diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzejbs" "b/data_all_eng_slimpj/shuffled/split2/finalzzejbs" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzejbs" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nGame Theory has nowadays become a frequent research tool to approach Network Science. Ideas like Nash equilibrium, price of anarchy and stability, cost allocation, transferable utilities, coalition formation, core, mechanism design, to name a few, are commonly used not only to understand the behavior of complex network phenomena but also to control and efficiently design such networks. See, for example, Chapter 10 in \\cite{MavronicolasSurvay} for a recent survey. Under the above setting, usually there exists an underlying \\emph{network} in which some \\emph{agents} (or otherwise stated, players, nodes, firms, etc.) are \\emph{linked} according to certain relationships and \\emph{compete} following specific rules in order to \\emph{optimize} a self-interested utility (price, bandwidth, network flow, etc.). However, as many of the real life problems map to large and complex networks \\cite{DorogovtsevMendes}, our bounded computational ability and known intractability results \\cite{SandhlomLesser,Sandhlometal} restrict our potential to fully unfold and understand coalition formation in networks. \n\nDue to this complex environment network agents are not able to base their behavior on the \"whole network status\" but have to follow certain \\emph{beliefs} as to how it is in their strategic interest to act. This behavior has gained a lot of attention and such agents are extensively used in Network science \\cite{SandhlomLesser,Sandhlometal,Sundararajan,MarginalContribution,Rubinstein,Survey_Jackson,AGT}. These agents constitute the main interest of this paper. To this end, we quantify and characterize the set of beliefs that support cooperation of such agents in a network. In order to illustrate our ideas, we force the agents to compete under a Cournot competition. We also agree that in order to conceive of a group of fully cooperating agents we will use a complete graph (clique). We suppose that at the begining of the competition all the agents, say $N$, are fully cooperating and form a large clique or in game theoretic terms, the \\emph{grand coalition}. We then try to answer the following\\\\\n\n\\begin{centering}\\fbox {\n\\parbox{\\linewidth}{\n\\textbf{Basic Question}: \\textit{Which are the coalitional beliefs of the agents that support network stability?} \n}}\n\\end{centering}\\\\\n\nBy network stability we mean the situation in which every agent cooperates with every other agent and no relationship between any two is broken (thus no edge is missing from the graph - network).\n\nIn the rest of the paper we proceed as follows: In section 2 we present our model, in sections 3, 4 we present our results and in section 5 we conclude.\n\n\n\\section{The model}\n\nConsider a set of agents $N=\\{1,2,\\cdots,n\\}$ which are the nodes of a graph $G$. Suppose that these agents are engaged in a cooperative game which we will soon define. Whenever two agents $l,m$, $1 \\leq l,m \\leq n$, cooperate, we draw an edge between\nthe corresponding nodes in $G$. The graph describing all the cooperations between all the agents in $N$ forms the network. It is clear that when every agent cooperates with every other, then the network is modeled by the complete graph (clique) of $n$ nodes. In this case we say that the\nagents form the grand coalition. It is also clear that, unless otherwise stated, every non empty subset of agents $S \\subset N$ that fully cooperates also forms a complete subgraph, of $G$, with $s=|S|$ nodes. Let ${\\cal{P}}_j$ denote a partition of the \\emph{outsiders}, $N \\setminus S$, such that $j \\in \\{1,2,\\cdots,n-s\\}$ and ${\\cal{P}}_j=\\{S_{1},S_{2},\\cdots,S_j\\}$, where $\\bigcup\\limits_{1 \\leq i\\leq j} S_i={\\cal{P}}_j$ and $S_l \\cap S_m = \\emptyset$, $\\forall \\, l \\neq m$, $1 \\leq l,m \\leq j$. We call ${\\cal{P}}_j$, a \\emph{coalition structure} with $j$ coalitions. Let $s_r=|S_r|$, $r=1,2,\\cdots,j$, denote the number of agents in $S_r$. \n\nThe $N$ agents are put together in order to cooperate under a Cournot competition, in which everyone must decide how much \"product\" to produce, all decisions have to be made at the same time, and each one must take its competitors into account. Each agent $i \\in N$, produces with the cost function $C(q_i)=c \\cdot q_i$ and with price function $P_i=a-q_i-\\gamma\\sum\\limits_{j=1,j \\neq i}^n q_j$, where $q_i$ is the quantity produced. The form of $P_i$ implies that we use product differentiation in the Cournot competition where $a > 0$ is a constant, $c$, $0 \\frac{-1}{n-1}$, then there exist real numbers $\\lambda_0,\\cdots,\\lambda_j$ such that the worth of $S$ is \n\n\\begin{equation}\n\\label{value final} v^{{\\cal{P}}_j}_j(S)=s(1+\\gamma s -\\gamma)\\left(\\frac{a-c}{C_0}\\right) ^2\n\\end{equation}\n\nwhere ${C_0=2[1+\\gamma(s-1)]+\\gamma\n\\sum\\limits_{k=1}^j\\frac{s_k}{\\lambda_k}\\lambda_0}$.\n\n\\end{theorem}\n\n\\noindent \\textbf{Proof:} For the proof we will need the condition $K : \\gamma > \\frac{-1}{n-1}$ which guarantees the existence of Cournot equilibrium when $\\gamma \\in (-1,0)$. The first-order conditions for the optimal quantities of the members of $S$ read\n\n\\begin{equation}\n2q_i=a-c-2\\gamma\\sum\\limits_{l\\in S,l\\neq i} q_l-\\gamma\\sum\\limits_{k=s+1}^nq_k, \\hspace{0.2cm} i\\in S\n\\nonumber\\end{equation}\n\nand for each $S_k$, $k=1,2,...,j$, \n\n\\begin{equation}\n2q_h=a-c-2\\gamma\\sum\\limits_{t\\in S_k,t\\neq h} q_t-\\gamma\\sum\\limits_{r\\notin S_k}q_r, \\hspace{0.2cm} h\\in S_k\n\\nonumber\\end{equation}\n\nNotice that due to (intra-coalitional) symmetry, the following must hold\n\n\\begin{equation}\n\\label{multiplicities}\n \\begin{pmatrix}\n q_1=q_2=\\cdots=q_s\\equiv y_0 \\\\\n q_{s+1}=q_{s+2}=\\cdots=q_{s+s_1}\\equiv y_1 \\\\\n \\vdots \\\\\n q_{i^*+1}=\\cdots=q_n\\equiv y_j\n \\end{pmatrix}\n\\end{equation}\n\n\nwhere $i^*=s+\\sum\\limits_{w=1}^{j-1}s_w$. Letting{\\footnote{From now on we will use both $s$ and $s_0$ to\ndenote the number of agents in $S$.}} $s=s_0,$ the system to solve for is\n\n\\begin{equation}\n\\label{system}\n \\begin{pmatrix}\n 2y_0=a-c-2\\gamma(s_0-1)y_0-\\gamma(s_1y_1+s_2y_2+\\cdots+s_jy_j) \\\\\n 2y_1=a-c-2\\gamma(s_1-1)y_1-\\gamma(s_0y_0+s_2y_2+\\cdots+s_jy_j) \\\\\n \\vdots \\\\\n 2y_j=a-c-2\\gamma(s_j-1)y_j-\\gamma(s_0y_0+s_1y_1+\\cdots+s_{j-1}y_{j-1})\n \\end{pmatrix}\n\\end{equation}\n\nConsidering $i$th and $k$th equations of (\\ref{system}) we have\n\n\\begin{align}\n2(y_i-y_k)=\\gamma (s_k-2)y_k-\\gamma(s_i-2)y_i \\nonumber \\Rightarrow y_k=\\frac{y_i(2+\\gamma s_i-2\\gamma)}{\\gamma s_k-2\\gamma+2} \\\\\n\\label{sumjskyk}\\Rightarrow \\sum\\limits_{k=0,k\\neq i}^js_ky_k=\\sum\\limits_{k=0,k\\neq i}^j\\frac{s_ky_i(2+\\gamma s_i-2\\gamma)}{\\gamma s_k-2\\gamma+2}\n\\end{align}\n\nMultiplying by $\\gamma$, using $k$th equation of (\\ref{system}) and the first equation of (\\ref{sumjskyk}) we get, \n\n\\begin{equation}\n\\label{yyii} y_i=\\frac{a-c}{2[1+\\gamma(s_i-1)]+\\gamma A_i}, \\hspace{0.2cm} i=0,1,...,j\n\\end{equation}\n\nwhere ${A_i=\\sum\\limits_{k=0,k\\neq i}^j\\frac{s_k(2+\\gamma s_i-2\\gamma)}{\\gamma s_k-2\\gamma+2}}$. Observe\nthat $A_i>0$ (due to condition $K$).\n\nLet us now compute the worth of $S$. Recall that\n\n\\begin{align}\nv_j^{{\\cal{P}}_j}(S) &=\\sum_{i\\in S}(a-q_i-\\gamma(\\sum\\limits_{l\\in S,l\\neq i}q_l+\\sum\\limits_{r=s+1}^nq_r)-c)q_i \\nonumber \\\\\n\\label{compu0} &=s_0(a-y_0-\\gamma(s_0-1)y_0-\\gamma\\sum\\limits_{k=1}^js_ky_k-c)y_0\n\\end{align}\n\nDefine $\\lambda_k=\\gamma s_k-2\\gamma+2,$ $k=0,1,...,j$. By using (\\ref{sumjskyk}) and (\\ref{yyii}) for $i=0$, and the multiplicities of $q$'s for each $y_i$ in (\\ref{multiplicities}) we get the following two equations\n\n\\begin{align}\n&\\label{sumjskykallo} \\sum\\limits_{k=1}^j s_k y_k = \\sum\\limits_{r=s+1}^n q_r = \\sum\\limits_{k=1}^j \\frac{s_k}{\\lambda_k}y_0\\lambda_0=\\sum\\limits_{k=1}^j \\frac{s_k}{\\lambda_k}\\frac{a-c}{C_0}\\lambda_0\\\\\n&\\label{sumy0} \\sum\\limits_{l \\in S \\setminus \\{0\\}} y_0=\\sum \\limits_{l \\in S \\setminus \\{0\\}}q_l=(s_0-1)\\frac{a-c}{C_0}\n\\end{align}\n\nwhere ${C_0=2[1+\\gamma(s_0-1)]+\\gamma \\sum\\limits_{k=1}^j\\frac{s_k}{\\lambda_k}\\lambda_0}$. Using (\\ref{compu0},\\ref{sumjskykallo},\\ref{sumy0}) we get the theorem. \\qed\n\n~\\\n\nUsing Theorem 1 we have the following\\\\\n\n\\begin{corollary} When the grand coalition forms, the total worth of the network is\n\\begin{equation}\nv(N)=\\frac{n(a-c)^2}{4(1+\\gamma(n-1))} \\; \\text{.}\n\\nonumber\\end{equation}\n\\end{corollary}\n\nCorollary 1 can be very useful in the following sense. Since, upon cooperation, the agents split the profit equally, when the grand coalition forms each of them will receive $\\frac{v(N)}{n}$. This means that no agent would accept deviating from the grand coalition and join a smaller one if he did not first ensure that he would receive at least more than $\\frac{v(N)}{n}$ with this move. So a first approach to the Basic Question is the following\\\\\n\n\\begin{centering}\\fbox {\n\\parbox{\\linewidth}{\n\\textbf{Answer 1:} \\textit{If no agent assures a profit of at least more than $\\frac{(a-c)^2}{4(1+\\gamma(n-1))}$, then no agent likes to deviate and thus the grand coalition forms and the network does not break.}}}\n\\end{centering}\\\\\n\nWhen the grand coalition forms it is said that the core of the game is non-empty and since we would like the whole network to be stable and not to break into disconnected components, we have to characterize the set of coalitional beliefs that support core non-emptiness. So it is of special interest to study the core of the above game. This is the subject of the following section.\n\n\\section{The core}\n\nBefore saying anything about the core of the game, let us examine the impact of the distribution of $(s_1,\\cdots,s_j)$ on $v^{{\\cal{P}}_j}_j(S)$. This is useful in case agents in $S$ need to follow a trend in their reasoning. When $\\gamma \\in (0,1)$ we have the following\\\\\n\n\\begin{proposition} For a fixed $j$, the optima of $v^{{\\cal{P}}_j}_j(S)$, w.r.t. ($s_1,\\cdots,s_j$), are\n\n\\noindent (i) min when $s_i=\\frac{n-s}{j},1\\leq i \\leq j$\n\n\\noindent (ii) max when $s_i=1$, in all but one value $h$, and $s_h=n-s-(j-1)$, $h \\in \\{1,\\cdots,j\\}$ .\n\n\\end{proposition}\n\n\\noindent \\textbf{Proof:} From (\\ref{value final}), $v^{{\\cal{P}}_j}_j(S)$ is min when $C_0$ is max which happens when $\\sum\\limits_{k=1}^j \\frac{s_k}{\\lambda_k}$ is max which happens when $s_i=\\frac{n-s}{j},1\\leq i \\leq j$, because\n\n\\begin{align}\n\\sum\\limits_{k=1}^j\\frac{1}{\\gamma s_k-2\\gamma+2}=\n&\\sum\\limits_{k=1}^{j-1}\\frac{1}{\\gamma s_k-2\\gamma+2}+\\frac{1}{\\gamma s_j-2\\gamma+2} \\nonumber \\\\\n=\\label{max1} &\\sum\\limits_{k=1}^{j-1}\\frac{1}{\\gamma s_k-2\\gamma+2}+\\frac{1}{\\gamma (n-s-\\sum\\limits_{k=1}^{j-1}s_k)-2\\gamma+2}\n\\end{align}\n\nLet $h(s_1,s_2,\\cdots,s_{j-1})$ denote the right hand side of (\\ref{max1}). To find the vector of $(s_1,...,s_{j-1})$ that minimizes (\\ref{max1}) we need to solve the system\n\n\\begin{equation}\n\\frac{\\partial{h(s_1,s_2,...,s_{j-1})}}{\\partial{s_k}}=0, \\hspace{0.2cm}k=1,2,\\cdots,j-1\n\\end{equation}\n\nStraightforward calculations lead to the solution $s_1=\\cdots=s_{j-1}=\\frac{n-s}{j}$ (and hence $s_j=\\frac{n-s}{j}$), so we have (i). In a similar way we can also prove (ii). \\qed\n\n~\\\n\nProposition 1 gives the deviating agents a first belief about their potential worth\\\\\n\n\\begin{centering}\\fbox {\n\\parbox{\\linewidth}{\n\\textbf{Answer 2:} If the number of coalitions that outsiders form is fixed, say $j$, then the worth of the deviating agents, $s$, is minimized (hence the network is less likely to break) when the $n-s$ agents split equally among the $j$ coalitions; and the worth of $s$ is maximized when $j-1$ coalitions have one member and one coalition has $n-s-(j-1)$ members.}}\n\\end{centering}\\\\\n\nThe above answer gives in a way an incentive to the deviating agents, meaning that a coalition of agents has a lower incentive to deviate when its opponents are split equally among $j$ (see Fig. \\ref{fig:MaxMin}). \n\nAnother look at how agents in $S$ might reason is given by the following\n\n\\begin{proposition} When $\\gamma = 1$, if a coalition $S$ believes that the $n-s$ outsiders will form more than $j^*(n,s)=2(\\sqrt{\\frac{n}{s}}-1)$ coalitions, then $S$ will not deviate from the grand coalition.\n\\end{proposition}\n\n\\noindent \\textbf{Proof:} Before giving the proof we make the following observation. As said earlier, since the agents upon cooperation split the profit equally, they know that when the network does not break they are going to receive $\\frac{v(N)}{n}$ each. On the other hand, agents belonging to a deviating coalition $S$, after calculating their worth, know that upon deviation they are going to receive $\\frac{v(S)}{s}$ each. So if $\\frac{v(N)}{n} \\geq \\frac{v(S)}{s}$, then no agent in $S$ would like to deviate. Now returning to the proof, from (\\ref{yyii}) for $\\gamma=1$ we get $y_i=\\frac{a-c}{s_i(2+j)}$, $0 \\leq i \\leq j$. Hence the value of $S$ is independent of the size of coalitions of the outsiders but depends only on their number, i.e. $v^{{\\cal{P}}_j}_j(S)=\\left(\\frac{a-c}{j+2}\\right)^2$. The core is then non empty if for all $S$, the inequality $\\frac{v(N)}{n}\\geq \\frac{v(S)}{s}$ holds, or if $\\frac{1}{4n} \\geq \\frac{1}{s(j+2)^2}$. The last inequality holds iff $j>j^*(n,s)=2(\\sqrt{\\frac{n}{s}}-1)$. \\qed\n\n~\\\n\nProposition 2 implicitly provides agents with another belief in the special case where $\\gamma=1$\\\\\n\n\\begin{centering}\\fbox {\n\\parbox{\\linewidth}{\n\\textbf{Answer 3:} \\textit{In the special case where $\\gamma=1$, agents do not have to worry about what the rest will do, meaning that they do not have to worry about how the rest are going to split. They only have to worry about how many of them there are, i.e. their total number.}\n}\n}\n\\end{centering}\\\\\n\nA more thorough look at how agents in $S$ might think is given by the following\n\n\\begin{theorem} Assume $\\gamma \\in (0,1]$. There exists a $j^*(n,s,\\gamma)$, such that the core is non-empty for all $j>j^*(n,s,\\gamma)$ and all corresponding ${\\cal{P}}_j$.\n\\end{theorem}\n\n\\noindent \\textbf{Proof:} We seek conditions for the inequality $\\frac{v(N)}{n}\\geq \\frac{v_j^{{\\cal{P}}_j}(S)}{s}$ to hold:\n\n\\begin{align}\n&C_0 \\geq 2\\sqrt{(1+\\gamma n-\\gamma)(1+\\gamma s-\\gamma)} \\nonumber \\\\\n\\text{or} \\; &2(1+\\gamma s-\\gamma)+\\gamma \\sum_{k=1}^j \\frac{s_k}{\\lambda_k}\\lambda_0 \\geq 2\\sqrt{(1+\\gamma n-\\gamma)(1+\\gamma s-\\gamma)} \\nonumber\\\\ \n\\text{or} \\; &\\sum_{k=1}^j \\frac{\\gamma s_k}{\\gamma s_k+2(1-\\gamma)} \\geq 2\\left(\\sqrt{\\frac{1+\\gamma n-\\gamma}{1+\\gamma s-\\gamma}}-1\\right)\/(1+\\frac{1-\\gamma}{1+\\gamma s -\\gamma}) \\equiv \\zeta \\nonumber\n\\end{align} \n\nSince $0 < \\gamma \\leq 1$ the sum can not exceed $j$, so for $j^*(n,s,\\gamma)=\\zeta$ the core is non empty for all $j>j^*(n,s,\\gamma)$.\n\nWe now have to show that $n-s > \\zeta$. But it can be easily proven by induction on $s, 01$ and $\\gamma \\in (0,1]$, $\\zeta < n-s$. And since for every $s 0$, we have the theorem. \\qed\n\n~\\\n\nTheorem 2 gives, in a computational efficient way, a belief as to how agents in $S$ might think when considering deviation from the Network and thus constitutes another answer to our Basic Question (see also Fig. \\ref{fig:MaxMin}):\\\\\n\n\\begin{centering}\\fbox {\n\\parbox{\\linewidth}{\n\\textbf{Answer 4:} \\textit{When $\\gamma \\in (0,1]$, the network does not break provided that agents in $S$ believe the outsiders will form a structure with a sufficiently large number of coalitions ($>j^*$)}.}\n}\n\\end{centering}\\\\\n\nUsing the formula for $j^*(n,s,\\gamma)$ we can prove proposition 2 in a different way by plugging $\\gamma=1$ and get $j^*(n,s,1)=\\zeta|_{\\gamma=1}=2(\\sqrt{\\frac{n}{s}}-1)$ as expected. \n\nAnother perspective of the beliefs of agents in $S$ is given by the following\n\n\\begin{theorem}\nIf $\\gamma \\in (-1,0)$ and condition $K$ holds, then the core is non-empty for all $j=1,2,\\cdots,n-s$ and all ${\\cal{P}}_j$.\n\\end{theorem}\n\n\\noindent \\textbf{Proof:} The inequality $\\frac{v(N)}{n}\\geq \\frac{v_j^{{\\cal{P}}_j}(S)}{s}$ holds iff\n\n\\begin{equation}\n2+\\gamma(1+\\frac{1-\\gamma}{1+\\gamma s -\\gamma}) \\sum_{k=1}^j \\frac{s_k}{\\lambda_k} \\geq 2\\frac{{\\sqrt{(1+\\gamma n-\\gamma)(1+\\gamma s-\\gamma)}}}{{(1+\\gamma s-\\gamma)}}\n\\nonumber\\end{equation}\n\nor using the definitions $\\sigma \\equiv 1+\\gamma(s-1)$, $\\nu \\equiv 1+\\gamma(n-1)$, it holds iff\n\n\\begin{align}\n2+\\frac{\\gamma \\lambda_0}{\\sigma} \\sum_{k=1}^j \\frac{s_k}{\\lambda_k} \\geq 2\\sqrt{\\frac{\\nu}{\\sigma}}\n\\; \\; \\; \\text{or} \\; \\; \\;\n\\gamma \\geq \\frac{2(\\sqrt{\\nu}-\\sqrt{\\sigma})\\sqrt{\\sigma}}{\\lambda_0\\sum_{k=1}^j \\frac{s_k}{\\lambda_k}}\n\\nonumber\\end{align}\n\nSince $\\gamma > \\frac{-1}{n-1}$, it suffices to show that\n\n\\begin{align}\n\\frac{-1}{n-1}>\\frac{2(\\sqrt{\\nu}-\\sqrt{\\sigma})\\sqrt{\\sigma}}{\\lambda_0\\sum_{k=1}^j \\frac{s_k}{\\lambda_k}}\n\\; \\; \\; \\text{or} \\; \\; \\;\n\\frac{2(n-1)(\\sigma-\\sqrt{\\sigma \\nu})}{\\lambda_0} > \\sum \\limits_{k=1}^{j} \\frac{s_k}{\\lambda_k}\n\\nonumber\\end{align}\n\nThe maximum value of the term ${\\sum \\limits_{k=1}^{j} \\frac{s_k}{\\lambda_k}}$ appears at\n$s_h=1$ for all but one $h$ (remember that now $\\gamma<0$). It suffices then to show that the left hand side of the last inequality is greater than the corresponding maximum value of the sum. So we must show\n\n\\begin{equation}\n\\frac{2(n-1)(\\sigma-\\sqrt{\\sigma \\nu)}}{\\lambda_0} > \\frac{j-1}{-\\gamma+2}+\\frac{n-s-(j-1)}{\\gamma(n-s-(j-1))-2\\gamma+2}\n\\nonumber\\end{equation}\n\n\\begin{align} \\text{But} \\; \\; \\lambda_0=\\gamma s -2\\gamma + 2 < 2 - 2\\gamma \\; \\; \\text{so} \\; \\; \\frac{2(n-1)(\\sigma-\\sqrt{\\sigma \\nu})}{\\lambda_0}>\\frac{2(n-1)(\\sigma-\\sqrt{\\sigma \\nu})}{2-2\\gamma}\n\\nonumber\\end{align}\n\n\\begin{align} \\text{or that} \\; \\; \\label{++}\\frac{{2}(n-1)(\\sigma-\\sqrt{\\sigma \\nu)}}{{2}(1 -\\gamma)} > \\frac{j-1}{2-\\gamma} + \\frac{n-s-(j-1)}{\\gamma(n-s-(j-1))-2\\gamma+2}\n\\end{align}\n\nSince $\\frac{j-1}{1-\\gamma}>\\frac{j-1}{2-\\gamma}$, instead of inequality (\\ref{++}) we can prove that\n\n\\begin{equation}\n\\label{A+} \\frac{(n-1)(\\sigma-\\sqrt{\\sigma \\nu)}-(j-1)}{1 -\\gamma} > \\frac{n-s-(j-1)}{\\gamma(n-s-(j-1))-2\\gamma+2}\n\\end{equation}\n\n\\begin{align}\n\\text{But} \\; \\; \\gamma(n-s-(j-1))>-1 \\; \\; \\text{so} \\; \\; \\frac{n-s-(j-1)}{-1-2\\gamma+2} > \\frac{n-s-(j-1)}{\\gamma(n-s-(j-1))-2\\gamma+2}\n\\nonumber\\end{align}\n\n\\begin{align}\n\\text{Instead of (\\ref{A+}) we can prove} \\; \\; &\\frac{(n-1)(\\sigma-\\sqrt{\\sigma \\nu)}-(j-1)}{1 -\\gamma} > \\frac{n-s-(j-1)}{1-2\\gamma}\\nonumber\\\\\n\\text{or} \\; \\; \\; &\\frac{(n-1)(\\sigma-\\sqrt{\\sigma \\nu)}-{(j-1)}}{{1 -\\gamma}} > \\frac{n-s-{(j-1)}}{{1-\\gamma}}\\nonumber\\\\\n\\text{because} \\; \\; \\; &\\frac{n-s-(j-1)}{1-\\gamma} > \\frac{n-s-(j-1)}{1-2\\gamma}\\nonumber\n\\end{align}\n\n\\begin{align}\n\\text{So must show that} \\; \\; &\\sigma-\\sqrt{\\sigma \\nu} > \\frac{n-s}{n-1}=\\frac{n-s+1-1}{n-1}=1-\\frac{s-1}{n-1}\\nonumber\\\\\n\\text{or} \\; \\; &1+\\gamma(s-1)-\\gamma\\sqrt{(\\frac{1}{\\gamma}+(s-1))(\\frac{1}{\\gamma}+(n-1))} > 1-\\frac{s-1}{n-1}\\nonumber\\\\\n\\text{or} \\; \\; &\\gamma(s-1)-\\gamma\\sqrt{(\\frac{1}{\\gamma}+(s-1))(\\frac{1}{\\gamma}+(n-1))} > -\\frac{s-1}{n-1}\\nonumber\n\\end{align}\n\nSince $\\gamma (s-1) > -\\frac{s-1}{n-1}$, it suffices to show that\n\n\\begin{align}\n\\gamma(s-1)-\\gamma\\sqrt{(\\frac{1}{\\gamma}+(s-1))(\\frac{1}{\\gamma}+(n-1))} > \\gamma(s-1) \\; \\; \\text{which holds. \\qed}\n\\nonumber\\end{align}\n\nTheorem 3 gives us another answer to the Basic Question (see also Fig. \\ref{fig:j*}):\\\\\n\n\\begin{centering}\\fbox {\n\\parbox{\\linewidth}{\n\\textbf{Answer 5:} \\textit{The network never breaks when $\\gamma \\in (-1,0)$ and $\\gamma > \\frac{-1}{n-1}$, so no group of agents will consider deviating from the grand coalition under these conditions.}}\n}\n\\end{centering}\n\n\\section{Conclusions}\n\nIn this paper we forced a set of $n$ agents to compete under a differentiated Cournot competition and by imagining their full cooperation as a complete graph we studied the various beliefs that support core non-emptiness and thus network stability. Fixing the number of coalitions that outsiders form to say $j$ coalitions, we proved that the worth of the deviating agents, $s$, is minimized (hence the network is less likely to break) when the $n-s$ agents split equally among the $j$ coalitions; and the worth of $s$ is maximized when $j-1$ coalitions have one member and one coalition has $n-s-(j-1)$ members. Given the above, we proved that when $\\gamma > 0$ the network does not break, provided the agents of a deviant coalition believe their opponents will form a sufficiently large number of coalitions. On the other hand, if $\\gamma < 0$ and $\\gamma > \\frac{-1}{n-1}$, the network does not break irrespective of the beliefs of the deviant coalition, so no agent will consider deviating. Finally, we proved that when $\\gamma=1$, the deviating agents, in order to calculate their worth and thus decide what to do, must only worry about how many coalitions are going to be formed by the outsiders. \n\n\\subsection*{Acknowledgements} I am greatly indebted to Professor Giorgos Stamatopoulos for advising and encouraging me to finish this work.\n\n\\newpage \n\n\\section*{Appendix}\n\\begin{center}\n\\includegraphics[scale=0.65]{maxmin.eps}\n\\end{center}\n\\begin{figure}[htbp]\n\\caption{On the left, when $\\gamma \\in (0,1)$ and for a fixed $j$, the worth of $S$ is maximized when the outsiders, $N \\setminus S$, split in all coalitions but one into singletons. In this figure we have the case where $n=46$, $s=4$, and $j=6$. Respectively, on the right the worth is minimized when they split equally among the $j=6$ coalitions, having $\\frac{n-s}{j}=7$ agents each.} \\label{fig:MaxMin}\n\\end{figure}\n\\begin{center}\n\\includegraphics[scale=0.65]{join.eps}\n\\end{center}\n\\begin{figure}[htbp]\n\\caption{On the left, when $\\gamma \\in (0,1]$ the network does not break provided that agents in $S$ believe the outsiders will form a large $>j^*$ number of coalitions. In this figure we illustrate the case where $n=46$, $s=4$, and $\\gamma=0.9$, for which $j^*(46,4,0.9) \\approx 4.57$. So when the outsiders split into 5 coalitions the network does not break. On the right, when $\\gamma \\in (-1,0)$, the network does not break irrespective of what $S$ might think.} \\label{fig:j*}\n\\end{figure}\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe idea that the Universe is homogeneous, isotropic and that space-time is Lorentz invariant are important pillars of theoretical physics. Whereas the cosmological principal assumes the Universe to be homogeneous and isotropic, Lorentz invariance is required to be a symmetry of any relativistic quantum field theory. These requirements have robust footings, but there can possibly be scenarios where these ideas are not sufficient to describe the dynamics of a system. Temperature fluctuations in the Cosmic Microwave Background (CMB) radiation indicate that the assumptions made by the cosmological principal are not perfect. There is no conclusive evidence of Lorentz violation to date but this has been a topic of considerable interest and the Standard Model Extension (SME) has been constructed which includes various terms that preserve observer Lorentz transformations but violate particle Lorentz transformations \\cite{Colladay:1998fq}.\n Limits have been placed on the coefficients of various terms in the SME as well \\cite{Kostelecky:2008ts}. Another important question is the matter-antimatter asymmetry of the Universe which is not completely resolved. Sakharov, in 1967 derived three conditions (baryon violation, C and CP violation and out of thermal equilibrium) for a theory to satisfy in order to explain the baryon asymmetry of the Universe.\n \n Origin of fermion masses is also one of the most intriguing questions which is now close to be answered by the ATLAS and CMS experiments at the Large Hadron Collider. Hints of the Higgs boson have been seen and we will know for sure soon whether it exists or not. The formalism presented in this article might also help us answer these two important questions, namely, the baryon asymmetry of the Universe and the origin of fermion masses.\n \n We intend here to describe the evolution of a theory that violates Lorentz invariance to a theory that preserves it. The fields that are involved in the Lorentz violating theory can be viewed in analogy with fields traveling in an anisotropic medium. When the system evolves from the anisotropic to isotropic phase the symmetry of the theory is restored and the partition function formalism can be used to better understand how this transition takes place. This formalism, we propose, can help explain the matter-antimatter asymmetry of the Universe.\n \nThe paper is organized as follows: In section \\ref{transformations} and \\ref{visualize}, we describe these transformations and propose a way to interpret them as plane wave transitions into anisotropic media. In section \\ref{partition}, the partition function is used to get a better insight into how the transformations in section \\ref{transformations} occur. Section \\ref{interaction} illustrates how some interaction terms lead to Lorentz violating operators which are suppressed due to the transition of the system to a Lorentz symmetric phase. We conclude in section \\ref{conclusion}.