v,k,\\lambda)$.\n\nThe \\emph{incidence graph} of a $BIBD$ is the bipartite graph on  $b+v$ vertices (correspond to the blocks and elements of the design) with two vertices adjacent if and only if one corresponds to a block and the other corresponds to an element contained in that block. As   shown in \\cite{Cvetkovic},  the incidence graph has spectrum $\\big\\{[\\sqrt{rk}]^1,[\\sqrt{r-\\lambda}]^{v-1},[0]^{b-v},[-\\sqrt{r-\\lambda}]^{v-1},$ $[-\\sqrt{rk}]^1\\big\\}$. In particular, if the design is symmetric,  then the incidence graph is a $k$-regular bipartite graph with spectrum $\\big\\{[k]^1,[\\sqrt{k-\\lambda}]^{v-1},[-\\sqrt{k-\\lambda}]^{v-1},[-k]^1\\big\\}$.\n\nThe following lemma gives a characterization of regular bipartite graphs with four distinct eigenvalues.\n\\begin{lem}\\label{lem-4}(See \\cite{Brouwer,Cvetkovic}.)\nA connected regular bipartite  graph $G$ with four distinct eigenvalues\nis the incidence graph of a symmetric BIBD.\n\\end{lem}\n\n\\begin{figure}[t]\n\\centering\n\\begin{center}\n\\unitlength 1.75mm\n\\linethickness{0.8pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi\n\\begin{picture}(52,27)(0,0)\n\\put(2,11){\\circle*{1}}\n\\put(10,11){\\circle*{1}}\n\\put(18,11){\\circle*{1}}\n\\put(2,5){\\circle*{1}}\n\\put(10,5){\\circle*{1}}\n\\put(18,5){\\circle*{1}}\n\\put(2,8){\\oval(4,10)[]}\n\\put(10,8){\\oval(4,10)[]}\n\\put(18,8){\\oval(4,10)[]}\n\\put(25,8){\\circle*{1}}\n\\put(20,8){\\line(1,0){5}}\n\\put(4,8){\\line(1,0){4}}\n\\put(12,8){\\line(1,0){4}}\n\\put(36,11){\\circle*{1}}\n\\put(44,11){\\circle*{1}}\n\\put(36,5){\\circle*{1}}\n\\put(44,5){\\circle*{1}}\n\\put(36,8){\\oval(4,10)[]}\n\\put(44,8){\\oval(4,10)[]}\n\\put(38,8){\\line(1,0){4}}\n\\put(36,13){\\line(0,1){4}}\n\\put(44,13){\\line(0,1){4}}\n\\put(36,25){\\circle*{1}}\n\\put(44,25){\\circle*{1}}\n\\put(36,19){\\circle*{1}}\n\\put(44,19){\\circle*{1}}\n\\put(36,22){\\oval(4,10)[]}\n\\put(44,22){\\oval(4,10)[]}\n\\put(38,22){\\line(1,0){4}}\n\\put(46,22){\\line(1,0){5}}\n\\put(46,8){\\line(1,0){5}}\n\\put(51,22){\\circle*{1}}\n\\put(51,8){\\circle*{1}}\n\\put(2.1,8.5){\\makebox(0,0)[cc]{$\\vdots$}}\n\\put(10.1,8.5){\\makebox(0,0)[cc]{$\\vdots$}}\n\\put(18.1,8.5){\\makebox(0,0)[cc]{$\\vdots$}}\n\\put(36.1,22.5){\\makebox(0,0)[cc]{$\\vdots$}}\n\\put(36.1,8.5){\\makebox(0,0)[cc]{$\\vdots$}}\n\\put(36.1,22.5){\\makebox(0,0)[cc]{$\\vdots$}}\n\\put(44.1,8.5){\\makebox(0,0)[cc]{$\\vdots$}}\n\\put(44.1,22.5){\\makebox(0,0)[cc]{$\\vdots$}}\n\\put(1,11){\\makebox(0,0)[cc]{\\scriptsize$1$}}\n\\put(1,5){\\makebox(0,0)[cc]{\\scriptsize$l$}}\n\\put(9,11){\\makebox(0,0)[cc]{\\scriptsize$1$}}\n\\put(9,5){\\makebox(0,0)[cc]{\\scriptsize$m$}}\n\\put(17,11){\\makebox(0,0)[cc]{\\scriptsize$1$}}\n\\put(17,5){\\makebox(0,0)[cc]{\\scriptsize$n$}}\n\\put(35,11){\\makebox(0,0)[cc]{\\scriptsize$1$}}\n\\put(35,5){\\makebox(0,0)[cc]{\\scriptsize$m$}}\n\\put(43,11){\\makebox(0,0)[cc]{\\scriptsize$1$}}\n\\put(43,5){\\makebox(0,0)[cc]{\\scriptsize$p$}}\n\\put(35,25){\\makebox(0,0)[cc]{\\scriptsize$1$}}\n\\put(35,19){\\makebox(0,0)[cc]{\\scriptsize$l$}}\n\\put(43,25){\\makebox(0,0)[cc]{\\scriptsize$1$}}\n\\put(43,19){\\makebox(0,0)[cc]{\\scriptsize$n$}}\n\\put(11,0){\\makebox(0,0)[cc]{\\footnotesize$A(l,m,n)$}}\n\\put(41,0){\\makebox(0,0)[cc]{\\footnotesize$B(l,m,n,p)$}}\n\\end{picture}\n\\end{center}\n\\vspace{-0.4cm}\n\\caption{\\footnotesize{The graphs $A(l,m,n)$ and $B(l,m,n,p)$.}}\\label{fig-1}\n\\end{figure}\nDenote by $A(l,m,n)$ and $B(l,m,n,p)$ ($l,m,n,p\\geq 1$) the two graphs from Fig.\\ref{fig-1}, where the vertices contained in an ellipse form an independent set, and any two ellipses or a vertex and an ellipse joined with one  line denote a complete bipartite graph.\n\\begin{lem}\\label{lem-6} (See \\cite{Torgasev}.)\nThe second least eigenvalue of a connected graph $G$ is  greater than $-1$ if and only if\n\\begin{enumerate}[(i)]\n\\vspace{-0.3cm}\n\\item $G=K_{m,n}$ with $m,n\\geq 1$, or\n\\vspace{-0.3cm}\n\\item $G=A(l,m,n)$ with one of the following cases occours: $n=1$ and $m,l\\geq 1$;  $n=l=2$ and $m\\geq 1$;  $n\\geq 2$ and $l=m=1$; $n\\geq 2$, $l=1$ and $m\\geq 1$, or\n\\vspace{-0.3cm}\n\\item $G=B(l,m,n,p)$ with $(p+l-pl)(m+n-mn)>(p-1)(n-1)$.\n\\end{enumerate}\n\\end{lem}\n\n\n\\section{Main results}\\label{s-3}\n\nLet $A$ be a real symmetric matrix whose all distinct eigenvalues are $\\lambda_1,\\ldots,\\lambda_s$. Then $A$ has the \\emph{spectral decomposition} $A=\\lambda_1P_1+\\cdots+\\lambda_sP_s$, where $P_i=\\mathbf{x}_1\\mathbf{x}_1^T+\\cdots+\\mathbf{x}_{d}\\mathbf{x}_{d}^T$ if the eigenspace $\\mathcal{E}(\\lambda_i)$ has $\\{\\mathbf{x}_1,\\ldots,\\mathbf{x}_d\\}$ as an orthonormal basis. It is easy to see that  for any real polynomial $f(x)$, we have $f(A)=f(\\lambda_1)P_1+\\cdots+f(\\lambda_s)P_s$.\n\nFirst of all, we will prove the non-existence of connected regular graphs with four distinct eigenvalues in which at least three eigenvalues are simple.\n\n\\begin{thm}\\label{thm-main-0}\nThere are no connected  $k$-regular graphs on $n$ ($n\\geq 4$) vertices with spectrum $\\big\\{[k]^1,[\\lambda_2]^1,[\\lambda_3]^1,[\\lambda_4]^{n-3}\\big\\}$.\n\\end{thm}\n\n\\begin{proof}\nSuppose that $G$ is a connected $k$-regular graph on $n$ vertices  with adjacency matrix $A$ and spectrum $\\big\\{[k]^1,[\\lambda_2]^1,$ $[\\lambda_3]^1,[\\lambda_4]^{n-3}\\big\\}$. Then $G$ has minimal polynomial $p(x)=(x-k)(x-\\lambda_2)(x-\\lambda_3)(x-\\lambda_4)$. By Lemmas \\ref{lem-5} and \\ref{lem-2}, $\\lambda_2$ and $\\lambda_3$ are integers,  so $\\lambda_4$ is also an integer. Furthermore, by Corollary \\ref{cor-1}, we may assume that $\\lambda_2$ and $\\lambda_3$, respectively, have orthonormal eigenvectors as follows:\n\\begin{equation*}\n\\mathbf{x}_2=\\frac{1}{\\sqrt{n}}(\\mathbf{e}_\\frac{n}{2}^T,-\\mathbf{e}_\\frac{n}{2}^T)^T\\ \\mbox{ and }\\ \\mathbf{x}_3=\\frac{1}{\\sqrt{n}}(\\mathbf{e}_\\frac{n}{4}^T,-\\mathbf{e}_\\frac{n}{4}^T,\\mathbf{e}_\\frac{n}{4}^T,-\\mathbf{e}_\\frac{n}{4}^T)^T.\n\\end{equation*}\nTaking $f(x)=x-\\lambda_4$, by the spectral decomposition of $f(A)$ we get\n\\begin{equation*}\nA-\\lambda_4I_n=\\frac{1}{n}(k-\\lambda_4)\\mathbf{e}_n\\mathbf{e}_n^T+(\\lambda_2-\\lambda_4)\\mathbf{x}_2\\mathbf{x}_2^T+(\\lambda_3-\\lambda_4)\\mathbf{x}_3\\mathbf{x}_3^T,\n\\end{equation*}\nor equivalently,\n\\begin{equation}\\label{equ-1}\nn(A-\\lambda_4I_n)=(k-\\lambda_4)J_n+(\\lambda_2-\\lambda_4)\n\\left(\\begin{matrix}\nJ_\\frac{n}{2}&-J_\\frac{n}{2}\\\\\n-J_\\frac{n}{2}&J_\\frac{n}{2}\n\\end{matrix}\\right)\n+(\\lambda_3-\\lambda_4)\n\\left(\\begin{smallmatrix}J_\\frac{n}{4}&-J_\\frac{n}{4}&J_\\frac{n}{4}&-J_\\frac{n}{4}\\\\\n-J_\\frac{n}{4}&J_\\frac{n}{4}&-J_\\frac{n}{4}&J_\\frac{n}{4}\\\\\nJ_\\frac{n}{4}&-J_\\frac{n}{4}&J_\\frac{n}{4}&-J_\\frac{n}{4}\\\\\n-J_\\frac{n}{4}&J_\\frac{n}{4}&-J_\\frac{n}{4}&J_\\frac{n}{4}\\\\\n\\end{smallmatrix}\\right).\n\\end{equation}\nOn the other hand, by considering the traces of $A$ and $A^2$, respectively, we obtain\n\\begin{align}\n&k+\\lambda_2+\\lambda_3+(n-3)\\lambda_4=0,\\label{equ-2}\\\\\n&k^2+\\lambda_2^2+\\lambda_3^2+(n-3)\\lambda_4^2=kn.\\label{equ-3}\n\\end{align}\n\nNow we partition $V(G)$ the same way as we partition the matrix $\\mathbf{x}_3\\mathbf{x}_3^T$ in (\\ref{equ-1}), and denote by $V_1,V_2,V_3,V_4$ the corresponding vertex subsets, respectively. By considering the block matrix $A(V_1,V_4)$ in (\\ref{equ-1}),  we have\n\\begin{equation}\\label{equ-4}\nnA(V_1,V_4)=((k-\\lambda_4)-(\\lambda_2-\\lambda_4)-(\\lambda_3-\\lambda_4))J_{\\frac{n}{4}}.\n\\end{equation}\n\nFirst suppose that $\\lambda_4=0$. From (\\ref{equ-2}) and (\\ref{equ-4}) we know that $nA(V_{1},V_{4})=(k-\\lambda_2-\\lambda_3)J_{\\frac{n}{4}}=2kJ_{\\frac{n}{4}}$. Thus $n=2k$ and $A(V_{1},V_{4})=J_{\\frac{n}{4}}$ because $k\\neq 0$. Putting $n=2k$ in (\\ref{equ-3}), we get $\\lambda_2^2+\\lambda_3^2=k^2$, and so $\\lambda_2\\lambda_3=0$ by (\\ref{equ-2}). This implies that $\\lambda_2=0$ or $\\lambda_3=0$, which is a contradiction because $\\lambda_2$, $\\lambda_3$ and $\\lambda_4$ are distinct.\n\nNow we can assume that  $\\lambda_4\\neq0$. For the block matrix $A(V_{1},V_{1})$,  from (\\ref{equ-1}) and (\\ref{equ-2})   it is seen that\n$n(A(V_{1},V_{1})-\\lambda_4I_{\\frac{n}{4}})=((k-\\lambda_4)+(\\lambda_2-\\lambda_4)+(\\lambda_3-\\lambda_4))J_{\\frac{n}{4}}=-n\\lambda_4J_{\\frac{n}{4}}$,\nthat is, $A(V_{1},V_{1})=-\\lambda_4(J_{\\frac{n}{4}}-I_{\\frac{n}{4}})$.\nIf $n\\geq 8$, then $J_{\\frac{n}{4}}-I_{\\frac{n}{4}}\\neq 0$. Thus $\\lambda_4=-1$  because $\\lambda_4\\neq 0$. Putting $\\lambda_4=-1$ in (\\ref{equ-4}), we get $nA(V_{1},V_{4})=(k-\\lambda_2-\\lambda_3-1)J_{\\frac{n}{4}}$. Then $A(V_{1},V_{4})=0$ or $A(V_{1},V_{4})=J_{\\frac{n}{4}}$. If $A(V_{1},V_{4})=0$, we have $k-\\lambda_2-\\lambda_3-1=0$. Then from (\\ref{equ-2}) and (\\ref{equ-3}), we get $\\lambda_2=k$ and $\\lambda_3=-1$, or $\\lambda_2=-1$ and $\\lambda_3=k$, a contradiction.\nThus $A(V_{1},V_{4})=J_{\\frac{n}{4}}$, and so  $k-\\lambda_2-\\lambda_3-1=n$. Again from (\\ref{equ-2}), we get $k=n-1$, which implies that $G$ is a complete graph, a contradiction. If $n<8$, from the above arguments we see that $\\frac{n}{4}$ is an integer, so $n=4$. Then $G=K_4$ or $G=C_4$ because $G$ is a connected regular graph. In both cases, $G$ has at most three distinct eigenvalues.\n\nWe complete the proof.\n\\end{proof}\n\n\nRecall that $\\mathcal{G}(4,2)$ denotes the set of connected regular graphs with four distinct eigenvalues in which exactly two eigenvalues are simple. The following lemma provides a necessary  condition for the graphs belonging to $\\mathcal{G}(4,2)$.\n\n\\begin{lem}\\label{lem-3}\nIf $G$ is a connected $k$-regular graph on $n$ vertices with spectrum $\\big\\{[k]^1,[\\lambda_2]^1,$ $[\\lambda_3]^{m}, [\\lambda_4]^{n-2-m}\\big\\}$ ($2\\leq m\\leq n-4$), then $G$ admits a  regular partition $V(G)=V_1\\cup V_2$ into halves  with degrees $(\\frac{1}{2}(k+\\lambda_2),\\frac{1}{2}(k-\\lambda_2))$ such that\n\\begin{equation*}\n|N_G(u) \\cap N_G(v)|\\!=\\!\\left\\{\n\\begin{array}{ll}\n\\lambda_3\\!+\\!\\lambda_4\\!+\\!\\frac{1}{n}[(k\\!-\\!\\lambda_3)(k\\!-\\!\\lambda_4)\\!+\\!(\\lambda_2\\!-\\!\\lambda_3)(\\lambda_2\\!-\\!\\lambda_4)]&\\mbox{if  $u\\sim v$ in $V_1$ or $V_2$};\\\\\n\\frac{1}{n}[(k\\!-\\!\\lambda_3)(k\\!-\\!\\lambda_4)\\!+\\!(\\lambda_2\\!-\\!\\lambda_3)(\\lambda_2\\!-\\!\\lambda_4)]&\\mbox{if  $u\\nsim v$ in $V_1$ or $V_2$};\\\\\n\\lambda_3\\!+\\!\\lambda_4\\!+\\!\\frac{1}{n}[(k\\!-\\!\\lambda_3)(k\\!-\\!\\lambda_4)\\!-\\!(\\lambda_2\\!-\\!\\lambda_3)(\\lambda_2\\!-\\!\\lambda_4)]&\\mbox{if  $u\\sim v$, $u\\in V_1$, $v\\in V_2$};\\\\\n\\frac{1}{n}[(k\\!-\\!\\lambda_3)(k\\!-\\!\\lambda_4)\\!-\\!(\\lambda_2\\!-\\!\\lambda_3)(\\lambda_2\\!-\\!\\lambda_4)]&\\mbox{if  $u\\nsim v$, $u\\in V_1$, $v\\in V_2$}.\n\\end{array}\n\\right.\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nSince $\\lambda_2$ ($\\neq k$) is a simple eigenvalue of $G$,  by Lemmas \\ref{lem-5}, \\ref{lem-2} and Corollary \\ref{cor-1} we know that $n$ is even, $\\lambda_2$ is an integer and $\\mathbf{x}_2=\\frac{1}{\\sqrt{n}}(\\mathbf{e}_\\frac{n}{2}^T,-\\mathbf{e}_\\frac{n}{2}^T)^T$ is an eigenvector of $\\lambda_2$. Putting $V_1=\\{v\\in V\\mid \\mathbf{x}_{2}(v)=1\\}$ and $V_2=\\{v\\in V\\mid \\mathbf{x}_{2}(v)=-1\\}$, again by Corollary \\ref{cor-1} we  see that $V(G)=V_1\\cup V_2$ is a regular  partition  of $G$ into halves with degrees $(\\frac{1}{2}(k+\\lambda_2),\\frac{1}{2}(k-\\lambda_2))$. Furthermore, the matrix $(A-\\lambda_3I_n)(A-\\lambda_4I_n)$ has the spectral decomposition\n\\begin{equation*}\n(A-\\lambda_3I_n)(A-\\lambda_4I_n)=\\frac{1}{n}(k-\\lambda_3)(k-\\lambda_4)\\mathbf{e}_n\\mathbf{e}_n^T+(\\lambda_2-\\lambda_3)(\\lambda_2-\\lambda_4)\\mathbf{x}_2\\mathbf{x}_2^T,\n\\end{equation*}\nthat is,\n\\begin{equation}\\label{equ-5}\nA^2-(\\lambda_3+\\lambda_4)A+\\lambda_3\\lambda_4I_n=\\frac{1}{n}(k-\\lambda_3)(k-\\lambda_4)J_n+\\frac{1}{n}(\\lambda_2-\\lambda_3)(\\lambda_2-\\lambda_4)\n\\left(\\begin{matrix}\nJ_\\frac{n}{2}&-J_\\frac{n}{2}\\\\\n-J_\\frac{n}{2}&J_\\frac{n}{2}\n\\end{matrix}\\right).\n\\end{equation}\nNote that the $(u,v)$-entry of $A^2$ is equal to $|N_G(u) \\cap N_G(v)|$. Then (\\ref{equ-5}) implies that\n\\begin{equation*}\n|N_G(u) \\cap N_G(v)|\\!=\\!\\left\\{\n\\begin{array}{ll}\n\\lambda_3\\!+\\!\\lambda_4\\!+\\!\\frac{1}{n}[(k\\!-\\!\\lambda_3)(k\\!-\\!\\lambda_4)\\!+\\!(\\lambda_2\\!-\\!\\lambda_3)(\\lambda_2\\!-\\!\\lambda_4)]&\\mbox{if  $u\\sim v$ in $V_1$ or $V_2$};\\\\\n\\frac{1}{n}[(k\\!-\\!\\lambda_3)(k\\!-\\!\\lambda_4)\\!+\\!(\\lambda_2\\!-\\!\\lambda_3)(\\lambda_2\\!-\\!\\lambda_4)]&\\mbox{if  $u\\nsim v$ in $V_1$ or $V_2$};\\\\\n\\lambda_3\\!+\\!\\lambda_4\\!+\\!\\frac{1}{n}[(k\\!-\\!\\lambda_3)(k\\!-\\!\\lambda_4)\\!-\\!(\\lambda_2\\!-\\!\\lambda_3)(\\lambda_2\\!-\\!\\lambda_4)]&\\mbox{if  $u\\sim v$, $u\\in V_1$, $v\\in V_2$};\\\\\n\\frac{1}{n}[(k\\!-\\!\\lambda_3)(k\\!-\\!\\lambda_4)\\!-\\!(\\lambda_2\\!-\\!\\lambda_3)(\\lambda_2\\!-\\!\\lambda_4)]&\\mbox{if  $u\\nsim v$, $u\\in V_1$, $v\\in V_2$}.\n\\end{array}\n\\right.\n\\end{equation*}\n\nThis completes the proof.\n\\end{proof}\n\nThe \\emph{Kronecker product} $A\\otimes B$ of matrices $A=(a_{ij})_{m\\times n}$ and $B=(b_{ij})_{p\\times q}$ is the $mp\\times nq$ matrix obtained from $A$ by replacing each element $a_{ij}$ with the block $a_{ij}B$. Given a graph $G$ on  $n$  vertices with adjacency matrix $A$, we denote by $G\\circledast J_m$ the graph with adjacency matrix $A\\circledast J_m=(A+I_n)\\otimes J_m-I_{nm}$ (see \\cite{Dam1}).  By the definition, $G\\circledast J_m$ is just the graph obtained from $G$ by replacing every vertex of $G$ with a clique $K_m$, and two such cliques are joined if and only if their corresponding vertices are adjacent in $G$. It is easy to see that the spectra of $G$ and $G\\circledast J_m$ are determined by each other, that is,\n\\begin{equation}\\label{equ-6}\n\\begin{array}{rl}\n&\\mathrm{Spec}(G)=\\left\\{[\\lambda_1]^{m_1},\\ldots,[\\lambda_t]^{m_t}\\right\\}\\\\\n\\Leftrightarrow &\\mathrm{Spec}(G\\circledast J_m)=\\big\\{[m\\lambda_1+m-1]^{m_1},\\ldots,[m\\lambda_t+m-1]^{m_t},[-1]^{nm-n}\\big\\}.\n\\end{array}\n\\end{equation}\n\nRecall that $\\mathcal{G}(4,2,-1)$ denotes the set of graphs belonging to $\\mathcal{G}(4,2)$ with $-1$ as an  eigenvalue.  The following result gives a partial characterization of the graphs in $\\mathcal{G}(4,2,-1)$.\n\\begin{thm}\\label{thm-main-1}\nLet $G\\in \\mathcal{G}(4,2,-1)$. Then $-1$ is a non-simple eigenvalue of $G$ if and only if  $G=K_{s,s}\\circledast J_{t}$ with $s,t\\geq 2$, or $G=K_{s,s}^{-}\\circledast J_{t}$ with $s\\geq 3$ and $t\\geq 1$.\n\\end{thm}\n\\begin{proof}\nBy the assumption, let $G$ be a connected $k$-regular graph with $\\mathrm{Spec}(G)=\\big\\{[k]^1,[\\alpha]^1,$ $[\\beta]^{n-2-m},[-1]^{m}\\big\\}$ ($2\\leq m\\leq n-4$). By considering the traces of $A(G)$ and $A^2(G)$, we get\n\\begin{equation}\\label{equ-main-0}\n\\left\\{\\begin{aligned}\n&k+\\alpha+(n-2-m)\\beta-m=0,\\\\\n&k^2+\\alpha^2+(n-2-m)\\beta^2+m=kn.\n\\end{aligned}\\right.\n\\end{equation}\nOwing to  $\\beta\\neq -1$ and $k+\\alpha+(n-2-m)\\beta-m=0$, we have $n-k-\\alpha-2\\neq 0$. Then from (\\ref{equ-main-0}) we can deduce that\n\\begin{equation}\\label{equ-main-1}\n\\left\\{\\begin{aligned}\n&\\beta=\\frac{kn-k^2-k-\\alpha^2-\\alpha}{n-k-\\alpha-2},\\\\\n&m=n-1+\\frac{(n-k-1)(n-2\\alpha-2)}{k^2 + (2 - n)k + \\alpha^2 + 2\\alpha - n + 2}.\n\\end{aligned}\\right.\n\\end{equation}\nSince $\\alpha$ ($\\neq k$) is a simple eigenvalue of $G$, by Lemma \\ref{lem-3} we know that\n$G$ admits a regular partition $V(G)=V_1\\cup V_2$ into halves with degree $(\\frac{1}{2}(k+\\alpha),\\frac{1}{2}(k-\\alpha))$,  and that if $u,v\\in V_i$ ($i=1,2$) are adjacent, then\n\\begin{equation}\\label{equ-main-2}\n|N_G(u) \\cap N_G(v)|=\\begin{aligned}\n\\beta-1+\\frac{1}{n}[(k-\\beta)(k+1)+(\\alpha-\\beta)(\\alpha+1)].\n\\end{aligned}\n\\end{equation}\nCombining (\\ref{equ-main-1}) and (\\ref{equ-main-2}),  we  have\n$|N_G(u) \\cap N_G(v)|=k-1$ by simple computation, so $u$ and $v$ have the same neighbors, that is, $N_G(u)\\backslash \\{v\\}=N_G(v)\\backslash \\{u\\}$. Since $u$ has exactly $\\frac{k+\\alpha}{2}$ neighbors in $G[V_i]$, each of them has the same neighbors (in $G$) as $u$, we see that such $\\frac{k+\\alpha}{2}$ neighbors together with $u$ induce a clique $K_{\\frac{k+\\alpha+2}{2}}$ which is totally included in $G[V_i]$ itself.  Furthermore, again by $N_G(u)\\backslash \\{v\\}=N_G(v)\\backslash \\{u\\}$,  there are no edges between any two such cliques in $V_i$, and if $v_1\\in V_1$ is adjacent to $v_2\\in V_2$, then the clique containing $v_1$ must be joined with the clique containing $v_2$.  Moreover, both $G[V_1]$ and $G[V_2]$ consist of the disjoint union of $\\frac{n}{k+\\alpha+2}$ copies of $K_{\\frac{k+\\alpha+2}{2}}$ because $|V_1|=|V_2|=\\frac{n}{2}$.\nHence, there exists a regular bipartite graph $H$ on $\\frac{2n}{k+\\alpha+2}$ vertices  such that $G=H\\circledast J_{\\frac{k+\\alpha+2}{2}}$. Then from (\\ref{equ-6}) we  deduce that\n\\begin{equation*}\n\\mathrm{Spec}(H)=\\Bigg\\{\\bigg[\\frac{k-\\alpha}{k+\\alpha+2}\\bigg]^1,\n\\bigg[-\\frac{k-\\alpha}{k+\\alpha+2}\\bigg]^1,\n\\bigg[\\frac{2\\beta-k-\\alpha}{k+\\alpha+2}\\bigg]^{n-m-2},\n[-1]^{m-\\frac{k+\\alpha}{k+\\alpha+2}n}\\Bigg\\}.\n\\end{equation*}\nSince $\\frac{k-\\alpha}{k+\\alpha+2}$ is the  maximum eigenvalue of $H$ which is simple, $H$ must be a connected $\\big(\\frac{k-\\alpha}{k+\\alpha+2}\\big)$-regular bipartite graph. Clearly,  $-\\frac{k-\\alpha}{k+\\alpha+2}\\neq -1$ since otherwise we have $\\alpha=-1$, a contradiction.  Thus $-\\frac{k-\\alpha}{k+\\alpha+2}<-1$ because  it is an integer, so $\\alpha<-1$. We consider the following two situations.\n\n\\textbf{Case 1.} $m-\\frac{k+\\alpha}{k+\\alpha+2}n=0$;\n\nSince $H$ is  bipartite,  we get $\\frac{2\\beta-k-\\alpha}{k+\\alpha+2}=0$, and then $\\beta=\\frac{k+\\alpha}{2}$. Putting $\\beta=\\frac{k+\\alpha}{2}$ in (\\ref{equ-main-1}) and  considering  $\\alpha<k$,  we  deduce that $\\alpha=k-n$, $\\beta=k-\\frac{n}{2}$, $m=\\frac{2kn-n^2}{2k-n+2}$, and $\\frac{n}{2}+1\\leq k\\leq\\frac{3n}{4}-1$ because $\\alpha\\geq-k$ and $2\\leq m\\leq n-4$. Thus\n\\begin{equation*}\n\\mathrm{Spec}(H)=\\Bigg\\{\\bigg[\\frac{n}{2k-n+2}\\bigg]^1,\\bigg[-\\frac{n}{2k-n+2}\\bigg]^1,\n[0]^{\\frac{2n}{2k-n+2}-2}\\Bigg\\}.\n\\end{equation*}\nThen $H$ is a connected $\\big(\\frac{n}{2k-n+2}\\big)$-regular bipartite graph with three distinct eigenvalues. Therefore, $H=K_{\\frac{n}{2k-n+2},\\frac{n}{2k-n+2}}$ because  complete bipartite graphs are the only connected  bipartite graphs with  three distinct eigenvalues. Hence, $G=H\\circledast J_{\\frac{k+\\alpha+2}{2}}= K_{\\frac{n}{2k-n+2},\\frac{n}{2k-n+2}}\\circledast J_{\\frac{2k-n+2}{2}}$. If we set $s=\\frac{n}{2k-n+2}$ and $t=\\frac{2k-n+2}{2}$, then $G=K_{s,s} \\circledast J_{t}$ with $s,t\\geq 2$ from the above arguments.\n\n\\textbf{Case 2.} $m-\\frac{k+\\alpha}{k+\\alpha+2}n\\geq 1$.\n\nSince $H$ is bipartite, we have $\\frac{2\\beta-k-\\alpha}{k+\\alpha+2}=-(-1)=1$, and so $\\beta=k+\\alpha+1$. Combining  this with (\\ref{equ-main-1}) we obtain that\n$(\\alpha+1)(n-2k-2)=0$, which implies that $k=\\frac{n}{2}-1$ because $\\alpha\\neq -1$. Then $\\beta=\\frac{n}{2}+\\alpha$, $m=n-1-\\frac{2n}{n+2\\alpha+2}$ by  (\\ref{equ-main-1}), and so $-\\frac{n}{2}+1\\leq\\alpha\\leq-\\frac{n}{6}-1$ because $2\\leq m\\leq n-4$. Furthermore, $n-m-2=m-\\frac{k+\\alpha}{k+\\alpha+2}n=\\frac{n-2\\alpha-2}{n+2\\alpha+2}$.  Thus we get\n\\begin{equation*}\n\\mathrm{Spec}(H)=\\Bigg\\{\\bigg[\\frac{n-2\\alpha-2}{n+2\\alpha+2}\\bigg]^1,\\bigg[-\\frac{n-2\\alpha-2}{n+2\\alpha+2}\\bigg]^1,[1]^{\\frac{n-2\\alpha-2}{n+2\\alpha+2}},[-1]^{\\frac{n-2\\alpha-2}{n+2\\alpha+2}}\\Bigg\\}.\n\\end{equation*}\nThen $H$ is a connected $\\big(\\frac{n-2\\alpha-2}{n+2\\alpha+2}\\big)$-regular bipartite graph with four distinct eigenvalues. By Lemma \\ref{lem-4}, we may conclude that $H$ is the incidence graph of  a symmetric BIBD with parameters $\\big(\\frac{n-2\\alpha-2}{n+2\\alpha+2}+1,\\frac{n-2\\alpha-2}{n+2\\alpha+2},\\frac{n-2\\alpha-2}{n+2\\alpha+2}-1\\big)$. It is well known that such a BIBD is unique (see \\cite{Dam1}), and the corresponding incidence  graph can be obtained by removing a perfect matching from the  complete bipartite graph $K_{s,s}$ , where $s=\\frac{n-2\\alpha-2}{n+2\\alpha+2}+1=\\frac{2n}{n+2\\alpha+2}$. Hence, $G=H\\circledast J_{\\frac{k+\\alpha+2}{2}}= K_{\\frac{2n}{n+2\\alpha+2},\\frac{2n}{n+2\\alpha+2}}^{-}\\circledast J_{\\frac{n+2\\alpha+2}{4}}$. If we put $s=\\frac{2n}{n+2\\alpha+2}$ and $t=\\frac{n+2\\alpha+2}{4}$, then $G=K_{s,s}^{-}\\circledast J_t$ with $s\\geq 3$ and $t\\geq 1$ from the above arguments.\n\n\nConversely,  by simple computation we obtain that\n\\begin{equation}\\label{equ-main-3}\n\\begin{aligned}\n&\\mathrm{Spec}(K_{s,s}\\circledast J_t)=\\{[st+t-1]^1,[-st+t-1]^1,[t-1]^{2s-2},[-1]^{2s(t-1)}\\};\\\\\n&\\mathrm{Spec}(K_{s,s}^{-}\\circledast J_t)=\\{[st-1]^1,[-st+2t-1]^1,[2t-1]^{s-1},[-1]^{2st-s-1}\\},\n\\end{aligned}\n\\end{equation}\nand our result follows.\n\\end{proof}\n\nBy Theorem \\ref{thm-main-1}, we obtain the following corollary immediately.\n\\begin{cor}\\label{cor-2}\nThe graphs $K_{s,s}\\circledast J_{t}$ ($s,t\\geq 2$) and $K_{s,s}^{-}\\circledast J_{t}$ ($s\\geq 3$, $t\\geq 1$) are DS.\n\\end{cor}\n\\begin{proof}\nNote that any graph cospectral with $K_{s,s}\\circledast J_{t}$  or $K_{s,s}^{-}\\circledast J_{t}$ must be connected. By Theorem \\ref{thm-main-1}, it suffices to prove that $K_{s_1,s_1}\\circledast J_{t_1}$ ($s_1,t_1\\geq 2$) and $K_{s_2,s_2}^{-}\\circledast J_{t_2}$ ($s_2\\geq 3$, $t_2\\geq 1$) cannot share the same spectrum. On the contrary, assume that  $\\mathrm{Spec}(K_{s_1,s_1}\\circledast J_{t_1})=\\mathrm{Spec}(K_{s_2,s_2}^-\\circledast J_{t_2})$. Then  $s_1t_1=s_2t_2$ and $s_1t_1+t_1-1=s_2t_2-1$ because they are regular graphs which have the same number of vertices and share the common degree. This implies that $t_1=0$, which is impossible because $t_1\\geq 2$.\n\\end{proof}\n\\begin{remark}\\label{rem-1}\n\\emph{It is worth mentioning that the DS-properties of $K_{s,s}\\circledast J_{t}$ and $K_{s,s}^{-}\\circledast J_{t}$ have been mentioned in  \\cite{Dam6} and \\cite{Dam1}, respectively.}\n\\end{remark}\nRecall that $\\mathcal{G}(4,\\geq -1)$ denotes the set of connected regular graphs with four distinct eigenvalues and second least eigenvalue  not less than $-1$.  Petrovi\\'{c} in \\cite{Petrovic} determined all the connected graphs whose second least eigenvalue is not less than $-1$. However, it is a difficult work to pick out all the graphs belonging to $\\mathcal{G}(4,\\geq -1)$ from their characterization. Here, from Theorem \\ref{thm-main-1} and Lemma \\ref{lem-6}, we can easily give a complete characterization of the graphs in $\\mathcal{G}(4,\\geq -1)$.\n\\begin{thm}\\label{thm-main-2}\nA connected graph $G\\in \\mathcal{G}(4,\\geq -1)$  if and only if $G=K_{s,s}\\circledast J_{t}$ with $s,t\\geq 2$, or $G=K_{s,s}^{-}\\circledast J_{t}$ with $s\\geq 3$ and $t\\geq 1$.\n\\end{thm}\n\\begin{proof}\nSuppose that $G\\in \\mathcal{G}(4,\\geq -1)$, and  $\\alpha$, $\\beta$ are the least and second least eigenvalues of $G$, respectively.  If $\\beta>-1$, then  $G=K_{m,n}$, $A(l,m,n)$ or $B(l,m,n,p)$ with proper parameters $l,m,n,p$ by Lemma \\ref{lem-6}. It is seen that $A(l,m,n)$ and  $B(l,m,n,p)$ cannot be regular, so $G=K_{n,n}$ with $n\\geq 2$. Note that $K_{n,n}$ has  only three distinct eigenvalues,  so there are no graphs in $\\mathcal{G}(4,\\geq -1)$ with  $\\beta>-1$, and thus $\\beta=-1$ because $G\\in \\mathcal{G}(4,\\geq -1)$. We claim that $\\alpha<\\beta=-1$ since otherwise $G$ will be a complete graph due to the least eigenvalue of $G$ is  $-1$, and thus $\\alpha$ is a simple eigenvalue of $G$. As a consequence, $G$ will belong to $\\mathcal{G}(4,2,-1)$ and $-1$ is a non-simple eigenvalue of $G$  by Theorem \\ref{thm-main-0}. Hence, $G=K_{s,s}\\circledast J_{t}$ with $s,t\\geq 2$, or $G=K_{s,s}^{-}\\circledast J_{t}$ with $s\\geq 3$ and $t\\geq 1$ by Theorem \\ref{thm-main-1}.\n\nConversely, from (\\ref{equ-main-3}) we obtain the required result immediately.\n\\end{proof}\n\nNow we continue to  consider the graphs in $\\mathcal{G}(4,2,-1)$. The following result excludes the existence of such graphs belonging to (2b) (see Section \\ref{s-1}).\n\\begin{thm}\\label{thm-main-3}\nThere are no connected $k$-regular graphs with spectrum $\\{[k]^1,[-1]^1,[\\alpha]^{m},$ $[\\beta]^{n-2-m}\\}$, where $\\alpha$ and $\\beta$ are not integers and $2\\leq m\\leq n-4$.\n\\end{thm}\n\\begin{proof}\nOn the conrary, assume that $G$ is such a graph.  Then $G$ will be a connected $k$-regular graphs having exactly two integral eigenvalues. By Lemma \\ref{lem-1}, the  eigenvalues $\\alpha$ and $\\beta$ are of the form $\\frac{1}{2}(a\\pm\\sqrt{b})$ ($a,b\\in \\mathbb{Z}$, $b>0$) and have the same multiplicity, i.e., $m=\\frac{1}{2}(n-2)$. Then $\\alpha$ and $\\beta$ satisfy  the following two equations:\n\\begin{equation*}\n\\left\\{\\begin{aligned}\n&k-1+\\frac{1}{2}(n-2)\\alpha+\\frac{1}{2}(n-2)\\beta=0,\\\\\n&k^2+1+\\frac{1}{2}(n-2)\\alpha^2+\\frac{1}{2}(n-2)\\beta^2=kn.\n\\end{aligned}\\right.\n\\end{equation*}\nBy simple computation, we obtain\n\\begin{equation*}\n\\left\\{\\begin{aligned}\n&\\alpha=\\frac{-k+1+\\sqrt{(kn-k-1)(n-k-1)}}{n-2},\\\\\n&\\beta=\\frac{-k+1-\\sqrt{(kn-k-1)(n-k-1)}}{n-2}.\n\\end{aligned}\\right.\n\\end{equation*}\nConsidering that $\\alpha$ and $\\beta$ are of the form $\\frac{1}{2}(a\\pm\\sqrt{b})$ ($a,b\\in \\mathbb{Z}$), we claim that  $\\frac{-k+1}{n-2}=\\frac{a}{2}$, and so  $a=-1$ because $a\\in \\mathbb{Z}$ and $1<k<n-1$. Thus $n=2k$, and then\n\\begin{equation}\\label{equ-main-4}\n\\begin{aligned}\n\\alpha=\\frac{1}{2}(-1+\\sqrt{2k + 1}),~~~~\\beta=\\frac{1}{2}(-1-\\sqrt{2k + 1}).\n\\end{aligned}\n\\end{equation}\nFurthermore,  by Lemma \\ref{lem-3}, the graph $G$ admits a  regular partition $V(G)=V_1\\cup V_2$ into halves  with degree $(\\frac{1}{2}(k-1),\\frac{1}{2}(k+1))$ such that\n\\begin{equation*}\n|N_G(u) \\cap N_G(v)|=\\left\\{\n\\begin{aligned}\n&\\alpha+\\beta+\\frac{1}{n}[(k-\\alpha)(k-\\beta)+(-1-\\alpha)(-1-\\beta)]&\\mbox{if  $u\\sim v$ in $V_i$};\\\\\n&\\frac{1}{n}[(k-\\alpha)(k-\\beta)+(-1-\\alpha)(-1-\\beta)]&\\mbox{if  $u\\nsim v$ in $V_i$}.\n\\end{aligned}\n\\right.\n\\end{equation*}\nPutting (\\ref{equ-main-4}) and $n=2k$  in the above equation, we get\n\\begin{equation*}\n|N_G(u) \\cap N_G(v)|=\\left\\{\n\\begin{aligned}\n&\\frac{k}{2}-1&\\mbox{if  $u\\sim v$ in $V_i$};\\\\\n&\\frac{k}{2}&\\mbox{if  $u\\nsim v$ in $V_i$},\n\\end{aligned}\n\\right.\n\\end{equation*}\nwhich is impossible because $k$ is  odd  due to $\\frac{1}{2}(k-1)$ is an integer.\n\nThis completes the proof.\n\\end{proof}\n\nBy Theorems \\ref{thm-main-1} and  \\ref{thm-main-3}, in order to determine all the graphs in  $\\mathcal{G}(4,2,-1)$, it remains to consider such graphs belonging to (2a) (see Section \\ref{s-1}), i.e., the $k$-regular graphs with spectrum $\\{[k]^1,[-1]^1,[\\alpha]^{m},[\\beta]^{n-2-m}\\}$, where $\\alpha$ and $\\beta$ are  integers and  $2\\leq m\\leq n-4$.  By Appendix A in \\cite{Dam2}, we find that there are no such graphs up to $30$ vertices. In what follows, we will prove the non-existence of such graphs for arbitrary $n$.\n\n\\begin{thm}\\label{thm-main-4}\nThere are no connected $k$-regular graphs with spectrum $\\{[k]^1,[-1]^1,[\\alpha]^{m},$ $[\\beta]^{n-2-m}\\}$, where $\\alpha$ and $\\beta$ are integers and $2\\leq m\\leq n-4$.\n\\end{thm}\n\\begin{proof}\nOn the contrary, assume that $G$ is such a graph with adjacency matrix $A$.  Firstly, we assert that $3\\leq k\\leq n-2$. In fact,  if $k=2$, then $G=C_n$, where $n$ is even by Lemmas \\ref{lem-5} and \\ref{lem-2}, and so  $G$ is a bipartite graph, which is impossible because $2\\leq m\\leq n-4$; if $k=n-1$, then $G$ is a complete graph, which has exactly two distinct eigenvalues, a contradiction.\n\nNow suppose that $\\alpha\\beta\\geq 0$, then $\\alpha,\\beta\\geq 0$ or $\\alpha,\\beta\\leq 0$. In the former case, we see that $G$ is a complete graph because $-1$ is the least eigenvalue of $G$, which is a contradiction. In the later case, we claim that $G$ is a  complete multipartite graph because $G$ has only one positive eigenvalue, thus $G=K_{n-k,n-k,\\ldots,n-k}$ due to $G$ is $k$-regular. Then $\\mathrm{Spec}(G)=\\big\\{[k]^1,[n-k-2]^{\\frac{n}{n-k}-1},[0]^{n-\\frac{n}{n-k}}\\big\\}$, a contradiction. Thus $\\alpha\\beta<0$. Without loss of generality, we may assume that $\\alpha\\geq 1$ and $\\beta\\leq -2$. By considering the traces of $A$ and $A^2$, we get\n\\begin{equation*}\n\\left\\{\\begin{aligned}\n&k-1+m\\alpha+(n-2-m)\\beta=0,\\\\\n&k^2+1+m\\alpha^2+(n-2-m)\\beta^2=kn.\n\\end{aligned}\\right.\n\\end{equation*}\nCanceling out $m$ in the above two equations, we have\n\\begin{equation}\\label{equ-main-5}\n\\beta=-\\frac{k(n-k)+(k-1)\\alpha-1}{(n-2)\\alpha+k-1}.\n\\end{equation}\nBy Lemmas \\ref{lem-5} and \\ref{lem-2}, it is seen that $n$ is a divisor of $(k-\\alpha)(k-\\beta)+(-1-\\alpha)(-1-\\beta)$ and $(k-\\alpha)(k-\\beta)-(-1-\\alpha)(-1-\\beta)$, and so  a divisor of $-2(-1-\\alpha)(-1-\\beta)=-2(1+\\alpha)(1+\\beta)>0$ because $\\alpha\\geq 1$ and $\\beta\\leq -2$ are integers. Thus  $c=\\frac{-2(1+\\alpha)(1+\\beta)}{n}$ is a positive integer.  Combining this with (\\ref{equ-main-5}), we get\n\\begin{eqnarray*}\nc&=&\\frac{2(k-\\alpha)(\\alpha+1)(n-k-1)}{n(n-2)\\alpha+n(k-1)}\n<\\frac{2(k-\\alpha)(\\alpha+1)(n-k-1)}{n(n-2)\\alpha}=\\frac{2(n-k-1)}{n\\cdot\\frac{n-2}{k-\\alpha}\\cdot\\frac{\\alpha}{\\alpha+1}}\\\\\n&<&\\frac{4(n-k-1)}{n}<4.\n\\end{eqnarray*}\nIt suffices to consider the following three situations.\n\n\\textbf{Case 1.} $c=1$;\n\nIn this case, we get $\\frac{2(k-\\alpha)(\\alpha+1)(n-k-1)}{(n-2)\\alpha+k-1}=n$, that is,\n\\begin{equation}\\label{equ-main-6}\n2(n-k-1)\\alpha^2+(n^2-2kn+2k^2-2)\\alpha+(2k-n)(k+1)=0,\n\\end{equation}\nwhich implies that $n>2k$ because $n-k-1>0$, $n^2-2kn+2k^2-2>0$ and $\\alpha\\geq 1$. Solving  (\\ref{equ-main-6}), we get\n\\begin{equation}\\label{equ-main-7}\n\\alpha=\\frac{-(n^2\\!-\\!2kn\\!+\\!2k^2\\!-\\!2)+\\!\\sqrt{(n^2-2kn+2k^2-2)^2+8(n-k-1)(n-2k)(k+1)}}{4(n-k-1)}.\n\\end{equation}\nAgain from (\\ref{equ-main-5}) we obtain\n\\begin{equation}\\label{equ-main-8}\n\\beta=\\frac{-(n^2\\!-\\!2kn\\!+\\!2k^2\\!-\\!2)-\\!\\sqrt{(n^2-2kn+2k^2-2)^2+8(n-k-1)(n-2k)(k+1)}}{4(n-k-1)}.\n\\end{equation}\nThen, by Lemma \\ref{lem-3}, $G$ admits a  regular partition $V(G)=V_1\\cup V_2$ into halves  with degrees $(\\frac{1}{2}(k-1),\\frac{1}{2}(k+1))$ such that\n\\begin{equation}\\label{equ-main-9}\n|N_G(u) \\cap N_G(v)|=\\alpha+\\beta+\\frac{1}{n}[(k-\\alpha)(k-\\beta)+(-1-\\alpha)(-1-\\beta)]\\mbox{ for   $u\\sim v$ in $V_i$}.\n\\end{equation}\nCombining (\\ref{equ-main-7}), (\\ref{equ-main-8}) and (\\ref{equ-main-9}), we thus have $|N_G(u)\\cap N_G(v)|=k-\\frac{n}{2}-1$. This implies that\n$n\\leq 2k-2$ because $|N_G(u)\\cap N_G(v)|\\geq 0$, contrary to $n>2k$.\n\n\n\\textbf{Case 2.} $c=2$;\n\nIn this case, we have $\\frac{(k-\\alpha)(\\alpha+1)(n-k-1)}{(n-2)\\alpha+k-1}=n$. This implies that\n\\begin{equation*}\n\\left\\{\\begin{aligned}\n&\\alpha=\\frac{-(n^2\\!-\\!(k\\!+\\!1)n\\!+\\!k^2\\!-\\!1)+\\!\\sqrt{(n^2\\!-\\!(k\\!+\\!1)n\\!+\\!k^2\\!-\\!1)^2\\!-\\!4(n\\!-\\!k\\!-\\!1)(k^2\\!+\\!k\\!-\\!n)}}{2(n-k-1)},\\\\\n&\\beta=\\frac{-(n^2\\!-\\!(k\\!+\\!1)n\\!+\\!k^2\\!-\\!1)-\\!\\sqrt{(n^2\\!-\\!(k\\!+\\!1)n\\!+\\!k^2\\!-\\!1)^2\\!-\\!4(n\\!-\\!k\\!-\\!1)(k^2\\!+\\!k\\!-\\!n)}}{2(n-k-1)}.\n\\end{aligned}\\right.\n\\end{equation*}\nAgain from (\\ref{equ-main-9}) we deduce that $|N_G(u)\\cap N_G(v)|=k-n-1<0$ for $u\\sim v$ in $V_i$ ($i=1,2$), which is a contradiction.\n\n\\textbf{Case 3.} $c=3$.\n\nIn this case, we have $\\frac{2(k-\\alpha)(\\alpha+1)(n-k-1)}{(n-2)\\alpha+k-1}=3n$, that is,\n\\begin{equation*}\n2(n-k-1)\\alpha^2+[3n^2-(2k+4)n+2k^2-2]\\alpha+(k-3)n+2k^2+2k=0,\n\\end{equation*}\nwhich is  impossible because $n-k-1>0$, $3n^2-(2k+4)n+2k^2-2>0$, $(k-3)n+2k^2+2k>0$ due to $k\\geq 3$, and  $\\alpha >0$.\n\nWe complete this proof.\n\\end{proof}\n\nCombining Theorems \\ref{thm-main-1}, \\ref{thm-main-3} and \\ref{thm-main-4}, we obtain the  main result of this paper.\n\\begin{thm}\\label{thm-main-5}\nA connected graph $G\\in \\mathcal{G}(4,2,-1)$ if and only if $G=K_{s,s}\\circledast J_{t}$ with $s,t\\geq 2$, or $G=K_{s,s}^{-}\\circledast J_{t}$ with $s\\geq 3$ and $t\\geq 1$.\n\\end{thm}\nRecall that  $\\mathcal{G}(4,2,0)$ denotes the set of  graphs belonging to $\\mathcal{G}(4,2)$ with $0$ as an  eigenvalue. Since the spectrum of a regular graph could be deduced from its complement,  we can easily characterize all the graphs in $\\mathcal{G}(4,2,0)$ by Theorems \\ref{thm-main-1}, \\ref{thm-main-3} and \\ref{thm-main-4}.\n\\begin{thm}\\label{thm-main-6}\nA connected graph $G\\in \\mathcal{G}(4,2,0)$ if and only if $G=\\overline{K_{s,s}^{-}\\circledast J_{t}}$ with $s\\geq 3$ and $t\\geq 1$.\n\\end{thm}\n\\begin{proof}\n Let $G\\in\\mathcal{G}(4,2,0)$ be a connected $k$-regular graph with adjacency matrix $A$, and let $\\overline{G}$ be the complement of $G$.\n\nIf $0$ is a non-simple eigenvalue of $G$, suppose that $\\mathrm{Spec}(G)=\\big\\{[k]^1,[\\alpha]^1,[\\beta]^{n-2-m},[0]^{m}\\big\\}$ ($2\\leq m\\leq n-4$). Then $\\overline{G}$ is a $(n-k-1)$-regular graph with $\\mathrm{Spec}(\\overline{G})=\\big\\{[n-k-1]^1,[-1-\\alpha]^1,[-1-\\beta]^{n-2-m},[-1]^{m}\\big\\}$.\nWe claim that $\\alpha>0$ and $\\beta<0$ or $\\alpha<0$ and $\\beta>0$, since otherwise $G$ will be  a regular complete multipartite graph (which  has only three distinct eigenvalues) or does not exist. Assume that $\\overline{G}$ is  disconnected.  If $\\alpha>0$ and $\\beta<0$,  we have $n-k-1=-1-\\beta$, i.e., $\\beta=k-n$. By considering the traces of $A$ and $A^2$, we obtain\n\\begin{equation*}\n\\left\\{\\begin{aligned}\n&k+\\alpha+(n-2-m)(k-n)=0,\\\\\n&k^2+\\alpha^2+(n-2-m)(k-n)^2=kn,\\\\\n\\end{aligned}\\right.\n\\end{equation*}\nand so $\\alpha=0$ or $\\alpha=k-n$, which are impossible due to $\\alpha>0$. If $\\alpha<0$ and $\\beta>0$, similarly,  we have $\\alpha=k-n$ because $\\overline{G}$ is disconnected and regular, and so $n=2k$, or $n\\neq 2k$ and $\\beta=k-n$ by considering the traces of $A$ and $A^2$. In both cases, we can deduce a contradiction  because $G$ cannot be a bipartite graph and $\\alpha\\neq\\beta$. Therefore, $\\overline{G}$ must be connected, and thus $\\overline{G}\\in\\mathcal{G}(4,2,-1)$ with $-1$ as a non-simple eigenvalue. By Theorem \\ref{thm-main-1}, we may conclude that $\\overline{G}=K_{s,s}\\circledast J_{t}$ with $s,t\\geq 2$, or $\\overline{G}=K_{s,s}^{-}\\circledast J_{t}$ with $s\\geq 3$ and $t\\geq 1$. Hence, $G=\\overline{K_{s,s}^{-}\\circledast J_{t}}$ with $s\\geq 3$ and $t\\geq 1$ because $G$ is connected.\n\nIf $0$ is a simple eigenvalue of $G$, we suppose that $\\mathrm{Spec}(G)=\\big\\{[k]^1,[0]^1,[\\alpha]^{m},[\\beta]^{n-2-m}\\big\\}$. Then $\\mathrm{Spec}(\\overline{G})=\\big\\{[n-k-1]^1,[-1]^1,[-1-\\alpha]^{m},[-1-\\beta]^{n-2-m}\\big\\}$. As above, one can easily deduce that $\\overline{G}$ is connected, and so $\\overline{G}\\in\\mathcal{G}(4,2,-1)$ with $-1$ as a simple eigenvalue. Then, by Theorems \\ref{thm-main-3} and \\ref{thm-main-4}, $\\overline{G}$ does not exist and so is $G$.\n\nConsequently, if $G\\in\\mathcal{G}(4,2,0)$ then $G=\\overline{K_{s,s}^{-}\\circledast J_{t}}$ with $s\\geq 3$ and $t\\geq 1$. Obviously, $\\overline{K_{s,s}^{-}\\circledast J_{t}}\\in \\mathcal{G}(4,2,0)$ because $\\mathrm{Spec}(\\overline{K_{s,s}^{-}\\circledast J_{t}})=\\{[st]^1,[st-2t]^1,[-2t]^{s-1},[0]^{2st-s-1}\\}$. Our result follows.\n\\end{proof}\n\n\\section*{Acknowledgements}\nThe authors are grateful to Professor Richard A. Brualdi  and the referees for their valuable comments and  helpful suggestions, which have considerably improved the presentation of this paper.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe X-ray spectra of Seyfert Galaxies and Galactic Black Hole\nCandidates (GBHCs) indicate that the reflection and reprocessing of\nincident X-rays into lower frequency radiation is an ubiquitous and\nimportant process. For Seyfert Galaxies, the X-ray spectral index\nhovers near a ``canonical value'' ($\\sim 0.95$; Pounds et al. 1990,\nNandra \\& Pounds 1994; Zdziarski et al. 1996, but also see Brandt \\&\nBoller 1998), after the reflection component has been subtracted out\nof the observed spectrum. It is generally believed that the\nuniversality of this X-ray spectral index may be attributed to the\nfact that the reprocessing of X-rays within the cold accretion disk\nleads to an electron cooling rate that is roughly proportional to the\nheating rate inside the active regions (AR) (Haardt \\& Maraschi 1991,\n1993; Haardt, Maraschi \\& Ghisellini 1994; Svensson 1996).\n\n\nAlthough the X-ray spectra of GBHCs are similar to that of Seyfert\ngalaxies, they are considerably harder (most have a power-law index of\n$\\Gamma \\sim 0.7$), and the reprocessing features are less prominent\n(Zdziarski et al. 1996, Dove et al. 1997). It is the relatively hard\npower law (and therefore the required large coronal temperature) and\nthe weak reprocessing\/reflection features that led Dove et al.  (1997,\n1998), Gierlinski et al. (1997) and Poutanen, Krolik \\& Ryde (1997) to\nconclude that the two-phase accretion disk corona (ADC) model, in both\npatchy and slab corona geometry cases, does not apply to Cygnus X-1.