\n\\section{Transformations leading to Covariant Dirac equation}\\label{transformations}\nIn this section we outline a set of transformations that lead to the Dirac equation for a QED (Quantum Electrodynamics) like theory with no interaction terms. The interaction terms will be discussed in section \\ref{interaction}. We start with a Dirac-like equation which involves four massless fields ($\\chi_a,\\chi_b,\\chi_c,\\chi_d$). These fields can be redefined in a simple way such that the covariant form of the Dirac equation is restored along with a mass term. In this section we will just consider the kinetic terms for the fields in the underlying theory so as to get the free Dirac equation in covariant form. \n\n If we start with the following equation ($\\hbar=c=1$):\n\\begin{eqnarray}\ni \\bar{\\chi_a} \\gamma^0 \\partial_0 \\chi_a + i\\bar{\\chi_b} \\gamma^1 \\partial_1 \\chi_b\n+i\\bar{\\chi_c} \\gamma^2 \\partial_2 \\chi_c\n+i\\bar{\\chi_d} \\gamma^3 \\partial_3 \\chi_d\n=0,\n\\label{eq1}\n\\end{eqnarray}\nand transform each of the $\\chi$ fields in the following manner,\n\\begin{eqnarray}\n\\chi_a(x) \\rightarrow e^{i\\alpha m \\gamma^{0} x_{0}} \\psi(x) \\nonumber ,\\ \\\n\\chi_b(x) \\rightarrow e^{i\\beta m \\gamma^{1} x_{1}} \\psi(x)\\nonumber \\\\\n\\chi_c(x) \\rightarrow e^{i\\delta m \\gamma^{2} x_{2}} \\psi(x) , \\ \\\n\\chi_d(x) \\rightarrow e^{i\\sigma m \\gamma^{3} x_{3}} \\psi(x),\n\\label{trans1}\n\\end{eqnarray}\nwe get the Dirac equation in covariant form, along with a mass term (using, for e.g., $e^{i\\beta m \\gamma^{1} x_{1}} \\gamma_0=\\gamma_0 e^{-i\\beta m \\gamma^{1} x_{1}} $),\n\\begin{eqnarray}\n\\overline{\\psi}(i \\gamma^{\\mu} \\partial_{\\mu}-(\\alpha+\\beta+\\delta+\\sigma)m)\\psi =0,\n\\label{eq2}\n\\end{eqnarray} \nwhere $\\alpha,\\ \\beta,\\ \\delta \\ \\text{and}\\ \\sigma$ are real positive constants. For plane wave solution for particles, $\\psi=e^{-i p.x} u(p)$, the above redefinition for the field $\\chi_a$, for example, is a solution of the following equation:\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t}\\chi_a(x)= -i( E- \\alpha m \\gamma_{0}) \\chi_a(x) ,\\ \\\n\\label{trans2}\n\\end{eqnarray}\nwith similar equations for the other fields. Equation (\\ref{trans2}), is similar to equation (27) in reference \\cite{braga} which is a solution of the differential equation governing linear elastic motions in an anisotropic medium (with a constant matrix, see section III of the reference). With $\\alpha=0$ the left hand side is just the Hamiltonian, with the plane wave its eigenstate. \n\nNote that the manner in which we can transform equation (\\ref{eq1}) to (\\ref{eq2}) is not unique and there are various ways to do this with different combinations of the $\\chi$ fields along with the field $\\psi$. A mass term ($m \\overline{\\chi} \\chi$) for the $\\chi$ fields could have been added to equation (\\ref{eq1}), but the transformations (\\ref{trans1}) can be used to eliminate it. So, if we want our resulting equation to describe a massive fermion, these fields should be massless or cannot have mass term of the form $m \\overline{\\chi} \\chi$. This argument will be further corroborated with the results we present in section \\ref{partition}. The transformation matrices in equation (\\ref{trans1}) are not all unitary, the matrix $e^{i\\alpha m \\gamma^{0} x_{0}}$ is unitary while the rest ($e^{i\\beta m \\gamma^{i} x_{i}}$) are hermitian. \n\n \n The fields in equation (\\ref{eq1}) can be considered as independent degrees of freedom satisfying equation (\\ref{trans2}) in an underlying theory that violates Lorentz invariance. The transformations (\\ref{trans1}) can, therefore, be seen as reducing the degrees of freedom of the theory from four to one. In such an underlying theory, various interaction terms can be written for these fields. Since we intend to obtain the free Dirac equation, we have considered only kinetic terms involving the fields $\\chi$. A quadratic term involving different $\\chi$ fields ($m \\overline{\\chi}_i \\chi_j$) can be added to equation (\\ref{eq1}) but this leads to a term that violates Lorentz invariance in the resulting Dirac equation. A quartic term ($c \\overline{\\chi}_i \\chi_i \\overline{\\chi}_j \\chi_j$) is possible and would result in a dimension 6 operator for the field $\\psi$ with the constant $c$ suppressed by the square of a cutoff scale. So, with the restriction that the resulting Dirac equation only contains terms that are Lorentz scalars the number of terms we can write for the $\\chi$ fields can be limited. In other words we impose Lorentz symmetry in the resulting equation so that various terms vanish or have very small coefficients. The interaction terms are further discussed in section \\ref{interaction} of the paper.\n\n \n\n\n\\section{Visualizing field Redefinitions} \\label{visualize}\n\n\nWe can visualize a global and a local transformation as transitions of plane waves to different types of media. The wave function of a particle, for example, which comes across a potential barrier $(E>V)$ of a finite width and height undergoes a phase rotation ($e^{i k l} \\psi$) upon transmission. If the width of the barrier extends to infinity, the wave function can be viewed as undergoing a position dependent phase rotation ($e^{i k x} \\psi$). The transformations (\\ref{trans1}) can similarly be seen as a plane wave entering an anisotropic medium. A phenomenon in optics called birefringence can be used to explain why these four fields map on to the same field $\\psi$. Birefringence results in a plane wave splitting into two distinct waves inside a medium having different refractive indices along different directions in a crystal. These analogies can serve as crude sketches to visualize how the transformations in equation (\\ref{trans1}) can occur.\n\nWhen a polarized electromagnetic plane wave enters a birefringent material, the wave splits into two distinct waves. This can be also be seen as a change in the coordinate system of the wave. Similarly, transformations (\\ref{trans1}) can be seen as rotation of the field $\\psi$ in spinor space or the transition of a wave in a material that splits the field $\\psi$ into four distinct waves and is anisotropic. Based on the latter view, we propose a transition from an anisotropic to isotropic phase in the early Universe whereby it became Lorentz symmetric. So the form of equation (\\ref{eq1}) is not a bizarre choice of basis for writing the equation but a possible form of the equation in an anisotropic space-time. The difference also comes from the interaction terms of these fields, which we shall discuss in section \\ref{interaction}. The interaction terms of the fields $\\chi$ that are due to the anisotropic character of space-time get enhanced in the anisotropic phase. As the transition takes place these terms become suppressed. This would not be the case if we just choose a different basis to write the equation. The suppression of the anisotropy is interpreted as the reduction of entropy or increase in order of the system discussed in section \\ref{partition}.\n\nSpace-time dependent field redefinitions in the usual Dirac Lagrangian result in violation of Lorentz invariance. For example, the field redefinition $\\psi \\rightarrow e^{-i a^{\\mu} x_{\\mu}} \\psi$ leads to the Lorentz violating terms in the Lagrangian \\cite{Colladay:1998fq}. This particular redefinition, however, would not lead to physically observable effects for a single fermion. A transformation of this type amounts to shifting the four momentum of the field. It can also be viewed in analogy with plane waves entering another medium of a different refractive index which results in a change in the wave number of the transmitted wave. Similarly, transformations (\\ref{trans1}) can be interpreted as transitions of a wave from an anisotropic to isotropic medium or vice versa as done in the Stroh's matrix formalism \\cite{braga}.\n\n For plane wave solutions of $\\psi$, the $\\chi$ fields have propagative, exponentially decaying and increasing solutions (for example, $e^{\\pm i m x}, \\ e^{\\pm m x}$). This wave behavior is similar to that in an anisotropic medium or a medium made of layers of anisotropic medium. The eigenvalues of the Dirac matrices being the wave numbers of these waves in this case. The coefficients in the exponent relates to how fast the wave oscillates, decays and\/or increases exponentially. The transfer matrix in Stroh's formalism describe the properties of the material and in this case can possibly represent the properties of the anisotropic phase from which the transition to the isotropic phase occurs.\n\n\n\n\nIn the usual symmetry breaking mechanism a Higgs field acquires a vacuum expectation value (VEV) and the resulting mass term does not respect the symmetry of the underlying group. For example, in the Standard Model, due to its chiral nature, a Higgs field is introduced in order to manifest gauge invariance. Once the Higgs field acquires a VEV the mass term only respects the symmetry of the resulting group which is $\\mathrm{U(1)_{EM} }$. In our case the mass term arises after symmetry of the Dirac equation is restored. Consider the simple case where we have one field $\\chi_a$ in addition to the field $\\psi$:\n\\begin{eqnarray} \ni \\bar{\\chi_a} \\gamma^0 \\partial_0 \\chi_a + i\\bar{\\psi} \\gamma^i \\partial_i \\psi\n=0,\n\\label{eq3} \n\\end{eqnarray}\n and this field transforms to the field $\\psi$ as $\\chi_a(x) \\rightarrow e^{i\\alpha m \\gamma^{0} x_{0}} \\psi(x),$ leading to the Dirac equation. In order to discuss the symmetries of the above equation let's assume that the two independent degrees of freedom are described by the above equation. Equation (\\ref{eq3}) then has two independent global U(1) symmetries and the resulting equation has one. In fact, there is a list of symmetries of equation (\\ref{eq3}) not possessed by (\\ref{eq2}), for example invariance under local transformations, $\\chi_a \\rightarrow e^{i b^i \\theta(x_i)} \\chi'_a$ ($i,j=1,2,3$), where $b_i$ can be a constant vector, the matrix $\\gamma_0$ or any matrix that commutes with $\\gamma_0$ (e.g., $\\sigma_{ij},\\ \\gamma_5 \\gamma_i$). This implies invariance under global and local SO(3) transformations (rotations of the fields $\\chi_a$ but not boosts). Similarly, $\\psi \\rightarrow e^{i A \\ \\theta(t)} \\psi'$ is a symmetry, where $A$ can be a constant or the matrix $i \\gamma_0 \\gamma_5$ which commutes with the three Dirac matrices $\\gamma_i$. After the transformation $\\chi_a \\rightarrow e^{i m \\gamma_0 t} \\psi$ the equation is no more invariant under these symmetries and the SO(1,3) symmetry of the Dirac equation is restored along with a global U(1) symmetry.\n\\section{Partition Function as a Transfer Matrix}\\label{partition}\n\nIn the early Universe, a transition from a Lorentz asymmetric to a symmetric phase could possibly induce transformations of the form (\\ref{trans1}). Let's again consider the simple example in equation (\\ref{eq3}). For this case the eigenvalues of the Dirac matrix $\\gamma_0$ define the wave numbers of the waves traveling in the anisotropic medium. The direction of anisotropy in this case is the temporal direction, which means that the time evolution of these waves is not like usual plane waves.\nIt is not straight forward to visualize the fields, the dynamics of whom are described by the anisotropy of space time, but we can use the partition function method to get a better insight into this. We can, by using this formalism, calculate the temperature at which the transformations in equation (\\ref{trans1}) occur.\n\n We next perform a transition to a thermodynamics system by making the transformation $i t \\rightarrow \\beta$, where $\\beta = 1\/k_{B} T$ \\cite{zee}. The partition function is then given by the trace of the transformation matrix $e^{im \\gamma_0 t}$,\n\\begin{eqnarray} \nZ=\\mathrm{Tr}(e^{m \\beta \\gamma_0 })=2 e^{\\beta m} +2 e^{-\\beta m}.\n\\label{eq5}\n\\end{eqnarray}\nIn order to represent the transition of the system with the above partition function the temporal transfer matrix should be unitary. This partition function is similar to that of a two-level system of spin 1\/2 particles localized on a lattice and placed in a magnetic field with each state, in this case, having a degeneracy of two. The lower energy state corresponding to spin parallel to the field ($E=-m,Z_1=e^{\\beta m}$). In this case the doubly degenerate states correspond to spins up and down of the particle or anti-particle. For $N$ distinguishable particles the partition function is $Z^N$, $N$ here is the total number of particles and antiparticles of a particular species. So, we are modeling our system as being on a lattice with the spin along the field as representing a particle and spin opposite to the field representing an antiparticle. \n \n The evolution of this system with temperature represents the time evolution of the system in equation (\\ref{eq1}). In other words the partition function describes the evolution of these waves from anisotropic to isotropic phase as the temperature decreases. \n For a two level system the orientation of the dipole moments becomes completely random for large enough temperatures so that there is no net magnetization. In our case we can introduce another quantity, namely a gravitational dipole, which would imply that the four states (particle\/antiparticle, spin up\/down) of $N$ such particles at high enough temperatures orient themselves in a way that the system is massless. This just serves as an analogy and does not mean that the masses are orientating themselves the same way as dipoles would do in space. The anisotropic character can be seen as mimicking the behavior of the field in a two level system. The population of a particular energy level is given by,\n\\begin{eqnarray}\nn_{p(\\overline{p})}= \\frac{N e^{\\pm \\beta m} }{e^{\\beta m} + e^{-\\beta m}}.\n\\label{eq5b}\n\\end{eqnarray}\n Which shows that the number density of particles and antiparticles vary in a different way with respect to temperature. In the early Universe, therefore the anisotropic character of space-time seems to play an important role such that particles and anti-particles behave in different manners. As the temperature decreases the number density of the anti-particles decreases and is vanishingly small for small temperatures ($\\sim e^{-2\\beta m}$). \nWhen the decoupling temperature is attained there is a difference in the number density of the particles and antiparticles as described by equation (\\ref{eq5b}). This leads to an excess of particles over antiparticles. The decoupling temperature of a particular species of particle with mass $m$ and which is non-relativistic is given by, $k_B T \\lesssim 2 m$. Below this temperature the particles annihilate to photons but the photons do not have enough energy to produce the pair. This can be used to get the ratio of antiparticles over particles (matter radiation decoupling). For $\\beta m \\approx 0.5$, we get,\n\\begin{eqnarray}\n\\frac{n_p-n_{\\overline{p}}}{n_{p}}\\approx 0.6 \\ .\n\\label{eq5c}\n\\end{eqnarray}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=1.1]{plot-cv.eps}\n\n\\caption{Plot of heat capacity $C_V$ for the mass of electron, up quark, neutrino and W boson. The maximum of the heat capacity of the electron occurs at $4.8 \\times 10^9 \\mathrm{K}$, for the up quarks is $1.9 \\times 10^{13} \\mathrm{K}$, for neutrinos is $291 \\mathrm{K}$ and for the W bosons is $7.8 \\times 10^{14} \\mathrm{K}$. We use $k_B=8.6 \\times 10^{-5} \\mathrm{\\ eV}\/\\mathrm{K}$ and $m_{\\nu}=0.03 \\mathrm{\\ eV}$.\n\\label{fig1}}\n\\end{center}\n\\end{figure}\nWhich implies an excess of particles over antiparticles and thus can serve as another possible way to explain the matter antimatter asymmetry of the Universe. This number is very large compared to the one predicted by standard cosmology ($\\sim 10^{-9}$). The above expression yields this order for $\\beta m \\approx 10^{-9}$ which implies a large temperature. For electrons this would imply a temperature of the order $10^{18} $K which is large and the electrons are relativistic. So if we assume that the decoupling takes place at a higher temperature, the baryon asymmetry can be explained. Even without this assumption the conditions proposed by Sakharov can also enhance the number of particles over the antiparticles. Sakharov's conditions involve the interaction dynamics of the fields in the early Universe whereas in our case the statistical system serves more as a model describing the dynamics of space-time to a more ordered phase. \n\nStatistical mechanics, therefore, enables us to visualize this transition in a rather lucid way. In a two level system the net magnetization at any given temperature is analogous to the excess of particles over antiparticles in the early Universe.\nThe time evolution of this anisotropic to isotropic transition is modeled on the evolution of a statistical thermodynamics system with particles on a lattice placed in a magnetic field. The particles on the lattice are localized, static and have no mutual interaction. The free energy of the system is given by:\n\\begin{eqnarray} \nF= -N k_B T \\mathrm{\\ ln \\{4 \\mathrm{cosh}\\left[m \\beta \\right]\\} }\n\\label{eq6}\n\\end{eqnarray}\nFrom this we can calculate the entropy $S$, heat capacity $C_V$ and mean energy ${U}$ of the system:\n\\begin{eqnarray}\nS &=& - \\left(\\frac{\\partial F}{\\partial T}\\right)_V \\nonumber \\\\\n &=& N {k_B} \\text{ln}\\left \\{4 \\ \\mathrm{cosh}\\left[m \\beta \\right]\\right \\}-m k_B \\beta \\ \\text{tanh}\\left[m \\beta \\right] \\label{eq8} \\\\\nU &=& F+TS\n = -N m \\ \\text{tanh}\\left[m \\beta \\right] \\label{eq9} \\\\\nC_V &=& \\left(\\frac{\\partial U}{\\partial T} \\right)_V \n= N k_B m^2 \\ \\beta^2 \\ \\text{sech}^2\\left[m \\beta \\right]\n\\label{eq10}\n\\end{eqnarray}\nIn Fig.\\ref{fig1}, the peaks in the heat capacity represent phase transition of a particular particle species. These are second order phase transitions and the peak in the heat capacity is usually referred to as the Schottky anomaly \\cite{Pathria}. Note that the phase transition we model our system on is a magnetic one. So, modeling the complex system in the early Universe on a lattice with spin 1\/2 particles can reduce the complications of the actual system by a considerable amount.\n\n\nThe Schottky anomaly of such a magnetic system, therefore, represents phase transitions in the early Universe. For a particular species of particles the Schottky anomaly shows a peak around $mc^2\\approx kT$. The phase transition for the electrons occurs at the temperature where nuclei start forming in the early Universe. For the quarks the transition temperature refers to confinement into protons and neutrons. Similarly, W boson's transition occurs at the electroweak breaking scale. The W boson, being a spin 1 particle, is not described by the Dirac equation, but the heat capacity entails this feature of showing a phase transition for \nthe energy scale relevant to the mass of a particle.\n\nThe curve for neutrinos implies that the transition temperature for neutrinos is around 291 K, which means that the density of antineutrinos from the big bang for present neutrino background temperatures ($\\sim$ 2 K) is not negligible. The ratio of antineutrinos over neutrinos for $T=2 \\ \\mathrm{K}$, is $n_{\\overline{\\nu}}\/n_{\\nu}= e^{-2\\beta m_{\\nu}}\\sim 10^{-15000}$ ($m_{\\nu}=2 \\ \\mathrm{eV}$). For an even smaller neutrino mass, $m_{\\nu}=1 \\times 10^{-4} \\ \\mathrm{eV}$, the ratio is $n_{\\overline{\\nu}}\/n_{\\nu} \\sim 0.3$, which for other more massive particles is much smaller. A cosmic neutrino and antineutrino background is one of the predictions of standard cosmology but is still unobserved. This model predicts an antineutrino background much less than the neutrino one. \n\n In Fig.\\ref{fig2}, the plots of mean energy and entropy are shown in dimensionless units. In the massless limit for fermions, the entropy attains its maximum value of $N k_B \\mathrm{ln 4}$. The plots show that the energy of the system approaches zero as the temperature approaches infinity. This situation is analogous to the spins being completely random at high temperatures for the two level system. The same way that the magnetic energy of the system on the lattice is zero at high temperatures, the mass of this system is zero in the very early Universe. As the temperature decreases the energy of the system attains it minimum value ($U=-Nm$) and the particles become massive at the temperature less than the value given by the peak of the heat capacity. The entropy for high temperatures asymptotically approaches its maximum value of $N k_B \\mathrm{ln 4}$. \n \n The value of the parameter $\\mathrm{k_B T\/m}$ at the peak of the heat capacity curves gives the temperature at which the transition takes place for a particular species. The transition for each field, therefore, depends on the energy scale relevant to its mass. So, this means that each field was experiencing the anisotropy of space time in a different manner whereas space-time itself was expanding towards a Lorentz symmetric phase. This can be understood if we imagine a material with anisotropies. Plane waves of different wavelengths inside the material experience the anisotropies in different ways. So the analogy discussed in section \\ref{visualize} tells us that the more massive the field, the faster it will oscillate, exponentially increase or decrease. A less massive field having a larger wavelength ($\\lambda= \\hbar\/mc$) therefore would experience the anisotropies when their scale is much larger.\n\n\n \n\nAccording to the statistical thermodynamics model that describes this transition, as this phase transition occurs antiparticles will start changing into particles and as can be seen from the figure the system will move towards all spins aligned parallel with the ``field\", i.e., towards being particles. From Fig.\\ref{fig2} we can see that the energy of the system starts attaining the minimum value as the temperature decreases where all particles are aligned with the field and are ``particles\". The plot of entropy vs. temperature also represents an important feature of these transformations. The entropy decreases with decreasing temperature and this represents the transition to a more ordered phase using equations (\\ref{trans1}). The plots of energy of the system U in Fig.\\ref{fig2} show that the system will eventually settle down to the lowest energy state which in this case means that the system will have almost all particles with negligible number of antiparticles. In short, the plot of the heat capacity reflects the phase transitions, the plot of energy U represents the transition from massless to massive states and the plot of entropy represents the transition of space time to a more ordered phase.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[clip=true,trim=0mm 23mm 0mm 23mm,scale=1.3]{plot-u.eps}\n\n\\caption{Plot of entropy and energy for a particle of mass m. For large enough temperatures the energy of the system approaches zero and the entropy approaches the limiting value of $N k_B \\mathrm{ln 4}$.\n\\label{fig2}}\n\\end{center}\n\\end{figure}\n\n The Big Bang theory is one of the most promising candidates to describe how the Universe began. According to this theory, the Universe expanded from a singularity where curved space-time, being locally Minkowskian, eventually became flat. It is possible that there even was a transition to the Minkowski space from a non-Minkowski one. If the Universe began with a state of maximum entropy than we can very well assume that space-time was not Minkowskian even locally. The fields that dwell in space-time are representation of the symmetry group that describes it. The $\\chi$ fields in the underlying theory, described by equation (\\ref{eq1}), are therefore, not representations of the Lorentz group. The CPT theorem assumes symmetries of Minkowski space-time in implying the similarities between particles and antiparticles. If the underlying theory is not Minkowskian than particles and antiparticles can behave differently and this is what the model described in this section implies. \n\n \nThe occurrence of the Schottky anomaly has motivated the study of negative temperatures \\cite{Ramsey}. Note that the partition functions is invariant under the transformation $T \\rightarrow -T$ but the equations for the free energy, entropy and energy are not. The existence of negative temperatures has been observed in experiments. Negative temperatures, for example, can be realized in a system of spins if the direction of the magnetic field is suddenly reversed for a system of spins initially aligned with the magnetic field \\cite{Pathria}. Similarly, as described in reference \\cite{Ramsey} the allowed states of the system must have an upper limit. Whereas this is not the case for the actual particles in the early Universe, the statistical mechanics system on which it can be modeled on has this property. A negative temperature system would eventually settle down to the lower energy state ($U=Nm$) which in our case would mean that the Universe would ends up having more antiparticles than particles. This is yet another interesting insight we get by modeling the early Universe on a two state system.\n\n The system on the lattice eventually has only spins aligned with the field and which are interpreted as particles. This means that the partition function can be written as,\n\\begin{eqnarray}\nZ=e^{\\beta m}=e^{(-\\beta) (-m)} \\nonumber .\n\\end{eqnarray}\nWhich implies that the system can eventually be seen as spins aligned towards the field evolving in a positive temperature system or spins aligned opposite to the field evolving in a negative temperature system. This gives a correlation of particles with positive temperature systems and antiparticles with negative temperature systems. This view in a way coincides with the fact that the resulting space-time configuration describes both particles and antiparticles in a similar but independent way.\n\n \nWe can also notice that in the statistical system modeled for the transition, the reason due to which entropy of the system decreases is the presence of an external field. As in a paramagnet, the spins remain randomly align in the absence of a magnetic field. In the system considered, spins interact with the magnetic field and not with each other. It is difficult to say what played the role of this field in the early Universe, but since the field is the reason entropy reduces, there should be a physical quantity present in the early Universe playing an analogous role. This quantity, for instance, can be speculated to be the expansion of the Universe.\n\\section{Interaction terms and the SME} \\label{interaction}\n In section \\ref{transformations} we only considered terms in the underlying theory that lead to the free Dirac equation. In this section we shall include some interaction terms and see how they lead to LV operators one of which is considered in the SME. Since we do not have Lorentz symmetry, the underlying theory can have a large number of terms. We will therefore restrict ourselves to interaction terms that lead to dimension 4 operators and only involve the gamma matrices. Also, the form of the Dirac equation restricts the number of ways we can start with an underlying LV theory. \n In the following we add a few interaction terms to equation (\\ref{eq1}): \n\\begin{eqnarray}\ni \\bar{\\chi_a} \\gamma^0 \\partial_0 \\chi_a + i \\bar{\\chi_b} \\gamma^1 \\partial_1 \\chi_b\n+i \\bar{\\chi_c} \\gamma^2 \\partial_2 \\chi_c\n+i \\bar{\\chi_d} \\gamma^3 \\partial_3 \\chi_d + I^{int}_1 + I^{int}_2\n=0,\n\\label{eq2b}\n\\end{eqnarray}\nwhere $I^{int}_1$ and $I^{int}_2$ include operators that lead to dimension 4 LV operators after the transformations,\n\\begin{eqnarray}\n I^{int}_1 &=& c_1 \\overline{\\chi}_a \\gamma^0 \\chi_a a_0 \n+ c_3 \\overline{\\chi}_b \\gamma^1 \\chi_b a_1 \n+ c_5 \\overline{\\chi}_c \\gamma^2 \\chi_c a_2\n+ c_7 \\overline{\\chi}_d \\gamma^3 \\chi_d a_3 \\nonumber \\\\\n\\vspace{2mm}\n&+& c_{9} \\overline{\\chi}_a \\gamma^0 \\gamma_5 \\chi_a b_0 \n+ c_{11} \\overline{\\chi}_b \\gamma^1 \\gamma_5 \\chi_b b_1\n+ c_{13} \\overline{\\chi}_c \\gamma^2 \\gamma_5 \\chi_c b_2\n+ c_{15} \\overline{\\chi}_d \\gamma^3 \\gamma_5 \\chi_d b_3+.... \\ , \\nonumber\n\\label{eq2c}\n\\end{eqnarray}\n\\begin{eqnarray}\n I^{int}_2 &=& c_{2} m \\overline{\\chi}_a \\chi_b + c_6 \\overline{\\chi}_a \\gamma^0 \\chi_b a_0 \n+ c_4 \\overline{\\chi}_b \\gamma^1 \\chi_c a_1 \n+ c_6 \\overline{\\chi}_c \\gamma^2 \\chi_d a_2\n+ c_8 \\overline{\\chi}_d \\gamma^3 \\chi_a a_3+... \\ , \\nonumber \n\\vspace{2mm}\n\\label{eq2d}\n\\end{eqnarray}\nwhere $I^{int}_1$ includes interaction terms of the same fields and $I^{int}_2$ involve different fields. As the system transits to a more ordered phase using (\\ref{trans1}) the coefficients of the terms leading to LV operators become suppressed. For example, after the transformations, the terms in $I^{int}_1$ would lead to the term in the SME with coefficient $a_{\\mu}$. The term $\\overline{\\psi}\\gamma^{\\mu}\\psi a_{\\mu}$ in the SME do not have any physical effects for a single fermion since the term can be canceled with a redefinition of the field $\\psi \\rightarrow e^{i a^{\\mu}x_{\\mu}}\\psi$, but can have physical effects if the theory has more than one fermions. Note that the SME term $\\overline{\\psi}\\gamma^{\\mu}\\gamma_5\\psi b_{\\mu}$ with a constant $b_{\\mu}$ is not generated from $I^{int}_1$. \n\nThe coefficients $c_i$ are dimensionless quantities which measure the anisotropies of the system and therefore can be taken to be proportional to the entropy $\\mathrm{S\/k_B}$ of the system in equation (\\ref{eq8}). As the system moves towards a Lorentz symmetric phase the entropy approaches zero and therefore the coefficients become vanishingly small. We can also impose four global U(1) symmetries in equation (\\ref{eq2b}) and this would forbid the terms in $I^{int}_2$. The symmetries can then be gauged to get interactions of each of the $\\chi$ fields with each component of the gauge field (e.g. $\\overline{\\chi}_a \\gamma^0 \\chi_a A_0(x)$) and this would convert to the usual gauge interaction term in the resulting covariant Dirac equation.\n\n\nAs mentioned earlier, the properties of the gamma matrices and the form of the Dirac equation restricts the ways we can start with an underlying LV theory. The gamma matrices commute with only the identity matrix and anti-commute with $\\gamma_5$. The other possible way to start with an underlying theory is to use the $\\gamma_5$ matrix. If we just include the $\\gamma_5$ matrix with the gamma matrices in (\\ref{eq1}) than we do not retrieve the Dirac equation. This is also the case when the $\\gamma_5$ matrix is included in the transformations. However, if we start with both a $\\gamma_5$ in the equation (e.g. $\\overline{\\chi}_a \\gamma^0 \\gamma_5 \\partial_0\\chi_a $) and perform the transformations (\\ref{trans1}) with a $\\gamma_5$ (e.g. $\\chi_a \\rightarrow e^{i m \\gamma^0 \\gamma_5 t} \\psi$) than the Dirac equation is retrieved with a $\\gamma_5$ and this can be rotated away with a redefinition of the field $\\psi$ in the free Dirac equation. Since the temporal transfer matrix is not unitary in this case, the analogy made in section \\ref{partition} will not apply and we get an oscillating partition function. This would also effect the resulting LV terms we get in the SME. In this case we get a term $\\overline{\\psi}\\gamma^{\\mu}\\gamma_5\\psi b_{\\mu}$ with a constant four vector $b_{\\mu}$, whereas $a_{\\mu}$ is not constant in $\\overline{\\psi}\\gamma^{\\mu}\\psi a_{\\mu}$. \n\\section{Conclusions}\\label{conclusion}\nConsidering that the Dirac equation can be written in the form of an underlying theory that violates Lorentz invariance, we suggest that such a transition took place for fermions in the early Universe. We propose that space-time was not Lorentz symmetric and that a gradual transition to the Lorentz symmetric phase occurred. The fields in the underlying Lorentz violating theory are massless and transformations were performed that restore the Dirac equation to its covariant form along with a mass term for the fermions. \n\nThe underlying theory depicting the Lorentz violating phase has interaction terms of the fields. As the transition takes place, these interaction terms result in suppressed Lorentz violating terms some of which can be identified with terms in the SME (Standard Model Extension). The partition function formalism is then used to model these transformations on the evolution of a system of spin 1\/2 particles on a lattice placed in a magnetic field. Symmetry breaking in this case takes place on this lattice, whereas, it is restored in the Dirac equation. The transition to the Lorentz symmetric phase in the early Universe can be modeled on this thermodynamic system. \n\nThe behavior of the fields in the anisotropic phase is suggested to be similar to that of plane waves in anisotropic media. The eigenvalues of the transfer matrices give the wavenumbers of the waves in the anisotropic media. The wavelengths given by the temporal transfer matrix ($\\hbar \/ mc$) show that each fermion field experienced the anisotropy of space-time in different ways. The reason entropy decreases in the statistical system is the presence of an external magnetic field. The expansion of the Universe is interpreted as playing this role. The fields with small masses, and hence large wavelengths, undergo phase transitions later as the scale of the anisotropies get larger with the expansion.\n \n \n We showed that modeling the transition in such a manner can describe three important features of the early Universe: (1) The heat capacity shows occurrence of phase transitions. (2) The mean energy of the system shows how the particles became massive from being massless. (3) The plot of entropy depicts the occurrence of a transition to a more ordered phase interpreted as the Lorentz symmetric phase. At any given temperature the net magnetization measures the excess of particles over antiparticles. We suggest that this model can be used to explain the matter antimatter asymmetry of the Universe. Also, since space-time is not Minkowskian in the underlying theory, the CPT theorem does not hold, implying a difference in the behavior of particles and antiparticles. This is in agreement with the analogy created with the statistical system whereby spin up and down particles behave in different ways with the evolution of the system. This formalism can arguably serve as another possible way to explain the origin of fermion masses till the final results related to the Higgs boson are presented.\n \n \n\\section{Acknowledgements}\nThe author would like to express his deep gratitude to Alan Kostelecky and Dmitry Gorbunov for very fruitful discussions and suggestions. I would also like to thank Fariha Nasir, Hassnain Jaffari, Ilia Gogoladze and Matthew DeCamp for useful discussions and comments. \n \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sect:introduction}\n\nStars, as building blocks of galaxies, contribute most radiation of galaxies,\ndominate metal enrichment of galaxies, drive galactic outflows, and essentially\nconstruct galaxy structures. Therefore, star formation activity is one of the\nmost important evolutionary processes in galaxies \\citep{kennicutt12}. It has\nbeen found that molecular gas supplies the raw material of star formation,\nwhile the majority of molecular gas is not directly observable from the H$_2$\nemission \\citep{Bolatto13}. Molecular gas is often traced with the rotational\ntransitions of other molecules excited by collisions with H$_2$ molecules.\nAmong them, CO transitions mostly trace the bulk of molecular gas, due to its\nweak permanent dipole moment \\citep[$\\mu_{10}^e = 0.11~\\rm\nDebye$;][]{Solomon05} and low upper energy levels \\citep[5.5 K for\n$J=1\\rightarrow0$;][]{moleculardata2005}, while the dense gas tracers, e.g.,\nrotational transitions of HCN, HCO$^+$, CS, N$_2$H$^+$, etc., can trace denser\nmolecular clouds, due to their much higher dipole moments \\citep[$\\mu_{10}^e$ =\n2.98\\,Debye and 3.92\\,Debye for HCN, HCO$^+$\\,$J=1\\rightarrow0$,\nrespectively;][]{Papadopoulos2007}. \n\n\nThe star formation rates in galaxies, on the other hand, are usually traced\nwith ultraviolet continuum emission, optical line tracers such as H$\\rm \\alpha$\nand [O {\\sc II}], dust emission at infrared (IR) wavelengths, radio continuum\nemission, or X-ray emission \\citep{Kennicutt98a,kennicutt12}. Among them, the\nIR emission from dust normally traces the bolometric energy heated up by the\nyoung stars. The IR emission is in general extinction-free and often covered\nby multi-IR-wavelength space telescopes such as \\textrm{ Herschel}\n\\citep{Herschel}, \\textrm{ Spitzer} \\citep{Spitzer}, Infrared Astronomical\nSatellite \\citep[IRAS;][]{IRAS1984}, etc. Therefore, the total IR luminosity\nis widely adopted as a star-formation rate tracer for gas-rich galaxies\n\\citep[e.g.,][]{Galametz16,Galametz20}. \n\n\nA long-standing question remains: which gas is forming stars?\n\\cite{Kennicutt98} found a super-linear correlation between the surface\ndensities of star formation rate (SFR) and total gas mass, where the area was\ndefined by CO $J=1\\rightarrow0$\\ or IR images. This correlation, however,\nturns to a linear shape with a slope index of unity, when the sample is limited\nto nearby normal spiral galaxies beyond 500-pc scales\n\\citep[e.g.,][]{Bigiel2008}. On the other hand, \\cite{Gao2004b} found that the\ndense molecular gas traced by HCN\\,$J=1\\rightarrow0$\\ is the direct source of\nstar formation. This correlation is further connected to Galactic massive\nstar-forming regions on sub-pc scales \\citep{Wu2005,Wu2010}. Furthermore,\n\\citet{Heiderman10} also found a similar result by counting young stellar\nobjects in Galactic dense molecular clouds. \n\n\nHowever, the emission of HCN\\,$J=1\\rightarrow0$\\ and\nHCO$^+$\\,$J=1\\rightarrow0$\\ could be largely contaminated by diffuse gas with\nrelatively high column densities \\citep{Evans2020}. Although the high-$J$\ntransitions (e.g., $J=3\\rightarrow2$ and $J=4\\rightarrow3$) of HCN and\nHCO$^+$ \\citep{zhang14apj,Tan2018,Lifei3-2} are found to have linear\ncorrelations with star formation rate, these high-$J$ transitions are mainly\nfrom the densest molecular cores heated by massive stars, and leave out most\nemission from cold clumps. \n\n\n\n\n\nHCN\\,$J=2\\rightarrow1$\\ and HCO$^+$\\,$J=2\\rightarrow1$, which have moderate critical densities \\citep[$n^{\\rm HCN\n2-1}_{\\rm crit}$= $\\sim 1.6\\times 10^6~\\rm cm^{-3}$, $n^{\\rm HCO^+ 2-1}_{\\rm\ncrit}$= $\\sim 2.8\\times 10^5~\\rm cm^{-3}$, for conditions of a kinetic\ntemperature $T_{\\rm kin} \\sim$ 50\\,K, optically thin, and no\nbackground][]{Shirley2015} and suitable upper energy levels \\citep[$E^{\\rm HCN\n2-1}_{\\rm up}= 12.76 \\rm K$ and $E^{\\rm HCO^+ 2-1}_{\\rm up}= 12.84 \\rm K$,\nrespectively;][] {Shirley2015}, could avoid aforementioned disadvantages of\nother transitions. On the other hand, the rest frequency of HCN\\,$J=2\\rightarrow1$\\ is close\nto the H$_2$O 177.3-GHz line in the Earth atmosphere, so the observation needs\nexcellent weather conditions for nearby galaxies.\n\n\n\n\nIn this paper, we present Atacama Pathfinder Experiment (APEX) observations of\nHCN\\,$J=2\\rightarrow1$\\ and HCO$^+$\\,$J=2\\rightarrow1$\\ in a sample of 17 IR bright galaxies. In Sect.\\ref{sec:obs}\nwe describe observations with APEX 12-m telescope and ancillary data adopted in\nthis paper. In Sect.\\ref{sec:methods} we describe methods to derive line\nluminosities, photometry, and dust properties. In Sect.4 we present the\nobtained spectra, correlations between star formation rate and dense gas\ntracers, and line ratios. In Sect.5 we discuss the assumptions, caveats, and\nphysical implications of our results. In Sect.6 we summarize our work. We adopt\ncosmological parameters of $H_0=71~\\rm km\\,s^{-1}\\, Mpc^{-1}$, $\\Omega_{\\rm\nM}=0.27$, $\\Omega_\\Lambda=0.73$ throughout this work \\citep{Spergel2007}.\n\n\\section{Observation and data reduction}\\label{sec:obs}\n\\begin{deluxetable*}{cccccccccccc}\n\\tablenum{1}\n\\tablecaption{Basic Information} \\label{table:basicinfo}\n\\tablewidth{0pt}\n\\setlength\\tabcolsep{3pt}\n\\tablehead{\n\\colhead{Source name} & \\colhead{R.A.} & \\colhead{Dec.} & \\colhead{Redshift} &\n\\colhead{Distance $^a$} & \\colhead{Ref.} & \\colhead{HPBW$^{1.4\\rm GHz}$} & \\colhead{Ref.}& \\colhead{$S^{\\rm 1.4GHz}_{\\rm VLA}$ $^d$} & \\colhead{$S^{\\rm 1.4GHz}_{\\rm VLBI}$} & \\colhead{$S_{\\rm CO J=1-0}$} & \\colhead{Type} \\\\\n\\colhead{} & \\colhead{(h,m,s)} & \\colhead{(d,m,s)} & \\colhead{($z$)} & \\colhead{(Mpc)} & \\colhead{} & \\colhead{(arcsec)} & \\colhead{} & \\colhead{(mJy)} &\\colhead{(mJy)} & \\colhead{($\\rm Jy~km~s^{-1}$)} & \\colhead{} }\n\\startdata\nNGC\\,4945 & 13:05:27.51 & $-$49:28:06.0 & 0.00188 & 3.8 $\\pm$ 0.3 & 1 & $7.9 \\times 4.3$ & 9 & 6450 & 33 & 9887 $\\pm$ 28 & AGN \\\\\nNGC\\,1068 & 02:42:40.70 & $-$00:00:48.0 & 0.00379 & 10.1 $\\pm$ 2.0 & 1 & $9.2 \\times 2.2$ & 9 & 4848 & 2.6 & 1903 $\\pm$ 77 & AGN \\\\\nNGC\\,7552 & 23:16:10.70 & $-$42:35:05.0 & 0.00536 & 14.8 $\\pm$ 1.3 & 2 & $7.3 \\times 7.1$ & 9 & 280 & ~~... $^e$ & 652 $\\pm$ 88 & SF \\\\\nNGC\\,4418 & 12:26:54.61 & $-$00:52:39.0 & 0.00727 & 23.9 $\\pm$ 2.2 & 3 & $0.5$ & 10 & 41 & ~~... $^e$ & 132 $\\pm$ 28 & SF \\\\\nNGC\\,1365 & 03:33:36.40 & $-$36:08:25.0 & 0.00546 & 17.5 $\\pm$ 3.5 & 1 & $21.7 \\times 10.5$ & 9 & 376 & ~~ 4.57 $^f$ & 2166 $\\pm$ 102 & AGN \\\\\nNGC\\,3256 & 10:27:51.30 & $-$43:54:13.0 & 0.00935 & 37.4 $\\pm$ 6.0 & 4 & $12.2 \\times 9.3$ & 9 & 668 & ~~... $^e$ & 1223 $\\pm$ 8 & SF \\\\\nNGC\\,1808 & 05:07:42.30 & $-$37:30:47.0 & 0.00332 & 12.3 $\\pm$ 2.5 & 1 & $14 \\times 8.8$ & 9 & 528 & ~~... $^e$ & 1898 $\\pm$ 137 & SF \\\\\nIRAS\\,13120-5453 & 13:15:06.30 & $-$55:09:23.0 & 0.031249 & 129.3 $\\pm$ 9.1 & 1 & $1.5 \\times 0.5$ & 9 & 118 & ~~... $^e$ & 126 $\\pm$ 13 & SF \\\\\nIRAS\\,13242-5713 & 13:27:23.80 & $-$57:29:22.0 & 0.009788 & 37.6 $\\pm$ 2.6 & 5 & $6 \\times 3.7$ & 9 & 97 & ~~... $^e$ & ... & SF \\\\\nMRK\\,331 & 23:51:26.80 & $+$20:35:10.0 & 0.01848 & 53 $\\pm$ 5 & 3 & $2.53$ & 11 & 71 & $<7.5$ & 49.5 $\\pm$ 3.7 & SF \\\\\nNGC\\,6240A & 16:52:58.90 & $+$02:24:03.5 & \\multirow{2}{*}{0.02488 } & \\multirow{2}{*}{103 $\\pm$ 7} & \\multirow{2}{*}{ 6 } & $0.66 \\times 0.42$ & \\multirow{2}{*}{~~12 $^b$} & \\multirow{2}{*}{396} & \\multirow{2}{*}{5.4} & \\multirow{2}{*}{333.2 $\\pm$ 33} & \\multirow{2}{*}{AGN} \\\\\nNGC\\,6240B & 16:52:58.91 & $+$02:24:04.2 & & & & $1.03\\times 0.41$ & & & & & \\\\\nNGC\\,3628 & 11:20:17.00 & $+$13:35:23.0 & 0.00281 & 10.3 $\\pm$ 0.4 & 7 & $71$ & 13 & 476 & $<4.5$ & 1309 $\\pm$ 25 & SF \\\\\nNGC\\,3627 & 11:20:14.90 & $+$12:59:30.0 & 0.00243 & 10.7 $\\pm$ 0.5 & 1 & $160$ & 13 & 459 & $<3.39$ & 4477 $\\pm$ 75 & AGN \\\\\nIRAS\\,18293-3413 & 18:32:41.10 & $-$34:11:27.0 & 0.017996 & 74.8 $\\pm$ 5.3 & 5 & $5.9 \\times 4.9$ & 9 & 226 & ~~... $^e$ & 686.1 $\\pm$ 7.6 & SF \\\\\nNGC\\,7469 & 23:03:15.60 & $-$08:52:26.0 & 0.01632 & 59.7 $\\pm$ 1.6 & 3 & $3.3$ & 11 & 181 & 32.5 & 48.1 $\\pm$ 3.5 & AGN \\\\\nIRAS\\,17578-0400 & 18:00:31.90 & $-$04:00:53.0 & 0.013325 & 60 $\\pm$ 6 & 3 & $3 \\times 2.3$ & 9 & 80 & ~~... $^e$ & ... & SF \\\\\nIC\\,1623 & 01:07:47.20 & $-$17:30:25.0 & 0.02007 & 80.9 $\\pm$ 5.7 & 6 & $15$ & 11 & 249 & 4.7 & 291 $\\pm$ 45 & AGN \\\\\nArp\\,220A & 15:34:57.29 & $+$23:30:11.3 & \\multirow{2}{*}{0.01813} & \\multirow{2}{*}{84.1 $\\pm$ 5.9} & \\multirow{2}{*}{ 6 } & $0.27 \\times 0.24$ & \\multirow{2}{*}{~~12 $^c$} & \\multirow{2}{*}{515} & \\multirow{2}{*}{91.25} & \\multirow{2}{*}{515 $\\pm$ 51} & \\multirow{2}{*}{SF} \\\\\nArp\\,220B & 15:34:57.22 & $+$23:30:11.5 & & & & $ 0.49\\times 0.31$ & & & & & \\\\\nIRAS\\,19254-7245 & 19:31:21.40 & $-$72:39:18.0 & 0.06171 & 273 $\\pm$ 18 & 8 & $0.5$ & 14 & ... & ... & 58.8 $\\pm$ 5.9 & AGN \\\\\n\\enddata\n\\tablecomments{ \n$^a$ We adopt redshift-independent distances measures with Tully-Fisher\nrelation, Tip of the Red-Giant Branch (TRGB) stars, supernova Ia (SN Ia), and\nCepheids from NED for most of the galaxies. NGC\\,1068, NGC\\,7552, NGC\\,4418, NGC\\,3265, NGC\\,1808,\nMRK\\,331, NGC\\,3628, and IRAS\\,17578-0400 are measured using the Tully-Fisher\nrelation \\citep{Nasonova11,Russell02,Theureau07,Tully88,Tully13}. NGC\\,4945 and\nNGC\\,3627 are measured with TRGBs \\citep{Tully15,Jang17}. NGC\\,1365 is\nmeasured with Cepheid \\citep{Willick01}. NGC\\,7469 is measured with SN Ia\n\\citep{Koshida17}. We adopt Hubble Flow Distance of the rest galaxies in our sample\n\\citep{Karachentsev96,Vaucouleurs91,Mould00}.\\\\\n$^b$ Size of NGC\\,6240 is estimated from two-component Gaussian fitting of ALMA 480 GHz continuum observation (Project code: 2015.1.00717.S).\\\\\n$^c$ Measurement of Arp\\,220 uses data of combination of VLA A configuration and MERLIN from \\cite{Varenius16}. \\\\\n$^d$ Fluxes of 1.4-GHz continuum only have uniform error of the survey and the errors of individual galaxy is hard to determine.\\\\\n$^e$ These galaxies are all star-formation\ndominated, so their SMBHs would not affect 1.4\\,GHz continuum\nsize. \\citep{Varenius14NGC4418,Saikia1990NGC1808,Herrera2017IRAS17578}\\\\\n$^f$ High resolution 1.4\\,GHz flux of NGC\\,1365 comes from VLA\nobservation in \\cite{Sandqvist1995NGC1365} }\n\\tablerefs{(1) \\citet{Nasonova11}; (2) \\citet{Russell02}; (3) \\citet{Theureau07}; (4) \\citet{Tully88}; (5) \\citet{Vaucouleurs91}; (6) \\citet{Mould00}; (7) \\citet{Tully13}; (8) \\citet{distref8}; (9) \\citet{Condon2021}; (10) \\citet{Costagliola13}; (11) \\citet{liu2015}; (12) This work; (13) \\citet{Condon87}; (14) \\citet{Imanishi16}}\n\n\\end{deluxetable*}\n\n\\subsection{Sample selection}\n\nOur sample was selected from the survey of CS $J=7\\rightarrow6$, HCN\n$J=4\\rightarrow3$, and HCO$^+~J=4\\rightarrow3$ \\citep{zhang14apj}, which\ncontains nearby normal galaxies, luminous and ultra-luminous infrared galaxies\n(ULIRGs). These galaxies are originally selected from the \\textrm{ Infrared\nAstronomical Satellite (IRAS)} Revised Bright Galaxy Sample\n\\citep{sanders2003}. All galaxies have $S_{\\nu}(100\\,\\rm \\mu m)>100\\,\\rm Jy$,\nand declination $<20^{\\circ}$ to be accessible from APEX. We exclude three\ntargets without any detection of HCN $J=4\\rightarrow3$ and\nHCO$^+~J=4\\rightarrow3$. The final sample consists of 19 galaxies, which\ninclude 17 newly observed galaxies and two ULIRGs from the literature, Arp~220\n\\citep{Galametz16} and Superantennae \\citep{Imanishi22}.\n\nThe total IR luminosities range from $1.8\\times 10^{10}$ $\\rm L_\\odot$\\, to $1.8\\times\n10^{12}$ $\\rm L_\\odot$, which implies a range of SFR from $3.6~\\rm M_{\\odot}\\,yr^{-1}$\nto $360~\\rm M_{\\odot}\\,yr^{-1}$. The distance range is $\\sim$ 3.72 -- 273\nMpc. The molecular gas masses (estimated from CO $J=1\\rightarrow0$) range\nfrom $4\\rm \\times 10^8~M_{\\odot}$ to $2.1 \\rm \\times 10^{10}~M_{\\odot}$. We\nfurther divided the sample into AGN-dominated and Star formation (SF)-dominated\ngalaxies according to the classifications on NASA\/IPAC Extragalactic Database\n(NED)\\footnote{\\url{http:\/\/ned.ipac.caltech.edu\/}}. The final sample consists\nof eight AGN-dominated and eleven SF-dominated galaxies. The basic information\nof the sample is shown in Table \\ref{table:basicinfo}. \n\n\n\\subsection{HCN\\,$J=2\\rightarrow1$\\ and HCO$^+$\\,$J=2\\rightarrow1$\\ observations}\n\n\\begin{figure*}[ht]\n\\includegraphics[height=1.45in]{NGC4945.eps}\n\\includegraphics[height=1.45in]{NGC1068.eps}\n\\includegraphics[height=1.45in]{NGC7552.eps}\n\\includegraphics[height=1.45in]{NGC4418.eps}\n\\includegraphics[height=1.45in]{NGC1365.eps}\n\\includegraphics[height=1.45in]{NGC3256.eps}\n\\includegraphics[height=1.45in]{NGC1808.eps}\n\\includegraphics[height=1.45in]{IRAS13120-5453.eps}\n\\includegraphics[height=1.45in]{IRAS13242-5713.eps}\n\\includegraphics[height=1.45in]{MRK331.eps}\n\\includegraphics[height=1.45in]{NGC6240.eps}\n\\includegraphics[height=1.45in]{NGC3628.eps}\n\\includegraphics[height=1.45in]{NGC3627.eps}\n\\includegraphics[height=1.45in]{IRAS18293-3413.eps}\n\\includegraphics[height=1.45in]{NGC7469.eps}\n\\includegraphics[height=1.45in]{IRAS17578-0400.eps}\n\\includegraphics[height=1.45in]{IC1623.eps}\n\\caption{HCN\\,$J=2\\rightarrow1$\\ and HCO$^+$\\,$J=2\\rightarrow1$\\ spectra of 17 galaxies observed with the APEX 12-m\n telescope. Solid and dashed lines show line profiles of HCN\\,$J=2\\rightarrow1$\\ and\nHCO$^+$\\,$J=2\\rightarrow1$, respectively. We detect both lines in 12 galaxies. Five sources only\nhave one detected line with a velocity-integrated line intensity $>$\n3$\\sigma$. The non-detection are tagged as ``ND'' above the line labels.} \n\\label{spectrum}\n\\end{figure*}\n\n\nThe observation was conducted with the APEX 12-m telescope during 2016 July and\nAugust (Project ID: E-097.B-0986A-2016). The weather was in good (precipitable\nwater vapour; PWV $<$ 0.9 mm) conditions for ten sources, and in normal\nconditions (PWV $\\sim$ 1--1.4 mm) for the rest. The wobbler switching mode was\nadopted for all observations, with a switching frequency of 2\\,Hz and a beam\nthrow of 120$''$ at each side of the target. The beam size is $\\sim$34.7$''$\n(from 34.2$''$ to 35.2$''$) on average, with a slight variation between\ntargets, depending on the specific line and redshift. \n\nWe employed the Swedish-ESO PI Instrument for APEX (SEPIA) receivers\n\\citep{Belitsky18} to observe HCN\\,$J=2\\rightarrow1$\\ and HCO$^+$\\,$J=2\\rightarrow1$\\ simultaneously, which have rest\nfrequencies of 177.261 GHz and 178.375 GHz, respectively. The SEPIA-180\nreceiver offers two sidebands, double polarisations, and a 4-GHz bandwidth for\neach sideband. The targeted two lines are configured in the lower sideband.\nFocusing calibrations were conducted on Mars, Jupiter, IRAS~15194-5115, IK-Tau,\nand R-Dor, every 3--4\\,hrs. Pointing calibrations were conducted on Mars,\nJupiter, or nearby ($<10^\\circ$) carbon stars every hour.\n\n\n\n\nWe used the {\\sc class} package in {\\sc gildas}\n\\footnote{\\url{https:\/\/www.iram.fr\/IRAMFR\/GILDAS\/}} to reduce the spectral line\ndata. We checked the quality of all spectra by eye, fitted the baseline with a\nfirst-order polynomial profile, and averaged them together. Each sideband has\n104,851 channels with an initial velocity resolution of 0.0644 $\\rm\nkm\\,s^{-1}$. We smoothed each spectrum to a velocity resolution of 26\\,$\\rm\nkm\\,s^{-1}$, at which the final r.m.s. noise ranges from 1.5\\,mK to 2\\,mK. The\nmain beam temperature, $T_{\\rm mb}$, is converted from the antenna temperature,\n$T^*_{\\rm A}$, using $T_{\\rm mb}=T^*_{\\rm A}\\eta_{\\rm f}\/\\eta_{\\rm mb}$, where\n$\\eta_{\\rm f}=0.95$ is the forward efficiency, and $\\eta_{\\rm mb}$=0.73 is the\nmain beam efficiency. We adopt Kelvin to Jansky conversion factor 39$\\,\\rm\nJy\/K$ to convert $T^*_{\\rm A}$ to flux density from the beam-covered regions.\nThe observational results are shown in Table\n\\ref{table:obsresult}, including integrated flux, Gaussian fitting Full Width\nHalf Maximum (FWHM) velocity width, and Gaussian fitting Peak flux density. \n\n\n\n\n\\subsection{Infrared data}\n\nWe obtained multi-wavelength photometric data of the Photodetector Array Camera\nand Spectrometer \\citep[PACS; for 70\\,$\\rm \\mu m$, 100\\,$\\rm \\mu m$, 160\\,$\\rm \\mu m$;][]{PACS2010} and\nthe Spectral and Photometric Imaging REceiver \\citep[SPIRE; for 250\\,$\\rm \\mu m$,\n350\\,$\\rm \\mu m$, 500\\,$\\rm \\mu m$;][] {SPIRE2010}, on board the Herschel Space Observatory.\nThese data were processed to level 2.5 and downloaded from European Space\nAgency (ESA)\\footnote{\\url{http:\/\/archives.esac.esa.int\/hsa\/whsa\/}}. We also\ndownloaded available 24\\,$\\rm \\mu m$~data from the archival Spitzer Space Telescope\n\\citep[MIPS;][]{MIPS2004}, which were processed to level 2. The beam sizes, in\nHalf Power Beam Width (HPBW), are 13.9$''$, 6.4$''$, 5.7$''$, 7$''$, 11.2$''$,\n18.2$''$, 24.9$''$, and 36.1$''$ for WISE 22\\,$\\rm \\mu m$, MIPS 24\\,$\\rm \\mu m$, PACS 70\\,$\\rm \\mu m$,\n100\\,$\\rm \\mu m$, 160\\,$\\rm \\mu m$, SPIRE 250\\,$\\rm \\mu m$, 350\\,$\\rm \\mu m$, and 500\\,$\\rm \\mu m$, respectively. \n\n\nBecause some nearby galaxies can not be fully enclosed by the APEX beam, we further\nmeasured the emission size of the PACS 70\\,$\\rm \\mu m$\\ images with the diameter which\nencloses 90\\% flux of the entire galaxy and listed them in Table\n\\ref{table:obsresult}. From the PACS 70 $\\rm \\mu m$\\ images, seven galaxies (NGC\\,4945,\nNGC\\,1068, NGC\\,7552, NGC\\,1365, NGC\\,1808, NGC\\,3627, and NGC\\,3628) have\nsizes (in diameter) larger than the APEX beam or can not be fully covered by\nthe APEX beam, while the other 12 galaxies are fully enclosed by the APEX beam\nof 34.7$''$ (FWHM). For these 12 point-like galaxies, we adopt the 25\\,$\\rm \\mu m$,\n60\\,$\\rm \\mu m$, and 100\\,$\\rm \\mu m$\\ fluxes from the RBGS survey \\citep{sanders2003}, which\nwas obtained with angular resolutions of 0.7$'$, 1.7$'$, $\\sim$ 3$'$,\nrespectively. For the seven extended galaxies, we adopt IRAS fluxes scaled from\nother wavelengths, as described in Section \\ref{section:photometry}. \n\n\n\n\\subsection{Radio continuum}\n\\label{radiocontinuum}\n\nWe assume that the 1.4\\,GHz radio continuum emission is dominated by\nsynchrotron radiation from supernova remnants \\citep{White1985} and free-free\nradiation from {H\\\/{\\sc ii}} regions \\citep{essential2016}, both would trace\nrecent star-forming activities. We use the 1.4\\,GHz radio continuum to measure\nthe size of star-forming region, assuming that star formation contributes\nmajority of the radio flux. To verify this assumption and to check possible AGN\ncontamination, very high-resolution radio data is needed. Therefore, we\ncollected very long baseline interferometric (VLBI) data in the literature,\nwhich are only available for a few targets.\n\n\nFor galaxies with multiple measurements of the radio size, we adopt the ones\nobserved with highest angular resolutions (Table \\ref{table:basicinfo}). Ten\ngalaxies have sizes measured from MeerKAT data, which has an angular resolution\nof $\\sim$7.5$''$ \\citep{Condon2021}. Sizes of NGC\\,3627 and NGC\\,3628 were\nmeasured using NRAO VLA Sky Survey \\citep[NVSS;][]{Condon87}, which has an\nangular resolution of 45$''$. The size of NGC\\,4418 is estimated from MERLIN\nobservation at a resolution of $0''.35\\times 0''.16$ \\citep{Costagliola13}. We\nadopt the 250-GHz ALMA continuum size for IRAS\\,19254-7245, which was measured\nby \\cite{Imanishi16}.\n\n\n\n\nFor other galaxies, we measured their sizes using data downloaded from ALMA\ndata archive and from the literature. We fit the maps with a 2-D Gaussian\nprofile, using task {\\sc imfit} in {casa} \\citep{CASAdoc}. The 1.4\\,GHz\ncontinuum data of Arp\\,220 was combined with MERLIN data and VLA\nA-configuration data in the literature \\citep{Varenius16}. NGC\\,6240 and\nIRAS\\,19254-7245 have strong AGN contribution to the 1.4 GHz continuum, we\nestimate their sizes using ALMA dust continuum data. We download ALMA 480-GHz\ndata of NGC~6240 (Project code: 2015.1.00717.S) and fit its sizes with a\ntwo-component Gaussian model (NGC~6140A and NGC~6240B in Table\n\\ref{table:basicinfo}). \n\nThe sizes and fluxes of the radio continuum are from the whole galaxies. When\ncomputing luminosity surface densities in Section \\ref{sec:surfacedensity}, we\nadopt the radio area for galaxies with radio sizes smaller than the APEX beam.\nFor galaxies with radio sizes larger than the APEX beam, we adopt the\nAPEX beam area to compute the surface densities.