\n\nOne of the main problems with the global slab-geometry ADC model is\nthat, for a given coronal optical depth, no self-consistent coronal\ntemperature is high enough to produce a spectrum both as hard and with\nan exponential cutoff at an energy as high as that of Cyg~X-1 (Dove,\nWilms, \\& Begelman 1997). However, this result is sensitive to the\nassumption that the accretion disk is relatively cold, such that $\\sim\n90$\\% of the reprocessed coronal radiation is re-emitted by the disk\nas thermal radiation (with a temperature $\\sim 150$ eV). It is this\nthermal radiation that dominates the Compton cooling rate within the\ncorona.  If the upper layers of the accretion disk were highly\nionized, creating a ``transition layer,'' a smaller fraction of the\nincident coronal radiation would be reprocessed into thermal radiation\n(i.e., the albedo of the disk would be increased), and therefore the\nCompton cooling rate in the corona would be reduced.  Furthermore, as\nshown by Ross, Fabian \\& Brandt (1996; RFB96 hereafter), the high\nionization state of the outer atmosphere of the accretion disk in\nGBHCs may explain the weakness of observed iron line features in Cyg\nX-1.\n\n\nRecently, Nayakshin \\& Melia (1997) investigated (via simple qualitative\nconsiderations) the structure of the X-ray reflecting material in AGNs\nassuming that the ARs are magnetic flares above the disk (Haardt et\nal. 1994; see also Galeev, Rosner \\& Vaiana 1979).  They showed that the\npressure and energy equilibrium conditions for the X-ray illuminated upper\nlayer of the disk require the gas temperature to be $T\\sim$ few $\\times\n10^5$ K, leading the upper atmosphere of the disk to a low ionization\nstate.\n\nIn this paper, we consider the reprocessing of X-rays from ARs for\nGBHCs. In \\S2, we show that there should be a thermal instability at\nthe surface of the cold disk, causing the temperature to climb up to\n$T\\sim$ a few $\\times 10^7$ K. This temperature is roughly the Compton\ntemperature with respect to the coronal radiation field.  At this\ntemperature the transition layer turns out to be almost completely\nionized. Recently, B\\\"ottcher, Liang, and Smith (1998), using an\niterative method, a linear Comptonization algorithm, and the\nphotoionization model XSTAR, also found that a highly ionized\ntransition layer should form for moderate ionization parameters,\ni.e., ionization parameters thought to be appropriate for GBHCs.\nIn \\S3, we explore the ramifications of this highly ionized transition\nlayer on the energetics of the corona, and investigate how it alters\nthe spectrum of escaping radiation. We also discuss whether slab\ngeometry ADC models, when transition layers are included, can account\nfor the observed spectra of BHCs. In \\S4, we give our conclusions.\n\n\n\n\\section{The Formation of a Transition Layer}\n\n\\subsection{Physical Conditions in X-ray Skin Near Magnetic Flares}\n\\label{sect:pressure}\n\n\\begin{figure*}[t]\n\\centerline{\\psfig{file=f1.eps,width=.72\\textwidth}}\n\\caption{The geometry of the active region (AR: magnetic flare) and\nthe transition layer. Magnetic fields, containing AR and supplying it\nwith energy are not shown. Transition region is defined as the  upper layer\nof the disk with Thomson depth of $\\sim$ few, where the incident X-ray flux\nis substantially larger than the intrinsic disk flux}\n\\label{fig:geometry}\n\\end{figure*}\n\nThe two-phase model was put forward by Haardt \\& Maraschi (1991, 1993)\nto explain the spectra of Seyfert Galaxies. Haardt et al. (1994)\npointed out that observations are inconsistent with a uniform corona\nand introduced a patchy corona, where each ``patch'' is a magnetic\nflare (also referred as an active region). The key assumptions of the\nmodel are (1) during the flare, the X-ray flux from the active region\ngreatly exceeds the disk intrinsic flux, and (2) the compactness\nparameter $l$ of the active region is large, so that the dominant\nradiation mechanism is Comptonization of the disk thermal\nradiation. We wish to apply the same model to the X-ray spectra of\nGBHCs, and we employ the same assumptions.\n\nAs discussed below, we argue for the existence of a transition layer\nin the vicinity of an active coronal region (see Figure 1). Since the\nflux of ionizing radiation is proportional to $1\/d^2 \\times \\cos i\n\\propto d^{-3}$, where $i$ is the angle between the normal of the disk\nand the direction of the incident radiation field and $d$ is the\ndistance between the active region and the position on the disk, the\nionization state of the disk surface will vary across the disk.\nConsequently, only the gas near the active regions (with a radial size\n$\\sim$ a few times the size of the active region, situated directly\nbelow it) may be highly ionized. Most reprocessing of coronal\nradiation will take place in the transition regions; in addition, most\nradiation emitted by the disk that propagates through the active\nregions will have been emitted in the vicinity of the transition\nregions. Therefore, in this paper, we will only consider a one-zone\nmodel, consisting of the active region, the transition layer, and the\nunderlying cold disk.\n\nThe structure of the transition layer, i.e., its temperature, density and\nionization state are determined by solving the energy, ionization and\npressure balance conditions. The first two conditions have been extensively\ntreated in the literature, and we will follow these standard methods here\n(see \\S \\ref{sect:instability}). The pressure equilibrium condition is\ntypically replaced by the constant gas density assumption (e.g., RFB, Zycki\net al. 1994 and references therein). For the problem at hand, however, the\nequilibrium state of the transition layer is sensitive to the pressure\nbalance, and thus we will attempt to take it into account. Accounting for\nthe pressure balance leads to results that are substantially different from\nprevious work, as discussed below.\n\n\nThe X-ray radiation pressure on the transition layer is equal to $\\fx\/c$,\nwhere $\\fx$ is the flux produced by the magnetic flare. The compactness\nparameter of the active region, $l$, is defined as\n\\begin{equation}\nl\\equiv {\\fx\\sigma_T \\Delta R\\over m_e c^3},\n\\label{compact}\n\\end{equation}\nand is expected to be larger than unity (e.g., Poutanen \\& Svensson\n1996, Poutanen, Svensson \\& Stern 1997). The size of the active region\n$\\Delta R$ is thought to be of the order of the accretion disk height\nscale $H$ (e.g., Galeev et al.  1979). Here, $H\/R$ is estimated from\nthe gas pressure dominated solution of Svensson \\& Zdziarski (1994;\nSZ94 hereafter),\n\\begin{equation}\n\\frac{H}{R} = 7.5\\times 10^{-3} (\\alpha M_1)^{-1\/10} r^{1\/20}\n [\\dot{m}J(r)]^{1\/5}[(1-f)]^{1\/10},\n\\end{equation}\nwhere $\\alpha$ is the viscosity parameter, $M_1\\equiv M\/10{\\,M_\\odot}$ is\nthe mass of the black hole, $f$ is the fraction of accretion power\ndissipated into the corona (averaged over the whole disk), $r$ is the\nradius relative to the Schwarzschild radius ($r\\equiv R\/R_g$), $J(r) =\n1-(3\/r)^{1\/2}$.  For the case of the hard state of Cyg X-1, most of\nthe bolometric luminosity is in the hard X-ray band (e.g., Gierlinski\net al. 1997). Thus, most of the accretion energy must be dissipated\ndirectly in the corona, i.e., $f\\sim 1$ (Haardt and Maraschi 1991;\nStern et al. 1995). The dimensionless accretion rate $\\dm =\n\\eta\\dot{M}c^2\/L_{\\rm Edd}$ for Cyg~X-1 seems to be around $0.05$. Here,\n$\\dot{M}$ is the accretion rate, $\\eta =0.056$ is the efficiency for\nthe standard Shakura-Sunyaev disk, and $L_{\\rm Edd}$ is the Eddington\nluminosity. Note that this definition of $\\dm$ is different by factor\n$\\eta$ from that used by SZ94 (i.e., $\\dm \\simeq 17 \\times \\dm_{\\rm\nSZ94}$). Finally, for $r = 6$, the X-ray flux is\n\\begin{equation}\n\\fx \\,\\simeq \\, 4 \\times 10^{23} \\, l\\,\\alpha^{1\/10} M_1^{-9\/10}\n\\left({\\dm\\over 0.05}\\right)^{-1\/5} (1-f)^{-1\/10} {\\rm erg \\\ncm}^{-2}\\,{\\rm sec}^{-1}.\n\\label{xflux}\n\\end{equation}\n\nTo check whether assumption (1) of the patchy two-phase model is consistent\nfor Cyg~X-1 parameters, we estimate the intrinsic flux of the disk,\n\\begin{equation}\n\\fdisk = 1.0 \\times 10^{22} M_1^{-1} \\left({\\dm\\over 0.05}\\right)\n(1-f)\\;{\\rm erg \\ cm}^{-2}\\,{\\rm sec}^{-1}.\n\\label{diskflux}\n\\end{equation}\nIt is seen that the X-ray flux is indeed much larger than the\nintrinsic disk emission if $1-f\\ll 1$ and the compactness parameter\n$l\\gg 0.01$. \n\nThe pressure of the disk surface layer before the occurrence of a\nflare (or, equivalently, far enough from the flare), assuming that the\nupper layer of the disk with Thomson optical depth $\\tau_x\\sim $ few\nis in the hydrostatic equilibrium, is\n\\begin{eqnarray}\n  P_0 &\\simeq& {G M m_p\\over R^2} \\tau_x\\, {H\\over R} = 6.2\\times 10^{10}\\;\n  M_1^{-11\/10} \\alpha^{-1\/10}\\nonumber \\\\ & &\\times \\tau_x\\,\\left({\\dm\\over\n      0.05}\\right)^{1\/5} (1-f)^{1\/10} \\;{\\rm erg \\ cm}^{-3},\n\\label{p0}\n\\end{eqnarray}\nwhere $r=6$ (SZ94). Near an active magnetic flare, the\nratio of the incident radiation pressure to the unperturbed accretion\ndisk atmosphere pressure is\n\\begin{equation}\n{\\fx\\over c P_0}\\, = \\, 2. \\times 10^2 \\; l \\,\\tau_x^{-1}\\,\\left(\\alpha\nM_1\\right)^{1\/5} \\,\\left({\\dm\\over 0.05}\\right)^{-2\/5}\n(1-f)^{-1\/5},\n\\label{prat}\n\\end{equation}\nSince the radiation pressure from the active region greatly exceeds the\nunperturbed thermal pressure, the transition layer will contract until\nthe gas pressure $P$ \n\\begin{equation}\nP \\sim \\fx\/c \\; .\n\\label{pestm}\n\\end{equation}\nA better treatment would solve for the gas opacity in the transition layer\nand thus the radiation force self-consistently, rather than simply using the\nram pressure $\\fx\/c$ (Nayakshin 1998). However, inequality (\\ref{pestm})\nturns out to be sufficient to prove the main point of this paper. We thus\nmove on to solve the energy and ionization balance equations for the\ntransition layer.\n\n\n\\subsection{The Thermal Instability}\\label{sect:instability}\n\nA general condition for a thermal instability was discovered by Field\n(1965). He argued that a physical system is usually in pressure\nequilibrium with its surroundings. Thus, any perturbations of the\ntemperature $T$ and the density $n$ of the system should occur at a\nconstant pressure.  The system is unstable when\n\\begin{equation}\n\\left({\\partial \\Lambda_{\\rm net}\\over \\partial T}\\right)_{P} < 0,\n\\label{field}\n\\end{equation}\nwhere the ``cooling function,'' $\\lnet$, is the\ndifference between cooling and heating rates per unit volume, divided by\nthe gas density $n$ squared. \n\n\n\n\n\nIn ionization balance studies, it turns out convenient to define two\nparameters. The first one is the ``density ionization parameter'' $\\xi$, \nequal to (Krolik, McKee \\& Tarter 1981)\n\\begin{equation}\n\\xi = {4\\pi \\fx\\over n},\n\\label{xid}\n\\end{equation}\nThe second one is the  ``pressure ionization parameter'', defined as\n\\begin{equation}\n\\Xi \\equiv \\frac{\\prad}{P},\n\\label{xip}\n\\end{equation}\nwhere $P$ is the full isotropic pressure due to neutral atoms, ions,\nelectrons and the trapped line radiation. (This definition of $\\Xi$ is\nthe one used in the ionization code XSTAR, and is more appropriate for\nstudies of the instabilities than the original definition of $\\Xi$\ngiven in Krolik et al. 1981, who used $n k T$ instead of the full\npressure $P$. For temperatures typical of GBHCs, however, we found\nthat the trapped line radiation was always a small fraction, e.g.,\n$\\sim$ few percent of the total pressure, and therefore the two\ndefinitions of $\\Xi$ are nearly identical).  Krolik et al. (1981)\nshowed that the instability criterion (\\ref{field}) is equivalent to\n\\begin{equation}\n\\left( {d\\Xi\\over d T}\\right)_{\\lnet=0}\\, < 0  \\; ,\n\\label{fcond}\n\\end{equation}\nwhere the derivative is taken with the condition $\\lnet = 0$\nsatisfied, i.e., when the energy balance is imposed. In this form, the\ninstability can be easily seen when one plots temperature $T$ versus\n$\\Xi$.\n\n\\begin{figure*}\n\\centerline{\\psfig{file=f2.eps,width=.5\\textwidth,angle=0}}\n\\caption{Gas temperature versus pressure ionization parameter $\\Xi$ --\n  the ionization equilibrium curves for parameters appropriate for\n  GBHCs.  The incident spectrum is approximated by a power law of\n  photon index $\\Gamma$, exponentially cutoff at $100$ keV, and the\n  reflected blackbody with equal flux and temperature $\\Tmin$\n  (equation \\ref{tmincyg}). Values of the parameters are: $\\Gamma =$\n  1.5, 1.75, 1.75, 1.7 and $k \\Tmin = $ 200, 100, 200, 400 eV,\n  corresponding to the fine solid, thick solid, dotted and dash-dotted\n  curves, respectively. Because the ionization equilibrium is unstable\n  when the curve has a negative slope, and because there exists no\n  solution below $\\Tmin$ (see text), the only stable solution is the\n  uppermost branch of the curve with $T\\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$} 10^7$ K.}\n\\label{fig:scurve}\n\\end{figure*}\n\nWe now apply the X-ray ionization code XSTAR (Kallman \\& McCray 1982,\nKallman \\& Krolik 1986), to the problem of the transition layer. A\ntruly self-consistent treatment would involve solving radiation\ntransfer in the optically thick transition layer, and, in addition,\nfinding the distribution of the gas density in the transition layer\nthat would satisfy pressure balance. Since radiation force acting on\nthe gas depends on the opacity of the gas, this is a difficult\nnon-linear problem. Thus, we defer such a detailed study to future\nwork, and simply solve (using XSTAR) the local energy and ionization\nbalance for {\\em an optically thin layer} of gas in the transition\nregion. We assume that the ionizing spectrum consists of the incident\nX-ray power law with the energy spectral index typical of GBHCs in the\nhard state, i.e., $\\Gamma = 1.5-1.75$, exponentially cutoff at 100\nkeV, and the blackbody spectrum from the cold disk below the\ntransition layer. If the energy and ionization balance is found to be\nunstable for this setup, the transition layer will also be unstable,\nbecause the instability is local in character.\n\nNote that it is not possible for the transition region to have a\ntemperature lower than the effective temperature of the X-radiation, i.e.,\n$\\Tmin = (\\fx\/\\sigma)^{1\/4}$. The reason why the simulations may give\ntemperatures lower than $\\Tmin$ for low values of $\\xi$ is that in this\nparameter range XSTAR neglects certain non-radiative de-excitation\nprocesses, which leads to an overestimate of the cooling rate (Zycki et\nal. 1994; see their section 2.3). Since we are using a one-zone model for\nthe transition layer, we use an attenuated X-ray flux $\\mean{\\fx}$ which\nrepresents the surface-averaged value as seen by the transition region,\n$\\mean{\\fx} = \\fx\/b = 0.1 \\fx\/b_1$, where $b$ is a dimensionless number of\norder 10, $b_1 = b\/10$, and $\\fx$ is the X-ray flux leaving the active\nregion (see figure \\ref{fig:geometry}). Using equation (\\ref{xflux}),\n\\begin{equation}\n\\Tmin\\simeq 5.0 \\times 10^6\\; l^{1\/4} \\, b_1^{-1\/4}\\, \\left({\\dm\\over\n0.05}\\right)^{-1\/20}\\, \\alpha^{1\/40} \\, M_1^{-9\/40} \\,\n\\left[1-f\\right]^{-1\/40}\n\\label{tmincyg}\n\\end{equation}\n\n\n\nFigure \\ref{fig:scurve} shows results of our calculations for several\ndifferent X-ray ionizing spectra. A stable solution for the transition\nlayer structure will have a positive slope, and also satisfy the\npressure equilibrium condition. As discussed in \\S\n\\ref{sect:pressure}, $P \\leq \\fx\/c$ (i.e., $\\Xi\\geq 1$). In addition,\nif the gas is completely ionized, the absorption opacity is negligible\ncompared to the Thomson opacity. Because all the incident X-ray flux\nis eventually reflected, the net flux is zero, and so the net\nradiation force is zero (note that the momentum of the incident\nradiation is reflected deep inside the disk, so the radiation does\napply a ram pressure equal to $2\\fx\/c$ to the whole disk, but {\\em not\nto the transition layer}). In that case the pressure $P$ adjusts to\nthe value appropriate for the accretion disk atmosphere in the absence\nof the ionizing flux (see also Sincell \\& Krolik 1996), which is given\nby equation (\\ref{p0}). Therefore, the pressure ionization parameter\nshould be in the range\n\\begin{equation}\n1< \\Xi < 1\\times 10^2 \\,l \\left(\\alpha\n    M_1\\right)^{1\/5} \\,\\left({\\dm\\over 0.05}\\right)^{2\/5} (1-f)^{-1\/5}.\n\\label{xip}\n\\end{equation}\nWith respect to the ionization equilibria shown in Figure\n(\\ref{fig:scurve}), the gas is almost completely ionized on the upper\nstable branch of the solution (i.e., the one with $T\\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$} 10^7$ K),\nand thus the pressure equilibrium for such temperatures requires $\\Xi\n\\sim \\fx\/c P_0\\gg 1$, which is allowed according to equation\n(\\ref{xip}).\n\n\n\nIn addition to the stable Compton equilibrium state, there is a small\nrange in parameter space, with the temperature between $100$ and $200$\neV, in which a stable thermal state exists . The presence of this\nregion is explained by a rapid decrease in {\\em heating} (with $T$\nincreasing), rather than an increase in cooling (cf. equation\n\\ref{field}). The X-ray heating rate decreases in the temperature\nrange $100-200$ eV with increasing $T$ since higher temperatures lead\nto higher ionization rates of ion species with ionization energies\n$\\sim kT$. This larger degree of ionization reduces the X-ray opacity,\nand thus the heating rate as well. Note that it is highly unlikely that\nthe transition region will stabilize within the temperature range\n$100$ -- $200$ eV because the effective temperature, $\\Tmin$, is most\nlikely above this temperature range.\n\n\n\\begin{figure*}\n\\centerline{\\psfig{file=f3.eps,width=.6\\textwidth,angle=0}}\n\\caption{Same as Figure \\ref{fig:scurve}. The thick solid curve is\nsame as that in Figure \\ref{fig:scurve} and is appropriate for the\nhard state of a GBHC, whereas two other curves are relevant to the\nsoft state in GBHCs, or at large depth in the transition layer (see\nDiscussion). Values of the parameters are: $\\Gamma =$ 2.1, 2.1 and $k\n\\Tmin = $ 200, 400 eV for the dotted and dashed curves, respectively.}\n\\label{fig:ssoft}\n\\end{figure*}\n\nDue to the above considerations, it is very likely that the transition\nlayer is highly ionized in GBHCs {\\em in the hard state} for $\\tau_{\\rm x}\n\\la 1$. The upper limit of $\\tau_{\\rm x}$ can only be found by a more\nexact treatment.  