\n\n\n\\subsection{Ancillary CO data}\n\nMost of the velocity-integrated CO $J=1\\rightarrow0$\\ fluxes come from\n\\cite{Baan2008}, observed with the Institut de Radioastronomie Millimetrique\n(IRAM) 30-m telescope (HPBW $\\sim$21$''$) and the Swedish-ESO Submillimeter\nTelescope (SEST) 15-m telescope (HPBW $\\sim$45$''$). CO fluxes of\nIRAS\\,13120-5453 and NGC\\,4418 are from \\cite{sliwa17} and\n\\cite{Papadopoulos12} observed with ALMA and IRAM 30 m, respectively. However,\nthere is no low-$J$ CO data available for IRAS\\,13242-5713 and\nIRAS\\,17578-0400. \n\n\\begin{deluxetable*}{cccccccc}\n\\tablenum{2}\n\\tablecaption{Observational Result}\\label{table:obsresult}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Source name} & \\multicolumn{3}{c}{HCN\\,$J=2\\rightarrow1$} & \\multicolumn{3}{c}{HCO$^+$\\,$J=2\\rightarrow1$} & \\colhead{Diameter$_{70}$ $^a$}\\\\\n\\colhead{} & \\colhead{Integrated flux} & \\colhead{FWHM} & \\colhead{Peak} & \\colhead{Integrated flux} & \\colhead{FWHM} & \\colhead{Peak} & \\\\\n\\colhead{} & \\colhead{($\\rm Jy~km~s^{-1}$)} & \\colhead{($\\rm km~s^{-1}$)} & \\colhead{(mJy)}& \\colhead{($\\rm Jy~km~s^{-1}$)} & \\colhead{($\\rm km~s^{-1}$)} & \\colhead{(mJy)} & \\colhead{(arcsec)}\n}\n\\startdata\nNGC\\,4945 & 885 $\\pm$ 11 & 328.2 $\\pm$ 1.9 & 3120 $\\pm$ 27 & 855 $\\pm$ 11 & 328.1 $\\pm$ 2.7 & 2917 $\\pm$ 35 & ~~ 85.2 $^b$ \\\\\nNGC\\,1068 & 243.5 $\\pm$ 6.4 & 257.9 $\\pm$ 6.3 & 1076 $\\pm$ 35 & 180 $\\pm$ 6 & 253.4 $\\pm$ 8.7 & 807 $\\pm$ 39 & ~~ 38.6 $^b$ \\\\\nNGC\\,7552 & 63.7 $\\pm$ 6.0 & 138.5 $\\pm$ 8.6 & 546 $\\pm$ 47 & 85.8 $\\pm$ 5.9 & 166.7 $\\pm$ 11.1 & 593 $\\pm$ 55 & ~~ 21.9 $^c$ \\\\\nNGC\\,4418 & 54.4 $\\pm$ 5.9 & 193 $\\pm$ 20 & 261 $\\pm$ 35 & 38.7 $\\pm$ 5.9 & 107 $\\pm$ 17 & 215 $\\pm$ 43 & 11.3 \\\\\nNGC\\,1365 & 95.7 $\\pm$ 8.7 & 328 $\\pm$ 29 & 343 $\\pm$ 39 & 71.1 $\\pm$ 8.5 & 241 $\\pm$ 24 & 351 $\\pm$ 47 & ~~ 19.2 $^c$ \\\\\nNGC\\,3256 & 57.9 $\\pm$ 7.6 & 180 $\\pm$ 18 & 398 $\\pm$ 55 & 106.1 $\\pm$ 7.6 & 181 $\\pm$ 11 & 683 $\\pm$ 59 & 19.5 \\\\\nNGC\\,1808 & 55.7 $\\pm$ 7.5 & 236 $\\pm$ 25 & 289 $\\pm$ 43 & 58.0 $\\pm$ 7.4 & 156 $\\pm$ 29 & 394 $\\pm$ 90 & ~~ 22.9 $^c$ \\\\\nIRAS\\,13120-5453 & 48 $\\pm$ 11 & 333 $\\pm$ 52 & 187 $\\pm$ 39 & 43 $\\pm$ 11 & 362 $\\pm$ 85 & 152 $\\pm$ 47 & 11.5 \\\\\nIRAS\\,13242-5713 & 57 $\\pm$ 11 & 223 $\\pm$ 42 & 300 $\\pm$ 78 & 108 $\\pm$ 11 & 194 $\\pm$ 18 & 636 $\\pm$ 82 & 16.7 \\\\\nMRK\\,331 & 30.4 $\\pm$ 8.3 & 214 $\\pm$ 83 & 101 $\\pm$ 27 & $<$ 32 & ... & ... & 11.0 \\\\\nNGC\\,6240 & 61 $\\pm$ 15 & 809 $\\pm$ 142 & 101 $\\pm$ 23 & 83 $\\pm$ 15 & 640 $\\pm$ 84 & 152 $\\pm$ 27 & 11.4 \\\\\nNGC\\,3628 & 41 $\\pm$ 11 & 235 $\\pm$ 58 & 187 $\\pm$ 59 & 55 $\\pm$ 11 & 210 $\\pm$ 30 & 289 $\\pm$ 59 & ~~ 42.5 $^b$ \\\\\nNGC\\,3627 & 46 $\\pm$ 12 & 332 $\\pm$ 62 & 179 $\\pm$ 43 & $<$ 54 & ... & ... & ~~ 63.8 $^b$ \\\\\nIRAS\\,18293-3413 & 35 $\\pm$ 10 & 332 $\\pm$ 105 & 117 $\\pm$ 51 & 60 $\\pm$ 10 & 418 $\\pm$ 96 & 164 $\\pm$ 47 & 12.3 \\\\\nNGC\\,7469 & $<$ 50.2 & ... & ... & 39.8 $\\pm$ 9.9 & 287 $\\pm$ 61 & 164 $\\pm$ 47 & 11.7 \\\\\nIRAS\\,17578-0400 & $<$ 38.2 & ... & ... & $<$ 33 & ... & ... & 11.7 \\\\\nIC\\,1623 & $<$ 32.1 & ... & ... & 30.6 $\\pm$ 8.8 & 208 $\\pm$ 47 & 187 $\\pm$ 55 & 14.0 \\\\\n\\enddata\n\\tablecomments{ \n$^a$ Diameter of PACS\\,70$\\rm \\mu m$\\ image which encloses 90\\% flux of the entire galaxy, which is convolved with PACS\\,70$\\rm \\mu m$\\ 5.7$''$ beam. \\\\\n$^b$ NGC\\,4945, NGC\\,1068, NGC\\,3628, and NGC\\,3627 have larger diameters than APEX beam.\\\\\n$^c$ NGC\\,7552, NGC\\,1365, and NGC\\,1808 have long structure disks which cannot be enclosed by APEX beam.\\\\\n}\n\\end{deluxetable*}\n\n\n\\section{Method and Analysis}\\label{sec:methods}\n\n\\subsection{HCN and HCO$^+$ $J=2\\rightarrow1$\\ line luminosity}\n\nWe calculate the velocity-integrated main beam temperature and the associated\nthermal noise following \\cite{trgreve2009}:\n\n\\begin{equation}\n\\centering\n\\sigma\\left(\\int _{\\Delta_ {\\rm v}}T_{\\rm mb}dv\\right)=\\sqrt{N_{\\Delta _{\\rm v}}}\\left(1+\\frac {N_{\\Delta _{\\rm v}}}{N_{\\rm bas}}\\right)^{1\/2} \\sigma (T_{\\rm mb,ch})\\Delta v_{\\rm ch} + 10\\,\\% \\int _{\\Delta_ {\\rm v}}T_{\\rm mb}dv,\n\\end{equation} \n\nwhere $v_{\\rm ch}~$= 26 $\\rm km\\,s^{-1}$\\, is the final velocity resolution,\n$N_{\\Delta_{\\rm v}}=(\\Delta v\/\\Delta v_{\\rm ch})$ is the number of channels\nthat cover the line, $N_{\\rm bas}$ is the number of the line-free channels, and\n$\\sigma (T_{\\rm mb,ch})$ is the channel-to-channel r.m.s. noise. We adopt\n10\\,\\% as the absolute flux calibration error during observation.\n\nThen we computed the line luminosities of the region observed in the whole\ngalaxy using the equation from \\cite{Gao2004a}:\n\n\\begin{equation}\nL^\\prime=\\pi\/(4{\\rm ln}2)\\theta_{\\rm mb}^2\\int_{\\Delta _{\\rm v}}T_{\\rm mb}dv~D_{\\rm L}^2(1+z)^{-3},\n\\end{equation}\n\nwhere luminosity distance, $D_{\\rm L}$, and redshift, $z$, are taken from NED\n(Table \\ref{table:basicinfo}). We further propagate uncertainties from distance\nand flux to the final error of line luminosities, using the following formula:\n\n\\begin{equation}\n\\centering\n\\sigma_{\\rm flux}(L^\\prime)=\\pi\/(4{\\rm ln}2)\\theta_{\\rm mb}^2(1+z)^{-3}D_{\\rm L}^2~\\sigma\\left(\\int _{\\Delta_ {\\rm v}}T_{\\rm mb}dv\\right),\n\\end{equation}\n\\begin{equation}\n\\centering\n\\sigma_{\\rm dist}(L^\\prime)=\\pi\/(4{\\rm ln}2)\\theta_{\\rm mb}^2(1+z)^{-3}\\times 2D_{\\rm L}\\int _{\\Delta_ {\\rm v}}T_{\\rm mb}dv~\\sigma\\left(D_{\\rm L}\\right),\n\\end{equation}\n\\begin{equation}\n\\centering\n\\sigma(L^\\prime)=\\sqrt{\\sigma^2_{\\rm flux}(L^\\prime)+\\sigma^2_{\\rm dist}(L^\\prime)},\n\\end{equation}\n\nwhere $\\sigma_{\\rm flux}(L^\\prime)$ is the luminosity error propagated from\nflux error, $\\sigma_{\\rm dist}(L^\\prime)$ is the luminosity error propagated\nfrom distance estimation, and $\\sigma (D_{\\rm L})$ is the distance error.\nBecause all luminosities adopt the same distance, their ratios do not include\nerrors of distances.\n\n\n\\begin{deluxetable*}{cccccccccc}\n\\tablenum{3}\n\\tablecaption{Infrared Fluxes inside the APEX beam} \\label{infrared}\n\\tablewidth{0pt}\n\\setlength\\tabcolsep{3pt}\n\\tablehead{\n\\colhead{Source name} & \\colhead{MIPS 24} & \\colhead{IRAS 25} & \\colhead{IRAS 60} & \\colhead{PACS 70} & \\colhead{PACS 100} & \\colhead{PACS 160} & \\colhead{SPIRE 250} & \\colhead{SPIRE 350} & \\colhead{SPIRE 500} \\\\\n\\colhead{} & \\colhead{(Jy beam$^{-1}$)} & \\colhead{(Jy)} & \\colhead{(Jy)} & \\colhead{(Jy beam$^{-1}$)} & \\colhead{(Jy beam$^{-1}$)} & \\colhead{(Jy beam$^{-1}$)} & \\colhead{(Jy beam$^{-1}$)} & \\colhead{(Jy beam$^{-1}$)} & \\colhead{(Jy beam$^{-1}$)}\n}\n\\startdata\nNGC\\,4945 & 9.7 $\\pm$ 1.0 & 42.3$\\pm$ 4.3 & 625.5 $\\pm$ 62.7 & 734.5 $\\pm$ 73.5 & 1060 $\\pm$ 106 & 883.8 $\\pm$ 88.4 & 333 $\\pm$ 33 & 124 $\\pm$ 12 & 33.1$\\pm$ 3.3 \\\\\nNGC\\,1068 & 31.4 $\\pm$ 3.2 & 87.6$\\pm$ 8.9 & 196.4 $\\pm$ 19.7 & 180.7 $\\pm$ 18.1 & ... & 146.4 $\\pm$ 14.7 & 51.1 $\\pm$ 5.1 & 18.3 $\\pm$ 1.8 & 5.0 $\\pm$ 0.51 \\\\\nNGC\\,7552 & ... & 11.9$\\pm$ 1.2 & 77.4 $\\pm$ 7.8 & 82.5 $\\pm$ 8.3 & 95.6 $\\pm$ 9.6 & 67.1 $\\pm$ 6.9 & 22.5 $\\pm$ 2.3 & 7.6 $\\pm$ 0.8 & 2.0 $\\pm$ 0.21 \\\\\nNGC\\,4418 & 5.9 $\\pm$ 0.7 & 9.7 $\\pm$ 1.1 & 43.9 $\\pm$ 4.5 & 41.5 $\\pm$ 4.2 & 33.0 $\\pm$ 3.3 & 18.1 $\\pm$ 1.8 & 6.2 $\\pm$ 0.6 & 2.3 $\\pm$ 0.2 & 0.7 $\\pm$ 0.08 \\\\\nNGC\\,1365 & 8.2 $\\pm$ 0.8 & 14.3$\\pm$ 1.5 & 94.3 $\\pm$ 9.5 & 95.7 $\\pm$ 9.6 & 132.4 $\\pm$ 13.3 & 111.4 $\\pm$ 11.2 & 46.5 $\\pm$ 4.7 & 17.8 $\\pm$ 1.8 & 5.1 $\\pm$ 0.52 \\\\\nNGC\\,3256 & 11.1 $\\pm$ 1.1 & 15.7$\\pm$ 1.6 & 102.6 $\\pm$ 10.3 & 111.3 $\\pm$ 11.1 & 120.7 $\\pm$ 12.1 & 82.5 $\\pm$ 8.3 & 26.9 $\\pm$ 2.7 & 8.7 $\\pm$ 0.9 & 2.5 $\\pm$ 0.26 \\\\\nNGC\\,1808 & 9.7 $\\pm$ 1.0 & 17.0$\\pm$ 1.7 & 105.5 $\\pm$ 10.6 & 113.1 $\\pm$ 11.3 & 130.5 $\\pm$ 13.1 & 94.7 $\\pm$ 9.5 & 33.6 $\\pm$ 3.4 & 11.6 $\\pm$ 1.2 & 3.1 $\\pm$ 0.32 \\\\\nIRAS\\,13120-5453 & 2.4 $\\pm$ 0.2 & 3.0 $\\pm$ 0.3 & 41.1 $\\pm$ 4.2 & 47.7 $\\pm$ 4.8 & 52.3 $\\pm$ 5.2 & 33.7 $\\pm$ 3.4 & 11.7 $\\pm$ 1.2 & 4.1 $\\pm$ 0.4 & 1.0 $\\pm$ 0.12 \\\\\nIRAS\\,13242-5713 & 5.5 $\\pm$ 0.6 & 7.6 $\\pm$ 0.8 & 81.4 $\\pm$ 8.2 & 91.5 $\\pm$ 9.2 & 98.1 $\\pm$ 9.8 & 66.0 $\\pm$ 6.7 & 23.0 $\\pm$ 2.4 & 7.9 $\\pm$ 0.9 & 2.2 $\\pm$ 0.25 \\\\\nMRK\\,331 & 1.9 $\\pm$ 0.2 & 2.5 $\\pm$ 0.3 & 18.0 $\\pm$ 1.8 & 20.0 $\\pm$ 2.0 & 22.5 $\\pm$ 2.3 & 16.3 $\\pm$ 1.7 & 5.8 $\\pm$ 0.6 & 2.1 $\\pm$ 0.2 & 0.5 $\\pm$ 0.06 \\\\\nNGC\\,6240 & 2.9 $\\pm$ 0.3 & 3.5 $\\pm$ 0.4 & 22.9 $\\pm$ 2.3 & 25.0 $\\pm$ 2.5 & 26.2 $\\pm$ 2.6 & 17.2 $\\pm$ 1.9 & 5.8 $\\pm$ 0.6 & 2.0 $\\pm$ 0.2 & 0.6 $\\pm$ 0.07 \\\\\nNGC\\,3628 & 2.1 $\\pm$ 0.4 & 4.8 $\\pm$ 0.5 & 54.8 $\\pm$ 5.6 & 50.6 $\\pm$ 5.1 & 52.0 $\\pm$ 5.2 & 67.8 $\\pm$ 6.8 & 29.1 $\\pm$ 2.9 & 10.8 $\\pm$ 1.1 & 3.1 $\\pm$ 0.32 \\\\\nNGC\\,3627 & ... & 8.6 $\\pm$ 0.9 & 66.3 $\\pm$ 6.7 & 15.2 $\\pm$ 1.5 & 22.8 $\\pm$ 2.3 & 19.0 $\\pm$ 1.9 & 7.0 $\\pm$ 0.7 & 2.5 $\\pm$ 0.3 & 0.8 $\\pm$ 0.08 \\\\\nIRAS\\,18293-3413 & 3.1 $\\pm$ 0.3 & 4.0 $\\pm$ 0.4 & 35.7 $\\pm$ 3.6 & 42.8 $\\pm$ 4.3 & 55.1 $\\pm$ 5.5 & 42.6 $\\pm$ 4.3 & 15.0 $\\pm$ 1.5 & 5.6 $\\pm$ 0.6 & 1.5 $\\pm$ 0.16 \\\\\nNGC\\,7469 & 4.6 $\\pm$ 0.5 & 6.0 $\\pm$ 0.6 & 27.3 $\\pm$ 2.8 & 28.9 $\\pm$ 3.1 & 32.9 $\\pm$ 3.7 & 23.3 $\\pm$ 2.8 & 8.2 $\\pm$ 1.0 & 2.9 $\\pm$ 0.4 & 0.7 $\\pm$ 0.10 \\\\\nIRAS\\,17578-0400 & 0.7 $\\pm$ 0.1 & 1.1 $\\pm$ 0.2 & 27.7 $\\pm$ 3.0 & 31.4 $\\pm$ 3.3 & 34.5 $\\pm$ 3.7 & 23.3 $\\pm$ 2.6 & 8.5 $\\pm$ 1.0 & 3.0 $\\pm$ 0.4 & 0.8 $\\pm$ 0.10 \\\\\nIC\\,1623 & 2.7 $\\pm$ 0.3 & 3.6 $\\pm$ 0.4 & 22.9 $\\pm$ 2.4 & 24.6 $\\pm$ 2.5 & 27.6 $\\pm$ 2.8 & 20.3 $\\pm$ 2.4 & 7.3 $\\pm$ 0.7 & 2.7 $\\pm$ 0.3 & 0.7 $\\pm$ 0.08 \\\\\nArp\\,220 & 5.3 $\\pm$ 0.5 & 8.0 $\\pm$ 0.8 & 104.1 $\\pm$ 10.5 & 132.8 $\\pm$ 13.4 & ... & 83.3 $\\pm$ 8.4 & 30.6 $\\pm$ 3.1 & 10.7 $\\pm$ 1.1 & 3.7 $\\pm$ 0.39 \\\\\nIRAS\\,19254-7245 & 1.1 $\\pm$ 0.1 & 1.4 $\\pm$ 0.2 & 5.2 $\\pm$ 0.6 & 5.5 $\\pm$ 0.6 & 5.8 $\\pm$ 0.6 & 4.1 $\\pm$ 0.4 & 1.5 $\\pm$ 0.2 & 0.5 $\\pm$ 0.1 & 0.1 $\\pm$ 0.02 \\\\\n\\enddata\n\\tablecomments{Flux densities of MIPS, PACS, and SPIRE maps are extracted at\n the same positions as the APEX pointings. We first convolve their\n native angular resolutions to the same beamsize of our APEX\n observations and scale the map unit to Jy beam$^{-1}$, by a factor of $\\rm\n \\pi\/4ln\\,2 \\times (HPBW_{beam}\/pixelsize)^2$. The IRAS data are taken\n from \\cite{sanders2003} as flux densities of the entire galaxies.}\n\\end{deluxetable*}\n\n\\subsection{Photometry, Infrared Luminosity, and Dust Temperature}\n\\label{section:photometry}\n\nBecause our APEX observations have different beam sizes compared to that of the\nIR photometric data, it is not possible to directly compare dense gas tracers\nwith IR data. Therefore, we convolve the Spitzer and Herschel maps to match\nthe beam size of the APEX 12-m data, a 34.7$''$ (diameter) Gaussian beam, using\nthe convolution kernels provided by \\cite{Aniano2011}. Compared to aperture\nphotometry, this method is more robust and is much less affected by the spatial\ndistribution of the target \\citep[e.g.,][]{Tan2018}. Details of the method were\ndescribed in \\cite{Tan2018}. \n\n\nTo measure the photometric flux at each wavelength, and to estimate their\nassociated noise levels, our photometry procedures are listed as follows:\nFirst, we subtract the background of an annulus region from 1.5 to 2 $\\times$\nthe maximum source size, which is estimated using the curve of growth (see\nAppendix \\ref{app:bkg} for details). Second, we scale the image units to\nJy\\,beam$^{-1}$ by a factor of $\\rm \\pi\/4ln\\,2 \\times\n(HPBW_{beam}\/pixelsize)^2$. Third, we measure the value of the central pixel to\nobtain the flux density at each IR band. Last, the error is estimated from the\nbackground regions. The photometric results are listed in Table \\ref{infrared}.\nThe flux error consists of photometric error, flux calibration error, and\nsystematic error. We adopt an absolute flux calibration error of 7\\,\\% and a\nsystematic error of 3\\,\\%, following \\cite{Balog14}. \n\nSince {\\it IRAS} and {\\it Herschel} SPIRE 500\\,$\\rm \\mu m$~data have lower resolutions\nthan our APEX data, we scale the IRAS\\,25\\,$\\rm \\mu m$, IRAS\\,60\\,$\\rm \\mu m$, and\nSPIRE\\,500\\,$\\rm \\mu m$\\ data to obtain the beam-matched fluxes, using aperture\ncorrection factors obtained from MIPS\\,24\\,$\\rm \\mu m$, PACS\\,70\\,$\\rm \\mu m$, and\nSPIRE\\,350\\,$\\rm \\mu m$\\ maps, respectively. For example, we measure the flux of entire galaxy ($S^{\\rm total}_{70}$) and flux in the 34.7$''$ gaussian beam of the PACS\\,70\\,$\\rm \\mu m$\\ ($S^{\\rm beam}_{70}$). Then we adopt $S^{\\rm total}_{60}*S^{\\rm beam}_{70}\/S^{\\rm total}_{70}$ as the IRAS\\,60\\,$\\rm \\mu m$\\ flux in the beam.\n\nWe then build dust spectral energy distribution (SED) for each galaxy and fit\ntotal IR luminosity and dust temperature. We fit the data with a two-component\nmodified blackbody (MBB) dust model, following \\cite{Galametz2013}. Since the\nwarm dust component contributes a non-negligible fraction to the 70\\,$\\rm \\mu m$~and\n100\\,$\\rm \\mu m$\\ emission, a single MBB model would not match the Wien side of the SED\nand would overestimate the cold dust component. The two-component MBB model is\ndescribed as follows:\n\n\\begin{equation}\nS_\\nu = A_{\\rm w}~\\lambda^{-2}B_\\nu(T_{\\rm w}) + A_{\\rm c}~\\lambda^{-\\beta_{\\rm c}}B_\\nu(T_{\\rm c}),\n\\end{equation}\n\nwhere $S_\\nu$ is the flux density obtained from photometry; $B_\\nu$ is the\nPlanck function; $\\beta_{\\rm c}$ is the emissivity index of the cold component;\n$T_{\\rm w}$ and $T_{\\rm c}$ are the temperature of warm and cold components,\nrespectively; $A_{\\rm c}$ and $A_{\\rm w}$ describe the peaks of the two\ncomponents, respectively. To limit the number of free parameters, we fix\n$\\beta_{\\rm w}= 2.0$, as a good approximation of dust model of\n\\cite{LiDraine2001}. So, the estimate of $T_{\\rm w}$ has an additional\nsystematic error compared to $T_{\\rm c}$.\n\nWe fit the SEDs and estimate the uncertainties using the Markov Chain Monte\nCarlo ({\\sf MCMC}) package, {\\sc emcee} \\citep{emcee2013}. Details of the dust\nSED fitting are shown in Appendix \\ref{app:SED_fitting}. Parameters obtained\nfrom the SED fitting are listed in Table \\ref{table:fittingresult}. Then we\ncompute the total IR luminosity by integrating from 3\\,$\\rm \\mu m$~to 1000\\,$\\rm \\mu m$, as\n$L_{\\rm IR}$, which is adopted as the SFR tracer. The beam-matched $L_{\\rm IR}$\naccounts for $\\sim 10\\%-100\\%$ of $L^{\\rm whole}_{\\rm IR}$ from the whole\ngalaxy provided by \\cite{sanders2003}. In Appendix \\ref{app:compare_lum}, we\npresent $L^\\prime_{\\rm HCN}$-$L_{\\rm IR}$ correlations between Far-IR\nluminosity (from 100\\,$\\rm \\mu m$~to 1000\\,$\\rm \\mu m$), near-to-mid-IR luminosity (from\n3\\,$\\rm \\mu m$~to 100\\,$\\rm \\mu m$) , and IR luminosity of the warm dust component. \n\n\n$L_{\\rm IR}$ within the 34.7$''$ beam area ranges from 2.7$\\,\\times 10^9 \\, \\rm\nL_{\\odot}$ to 1.7$\\,\\times 10^{12} \\, \\rm L_{\\odot}$. Cold dust temperature\n$T_{\\rm c}$ ranges from 16.9 K to 32.3 K, with an average value of 24 K.\nCompared with the result of the overlapped galaxies in \\cite{Uvivian12}, our\ndust temperature is lower, possibly because of the inclusion of the\nHerschel\/SPIRE data, which would better constrain the cold dust component. Our\nfitting result is consistent with \\cite{Galametz2013}.\n\n\\subsection{Molecular gas mass and SFR estimation}\n\nIn this work, we derive dense gas mass from $L'_{\\rm HCN {\\it\nJ}=2\\rightarrow1}$ and $L'_{\\rm HCO^+ {\\it J}=2\\rightarrow1}$ following\n\\cite{Gao2004a}: \n\n\\begin{equation}\nM_{\\rm dense}({\\rm H_2}) \\approx 2.1 \\frac{\\langle \\, n({\\rm H_2}) \\rangle ^{1\/2}}{T_{\\rm b}}L^\\prime_{\\rm HCN\\,1-0}\\sim 10~L^\\prime_{\\rm HCN\\,1-0}~\\rm M_{\\odot}\\,(K\\,km\\,s^{-1}\\,pc^2)^{-1},\n\\end{equation}\n\nwhere $\\langle \\, n({\\rm H_2}) \\rangle$ is the average density of the dense gas\nand $T_{\\rm b}$ is the brightness temperature of the dense gas tracer. The\nestimation of dense gas mass relies on both brightness temperature and density,\nwhere we adopt $T_{\\rm b}\\sim 35~\\rm K$ and $\\langle \\, n({\\rm H_2}) \\rangle =\n3 \\times 10^4~\\rm cm^{-3}$ for typical conditions of Galactic virialized cloud\ncores \\citep{Radford1991a}. There are few HCN\\,$J=2\\rightarrow1$\\ and HCO$^+$\\,$J=2\\rightarrow1$\\ observations in\nthe literature \\citep[e.g.,][]{Immer2016}. Therefore, we can derive an average\nHCN\\,$J=2\\rightarrow1$\/HCN\\,$J=1\\rightarrow0$\\ $T_{\\rm b}$ from\nHCN\\,$J=3\\rightarrow2$\/HCN\\,$J=1\\rightarrow0$ ratios with RADEX, by assuming\nthe above physical conditions. We adopt the average\nHCN\\,$J=3\\rightarrow2$\/HCN\\,$J=1\\rightarrow0$ $T_{\\rm b}$ ratio of $\\sim$\n0.26$\\pm$0.10 found in Galactic dense cores \\citep{Wu2010}, and the derived\nHCN\\,$J=2\\rightarrow1$\/HCN\\,$J=1\\rightarrow0$\\ $T_{\\rm b}$ ratio is $\\sim 0.67$, which is further\nadopted to convert from luminosity to the dense gas mass. We derive the dense\ngas mass traced by HCO$^+$\\ with the same method, by assuming that the two lines\nhave similar excitations. The derived dense gas mass ranges from\n$4\\times10^7~\\rm M_\\odot$ to $10^{10}~\\rm M_\\odot$. \n\n\nFollowing \\cite{Tan2018}, we adopt the SFR conversion calibrated by\n\\cite{Kennicutt98a} and \\cite{Murphy2011}:\n\n\\begin{equation}\n\\left(\\frac{\\rm SFR}{\\rm M_{\\odot}yr^{-1}}\\right)=1.50\\times 10^{-10}\\left(\\frac{L_{\\rm IR}}{\\rm L_{\\odot}}\\right)\n\\end{equation}\n\nThe SFR is calculated based on \\cite{Kroupa2001} IMF. The derived gas mass and SFR are shown in Table \\ref{table:fittingresult}. \n\n\\begin{figure*}[ht]\n\\includegraphics[height=3.5in,width=7in]{.\/relation_mcmc.pdf}\n\\caption{{\\it Left:} Correlation between $L^\\prime_{\\rm HCN\\,{\\it\nJ}=2\\rightarrow1}$ and $L_{\\rm IR}$. {\\it Right:} Correlation between\n$L^\\prime_{\\rm HCO^+\\,{\\it J}=2\\rightarrow1}$ and $L_{\\rm IR}$. AGN-dominated\nand star-formation dominated galaxies are shown in blue and red points,\nrespectively. The fitting results of Orthogonal Least Squares ({\\sf OLS}) and\nMarkov chain Monte Carlo ({\\sf MCMC}) are shown in black and cyan lines,\nrespectively. The green-dashed line shows a linear relation for reference.\nThe insets present the probability density distributions of the slope\nsampling.}\n\\label{SFLrelation}\n\\end{figure*}\n\n\\begin{figure*}[ht]\n\\includegraphics[height=3.5in,width=7in]{.\/relation_des_mcmc.pdf}\n\\caption{{\\it Left:} Correlation between $\\Sigma_{L^\\prime_{\\rm HCN\\,{\\it\nJ}=2\\rightarrow1}}$ and $\\Sigma_{L_{\\rm IR}}$. {\\it Right:} Correlation between\n$\\Sigma_{L^\\prime_{\\rm HCO^+\\,{\\it J}=2\\rightarrow1}}$ and $\\Sigma_{L_{\\rm\nIR}}$. AGN-dominated and star-formation dominated galaxies are shown in blue\nand red points, respectively. The fitting results of Orthogonal Least Squares\n({\\sf OLS}) and Markov chain Monte Carlo ({\\sf MCMC}) are shown in black and\ncyan lines, respectively. The green-dashed line shows a linear relation for\nreference. The insets present the probability density distributions of the\nfitted slopes. } \\label{densityrelation}\n\\end{figure*}\n\n\\begin{deluxetable*}{cccccccccc}\n\\tablenum{4}\n\\tablecaption{Derived molecular Properties and Fitting Results} \\label{table:fittingresult}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Source name} & \\colhead{$L'_{\\rm HCN\\,2-1}$} & \\colhead{$L'_{\\rm HCO^+\\,2-1}$} &\\colhead{$L_{\\rm IR}$}&\\colhead{$T_{\\rm c}$} & \\colhead{$\\beta_{c}$} & \\colhead{$T_{\\rm w}$} & \\colhead{$M^{\\rm HCN}_{\\rm dense}$} & \\colhead{$M^{\\rm HCO^+}_{\\rm dense}$} &\\colhead{$\\rm SFR_{IR}$} \\\\\n\\colhead{}& \\multicolumn{2}{c}{($\\rm 10^7\\,K\\,km\\,s^{-1}\\,pc^{2}$)}& \\colhead{($10^9\\,\\rm L_{\\rm \\odot}$)} & \\colhead{(K)}&\\colhead{}&\\colhead{(K)}&\\colhead{($\\rm 10^8~M_\\odot$)}&\\colhead{($\\rm 10^8~M_\\odot$)}&\\colhead{($\\rm M_\\odot\\,yr^{-1}$)}\\\\\n\\colhead{(1)}&\\colhead{(2)}&\\colhead{(3)}&\\colhead{(4)}&\\colhead{(5)}&\\colhead{(6)}&\\colhead{(7)}&\\colhead{(8)}&\\colhead{(9)}&\\colhead{(10)}\n}\n\\startdata\nNGC\\,4945 & 1.7 $\\pm$ 0.4 & 1.6 $\\pm$ 0.4 & 19.0 $\\pm$ 0.8 & 23 $\\pm$ 1 & 2.4 $\\pm$ 0.2 & 49 $\\pm$ 2 & 1.8 $\\pm$ 0.4 & 1.8 $\\pm$ 0.4 & 3.5 $\\pm$ 0.4 \\\\\nNGC\\,1068 & 3.2 $\\pm$ 1.6 & 2.4 $\\pm$ 1.2 & 42.3 $\\pm$ 3.1 & 28 $\\pm$ 1 & 2.1 $\\pm$ 0.1 & 74 $\\pm$ 10 & 3.4 $\\pm$ 1.8 & 2.5 $\\pm$ 1.3 & 7.3 $\\pm$ 1.1 \\\\\nNGC\\,7552 & 1.8 $\\pm$ 0.5 & 2.4 $\\pm$ 0.7 & 40.0 $\\pm$ 2.3 & 23 $\\pm$ 2 & 2.5 $\\pm$ 0.2 & 62 $\\pm$ 2 & 1.9 $\\pm$ 0.6 & 2.7 $\\pm$ 0.7 & 7.9 $\\pm$ 1.2 \\\\\nNGC\\,4418 & 4.0 $\\pm$ 1.3 & 2.8 $\\pm$ 1.0 & 55.2 $\\pm$ 2.7 & 30 $\\pm$ 4 & 1.8 $\\pm$ 0.2 & 62 $\\pm$ 2 & 4.3 $\\pm$ 1.3 & 3.1 $\\pm$ 1.0 & 9.6 $\\pm$ 1.4 \\\\\nNGC\\,1365 & 3.8 $\\pm$ 1.9 & 2.8 $\\pm$ 1.5 & 69.7 $\\pm$ 3.3 & 22 $\\pm$ 1 & 2.3 $\\pm$ 0.2 & 58 $\\pm$ 2 & 4.2 $\\pm$ 2.1 & 3.0 $\\pm$ 1.6 & 12.2 $\\pm$ 1.7 \\\\\nNGC\\,3256 & 10.4 $\\pm$ 4.6 & 19.0 $\\pm$ 8.1 & 351 $\\pm$ 18 & 25 $\\pm$ 2 & 2.4 $\\pm$ 0.2 & 62 $\\pm$ 3 & 11.2 $\\pm$ 5.1 & 20.6 $\\pm$ 8.8 & 53.3 $\\pm$ 8.1 \\\\\nNGC\\,1808 & 1.1 $\\pm$ 0.6 & 1.1 $\\pm$ 0.6 & 20.7 $\\pm$ 0.8 & 23 $\\pm$ 2 & 2.5 $\\pm$ 0.2 & 61 $\\pm$ 2 & 1.2 $\\pm$ 0.6 & 1.2 $\\pm$ 0.6 & 3.5 $\\pm$ 0.5 \\\\\nIRAS\\,13120-5453 & 97 $\\pm$ 36 & 87 $\\pm$ 35 & 1517 $\\pm$ 79 & 25 $\\pm$ 2 & 2.4 $\\pm$ 0.2 & 54 $\\pm$ 2 & 106 $\\pm$ 39 & 94 $\\pm$ 37 & 243.5 $\\pm$ 36.7 \\\\\nIRAS\\,13242-5713 & 10.3 $\\pm$ 3.4 & 19.5 $\\pm$ 5.3 & 258 $\\pm$ 14 & 24 $\\pm$ 2 & 2.4 $\\pm$ 0.2 & 55 $\\pm$ 2 & 11.2 $\\pm$ 3.7 & 21.2 $\\pm$ 5.7 & 41.4 $\\pm$ 6.2 \\\\\nMRK\\,331 & 10.7 $\\pm$ 4.6 & $<$ 11.1 & 125.0 $\\pm$ 6.5 & 23 $\\pm$ 2 & 2.5 $\\pm$ 0.2 & 59 $\\pm$ 2 & 11.6 $\\pm$ 4.9 & $<$ 11.9 & 21.3 $\\pm$ 3.1 \\\\\nNGC\\,6240 & 80 $\\pm$ 30 & 107 $\\pm$ 36 & 586 $\\pm$ 35 & 25 $\\pm$ 2 & 2.3 $\\pm$ 0.2 & 60 $\\pm$ 2 & 87 $\\pm$ 32 & 117 $\\pm$ 39 & 95 $\\pm$ 15 \\\\\nNGC\\,3628 & 0.6 $\\pm$ 0.2 & 0.8 $\\pm$ 0.2 & 11.7 $\\pm$ 0.5 & 17 $\\pm$ 1 & 2.8 $\\pm$ 0.2 & 51 $\\pm$ 1 & 0.6 $\\pm$ 0.3 & 0.9 $\\pm$ 0.3 & 2.2 $\\pm$ 0.3 \\\\\nNGC\\,3627 & 0.7 $\\pm$ 0.3 & $<$ 0.8 & 2.7 $\\pm$ 0.1 & 23 $\\pm$ 1 & 2.4 $\\pm$ 0.1 & 57 $\\pm$ 3 & 0.7 $\\pm$ 0.3 & $<$ 0.9 & 0.7 $\\pm$ 0.1 \\\\\nIRAS\\,18293-3413 & 25 $\\pm$ 10 & 42 $\\pm$ 13 & 521 $\\pm$ 25 & 23 $\\pm$ 2 & 2.4 $\\pm$ 0.2 & 58 $\\pm$ 2 & 26.7 $\\pm$ 11.3 & 46 $\\pm$ 15 & 83 $\\pm$ 12 \\\\\nNGC\\,7469 & $<$ 22.4 & 17.8 $\\pm$ 6.3 & 250 $\\pm$ 16 & 23 $\\pm$ 2 & 2.4 $\\pm$ 0.2 & 63 $\\pm$ 3 & $<$ 24.3 & 19.3 $\\pm$ 6.9 & 41.8 $\\pm$ 6.4 \\\\\nIRAS\\,17578-0400 & $<$ 17.4 & $<$ 14.9 & 205 $\\pm$ 10 & 23 $\\pm$ 2 & 2.4 $\\pm$ 0.2 & 48 $\\pm$ 1 & $<$ 18.8 & $<$ 16.1 & 32.8 $\\pm$ 4.8 \\\\\nIC\\,1623 & $<$ 26.0 & 25 $\\pm$ 10 & 350 $\\pm$ 19 & 23 $\\pm$ 2 & 2.4 $\\pm$ 0.2 & 60 $\\pm$ 2 & $<$ 28.2 & 27.0 $\\pm$ 11.3 & 59.1 $\\pm$ 8.9 \\\\\nArp\\,220 & 138 $\\pm$ 36 & 70 $\\pm$ 19 & 1650 $\\pm$ 65 & 31 $\\pm$ 3 & 1.7 $\\pm$ 0.1 & 56 $\\pm$ 4 & 151 $\\pm$ 39 & 76 $\\pm$ 21 & 258 $\\pm$ 39 \\\\\nIRAS\\,19254-7245 & 58.9 $\\pm$ 6.3 & 37.5 $\\pm$ 4.3 & 1153 $\\pm$ 97 & 22 $\\pm$ 2 & 2.6 $\\pm$ 0.2 & 64 $\\pm$ 2 & 64 $\\pm$ 15 & 40.7 $\\pm$ 9.9 & 171 $\\pm$ 27 \\\\\n\\enddata\n\\tablecomments{ Column 1: galaxy name. Column 2: HCN\\,$J=2\\rightarrow1$\\ line luminosity.\n Column 3: HCO$^+$\\,$J=2\\rightarrow1$\\ line luminosity. Column 4: total infrared luminosity.\n Column 5: cold component dust temperature. Column 6: cold component\n dust emissivity index. Column 7: warm component dust temperature.\n Column 8: dense gas mass derived from HCN\\,$J=2\\rightarrow1$. Column 9: dense gas mass\n derived from HCO$^+$\\,$J=2\\rightarrow1$. Column 10: star formation rate derived from\n infrared luminosity.}\n\\end{deluxetable*}\n\n\n\\section{Results}\n\\subsection{Spectra}\n\nWe present APEX spectra of HCN\\,$J=2\\rightarrow1$\\, and HCO$^+$\\,$J=2\\rightarrow1$\\, in Figure \\ref{spectrum}. In\ntotal, we detect 14 HCN\\,$J=2\\rightarrow1$\\, and 14 HCO$^+$\\,$J=2\\rightarrow1$\\, lines, with their\nvelocity-integrated line intensities, $I > 3\\,\\sigma$. Twelve galaxies have\ndetections of both HCN\\,$J=2\\rightarrow1$\\, and HCO$^+$\\,$J=2\\rightarrow1$. IRAS\\,17578-0400\nwas not detected in either of the two lines. NGC\\,7469 and IC\\,1623 were not\ndetected only in HCN\\,$J=2\\rightarrow1$, possibly because of their large distances. On the\nother hand, HCO$^+$\\,$J=2\\rightarrow1$\\ is only marginally detected on $\\sim$ 3-$\\sigma$ levels in\nthese two galaxies. MRK\\,331 and NGC\\,3627 have non-detections of HCO$^+$\\,$J=2\\rightarrow1$. The\nvelocity-integrated fluxes of all galaxies are shown in Table\n\\ref{table:obsresult}. We show 3-$\\sigma$ upper limits for those non-detected\nlines. \n\n\n\\subsection{Correlation between luminosities of dense gas tracers and infrared emission}\n\nIn Figure \\ref{SFLrelation}, we present the $L_{\\rm IR}-L'_{\\rm dense}$\ncorrelation using HCN\\,$J=2\\rightarrow1$\\, and HCO$^+$\\,$J=2\\rightarrow1$\\, line luminosities, which trace total\ndense gas mass. We fit linear regressions with both methods of Orthogonal Least\nSquares ({\\sf OLS}) and {\\sf MCMC}, to avoid possible bias from the fitting algorithm. The\nOLS was fitted with an IDL Astrolibrary {\\sc sixlin},\nwhich adopts orthogonal distances from data points to the fit line. The code\nfirst assumes a linear slope to compute orthogonal distance, and then performs\nlinear regression steps to compare with previous slopes, until it reaches\nsteady state. We perform {\\sf MCMC} method using a Python package {\\sc emcee}:\nwe first employ 64 walkers and each of them samples 5,000 steps. This would\nensure that most sampling are convergent. Then we burn in them, and perform\nanother 10,000 steps for each walker. This will offer a final sampling number\nof 640,000 for statistics. We adopt the range encompassing 68\\% of the data\nabout the median of the posterior probability density distribution to represent\napproximate upper and lower 1-$\\sigma$ limits for Gaussian-like distributions.\nThese fitted results from the two methods are shown in Figure\n\\ref{SFLrelation}, without significant difference between each other. The\nslope obtained from OLS fitting is within the 1-$\\sigma$ range of that\nfrom the Bayesian fitting.\n\nWe find linear correlations between $L_{\\rm IR}$ with $L'_{\\rm HCN {\\it\nJ}=2\\rightarrow1}$ and $L'_{\\rm HCO^+ {\\it J}=2\\rightarrow1}$. The Pearson\ncorrelation efficiencies, which quantifies linear correlations, are 0.96 and\n0.97 with $p$-values of $1.7 \\times 10^{-9}$ and $2.0 \\times 10^{-10}$,\nrespectively. The Spearman correlation efficiencies, which quantifies monotone\ncorrelations, are 0.98 and 0.96, with $p$-values of $1.0 \\times 10^{-10}$ and\n$4.8 \\times 10^{-9}$, respectively. AGN-dominated and SF-dominated galaxies are\nshown in blue and red points, respectively. The {\\sc mcmc} fitting results are\nlisted below:\n\n\\begin{equation}\n{\\rm log}L_{\\rm IR}=1.034^{+0.055}_{-0.051}~{\\rm log}L'_{\\rm HCN\\,2-1}+2.91^{+0.4}_{-0.4},\n\\end{equation}\n\\begin{equation}\n{\\rm log}L_{\\rm IR}=1.000^{+0.058}_{-0.054}~{\\rm log}L'_{\\rm HCO^+\\,2-1}+3.21^{+0.43}_{-0.47},\n\\end{equation}\n\nwhere the upper and lower errors are from probability density distributions of\nparameters. Non-detections are not included during fitting. We find no\ndefinite difference between AGN-dominated and SF-dominated galaxies, which\nessentially follow the same trend of $L_{\\rm IR}$ and $L'_{\\rm dense}$. \n\n\\subsection{Correlations of Luminosity Surface Densities}\n\\label{sec:surfacedensity}\n\n\nHere we derive luminosity surface densities by adopting the area measured from\nthe 1.4-GHz radio continuum images, which can eliminate the degeneracy\nintroduced by the distance in luminosity correlations. The 1.4-GHz radio\nemission, which is contributed both by the synchrotron emission from supernova\nremnants and by the free-free emission from H{\\sc ii} regions, originates from\nthe same star-forming regions as the IR emission \\citep{Bell2003}. There exists\nrich archival radio data at high angular resolutions, which would help\ndetermine the 1.4\\,GHz continuum sizes (see details in section\n\\ref{radiocontinuum}). \n\n\nThe correlations between the surface densities of IR luminosity ($\\Sigma_{\nL_{\\rm IR}}$) and dense gas tracer line luminosity ($\\Sigma_{L'_{\\rm dense}}$)\nare shown in Figure \\ref{densityrelation}. The {\\sf MCMC} fitting results are\nshown below:\n\n\\begin{equation}\n{\\rm log}\\Sigma_{L_{\\rm IR}}=0.987^{+0.032}_{-0.030}~{\\rm log}\\Sigma_{L'_{\\rm HCN\\,2-1}}+3.21^{+0.09}_{-0.09}\n\\end{equation}\n\\begin{equation}\n{\\rm log}\\Sigma_{L_{\\rm IR}}=1.017^{+0.035}_{-0.034}~{\\rm log}\\Sigma_{L'_{\\rm HCO^+\\,2-1}}+3.17^{+0.09}_{-0.10}\n\\end{equation}\n\nwhere $\\Sigma_{L}=L\/(2\\pi r^2_{\\rm RC})$, is the luminosity surface density.