In addition, the transition layer may be heated by the\nsame process that heats the corona above it, albeit with a smaller heating\nrate.  Furthermore, Maciolek-Niedzwiecki, Krolik \\& Zdziarski (1997) have\nrecently shown that the thermal conduction of energy from the corona to the\ndisk below may become important for low coronal compactness parameters and\nsubstantially contribute to the heating rate of the transition layer. Thus,\nthe transition layer may be even hotter than found by photoionization\ncalculations.\n\n\nEventually, the X-rays are down-scattered and the radiation spectrum becomes\nsofter as one descends from the top of the transition layer to its bottom.\nWe can qualitatively test the gas ionization stability properties by\nallowing the ionizing spectrum to be softer than the observed spectrum of\nGBHCs in the hard state. In Figure \\ref{fig:ssoft} we show two examples\nof such calculations. The slope of the ionization equilibrium curve becomes\npositive everywhere above $k T\\sim$ 100 eV, so that these equilibria are\nstable, and thus the gas temperature may saturate at $T\\sim \\Tmin$, far\nbelow few keV, the appropriate temperature for the uppermost layer of the\ntransition region.  Thus, we know (see also \\S \\ref{sec:discussion}) that the\ntransition layer should terminate at some value of $\\tau_x \\sim$ few.\n\nWe point out the similarity of our results to those of Krolik, McKee\n\\& Tarter (1981) for the emission line regions in quasars. These\nauthors solved the ionization and energy balance equations for\noptically thin clouds illuminated by a broad band quasar spectrum, and\nshowed that the thermal instability exists if the pressure ionization\nparameter is close to unity.  They found that there are two stable\nstates for the line emitting clouds, one cold and one hot. The cold\nstate corresponds to the gas temperature $T\\lesssim $ few $\\times\n10^4$ K, where collisional cooling balances ionization heating. The\nhot state corresponds to temperatures $T\\gtrsim 10^7$ K, when\nionization heating decreases due to almost complete ionization of the\nelements and the Compton heating decreases since the gas is close to\nthe Compton temperature $\\tcomp$. Here, we found a similar instability\nfor the X-ray skin near an active magnetic flare in accretion disks of\nGBHCs. The lower temperature equilibrium state is not allowed,\nhowever, since the gas density and the X-ray ionizing flux in GBHCs is\nlarger than these quantities in quasar emission line regions by some\n$\\sim 10 -12$ orders of magnitude. Finally, we note that the thermal\ninstability is not apparent in studies where the gas density is fixed\nto a constant value. Following Field (1965), we argue that the\nassumption of the constant gas density is not justified for real\nphysical systems, and that one always should use the pressure\nequilibrium arguments to determine the actual gas density and the\nstability properties of the system.\n\n\n\\section{Global ADC Models with a Transition Layer} \n\\subsection{Physical Setup}\n\\begin{figure*}\n\\centerline{\\psfig{file=f4.eps,width=.6\\textwidth,angle=0}}\n\\caption{The maximum temperature of the corona as a function of the optical\n  depth of the transition layer. For all models, we assume that the\n  intrinsic compactness parameter of the disk is $l_{\\rm bb} = 0.01$ and the disk\n  temperature is $kT_{\\rm bb} = 150$ eV. For all models, the maximum temperature\n  is reached for $l_{\\rm c} \\sim 2-5$.}\n\\label{fig:Tmax-vs-tautrans}\n\\end{figure*}\n\nWe now explore how a transition layer affects the physical properties of\nthe coronal gas and the spectrum of escaping radiation.  Since a transition\nlayer may occur for both a global ADC model and for a patchy coronal model,\nboth models should be explored in detail. In this paper, however, we only\nstudy the global ADC model, and defer a self-consistent treatment of the\npatchy corona to a future paper.  We expect, however, that the systematic\ntrends found for the global ADC models should also occur for the patchy\nmodel, with the main difference being the maximum coronal temperature of\nthe patchy model being higher.\n\n\n\nA self consistent treatment of the transition layer would solve for the\nionization layer structure as a function of optical depth.  However, we\nwill make the simplifying assumption that the transition layer is\ncompletely ionized. The optical depth of the transition layer will be kept\nas a free parameter, and we will explore how sensitive the spectrum of\nescaping radiation is to $\\tau_{\\rm tr}$. The two issues we will address are (1)\nfor a given coronal optical depth (e.g., one thought to be appropriate for\nfitting the spectra of GBHCs), how does the maximum coronal temperature\ndepend on the optical depth of the transition layer, and (2) how does the\nspectrum of escaping radiation, most importantly the reprocessing features,\nvary with the optical depth of the transition layer.  Of particular\nimportance is the parameter range of $\\tau_{\\rm tr}$ such that the predicted\nspectrum is consistent with that of GBHCs, and whether these values of\n$\\tau_{\\rm tr}$ are consistent with the assumption of the transition layer being\ncompletely ionized.\n\\begin{figure*}\n\\centerline{\\psfig{file=f5.eps,width=.6\\textwidth}}\n\\caption{The predicted spectrum for various values of\nthe transition layer optical depth. From top to bottom, $\\tau_{\\rm tr} = 10, 5,\n2.5,$ and $1.0$.}\n\\label{fig:sequence-of-spectra}\n\\end{figure*}\n\nThe model contains three regions: (1) A cold accretion disk, assumed here\nto have a temperature $kT_{\\rm bb} = 150$ keV, (2) the transition layer,\nsituated directly above the cold disk, and (3) the corona, situated\ndirectly above the transition layer. Plane parallel geometry is assumed.\n\nWe use the slab-geometry ADC model of Dove, Wilms, \\& Begelman (1997),\nwhich uses a non-linear Monte Carlo (NLMC) routine to solve the radiation\ntransfer problem of the system.  The free parameters of the model are the\nseed optical depth $\\tau_{\\rm e}$, (the optical depth of the corona excluding the\ncontribution from electron-positron pairs), the blackbody temperature of\nthe accretion disk and its compactness parameter, $l_{\\rm bb}$, and the heating\nrate (i.e., the compactness parameter), $l_{\\rm c}$, of the ADC.  The temperature\nstructure of the corona is determined numerically by balancing Compton\ncooling with heating, where the heating rate is assumed to be uniformly\ndistributed.  The $e^-e^+$-pair opacity is given by balancing photon-photon\npair production with annihilation.  Reprocessing of coronal radiation in\nthe cold accretion disc is also treated numerically. For a more thorough\ndiscussion of the NLMC routine, see Dove, Wilms, \\& Begelman (1997). The\ntransition layer is treated identically to the corona, accept here the\nheating rate is set to zero. Therefore, the transition layer, numerically\nmodeled using 8 shells, each with equal optical depth ${\\rm d}\\tau =\n\\tau_{\\rm tr}\/8$, will obtain the Comptonization temperature due to the\nradiation field from both the corona and the accretion disk.\n  \n\\subsection{Maximum Coronal Temperature}\n\nAs discussed by Dove, Wilms, \\& Begelman (1997), for a given total\noptical depth, there exists a maximum coronal temperature, above which\nno self-consistent solution exists. Raising the compactness parameter\nof the corona to a value higher than that corresponding to the maximum\ntemperature gives rise to a higher optical depth (due to pair\nproduction), causing more reprocessing, subsequent Compton cooling,\nand ultimately a {\\em lower} coronal temperature. Here, as much as\n80-90\\% of the incident X-ray flux is re-radiated at the disk\ntemperature, i.e., the X-ray integrated albedo is 0.1-0.2.  A\ncompletely ionized transition layer will increase the albedo of the\ncold disk.  Accordingly, the amount of energy in the reprocessed\nblackbody (or Comptonized blackbody) should become smaller with\nincreasing Thomson optical depth of the transition layer, and this\nCompton cooling via reprocessed radiation should be less efficient and\ntherefore higher maximum coronal temperatures (as compared to models\nwith no transition layer) should be allowed.\n\n\\begin{figure*}\n\\centerline{\\psfig{file=f6.eps,width=.8\\textwidth}}\n\\caption{Internal spectrum for several shells within the transition disk\n  and corona. From bottom to top, $\\tau_{\\rm tr}(z) = 1.25, 2.5, 3.75,$ and $\n  5.0$, where $\\tau_{\\rm tr}(z)$ is the Thomson optical depth from the\n  cold-disk\/transition layer interface ($z=0$) to a height $z$. The\n  uppermost spectrum is the internal spectrum within the corona. For this\n  model, $\\tau_{\\rm tr} = 5.0$ and $\\tau_{\\rm c} = 0.3$.}\n\\label{fig:spec-vs-tau}\n\\end{figure*}\n\nIn Figure \\ref{fig:Tmax-vs-tautrans}, we show how the average coronal\ntemperature varies with the optical depth of the transition layer,\n$\\tau_{\\rm tr}$. In Figure \\ref{fig:sequence-of-spectra}, we also show the\nresulting broad band spectra for the model parameters tested. For all\nmodels, we chose $l_{\\rm bb} = 0.01$ in order to be consistent with the\ndefinition of the disk compactness parameter, i.e., $l_{\\rm bb}\\equiv\n\\fdisk\\sigma_T H\/ ( m_e c^3)$, according to which\n\\begin{equation}\nl_{\\rm bb} = 0.03 \\left({\\dm\\over 0.05}\\right)^{6\/5}\n\\left(1-f\\right)^{11\/10} \\left(\\alpha M_1\\right)^{-1\/10}\\; .\n\\label{lbb}\n\\end{equation}\nFurther, other parameter values are $l_{\\rm c} = 2$, $kT_{bb} = 150$ keV,\nand $\\tau_c = 0.3$. These parameters correspond to the model producing\nthe maximum coronal temperature. In contrast to models in which\n$\\tau_{\\rm tr} = 0$, the coronal temperature for a given value of\n$\\tau_c$ is not simply a function of $l_{\\rm c}\/l_{\\rm bb}$. To see this, consider\nthe case where $\\tau_{\\rm tr} \\gg 1$. Here, the albedo of the disk is\nessentially unity, and therefore all of the soft photons emitted will\nbe from the intrinsic flux of the disk (no reprocessing). Therefore,\nsetting $l_{\\rm bb} \\ll 1$ yields the maximum coronal temperatures possible.\nNote that the maximum temperature levels out as $\\tau_{\\rm tr} \\rightarrow\n\\infty$. Although, in this limit, there is no reprocessing of hard\nX-rays in the cold disk, there is still ``reprocessing'' in the\ntransition layer.  As $\\tau_{\\rm tr}$ increases, more coronal radiation is\ndown-scattered to the Compton temperature of the transition layer,\nwhich is $kT_{tr} \\sim 1 - 4$ keV. Even at these temperatures, Compton\ncooling of this ``reprocessed'' radiation in the corona is very\nefficient.\n\n\nIt is interesting to note that, only for $\\tau_{\\rm tr} \\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$} 10$,\nthe coronal temperature is high enough such that the corresponding\nspectrum of escaping radiation is hard enough to describe the\nobservations of Cyg~X-1.  (The canonical value of the photon power law\nof Cyg~X-1 is $\\Gamma = 1.7$; for $\\tau_c = 0.3$, this power law\ncorresponds to $kT_c \\sim 150$ keV). It is probably unphysical,\nhowever, to assume the transition layer is completely ionized for such\nlarge optical depths. In fact, the numerical model for $\\tau_{\\rm tr}\n= 10$ predicts a temperature of $kT_{\\rm tr} \\sim 500$ eV near the\nbottom of the layer. Furthermore, as is seen in Figure\n\\ref{fig:spec-vs-tau}, the radiation spectra do become substantially\nsofter deeper in the transition layer, and, using our simple\nexperimentation with softer ionizing spectra, shown in Figure\n\\ref{fig:ssoft}, we can expect that the gas will become stable and may\nbe at a temperature lower than the Compton temperature for $\\tau_{\\rm\ntr}$ as small as $2-3$. Therefore, even with the advent of transition\nlayers, it still appears unlikely that a global slab geometry ADC\nmodel can have self-consistent temperatures high enough to reproduce\nthe observed hard spectra of Cyg X-1 and other similar BHCs. This does\nnot rule out ADC models in which the corona is patchy, e.g.,\ncontaining several localized active regions such as magnetic flares.\nThe patchy geometry leads to higher coronal temperatures due to less\nreprocessed soft flux re-entering active regions (Poutanen \\& Svensson\n1996), so that lower $\\tau_{\\rm tr}$ may be sufficient to explain the\nCyg X-1 spectrum.\n\n\\subsection{Reprocessing Features in the Spectrum of Escaping Radiation}\n\nAs shown in Figure \\ref{fig:sequence-of-spectra}, the reprocessing\nfeatures in the spectrum of escaping radiation depend sensitively on\nthe optical depth of the transition layer. In Figure\n\\ref{fig:spec-vs-tau}, we show how the internal spectrum (averaged\nover all angles) varies throughout the transition layer and corona.\nNote that most of the hard X-rays do not penetrate through the\ntransition layer, and that the spectrum gets softer as it approaches\nthe cold disk. On the other hand, the thermal radiation emitted by the\ndisk and the 10-50 keV Compton reflection hump are broadened by\nCompton scatterings as the disk radiation diffuses upward through the\ntransition layer and the corona. The Iron K$\\alpha$ line, small to\nstart with due to the small amount of reprocessing of coronal\nradiation, is completely smeared out by the time the radiation escapes\nthe system.  No line photons are created in the transition layer\nitself, because we found that the Compton equilibrium state typically\nresides at the ionization parameter $\\xi \\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$} 10^4$, whereas no\nfluorescent iron line emission is produced for $\\xi \\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$} 5\\times\n10^3$ (Matt, Fabian \\& Ross 1993, 1996).\n\nTherefore, in agreement with B\\\"ottcher, Liang, and Smith (1998), we\nconclude that a highly ionized transition layer can reduce the\nreprocessing features in the spectrum of escaping radiation emanating\nfrom slab geometry ADC models. Previous work has used the lack of\nobserved reprocessing features to argue against the applicability of\nslab geometry ADC models, and argued for a disk\/corona configuration\nwhere the disk has a small solid angle relative to the coronal region,\nsuch as a ``sphere$+$disk model (Dove et al. 1998, Gierlinski et\nal. 1997, Poutanen, Krolik \\& Ryde 1997). A slab geometry ADC model\ncontaining a transition layer, however, is another possibility.\nAlthough, as discussed above, these models could not obtain high\nenough temperatures to be able to predict spectra as hard as the\nobserved spectra of GBHCs unless $\\tau_{\\rm tr} \\ga 10$, a model with a\npatchy corona and underlying transition layers appears likely to be\nable to predict hard enough spectra with a relatively small amount of\nreprocessing features. In a forthcoming paper, we plan to investigate\nsuch a model.\n\n\n\\section{Discussion}\\label{sec:discussion}\n\nWe have shown that the transition layer in the vicinity of transient\nflares for ADC models of GBHCs must be highly ionized and very\nhot. Specifically, due to a thermal instability, the only stable\ntemperature of the transition layer is the local Compton temperature\n($k T\\sim$ few keV).  In fact, even for global ADC models of GBHCs,\nsuch a transition layer is found to be likely. However, we have not\naccurately modeled the full radiative transfer problem through the\ntransition layer at this time, and therefore could not determine its\nvertical optical depth, which was treated as a free parameter.\n\nDue to the transition layer, a larger fraction of incident X-rays are\nCompton reflected back into the corona without being reprocessed by the\ncold disk. In addition, for $\\tau_{\\rm trans}\\gg 1$, the predicted\nreprocessing features as well as the thermal excess should be substantially\nsmaller (as is found in this paper for the global ADC model) than that of\nprevious ADC models in which the transition layer was not considered.  This\nreduction of the reprocessing features is crucial for the model being\nconsistent with the observations of GBHCs (e.g., Gierlinski et al.  1997,\nDove et al. 1998).\n\nThis reduction in reprocessing yields a lower Compton cooling rate within\nthe corona, and higher coronal temperatures than previous ADC models are\nallowed.  For global ADC models with $\\tau_{\\rm c} \\sim 0.3$, we find that the\ncoronal temperature can be as high as $\\sim 150$ keV {\\em if} the optical\ndepth of the transition layer is $\\tau_{\\rm tr} \\ga 10$.  However, a completely\nionized transition layer with $\\tau_{\\rm tr} \\ga 10$ is most likely physically\ninconsistent since the bottom layer was found to be too cold to\nrealistically stay highly ionized. Therefore, global slab-geometry ADC\nmodels are still problematic in explaining the observations of GBHCS.\nNevertheless, these global ADC results are very encouraging, and a slab\ngeometry ADC model containing a patchy corona (e.g., individual ARs) with\nunderlying transition layers should be rigorously studied. Here, due to the\nlower amount of reprocessed radiation within the ARs as compared to the\nglobal model, models with coronae hot enough to reproduce the observed\nspectra of GBHCs may be allowed for more reasonable transition layer\noptical depths.\n\n\n\nWe have considered only radiative cooling mechanisms for the\ntransition layer in this paper.  It is possible that a wind is induced\nby the X-ray heating.  However, the maximum gas temperature obtained\ndue to the X-ray heating is the Compton temperature ($\\lesssim 10^8$\nK). Therefore, as shown by Begelman, McKee \\& Shields (1983), a large\nscale outflow cannot occur for $R\\lesssim 10^4 R_g$. On the other\nhand, a local uprising of the gas is still possible. The maximum\nenergy flux due to this process is $F_{\\rm ev} \\sim P c_s$, where\n$c_s$ is the sound speed in the transition region. Since $P\\lesssim\n\\fx\/c$, and $c_s \\lesssim 3\\times 10^{-3}\\, c$, we have ${F_{\\rm\nev}\/\\fx} \\lesssim 3 \\times 10^{-3}$. Therefore, a wind {\\em or any\nother mechanical process} cannot cool the gas efficiently, and thus it\nis justifiable to consider cooling via emission of radiation only.\n\n\n\nIn a forthcoming paper, we will discuss the implications of the\ninstability for the case of AGN. As shown in Figures \\ref{fig:scurve}\nand \\ref{fig:ssoft}, there is a stable ionization equilibrium state\nbelow $T\\sim 3\\times 10^5$ Kelvin. Furthermore, due to increasing\nopacity to the soft disk radiation, we find that equilibria below\n$T\\sim 10^5$ K are not possible.  We found that the existence of this\nstable region, and the unstable region above $3\\times 10^5$ K, is very\nmuch independent of the details of the incident spectra (indeed,\nFigures \\ref{fig:scurve} and \\ref{fig:ssoft} were not even intended\nfor the AGN case). Thus, it is possible that the differences in the\nintrinsic X-ray spectra, the ionization state of the reflector and the\nstrength of the iron line in AGNs and GBHCS can be explained by the\ndifferent end points of the thermal ionization instability described\nhere.\n\n\n\n\\section{Acknowledgments}\n\nSN acknowledges the contributions, guidance and financial support \nthrough NASA grant NAG 5-3075 by Prof. Fulvio Melia. JD acknowledges the\nuseful discussions with P. Maloney, J. Wilms, and M. Nowak.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{G{\\footnotesize \u0130}r{\\footnotesize \u0130}\u015f}\n\n\u00c7izge renklendirme probleminde bir \u00e7izgenin noktalar\u0131, her bir nokta kom\u015fular\u0131ndan farkl\u0131 renkte olacak \u015fekilde ve en az renk kullan\u0131larak renklendirilmelidir. Kom\u015fuluk ili\u015fkisi tek bir kenar \u00fczerinden tan\u0131mland\u0131\u011f\u0131ndan problemin bu hali {\\em 1-uzakl\u0131k \u00e7izge renklendirme} olarak da adland\u0131r\u0131l\u0131r. Problemin bu basit halinin bile NP-Zor oldu\u011fu kan\u0131tlanm\u0131\u015ft\u0131r~\\cite{matula_SL,Zuckerman}. Literat\u00fcrde daha genel renklendirme problemleri de incelenmi\u015ftir. \u00d6rne\u011fin kom\u015fuluk ili\u015fkisi $\\ell$ kenar ile tan\u0131mland\u0131\u011f\u0131nda, problem {\\em $\\ell$-uzakl\u0131k \u00e7izge renklendirme} problemi olarak adland\u0131r\u0131l\u0131r. Bu problemde, iki nokta aras\u0131nda $\\ell$ ya da daha az kenar kullanan bir yol varsa, bu iki nokta farkl\u0131 renklerde olmal\u0131d\u0131r. \n\n\u00c7izgeler ger\u00e7ek hayatta bir\u00e7ok farkl\u0131 veriyi ve problemi modellemek i\u00e7in kullan\u0131lmaktad\u0131r. Bu nedenle, \u00e7izge renklendirme problemlerinin pratikte bir\u00e7ok uygulamas\u0131 bulunmaktad\u0131r. \u00d6rne\u011fin, kablosuz bir a\u011fdaki cihazlar \u00e7izge \u00fczerindeki noktalar, cihazlar aras\u0131ndaki potansiyel parazitler \u00e7izgede birer kenar olarak modellenebilir. Bu a\u011f \u00fczerindeki kanal atama probleminde, cihazlar kanallara birbirleri ile parazit olu\u015fturan (ya da \u00e7izge \u00fczerinde kenar payla\u015fan) iki cihaz farkl\u0131 kanallara atanacak \u015fekilde yerle\u015ftirilmelidir. Bu atama yap\u0131l\u0131rken b\u00fct\u00fcn cihazlar\u0131n (noktalar\u0131n) kapsanmas\u0131 ve kanal (renk) say\u0131s\u0131n\u0131n en az olmas\u0131 istenmektedir. Bu da bir \u00e7izge renklendirme problemidir~\\cite{Balasundaram06graphdomination}. \u00c7izge renklendirme ileti\u015fim a\u011flar\u0131 \u00fczerinde parazit azaltma, i\u00e7erik iletimi, \u00f6nbellek yap\u0131s\u0131n\u0131n ayarlanmas\u0131 gibi problemler i\u00e7in de kullan\u0131lm\u0131\u015ft\u0131r~\\cite{hassan11, 8080217,DBLP:journals\/corr\/abs-1801-00106}. Bunun yan\u0131nda n\u00fckleik asit dizisi tasar\u0131m\u0131~\\cite{7498232}, hava trafik ak\u0131\u015f y\u00f6netimi~\\cite{Barnier2004}, sosyal a\u011flarda topluluk tespiti~\\cite{6921552} ve paralel hesaplama~\\cite{10.1145\/2513109.2513110} alanlar\u0131nda problemin farkl\u0131 t\u00fcrleri kar\u015f\u0131m\u0131za \u00e7\u0131kmaktad\u0131r. \n\nRenklendirme problemi NP-Zor oldu\u011fundan literat\u00fcrde bu problem i\u00e7in a\u00e7g\u00f6zl\u00fc algoritmalar \u00f6nerilmi\u015ftir. \nBu algoritmalar \u00e7izge \u00fczerindeki noktalar\u0131 belli bir s\u0131rada gezer ve her noktay\u0131 o an i\u00e7in kom\u015fulu\u011funda kullan\u0131lmayan bir renge boyar. Bu \u00e7al\u0131\u015fmada kullan\u0131lan algoritma, noktay\u0131 kullan\u0131lmayan ilk renge boyamaktad\u0131r. \nA\u00e7g\u00f6zl\u00fc algoritmalar\u0131n kulland\u0131\u011f\u0131 nokta s\u0131ras\u0131 olduk\u00e7a \u00f6nemlidir. \n\u00d6rne\u011fin, verilen bir \u00e7izge i\u00e7in elimizde en az say\u0131da rengi kullanan bir renklendirmenin oldu\u011funu varsayal\u0131m. \nNoktalar\u0131 renklerine g\u00f6re s\u0131ralay\u0131p (ve renkleri silip) a\u00e7g\u00f6zl\u00fc algoritmaya verdi\u011fimizde algoritma bize en iyi sonucu verecektir. Fakat k\u00f6t\u00fc bir s\u0131ralama, renk say\u0131s\u0131n\u0131 artt\u0131racakt\u0131r. \nBu makalede sosyal a\u011f analizinde kullan\u0131lan metriklerin, a\u00e7g\u00f6zl\u00fc renklendirme algoritmalar\u0131n\u0131n nokta s\u0131ralamas\u0131n\u0131n bulunmas\u0131 \u00fczerine \u00e7al\u0131\u015f\u0131lm\u0131\u015ft\u0131r.\nYap\u0131lan deneyler g\u00f6stermi\u015ftir ki, bir sosyal a\u011f metri\u011fi olan {\\em yak\u0131nl\u0131k merkeziyeti} ile noktalar s\u0131raland\u0131\u011f\u0131nda, a\u00e7g\u00f6zl\u00fc algoritman\u0131n kulland\u0131\u011f\u0131 renk say\u0131s\u0131 azalmaktad\u0131r. \n\n\\section{Y\u00f6ntem}\\label{sec:yontem}\n$G = (V, E)$ y\u00f6ns\u00fcz bir \u00e7izge olsun ve $V$ k\u00fcmesindeki b\u00fct\u00fcn $v$ noktalar\u0131 i\u00e7in ${\\tt nbor}$($v$) $\\subset V$ fonksiyonu $v$'nin kom\u015fuluk k\u00fcmesini belirtiyor olsun. Algoritma anlat\u0131l\u0131rken renkler, do\u011fal say\u0131lar ile g\u00f6sterilecektir ve say\u0131 {\\bf -1} oldu\u011funda nokta hen\u00fcz renklendirilmemi\u015f olarak d\u00fc\u015f\u00fcn\u00fclecektir. A\u00e7g\u00f6zl\u00fc algoritman\u0131n s\u00f6zde kodu Algoritma~\\ref{alg:color}'de g\u00f6sterilmi\u015ftir. Algoritma ${\\tt nbor}$ fonksiyonunun tan\u0131m\u0131na g\u00f6re farkl\u0131 t\u00fcrdeki renklendirme problemleri i\u00e7in kullan\u0131labilir. Kom\u015fuluk fonksiyonu $${\\tt nbor}(v) = \\{u : \\{v,u\\} \\in E\\}$$ olarak tan\u0131mland\u0131\u011f\u0131nda algoritma 1-uzakl\u0131k renklendirme problemi i\u00e7in bir renklendirme \u00fcretecektir. \n\nKom\u015fuluk fonksiyonundan ba\u011f\u0131ms\u0131z olarak, a\u00e7g\u00f6zl\u00fc algoritma ilk noktalarda renk se\u00e7imi a\u00e7\u0131s\u0131ndan daha rahat olacak, fakat noktalar boyand\u0131k\u00e7a renk se\u00e7iminde zorlanmaya ba\u015flayacakt\u0131r. Dolay\u0131s\u0131yla noktalar\u0131n s\u0131ralamas\u0131 olduk\u00e7a \u00f6nemlidir. Rastgele bir s\u0131ralama her zaman iyi sonu\u00e7 vermeyebilir. Literat\u00fcrde bu algoritman\u0131n s\u0131ralama \u00f6zelinde analizi yap\u0131lm\u0131\u015f, rastgele s\u0131ralamalar\u0131n y\u00fcksek olas\u0131l\u0131kla minimum renk say\u0131s\u0131ndan \u00e7ok daha fazla renk kulland\u0131\u011f\u0131 \u00e7izge \u00e7e\u015fitlerinin varl\u0131\u011f\u0131 g\u00f6sterilmi\u015ftir~\\cite{KUCERA1991674,DBLP:journals\/corr\/Husfeldt15}.  \n \n\\begin{algorithm}\n\\small\n\\caption{\\textsc{\u00c7izgeRenklendirme}}\n\\algorithmicrequire{ $G = (V, E)$, $V$: renklendirilecek noktalar, $E$: kenarlar\\\\\\hspace*{7ex}${\\tt nbor}$($.$): kom\u015fuluk fonksiyonu}\\\\\n\\algorithmicensure{ $c[.]$:  renklendirme listesi.}\n\\begin{algorithmic}[1]\n\\For{her $v \\in V$ noktas\u0131 i\u00e7in} \n\\State{$F \\leftarrow \\emptyset$} \n\\For{her $u \\in$ ${\\tt nbor}$($v$)}\n\\If{$c[u] \\neq {\\mathbf {-1}}$}\n\\State{$F \\leftarrow F \\cup \\{c[u]\\}$}\n\\EndIf\n\\EndFor\n\\State{$col \\leftarrow 0$} \\Comment{ilk uygun renk y\u00f6ntemi}\n\\While{$col \\in F$}\n\\State{$col \\leftarrow col + 1$}\n\\EndWhile\n\\State{$c[v] \\leftarrow col$}\n\\EndFor\n\\end{algorithmic}\n\\label{alg:color}\n\\end{algorithm}\n\nLiterat\u00fcrde noktalar\u0131n s\u0131ralanmas\u0131 i\u00e7in farkl\u0131 y\u00f6ntemler denenmi\u015ftir. S\u0131k kullan\u0131lan bir y\u00f6ntem,\nnoktalar\u0131n ba\u011fl\u0131 olduklar\u0131 kenar say\u0131lar\u0131 (dereceleri) kullan\u0131larak s\u0131ralanmas\u0131d\u0131r. \nNoktalar azalan kenar say\u0131s\u0131na g\u00f6re \ns\u0131raland\u0131\u011f\u0131nda daha riskli noktalar \u00f6nce boyanacak ve algoritma daha k\u0131s\u0131tlanm\u0131\u015f oldu\u011fu son ad\u0131mlarda \nrenk say\u0131s\u0131n\u0131 artt\u0131rmadan daha bir \u00e7\u00f6z\u00fcm elde edebilecektir. Bu s\u0131ralamaya Welsch-Powell y\u00f6ntemi de denmektedir~\\cite{wwws}. \nNoktalar derecelerine, $|{\\tt nbor}_1(.)|$ de\u011ferlerine g\u00f6re s\u0131raland\u0131klar\u0131nda a\u00e7g\u00f6zl\u00fc algoritman\u0131n kullanaca\u011f\u0131 maksimum renk say\u0131s\u0131 ${{\\tt max}_{v \\in V}}\\{d_v\\} + 1$ olacakt\u0131r. \n\nBu \u00e7al\u0131\u015fmada kullan\u0131lan s\u0131ralama y\u00f6ntemleri a\u015fa\u011f\u0131da verilmi\u015ftir.\n\n\\begin{itemize}\n    \\item {\\bf 1-kom\u015fuluk}: Bu y\u00f6ntem yukar\u0131da anlat\u0131lan, azalan $|{\\tt nbor}(.)|$ de\u011ferlerini kullanan s\u0131ralama y\u00f6ntemidir. \n   \n    \\item {\\bf 2-kom\u015fuluk}: Bu y\u00f6ntem noktalar\u0131 azalan $|{\\tt nbor}_2(.)|$ de\u011ferlerine g\u00f6re s\u0131ralar. \u00c7izge \u00fczerindeki herhangi iki $v$ ve $u$ noktas\u0131 aras\u0131ndaki en k\u0131sa uzakl\u0131k $d(v,u)$ ile g\u00f6sterilsin. Bir $v \\in V$ noktas\u0131n\u0131n 2-kom\u015fulu\u011fu, $${\\tt nbor}_2(v) = \\{u \\in V: d(v,u) = 2\\}$$ olarak hesaplan\u0131r.\n   \n    \\item {\\bf 3-kom\u015fuluk}: Bu y\u00f6ntem noktalar\u0131 azalan $|{\\tt nbor}_3(.)|$ de\u011ferlerine g\u00f6re s\u0131ralar. Bir $v \\in V$ noktas\u0131 i\u00e7in ${\\tt nbor}_3(v)$ k\u00fcmesi $${\\tt nbor}_3(v) = \\{u \\in V: d(v,u) = 3\\}$$ olarak hesaplan\u0131r.\n   \n    \\item {\\bf Yak\u0131nl\u0131k merkeziyeti}: \u00c7izge \u00fczerindeki herhangi iki $v$ ve $u$ noktas\u0131 aras\u0131ndaki en k\u0131sa uzakl\u0131k $d(v,u)$ ile g\u00f6sterilsin. Bir $v$ noktas\u0131n\u0131n yak\u0131nl\u0131k merkeziyeti $${\\tt ym}(v) = \\frac{1}{\\sum_{u \\in V} d(v,u)}$$ olarak hesaplan\u0131r~\\cite{Sabidussi1966,doi:10.1121\/1.1906679}. Bu y\u00f6ntem noktalar\u0131 azalan yak\u0131nl\u0131k merkeziyeti de\u011ferlerine g\u00f6re s\u0131ralar. Y\u00f6ntemin amac\u0131 \u00e7izgeyi {\\em i\u00e7eriden d\u0131\u015far\u0131ya} do\u011fru, \u00f6ncelikle i\u00e7erideki yo\u011fun k\u0131sm\u0131n \u00fczerinden ge\u00e7erek renklendirmektir. Dolay\u0131s\u0131yla \u00e7izgedeki zor ve riskli noktalar daha \u00f6nce renklendirilecek, d\u0131\u015far\u0131da kalan, kullan\u0131lan renk say\u0131s\u0131n\u0131 artt\u0131rmas\u0131 daha d\u00fc\u015f\u00fck olas\u0131l\u0131\u011fa sahip u\u00e7 noktalar algoritman\u0131n k\u0131s\u0131tland\u0131\u011f\u0131 son ad\u0131mlarda renklendirilecektir. \n   \n     \\item {\\bf K\u00fcmeleme katsay\u0131s\u0131}: Bu y\u00f6ntemde noktalar azalan {\\em k\u00fcmeleme katsay\u0131s\u0131} de\u011ferlerine g\u00f6re s\u0131ralan\u0131r. K\u00fcmeleme katsay\u0131s\u0131, noktan\u0131n kom\u015fular\u0131n\u0131n birbirine ne kadar ba\u011fl\u0131 oldu\u011funu g\u00f6sterir~\\cite{Watts1998Collective,doi:10.1177\/104649647100200201}. \u00c7izge \u00fczerindeki bir $v$ noktas\u0131 i\u00e7in $${\\tt kk}(v) = \\frac{|\\{\\{u,w\\} \\in E: u,w \\in {\\tt nbor}(v)\\}| }{|{\\tt nbor}(v)| (|{\\tt nbor}(v)| - 1)}$$ noktan\u0131n k\u00fcmeleme katsay\u0131s\u0131n\u0131 vermektedir. Bu de\u011fer y\u00fcksek oldu\u011funda nokta neredeyse birbirine tam ba\u011fl\u0131 bir alt \u00e7izge i\u00e7erisinde yer almaktad\u0131r. Bu da noktay\u0131 a\u00e7g\u00f6zl\u00fc algoritma i\u00e7in riskli bir nokta yapmaktad\u0131r. Bu s\u0131ralama y\u00f6ntemi bu t\u00fcr noktalar\u0131 ilk s\u0131ralara koyarak renk say\u0131s\u0131n\u0131 azaltmay\u0131 hedeflemektedir. \n   \n     \\item {\\bf PageRank}: PageRank sosyal a\u011f analizinde s\u0131kl\u0131kla kullan\u0131lan bir merkeziyet metri\u011fidir~\\cite{Pageetal98}. \u00c7izge \u00fczerindeki bir noktan\u0131n PageRank de\u011feri, kenarlar \u00fczerinden rastgele nokta gezintisi yapan bir ki\u015finin belli bir anda o noktada bulunma olas\u0131l\u0131\u011f\u0131n\u0131 vermektedir.\n     Bu s\u0131ralama y\u00f6ntemi azalan PageRank de\u011ferlerini kullan\u0131r. Bu y\u00f6ntemin temel motivasyonu, yak\u0131nl\u0131k merkeziyeti i\u00e7in kullan\u0131lan y\u00f6ntemin motivasyonu ile ayn\u0131d\u0131r ve \u00e7izgenin merkezinden ba\u015flayarak, renk say\u0131s\u0131n\u0131 artt\u0131rma olas\u0131l\u0131\u011f\u0131 y\u00fcksek olan noktalar\u0131 ilk s\u0131rada renklendirmektir. \u00c7izge \u00fczerindeki noktalar\u0131n PageRank de\u011ferleri yinelemeli bir algoritma ile hesaplan\u0131r. Bir $v$ noktas\u0131n\u0131n ba\u015flang\u0131\u00e7taki PageRank degeri ${\\tt pr}_0(v) = \\frac{1}{|V|}$ olarak kabul edilir. Hesaplama esnas\u0131nda $i$ ad\u0131m\u0131ndaki PageRank de\u011feri ise\n$${\\tt pr}_i(v) = \\frac{1-\\alpha}{|V|} + \\alpha \\sum_{u \\in {\\tt nbor}_1(v)} \\frac{{\\tt pr}_{i-1}(u)}{|{\\tt nbor}_1(u)|}\n$$ olur. \n\n     \\item {\\bf Rastgele}: Bu y\u00f6ntem noktalar\u0131 rastgele, geli\u015fig\u00fczel bir \u015fekilde s\u0131ralamaktad\u0131r. \n\\end{itemize}\n\n\\section{Deney Sonu\u00e7lar\u0131}\nBu \u00e7al\u0131\u015fmada yap\u0131lan deneyler, 512GB RAM'e sahip, 64 bit CentOS 6.5 ile \u00e7al\u0131\u015fan, her bir soketin 15 \u00e7ekirde\u011fe sahip oldu\u011fu (toplamda 60), 2.30 GHz ile \u00e7al\u0131\u015fan Intel Xeon E7-4870 v2'de yap\u0131ld\u0131. Yaz\u0131lan b\u00fct\u00fcn kodlar {\\tt gcc 8.2.0} ile optimizasyon parametresi {\\tt -O3} kullan\u0131larak derlendi.\n\nDeneylerde kullan\u0131lan b\u00fct\u00fcn \u00e7izgeler SuiteSparse\\footnote{\\url{http:\/\/faculty.cse.tamu.edu\/davis\/suitesparse.html}} seyrek matris k\u00fct\u00fcphanesinden indirildi. \u0130lk k\u0131s\u0131m deneyler i\u00e7in nokta say\u0131s\u0131 $10^5$ ila $10^6$ aras\u0131ndaki b\u00fct\u00fcn simetrik matrisler kullan\u0131ld\u0131. Deneyler yap\u0131l\u0131rken, k\u00fct\u00fcphanede bu \u00f6zelliklere sahip 204 matris bulunmaktayd\u0131. Her bir matris, noktalar matrisin sat\u0131r ve s\u00fctunlar\u0131, kenarlar da matris i\u00e7erisindeki s\u0131f\u0131rdan farkl\u0131 say\u0131lar olmak \u00fczere bir \u00e7izge \u015feklinde modellendi. Bir s\u0131ralama y\u00f6nteminin bir matris \u00f6zelindeki performans\u0131 \u00f6l\u00e7\u00fcl\u00fcrken, o y\u00f6ntem  ile elde edilen renk say\u0131s\u0131n\u0131n, 1-kom\u015fuluk y\u00f6ntemi ile elde edilen renk say\u0131s\u0131na oran\u0131 kullan\u0131ld\u0131. Performans sonu\u00e7lar\u0131 \u00f6l\u00e7\u00fcl\u00fcrken, PageRank i\u00e7in 20 yineleme yap\u0131lm\u0131\u015ft\u0131r. Rastgele s\u0131ralama y\u00f6nteminin performans\u0131 ise 5 rastgele perm\u00fctasyonun ortalamas\u0131 al\u0131narak \u00f6l\u00e7\u00fclm\u00fc\u015ft\u00fcr. Yak\u0131nl\u0131k merkeziyeti i\u00e7in ise sonucu kesin olmasa da \u00e7ok h\u0131zl\u0131 \u015fekilde veren bir yak\u0131nsama algoritmas\u0131 kullan\u0131lm\u0131\u015ft\u0131r~\\cite{doi:10.1142\/S0218127407018403}. \n\nTablo~\\ref{tab:big_graphs}'de, B\u00f6l\u00fcm~\\ref{sec:yontem}'de anlat\u0131lan s\u0131ralama y\u00f6ntemlerinin b\u00fct\u00fcn matrisler i\u00e7in hesaplanan performanslar\u0131n\u0131n geometrik ortalamalar\u0131 verilmi\u015ftir. Tabloda da g\u00f6r\u00fcld\u00fc\u011f\u00fc \u00fczere, Yak\u0131nl\u0131k Merkeziyeti tabanl\u0131 s\u0131ralama di\u011fer t\u00fcm s\u0131ralamalardan daha etkili bir sonu\u00e7 vermi\u015ftir. Bu sonu\u00e7lar\u0131n daha detayl\u0131 incelemesi i\u00e7in \u015eekil~\\ref{fig:d1d2}'e bak\u0131labilir. G\u00f6r\u00fcld\u00fc\u011f\u00fc gibi, yak\u0131nl\u0131k merkeziyeti tabanl\u0131 s\u0131ralama kimi \u00e7izgeler i\u00e7in 4.5 kat daha az renk kullan\u0131rken, bu \u00e7izgeler \u00fczerinde en fazla $\\%25$ fazla renk kullanmaktad\u0131r. \n\\vspace*{-2ex}\n\\begin{table}[H]\n    \\centering\n    \\fontsize{9}{9}\\selectfont\n    \\caption{1-UZAKLIK RENKLEND\u0130RME \u0130\u00c7\u0130N SIRALAMA ALGOR\u0130TMALARININ PERFORMANSLARININ GEOMETR\u0130K ORTALAMASI}\n    \\def1.3{1.3}\n    \\scalebox{0.99}{\n    \\begin{tabular}{l|r}\n       \n        {\\bf S\u0131ralama} & {\\bf1-uzakl\u0131k} \\\\\n        \\hline\n        1-kom\u015fuluk & 1.00 \\\\ \n        2-kom\u015fuluk & 1.02 \\\\ \n        3-kom\u015fuluk & 1.07 \\\\ \n        Yak\u0131nl\u0131k Merkeziyeti & {\\bf 0.97} \\\\ \n        K\u00fcmelenme Katsay\u0131s\u0131 & 1.03 \\\\ \n        PageRank & 1.02  \\\\ \n        Rastgele S\u0131ralama & 1.08 \\\\\n\n    \\end{tabular}\n    }\n    \\label{tab:big_graphs}\n\\end{table}\n\n\\begin{figure}[H]\n    \\centering\n      \\shorthandoff{=}\n    \\includegraphics[width=0.95\\linewidth]{d1d2}\n    \\caption{Yak\u0131nl\u0131k merkeziyeti metri\u011finin renklendirme problemi i\u00e7in 204 \u00e7izge \u00fczerindeki performans\u0131.}\n    \\label{fig:d1d2}\n\\end{figure}\n\nHer ne kadar yak\u0131nl\u0131k merkeziyeti tabanl\u0131 s\u0131ralama b\u00fct\u00fcn s\u0131ralama y\u00f6ntemlerinden daha iyi \u00e7al\u0131\u015fm\u0131\u015f olsa da, performans sadece 1-kom\u015fuluk s\u0131ralama y\u00f6ntemine g\u00f6re de\u011ferlendirilmi\u015ftir. Bunun sebebi, kullan\u0131lan \u00e7izgelerin en iyi \u00e7\u00f6z\u00fcm\u00fc h\u0131zl\u0131 bir \u015fekilde elde edemeyece\u011fimiz kadar b\u00fcy\u00fck olmas\u0131d\u0131r. Performans\u0131n en iyi \u00e7\u00f6z\u00fcme g\u00f6re de\u011ferlendirilmesi i\u00e7in ikinci, g\u00f6rece daha k\u00fc\u00e7\u00fck \u00e7izgelerden olu\u015fan bir \u00e7izge k\u00fcmesi olu\u015fturulmu\u015ftur. Bu k\u00fcmeyi olu\u015fturmak i\u00e7in SuiteSparse k\u00fct\u00fcphanesinden nokta say\u0131s\u0131 100 ila 500 aras\u0131, 260 simetrik matris se\u00e7ilmi\u015f ve \u00e7izge olarak modellenmi\u015ftir. B\u00fct\u00fcn bu \u00e7izgeler i\u00e7in en az say\u0131da renk kullanan \u00e7\u00f6z\u00fcmler, lineer programlama y\u00f6ntemi ve {\\em CPLEX} k\u00fct\u00fcphanesi kullanarak elde edilmi\u015ftir. \n\nBu k\u00fcme \u00fczerindeki performans sonu\u00e7lar\u0131 Tablo~\\ref{tab:small_graphs}'de verilmi\u015ftir. De\u011ferlerden g\u00f6r\u00fclece\u011fi \u00fczere, yak\u0131nl\u0131k merkeziyeti b\u00fcy\u00fck \u00e7izgelerde oldu\u011fu gibi daha \u00f6nce anlat\u0131lan b\u00fct\u00fcn s\u0131ralama y\u00f6ntemlerinden daha ba\u015far\u0131l\u0131 bir performans elde etmi\u015ftir. Yak\u0131nl\u0131k merkeziyeti, en iyi sonuca g\u00f6re ortalama $\\%23$ daha fazla renk kullanmaktad\u0131r. 1-kom\u015fuluk tabanl\u0131 s\u0131ralama ise en iyi sonuca g\u00f6re $\\%36$ daha fazla renkle \u00e7izge  renklendirme yapabilmektedir. \n\n\\begin{table}[H]\n    \\centering\n    \\fontsize{9}{9}\\selectfont\n    \\caption{1-UZAKLIK RENKLEND\u0130RME \u0130\u00c7\u0130N SIRALAMA ALGOR\u0130TMALARININ EN \u0130Y\u0130 \u00c7\u00d6Z\u00dcME G\u00d6RE PERFORMANSLARININ GEOMETR\u0130K ORTALAMASI}\n    \\def1.3{1.3}\n    \\scalebox{0.99}{\n    \\begin{tabular}{l|r}\n       \n        {\\bf \u00d6l\u00e7\u00fct} & {\\bf1-uzakl\u0131k} \\\\\n        \\hline\n        1-kom\u015fuluk & 1.36  \\\\ \n        2-kom\u015fuluk & 1.35 \\\\ \n        3-kom\u015fuluk & 1.40 \\\\ \n        Yak\u0131nl\u0131k Merkeziyeti & {\\bf 1.23} \\\\ \n        K\u00fcmelenme Katsay\u0131s\u0131 & 1.40 \\\\\n        PageRank & 1.34 \\\\ \n        Rastgele S\u0131ralama & 1.56 \\\\\n        \\hline\n        A\u011f\u0131rl\u0131kl\u0131 S\u0131ralama & {\\bf 1.16} \\\\ \n        Tekd\u00fcze S\u0131ralama & 1.53 \\\\\n       \n    \\end{tabular}}\n    \\label{tab:small_graphs}\n\\end{table}\n    \nBu k\u00fcme \u00fczerindeki performans sonu\u00e7lar\u0131n\u0131n olu\u015fturulmas\u0131 s\u00fcrecinde b\u00fcy\u00fck \u00e7izgelerde kullan\u0131lan s\u0131ralama algoritmalar\u0131na ek olarak {\\em tekd\u00fcze} ve {\\em a\u011f\u0131rl\u0131kl\u0131} s\u0131ralama y\u00f6ntemleri de denenemi\u015ftir. Bu s\u0131ralama y\u00f6ntemleri normalize edilmi\u015f de\u011ferleri kullan\u0131r. Bunun i\u00e7in b\u00fct\u00fcn nokta de\u011ferlerinden o de\u011ferin ortalamas\u0131 \u00e7\u0131kart\u0131l\u0131r ve elde edilen de\u011fer standart sapmaya b\u00f6l\u00fcn\u00fcr. {\\em Tekd\u00fcze s\u0131ralama} bir $v$ noktas\u0131n\u0131n normalize edilmi\u015f b\u00fct\u00fcn de\u011ferlerini~(1-kom\u015fuluk, 2-kom\u015fuluk, 3-kom\u015fuluk, yak\u0131nl\u0131k merkeziyeti, k\u00fcmelenme katsay\u0131s\u0131 ve PageRank) toplayarak elde edilen de\u011fer i\u00e7in azalan s\u0131raya g\u00f6re noktalar\u0131 s\u0131ralar. Tablo~\\ref{tab:small_graphs}'de g\u00f6r\u00fcld\u00fc\u011f\u00fc gibi bu s\u0131ralama rastgele bir s\u0131ralama kadar k\u00f6t\u00fc sonu\u00e7lar vermi\u015ftir.