\nThe fitted slope indices of HCN\\,$J=2\\rightarrow1$\\ and HCO$^+$\\,$J=2\\rightarrow1$\\ are 0.99 and 1.02,\nrespectively. Both OLS and MCMC methods give identical fitting results. The\nPearson correlation efficiencies are 0.99 and 0.99 with $p$-values of $2.7\n\\times 10^{-12}$ and $1.8 \\times 10^{-13}$, for HCN\\,$J=2\\rightarrow1$\\ and HCO$^+$\\,$J=2\\rightarrow1$,\nrespectively. The Spearman correlation efficiencies of these two lines are 0.99\nand 0.98, with $p$-values of $8.1 \\times 10^{-13}$ and $1.0 \\times 10^{-10}$,\nrespectively. Both correlation coefficients are higher than those obtained\nfrom the luminosity relations, showing much tighter correlations in the surface\ndensity relations. \n\nScatters in the surface density correlations are 0.150 and 0.113 dex for $\\rm\nHCN$ and HCO$^+$\\,$J=2\\rightarrow1$, respectively. These values are very close to the scatters in\nthe luminosity correlations (0.147 and 0.113 dex). This is likely because the\nsurface density correlations have larger dynamic ranges, and both types of\ncorrelations share the same physical origin. \n\n\n\\subsection{Star formation efficiency of dense molecular gas}\n\n\nUsing HCN and HCO$^+$ $J=4\\rightarrow3$ transitions in nearby star-forming\ngalaxies, \\cite{Tan2018} found that $L_{\\rm IR}\/L^\\prime_{ \\rm dense}$, which\nis a proxy of dense gas star formation efficiency (SFE), increases with $L_{\\rm\nIR}$ within individual galaxies, but not for galaxy-integrated ratios.\nFurthermore, there seems to be also a correlation between $L_{\\rm IR}\/L^\\prime_{ \\rm\ndense}$ and PACS 70\/100$\\rm \\mu m$~ratio (as a proxy of warm dust temperature). \n\nTo further verify these trends, we plot SFE (derived from $L_{\\rm IR}\/L'_{\\rm\nHCN}$ and $L_{\\rm IR}\/L'_{\\rm HCO^+}$) as a function of IR luminosity In Figure\n\\ref{SFE}. Our data do not show any statistical correlation in the SFE-$L_{\\rm\nIR}$ diagram, with Spearman correlation $p$-values of 0.99 and 0.85 for HCN\\,$J=2\\rightarrow1$\\\nand HCO$^+$\\,$J=2\\rightarrow1$, respectively (see Table \\ref{table:fittingresult}), meaning that\nthe data has a high chance to distribute randomly. These are consistent with\nthe integrated results of \\cite{Tan2018}. \n\nIn general, SFE obtained from HCN and HCO$^+$\\ seem to be both confined within a\nsmall range, with scatters of only 0.19 and 0.16. These small scatters are even\nless than that of HCN $J=1\\rightarrow0$\\ \\citep[$\\sim 0.25$ found in $L_{\\rm\nIR}\/L_{\\rm HCN 1-0}$][]{Gao2004b}. This indicates that the $J=2\\rightarrow1$\\\ntransition of HCN and HCO$^+$\\ are more robust in tracing the dense gas mass in\ngalaxies. However, the $L_{\\rm IR}\/L_{\\rm HCN 4-3}$ ratios have a much more\ndiverged range of almost two orders of magnitude\\citep{Tan2018}. \n\n\nUnlike \\cite{Tan2018}, on the other hand, the SFE-$T_{\\rm c}$ diagram also\ndoes not show any correlations, with Spearman correlation $p$-values of 0.46\nand 0.91 for HCN\\,$J=2\\rightarrow1$\\ and HCO$^+$\\,$J=2\\rightarrow1$, respectively. The $J=4\\rightarrow3$\ntransitions are more sensitive to the dust temperature, because they need very\nhigh density and relatively high temperature to be excited. These makes them to\ntrace the dense gas close to young massive stars, instead of the global\nproperties cold dense cores traced by the $J=2\\rightarrow1$\\ and\n$J=1\\rightarrow0$\\ transitions. \n\n\n\\begin{figure*}[ht]\n\\includegraphics[height=3.5in]{.\/r_SFR_HCN_IR.pdf}\n\\includegraphics[height=3.5in]{.\/r_SFR_HCO_IR.pdf}\n\\includegraphics[height=3.5in]{.\/r_SFR_HCN_Tc.pdf}\n\\includegraphics[height=3.5in]{.\/r_SFR_HCO_Tc.pdf}\n\\caption{{\\it Top left:} SFR\/$M_{\\rm dense}$ traced by HCN\\,$J=2\\rightarrow1$\\ as a function of\nIR luminosity. {\\it Top right} SFR\/$M_{\\rm dense}$ traced by HCO$^+$\\,$J=2\\rightarrow1$\\ as a\nfunction of IR luminosity. {\\it Bottom left:} SFR\/$M_{\\rm dense}$ traced by\nHCN\\,$J=2\\rightarrow1$\\ as a function of cold dust temperature. {\\it Bottom right:} SFR\/$M_{\\rm\ndense}$ traced by HCN\\,$J=2\\rightarrow1$\\ as a function of cold dust temperature. AGN-dominated\nand star-formation dominated galaxies are shown in blue and red points,\nrespectively. } \\label{SFE}\n\\end{figure*}\n\n\\subsection{Ratios of $L'_{\\rm HCN\\, {\\it J}=2\\rightarrow1}$\/$L'_{\\rm HCO^+\\, {\\it J}=2\\rightarrow1}$ }\n\nFigure \\ref{lineratio} presents the line ratio of $L'_{\\rm HCN}\/L'_{\\rm HCO^+}$\nas a function of IR luminosity surface density and cold dust temperature. We\nalso overlay HCN and HCO$^+$ $J=1\\rightarrow0$\\ ratios collected from the\nliterature \\citep{Wang2004,Baan2008,Gao2004a,Garcia06,Nguyen1992,Solomon1992,Krips08}.\nThere seems no systematic difference between the ratios of both HCN\/HCO$^+$\n$J=1\\rightarrow0$\\ and HCN\/HCO$^+$ $J=2\\rightarrow1$. The average $J=1\\rightarrow0$\\ ratio is 1.25 $\\pm$ 0.42\nand $J=2\\rightarrow1$\\ ratio is 1.05 $\\pm$ 0.42. $J=1\\rightarrow0$\\ ratio is higher, the difference is\nsmaller than the scatter.\n\nThe ratio of $L'_{\\rm HCN}\/L'_{\\rm HCO^+}$ may be changed by variation from\nastrochemistry \\citep{Imanishi07,Lintott06}, molecular excitation\n\\citep{Garcia06}, radiative transfer \\citep{Knudsen07}, metallicities\n\\citep{Liang06}, and many other processes. Temperature and IR luminosity of\ndust grains would reflect the energy from young stars and their surrounding\ndense gas. Therefore, here we try to find possible dependence of this ratio\non star formation intensity (traced by the IR luminosity surface density) and on\nof the cold dust temperature. We choose the temperature of the\ncold dust component, because 1). It comes from the bulk of the galactic dust\ncomposition, taking the majority of the dust mass (>95\\%; e.g., Appendix\n\\ref{app:SED_fitting}); 2). It is well sampled in our SED fitting (e.g., Fig.\n\\ref{SED_plot}), and 3). It contributes the majority of the far-IR luminosity,\nwhich has an excellent correlation with the total IR luminosity\n\\citep[e.g.,][]{Zhu2008}. \n\nThe average ratios of HCN\/HCO$^+$ $J=2\\rightarrow1$\\ are 1.15$\\,\\pm\\,$0.26 and\n0.98$\\,\\pm\\,$0.42 for AGN-dominated galaxies and SF-dominated galaxies,\nrespectively. Though it seems that AGN-dominated galaxies may systematically\nhave higher HCN\/HCO$^+$ line ratios, the difference is still within\n1\\,$\\sigma$. There seems to also exist a weak trend between line ratio and dust\ntemperature. But if we remove the data points of NGC\\,3628 and Arp\\,220, this\ntrend disappears.\n\n\\begin{figure*}[ht]\n\\includegraphics[width=3.5in]{.\/r_HCN_HCOp_IR_des.pdf}\n\\includegraphics[width=3.5in]{.\/r_HCN_HCOp_Tc.pdf}\n\\caption{{\\it Left:} Ratio of $L'_{\\rm HCN}\/L'_{\\rm HCO^+}$ $J=2\\rightarrow1$\\\nas a function of infrared luminosity surface density. {\\it Right:} Ratio of\n$L'_{\\rm HCN}\/L'_{\\rm HCO^+}$ $J=2\\rightarrow1$\\ as a function of cold\ncomponent dust temperature. The sample in this work is shown in circle, while\nthe ALMA-ULIRG sample from \\citet{Imanishi22} is shown in triangle.\nAGN-dominated galaxies are shown in blue filled points and SF-dominated\ngalaxies are shown in red empty points. Grey square points are HCN and HCO$^+$\n$J=1\\rightarrow0$\\ from the literature for comparison\n\\citep{Wang2004,Baan2008,Gao2004a,Garcia06,Nguyen1992,Solomon1992,Krips08}.}\n\\label{lineratio}\n\\end{figure*}\n\n\n\\subsection{Dense gas fraction in galaxies}\n\nThe ratios of $L'_{\\rm HCN}\/L'_{\\rm CO}$ and $L'_{\\rm HCO^+}\/L'_{\\rm CO}$ would\nroughly trace the dense gas fraction in galaxies, albeit the large\nuncertainties of the conversion factors both in CO and dense gas tracers.\nSimilar to what was found in the $J$=1$\\rightarrow$0 transition of HCN in\n\\citet{Gao2004b}, the ratios of $L'_{\\rm HCN}\/L'_{\\rm CO}$ and $L'_{\\rm\nHCO^+}\/L'_{\\rm CO}$ both show increasing trends as a function of $L_{\\rm IR}$\nand $\\Sigma_{L_{\\rm IR}}$ (Figure \\ref{fraction}). The Pearson correlation\ncoefficient between $L'_{\\rm HCN~J=2\\rightarrow1}\/L'_{\\rm CO~J=1\\rightarrow0}$\nratio and IR surface density is 0.81 ($p$-value=2.4$\\times 10^{-4}$), which is higher than that of the $L'_{\\rm\nHCO^+~J=2\\rightarrow1}\/L'_{\\rm CO~J=1\\rightarrow0}$ ratio (0.49, $p$-value=0.064), indicating\nthat HCN\/CO ratios might be more robust in tracing dense gas fractions.\n\nThese are also consistent with the positive correlations between $L'_{\\rm\nHCN\\,1-0,3-2}\/L'_{\\rm CO\\,1-0}$ ratio and $L_{\\rm IR}$\n\\citep[e.g.,][]{Juneau2009}, who found a better correlation for the\n$J=3\\rightarrow2$ transition than that of $J=1\\rightarrow0$\\, indicating an\nincreased molecular gas density in more IR-bright galaxies. \n\n\n\nWe summarize the statistics parameters of correlations and list all of them in\nTable \\ref{table:fittingparam}, including {\\sf MCMC} fitting results, {\\sf OLS}\nfitting results, Pearson (linear relation) ordered correlation coefficient and\n$p$-value, Spearman (monotonic relation) ordered correlation coefficient and\n$p$-value, and scatter of diversions from the {\\sf OLS} fitting. \n\nCorrelations of luminosities ($L'_{\\rm dense}-L_{\\rm IR}$) and luminosity\nsurface densities ($\\Sigma_{L'_{\\rm dense}}-\\Sigma_{L_{\\rm IR}}$) all show\nsignificant correlation (Spearman $p$-value<10$^{-8}$). The correlations of\n$L^\\prime_{\\rm HCN}\/L^\\prime_{\\rm CO}-\\Sigma_{L_{\\rm IR}}$, $L^\\prime_{\\rm\nHCN}\/L^\\prime_{\\rm HCO^+}-\\Sigma_{L_{\\rm IR}}$, and $T_{\\rm c}-\\Sigma_{L_{\\rm\nIR}}$, are also statistically valid with $p$-values $<$ 0.05 for both Pearson\nand Spearman rank. Those non-correlated relations show large error bars of\nlinear fitting and large $p$-values ($>$ 0.05), and sometimes also show large\ndifference between the results from {\\sf MCMC} and {\\sf OLS}. \n\n\n\\begin{figure*}[ht]\n\\includegraphics[height=3.5in,width=3.5in]{r_HCN_CO_to_IR_des.pdf}\n\\includegraphics[height=3.5in,width=3.5in]{r_HCOp_CO_to_IR_des.pdf}\n\\caption{ {\\it Left:} $L'_{\\rm HCN}\/L'_{\\rm CO}$ as a function of IR surface\ndensity. {\\it Right:} $L'_{\\rm HCO^+}\/L'_{\\rm CO}$ as a function of IR surface\ndensity. AGN-dominated and SF-dominated galaxies are shown in blue filled\npoints and red empty points, respectively. The best-fitted results are shown\nwith solid lines. Non-detections are not included in the fitting.}\n\\label{fraction}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=1\\textwidth]{relation_app.eps}\n\\includegraphics[width=1\\textwidth]{relation_des_app.eps}\n\\caption{ {\\it Top:} Correlations of $L'_{\\rm HCN}-L_{\\rm IR}$ ({\\it top left})\nand $L'_{\\rm HCO^+}-L_{\\rm IR}$ ({\\it top right}). {\\it Bottom:} Correlations\nof $\\Sigma_{L^\\prime_{\\rm HCN}}-\\Sigma_{L_{\\rm IR}}$ ({\\it bottom left}) and\n$\\Sigma_{L^\\prime_{\\rm HCO^+}}-\\Sigma_{L_{\\rm IR}}$ ({\\it bottom right}). The\ncorrelations include dense molecular clouds from Magellanic Clouds \\citep{Galametz20},\nwhich are shown in orange circles. The total luminosity and the averaged\nluminosity surface density of detected targets in the Magellanic Clouds are\nalso shown for comparison. Luminosity upper limits of non-detection clouds are\nshown in grey arrows. The purple dashed lines are taken from Figures\n\\ref{SFLrelation} and \\ref{densityrelation} and present fitting results of\ngalaxies.}\n\\label{relation_append}\n\\end{figure*}\n\n\\begin{deluxetable*}{cccccccccc}\n\\tablenum{5}\n\\tablecaption{Statistic parameters of different fittings}\\label{table:fittingparam}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{} & \\multicolumn{2}{c}{\\sf MCMC} & \\multicolumn{2}{c}{\\sf OLS} & \\multicolumn{2}{c}{Pearson} & \\multicolumn{2}{c}{Spearman} & \\colhead{} \\\\\n\\colhead{Fitting name} & \\colhead{slope} & \\colhead{intercept} &\\colhead{slope}&\\colhead{intercept} & \\colhead{$r_{\\rm xy}$} & \\colhead{$p$-value} &\\colhead{$r_{\\rm xy}$} & \\colhead{$p$-value} & \\colhead{scatter}\\\\\n\\colhead{(1)}&\\colhead{(2)}&\\colhead{(3)}&\\colhead{(4)}&\\colhead{(5)}&\\colhead{(6)}&\\colhead{(7)}&\\colhead{(8)}&\\colhead{(9)}&\\colhead{(10)}\n}\n\\startdata\n$L_{\\rm IR}-L^\\prime_{\\rm HCN}$ & $1.03 _{-0.05 } ^{+0.05 }$ & $2.9 _{-0.4 } ^{+0.4 }$ & 1.05 $\\pm$ 0.04 & 2.8 $\\pm$ 0.3 & 0.96 & 1.7 $\\times 10^{-9}$ & 0.98 & 1.0 $\\times 10^{-10}$ & 0.15 \\\\\n$L_{\\rm IR}-L^\\prime_{\\rm HCO^+}$ & $1.00 _{-0.05 } ^{+0.06 }$ & $3.2 _{-0.5 } ^{+0.4 }$ & 1.00 $\\pm$ 0.03 & 3.2 $\\pm$ 0.2 & 0.97 & 2.0 $\\times 10^{-10}$ & 0.96 & 4.8 $\\times 10^{-9}$ & 0.11 \\\\\n$\\Sigma_{L_{\\rm IR}}-\\Sigma_{L^\\prime_{\\rm HCN}}$ & $0.99 _{-0.03 } ^{+0.03 }$ & $3.21 _{-0.09} ^{+0.09}$ & 0.99 $\\pm$ 0.02 & 3.20 $\\pm$ 0.07 & 0.99 & 2.7 $\\times 10^{-12}$ & 0.99 & 8.1 $\\times 10^{-13}$ & 0.15 \\\\\n$\\Sigma_{L_{\\rm IR}}-\\Sigma_{L^\\prime_{\\rm HCO^+}}$ & $1.02 _{-0.03 } ^{+0.04 }$ & $3.17 _{-0.10} ^{+0.09}$ & 1.02 $\\pm$ 0.01 & 3.17 $\\pm$ 0.03 & 0.99 & 1.8 $\\times 10^{-13}$ & 0.98 & 1.0 $\\times 10^{-10}$ & 0.11 \\\\\n$L^\\prime_{\\rm HCN}\/L^\\prime_{\\rm HCO^+}-\\Sigma_{L_{\\rm IR}}$ & $0.33 _{-0.04 } ^{+0.04 }$ & $-0.9 _{-0.2 } ^{+0.2 }$ & 0.30 $\\pm$ 0.03 & -0.7 $\\pm$ 0.2 & 0.65 & 0.013 & 0.59 & 0.027 & 0.32 \\\\\n$L^\\prime_{\\rm HCN}\/L^\\prime_{\\rm HCO^+}-T_{\\rm c}$ & $0.11 _{-0.26 } ^{+0.03 }$ & $-1.6 _{-0.7 } ^{+6.1 }$ & 0.06 $\\pm$ 0.01 & -0.4 $\\pm$ 0.3 & 0.53 & 0.053 & 0.21 & 0.47 & 0.36 \\\\\nSFR\/$M^{\\rm HCN}_{\\rm dense}-L_{\\rm IR}$ & $0.02 _{-0.05 } ^{+0.05 }$ & $-7.7 _{-0.5 } ^{+0.6 }$ & 0.03 $\\pm$ 0.03 & -7.8 $\\pm$ 0.3 & 0.14 & 0.62 & -0.003 & 0.99 & 0.19 \\\\\nSFR\/$M^{\\rm HCO^+}_{\\rm dense}-L_{\\rm IR}$ & $0.01 _{-0.05 } ^{+0.05 }$ & $-7.5 _{-0.6 } ^{+0.6 }$ & -0.01 $\\pm$ 0.02 & -7.3 $\\pm$ 0.3 & -0.13 & 0.62 & -0.053 & 0.85 & 0.16 \\\\\nSFR\/$M^{\\rm HCN}_{\\rm dense)}-T_{\\rm c}$ & $-0.02 _{-0.01 } ^{+0.01 }$ & $-7.1 _{-0.3 } ^{+0.3 }$ & -0.016 $\\pm$ 0.002 & -7.07 $\\pm$ 0.06 & -0.26 & 0.33 & -0.20 & 0.46 & 0.18 \\\\\nSFR\/$M^{\\rm HCO^+}_{\\rm dense}-T_{\\rm c}$ & $0.00 _{-0.01 } ^{+0.01 }$ & $-7.5 _{-0.3 } ^{+0.3 }$ & 0.003 $\\pm$ 0.003 & -7.52 $\\pm$ 0.08 & 0.052 & 0.85 & -0.029 & 0.91 & 0.16 \\\\\nSFR\/$M^{\\rm HCN}_{\\rm dense}-T_{\\rm w}$ & $-0.005 _{-0.08 } ^{+0.008 }$ & $-7.8 _{-0.5 } ^{+0.5 }$ & 0.003 $\\pm$ 0.002 & -7.63 $\\pm$ 0.14 & 0.050 & 0.85 & 0.091 & 0.74 & 0.19 \\\\\nSFR\/$M^{\\rm HCO^+}_{\\rm dense}-T_{\\rm w}$ & $0.007 _{-0.007 } ^{+0.008 }$ & $-7.9 _{-0.5 } ^{+0.4 }$ & 0.006 $\\pm$ 0.002 & -7.79 $\\pm$ 0.08 & 0.17 & 0.53 & 0.35 & 0.18 & 0.16 \\\\\n$L^\\prime_{\\rm HCN}\/L^\\prime_{\\rm CO}-\\Sigma_{L_{\\rm IR}}$ & $0.27 _{-0.03 } ^{+0.03 }$ & $-2.8 _{-0.2 } ^{+0.2 }$ & 0.29 $\\pm$ 0.02 & -2.89 $\\pm$ 0.09 & 0.81 & 2.4 $\\times 10^{-4}$ & 0.84 & 8.0 $\\times 10^{-5}$ & 0.27 \\\\\n$L^\\prime_{\\rm HCO^+}\/L^\\prime_{\\rm CO}-\\Sigma_{L_{\\rm IR}}$ & $0.10 _{-0.03 } ^{+0.03 }$ & $-1.8 _{-0.2 } ^{+0.2 }$ & 0.13 $\\pm$ 0.02 & -2.0 $\\pm$ 0.1 & 0.49 & 0.064 & 0.58 & 0.023 & 0.33 \\\\\n$^*L_{\\rm IR}-L^\\prime_{\\rm HCN}$ & $1.01 _{-0.01 } ^{+0.01 }$ & $3.08 _{-0.06} ^{+0.06}$ & 1.010 $\\pm$ 0.006 & 3.11 $\\pm$ 0.05 & 0.997 & 4.2 $\\times 10^{-26}$ & 0.97 & 2.3 $\\times 10^{-14}$ & 0.17 \\\\\n$^*L_{\\rm IR}-L^\\prime_{\\rm HCO^+}$ & $1.082 _{-0.008} ^{+0.008}$ & $2.56 _{-0.04} ^{+0.04}$ & 1.081 $\\pm$ 0.005 & 2.57 $\\pm$ 0.04 & 0.997 & 1.2 $\\times 10^{-38}$ & 0.95 & 3.6 $\\times 10^{-18}$ & 0.20 \\\\\n$^*\\Sigma_{L_{\\rm IR}}-\\Sigma_{L^\\prime_{\\rm HCN}}$ & $1.00 _{-0.01 } ^{+0.02 }$ & $3.18 _{-0.04} ^{+0.04}$ & 0.993 $\\pm$ 0.008 & 3.19 $\\pm$ 0.02 & 0.99 & 1.4 $\\times 10^{-21}$ & 0.99 & 7.1 $\\times 10^{-18}$ & 0.16 \\\\\n$^*\\Sigma_{L_{\\rm IR}}-\\Sigma_{L^\\prime_{\\rm HCO^+}}$ & $1.10 _{-0.01 } ^{+0.01 }$ & $2.90 _{-0.02} ^{+0.02}$ & 1.101 $\\pm$ 0.009 & 2.92 $\\pm$ 0.02 & 0.99 & 1.0 $\\times 10^{-29}$ & 0.97 & 2.6 $\\times 10^{-21}$ & 0.20 \\\\\n$T_{\\rm c}-\\Sigma_{L_{\\rm IR}}$ & $1.4 _{-0.3 } ^{+0.3 }$ & $16 _{-2 } ^{+2 }$ & 3.5 $\\pm$ 0.6 & 6 $\\pm$ 3 & 0.67 & 0.002 & 0.55 & 0.015 & 0.87 \\\\\n$T_{\\rm w}-\\Sigma_{L_{\\rm IR}}$ & $0.9 _{-0.4 } ^{+0.4 }$ & $52 _{-2 } ^{+2 }$ & 25 $\\pm$ 50 & -75 $\\pm$ 260 & 0.13 & 0.58 & 0.091 & 0.71 & 1.20 \\\\\n\\enddata\n\\tablecomments{ Column 1: fitting name. Column 2: slope fitted with Markov Chain Monte\nCarlo ({\\sf MCMC}). Column\n3: intercept fitted with {\\sf MCMC}. Column 4: slope fitted with Orthogonal\nLeast Squares ({\\sf OLS}). Column 5: intercept fitted with {\\sf OLS}. Column 6:\nPearson rank-order correlation coefficient. Column 7: Pearson rank-order\n$p$-value. Column 8: Spearman rank-order correlation coefficient. Column 9:\nSpearman rank-order $p$-value. Column 10: scatter of diversion from the\nfitting.\\\\ $^*$ The fitting includes molecular clouds in LMC and SMC.\n}\n\\end{deluxetable*}\n\n\\section{Discussion}\n\n\n\n\nWe adopt the total IR luminosity (integrated from 3\\,$\\rm \\mu m$~to 1000\\,$\\rm \\mu m$) as the\nSFR tracer of galaxies. $L_{\\rm IR}$ has been widely used as SFR tracer, while\nit still has some limitations. AGN may contribute to mid-infrared emission by\ntheir hot dusty tori \\citep{Padovani2017} and thus $L_{\\rm IR}$ may\noverestimate SFR in strong AGN-dominated galaxies. On the other hand, $L_{\\rm\nIR}$ traces SFR on relatively long timescales up to 200 Myr \\citep{kennicutt12}\nand it may overestimate significantly the instantaneous SFR of recent-quenched\ngalaxies \\citep{Hayward2014}. When converting from gas luminosity to dense gas\nmass, we assume that all molecular lines are collisionally excited. However,\nIR-pumping mechanism may also contribute to the line excitation and could\nlargely enhance HCN and HCO$^+$ emission. If the observed HCN and HCO$^+$\nfluxes are dominated by IR-pumping, the dense gas mass would be overestimated.\nFurthermore, variation in the stellar initial mass function (IMF) between\ndifferent galaxies would also bias the SFR calculation from $L_{\\rm IR}$\n\\citep[e.g.,][]{Jerabkova2017,Zhang2018}.\n\n\n\nThe sizes of star-forming regions may be biased by the compact radio emission\ncontributed by AGN. Therefore, we collect VLBI high-resolution data from the\nliterature\n\\citep{Lenc2009NGC4945,Roy1998NGC1068,Varenius14NGC4418,Smith1998MRK331IC1623arp220,Gallimore2004NGC6240,Cole1998NGC3628,Deller2014NGC3627,Lonsdale2003NGC7469}\nand determine if our measured sizes are severely biased by AGN (see Table\n\\ref{table:basicinfo}). For most galaxies, the central compact sources only\ncontribute $\\lesssim$ 10\\% of the total radio fluxes. Among all targets,\nNGC~7469 has the highest AGN contribution, which is $\\sim$ 18\\%. For the rest\nof the sample, most of them are classified as star-formation dominated galaxies\nso AGN could not contribute much radio emission. For IRAS\\,19254-7245 and\nNGC\\,6240, we adopt ALMA 250 GHz and 480 GHz continuum to estimate star-forming\nregion sizes, respectively.\n\n\n\n\\subsection{Star formation relation for dense gas tracers}\n\nOur results show that SFR follows tight linear correlations with dense gas mass\ntraced by both HCN\\,$J=2\\rightarrow1$\\ and HCO$^+$\\,$J=2\\rightarrow1$, with Slopes of 1.03 $\\pm$ 0.05 and 1.00\n$\\pm$ 0.06, Pearson coefficients of 0.96 and 0.97, and dispersions of 0.15 and\n0.11 dex, respectively (see Figure \\ref{SFLrelation}). The star formation\nefficiencies are roughly constant (Pearson coefficients: 0.14 and -0.13)\nagainst SFR, with scatters of 0.19 dex and 0.16 dex (see Figure \\ref{SFE}) for\nHCN\\,$J=2\\rightarrow1$\\ and HCO$^+$\\,$J=2\\rightarrow1$, respectively. These suggest that dense gas clouds are\ndirect sources of star formation, while HCN\\,$J=2\\rightarrow1$\\ and HCO$^+$\\,$J=2\\rightarrow1$\\ can be good tracers\nof dense molecular gas. On the other hand, we find no systematic variation\nbetween the slopes derived from the $J=2\\rightarrow1$\\ (1.03$\\pm$0.05) and $J=1\\rightarrow0$\\\n(1.00$\\pm$0.05) transitions. \n\n\nNGC\\,6240 and NGC\\,3627 are two outliers deviated from the relationship of IR\nand dense gas tracer luminosity. This makes their dense-gas star formation\nefficiencies, which are traced by infrared to dense gas luminosity ratio, about\nthree times lower than the average value of other galaxies. This may indicate\nthat star formation efficiency of these dense gas is suppressed, or the dense\ngas tracer emission is enhanced by the extra heating mechanisms for the same\nstar-forming activity. NGC\\,6240 is a starburst galaxy in final merger stage\n\\citep{Tacconi99,Iono07,Papadopoulos14,Kollatschny2020}. The enormous shock\ncondition across NGC\\,6240, shown both in the enhanced high-$J$ CO lines and in\nlarge scale shocked gas distribution \\citep{Meijerink2013,Cicone18, Lu15},\ncould contribute mechanical heating to the dense gas, along with cosmic-ray\nand far-ultraviolet (FUV) radiation from photon-dominated regions\n\\citep{Papadopoulos14}. These extreme conditions in NGC~6240 may also enhance\nthe HCN\\,$J=2\\rightarrow1$\\ and HCO$^+$\\,$J=2\\rightarrow1$\\ emission. Our observation of NGC\\,3627 is consistent\nwith \\cite{Murphy2015}, who found that star formation efficiency in the nuclear\nregion of NGC\\,3627 is several times lower than that in off-nuclear star-forming region. This could explain why NGC~3627 also shows offsets from both\nHCN and HCO$^+$\\,$J=2\\rightarrow1$\\ correlations. \n\n\n\\subsection{Connecting to star-forming clouds on small scales}\n\nThe correlations obtained on large scales may have contamination from\nnon-star-forming activities such as shock-enhanced line emission, diffuse gas\ncomponent, etc. Therefore, it is necessary to check if the correlations can be\nextended to pc scales that only contains star-forming dense molecular cores.\nUnfortunately, it is difficult to find systematic observations of the\n$J=2\\rightarrow1$\\ transition towards our Milky Way, while such observations\ndo exist towards the Magellanic Clouds. Therefore, we compare the galaxy survey\nwith the Magellanic Clouds data from \\cite{Galametz20}, who presented HCN\\,$J=2\\rightarrow1$\\\nand HCO$^+$\\,$J=2\\rightarrow1$\\ observations towards $\\sim$30 LMC and SMC molecular clouds using\nAPEX. \n\nAs shown in Figure \\ref{relation_append}, after including molecular clouds of\nthe Magellanic Clouds into our galaxy sample, the correlations between\n$L'_{\\rm HCN~J=2\\rightarrow1}$--$L_{\\rm IR}$ and $L'_{\\rm\nHCO^+~J=2\\rightarrow1}$--$L_{\\rm IR}$ still hold with semi-linear slopes of\n1.021 and 1.088, respectively. The upper limits of non-detections seem off the\ncorrelation, possibly due to different detection criteria in \\cite{Galametz20},\nwho consider a 3-$\\sigma$ peak Gaussian fitting (instead of integrated flux) as\nthe detection threshold. We also include data of Magellanic Clouds into the\ncorrelations of surface densities, by adopting the area from\n\\citep[e.g.,][]{Wong2011,Muller2010}. The obtained slopes are 1.00 and 1.10\nfor $\\Sigma_{L'_{\\rm HCN}}$--$\\Sigma_{L_{\\rm IR}}$ and $\\Sigma_{L'_{\\rm\nHCO^+}}$--$\\Sigma_{L_{\\rm IR}}$, respectively. Both HCN and HCO$^+$\\ data\naveraged (summed) across the Magellanic Clouds are shown in Figure\n\\ref{relation_append} as individual data points, which seem to also follow the\ncorrelations of $\\Sigma_{L'_{\\rm HCN}}$--$\\Sigma_{L_{\\rm IR}}$ ($L'_{\\rm\nHCN~J=2\\rightarrow1}$--$L_{\\rm IR}$) and $\\Sigma_{L'_{\\rm\nHCO^+}}$--$\\Sigma_{L_{\\rm IR}}$ ($L'_{\\rm HCO^+~J=2\\rightarrow1}$--$L_{\\rm\nIR}$) found in galaxies. \n\nOn the other hand, we also simply overlay the Magellanic Clouds data with our\nfitted galaxy-only correlations. The HCN data from Magellanic Clouds well match\nwith the correlation fitted in the galaxy-only sample, while the HCO$^+$\\ data\nfrom Magellanic Clouds systematically lay below the fitted lines from galaxies,\nindicating a systematic HCN\/HCO$^+$ ratio variation possibly due to different\nmetallicities. \n\n\n\\subsection{$L'_{\\rm HCN}\/L'_{\\rm HCO^+}J=2\\rightarrow1$ ratio and its origins} \n\n\n\n\nThe average of HCN-to-HCO$^+$ $J=2\\rightarrow1$\\ flux ratio of our sample is\n1.15$\\pm$0.26 and 0.98$\\pm$0.42 for AGN-hosting and SF-dominated galaxies,\nrespectively. Though it seems that the AGN-hosting galaxies may have relatively\nhigher ratios than SF-dominated galaxies, the difference is still within 1\n$\\sigma$. So, we could not use this ratio to separate populations with AGNs.\nOur result of $J=2\\rightarrow1$\\ is consistent with that found in the $J=1\\rightarrow0$\\ lines\n\\citep{Privon15, Lifei20}. \n\n\n\n\\cite{Kohno2001} suggested that enhanced HCN emission originate from X-ray\ndominated region and can be used to search for pure AGNs, which is also\nsupported by \\cite{Imanishi2004,Imanishi07}. \\cite{Imanishi22} found\nsignificantly higher HCN-to-HCO$^+$ $J=2\\rightarrow1$\\ flux ratios in a high fraction of,\nbut not all, AGN-important ULIRGs than that in starburst-classified sources.\nSome starburst-dominated galaxies may have HCN enhancement, which seems not\ndriven by a single process. We include their flux ratios (Figure\n\\ref{lineratio}). The ALMA-ULIRG sample\\citep{Imanishi22} has extreme\ncontribution from AGNs, as discussed in last paragraph of Section\n\\ref{section:Imanishi}. In such conditions, the HCN\/HCO$^+$ line ratios might\nbe enhanced by the chemistry from X-ray dominated regions\n\\citep{Kohno2001,Harada2015}. \n\n\nThe average HCN-to-HCO$^+$ $J=2\\rightarrow1$\\ flux ratio in Magellanic Clouds is\n0.49$\\pm$0.34, which seems to be systematically lower than the ratios found in\ngalaxies. This is consistent with \\cite{Braine17}, who also found that\nHCN\/HCO$^+$\\,$J=1\\rightarrow0$\\ is lower in low-metallicity environment. \n\nIn both the Milky Way and external galaxies, the [N\/O] abundance ratio has a\npositive correlation with [O\/H] and [Fe\/H], because nitrogen mostly originates\nfrom low-mass stars and its nuclear production is a long-time\nprocess\\citep{Pilyugin03,Liang06}. Therefore, the HCN\/HCO$^+$ abundance ratio\nwould also decrease with [N\/O] in low metallicity environments such as the\nMagellanic Clouds, compared to normal metal-rich star-forming galaxies. \n\n\n\n\\subsection{Influence of infrared pumping}\n\nVibrational transitions of HCN and HCO$^+$ are difficult to be excited by\ncollision, but they can be excited by absorbing infrared photons at $\\sim$ 13\n-- 14\\,$\\rm \\mu m$, through infrared pumping \\citep{Imanishi17}. Then the vibrational\ntransitions can cascade to rotational transitions and enhance HCN\\,$J=2\\rightarrow1$\\ and\/or\nHCO$^+$\\,$J=2\\rightarrow1$\\ lines. If infrared pumping becomes a dominant process, the dense gas\nmass estimated from HCN\\,$J=2\\rightarrow1$\\ and\/or HCO$^+$\\,$J=2\\rightarrow1$\\ would be over-estimated.