\n\n{\\em A\u011f\u0131rl\u0131kl\u0131 s\u0131ralama}, Tablo~\\ref{tab:small_graphs}'de verilen metriklerin farkl\u0131 kombinasyonlar\u0131n\u0131n kullan\u0131lmas\u0131 ile en fazla ne kadar iyi bir sonu\u00e7 al\u0131nabilece\u011fini \u00f6l\u00e7mek i\u00e7in tasarlanm\u0131\u015ft\u0131r. S\u0131ralamalar, birbirleri aras\u0131ndaki \u00f6l\u00e7ek fark\u0131n\u0131 gidermek i\u00e7in normalize edildikten sonra, t\u00fcm lineer kombinasyonlar aras\u0131ndan en iyi ortalama sonucu veren a\u011f\u0131rl\u0131k kombinasyonu bulunmu\u015ftur. Elde edilen a\u011f\u0131rl\u0131klar Tablo~\\ref{tab:weights}'te verilmi\u015ftir.  \n\n\\begin{table}[H]\n    \\centering\n    \\fontsize{9}{9}\\selectfont\n    \\caption{A\u011eIRLIKLI SIRALAMA ALGOR\u0130TMASI \u0130\u00c7\u0130N A\u011eIRLIK DE\u011eERLER\u0130}\n    \\def1.3{1.3}\n    \\scalebox{0.99}{\n    \\begin{tabular}{l|r}\n       \n        {\\bf \u00d6l\u00e7\u00fct} & {\\bf1-uzakl\u0131k} \\\\\n        \\hline\n        1-kom\u015fuluk & 0.10  \\\\ \n        2-kom\u015fuluk & 0.05 \\\\ \n        3-kom\u015fuluk & 0.10 \\\\ \n        Yak\u0131nl\u0131k Merkeziyeti & {\\bf 0.70} \\\\ \n        K\u00fcmelenme Katsay\u0131s\u0131 & 0.05 \\\\ \n        PageRank & 0.00 \\\\\n    \\end{tabular}}\n    \\label{tab:weights}\n\\end{table}\n\nKatsay\u0131lar incelendi\u011finde, yak\u0131nl\u0131k merkeziyeti de\u011ferinin di\u011fer a\u011f\u0131rl\u0131klardan daha y\u00fcksek oldu\u011fu g\u00f6r\u00fcl\u00fcr. Ba\u015fka bir deyi\u015fle, yak\u0131nl\u0131k merkeziyeti, en iyi kombinasyon i\u00e7erisinde di\u011fer metriklerden daha b\u00fcy\u00fck bir etkiye sahiptir. Bu s\u0131ralama algoritmas\u0131 ile en iyi sonu\u00e7tan sadece $\\%16$ k\u00f6t\u00fc sonu\u00e7 elde edilmi\u015ftir. Yine Tablo~\\ref{tab:small_graphs}'de g\u00f6r\u00fcld\u00fc\u011f\u00fc gibi, yak\u0131nl\u0131k merkeziyeti, k\u00fc\u00e7\u00fck \u00e7izgeler i\u00e7in s\u0131ralama algoritmalar\u0131yla olu\u015fturulabilecek en iyi renklendirmeden ortalama sadece $\\%7$ daha fazla renk ile \u00e7izgeleri renklendirebilmektedir.\n\n\\begin{figure}[H]\n    \\centering\n      \\shorthandoff{=}\n    \\includegraphics[width=0.95\\linewidth]{yw}\n    \\caption{A\u011f\u0131rl\u0131kland\u0131r\u0131lm\u0131\u015f s\u0131ralaman\u0131n ve yak\u0131nl\u0131k merkeziyeti metri\u011finin renklendirme problemi i\u00e7in 204 \u00e7izge \u00fczerindeki performans\u0131.}\n    \\label{fig:yw}\n\\end{figure}\n\nA\u011f\u0131rl\u0131kland\u0131r\u0131lm\u0131\u015f s\u0131ralama ilk deneydeki \u00e7izge k\u00fcmesi \u00fczerinde kullan\u0131ld\u0131\u011f\u0131nda, yak\u0131nl\u0131k merkeziyeti tabanl\u0131 s\u0131ralaman\u0131n iyile\u015ftiremedi\u011fi \u00e7ok say\u0131da \u00e7izgeyi iyile\u015ftirdi\u011fi g\u00f6r\u00fclm\u00fc\u015ft\u00fcr (\u015eekil~\\ref{fig:yw}). A\u011f\u0131rl\u0131kland\u0131r\u0131lm\u0131\u015f s\u0131ralama y\u00f6ntemi, 204 \u00e7izgenin sadece 45'inde yak\u0131nl\u0131k merkeziyeti tabanl\u0131 s\u0131ralamadan daha iyi sonu\u00e7 elde edememi\u015ftir. Dolay\u0131s\u0131yla, a\u011f\u0131rl\u0131kland\u0131r\u0131lm\u0131\u015f y\u00f6ntemin ba\u015far\u0131l\u0131 bir s\u0131ralama yaratma potansiyelinin oldu\u011fu d\u00fc\u015f\u00fcn\u00fclmektedir.  \n\n\\section{Sonu\u00e7}\n\nBu \u00e7al\u0131\u015fmada \u00e7izge renklendirme problemi i\u00e7in kullan\u0131lan a\u00e7g\u00f6zl\u00fc algoritmalardaki nokta s\u0131ralamas\u0131 i\u00e7in y\u00f6ntemler \u00fczerine yo\u011funla\u015f\u0131lm\u0131\u015ft\u0131r. Sosyal a\u011f analizi i\u00e7in kullan\u0131lan metriklerin bu s\u0131ralamalar\u0131n bulunmas\u0131ndaki ba\u015far\u0131lar\u0131 ara\u015ft\u0131r\u0131lm\u0131\u015ft\u0131r. Yap\u0131lan deneyler g\u00f6stermi\u015ftir ki, yak\u0131nl\u0131k merkeziyeti metri\u011fi, s\u0131ralama i\u00e7in olduk\u00e7a ba\u015far\u0131l\u0131 bir metriktir. \n\nBu \u00e7al\u0131\u015fmada sadece statik s\u0131ralama y\u00f6ntemleri \u00fczerinde durulmu\u015ftur. Algoritma \u00e7al\u0131\u015ft\u0131k\u00e7a noktalar renklendirildi\u011finden, her ad\u0131mda farkl\u0131 bir s\u0131ralaman\u0131n en iyi olma olas\u0131l\u0131\u011f\u0131 olduk\u00e7a fazlad\u0131r. \u0130leride, s\u0131ralaman\u0131n noktalar\u0131n kom\u015fuluklar\u0131ndaki farkl\u0131 renk say\u0131s\u0131 gibi dinamik metrikler de kullan\u0131lrarak, s\u00fcrekli de\u011fi\u015fen s\u0131ralamalar \u00fczerinde \u00e7al\u0131\u015f\u0131lacakt\u0131r. Verilen bir \u00e7izge i\u00e7in en iyi s\u0131ralama y\u00f6nteminin tahmin edilmesi de olduk\u00e7a ilgin\u00e7 bir problemdir. \u0130leride bu problem \u00fczerinde de \u00e7al\u0131\u015f\u0131lmas\u0131 planlanmaktad\u0131r.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWith three distinct superconducting phases {UPt$_3$} has attracted significant attention~\\cite{Joynt:2002wt}, but despite decades of experimental and theoretical studies the unconventional superconductivity in this material is still not fully understood.\nFigure~\\ref{VLrotation}(b) shows the {UPt$_3$} phase diagram, indicting the extent of the superconducting A, B and C phases.\nThe presence of two distinct zero-field superconducting transitions suggests that the order parameter belongs to one of the two-dimensional representations of the D$_{6h}$ point group~\\cite{Hess:1989wt}.\nHere, $f$-wave pairing states with the $E_{2u}$ irreducible representation are the most likely~\\cite{Sauls:1994wd}.\nIn such a scenario the B phase breaks time reversal and mirror symmetries while the A and C phases are time-reversal symmetric.\nExperimental support comes from the $H$-$T$ phase diagram~\\cite{Adenwalla:1990we,Sauls:1994wd,Shivaram:1986wo,Choi:1991wa}, and thermodynamic and transport studies~\\cite{Taillefer:1997eq,Graf:2000ua}.\nBroken time-reversal symmetry in the B phase is supported by phase-sensitive Josephson tunneling~\\cite{Strand:2009eq}, the observation of polar Kerr rotation~\\cite{Schemm:2014fv}, and a field history-dependent vortex lattice (VL) configuration~\\cite{Avers:2020wx}.\nFinally, the linear temperature dependence of the London penetration depth is consistent with a quadratic dispersion of the energy gap at the polar nodes structure, which is a characteristic of the $E_{2u}$ model~\\cite{Signore:1995hu,Schottl:1999ks,Gannon:2015ct}.\n\nA key component in the understanding of superconductivity in {UPt$_3$} is the presence of a symmetry breaking field (SBF) that couples to the $E_{2u}$ superconducting order parameter~\\cite{Hayden:1992gp}.\nThe SBF lifts the degeneracy of the multi-dimensional representation, splitting the zero-field transition and leading to the multiple superconducting phases~\\cite{Sauls:1994wd}.\nHowever, the origin of the SBF is an outstanding issue, with possible candidates that include a quasi-static antiferromagnetic state that develops at 5~K above the superconducting transition~\\cite{Aeppli:1988ur,Aeppli:1988gw,Hayden:1992gp}, a distortion of the hexagonal crystal structure~\\cite{Walko:2001fm}, or prismatic plane stacking faults~\\cite{Hong:1999ud,Gannon:2012fu}.\n\nVortices provide a highly sensitive probe of the host superconductor.\nThis includes anisotropies in the screening current plane perpendicular to the applied magnetic field which affect the VL symmetry and orientation.\nSuch anisotropies may arise from the Fermi surface~\\cite{Kogan:1981vl,Kogan:1997vm}, and nodes in or distortions of the superconducting gap~\\cite{Huxley:2000aa,Avers:2020wx}.\nAs an example one can consider the ``simple'' superconductor niobium that displays a rich VL phase diagram when the applied field is along the [100] crystalline direction and the Fermi surface anisotropy is incommensurate with an equilateral triangular VL~\\cite{Laver:2006bn,Laver:2009bk,Muhlbauer:2009hw}.\nEven in materials with a hexagonal crystal structure VL rotations may occur due to competing anisotropies, as observed in MgB$_2$ when the applied field is perpendicular to the basal plane~\\cite{Cubitt:2003ab,Das:2012cf}.\n\nWe have used small-angle neutron scattering (SANS) to determine the VL phase diagram in {UPt$_3$}.\nThis extends our previous studies at low temperature, where the VL was found to undergo a field-driven, non-monotonic rotation transition~\\cite{Avers:2020wx}.\nWe discuss how the VL phase diagram and the existence of the VL rotation transition can be directly attributed to the SBF.\n\n\\begin{figure*}\n    \\includegraphics[width = 168mm]{Fig1.jpg}\n    \\caption{\\label{DifPat}\n        SANS VL diffraction patterns obtained at $H = 0.6$~T and $T = 100$~mK (a), 200~mK (b) and 300~mK (c).\n        The Bragg peak splitting ($\\omega$) is indicated in (a) and crystallographic directions within the scattering plane in (b).\n        Only peaks at the top of the detector were imaged.\n        Zero field background scattering is subtracted, and the detector center near $Q = 0$ is masked off.\n       \n        }\n\\end{figure*}\n\n\n\\section{Experimental Details}\nSmall-angle neutron scattering studies of the VL are possible due to the periodic field modulation from the vortices~\\cite{Muhlbauer:2019jt}.\nThe scattered intensity depends strongly on the superconducting penetration depth, and for {UPt$_3$} with a large in-plane $\\lambda_{ab} \\sim 680$~nm~\\cite{Gannon:2015ct} necessitates a large sample volume.\nFor this work we used a high-quality single crystal (ZR11), combined with previously published results obtained on a separate sample (ZR8)~\\cite{Avers:2020wx}.\nProperties of both single crystals are listed in table~\\ref{Xtals},\n\\begin{table}[b]\n    \\begin{tabular}{lcccc}\n    \\hline \\hline\n    Sample \\hspace{0.1cm} & \\hspace{0.1cm} mass~(g) \\hspace{0.1cm} & \\hspace{0.1cm} RRR \\hspace{0.1cm} & \\hspace{0.1cm} $T_{\\rm c}$~(mK) \\hspace{0.1cm} & \\hspace{0.1cm} $\\Delta T_{\\rm c}$~(mK) \\\\ \\hline\n    ZR8  & 15 & $> 600$ & $560 \\pm 2$ & 10 \\\\\n    ZR11 &  9 & $> 900$ & $557 \\pm 2$ &  5 \\\\ \\hline \\hline\n    \\end{tabular}\n    \\caption{\\label{Xtals}\n        Properties of the two {UPt$_3$} single crystals used for the SANS experiments.}\n\\end{table}\ndetermined from resistive measurements performed on smaller samples cut from the main crystals.\nHere, RRR is the residual resistivity ratio, $T_{\\rm c}$ is the superconducting transition temperature and $\\Delta T_{\\rm c}$ is the width of the transition.\n\nFor the SANS measurements each long, rod-like crystal was cut into two pieces, co-aligned and fixed with silver epoxy (EPOTEK E4110) to a copper cold finger.\nThe sample assembly was mounted onto the mixing chamber of a dilution refrigerator and placed inside a superconducting magnet, oriented with the crystalline $\\textbf{a}$ axis vertical and the $\\textbf{c}$ axis horizontally along the magnetic field and the neutron beam.\nThe neutron beam was masked off to illuminate a $7 \\times 11$~mm$^2$ area.\n\nThe SANS experiment was performed at the GP-SANS beam line at the High Flux Isotope Reactor at Oak Ridge National Laboratory~\\cite{Heller:2018gq}.\nAll measurements were carried out in a ``rocked on'' configuration, satisfying the Bragg condition for VL peaks at the top of the two-dimensional position sensitive detector, as seen in Fig.~\\ref{DifPat}.\nBackground measurements, obtained either in zero field or above $H_{\\text{c2}}$, were subtracted from both the field reduction and field reversal data.\n\nMeasurements were performed at temperatures between 100~mK and 300~mK and fields between 0.4~T and 1.2~T.\nPrior to the SANS measurements the field was reduced from above the B-C phase transition at base temperature.\nThe sample was then heated to the measurement temperature and a damped field oscillation with an initial amplitude of 20~mT was applied to obtain a well ordered VL with a homogeneous vortex density~\\cite{Avers:2020wx}.\nFurthermore, a 5~mT field oscillation was applied approximately every 60 seconds during the SANS measurements, in order to counteract VL disordering due to neutron induced fission of $^{235}$U~\\cite{Avers:2021wu}.\n\n\n\\section{Results}\nFigure~\\ref{DifPat} shows VL diffraction patterns obtained in an applied field of 0.6~T and temperature between 100~mK and 300~mK.\nAs previously reported, the VL in {UPt$_3$} has a triangular symmetry but is in general not oriented along a high symmetry direction of the hexagonal crystalline basal lattice ($\\bm{a}$ or $\\bm{a}^*$)~\\cite{Avers:2020wx}.\nThis causes the VL to break up into clockwise and counterclockwise rotated domains, and gives rise to the Bragg peak splitting in Figs.~\\ref{DifPat}(a) and \\ref{DifPat}(b).\nWith increasing temperature the splitting decreases, and the two peaks eventually merge as seen in Fig.~\\ref{DifPat}(c).\n\nTo quantify the VL rotation we define the peak splitting angle ($\\omega$) shown in Fig.~\\ref{DifPat}(a), determined from two-Gaussian fits to the diffraction pattern intensity.\nSpecific details of the fitting will be discussed in more detail later.\nThe temperature dependence of $\\omega$ is summarized in Fig.~\\ref{VLrotation}(a) for all the magnetic fields measured, together with results from our previous SANS studies obtained at base temperature~\\cite{Avers:2020wx}.\n\\begin{figure}\n    \\includegraphics[width = 66mm]{Fig2.jpg}\n    \\caption{\\label{VLrotation}\n        Vortex lattice rotation.\n        (a) VL peak splitting vs temperature for different magnetic fields.\n        The data at 50~mK (solid symbols) was previously obtained on the ZR8 crystal~\\cite{Avers:2020wx}.\n        Error bars represent one standard deviation.\n        (b) Constant $\\omega$ contours superimposed on the {UPt$_3$} phase diagram.\n        Values are obtained from the data in (a) by interpolation (open diamonds) and from the 50~mK field dependence in Ref.~\\onlinecite{Avers:2020wx} (solid diamonds).\n        The $30^{\\circ}$ data point (open circle) is from previous work by Huxley {\\em et al.}~\\cite{Huxley:2000aa}.}\n\\end{figure}\nAt all fields the temperature dependence of $\\omega$ appears to be linear within the measurement error, and extrapolate to zero well below the A-B phase transition.\nThe larger error bars at higher temperature is due to an increasing penetration depth and the resulting decrease in the scattered intensity~\\cite{Gannon:2015ct}.\n\nFigure~\\ref{VLrotation}(b) shows $\\omega$ equicontours superimposed on the {UPt$_3$} $H$-$T$ phase diagram.\nThe nonmonotonic behavior, previously reported at base temperature~\\cite{Avers:2020wx}, is clearly observed at higher temperatures, although with a decreasing amplitude.\nFurthermore, the splitting extrapolates to zero in the zero field limit, and also decreases upon approaching the B-C phase transition.\nHowever, once in the C phase the splitting remains at a fixed value of $\\sim 8^{\\circ}$~\\cite{Avers:2020wx}.\nAt all temperatures the maximal VL rotation is observed at 0.8~T.\nAlso indicated in Fig.~\\ref{VLrotation}(b) is the approximate temperature at 0.19~T at which $\\omega$ reaches $30^{\\circ}$ in the vicinity of the A phase, reported by Huxley {\\em et al.}~\\cite{Huxley:2000aa}.\n\nEnsuring a reliable determination of $\\omega$ requires a careful approach to the fitting.\nAt all fields and temperatures the radial position ($Q_R$) as well as the radial ($\\Delta Q_R$) and azimuthal ($\\Delta \\theta$) widths were constrained to be the same for both of the split peaks.\nFurthermore, the azimuthal width at each field was determined from fits at low temperature where the peaks are clearly separated, and then kept fixed at the higher temperature where they begin to overlap.\nTo justify this approach, we note that when the peaks are clearly separated, $\\Delta \\theta$ does not exhibit any systematic temperature dependence.\nThe azimuthal width does show a field dependence, however, with $\\Delta \\theta$ decreasing from $\\sim 11.5^{\\circ}$ FWHM at 0.4~T to $\\sim 6.5^{\\circ}$ FWHM at 1.2~T.\n\nThe VL density is reflected in $Q_R$ and $\\Delta Q_R$, shown in Fig.~\\ref{VLdensity}.\n\\begin{figure}\n    \\includegraphics[width = 67mm]{Fig3.jpg}\n    \\caption{\\label{VLdensity}\n        Vortex lattice density.\n        (a) Scattering vector magnitude normalized to the value expected for a triangular VL.\n        (b) Radial width of the VL Bragg peaks (FWHM) compared to the incident beam divergence (dashed line).\n        The inset indicates $\\Delta Q_R$ within the detector plane.}\n\\end{figure}\nThe magnitude of the scattering vector in Fig.~\\ref{VLdensity}(a) agrees to within a few percent with $Q_0 = 2\\pi (\\sqrt{3}\/2)^{1\/4} \\sqrt{B\/\\Phi_0}$ expected for a triangular VL and assuming that the magnetic induction ($B$) is equal to the applied magnetic field.\nHere $\\Phi_0 = h\/2e = 2069$~T~nm$^2$ is the flux quantum.\nThe small deviation between $Q_R$ and $Q_0$ is slightly greater at low fields consistent with earlier work~\\cite{Avers:2020wx}, but notably independent of temperature.\nSimilarly, there is no systematic temperature or field dependence in the radial width in Fig.~\\ref{VLdensity}(b).\nHowever, the values are systematically at or below the divergence of the incident beam, indicating a highly ordered VL which leads to a diffracted neutron beam that is more collimated than the incident one.  \n\n\\begin{figure*}\n    \\includegraphics[width = 165mm]{Fig4.jpg}\n    \\caption{\\label{OP}\n        Order parameter in the\n        (a) A phase ($\\eta_1 \\neq 0$, $\\eta_2 = 0$),\n        (b) B phase distorted by the SBF ($\\eta_1 = \\eta_2 = 1$, $\\epsilon = 0.1$), and\n        (c) C phase ($\\eta_1 = 0$, $\\eta_2 \\neq 0$).}\n\\end{figure*}\n\n\\section{Discussion}\nThe complex VL phase diagram in Fig.~\\ref{VLrotation}(b) reflects the presence of multiple competing effects.\nIn the following we discuss how, at the qualitative level, this phase diagram arises from the interplay between the SBF and the nodal configuration of the superconducting energy gap for the A and C phases.\nFirst, however, we note that $\\omega \\rightarrow 0$ in the limit $T = H = 0$.\nFor large vortex separations the order parameter has a vanishing effect on the VL, and the orientation with Bragg peaks along the $\\textbf{a}$ axis must be due to the Fermi surface anisotropy~\\cite{Huxley:2000aa}.\n\nIn momentum space the two-component $E_{2u}$ order parameter proposed for {UPt$_3$} is given by~\\cite{Sauls:1994wd}\n\\begin{equation}\n    \\label{E2uOP}\n    \\Delta(\\mathbf k) = \\left( \\eta_1 \\left( k_x^2 - k_y^2 \\right) \\pm 2i \\, \\eta_2 (1 - \\epsilon) \\, k_x \\, k_y \\right) k_z.\n\\end{equation}\nHere, $\\eta_1$ and $\\eta_2$ are real amplitudes which depend on temperature and magnetic field and $\\epsilon$ is due to the SBF.\nThe A and C phases correspond to a vanishing of $\\eta_2$ and $\\eta_1$ respectively.\nThe magnitude of the SBF determines the zero-field split in the superconducting transition ($\\Delta T_{\\text{AB}}$) and thus the width of the A phase.\nExperimentally, $\\Delta T_{\\text{AB}} \\approx 55$~mK which yields $\\epsilon \\sim \\tfrac{\\Delta T_{\\text{AB}}}{T_c} \\approx 0.1$.\nWithin the B phase both components of the order parameter are non-zero, although with different amplitudes.\nDue to the SBF this imbalance persists even in the low-temperature, low-field limit where both $\\eta_2$ and $\\eta_1$ approach unity~\\cite{Sauls:1994wd}.\nThe order parameter structure is illustrated in Fig.~\\ref{OP}.\n\nWithin the A phase SANS studies by Huxley {\\em et al.} found a VL with domains rotated by $\\pm 15^{\\circ}$ relative to the $\\textbf{a}$ axis ($\\omega = 30^{\\circ}$)~\\cite{Huxley:2000aa}.\nThe VL rotation was attributed to a competition between the sixfold Fermi surface anisotropy and the fourfold anisotropy of the nodal structure in the A phase.\nNotably, the rotation persists into the B phase as indicated in Fig.~\\ref{VLrotation}(b).\nThis is not surprising since the $\\eta_1\/\\eta_2 \\rightarrow \\infty$ upon approaching the A phase from low temperature, and the B phase order parameter therefore exhibit a substantial fourfold anisotropy.\nHowever, as $\\eta_2$ increases with decreasing temperature this ratio quickly decreases, causing an abrupt transition to $\\omega = 0$ around 425~mK~\\cite{Huxley:2000aa}.\n\nDue to the SBF the order parameter in the B phase preserves a degree of fourfold anisotropy, as shown in Fig.~\\ref{OP}.\nThis anisotropy is oriented in a manner similar to the A phase, with an effect on the vortex-vortex interactions which will increase with increasing field (vortex density).\nThe influence of the SBF anisotropy will increase further at low temperature as the superfluid density increases~\\cite{Gannon:2015ct}, even as $\\epsilon$ remains fixed.\nThis explains the initial increase of $\\omega$ with field at low temperatures, with an amplitude (0.8~T) that extrapolates to a value close to $30^{\\circ}$ for $T \\rightarrow 0$.\n\nAs the field is increased further and approaches the BC phase transition, $\\eta_1$ decreases and finally vanish.\nThe C phase order parameter is rotated by $45^{\\circ}$ about $k_z$ with respect to the B phase, as shown in Fig.~\\ref{OP}.\nThis will favor a VL oriented along the $\\textbf{a}$ axis, i.e. the same as the Fermi surface anisotropy, and explains the non-monotonic VL rotation as a function of field.\nOnce $\\eta_1$ has fully vanished no further VL rotation is expected, in agreement with the observed field-independence of $\\omega \\approx 8^{\\circ}$ in the C phase~\\cite{Avers:2020wx}.\n\n\n\\section{Conclusion}\nIn summary, the rotated VL phase at low temperatures and intermediate fields in Fig.~\\ref{VLrotation}(b) can be directly attributed to the SBF.\nTo our knowledge this is the first observation of such an effect at the microscopic level, and may provide further constraints on the nature of both the SBF and the order parameter in {UPt$_3$}.\nA quantitative understanding of $\\omega(T,H)$ will require a detailed theoretical analysis, taking into account the field and temperature dependence of the superfluid density as well as the complex Fermi surface of {UPt$_3$}.\nHere, the finite value of $\\omega$ in the C phase is somewhat surprising and not obviously consistent with the order parameter in Eq.~(\\ref{E2uOP}).\n\n\n\\section*{Conflict of Interest Statement}\nThe authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.\n\n\\section*{Author Contributions}\nKEA, WPH, and MRE conceived of the experiment.\nWJG and KEA grew and annealed the crystals.\nKEA, AWDL, and MRE performed the SANS experiments with assistance from LDS.\nKEA, WPH, and MRE wrote the paper with input from all authors.\n\n\\section*{Funding}\nThis work was supported by the the Northwestern-Fermilab Center for Applied Physics and Superconducting Technologies (KEA) and by the U.S. Department of Energy, Office of Basic Energy Sciences, under Awards No.~DE-SC0005051 (MRE: University of Notre Dame; neutron scattering) and DE-FG02-05ER46248 (WPH: Northwestern University; crystal growth and neutron scattering).\nA portion of this research used resources at the High Flux Isotope Reactor, a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory.\n\n\\section*{Acknowledgments}\nWe are grateful to J.~A.~Sauls for numerous discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nHomogeneous object clusters (HOC) are ubiquitous. From microscopic cells to gigantic galaxies, they tend to cluster together. Figure~\\ref{fig:scenario case} shows typical examples. Delineating individual homogeneous objects from their cluster gives an accurate estimate of the number of instances which further enables many important applications: for example, in medicine, various blood cell counts give crucial information on a patient's health.\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=\\linewidth]{images\/real_scenario}\n  \\caption{Typical homogeneous object clusters.}\n  \\label{fig:scenario case}\n  \\end{center}\n\\end{figure}\n\nInstance segmentation for HOCs is by no means a trivial task despite the uniformity of the target objects.  Directly applying current best performing instance segmentation methods~\\cite{he2017mask,li2016fully}, many of which are based on some kind of Deep Convolutional Neural Network (DCNN), meets a bottleneck: unaffordable annotation cost. All of these segmentation models require a large number of annotated images for training purpose. Unlike typical object segmentation (e.g., car, chair), homogeneous clustered objects are densely distributed in a given image with various degrees of occlusion. Pixel-wise labeling these objects is extremely time consuming. However, in many realistic scenarios (e.g., merchandise sold in a supermarket), we have tens of thousands of categories to process. Category-specific annotation is impractical to the instance segmentation problem we address in this paper. We need to automatically generate large amount training data (HOC images with segmentation annotation) with cheap cost.\n\nGenerative adversarial nets \\cite{goodfellow2014generative} has been widely used to generate images and is a seemingly promising direction. However, the GAN framework cannot generate images with pixel-level annotation, so the segmentation models mentioned above cannot be trained using such images. RenderGAN \\cite{sixt2016rendergan} is proposed to generate images from labels. However the method also cannot generate images with  pixel-level annotation. Image-to-image translation \\cite{isola2016image} based on conditional GAN \\cite{mirza2014conditional} can get the annotation from images but it requires a large collection of image-annotation pairs to train, and thus it still needs a lot of laborious annotation.\n\nDriven by the above considerations, in this paper, we propose a novel framework to tackle the challenging instance segmentation problem.\nInspired by~\\cite{oneshot}, our learning framework is \\textbf{one-shot} because it learns by looking only once the single sample captured in a single short video, which avoids the cumbersome collection of large-scale image datasets for training. Then, these single-object video frames are used to automatically synthesize realistic images of homogeneous object clusters. In doing so, we can acquire a large amount of training data automatically. Therefore, our framework is \\textbf{annotation-free}. However, generating visually realistic images is a non-trivial task, since structural constraint (i.e., cluster layout should look reasonable) and illumination should be taken into consideration. In this paper, we propose a novel image synthesis framework to capture key priors from real images depicting HOCs. Structure prior is captured by learning a structured likelihood function. We generate structurally realistic synthetic images of HOC by placing synthetic objects in a specific way that maximizes the structured likelihood given by our defined function. Illumination is simulated through an efficient illumination transformation method we develop to transform both synthetic images and real images to share a similar illumination condition.\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=\\linewidth]{images\/approach}\n  \\caption{Overview of our pipeline. Our system (1) takes single-object video as input, extracts the mask for each frame, (2) generates synthetic images of homogeneous objects in cluster, then (3) uses the synthetic images to train an instance segmentation model.}\n    \\label{fig:framework}\n  \\end{center}\n\\end{figure}\n\nTo benchmark our performance, a dataset is built that consists of 200 properly selected and annotated images of homogeneous clustered objects. The extensive experiments show that our approach significantly improves the segmentation performance over baseline methods on the proposed dataset.\n\nThe contributions of this paper are summarized as follows:\n\\begin{enumerate}\n\\item We propose an efficient framework for instance segmentation of HOCs.\nOur proposed method significantly reduces the cost of data collection\nand labeling.\n\\item We propose an efficient method to generate realistic synthetic training data which significantly improves instance segmentation performance.\n\\item We build a dataset consisting of HOC images.  The dataset is used to evaluate our proposed method. The dataset and codes will be published with the paper.\n\\end{enumerate}\n\n\\section{Related Work}\n\nInstance segmentation, which aims to assign class-aware and instance-aware label to each pixel of the image, has witnessed rapid progress recently in computer vision. A series of work reporting good performance has been proposed~\\cite{he2017mask,li2016fully}, in which all of them use some kind of DCNN trained on large datasets~\\cite{mscoco,cityscapes}. However, none of these current representative instance segmentation benchmarks has adequately addressed the task of segmenting HOCs at the instance level. Unlike the classical instance segmentation problem, the testing images for our problem have much more homogeneous instances on average, and exhibit a much higher degree of occlusion. Since currently all of the best models in performing instance segmentation require a large volume of pixel-level annotated images during training, and that no public dataset is available for our specific task, the traditional training framework is inadequate to address our problem of HOC instance segmentation.\n\nOur work is inspired by one-shot learning~\\cite{oneshot}. One-shot learning learns object categories from one training sample. Recently, it has been applied to solve image segmentation problem~\\cite{CaellesMPLCG16,rong2016one}. \\cite{CaellesMPLCG16} use intrinsic continuity to segment objects in a video, but pixel-level annotation for the first frame is required. The work of~\\cite{rong2016one} addresses the problem of segmenting gestures in video frames by learning a probability distribution vector (PDV) that describes object motions. However, these methods still require a number of images with pixel-level annotation.\n\nIn this paper we use single-object images to synthesize images of HOC as our training data. Recently, many studies have been done to use different methods to generate synthetic images to train a DCNN model. In~\\cite{papon2015semantic} realistic synthetic scenes are generated by placing object models at random in a virtual room, and uses the synthetic data to train a DCNN. In the area of object detection, the authors in \\cite{peng2015learning} propose to use 3D CAD models to generate synthetic training images automatically to train an object detector, while in 3D recognition,  the work in \\cite{su2015render} synthesizes images by overlaying images rendered from 3D model on top of real images.  Among these methods using synthetic data generation, \\cite{papon2015semantic} is most similar to ours (i.e., synthesizing images by placing objects on a background sequentially). While they simply place the synthesized objects randomly, we propose to learn how to place synthesized objects based on the knowledge learned from realistic images of HOCs.\n\n\\section{Our Method}\n\\subsection{Data Collection and Preprocessing}\n\\label{Our Method:Data Collecting and Preprocessing}\nWe aim to collect single-object images quickly and efficiently, so videos are the natural choice. We capture a short video for a single object, which is processed to extract individual frames alongside with the corresponding masks. The details are described in the following.\n\n\\subsubsection{One-Shot Video Collection}\nSuppose we want to segment each orange from a cluster of oranges. A common solution is to collect a large number of images of orange clusters in different layouts, followed by annotating all of them and then training an instance segmentation model using the annotated images. In this paper, instead of adopting such conventional data collection, we perform the following: put one single orange at the center of a contrastive background, and take a video of the orange at different angles and positions. Such video typically lasts for about 20 seconds. It takes only a few minutes to acquire the data we require, which significantly reduces the cost of data collection compared to the previous methods. Since in our framework this single short video capturing one single object is all we need for learning, our framework may be regarded as a close kin to \\textbf{one-shot learning} (i.e., learning by looking only once, one video in our case).\n\n\\subsubsection{Video Matting}\nOur goal here is to automatically obtain the mask of each single-object image (i.e., the video frames) in an annotation-free manner. Since the foreground and background are controlled, and the image object is at the center, we apply color and location prior in the first frame to automatically sample seeds of the object and background respectively. Then we take the seeds as input and apply KNN matting~\\cite{Chen:2012:KM} on the video sequence to produce the mask of each frame.  We take into consideration temporal coherence, and instead of producing a hand-drawn trimap (a pre-segmented image consisting of three regions: foreground, background and unknown) for every single frame, classical optical flow algorithm is applied for trimap generation and propagation. Figure~\\ref{fig:data preprocess} shows an example.\n\n\n\\begin{figure}[h]\n\\vspace{1mm}\n  \\begin{center}\n  \t\\includegraphics[width=\\linewidth]{images\/data_process}\n  \t\\caption{Example of our video processing method. For the first frame, seeds of foreground and background are automatically sampled based on color and location priors. Trimaps are interpolated across the video volume using optical flow. Matting technique uses the flowed trimaps to yield high-quality masks of the moving orange.}\n  \t\\label{fig:data preprocess}\n  \\end{center}\n\\end{figure}\n\n\\begin{figure*}[t]\n\\vspace{1mm}\n  \\begin{center}\n  \t\\includegraphics[width=\\linewidth]{images\/synthetic_process2}\n  \t\\caption{Synthetic images generation. $N$ objects are selected and placed on a white background sequentially in an iterative manner (\\textbf{part 1}). We start by placing one object directly on the background in iteration 1. For iteration $k, 1<k \\le N$, we use bayesian optimization (\\textbf{part 3}) to approximate an optimal placement that is guided by our structured likelihood function (\\textbf{part 2}). During bayesian optimization process, $M$ points $\\{\\Omega^1,...,\\Omega^M\\}$ are sampled, while we only plot three due to space constraint. Each point is scored by $f(\\Omega)$. The optimal placement is the point with the highest score. Then we apply illumination transformation (\\textbf{part 4}) on the output of iteration $N$ to obtain the final result.}\n  \t\\label{fig:synthetic overview}\n  \\end{center}\n\\end{figure*}\n\n\n\\subsection{Synthetic Data Generation}\n\\label{Our Method:Synthetic data generation}\nGiven the segmented single-object images from the previous stage, we generate synthetic images in realistic structure and illumination. Realistic images of HOC are generated in an iterative manner, followed by illumination transformation to make synthetic images and real images share a similar illumination condition.\n\n\\subsubsection{Structural Constraint}\n\\label{Our Method:Synthetic data generation:Structural constraints}\n\n\\paragraph{Structured Likelihood Function}\nThe problem of generating realistic images of HOC deals with finding a solution to properly place $N$ objects\n$\\{{O}_{1},{O}_{2},...,{O}_{N}\\}$\nsequentially in order to maximize its likelihood of being a real image of HOC. Let $\\textbf{I}_k$ denote the image we obtain after placing $k$ objects $\\{{O}_{1},...,{O}_{k}\\}, 1 < k \\le N$, and $P(\\cdot)$ denote the likelihood of being a real image. Our problem is equivalent to finding a solution to maximize $P(\\textbf{I}_N)$.