\n\n\\cite{Sakamoto2010,Sakamoto2021} found strong detections of HCN v$_2$=1\n$J=3\\rightarrow2$, $J=4\\rightarrow3$ in NGC\\,4418 and Arp\\,220 by ALMA\nobservation, indicating somewhat impact on the HCN rotational line. All of\nour spectra do not show any detections of HCN v$_2$=1 $J=2\\rightarrow1$\\ including NGC\\,4418\nand Arp\\,220, possibly due to limited S\/N of APEX observation. Both NGC\\,4418\nand Arp\\,220 show elevated $L'_{\\rm HCN}\/L'_{\\rm HCO^+}$ ratio, especially\namong SF-dominated galaxies (see Figure \\ref{lineratio}), indicating possible\ninfluence of IR pumping. NGC\\,4418 and Arp\\,220 are the most heavily influenced\ngalaxies by infrared pumping in our sample. However, both NGC\\,4418 and\nArp\\,220 do not show outliers in correlations of $L'_{\\rm dense}$-$L_{\\rm IR}$\nand $\\Sigma_{ L'_{\\rm dense}}$-$\\Sigma_{ L_{\\rm IR}}$, indicating limited\nenhancement towards HCN\\,$J=2\\rightarrow1$. Therefore, we do not expect a significant influence\nby IR pumping in our sample. \n\n\n\n\\subsection{Comparison with other HCN and HCO$^+$\\,$J=2\\rightarrow1$\\ work}\n\\label{section:Imanishi}\n\n\n\\cite{Imanishi22} reported ALMA observations of HCN\\,$J=2\\rightarrow1$\\ and HCO$^+$\\,$J=2\\rightarrow1$\\ towards ten\nULIRGs, most of which contain AGNs. They also found positive correlations\nof $L_{\\rm IR}$-$L^\\prime_{\\rm HCN}$ and $L_{\\rm IR}$-$L^\\prime_{\\rm HCO^+}$. \n\nWe obtained the data (Project code: 2017.1.00022.S and 2017.1.00023.S) from the\nALMA archive, and processed HCN and HCO$^+$\\,$J=2\\rightarrow1$\\ datacubes with the standard\npipeline. Overall we got roughly consistent results compared to those in\n\\cite{Imanishi22}. We then adopted the size estimated from dust continuum\ngiven by \\cite{Imanishi22}. Details of the measured fluxes and sizes are shown\nin Appendix \\ref{app:Imanishmeasure} and Table \\ref{table:Imanishflux}.\n \nIn Figure \\ref{comparison}, we overplot the ALMA-ULIRG data on the HCN\\,$J=2\\rightarrow1$\\ and\nHCO$^+$\\,$J=2\\rightarrow1$\\ correlations. The ALMA-ULIRG data seem to be systemically above both\ncorrelations of $L'_{\\rm denseJ=2\\rightarrow1}\/L_{\\rm IR}$ and $\\Sigma_{L'_{\\rm\ndense}}-\\Sigma_{L_{\\rm IR}}$. For our APEX data, the orthogonal scatters on\nthe correlation lines are $0.15$ and $0.11$ dex for $\\rm HCN$ and $\\rm HCO^+$,\nrespectively. And the mean orthogonal offset of the ALMA-ULIRG galaxies are\n$0.24\\pm0.22$ and $0.27\\pm0.16$ dex for $\\rm HCN$ and $\\rm HCO^+$,\nrespectively. Only two and one ALMA-ULIRG galaxies lie in the 1-$\\sigma$ range\nof our $L_{\\rm IR}-L^\\prime_{\\rm HCN}$ and $L_{\\rm IR}-L^\\prime_{\\rm HCO^+}$\ncorrelation, respectively. Two and three galaxies of ALMA-ULIRGs are off by\n$>3\\,\\sigma$ from the correlation lines of $\\rm HCN$ and $\\rm HCO^+$,\nrespectively.\n\n\n\\begin{figure*}[ht]\n\\includegraphics[height=3.5in,width=7in]{relation_Ima_mcmc.pdf}\n\\includegraphics[height=3.5in,width=7in]{relation_Ima_des_mcmc.pdf}\n\\caption{ Correlations including ALMA-ULIRG sample from \\citep{Imanishi22}.\n{\\it Top:} Correlations of $L'_{\\rm HCN}-L_{\\rm IR}$ ({\\it top left}) and\n$L'_{\\rm HCO^+}-L_{\\rm IR}$ ({\\it top right}). {\\it Bottom:} Correlations of\n$\\Sigma_{L^\\prime_{\\rm HCN}}-\\Sigma_{L_{\\rm IR}}$ ({\\it bottom left}) and\n$\\Sigma_{L^\\prime_{\\rm HCO^+}}-\\Sigma_{L_{\\rm IR}}$ ({\\it bottom right}).\nAGN-dominated and star-formation dominated galaxies are shown in blue and red\npoints, respectively. The fitting results of Orthogonal Least Squares ({\\sf\nOLS}) and Markov chain Monte Carlo ({\\sf MCMC}) are shown in black and cyan\nlines, respectively. The green-dashed line shows a linear relation for\nreference. The insets present the probability density distributions of the\nfitted slopes. Data points from \\cite{Imanishi22} are shown in purple and not\nincluded into fitting. The purple dashed lines are the fitting results of $\\rm\nHCN$ $J=4\\rightarrow3$ and $\\rm HCO^+$ $J=4\\rightarrow3$ obtained from\n\\cite{Tan2018}.} \\label{comparison} \n\\end{figure*}\n\n\n\n\nThis offset is consistent with that found by \\cite{Imanishi22}, who found that\nthe ALMA-ULIRG $J=2\\rightarrow1$\\ data follow $J=4\\rightarrow3$ relation\ngiven in \\cite{Tan2018}. \\cite{Imanishi22} interpret that both HCN and HCO$^+$\nare thermalised, which would bring the $\\rm HCN$ and $\\rm HCO^+$\n$J=4\\rightarrow3$ line luminosities comparable to those of\n$J=2\\rightarrow1$\\ transitions. From our APEX data, the fitted linear lines\nare systematically lower than those obtained from the $J=4\\rightarrow3$ data\nfrom \\cite{Tan2018}, making the ALMA-ULIRGs above the $J=2\\rightarrow1$\\\nrelations. \n\n\nConsidering most of ALMA-ULIRGs are AGN-important ULIRGs, their $L_{\\rm IR}$\nmight be dominated by AGNs, similar to the hot dust-obscured galaxies\n(hot-DOGs) found with WISE mid-IR surveys \\citep{Wu2012,Tsai2015}. The IR\nemission of hot-DOGs is mainly from the hot dust heated up by AGNs, which on\naverage contribute $> 75\\%$ of the bolometric luminosity \\citep{Fan2016}. On\nthe other hand, the hot-DOGs also have high dust temperature ($\\langle T_{\\rm\ndust} \\rangle \\sim 72 \\,\\rm K$), which seems to be the same for the\nALMA-ULIRGs. The IRAS $f_{60\\rm \\mu m}\/f_{\\rm 100\\mu m}$ ratio of the\nALMA-ULIRGs are all higher than those of the APEX galaxies, except for\nNGC~4418, with average $f_{60\\rm \\mu m}\/f_{\\rm 100\\mu m}$ ratios of\n1.09$\\,\\pm\\,$0.28 and 0.77$\\,\\pm\\,$0.19 for the ALMA-ULIRGs and our APEX\nsample, respectively. Therefore, we suspect that the ALMA-ULIRGs are\nAGN-dominated, IR-overluminous galaxies, and they should behave similarly for\nthe $J=1\\rightarrow0$\\ lines of both HCN and HCO$^+$. Missing flux of ALMA observation may\nalso help to explain in a few targets (details see Appendix\n\\ref{app:Imanishmeasure}).\n\n\n\n\n\\section{Summary}\nWe present APEX observation towards HCN\\,$J=2\\rightarrow1$\\ and HCO$^+$\\,$J=2\\rightarrow1$\\ in 17 nearby\ninfrared-bright star-forming galaxies. Combining HCN\\,$J=2\\rightarrow1$\\ and HCO$^+$\\,$J=2\\rightarrow1$\\ data in\nthe literature, and with the total IR luminosity fitted from dust SED, we\ncorrelations slopes of $1.03\\pm 0.05$ and 1.00$\\pm 0.05$ for $L'_{\\rm\nHCO^+}-L_{\\rm IR}$ and $L'_{\\rm HCN}-L_{\\rm IR}$, respectively. \n\nTo obtain correlations of surface densities, which could eliminate the biases\nfrom uncertain distances, we use the size of 1.4\\,GHz radio continuum to\nnormalise the luminosities of both IR emission ($\\Sigma_{ L_{\\rm IR}}$) and\ndense gas tracers ($\\Sigma_{ L'_{\\rm HCN}}$ and $\\Sigma_{ L'_{\\rm HCO^+}}$).\nThese surface density correlations also show linear slopes of $0.99 \\pm 0.03$\nand $1.02 \\pm 0.03$, for HCN and HCO$^+ J=2\\rightarrow1$ lines, respectively.\nThe slope errors and $p$-values of the surface-density correlations are also\nsmaller than those of the luminosity correlations. The Spearman correlation\ncoefficients of the $J=2-1$ lines (0.98 and 0.96 for HCN\\,$J=2\\rightarrow1$\\ and HCO$^+$\\,$J=2\\rightarrow1$) are\nhigher than those obtained from the $J=1-0$ \\citep[0.94 for HCN\n$J=1\\rightarrow0$;][]{Gao2004b} and the $J=4-3$ \\citep[0.89 and 0.84 for HCN\n$J=4\\rightarrow3$ and HCO$^+$ $J=4\\rightarrow3$;][]{Tan2018} transitions,\nindicating the advantage of the $J=2\\rightarrow1$\\ transitions of HCN and HCO$^+$ in tracing\nthe star-forming gas. \n\n\nComparing with the HCN\\,$J=2\\rightarrow1$\\ and HCO$^+$\\,$J=2\\rightarrow1$\\ data from star-forming clouds of the\nMagellanic Clouds \\citep{Galametz20}, we find that the low-metallicity\nenvironment not only slightly deviates the data from the overall surface-density\ncorrelations, but also would bias the HCN\\,$J=2\\rightarrow1$\/HCO$^+$\\,$J=2\\rightarrow1$\\ line ratio to low ratios.\nThis is consistent with previous findings in low-metallicity galaxies\n\\citep[e.g.,][]{Braine17}. The systematically lower HCN\/HCO$^+$\\ ratios are\npossibly owing to the variation of [N\/O] elemental abundance variation in\nlow-metallicity environments. \n\n\nOn the other hand, when comparing with the ULIRGs sample from\n\\cite{Imanishi22}, those AGN-important galaxies lay systematically above both\ncorrelations of $L'_{\\rm dense}$-$L_{\\rm IR}$ and $\\Sigma_{ L'_{\\rm\ndense}}$-$\\Sigma_{ L_{\\rm IR}}$ found in our APEX sample. This is likely due to\nthe AGN contribution to the total IR luminosity for such compact objects, which\ncould heavily overestimate SFRs. Therefore, the contribution from AGNs may not\nbe negligible in such extreme conditions.\n\n\n\n\n\n\\begin{acknowledgments}\nWe thank Dr. Maud Galametz for providing HCN and HCO$^+$ data of Magellanic Clouds\nin \\cite{Galametz20}. Z.Y.Z. and J.Z. acknowledge the support of the National\nNatural Science Foundation of China (NSFC) under grants No. 12041305, 12173016,\nthe Program for Innovative Talents, Entrepreneur in Jiangsu, the science\nresearch grants from the China Manned Space Project with No. CMS-CSST-2021-A08\nand CMS-CSST-2021-A07. Chentao Yang acknowledges support from ERC Advanced Grant 789410. This\npublication is based on data acquired with the Atacama Pathfinder Experiment\n(APEX). APEX is a collaboration between the Max-Planck-Institut fur\nRadioastronomie, the European Southern Observatory, and the Onsala Space\nObservatory. $Herschel$ was an ESA space observatory with science instruments\nprovided by European-led Principal Investigator consortia and with important\nparticipation from NASA. PACS has been developed by a consortium of institutes\nled by MPE (Germany) and including UVIE (Austria); KU Leuven, CSL, IMEC\n(Belgium); CEA, LAM (France); MPIA (Germany); INAFIFSI\/OAA\/OAP\/OAT, LENS, SISSA\n(Italy); IAC (Spain). This development has been supported by the funding\nagencies BMVIT (Austria), ESA-PRODEX (Belgium), CEA\/CNES (France), DLR\n(Germany), ASI\/INAF (Italy), and CICYT\/MCYT (Spain). SPIRE has been developed\nby a consortium of institutes led by Cardiff University (UK) and including\nUniversity of Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI,\nUniversity of Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden);\nImperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and Caltech,\nJPL, NHSC, University of Colorado (USA). This development has been supported by\nnational funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS\n(France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC, UKSA (UK); and NASA\n(USA). This work is based [in part] on observations made with the Spitzer Space\nTelescope, which was operated by the Jet Propulsion Laboratory, California\nInstitute of Technology under a contract with NASA. This publication makes use\nof data products from the Wide-field Infrared Survey Explorer, which is a joint\nproject of the University of California, Los Angeles, and the Jet Propulsion\nLaboratory\/California Institute of Technology, funded by the National\nAeronautics and Space Administration. This paper makes use of the following\nALMA data: ADS\/JAO.ALMA\\#2015.1.00717.S. ALMA is a partnership of ESO\n(representing its member states), NSF (USA), and NINS (Japan), together with\nNRC (Canada) and NSC and ASIAA (Taiwan) and KASI (Republic of Korea), in\ncooperation with the Republic of Chile. The Joint ALMA Observatory is\noperated by ESO, AUI\/NRAO and NAOJ.\n\n\\end{acknowledgments}\n\\software{GILDAS\/CLASS \\citep{GILDAS}, Numpy \\citep{numpy1,numpy2}, emcee \\citep{emcee2013}, Photutils \\citep{photutils}}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n \n First detected as a radio source \\citep{1991ApJS...75....1B} by the NRAO Green Bank Telescope and in the Parkes-MIT-NRAO surveys \\citep{1995ApJS...97..347G}, RGB\\,J0152$+$017\\ was later identified as a BL\\,Lac object by \\citet{1998ApJS..118..127L}, who located it at $z=0.080$, and was claimed as an intermediate-frequency-peaked BL\\,Lac object by \\citet{1999ApJ...525..127L}. \\citet{1997A&A...323..739B} report the first detection of RGB\\,J0152$+$017\\ in X-rays in the {\\it ROSAT}-Green Bank (RGB) sample. The host is an elliptical galaxy with luminosity $M_R=-24.0$ \\citep{2003A&A...400...95N}. The source has high radio and X-ray fluxes, making it a good candidate for VHE emission \\citep{2002A&A...384...56C}, motivating its observation by the H.E.S.S. experiment.\n \n The broad-band SED of BL\\,Lac objects is typically characterised by a double-peaked structure, usually attributed to synchrotron radiation in the radio-to-X-ray domain and inverse Compton scattering in the $\\gamma$-ray domain, which is frequently explained by SSC models \\citep[see, e.g.,][]{2005A&A...442..895A}. However, since the flux of BL\\,Lac objects can be highly variable \\citep[e.g.][]{2000A&A...353...97K}, stationary versions of these models are only relevant for contemporaneous multi-wavelength observations of a non-flaring state. The contemporaneous radio, optical, X-ray, and VHE observations presented here do not show any significant variability, and thus enable the first SSC modelling of the emission of RGB\\,J0152$+$017.\n \n \n \\section{H.E.S.S. observations and results}\n \n \n \n RGB\\,J0152$+$017\\ was observed by the H.E.S.S. array consisting of four imaging atmospheric Cherenkov telescopes, located in the Khomas Highland, Namibia \\citep{2006A&A...457..899A}. The observations were performed from October 30 to November 14, 2007. The data were taken in {\\em wobble} mode, where the telescopes point in a direction typically at an offset of 0.5\\degr\\ from the nominal target position \\citep{2006A&A...457..899A}. After applying selection cuts to the data to reject periods affected by poor weather conditions and hardware problems, the total live-time used for analysis amounts to 14.7\\,h. The mean zenith angle of the observations is $26.9$\\degr. \n \n The data are calibrated according to \\citet{2004APh....22..109A}. Energies are reconstructed taking the effective optical efficiency evolution into account \\citep{2006A&A...457..899A}. The separation of $\\gamma$-ray-like events from cosmic-ray background events was made using the Hillas moment-analysis technique \\citep{1985ICRC....3..445H}. Signal extraction was performed using {\\em standard cuts} \\citep{2006A&A...457..899A}. The on-source events were taken from a circular region around the target with a radius of $\\theta=0.11$\\degr. The background was estimated using {\\em reflected regions} \\citep{2006A&A...457..899A} located at the same offset from the centre of the observed field as the on-source region. \n \n \n \n \n A signal of 173 $\\gamma$-ray events is found from the direction of RGB\\,J0152$+$017. The statistical significance of the detection is 6.6 $\\sigma$ according to \\citet{1983ApJ...272..317L}. The preliminary detection was reported by \\citet{2007ATel.1295....1N}. A two-dimensional Gaussian fit of the excess yields a position $\\alpha_{J2000}=1^{\\mathrm h}52^{\\mathrm m}33 \\fs 5 \\pm 5 \\fs 3_{\\rm stat}\\pm 1 \\fs 3_{\\rm syst}$, $\\delta_{J2000}=1^{\\circ}46' 40 \\farcs 3 \\pm 107''_{\\rm stat}\\pm 20''_{\\rm syst}$. The measured position is compatible with the nominal position of RGB\\,J0152$+$017\\ ($\\alpha_{J2000}=1^{\\mathrm h}52^{\\mathrm m}39\\fs78$, $\\delta_{J2000}=1^{\\circ}47'18\\farcs70$) at the 1$\\sigma$ level. Given this spatial coincidence, we identify the source of $\\gamma$-rays with RGB\\,J0152$+$017. The angular distribution of events coming from RGB\\,J0152$+$017, shown in Fig.~\\ref{fig:thetasq}, is compatible with the expectation from the Monte Carlo simulations of a point source.\n \n \n \\begin{figure}\n \\resizebox{\\hsize}{!}{\\includegraphics{fig1.eps}}\n \\caption{Angular distribution of excess events. The dot-dashed line shows the angular distance cut used for extracting the signal. The excess distribution is consistent with the H.E.S.S. point spread function as derived from Monte Carlo simulations (solid line).\n }\n \\label{fig:thetasq}\n \\end{figure}\n %\n %\n \n \n \n \n \\begin{figure}\n \\resizebox{\\hsize}{!}{\\includegraphics{fig2.eps}}\n \\caption{Differential spectrum of RGB\\,J0152$+$017. The spectrum obtained using {\\em spectrum cuts} (black closed circles) is compared with the one obtained by the {\\em standard cuts} (blue open circles). The black line shows the best fit by a powerlaw function of the former. The three points with the highest photon energy represent upper limits at 99\\% confidence level, calculated using \\citet{1998PhRvD..57.3873F}. All error bars are only statistical. The fit parameters of a powerlaw fit are $\\Gamma=2.95\\pm0.36_{\\mathrm{stat}}\\pm 0.20_{\\mathrm{syst}}$\\ and $\\Phi(1 \\mathrm{TeV}) = (5.7 \\pm 1.6_{\\mathrm{stat}}\\pm 1.1_{\\mathrm{syst}})\\times 10^{-13}$ \\difffluxunits\\ for the {\\em spectrum cuts}, and $\\Gamma=3.53\\pm0.60_{\\mathrm{stat}}\\pm 0.2_{\\mathrm{syst}}$\\ and $\\Phi(1 \\mathrm{TeV}) = (4.4 \\pm 2.0)\\times 10^{-13}$ \\difffluxunits\\ for the {\\em standard cuts}. The insert shows 1 and 2\\,$\\sigma$ confidence levels of the fit parameters.\n }\n \\label{fig:spectrum}\n \\end{figure}\n %\n %\n \n \n \n \n Figure~\\ref{fig:spectrum} shows the time-averaged differential spectrum. The spectrum was derived using {\\em standard cuts} with an energy threshold of 300\\,GeV. Another set of cuts, the {\\em spectrum cuts} described in \\citet{2006A&A...448L..19A}, were used to lower the energy threshold and improve the photon statistics (factor $\\sim$2 increase above the {\\em standard cuts}). Both give consistent results (see inlay in Fig.~\\ref{fig:spectrum} and caption). Because of the better statistics and energy range, we use the spectrum derived using {\\em spectrum cuts} in the following. Between the threshold of 240\\,GeV and 3.8\\,TeV, this differential spectrum is described well ($\\chi^2$\/d.o.f.=2.16\/4) by a power law ${\\rm d}N\/{\\rm d}E=\\Phi_0(E\/ 1\\mathrm{TeV})^{-\\Gamma}$ with a photon index $\\Gamma=2.95\\pm0.36_{\\mathrm{stat}}\\pm 0.20_{\\mathrm{syst}}$\\ and normalisation at 1\\,TeV of $\\Phi(1 \\mathrm{TeV}) = (5.7 \\pm 1.6_{\\mathrm{stat}}\\pm 1.1_{\\mathrm{syst}})\\times 10^{-13}$ \\difffluxunits. The 99\\% confidence level upper limits for the highest three bins shown in Fig.~\\ref{fig:spectrum} were calculated using \\citet{1998PhRvD..57.3873F}. \n \n \n \n \\begin{figure}\n \\resizebox{\\hsize}{!}{\\includegraphics{fig3.eps}}\n \\caption{Mean nightly integral flux from RGB\\,J0152$+$017\\ above 300\\,GeV. Only the statistical errors are shown. Upper limits at 99\\% confidence level are calculated when no signal is found (grey points). The dashed line shows a fit of a constant to the data points with $\\chi^2\/\\rm{d.o.f.}$ of $17.2\/12$. The fit was performed using all nights.}\n \\label{fig:lightcurve}\n \\end{figure}\n %\n %\n \n \n \n \n The integral flux above 300\\,GeV is $I = (2.70\\pm0.51_{\\mathrm{stat}}\\pm 0.54_{\\mathrm{syst}})\\times 10^{-12}$cm$^{-2}$\\,s$^{-1}$, which corresponds to $\\sim$2\\% of the flux of the Crab nebula above the same threshold as determined by \\citet{2006A&A...457..899A}. Figure~\\ref{fig:lightcurve} shows the nightly evolution of the $\\gamma$-ray flux above 300\\,GeV. There is no significant variability between nights in the lightcurve. The $\\chi^2\/\\rm{d.o.f.}$ of the fit to a constant is $17.2\/12$, corresponding to a $\\chi^2$ probability of 14\\%.\n \n All results were checked with independent analysis procedures and calibration chain giving consistent results.\n \n \n \\section{Multi-wavelength observations with \\textit{Swift}, \\textit{RXTE}, ATOM, and the Nan\\c{c}ay Radio Telescope}\n \n \\subsection{X-ray data from {\\it Swift} and {\\it RXTE}}\n \n Target of opportunity (ToO) observations of RGB\\,J0152$+$017\\ were performed with {\\it Swift} and {\\it RXTE} on November 13, 14, and 15, 2007 triggered by the H.E.S.S. discovery.\n \n The {\\it Swift}\/XRT \\citep{2005SSRv..120..165B} data (5.44\\,ks) were taken in photon-counting mode. The spectra were extracted with {\\tt xselect v2.4} from a circular region with a radius of 20 pixels ($0.8\\arcmin$) around the position of RGB\\,J0152$+$017, which contains $90\\%$ of the PSF at 1.5\\,keV. An appropriate background was extracted from a region next to the source with four times this area. The auxiliary response files were created with the script {\\tt xrtmkarf v0.5.6} and the response matrices were taken from the {\\it Swift} package of the calibration database {\\tt caldb v3.4.1}. Due to the low count rate of $0.3\\,\\rm{cts\/s}$, any pileup effect on the spectrum is negligible. We find no significant variability during any of the pointings or between the three subsequent days; hence, individual spectra were combined to achieve better photon statistics. The spectral analysis was performed using the tool {\\tt Xspec v11.3.2}. A broken powerlaw ($\\Gamma_1 =1.93 \\pm 0.20, \\Gamma_2 = 2.82 \\pm 0.13, E_\\mathrm{break} = 1.29 \\pm 0.12 \\, \\rm{keV}$) with a Galactic absorption of $2.72 \\times 10^{20}\\,\\rm{cm^{-2}}$ \\citep[LAB Survey,][]{2005A&A...440..775K} is a good description ($\\chi^2 \/ \\rm{d.o.f.} = 24 \/ 26$), and the resulting unabsorbed flux is $F_{0.5 - 2 \\,\\rm{keV}}\\sim 5.1 \\times10^{-12}\\,\\rm{erg \\,cm^{-2}\\, s^{-1}}$ and $F_{2 - 10 \\,\\rm{keV}}\\sim 2.7 \\times10^{-12}\\,\\rm{erg \\,cm^{-2}\\, s^{-1}}$.\n \n Simultaneous observations at higher X-ray photon energies were obtained with the {\\it RXTE}\/PCA \\citep{1996SPIE.2808...59J}. Only data of PCU2 and the top layer were taken to obtain the best signal-to-noise ratio. After filtering out the influence of the South Atlantic Anomaly, tracking offsets, and the electron contamination, an exposure of 3.2\\,ks remains. Given the low count rate of $0.7\\,\\rm{cts\/s}$, the ``faint background model'' provided by the {\\it RXTE} Guest Observer Facility was used to generate the background spectrum with the script {\\tt pcabackest v3.1}. The response matrices were created with {\\tt pcarsp v10.1}. Again no significant variations were found between the three observations, and individual spectra were combined to achieve better photon statistics. The PCA spectrum can be described by an absorbed single powerlaw with photon index $\\Gamma = 2.72 \\pm 0.08$ ($\\chi^2 \/ \\rm{d.o.f.} = 20\/16$) between 2 and 10\\,keV, using the same Galactic absorption as for {\\it Swift} data. The resulting flux $F_{2 - 10 \\,\\rm{keV}}\\sim 6.8 \\times10^{-12}\\,\\rm{erg \\,cm^{-2}\\, s^{-1}}$ exceeds the one obtained {\\em simultaneously} with {\\it Swift} by a factor of 2.5. We attribute this mostly to contamination by the nearby galaxy cluster Abell\\,267 ($44.6\\arcmin$ offset from RGB\\,J0152$+$017\\ but still in the field of view of the non-imaging PCA).\n\n A detailed decomposition is beyond the scope of this paper, so we exclude {\\it RXTE} data from broadband modelling. The {\\it RXTE} data-set confirms the absence of variability during November 2007, also in the energy range up to 10 keV. For the SED modelling, the average spectrum is treated as an upper limit. Further observations with {\\it RXTE} in December 2007 also show no indication of variability.\n\n \n \n \\subsection{Optical data}\n \n Optical observations were taken using the ATOM telescope \\citep{2004AN....325..659H} at the H.E.S.S site from November 10, 2007. No significant variability was detected during the nights between November 10 and November 20; R-band fluxes binned nightly show an RMS of 0.02\\,mag.\n \n Absolute flux values were found using differential photometry against stars calibrated by K.\\,Nilsson (priv.~comm.). We measured a total flux of $m_R = 15.25 \\pm 0.01$\\,mag (host galaxy + core) in an aperture of $4\\arcsec$ radius. The host galaxy was subtracted with galaxy parameters given in \\citet{2003A&A...400...95N}, and aperture correction given in Eq.~(4) of \\citet{1976AJ.....81..807Y}. The core flux in the R-band (640\\,nm) was found to be 0.62 $\\pm$ 0.08\\,mJy. This value was not corrected for Galactic extinction.\n\n \n \\subsection{Radio data}\n \n The Nan\\c{c}ay Radio Telescope (NRT) is a meridian transit telescope with a main spherical mirror of 300\\,m $\\times$ 35\\,m \\citep{2007A&A...465...71T}. The low-frequency receiver, covering the band 1.8--3.5\\,GHz was used, with the NRT standard filterbank backend.\n \n The NRT observations were obtained in two contiguous bands of 12.5\\,MHz bandwidth, centred at 2679 and 2691\\,MHz (average frequency: 2685\\,MHz). Two linear polarisation receivers were used during the 22 60-second drift scan observations on the source on November 12 and 14, 2007. The data have been processed with the standard NRT software packages NAPS and SIR. All bands and polarisations have been averaged, giving an RMS noise of 2.2\\,mJy. The source 3C\\,295 was observed for calibration, on November 11, 13, and 15, 2007.\n \n Taking into account a flux density for this source of $12.30 \\pm 0.06$\\,Jy using the spectral fit published by \\citet{1994A&A...284..331O}, we derived a flux density of $56 \\pm 6$\\,mJy at 2685\\,MHz for RGB\\,J0152$+$017. No significant variability was found in the radio data.\n \n \n \n \\section{Discussion}\n \\label{sec:modeling}\n\n\n \n \n \\begin{figure*}[tbh] \n \\centering\n \\begin{minipage}[c]{0.65\\textwidth}\n \\includegraphics[angle=-90,width=\\textwidth]{fig4.eps}\n \\end{minipage}%\n \\begin{minipage}[c]{0.35\\textwidth}\n \\caption{The spectral energy distribution of RGB\\,J0152$+$017. Shown are the H.E.S.S. spectrum ({\\it red filled circles and upper limits\\\/}), and contemporaneous {\\it RXTE} ({\\it blue open triangles\\\/}), {\\it Swift}\/XRT (corrected for Galactic absorption, {\\it magenta filled circles}), optical host galaxy-subtracted (ATOM) and radio (Nan\\c{c}ay) observations ({\\it large red filled squares\\\/}). The black crosses are archival data. The blue open points in the optical R-band correspond to the total and the core fluxes from \\citet{2003A&A...400...95N}. A blob-in-jet synchrotron self-Compton model (see text) applied to RGB\\,J0152$+$017\\ is also shown, describing the soft X-ray and VHE parts of the SED, with a simple synchrotron model shown at low frequencies to describe the extended part of the jet. The contribution of the dominating host galaxy is shown in the optical band. The dashed line above the solid line at VHE shows the source spectrum after correcting for EBL absorption. The left- and right-hand side inlays detail portions of the observed X-ray and VHE spectrum, respectively.\n }\n \\end{minipage}%\n \\label{fig:SED}\n \\end{figure*}\n %\n %\n \n \n Figure~\\ref{fig:SED} shows the SED of RGB\\,J0152$+$017\\ with the data from Nan\\c{c}ay, ATOM, {\\it Swift}\/XRT, {\\it RXTE}\/PCA, and H.E.S.S. Even though some data are not strictly simultaneous, no significant variability is found in the X-ray and optical bands throughout the periods covered; hence, a common modelling of the contemporaneous X-ray and VHE data appears justified.\n \n The optical part of the SED is mainly due to the host galaxy, which is detected and resolved in optical wavelengths \\citep{2003A&A...400...95N}. A template of the spectrum of such an elliptic galaxy is shown in the SED, as inferred from the code PEGASE \\citep{1997A&A...326..950F}. The host-galaxy-subtracted data point from the ATOM telescope might include several additional contributions---from an accretion disk, an extended jet (see below), or a central stellar population---so that it is considered as an upper limit in the following SSC model. A model including the optical ATOM data with possible additional contributions is beyond the scope of this paper.\n\n\n We applied a non-thermal leptonic SSC model \\citep{2001A&A...367..809K} to account for the contemporaneous observations by {\\it Swift} in X-rays and by H.E.S.S. in the VHE band. The radio data are assumed to originate in an extended region, described by a separate synchrotron model for the extended jet \\citep{2001A&A...367..809K} to explain the low-frequency part of the SED \\citep[as in, e.g.,][]{2005A&A...442..895A,2008A&A...477..481A}.\n\n We should emphasise that the aim of applying this model in this work is not to present a definitive interpretation for this source, but rather to show that a standard SSC model is able to account for the VHE and {\\it Swift} X-ray observations.\n \n For the SSC model, we describe the system as a small homogeneous spherical, emitting region (blob) of radius $R$ within the jet, filled with a tangled magnetic field $B$ and propagating with a Doppler factor $\\delta=\\left[ \\Gamma \\left( 1 -\\beta \\cos{\\theta} \\right) \\right]^{-1}$. Here $\\Gamma$ is the bulk Lorentz factor of the emitting plasma blob, $\\beta = v\/c$, and $\\theta$ is the angle of the velocity vector, with respect to the line-of-sight. The electron energy distribution (EED) is described by a broken powerlaw, with indices $n_1$ and $n_2$, between Lorentz factors $\\gamma_\\mathrm{min}$ and $\\gamma_\\mathrm{max}$, with a break at $\\gamma_\\mathrm{break}$ and density normalisation $K$.\n \n The model also accounts for the absorption by the extragalactic background light (EBL) with the parameters given in \\citet{2005AIPC..745...23P}. RGB\\,J0152$+$017\\ is too nearby ($z=0.08$) to add to the constraints on the EBL that were found by H.E.S.S. measurements of other blazars \\citep{2006Natur.440.1018A}. In all the models, we assume $H_0 = 70$\\,km\\,s$^{-1}$\\,Mpc$^{-1}$, giving a luminosity distance of $d_L = 1.078 \\times 10^{27}$\\,cm for RGB\\,J0152$+$017.\n \n The EED can be described by $K=3.1~\\times~10^4$\\,cm$^{-3}$, $\\gamma_\\mathrm{min}=1$, and $\\gamma_\\mathrm{max}=4~\\times~10^5$. The break energy is assumed at $\\gamma_\\mathrm{break}=7.0~\\times~10^4$ and is consistent with the {\\it Swift}\/XRT spectrum, while providing good agreement with the H.E.S.S. data. We assume the canonical index $n_1=2.0$ for the low-energy part of the EED, in accordance with standard Fermi-type acceleration mechanisms. The value $n_2=3.0$ for the high-energy part of the EED is constrained by the high-energy part of the X-ray spectrum. A good solution is found with the emitting region characterised by $\\delta=25$, $R=1.5~\\times~10^{15}$\\,cm, and $B=0.10$\\,G.\n \n For the extended jet, the data are described well by $R_\\mathrm{jet}=10^{16}$\\,cm, $\\delta_\\mathrm{jet}=7$, $K_\\mathrm{jet}=70$\\,cm$^{-3}$, $B_\\mathrm{jet}=0.05$\\,G, and $\\gamma_\\mathrm{break,\\;jet}=10^4$ at the base of the jet, and $L_\\mathrm{jet}=50$\\,pc \\citep[all the parameters are detailed in][]{2001A&A...367..809K}.\n \n Assuming additional contributions in the optical band, the multi-wavelength SED can thus be explained with a standard shock-acceleration process. The parameters derived from the model are similar to previous results for this type of source \\citep[see, e.g.,][]{2002A&A...386..833G}.\n\n From the current Nan\\c{c}ay radio data and the {\\it Swift} X-ray data, we obtain a broad-band spectral index $\\alpha_{rx} \\sim 0.56$ between the radio and the X-ray domains. The obtained SED, the corresponding location of the synchrotron peak, and the flux and shape of the {\\it Swift} spectrum lead us to conclude that RGB\\,J0152$+$017\\ can clearly be classified as an HBL object at the time of H.E.S.S. observations.\n\n \n \n \\section{Conclusion}\n \n The HBL RGB\\,J0152$+$017\\ was detected in VHE at energies $>~300$\\,GeV with the H.E.S.S. experiment. The contemporaneous {\\it Swift}, {\\it RXTE}, Nan\\c{c}ay, ATOM, and H.E.S.S. data allow the multi-wavelength SED for RGB\\,J0152$+$017\\ to be derived for the first time , and to clearly confirm its HBL nature at the time of the H.E.S.S. observations. In general, large variations of the VHE flux are expected in TeV blazars, making further monitoring of this source to detect high states of the VHE flux (flares) desirable.\n \n \n \\begin{acknowledgements}\n The support of the Namibian authorities and of the University of Namibia in facilitating the construction and operation of H.E.S.S. is gratefully acknowledged, as is the support by the German Ministry for Education and Research (BMBF), the Max Planck Society, the French Ministry for Research, the CNRS-IN2P3 and the Astroparticle Interdisciplinary Programme of the CNRS, the U.K. Science and Technology Facilities Council (STFC), the IPNP of the Charles University, the Polish Ministry of Science and Higher Education, the South African Department of Science and Technology and National Research Foundation, and by the University of Namibia. We appreciate the excellent work of the technical support staff in Berlin, Durham, Hamburg, Heidelberg, Palaiseau, Paris, Saclay, and in Namibia in the construction and operation of the equipment.\n \n This research made use of the NASA\/IPAC Extragalactic Database (NED). The authors thank the {\\it RXTE} team for their prompt response to our ToO request and the professional interactions that followed. The authors acknowledge the use of the publicly available {\\it Swift} data, as well as the public HEASARC software packages. This work uses data obtained at the Nan\\c{c}ay Radio Telescope. The authors also thank Dr.~Mira V\\'eron-Cetty for fruitful discussions.\n \\end{acknowledgements}\n \n \\bibliographystyle{aa}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\\label{sec:intro}\n\nLet $X$ be a one-dimensional time-homogeneous diffusion, and let $\\rho\n$ be a\nprobability measure on $\\R$. The Skorokhod embedding problem (SEP) for\n$\\rho$\nin $X$ is to find a stopping time $\\tau$ such that $X_\\tau\\sim\\rho$. Our\nmain goals in this article are firstly to construct a solution of the\nSkorokhod embedding problem, and secondly to discuss when does there\nexist a\nsolution which is finite, integrable or bounded in time, and when does our\nconstruction have these properties.\n\nOur construction is based on the observation that we can remove the\ndrift of\nthe time-homogeneous diffusion by changing the space variable via a\n\\textit{scale function}. We can thus simplify the embedding problem to the case where\n$X$ is a local martingale diffusion. We then consider a random variable that\nhas the distribution we want to embed and that can be represented as a\nBrownian martingale $N$ on the time interval $[0,1]$. Further, we set\nup an\nODE that uniquely determines a time-change along every path of $X$. We then\nshow, by drawing on a result of uniqueness in law for weak solutions of SDEs,\nthat the time-changed diffusion has the same distribution as the martingale\n$N$. Thus the time-change provides a solution of the SEP.\n\nOur solution is a generalization of Bass's solution of the SEP for\n\\textit{Brownian motion} (see \\cite{Bass1983}). Bass\nalso starts with the martingale\nrepresentation of a random variable with the given distribution. By changing\nthe martingale's clock, he obtains a Brownian motion and an associated\nembedding stopping time. The time-change is governed by an ODE, a\nspecial case\nof our ODE, which establishes an analytic link between Brownian paths and\nthe embedding stopping time. This link yields embedding stopping\ntimes for \\textit{arbitrary} Brownian motions.\n\nNow consider properties of solutions of the SEP.\nAs is well known from the literature, whether a distribution is embeddable\ninto the diffusion $X$ depends on the relation between the support of the\ndistribution and the state space of $X$ and the relation between the initial\nvalue $X_0$ and the first moment of the distribution. We include in our\nanalysis a general discussion of sufficient and necessary conditions\nfor the existence of \\emph{finite} embedding stopping\ntimes, with particular reference to our proposed construction.\n\nNext, we\nfully determine the collection\nof distributions that can be embedded in $X$ with \\textit{integrable} stopping\ntimes. The associated conditions involve an integrability condition on the\ntarget distribution which makes use of a function that also\nappears in Feller's test for explosions (see, e.g., \\cite{KS}).\n\n\n\nFinally, we address the question of whether a distribution can be\nembedded in\n\\textit{bounded} time. Recall that the Root solution (\\cite{Root1969}) of the\nSEP has the property that it minimises\n$\\EE[(\\tau- t)^+]$ uniformly in $t$. The Root solution $\\tau_R$ is\nof the\nform \\( \\tau_R = \\inf\\{ t \\dvtx (X_t, t) \\in\\cR_{\\rho} \\} \\) where\n$\\cR_\\rho$ is\na `barrier'; in particular $\\cR_{\\rho} = \\{ (x,t) \\dvtx t \\geq\\beta\n_\\rho(x) \\}\n\\subseteq\\R\\times\\Rp$ for some suitably regular function $\\beta\n_\\rho$\ndepending on the target law. Hence, a necessary and sufficient\ncondition for\nthere to be an embedding $\\tau$ with $\\tau\\leq T$ is that $\\beta\n_\\rho( \\cdot\n) \\leq T$. However, the Root barrier is non-constructive and difficult to\nanalyse (though for some recent progress in this direction see Cox and\nWang \\cite{CoxWang2011} and Oberhauser and Dos Reis \\cite{OberhauserDosReis}).\nFor this reason, instead of searching for a single set of necessary\n\\textit{and}\nsufficient conditions we limit ourselves to finding separate sets of necessary\nconditions and sufficient conditions.\n\nOur original motivation in developing a solution of the SEP for diffusions\nwas to study bounded stopping times with the aim of providing simple\nsufficient conditions for the existence of a bounded embedding.\nThe boundedness (finiteness) of an embedding is an important property\nof the embedding used to solve the gambling in contests problem of Seel and\nStrack \\cite{SeelStrack2008}, and is also relevant in the model-independent\npricing of variance swaps, see Carr and Lee \\cite{CarrLee},\nHobson \\cite{Hobson2011} and Cox and Wang \\cite{CoxWang2011}.\n\nConsider for a moment the case where $X$ is a real-valued Brownian\nmotion, null\nat 0. Then it is possible to embed any target probability measure $\\rho\n$ in\n$X$. Moreover, $\\rho$ can be embedded in integrable time if and only\nif $\\rho$\nis centred and in $L^2$, and then $\\EE[\\tau] = \\int x^2 \\rho(\\mathrm{d}x)$.\nThe case of\nembeddings in bounded time is more subtle.\nClearly a necessary condition for there to exist an embedding $\\tau$\nof $\\rho$\nin $X$ such that $\\tau\\leq1$ is that $\\rho$ is smaller that $\\mu_G$ in\nconvex order, where $\\mu_G$ is the law of a standard Gaussian. But\nthis is not\nsufficient. Let $\\mu_{\\pm a}$ be the uniform measure on $\\{-a, +a \\}$. Then\n$\\mu_{\\pm a}$ is smaller than $\\mu_G$ in convex order if and only if\n$a \\leq\n\\sqrt{2\/\\pi}$. But any embedding $\\tau$ of $\\mu_{\\pm a}$ has $\\tau\n\\geq\\min\n\\{ u \\dvtx |B_{u}| \\geq a\\}$, and thus does not satisfy $\\tau\\leq T$ for\nany $T$.\nHence, we would like to find sufficient conditions on $\\rho$ such that there\nexists $\\tau\\leq T$ with $X_\\tau\\sim\\rho$. The case where $X$ is Brownian\nmotion, possibly with drift, was considered in Ankirchner and\nStrack \\cite{as11}. Here we consider general time-homogeneous diffusions.\n\n\nThe paper is organized as follows. In Section~\\ref{sec:mg}, we\ndescribe our\nsolution method of the SEP for a diffusion \\textit{without} drift. In this\nsection, we assume that the initial value of the diffusion coincides\nwith the\nfirst moment of the distribution to embed (the centred case). In the following\nSection~\\ref{non-c case}, we briefly explain how to construct\nsolutions if the\nfirst moment does \\textit{not} match the initial value (the\nnon-centred case). In\nSection~\\ref{sec finite e}, we collect some general conditions which guarantee\nthat a distribution can be embedded into $X$ in finite time. We then consider\nintegrable embeddings in Section~\\ref{sec int emb}. Distinguishing between\nthe centred and non-centred case, we provide sufficient and necessary\nconditions for the existence of integrable solutions of the SEP.\nSection~\\ref{sec:bte} discusses bounded embeddings. In Section~\\ref{sec:diff}, we\nexplain how one can reduce the SEP for diffusions with drift to the case\nwithout drift. Finally, in Section~\\ref{sec:eg} we illustrate our\nresults with\nsome examples.\n\n\\section{The martingale case}\n\\label{sec:mg}\n\nWe will argue in Section~\\ref{sec:diff} below that the problem of\ninterest can\nbe reduced to the case in which the process is a continuous local martingale.\nIn this section, we describe a generalisation of the Bass \\cite{Bass1983}\nsolution of the SEP. The Bass solution is an embedding of $\\nu$ in Brownian\nmotion: we consider embeddings in a local martingale diffusion which\nmay be\nthought of as time-changed Brownian motion.\n\nConsider the time-homogeneous local martingale diffusion $M$, where $M$ solves\n\\begin{equation}\n\\label{sde} \\mathrm{d}M _s = \\eta(M_s)\\,\\mathrm{d}W\n_s, \\qquad\\mbox{with } M_0=m;\n\\end{equation}\nhere\n$m \\in\\R$ and $\\eta\\dvtx \\R\\to\\R_{+}$ is Borel-measurable. We assume\nthat the\nset of points $x\\in\\R$ with $\\eta(x) = 0$ coincides with the set of points\nwhere $\\frac{1}{\\eta}$ is \\textit{not} locally square integrable.\nThen a result\nby Engelbert and Schmidt implies that the SDE \\eqref{sde} possesses a weak\nsolution that is unique in law (see, e.g., Theorem~5.5.4 in \\cite\n{KS}). We define\n$l = \\sup\\{x \\leq m \\mid \\eta(x) = 0\\}$ and $r = \\inf\\{x \\geq m \\mid \\eta\n(x) =\n0\\}$ so that $-\\infty\\leq l \\leq r \\leq\\infty$ (to exclude\ntrivialities we\nassume $l < m < r$) and for $x \\in\\R$,\n\\begin{equation}\n\\label{eqn:qdef} q(x) = \\int_m^x \\,\\mathrm{d}y\n\\int_m^y \\frac{2}{\\eta^2(z)} \\,\\mathrm{d}z.\n\\end{equation}\nBy our assumption on $\\eta$, $q$ is infinite on $(-\\infty, l)$ and\n$(r, \\infty)$.\n\n\\begin{remark}\n\\label{rem:felkot}\nBy Feller's test, $\\PP[\\inf_{s \\le t} M_s\n\\le l] = 0$ for one, and then every, $t>0$ if and only if $q(l+) = \\lim_{x\n\\downarrow l} q(x) = \\infty$. Similarly, $\\PP[\\sup_{s \\le t} M_s \\ge\nr] = 0$\nif and only if $q(r-) = \\infty$ (see, e.g., Theorem~5.5.29 in \\cite{KS}).\nFurther, by results of Kotani \\cite{Kotani2006}, the local martingale\n$M$ is a\nmartingale \\emph{provided} either $-\\infty< l$ or $\\int_{-\\infty} |x|\n\\eta(x)^{-2} \\,\\mathrm{d}x = \\infty$ \\emph{and} either $r<\\infty$ or $\\int^\\infty x\n\\eta(x)^{-2} \\,\\mathrm{d}x = \\infty$.\n\\end{remark}\n\nNote that our assumption that $\\frac{1}{\\eta}$ is not locally square\nintegrable at $l$ and $r$ implies that $l$ and $r$ are absorbing\nboundaries if\nthey can be reached in finite time. Then without loss of generality we may\nassume that $\\eta= 0$ on $(-\\infty,l)$ and $(r, \\infty)$ and $\\eta$ is\npositive on $(l,r)$.\\looseness=-1\n\nWe want to embed a non-Dirac probability measure $\\nu$ with $ \\int x\n\\,\\mathrm{d}\\nu(x) =\nm$.\nLet $\\underline{v} = \\inf\\{ \\supp( \\nu) \\}$\nand $\\overline{v} = \\sup\\{ \\supp( \\nu) \\}$ be the extremes of the\nsupport of\n$\\nu$,\nand\nlet $F$ be the distribution function associated to the target law $\\nu$.\nMoreover, let $\\Phi\\dvtx \\R\\rightarrow[ 0, 1 ]$ be the cumulative\ndistribution\nfunction of the normal distribution and $\\phi= \\Phi'$ its density.\nDefine the\nfunction $h = F^{- 1} \\circ\\Phi$. Let $(\\Wtil_t)_{t\\geq0}$ be a Brownian\nmotion on a filtration $\\tilde{\\mathbb F} = (\\Ftil_t)_{t \\geq0}$.\nNotice that\n$h(\\Wtil_1)$ has the distribution $\\nu$. In particular, $h(\\Wtil_1)$ is\nintegrable and $\\EE[h(\\Wtil_1)]=m$.\n\nWe define the $\\tilde{\\mathbb F}$-martingale\n$N_t = \\EE[ h ( \\Wtil_1 ) \\mid \\tilde{\\cF}_t ]$ for $t \\in\n[0,1]$.\nNotice that $N_0=m$, $N_1$ has distribution $\\nu$ and $N_t = b(t,\\Wtil_t)$,\nwhere\n\\[\nb(t,x) = \\int_{\\R} h ( y ) \\phi_{1 - t} (x - y) \\,\n\\mathrm{d}y = (\\phi_{1-t} \\star h) (x),\n\\]\nand $\\phi_v$ is the density of the normal distribution with variance $v$.\n\nSince $\\nu$ is not a Dirac measure we have that $h$ is increasing\nsomewhere, and hence, for all $t \\in[0,1)$, the mapping $x \\mapsto b\n(t, x)$ is strictly increasing. Thus, we can define the inverse function\n$B \\dvtx [ 0, 1] \\times\\R\\rightarrow\\R$ implicitly by\n\\begin{equation}\n\\label{defi b} b \\bigl( t, B ( t, x ) \\bigr) = x, \\qquad\\mbox{for all } t \\in[ 0,\n1 ), x \\in\\R;\n\\end{equation}\nmoreover we set $B(1,x) = h^{-1}(x)$.\nThe process $N$ solves the SDE\n\\begin{equation}\n\\label{sde2} dN_t = b_x\\bigl(t,B(t,N_t)\n\\bigr) \\,\\mathrm{d}\\Wtil_t,\\qquad N_0 = \\int x \\,\n\\mathrm{d}\\nu(x) = m.\n\\end{equation}\nDefine\n\\begin{equation}\n\\label{eq:defrR} \\lambda(t,y)= \\frac{b_x(t,y)}{\\eta(b(t,y))},\\qquad \\Lambda(t,y)=\\lambda\n\\bigl(t, B(t,y)\\bigr) = \\frac{b_x(t,B(t,y))}{\\eta(y)}.\n\\end{equation}\nThe candidate embedding which we want to discuss is $\\delta(1)$ where\n$\\delta$\nsolves\n\\begin{equation}\n\\label{ode} \\delta'(t) = \\Lambda(t, M_{\\delta(t)})^2\n= \\frac\n{b_x(t,B(t,M_{\\delta(t)}))^2}{\\eta(M_{\\delta(t)})^2},\\qquad \\delta(0) = 0.\n\\end{equation}\nNote that $\\delta$ is increasing so that if $\\delta$ is defined on\n$[0,1)$ then\nwe can set $\\delta(1) = \\lim_{t \\uparrow1} \\delta(t)$.\n\n\\begin{theorem} \\label{thm:odesol}\nIf the ODE \\eqref{ode} has a\nsolution on\n$[0,1)$\nfor almost all paths of $M$, then $\\delta(1)$ embeds $F$ into $M$,\nthat is, the\nlaw of $M_{\\delta(1)}$ is $\\nu$.\n\\end{theorem}\n\n\\begin{pf}\nLet $Y_t = M_{\\delta(t)}$ for all $t \\in[0,1) $. By interchanging the\ntime-change and integration, see, for example, Proposition V.1.5 in\n\\cite{RY},\nwe get\n\\[\nY_t - m = \\int_0^{\\delta(t)}\n\\eta(M_s) \\,\\mathrm{d}W _s = \\int_0^t\n\\eta(M_{\\delta(s)}) \\,\\mathrm{d}W _{\\delta(s)} = \\int_0^t\n\\eta(Y_s) \\,\\mathrm{d}W _{\\delta(s)}.\n\\]\nLet $Z_t = \\int_0^t \\frac{1}{\\sqrt{\\delta'(s)}} \\,\\mathrm{d}W _{\\delta(s)}$,\nfor $t\n\\in[0,1]$. Notice that $\\langle Z,Z \\rangle_t = \\int_0^t\n\\frac{1}{\\delta'(s)} \\,\\mathrm{d}\\delta(s) = t$ (Proposition V.1.5 in\n\\cite{RY}) and then by L\\'evy's characterization theorem, $Z$ is a\nBrownian motion\non $[0,1]$.\nNext, observe that\n\\begin{eqnarray}\n\\label{Ysol} Y_t - m &=& \\int_0^t\n\\eta(Y_s) \\sqrt{\\delta'(s)} \\,\\mathrm{d}Z\n_s = \\int_0^t\n\\eta(Y_s) \\Lambda(s, M_{\\delta(s)})\\,\\mathrm{d}Z _s\n\\nonumber\n\\\\[-8pt]\n\\\\[-8pt]\n&=& \\int_0^t b_x\n\\bigl(s,B(s,Y_{s})\\bigr) \\,\\mathrm{d}Z _s,\n\\nonumber\n\\end{eqnarray}\nwhich shows that $Y$ solves the SDE \\eqref{sde2} with $W$ replaced by $Z$;\nin other words $(Y,Z)$ is a weak solution of \\eqref{sde2}.\n\nIt follows directly from Lemma~2(a) in Bass \\cite{Bass1983} that\n$b_x(t,B(t,x))$ is Lipschitz continuous in $x$, uniformly in $t$, on compact\nsubsets of $[0,1) \\times\\R$. Therefore, the SDE \\eqref{sde2} has at\nmost one\nstrong solution on $[0,1)$ and hence \\eqref{sde2} is pathwise unique, from\nwhich it follows (see, e.g., Section~5.3 in \\cite{KS}) that solutions of\n\\eqref{sde2} are unique in law. Hence, for $t<1$, $Y_t=M_{\\delta(t)}$\nhas the\nsame distribution as $N_t$, and in the limit $t$ tends to 1 we have\n$N_1$ and\nhence $Y_1$ has law $\\nu$.\n\\end{pf}\n\n\\begin{remark}\nNotice that the assumption that $\\int x \\nu(\\mathrm{d}x) = m$ is crucial\nfor the conclusion of Theorem~\\ref{thm:odesol}. Indeed, if $\\int x \\nu(\\mathrm{d}x)\n\\neq m$, then $Y$ and $N$ solve the same SDE, but with \\emph{different}\ninitial conditions. Hence, one cannot derive that $Y_1$ has the same\ndistribution as $N_1$.\n\\end{remark}\n\nWe next aim at showing that $\\delta(1)$ is a stopping time with\nrespect to\n${\\mathbb F}^M = (\\cF^M_t)_{t \\geq0}$, the smallest filtration\ncontaining the\nfiltration generated by the martingale $M$ and satisfying the usual\nconditions. To this end we consider, as in \\cite{Bass1983}, the ODE satisfied\nby the inverse of $\\delta(t)$. The ODE for the inverse is Lipschitz continuous\nand hence guarantees that Picard iterations converge to a unique solution.\n\n\\begin{lemma}\\label{lemma ode inv}\nLet $M$ be a path of the solution of \\eqref{sde}.\nThen \\eqref{ode} has a solution on $[0,1)$ if and only if there exists\n$a \\in\n\\Rp\\cup\\{\\infty\\}$ such that the ODE\n\\begin{equation}\n\\label{eq:Delta} \\Delta'(s) = \\frac{\\eta(M_s)^2}{b_x(\\Delta(s),\nB(\\Delta(s),M_s))^2}\n\\end{equation}\nhas a solution on $[0,a)$ with $\\lim_{s \\uparrow a} \\Delta(s) = 1$.\n\\end{lemma}\n\n\\begin{pf}\nAssume that there exists a solution of \\eqref{ode} on $[0,1)$. Set $a =\n\\delta(1)$ and define $\\Delta(s) = \\delta^{-1}(s)$ for all $s \\in\n[0,a]$. Then a straightforward calculation shows that $\\Delta$ satisfies\n\\eqref{eq:Delta}.\\looseness=-1\n\nThe reverse direction can be shown similarly.\n\\end{pf}\n\n\n\n\\begin{remark}\nIf $\\eta= 1$, then the ODE \\eqref{eq:Delta} is the ODE (1)\nof Bass'\npaper \\cite{Bass1983}.\n\\end{remark}\n\n\n\\begin{lemma}\nSuppose the ODE \\eqref{ode} has a solution on $[0,1)$ for almost all\npaths of\n$M$. Then $\\delta(t)$ is an ${\\mathbb F}^M$-stopping time, for all\n$t\\in[0,1]$.\n\\end{lemma}\n\n\\begin{pf}\nLet $C$ be a compact subset of $[0,1) \\times\\Rp$. By\nLemma~2 of Bass \\cite{Bass1983}, $b_x(t,x)$ and $B(t,x)$ are Lipschitz\ncontinuous on $C$. Moreover, on $C$ the function $b_x$ is bounded away from\nzero and bounded from above. This implies that $\\frac\n{1}{b_x(t,B(t,x))^2}$ is\nLipschitz continuous on $C$, too.\n\nDefine the mapping $\\gamma\\dvtx (t,y) \\mapsto\\frac{\\eta(M_t)^2}{b_x(y,\nB(y,M_t))^2}$. Now let $D$ be a compact subset of $\\Rp\\times[0,1)$.\nThen there exists an\n$L \\in\\Rp$ such that for all $(t,y)$ and $(t, \\tilde y) \\in D$ we have\n\\[\n\\bigl\\llvert \\gamma(t,y) - \\gamma(t,\\tilde y)\\bigr\\rrvert \\leq L\n\\eta(M_t)^2 \\llvert y - \\tilde y\\rrvert ,\n\\]\nthat is, $\\gamma$ is Lipschitz continuous in the second argument.\n\nWe define the Picard iterations $\\Delta_0(t) = 0$ and for $n \\geq0$,\n\\[\n\\Delta_{n+1}(t) = 1 \\wedge\\int_0^t\n\\frac{\\eta(M_s)^2}{b_x(\\Delta_n(s),\nB(\\Delta_n(s),M_s))^2} \\,\\mathrm{d}s.\n\\]\nWe have that $\\Delta_n(t) = 1$ after the first time where\n$\\Delta_n$ attains $1$.\nThe assumptions on $\\eta$ guarantee that $\\int_0^s \\eta(M_t)^2 \\,\\mathrm{d}t $\nis finite, a.s.\nfor each $sr$. Then for each $t<1$ we have $b(t, \\cdot) \\dvtx\n\\R\\to\n(\\underline{v},\\overline{v})$ and there exists a continuous function $y(t)$\nsuch\nthat $b(t,y) > r$ for $y> y(t)$. Then,\n$\\int_0^T \\lambda(t, \\tilde{W}_t)^2 \\,\\mathrm{d}t = \\infty$ for all $T<1$ such that\n$\\sup_{00$. Since the set $\\sup_{00$ has\npositive probability, $\\delta$ explodes before time 1 with positive\nprobability\nalso.\n\n\\begin{assumption}\nHenceforth, we will assume that $\\nu$ places no mass outside $[\\ell,r]$.\n\\end{assumption}\n\nRecall that we have assumed that we are given a diffusion with $M_0=m$, and\nthat the target measure $\\nu$ satisfies $\\nu\\in L^1$ and $m = \\int x\n\\nu(\\mathrm{d}x)$. We call this the centred case. In the next section, we\nconsider what\nhappens if we relax this assumption.\n\nIn the case where $\\nu\\in L^1$ but $m \\neq\\int x\n\\nu(\\mathrm{d}x)$, we introduce an embedding $\\delta^*$ which involves running the\nmartingale $M$ until it first hits $\\int x \\nu(\\mathrm{d}x)$ and then using the\nstopping time $\\delta(1)$ defined above, but for $M$ started at $\\int x\n\\nu(\\mathrm{d}x)$.\n\nThen in subsequent sections we will ask, when does there exist a finite\n(respectively \\{integrable, bounded\\}) embedding, and when does $\\delta\n(1)$ or\nmore generally $\\delta^*$ have this property.\n\n\\section{The non-centred case}\\label{non-c case}\n\nIn this section, we do not assume that $\\nu\\in L^1$ and that $m= \\int x\n\\nu(\\mathrm{d}x)$.\n\nWhen at least one of $\\int_{-\\infty}\n|x| \\nu(\\mathrm{d}x)$ and $\\int^\\infty x\\nu(\\mathrm{d}x)$ is finite we write\n${\\nu}^* = \\int x \\nu(\\mathrm{d}x) \\in\\bar{\\R}$.\nNote that we assume that $\\nu$ has support in the state space of $M$.\n\n\\begin{proposition}[(Pedersen and Peskir \\cite{PedersenPeskir2001}, Cox and\nHobson \\cite{CoxHobson2004})] \\label{prop:mneqbarnu}\nSuppose $-\\infty< l < m <\\allowbreak r < \\infty$. Then for there to be an\nembedding of\n$\\nu$ in $M$ we must have that $\\int x \\nu(\\mathrm{d}x) = m$. In this case $M$\nis a\nuniformly integrable martingale.\n\nSuppose $-\\infty=l < m < r < \\infty$. Then there exists an embedding\nof $\\nu$\nin $M$ if and only if ${\\nu}^* \\geq m$. Conversely, if $-\\infty< l <\nm < r\n= \\infty$ there exists an embedding of $\\nu$ in $M$ if and only if\n${\\nu}^*\n\\leq m$.\n\nFinally, suppose $-\\infty=l < m < r = \\infty$. Then we can embed any\ndistribution\n$\\nu$ in $M$.\n\\end{proposition}\n\n\\begin{pf}\nIn the bounded case, the fact that $M$ is a bounded local\nmartingale gives that it is a UI-martingale, and hence $\\int x \\nu(\\mathrm{d}x) =\n\\EE[M_\\tau] = M_0=m$.\n\nFor the second case, the upper bound on the state space means that $M$\nis a\nsubmartingale so that the condition $m \\leq{\\nu}^*$ is necessary. Then\nprovided $\\nu\\in L^1$ we can run $M$ until it first reaches ${\\nu}^*\n\\in\n[m,\\infty)$. Note that $M$ hits $\\nu^*$ in finite time by the\nargument in\nKaratzas, Shreve \\cite{KS}, Section~5.5 C. Then we can embed $\\nu$\nusing the\nlocal martingale $M$ started from ${\\nu}^*$ (using, for example, the time\n$\\delta(1)$ defined above, or the Az{\\'e}ma--Yor construction as in\nPedersen and\nPeskir \\cite{PedersenPeskir2001}). If ${\\nu}^*$ is infinite, then we\nneed a\ndifferent construction, see, for example, Cox and Hobson \\cite{CoxHobson2004}.\n\nFor the final case, any distribution can be embedded in $M$. If $\\nu\n\\in L^1$\nthen we can run $M$ until it hits ${\\nu}^*$ and then consider an\nembedding for\nthe local martingale started at the mean of the target distribution. If\n$\\nu\n\\notin L^1$, then we can use the construction in \\cite\n{CoxHobson2004}, but not\nthe one in this paper.\n\\end{pf}\n\nLet $H^M_{z}$ be the first hitting time of $z$ by\n$M$, and more generally let $H^X_x$ be the first hitting time of $x$ by a\nstochastic process $X$. Suppose $\\mu\\in L^1$ and let $\\delta_{\\nu\n^*}(1)$ be\nthe\nstopping\ntime $\\delta(1)$ constructed\nin the previous section to embed $\\nu$ in the time-homogeneous diffusion\nstarted at $M_0=\\nu^*$. Then let $\\delta^* = H^M_{\\nu^*} +\n\\delta_{\\nu^*}(1)$. By the results of the proposition, provided $\\nu\n\\in L^1$\nand both $\\nu^* \\leq m$ if $r<\\infty$ and $\\nu^* \\geq m$ if\n$l>-\\infty$, then\n$\\delta^*$ is an embedding of $\\nu$.\n\n\\section{Finite embeddings}\n\\label{sec finite e}\n\n\\subsection{The centred case}\nSuppose $\\nu\\in L^1$ and $m = \\int x \\nu(\\mathrm{d}x)$.\n\n\\begin{proposition}\n\\begin{enumerate}[(ii)]\n\\item[(i)]If $\\ell>-\\infty$, $M$ does not hit $\\ell$ in finite time and\n$\\nu(\\{l\\})>0$\nor if\n$r<\\infty$, $M$ does not hit $r$ in finite time and $\\nu(\\{r\\})>0$, then\nany embedding of $\\nu$ has $\\tau= \\infty$ with positive probability.\n\n\\item[(ii)]Otherwise, either $\\ell= -\\infty$, or $M$ does not hit $\\ell$\nin finite\ntime\nand $\\nu(\\{ \\ell\\}) = 0$ or $M$ can hit $\\ell$ in finite time \\emph{and}\neither $r = \\infty$, or $M$ does not hit $r$ in finite time and $\\nu\n(\\{ r \\})\n= 0$ or $M$ can hit $r$ in finite time. Then if $\\tau$ is an embedding of\n$\\nu$ we have that $\\bar{\\tau}= \\tau\\wedge H^M_\\ell\\wedge H^M_r$\nis also an\nembedding of $\\nu$ and $\\bar{\\tau}$ is finite almost surely.