\n\nThe image we get after placing $k$ objects, $\\textbf{I}_k$, is determined by three factors: the images we get after placing $k-1$ objects $\\textbf{I}_{k-1}$, the $k$-th chosen object $O_k$ and the way we place $O_k$ in $\\textbf{I}_{k-1}$. We model our object placement by a four-parameter operation set $\\Omega = \\{\\theta, \\gamma, x, y\\}$, in which $\\theta$ denotes the rotation, $\\gamma$ denotes the resize factor, and $x$ and $y$ denote the coordinates of the center of $O_k$ in $\\textbf{I}_{k-1}$. The relation between $\\textbf{I}_k$ and $\\textbf{I}_{k-1}$, $O_k$, $\\Omega_k$ can be written as:\n\\begin{equation}\n\t\\textbf{I}_k=g(\\textbf{I}_{k-1},O_k,\\Omega_k)\n\\end{equation}\n\n\nGiven a sequence of $\\{{O}_{1},{O}_{2},...,{O}_{N}\\}$, our goal is to find a $(\\Omega_1,...,\\Omega_N)$ that maximizes $P(\\textbf{I}_N)$. Since $\\{{O}_{1},{O}_{2},...,{O}_{N}\\}$ is given, finding a solution to maximize $P(\\textbf{I}_N)$ is equivalent to finding an optimal $(\\Omega_1,...,\\Omega_N)$ that maximizes $P(\\textbf{I}_N)$. Since searching for an optimal $(\\Omega_1,...,\\Omega_N)$ is intractable, we simplify the problem by applying greedy search algorithm, that is, in each iteration, the operation that maximizes $P(\\textbf{I}_k)$ is applied. Given $\\textbf{I}_{k-1}$ and $O_k$ $(1<k \\le N)$, we search for an optimal $\\Omega_k$ that maximizes $P(\\textbf{I}_k)$:\n\\begin{equation}\n\\begin{aligned}\n\t\\overline{\\Omega}_k &=\\arg\\max_{{\\Omega}_{k}\\in \\mathcal{D}} P(\\textbf{I}_k)\\\\\n\t&=\\arg\\max_{ {\\Omega}_{ k }\\in \\mathcal{D} } P(g(\\textbf{I}_{k-1},O_k,\\Omega_k))\n\\end{aligned}\n\\end{equation}\nwhere $\\mathcal{D}$ is the feasible set of object placement.  When $\\textbf{I}_{k-1}$ and $O_k$ are given, $P(g(\\textbf{I}_{k-1},O_k,\\Omega_k))$ is the function of $\\Omega_k$. Let $f(\\Omega_k)=P(g(\\textbf{I}_{k-1},O_k,\\Omega_k))$, then\n\\begin{equation}\n\t\\overline{\\Omega}_k=\\arg\\max_{ {\\Omega}_{ k }\\in  \\mathcal{D} } f(\\Omega_k;\\textbf{I}_{k-1},O_k)\n\\end{equation}\nWe aim to find an optimal solution $\\overline{\\Omega}_k$ that maximizes $f(\\Omega_k)$. Since $f(\\Omega_k;\\textbf{I}_{k-1},O_k)=P(I_k)$, $f(\\Omega_k;\\textbf{I}_{k-1},O_k)$ represents the likelihood of $\\textbf{I}_k$ being a real image given $\\textbf{I}_{k-1}$ and $O_k$. We turn to DCNN to judge whether $\\textbf{I}_{k-1}$ is real or otherwise. In detail, we build the classifier by finetuning a pre-trained ImageNet model, where we use object proposals of real images as positive samples, and object proposals of randomly synthesized images (i.e., images synthesized by placing objects at random) as negative samples. We take the tight bounding box~\\cite{su2012crowdsourcing} of $\\textbf{I}_{k}$ as the input of the learnt classifier. The softmax output of the real class is taken as the output of $f(\\Omega_k;\\textbf{I}_{k-1},O_k)$.\n\n\\paragraph{Bayesian optimization framework}\nSince $f(\\Omega_k)$ is a black-box function (i.e., we do not have a specific expression of the function), we cannot optimize $f$ by computing its derivative. We solve this problem by adopting bayesian optimization~\\cite{bayesian}. Bayesian optimization is a powerful strategy for optimization of black-box function.\n\nHere, we use bayesian optimization to optimize our continuous objective function $f(\\Omega_k)$. Bayesian optimization behaves in an iterative manner. At iteration $m$, given the observed point set $D_{m-1}=\\{(\\Omega^1_k,f(\\Omega^1_k)),...,(\\Omega^{m-1}_k,f(\\Omega^{m-1}_k))\\}$, we model a posterior function distribution by the observed points. Then, the acquisition function (i.e., a utility function constructed from the model posterior) is maximized to determine where to sample the next point $(\\Omega^{m}_k,f(\\Omega^{m}_k))$. $(\\Omega^{m}_k,f(\\Omega^{m}_k))$ is collected and the process is repeated. Iteration ends when $m=M$ ($M$ is a user-defined parameter). For more detail on optimization framework please refer to \\cite{bayesian}.\n\n\n\\paragraph{Overall algorithm}\nWe generate our realistic HOC images by placing objects sequentially in an iterative manner (see Figure \\ref{fig:synthetic overview}). We start by directly placing one object on a white background in iteration 1. In iteration $k$ ($1<k\\le N$), given $\\textbf{I}_{k-1}$ and $O_k$, bayesian optimization is applied to approximate an optimal $(\\theta_k,\\gamma_k,x_k,y_k)$ that maximizes the structured likelihood function $f(\\Omega_k;\\textbf{I}_{k-1},O_k)$. Iteration ends when $k=N$. $N$ is a prior given by user. User provides a range of how many instances per image in his (or her) case, thus, $N$ is a random integral in this range. For example, in our dataset, $N$ is a random integral between 10 and 30.\n\n\\subsubsection{Illumination Transformation}\n\\label{Illumination transformation}\nTo simulate lighting condition, we develop an efficient method to transform the synthetic and the real image so that they share a similar illumination condition.\n\nWe are inspired by~\\cite{leong2003correction} where an uneven illumination correction method was proposed using Gaussian smoothing. We first convert both the synthetic image and the real image from RGB color space to HSV space, where $V$ represents illumination information. Then, we implement detail removing, by using a large kernel Gaussian smoothing on both images to model general illumination condition. This general illumination condition (denoted as $blur(V_{real})$) is imposed on both real and synthetic images (after mean subtraction) to unify illumination.\n\\begin{equation}\n\tV_{syn}=V_{syn}-\\mbox{mean}(V_{syn})+blur(V_{real})\n\\end{equation}\n\\begin{equation}\n\tV_{real}=V_{real}-\\mbox{mean}(V_{real})+blur(V_{real})\n\\label{eq:illumination}\n\\end{equation}\nHere, $V_{syn}$ and $V_{real}$ respectively denote the V channel of the synthetic image and the real image, and $\\mbox{mean}(\\cdot)$ denotes the global mean value of 2-D matrix.\n\nFinally, we convert the synthetic image and real image from HSV space back to RGB space. Figure~\\ref{fig:illumination_result} shows that our illumination transformation algorithm makes real and synthetic images share the similar illumination condition.\n\n\\begin{figure}[!htp]\n  \\begin{center}\n  \t\\includegraphics[width=\\linewidth]{images\/illumination_transfer}\n  \\caption{Examples of our illumination transformation method. (a) real image before illumination transformation, (b) synthetic image before illumination transformation, (c) real image after illumination transformation, (d) synthetic image after illumination transformation.}\n  \\label{fig:illumination_result}\n  \\end{center}\n\\end{figure}\n\n\\vspace{-5mm} \n\\subsection{End-to-End Training}\nFinally, we use the synthetic images together with their corresponding annotations generated by the above method as our training data, and train an instance segmentation model in an end-to-end manner.\n\nIn this paper, we adapt Mask R-CNN~\\cite{he2017mask} to handle our task. We train the model using our synthetic data. In testing phase, the model works on real image. Before the testing image is fed to the model, we adjust its illumination using Eq. \\ref{eq:illumination} to ensure its illumination condition is similar to training images.\n\n\\begin{table*}[htp]\n\\centering\n\\caption{Results on mAP$^r$@0.5 on our dataset. All numbers are percentages \\%.}\n\\label{table1}\n\\tabcolsep=0.11cm\n\\begin{tabular}{c|cccccccccc|c}\n&badminton&battery&clothespin&grape&milk&hexagon nut&orange&ping pong&tissue&wing nut&mAP \\\\ \\hline\n\\textbf{Single} &12.1&23.2&4.4&19.3&8.2&17.5&17.6&14.2&13.1&21.1&15.1 \\\\\n\\textbf{Random} &40.6&50.9&38.4&50.8&26.7&52.9&63.8&67.2&83.9&32.9&50.8 \\\\\n\\textbf{Random+illumination} &44.6&48.7&34.3&41.6&26.0&46.3&54.9&64.2&68.9&39.4&46.9 \\\\\n\\textbf{Random+structure} &34.2&39.3&52.6&\\textbf{72.7}&31.3&62.8&\\textbf{90.3}&\\textbf{81.7}&\\textbf{90.7}&23.7&57.9 \\\\\n\n\\textbf{Ours} &\\textbf{53.0}&\\textbf{69.5}&\\textbf{67.7}&72.5&\\textbf{52.6}&\\textbf{73.6}&90.0&81.2&90.4&\\textbf{48.4}&\\textbf{69.9}\n\\end{tabular}\n\\centering\n\\caption{Results on mAP$^r$@[0.5:0.95] on our dataset. All numbers are percentages \\%.}\n\\label{table2}\n\\tabcolsep=0.11cm\n\\begin{tabular}{c|cccccccccc|c}\n&badminton&battery&clothespin&grape&milk&hexagon nut&orange&ping pong&tissue&wing nut&mAP \\\\\n\\hline\n\\textbf{Single} &8.7&21.2&2.0&16.8&5.0&15.2&16.0&11.8&9.9&18.8&12.5 \\\\\n\\textbf{Random} &33.1&44.7&29.3&44.8&20.9&43.4&56.3&59.5&68.4&27.7&42.8 \\\\\n\\textbf{Random+illumination} &36.5&43.2&28.4&37.2&20.8&38.3&50.4&56.4&56.8&35.5&40.4 \\\\\n\\textbf{Random+structure} &29.8&36.5&36.7&\\textbf{66.5}\n&22.0&45.2&83.8&\\textbf{76.4}&\\textbf{80.4}&20.7&49.8 \\\\\n\\textbf{Ours}  &\\textbf{44.2}&\\textbf{60.3}&\\textbf{46.2}&65.4&\\textbf{41.3}&\\textbf{54.8}&\\textbf{84.0}&75.6&\\textbf{80.4}&\\textbf{39.3}&\\textbf{59.2} \\\\\n\n\\end{tabular}\n\\end{table*}\n\n\n\\section{Experimental Evaluation}\n\\subsection{Dataset}\nOur task is totally new and none of current datasets is suitable for evaluating this problem. Our dataset is designed for benchmarking methods for instance segmentation of HOCs. Our dataset contains 10 types of objects; some are solid (\\textit{e.g.}, ping pong ball) while some are fluid (\\textit{e.g.}, bagged milk). For each type of object, 20 images are taken under different illumination conditions. We annotate all of the instances with less than 80\\% occlusion. Our dataset has $3,669$ instances in total, each image has 18.3 instances on average. We do not build an extremely larger dataset due to budget constraints, but it is sufficient to benchmark this problem.\n\n\\subsection{Implementation details}\n\n\\paragraph{Structured likelihood function} In the training step, we construct the likelihood function by finetuning the AlexNet~\\cite{alexnet} pre\\-trained model. For each type of object, we use \\cite{edgeBoxes} to generate object proposals for both real images and randomly synthesized images (i.e., images synthesized by placing objects at random), and filter out proposals whose area are less than 0.01 of the original image. Then, we randomly choose 30,000 proposals from real images as positive examples, and 30,000 proposals from synthetic images as negative examples.  We finetune the model for 30,000 iterations with a learning rate of 0.001, minibatch size of 32, momentum of 0.9 and weight decay of 0.01.\n\nIn the inference step, given $\\textbf{I}_{k-1}$, $O_k$ and $(\\theta_k,\\gamma_k,x_k,y_k)$, we rotate $O_k$ by $\\theta_k$ degrees, resize it by a factor of $\\gamma_k$, and place the center of $O_k$ after rotation and resizing at location $(x_k,y_k)$. When we place $O_k$, it can only be overlapped by previous objects. Then the tight bounding box of the new image we get after placing $O_k$ is taken as input of the finetuned model. The softmax output of the real class is taken as the output of our structured likelihood function.\n\n\\paragraph{Instance segmentation model} We adopt the Mask R-CNN~\\cite{he2017mask} as our instance segmentation model. We finetune the VGG-16 model~\\cite{vgg16} using the synthetic data generated by our method for 20,000 iterations with a learning rate of 0.002, minibatch size of 4, momentum of 0.9 and weight decay of 0.0001 on four Titan X GPU. Other settings are identical with~\\cite{he2017mask}.\n\n\\subsection{Results and analysis}\nWe follow the protocols used in \\cite{Hariharan2014Simultaneous,he2017mask,li2016fully} for evaluating our proposed method for solving the problem of instance segmentation of HOCs. We evaluate using standard COCO evaluation metric, mAP$^r$@[0.5:0.95], as well as traditional mAP$^r$@0.5 metric. Table \\ref{table1} and Table \\ref{table2} respectively show the results on the mAP$^r$@0.5 and the mAP$^r$@[0.5:0.95] metrics on our dataset. \n\n\\begin{enumerate}\n\\item \\textbf{Single} is our baseline method, which uses outputs from Section \\ref{Our Method:Data Collecting and Preprocessing} (i.e., images containing one single object and corresponding annotation) as training data to train the Mask R-CNN~\\cite{he2017mask}. The baseline method reports mAP$^r$@0.5 of $15.1\\%$ and mAP$^r$@[0.5:0.95] of $12.5\\%$. Since the training images of \\textbf{Single} only contain a single object each, while testing realistic images contain multiple instances, the poor result is within our expectation.\n\n\\item \\textbf{Random} is the result of training with randomly synthesized images (i.e., images synthesized by placing objects at random), which achieves mAP$^r$@0.5 of $50.8\\%$ and mAP$^r$@[0.5:0.95] of $42.8\\%$. This is a large improvement over \\textbf{Single}, although \\textbf{Single} is not designed for the task, while the comparison is indicative it is somewhat unfair.\n\n\\item \\textbf{Random+illumination} applies the illumination transformation method in Section \\ref{Illumination transformation} to transform synthetic images generated in \\textbf{Random} and real images to share similar illumination condition. It uses synthetic images after illumination transformation as training data to train Mask R-CNN~\\cite{he2017mask} model. The result shows a slight decrease in performance using the mAP$^r$@0.5 metric (of about $3.9\\%$) and mAP$^r$@[0.5:0.95] metric (of about $2.4\\%$) over \\textbf{Random}, indicating that illumination transformation does not help when training with randomly synthesized images.\n\n\\item \\textbf{Random+structure} uses the method proposed in Section~\\ref{Our Method:Synthetic data generation:Structural constraints} to generate structurally realistic synthetic images {\\em without} illumination transformation as training data. The result shows a $7.1\\%$ improvement on mAP$^r$@0.5 and a $7.0\\%$ improvement on mAP$^r$@[0.5:0.95] over \\textbf{Random}. The significant improvement demonstrates the efficacy of our method which synthesizes realistic HOC images.\n\n\\item \\textbf{Ours} is our method which trains the model using synthetic images generated under both the structural constraint and with illumination transformation. Our method reports $12.0\\%$ higher in mAP$^r$@0.5 and $9.4\\%$ higher in mAP$^r$@[0.5:0.95] over \\textbf{Random+structure}. While illumination transformation does not show effectiveness for training with randomly synthesized images, it indeed significantly improves performance for training with structurally realistic synthetic images.\n\\end{enumerate}\n\\begin{figure}[!htb]\n  \\begin{center}\n  \t\\includegraphics[width=\\linewidth]{images\/expe_result}\n  \t\\caption{Improvement in mAP$^r$ over \\textbf{Single} for a variety of overlap thresholds. Our method reports improvements on all thresholds.}\n  \t\\label{fig:experiment result}\n  \\end{center}\n\\end{figure}\n\\vspace{-3mm}\n \\begin{figure}[htp]\n  \\begin{center}\n  \t\\includegraphics[width=0.9\\linewidth]{images\/result}\n  \\end{center}\n   \\caption{Qualitative segmentation results on our dataset. Each color denotes one instance. Our method can segment more instances than other baseline methods.}\n  \\label{fig:qualitative result}\n\\end{figure}\n\nFollowing the evaluation in~\\cite{Hariharan2014Simultaneous}, Figure~\\ref{fig:experiment result} plots the improvements on mAP$^r$ over \\textbf{Single} at 9 different IoU thresholds. As shown in Figure \\ref{fig:experiment result}, the performance of our proposed method is superior to other methods at every threshold, which demonstrates the robustness of our algorithm. We also show some qualitative results in Figure \\ref{fig:qualitative result}, where we can observe that \\textbf{Random} (trained with images synthesized by placing objects at random) can only segment objects that are not occluded by other objects, while our method can segment out a much larger number of partially occluded instances. This demonstrates that our image synthesis method is able to learn the occlusion pattern of real images. Besides, as shown in Table \\ref{table1} and Table \\ref{table2}, the significant gap between \\textbf{Random+structure} and \\textbf{Ours} ($12.0\\%$ in mAP$^r$@0.5 and $9.4\\%$ in mAP$^r$@[0.5:0.95]) indicates the effectiveness of our illumination transformation algorithm.\n\n\\subsection{Comparison of training with existing dataset}\nTo further demonstrate the efficacy of our method which synthesizes HOC images is to compare our model with one that is trained on equal number of real HOC images with instance segmentation annotation. However, due to our limited budget, we cannot afford to build such a large dataset of HOC images with thorough annotation.  Alternatively, we validate the effectiveness of our proposed method by comparing with state-of-the-art an instance segmentation model trained on existing dataset. Since our proposed dataset of HOCs and the COCO dataset have \\textit{orange} class in common, we use Mask R-CNN trained on COCO to directly test for object categories in common (i.e., \\textit{orange}) on our dataset. Mask R-CNN trained on COCO achieves $86.4\\%$ mAP$^r$@0.5 and $72.3\\%$ mAP$^r$@[0.5:0.95] on \\textit{orange} class in our dataset while our method achieves $90.0\\%$ mAP$^r$@0.5 and $84.0\\%$ mAP$^r$@[0.5:0.95], which significantly outperforms the baseline method. The reason could be the baseline model does not train on large-scale HOC images, which are difficult to obtain due to expensive human annotation. By contrast, our proposed method can generate high-quality annotated HOC images with little effort which helps to achieve better performance.  \nFigure~\\ref{fig:qualitative result 2} shows some qualitative results. \n\n\\begin{figure}[!htb]\n  \\begin{center}\n  \t\\includegraphics[width=\\linewidth]{images\/result2}\n  \t\\caption{Comparison of COCO pre-trained model and our method. Our method is better at handling occlusion.}\n  \t\\label{fig:qualitative result 2}\n  \\end{center}\n\\end{figure}\n\n\\vspace{-5mm} \n\\section{Conclusion and Future Work}\nOur work is the first attempt in tapping into learning how to segment homogeneous objects in clusters via single-object samples. The key to our proposed framework is an image synthesis technique based on the intuition that images of homogeneous clustered objects can be synthesized based on priors from real images. We build a dataset consisting of 200 carefully selected and annotated images of homogeneous object of clusters to evaluate our algorithm. While it indeed shows promises, the problem is far from solved. Our current framework may show poor performance on deformable objects (\\textit{e.g.}, human) since the collected single-object samples may not cover adequately all the possible appearances of the object, and training with synthetic images generated by these samples makes DCNN prone to overfitting. Nonetheless, given its odds in its present form, we believe this is an interesting work that will spawn future work and useful applications. We plan to produce a larger dataset in the future.\n\n\\bibliographystyle{named}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}