\\looseness=1\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{pf}\n(i) Suppose $\\tau$ is any embedding of $\\nu$ in $M$. Then\n$\\tau= \\infty$ on the set where $M_{\\tau} \\in\\{ \\ell, r \\}$.\nMoreover, this set has positive probability by assumption.\n\n(ii) If $\\tau$ is an embedding of $\\nu$, then $M_{t \\wedge\\tau}$ converges\nalmost surely, even on the set $\\tau= \\infty$. However, if $(\\ell=\n-\\infty,r=\\infty)$ then by the Rogozin trichotomy (see \\cite\n{rogozin}), $- \\infty= \\liminf M_t <\n\\limsup M_t = \\infty$ and $(M_t)_{t \\geq0}$ does not converge. Hence,\nwe must\nhave $\\tau<\\infty$.\\looseness=1\n\nOtherwise, one or both of $\\{\\ell, r\\}$ is finite. Then\n$M$ converges and so if $\\tau= \\infty$ then either\n$M_{\\tau}=\\ell$ or $M_{\\tau}=r$.\n\nIf $\\ell$ or $r$ is finite but $M$ hits neither $\\ell$ nor $r$ in\nfinite time,\nthen $\\tau= \\infty$ is excluded outside a set of measure zero by the\nhypothesis that $\\nu(\\{\\ell\\})=0$ and $\\nu(\\{r\\})=0$. Hence, $\\tau<\n\\infty$\nalmost surely.\n\nFinally, if $M$ can hit either $\\ell$ or $r$ in finite time then it\nwill do so\nand $\\bar{\\tau}= H^M_\\ell\\wedge\\break H^M_r < \\infty$.\n\\end{pf}\n\n\\begin{coro}\n\\label{cor:finite}\nIf there exists an embedding $\\tau$ of $\\nu$ in $M$ which is\nfinite almost surely then $\\delta(1)$ is finite almost surely.\n\\end{coro}\n\n\n\\begin{pf}\nIf there is a finite embedding, then we must be in case (ii) of the\nproposition. Then $\\delta(1)\\wedge H^M_\\ell\\wedge H^M_r$ is finite almost\nsurely. But also $\\delta(1)\\leq H^M_\\ell\\wedge H^M_r$ so that\n$\\delta(1)<\\infty$ almost surely.\n\\end{pf}\n\n\\subsection{The non-centred case}\n\nSuppose $\\nu$ and $m$ are such that an embedding exists (recall\nProposition~\\ref{prop:mneqbarnu}). Necessarily we must have that at\nleast one\nof $\\ell$ and $r$ is infinite.\n\nSuppose $\\nu\\in L^1$ so that $\\nu^*$ and $\\delta^*$ are well\ndefined. Then\nsince $H^M_{\\nu^*}$ is finite almost surely, we have that $\\delta^*$\nis finite\nif and only if $\\delta_{\\nu^*}(1)$ is finite almost surely.\n\nThen the result for the non-centred case is identical to both the proposition\nand the corollary describing the results in the centred case, modulo the\nsubstitution of $\\delta^*$ for $\\delta(1)$ in Corollary~\\ref{cor:finite}.\n\n\\section{Integrable embeddings}\n\\label{sec int emb}\n\n\\subsection{The centred case}\n\nSuppose $\\nu\\in L^1$ and $m = \\int x \\nu(\\mathrm{d}x)$.\n\nIn this section, we provide an integrability condition on $\\nu$ that guarantees\nthat\n\\eqref{ode} has a solution on $[0,1]$ and that $\\delta(1)$ is integrable.\nNotice that $q$ is twice continuously differentiable on $(l,r)$. The second\nderivative\n\\[\nq''(x)= \\frac{2}{\\eta^2(x)}\n\\]\nis positive, which means that $q$ is convex. Moreover, $q$ is decreasing\non $(l,m)$ and increasing on $(m,r)$; in particular $q \\ge0$.\n\n\n\\begin{theorem}\\label{thm:intcondidelta}\nIf the function $q$ is integrable wrt $\\nu$, then the ODE \\eqref{ode}\nhas a\nsolution on $[0,1]$ for almost all paths of $M$ and $\\delta(1)$ is integrable.\nIn this case, $\\EE[\\delta(1)] = \\int q(x) \\nu(\\mathrm{d}x)$.\n\\end{theorem}\n\n\\begin{pf}\nAssume first that $q$ is integrable wrt $\\nu$. This means that the random\nvariable $q(N_1)$ is integrable. Let\n\\begin{equation}\n\\label{defi tau_n} \\tau_n = 1 \\wedge\\inf\\biggl\\{t \\ge0\\Bigm|\\int\n_0^t \\bigl |q'(N_s)\nb_x\\bigl(s,B(s,N_s)\\bigr)\\bigr |^2 \\,\\mathrm{d}s\n\\geq n\\biggr\\},\n\\end{equation}\nand observe that $(N_u)_{u \\leq s}$ is bounded away from $l$ and $r$\nfor any\n$s<1$,\nand hence\n$\\tau_n \\uparrow1$, a.s.\nBy It\\^o's formula, and using $q(N_0)=q(m) =0$, we get\n\\begin{eqnarray*}\nq(N_{\\tau_n}) &=& \\int_0^{\\tau_n}\nq'(N_s) b_x\\bigl(s,B(s,N_s)\n\\bigr) \\,\\dd \\Wtil_s + \\frac{1}{2} \\int_0^{\\tau_n}\nq''(N_s) b_x\n\\bigl(s,B(s,N_s)\\bigr)^2 \\,\\dd s\n\\\\\n&=&\\int_0^{\\tau_n} q'(N_s)\nb_x\\bigl(s,B(s,N_s)\\bigr) \\,\\dd\\Wtil_s + \\int\n_0^{\\tau_n} \\frac{b_x(s,B(s,N_s))^2}{\\eta^2(N_s)} \\,\\dd s.\n\\end{eqnarray*}\nTaking expectations, the martingale part disappears and we obtain\n\\begin{equation}\n\\label{eq aux 160612} \\EE \\biggl[ \\int_0^{\\tau_n}\n\\frac{b_x(s,B(s,N_s))^2}{\\eta^2(N_s)} \\,\\dd s \\biggr] = \\EE\\bigl[q(N_{\\tau_n})\\bigr].\n\\end{equation}\nNotice that Jensen's inequality implies that\n\\[\n0 \\le q(N_{\\tau_n}) \\le\\EE\\bigl[q(N_1) \\mid\n\\Ftil_{\\tau_n}\\bigr].\n\\]\nSince the family $(\\EE[q(N_1) \\mid \\Ftil_{\\tau_n}])_{n \\ge1}$ is uniformly\nintegrable, also $(q(N_{\\tau_n}))_{n \\ge1}$ is uniformly integrable.\nTherefore we can interchange the expectation operator and the limit $n\n\\to\n\\infty$ on the RHS of \\eqref{eq aux 160612}. By monotone convergence,\nwe can do\nso also on the LHS and hence we get\n\\[\n\\EE \\biggl[ \\int_0^1 \\frac{b_x(s,B(s,N_s))^2}{\\eta^2(N_s)} \\,\\dd\ns \\biggr] = \\EE\\bigl[q(N_1)\\bigr] < \\infty.\n\\]\nLemma~\\ref{aux lem 160612} implies that the ODE \\eqref{ode} has a\nsolution on\n$[0,1]$ for almost all paths of $M$ and that $\\delta(1)$ is integrable.\n\\end{pf}\n\nThe reverse statement of Theorem~\\ref{thm:intcondidelta} holds true as-well,\nthat is, if $\\delta(1)$ is integrable, then $q$ is integrable wrt $\\nu\n$. Indeed,\nwe next show that the existence of an \\textit{arbitrary} integrable\nsolution of\nthe SEP implies that $q$ is integrable wrt $\\nu$.\n\n\\begin{proposition}\\label{thm:intimplintofq}\nAny stopping time $\\tau$ that solves the SEP satisfies\n\\begin{equation}\n\\label{expectation sol sep} \\EE[\\tau] \\ge\\int q(x) \\,\\mathrm{d}\\nu(x).\n\\end{equation}\n\\end{proposition}\n\n\\begin{pf}\nRecall that $\\overline{\\tau}= \\tau\\wedge H_l \\wedge H_r$.\nSince $\\ell$ is absorbing if $H_{\\ell}<\\infty$ and similarly if $H_r\n< \\infty$\nthen $r$ is absorbing,\nwe have that\n$M_{\\overline{\\tau}} = M_\\tau$ and $\\overline{\\tau}$ is also an\nembedding of\n$\\nu$.\n\nLet $\\tau$ be an stopping time with $M_{\\tau} \\sim\\nu$. Suppose\nthat $\\tau$\nis integrable; else the statement is trivial. Let\n\\[\n\\sigma_n = n \\wedge\\inf\\biggl\\{t \\ge0\\Bigm| \\int\n_0^t \\bigl\\llvert q'(M_s)\n\\bigr\\rrvert ^2 \\eta(M_s)^2 \\,\\mathrm{d}s \\ge\nn\\biggr\\}.\n\\]\n\nObserve that $\\sigma_n \\uparrow H_l \\wedge H_r$, a.s. Using It\\^{o}'s formula,\nwe obtain\n\\begin{eqnarray}\n\\label{ito q m stopped} \\EE\\bigl[q(M_{\\tau\\wedge\\sigma_n})\\bigr] &=& q(M_0) + \\EE\n\\biggl[\\frac{1}{2} \\int_0^{\\tau\\wedge\\sigma_n}\nq''(M_s) \\,\\dd\\langle M, M\n\\rangle_s \\biggr]\n\\nonumber\n\\\\[-8pt]\n\\\\[-8pt]\n&=& \\EE[\\tau\\wedge\\sigma_n].\n\\nonumber\n\\end{eqnarray}\nThen Fatou's lemma implies\n\\[\n\\EE\\bigl[q(M_\\tau)\\bigr]=\\EE\\bigl[q(M_{\\overline{\\tau}})\\bigr] \\le\n\\liminf_n \\EE\\bigl[ q(M_{\\tau\n\\wedge\n\\sigma_n})\\bigr] \\le \\EE[\n\\overline{\\tau}] \\leq\\EE[\\tau],\n\\]\nand hence \\eqref{expectation sol sep}.\n\\end{pf}\n\n\\begin{remark}\nNotice that if $M$ attains the boundary $l$ with positive\nprobability in finite time, then the function $q$ is finite at $l$. In this\ncase $\\nu$ can have mass on $l$. If $M$ does not attain the boundary\n$l$ in\nfinite time, then obviously a distribution $\\nu$ with mass in $l$ can\nnot be\nembedded with an integrable stopping time. Similar considerations apply\nat the\nright boundary $r$.\n\\end{remark}\n\nTheorems~\\ref{thm:intcondidelta} and~\\ref{thm:intimplintofq} imply the\nfollowing\ncorollaries.\n\n\\begin{coro}\nSuppose $\\nu\\in L^1$ and $m = \\int x \\nu(\\mathrm{d}x)$. There\nexists an\nintegrable solution $\\tau$ of the SEP\nif and only\nif $q$ is integrable wrt $\\nu$. In this case, $\\tau$ satisfies\n\\eqref{expectation sol sep}.\n\\end{coro}\n\n\\begin{coro}\nSuppose $\\nu\\in L^1$ and $m = \\int x \\nu(\\mathrm{d}x)$.\nWhenever there exists an integrable solution of the SEP, then\n$\\delta(1)$ is also an integrable solution.\n\\end{coro}\n\n\\subsection{The non-centred case}\n\nSuppose we are given a local martingale diffusion $M$ started at\n$M_0=m$ and a\nmeasure $\\nu\\in L^1$ with $\\nu^* \\neq m$.\n\nRecall the definition of $q$ in (\\ref{eqn:qdef}). To emphasise the\nrole of the\ninitial point, write $q_m$ for this expression. More generally, for $n\n\\in\n(l,r)$ define\\vspace*{-1pt}\n\\begin{equation}\n\\label{eqn:qdefG} q_n(x) = \\int_n^x\n\\,\\mathrm{d}y \\int_n^y \\,\\mathrm{d}z\n\\frac{2}{\\eta(z)^2}.\n\\end{equation}\n\nThen $q_m(z) = q_n(z) + q_m(n) + q_m'(n)(z-n)$. As $q=q_m$, in particular\n\\[\nq(z) = q_{\\nu^*}(z) + q\\bigl(\\nu^*\\bigr) + q'\\bigl(\\nu^*\n\\bigr) \\bigl(z - \\nu^*\\bigr)\n\\]\n{\\spaceskip=0.185em plus 0.05em minus 0.02em and $\\int q(z) \\nu(\\mathrm{d}z) = \\int q_{\\nu^*}(z) \\nu(\\mathrm{d}z) + q(\\nu^*)$.\nHence,\n$\\int q(z) \\nu(\\mathrm{d}z)$ is finite if and only if $\\int q_{\\nu^*}(z)\\* \\nu\n(\\mathrm{d}z)$} is\nfinite.\n\nWe state the following theorem in the case $m> \\nu^*$ which necessitates\n$r=\\infty$, and then $\\ell\\in[-\\infty, m)$. There is a\ncorresponding result\nfor $m<\\nu^*$ in which the condition $\\lim_{n \\uparrow\\infty}\nq(n)\/n <\n\\infty$ is replaced by\n$\\lim_{n \\uparrow\\infty} q(-n)\/n < \\infty$.\nNote that the limit $\\lim_n q(n) \/ n$ is well defined because $q$ is convex.\n\n\\begin{theorem}\n\\label{thm:integrableNC} Suppose $m>\\nu^*$.\n\nSuppose $\\int q(z) \\nu(\\mathrm{d}z)<\\infty$ and $\\lim q(n)\/n < \\infty$. Then\n$\\delta^*$ is an integrable embedding of $\\nu$.\n\nConversely, suppose there exists an integrable embedding $\\tau$ of\n$\\nu$ in\n$M$.\nThen $\\int q(z) \\nu(\\mathrm{d}z)<\\infty$ and $\\lim q(n)\/n < \\infty$.\n\\end{theorem}\n\n\\begin{pf}\nConsider the first part of the theorem.\nBy the comments before the theorem, we may assume that $\\int q_{\\nu^*}(z)\n\\nu(\\mathrm{d}z) < \\infty$ and hence, for $M$ started at $\\nu^*$,\n$\\EE[\\delta_{\\nu^*}(1)]<\\infty$. Then, it is\nsufficient to show that $\\EE[H^M_{\\nu^*}]<\\infty$. But\n\\begin{eqnarray*}\n\\EE\\bigl[H^M_{\\nu^*}\\bigr] &=& \\lim_{n \\uparrow\\infty}\n\\EE \\bigl[H^M_{\\nu^*} \\wedge H^M_n\n\\bigr] = \\lim_{n \\uparrow\\infty} \\EE\\bigl[q(M_{H^M_{\\nu^*} \\wedge H^M_n})\\bigr]\n\\\\\n& = & q\\bigl(\\nu^*\\bigr) \\lim_{n\n\\uparrow\\infty} \\frac{n-m}{n - \\nu^*} + \\lim\n_{n \\uparrow\\infty} q(n) \\frac{m - \\nu^*}{n - \\nu^*}\n\\\\\n& = & q\\bigl(\\nu^*\\bigr) + \\bigl(m - \\nu^*\\bigr) \\lim_{n \\uparrow\\infty}\n\\frac{q(n)}{n},\n\\end{eqnarray*}\nwhich is finite under the assumption that $\\lim q(n)\/n < \\infty$.\n\nFor the converse, suppose that $\\tau$ is an integrable embedding.\nWithout loss\nof generality, we may assume that $\\tau$ is minimal; if not we may\nreplace it\nwith a smaller embedding which is also integrable.\nThen\n\\begin{eqnarray*}\n\\int q(x) \\nu(\\mathrm{d}x) &=& \\EE\\Bigl[\\liminf_{n \\to\\infty}\nq(M_{\\tau\\wedge H_{-n}^M \\wedge H_{n}^M})\\Bigr] \\leq \\liminf_{n \\to\\infty} \\EE\n\\bigl[q(M_{\\tau\\wedge H^M_{-n} \\wedge H^M_{n}})\\bigr]\n\\\\\n& = & \\lim_{n \\to\\infty} \\EE\\bigl[ \\tau\\wedge H^M_{-n}\n\\wedge H^M_{n}\\bigr]\n\\\\\n& = & \\EE[\\tau] < \\infty.\n\\end{eqnarray*}\n\nIt remains to show that $\\EE[H^M_{\\nu^*}] < \\infty$. Recall that\nthis is\nequivalent to the condition $\\lim_{n \\uparrow\\infty} q(n)\/n < \\infty$.\n\n\nRecall that by the Dubins--Schwarz theorem\n(Rogers and Williams \\cite{RogersWilliams}, page 64) we can write $M_t =\n\\hat{W}_{C_t}$\nfor a ${\\mathbb G}=({\\mathcal G}_t)_{t \\geq0}$-Brownian motion $\\hat{W}$\nwhere ${\\mathcal G}_s = {\\mathcal F}_{C^{-1}_s}$. Let $\\sigma= C_\\tau\n$. Then\n$M_\\tau= \\hat{W}_{C_\\tau} = \\hat{W}_{\\sigma}$ and $\\sigma$ embeds\n$\\nu$ in\n$\\hat{W}$.\n\nSince $\\sigma$ is a\nminimal embedding of $\\nu$ in $\\hat{W}$, by Theorem~5 of Cox and\nHobson \\cite{CoxHobson05}\n\\begin{equation}\n\\label{eqn:ch1} \\lim_n n \\PP\\bigl[\\sigma>\nH^{\\hat{W}}_{-n}\\bigr] = 0.\n\\end{equation}\nMoreover, by arguments in the proof of Lemma~11 of Cox and\nHobson \\cite{CoxHobson05}, for any stopping time $\\tilde{\\sigma}\n\\leq\\sigma$\n\\[\n\\EE \\bigl[ |\\hat{W}_{\\tilde{\\sigma}}| \\bigr] \\leq\\EE \\bigl[ |\\hat{W}_{\\sigma}| \\bigr] =\n\\int|z| \\nu(\\mathrm{d}z).\n\\]\nHence, $(\\hat{W}_{t \\wedge\\sigma})_{t \\geq0}$ is bounded in $L^1$,\nand then\nby Theorem~1\nof Az\\'{e}ma \\textit{et al.} \\cite{AzemaGundyYor}, $(\\hat{W}_{t \\wedge\\sigma\n})_{t \\geq\n0} $ is uniformly integrable if and only if\n$\\lim_n n \\PP[\\sigma> H^{\\hat{W}}_{-n} \\wedge H^{\\hat{W}}_{n}] = 0$.\nSince $\\nu$ is not centred and $(\\hat{W}_{t \\wedge\\sigma})_{t \\geq\n0} $ is not UI, it follows from (\\ref{eqn:ch1}) that $\\lim_n n \\PP\n[\\sigma>\nH^{\\hat{W}}_{n}] >0$.\nBut $( \\sigma> H^{\\hat{W}}_{n}) \\equiv(\\tau> H^M_n)$ so\n$\\limsup_n n \\PP[\\tau> H^M_{n}] > 0$.\n\nThen\n\\begin{eqnarray*}\n\\EE[\\tau] & = & \\lim_n \\EE\\bigl[q(M_{\\tau\\wedge H^M_n})\\bigr]\n\\\\\n& \\geq& \\lim_n \\EE\\bigl[q(n) ; {\\tau> H^M_n}\n\\bigr]\n\\\\\n& = & \\lim_n \\biggl( \\frac{q(n)}{n} n \\PP\\bigl[\\tau>\nH^M_n\\bigr] \\biggr)\n\\\\\n& \\geq& \\lim\\frac{q(n)}{n} \\cdot\\lim\\sup n \\PP\\bigl[\\tau>\nH^M_n\\bigr].\n\\end{eqnarray*}\nThen, if $\\EE[\\tau]<\\infty$ it follows that $\\lim\\frac{q(n)}{n} <\n\\infty$ and\n$\\EE[H^M_{\\nu^*}] < \\infty$.\n\\end{pf}\n\nFinally, we consider the case where $\\nu\\notin L^1$.\n\n\n\\begin{lemma}\nSuppose $\\nu\\notin L^1$. If $\\tau$ is an embedding of $\\nu$, then\n$\\tau$ is not integrable.\n\\end{lemma}\n\n\\begin{pf}\nObserve that $q(x)\\geq0$ and that if $\\nu\\notin L^1$ then since $q$\nis convex we must have\n$\\int q(x) \\nu(\\mathrm{d}x) = \\infty$. Then if $\\tau$ is an embedding of $\\nu$\n\\begin{eqnarray*}\n\\EE[\\tau] &=&\n\\lim_{n \\uparrow\\infty} \\EE\\bigl[\\tau\\wedge\nH^{q(M)}_n \\wedge H^{q(M)}_{-n} \\bigr] =\n\\lim_{n \\uparrow\\infty} \\EE\\bigl[q(M_{\\tau\\wedge\nH^{q(M)}_n\\wedge H^{q(M)}_{-n}})\\bigr]\n\\\\\n&\\geq& \\EE\\Bigl[ \\liminf_{n \\uparrow\\infty} q(M_{\\tau\\wedge\nH^{q(M)}_n\\wedge H^{q(M)}_{-n} })\\Bigr] =\n\\EE\\bigl[q(M_\\tau)\\bigr] = \\infty.\n\\end{eqnarray*}\n\\upqed\\end{pf}\n\n\\section{Bounded time embedding}\n\\label{sec:bte}\n\n\\subsection{The centred case}\n\nIn this section, we analyze the question under which conditions we can\nguarantee the stopping time $\\delta(1)$ to be bounded, that is,\n$\\delta(1)\n\\leq T \\in\\Rp$. Let us first state a necessary condition which places\na lower bound on how little mass must be embedded in each a\nneighbourhood of a point~$x$.\n\n\\begin{theorem}\n\\label{thm:bte}\nSuppose that $\\eta$ is locally bounded and denote by $\\eta^*$ its\nupper semicontinuous envelope.\nIf a distribution with distribution function $F$ can be embedded before\ntime $T > 0$,\nthen for all $x \\in\\R$ with $00$ define $B_\\epsilon(x)= \\{ y \\mid |y-x|<\\epsilon\\}$ and\n$\\bar{\\eta}(x,\\epsilon)=\\max\\{\\eta^*(z)\\mid z \\in\\bar B_\\epsilon\n(x)\\} $.\nNote that on $t \\geq t'$ the process $\\tilde{M}$ which solves the SDE\n$\\dd\\tilde{M}_t = \\tilde{\\eta}(M_t)\n\\,\\mathrm{d}W _t$ where\n\\[\n\\tilde{\\eta}(m) = \\bigl(1_{\\{m \\in B_\\epsilon(x)\\}} \\eta(m) + 1_{\\{m \\notin B_\\epsilon(x)\\}} \\bar{\n\\eta}(m,\\epsilon) \\bigr)\n\\]\nsubject to $\\tilde{M}_{t'} = M_{t'}=x$,\ncoincides with $M$ up to the first leaving time of $B_\\epsilon(x)$.\nMoreover, there exists a Brownian motion $\\tilde{W}$ such that on $t\n\\geq t'$,\n$\\tilde{M}_t = \\tilde{W}_{\\Gamma_t}$, where $\\Gamma(t) = \\int_{t'}^t \\tilde{\\eta}(M_s)^2 \\,\\dd s \\leq\n\\bar{\\eta}(x,\\epsilon)^2 t$.\nThen\n\\begin{eqnarray*}\n\\PP \\Bigl[ \\sup_{t' \\leq t \\leq T} \\llvert M_t -\nM_{t'} \\rrvert < \\epsilon \\Bigr] &=& \\PP \\Bigl[ \\sup\n_{t'\\leq t \\leq T} | \\tilde{M}_t -\\tilde {M}_{t'}| <\n\\epsilon \\Bigr]\n\\\\\n&=& \\PP \\Bigl[\\sup_{t' \\leq t \\leq T}\\llvert \\tilde{W}_{\\Gamma(t)} -\n\\tilde{W}_{\\Gamma(t')}\\rrvert < \\epsilon \\Bigr]\n\\\\\n&\\geq& \\PP \\Bigl[ \\sup_{0 \\leq s \\leq\\bar{\\eta}(x,\\epsilon)^2 T} | {W}_s| < \\epsilon\n\\Bigr].\n\\end{eqnarray*}\nThe probability for the absolute value of the Brownian motion ${W}$ to\nstay within the ball\n$B_\\epsilon(0)$ up\nto time $K T\\ge0$ is given by (see Section~5, Chapter X in Feller\n\\cite{feller2})\n\\begin{eqnarray*}\n\\PP \\Bigl[\\sup_{s \\in[0,K^2 T]} |W_s| < \\epsilon \\Bigr] &=&\n\\frac\n{4}{\\pi} \\sum_{n = 0}^\\infty\n\\frac{1}{2n+1} \\mathrm{e}^{-\\fraca{(2n+1)^2 \\pi^2}{(8\n\\epsilon^2)} K T} (-1)^{n}\n\\\\\n&\\ge& \\frac{4}{\\pi} \\mathrm{e}^{-\\fraca{\\pi^2}{(8 \\epsilon^2)} K T} - \\frac\n{4}{3 \\pi}\n\\mathrm{e}^{-\\fraca{9 \\pi^2}{(8 \\epsilon^2)}\nK T} \\geq\\frac{8}{3\\pi} \\mathrm{e}^{-\\fraca{\\pi^2}{(8 \\epsilon^2)} K T}.\n\\end{eqnarray*}\n\nAssume that there\nexists a stopping time $\\tau$ such that $M_{\\tau}$ has the\ndistribution $F$. Denote\nby $\\zeta= \\inf\\{t \\ge0\\dvtx M_t = x\\}$ the first time the process $M$\nhits $x$. Since $F(x) \\notin\\{0, 1\\}$,\nthe event $A = \\{ \\zeta< \\tau\\}$ occurs with positive probability.\n\nLet $\\FF_{\\zeta}$ be the $\\sigma$-field generated by $M$ up to\ntime\n$\\zeta$ and observe that $A \\in\\FF_{\\zeta}$. Note further\nthat the process $Z=(Z_h)_{h \\geq0}$ given by $Z_h = M_{h + \\zeta} -\nM_\\zeta$ is independent of\n$\\FF_{\\zeta}$.\n\nNow suppose $\\tau$ is bounded by $T$.\nThe mass of $F$ on the ball $B_\\eps(x)$ has to be at least as large as\nthe probability that $A$ occurs\nand that $X$ stays\nwithin the ball $B_\\eps(x)$ between $\\zeta$ and $T$. Therefore,\n\\begin{eqnarray*}\nF(x+\\eps) - F(x - \\eps) &\\ge& \\PP \\Bigl[A \\cap\\Bigl\\{\\sup_{\\zeta\\le s\n\\le T}\n|M_s - M_\\zeta| < \\eps\\Bigr\\} \\Bigr]\n\\\\\n&=& \\PP[A] \\PP \\Bigl[\\sup_{\\zeta\\le s \\le T} |M_s -\nM_\\zeta| < \\eps \\Bigr]\n\\\\\n&\\geq& \\PP[A] \\frac{8}{3\\pi} \\mathrm{e}^{-\\fraca{\\pi^2}{(8 \\epsilon^2)} \\bar\n{\\eta}(x,\\epsilon)^2 T}.\n\\end{eqnarray*}\nHence, we have\n\\[\n-\\eps^2 \\ln \\bigl(F(x+\\eps) - F(x - \\eps) \\bigr) \\le\n\\eps^2 \\ln \\frac{3\\pi}{8 \\PP[A]} + \\frac{\\pi^2}{8} \\bar{\\eta}(x,\n\\epsilon)^2 T,\n\\]\nwhich implies the result.\n\\end{pf}\n\nNow we turn to the converse, and sufficient conditions for these to\nexist an embedding of $\\nu$ in bounded\ntime. Suppose again that $\\nu\\in L^1$ and $M_0=m=\\int x \\nu(\\mathrm{d}x)$.\n\nRecall the definition of $r$ in \\eqref{eq:defrR}.\nThe first result is an immediate corollary of Theorem~\\ref{thm:odesol}.\n\n\\begin{coro}\\label{suff condi bound}\nIf $\\lambda(t,y)^2$ is bounded by $T \\in\\Rp$, for all $y \\in\\R$\nand $t\\in[0,1]$, then the stopping time\n$\\delta(1)$ is also bounded by $T$.\n\\end{coro}\n\n\\begin{proposition}\nAssume that $F$ is absolutely continuous and has compact support.\nSuppose $F$ has density $f$.\nIf $\\eta$ and $f$ are bounded away from zero, then the stopping time\n$\\delta(1)$ is bounded.\n\\end{proposition}\n\n\\begin{pf}\nNote that $h' = \\frac{\\phi}{f \\circ F^{-1} \\circ\\Phi}$ and\nthus it follows from $f$ bounded away from zero\nthat $h'$ is bounded. Hence, $b_x$ is bounded and thus $\\lambda(t,y)$\nis bounded.\n\\end{pf}\n\n\\begin{lemma}\n\\label{etaconcave}\nSuppose that $\\eta$ is concave on $(l,r)$.\nLet $F$ be an absolutely continuous distribution with\\vspace*{2pt} $\\supp(F)\n\\subseteq[l,r]$ and suppose that\n$\\sup_{x \\in[l,r]} \\frac{h'(x)}{\\eta(h(x))} \\leq\\sqrt{T} < \\infty\n$. Then $F$\nis embeddable in bounded time, and there exists an embedding $\\tau$\nwith $\\tau\\leq T$.\n\\end{lemma}\n\n\\begin{pf}\nWe have $b(t,x)= (\\phi_{1-t} \\star h)(x)$ and\n\\[\nb_x(t,x)= \\bigl(\\phi_{1-t} \\star h'\\bigr) (x)\n\\leq\\sqrt{T} \\bigl(\\phi_{1-t} \\star(\\eta\\circ h)\\bigr) (x) \\leq\\sqrt{T}\n\\eta\\circ(\\phi_{1-t} \\star h) (x) = \\sqrt{T} \\eta\\bigl(b(t,x)\\bigr)\n\\]\nand then $\\lambda(t,x)^2 \\leq T$ and the result follows from Corollary~\\ref{suff condi bound}.\n\\end{pf}\n\n\\begin{remark}\n\\label{rem:notconcave}\nMore generally for the existence of a bounded embedding it is\nsufficient that\nthere is a\nconcave function\n$\\xi$ and $\\epsilon$ in $(0,1)$ for which $\\epsilon\\xi\\leq\\eta\n\\leq\n\\epsilon^{-1} \\xi$. Then if $\\sup_{x \\in[l,r]} \\frac{h'(x)}{\\eta\n(h(x))} \\leq\\sqrt{T}$\n\\[\nb_x(t,x) \\leq\\sqrt{T} \\epsilon^{-1} \\bigl(\n\\phi_{1-t} \\star(\\xi\\circ h)\\bigr) (x) \\leq \\sqrt{T}\\epsilon^{-1}\n\\xi\\circ (\\phi_{1-t} \\star h) (x) \\leq \\sqrt{T} \\epsilon^{-2}\n\\eta\\bigl(b(t,x)\\bigr),\n\\]\nand $\\lambda(t,x)^2 \\leq T \\epsilon^{-4}$.\n\\end{remark}\n\n\n\\begin{remark}\nThe sufficient condition from Lemma~\\ref{etaconcave} implies a\nstronger version of the necessary condition of Theorem~\\ref{thm:bte}.\nIndeed, it implies that the limit superior of the left hand side of\nequation \\eqref{nece condi} is equal to zero. To show this, let $\\eta\n$ be locally bounded and assume that $F$ satisfies the assumptions of\nLemma~\\ref{etaconcave}. Then for all $z \\in(\\underline v, \\bar v)$,\nthe interior of $\\supp(F)$, we have\n\\[\nf(z) \\ge\\frac{1}{\\sqrt{T}} \\frac{\\phi\\circ\\Phi^{-1} \\circ\nF(z)}{\\eta(z)}.\n\\]\nLet $x \\in(\\underline v, \\bar v)$. Since $\\eta$ is locally bounded\nthere exists $B \\in\\R_+$ such that $\\eta(z) \\le B$ for $z$ close\nenough to $x$. Then, for $\\eps$ small we have\n\\begin{eqnarray*}\n\\ln\\bigl(F(x+\\eps) - F(x - \\eps)\\bigr) &=& \\ln \\biggl( \\int_{x-\\eps}^{x +\n\\eps}\nf(z)\\,\\mathrm{d}z \\biggr)\n\\\\\n&\\ge& \\ln \\biggl( \\frac{1}{\\sqrt{T}B}\\int_{x-\\eps}^{x + \\eps}\n\\phi\\circ\\Phi^{-1} \\circ F(z)\\,\\mathrm{d}z \\biggr),\n\\end{eqnarray*}\nand applying Jensen's inequality we obtain\n\\[\n\\ln\\bigl(F(x+\\eps) - F(x - \\eps)\\bigr) \\ge \\ln\\frac{2\\eps}{\\sqrt{T}B} -\n\\frac{1}{4\\eps} \\int_{x-\\eps}^{x + \\eps} \\bigl[\\bigl(\n\\Phi^{-1} \\circ F(z)\\bigr)^2 + \\ln2 \\pi\\bigr] \\,\\mathrm{d}z.\n\\]\nConsequently $\\limsup_{\\eps\\downarrow0} - \\eps^2 \\ln(F(x+\\eps) -\nF(x - \\eps)) = 0$.\n\\end{remark}\n\n\n\\subsection{The non-centred case}\n\\label{ss:Bnc}\n\nIf $M$ is a martingale and\nif $\\tau$ is bounded by $L$, then $M_{t \\wedge\\tau}$ is uniformly integrable\nand $\\EE[M_\\tau]=m$. Hence, there are no embeddings of $\\nu$ in $M$\nif $\\nu\n\\notin L^1$ or $\\nu^* \\neq m$.\n\nIf $M$ is a local martingale but not a martingale, then we may have\n$\\tau\\leq\nL$ and $\\EE[M_\\tau] \\neq M_0 = m$. However, $\\delta^*$ is not\nbounded since\n$\\delta^* > H^M_{\\nu^*}$ which is not bounded.\n\nFor example, let $m=1$ and $\\eta(x)=x^2$ so that $\\mathrm{d}M _t= M_t^2 \\,\\mathrm{d}B_t$,\nand $M$\nis the reciprocal of a 3-dimensional Bessel process. Let $\\nu=\n{\\mathcal\nL}(M_1)$. Then $\\nu\\in L^1$ and $\\nu^* \\leq1 = m$. Then, trivially,\n$\\tau\n\\equiv1 $ is a bounded embedding of $1$.\n\n\\section{General diffusions}\n\\label{sec:diff}\n\nLet $(X_t)_{t \\geq0}$ be a solution to\n\\[\n\\mathrm{d} X_t = \\beta(X_t)\\,\\mathrm{d}t +\n\\alpha(X_t)\\,\\mathrm{d}W _t, \\qquad\\mbox{with }\nX_0 = x_0,\n\\]\nwhere $x_0 \\in\\R$, $\\beta\\dvtx \\R\\rightarrow\\R$ and $\\alpha\\dvtx \\R\n\\rightarrow\\R$ are\nBorel-measurable. We assume that $X$ takes values only in an interval\n$[l,r]$ with\n$-\\infty\\le l < x_0 < r \\le\\infty$. Moreover, we assume that $\\alpha\n(x) \\neq 0$ for\nall $x \\in(l,r)$ and that $\\frac{1+ |\\beta|}{\\alpha^2}$ is locally\nintegrable on\n$(l,r)$.\n\nSuppose we want to embed $\\rho$ in $X$ with a stopping time $\\tau$.\n\nBy changing the space scale one can transform the diffusion $X$ into a\ncontinuous local martingale.\nTo this end, we define the \\textit{scale function} $s$ (cf. \\cite{RY},\nChapter VII, \\S3)\nvia\n\\[\ns ( x ) = \\int_{x_0}^x \\exp \\biggl( - \\int\n_{x_0}^y \\frac{2 \\beta\n( z )}{\\alpha( z )^2} \\,\\dd z \\biggr) \\,\\dd\ny, \\qquad x \\in(l,r).\n\\]\nNote that we are always free to choose the scale function such that\n$M_0=s(x_0)=0$, and we have done so.\n\nThen $s$ solves $\\beta(x) s'(x) + \\frac{1}2 \\alpha^2(x) s''(x) = 0$.\nNote that the scale function $s$ is strictly increasing and\ncontinuously differentiable.\nIt\\^o's formula implies that $M_t = s ( X_t )$ is a local martingale with\nintegral representation\n\\[\nM_t = \\int_0^t s' (\nX_s ) \\alpha( X_s ) \\,\\dd W_s.\n\\]\nThus $\\dd M_t = \\eta(M_t) \\,\\dd W_t$ where $\\eta\\equiv(s' \\alpha)\n\\circ s^{-1}$.\n\nNote that\n\\[\n\\int_{\\chi-\\epsilon}^{\\chi+\\epsilon} \\frac{1}{((s' \\alpha)\\circ\ns^{-1})^2(z)} \\,\\mathrm{d}z =\n\\int_{s^{-1}(\\chi-\\epsilon)}^{s^{-1}(\\chi+\\epsilon)} \\frac\n{1}{\\alpha^2(z)s'(z)} \\,\\mathrm{d}z.\n\\]\nSince $s$ is continuous, $\\frac{1}{\\eta}$ is locally square\nintegrable provided\n\\[\n\\frac{1}{\\alpha(y) \\sqrt{s'(y)}} = \\frac{1}{\\alpha(y)} \\exp \\biggl( -\\int_{x_0}^y\n\\frac{2 \\beta(z)}{ \\alpha(z)^2 } \\,\\mathrm{d}z \\biggr)^{-\\fraca{1}{2}} = \\frac{1}{\\alpha(y)} \\exp\n\\biggl( \\int_{x_0}^y \\frac{\\beta(z)}{ \\alpha(z)^2 } \\,\n\\mathrm{d}z \\biggr)\n\\]\nis locally square integrable which follows from our assumptions on the pair\n$(\\alpha, \\beta)$.\n\nLet $F_\\rho$ be the distribution function of $\\rho$. If $\\nu= \\rho\n\\circ s^{-1}$ so that\n$F(x) = F_\\rho(s^{-1}(x))$,\nthen $X_\\tau\\sim\\rho$ is equivalent to $M_\\tau\\sim\\nu$.\nThen $\\nu$ has mean zero if and only if\n\\begin{equation}\n\\label{meancondition} \\int_\\R s(x) \\rho(\\mathrm{d}x) = 0.\n\\end{equation}\n\nClearly the requirement that $\\tau$ is finite, integrable or bounded is\ninvariant under the change of scale. However, in the case of bounded\nembeddings we can give a simple sufficient condition in terms of data relating\nto the general diffusion $X$.\n\nDefine $g = F_\\rho^{-1} \\circ\\Phi$ and $h = s \\circ g = F^{-1}\n\\circ\\Phi$.\n\n\\begin{theorem}\n\\label{thm:gendiff}\nIf $ x \\mapsto- \\frac{2 \\beta(x)}{\\alpha(x)} + \\alpha'(x) $ is\nnon-increasing\nand $\\frac{g'}{\\alpha\\circ g}$ is bounded by $\\sqrt{T}$, then $\\rho\n$ can be\nembedded in $X$ in bounded time. In particular, there exists an embedding\n$\\tau$ with $\\tau\\leq T$.\n\\end{theorem}\n\n\\begin{pf}\nWe prove in the first step that $\\eta\\equiv(s' \\alpha) \\circ s^{-1}$\nis concave. We have\n\\[\n\\eta' = \\bigl( \\bigl(s' \\alpha\\bigr) \\circ\ns^{-1} \\bigr)' = \\frac{(s'' \\alpha+ s' \\alpha') \\circ s^{-1}}{s' \\circ s^{-1}} = \\biggl( -\n\\frac{2\\beta}{\\alpha} + \\alpha' \\biggr) \\circ s^{-1}\n\\]\nwhere we have used the fact that $s$ solves $\\alpha s'' = - 2\\beta\ns'\/\\alpha$.\nAs $s^{-1}$ is monotone increasing, under the first hypothesis of the\ntheorem we have that $ ( (s' \\alpha)\n\\circ\ns^{-1} )'$ is\nnon-increasing and hence $\\eta$ is concave.\n\nWe have $h = s \\circ g$ and hence, again by hypothesis,\n\\[\n\\frac{h'}{\\eta\\circ h} = \\frac{(s' \\circ g) g'}{(s'\\alpha) \\circ g\n} = \\frac{g'}{\\alpha\n\\circ g} \\le\\sqrt{T}.\n\\]\nLemma~\\ref{etaconcave} implies that $\\nu$ can be embedded in $M$\nwith a stopping time $\\tau$ satisfying $\\tau\\leq T$, and the same stopping\ntime embeds $\\rho$ in $X$.\n\\end{pf}\n\n\\section{Examples}\n\\label{sec:eg}\n\n\\subsection{Brownian motion with drift}\n\nLet $X$ be a Brownian motion with drift, that is,\n\\[\nX_t = x_0 + \\gamma t + \\theta W_t,\n\\]\nwhere $\\gamma\\in\\R$, $\\theta> 0$ and $x_0 = 0$.\nThe scale function equals\n\\[\ns(x) =\n\\cases{ \\displaystyle\\frac{1}{\\kappa} \\bigl(1-\\exp( - \\kappa x ) \\bigr)\n&\\quad$\\mbox{for } \\kappa\\neq0$,\\vspace*{2pt}\n\\cr\nx &\\quad$\\mbox{for } \\kappa= 0$, }\n\\]\nwith $\\kappa=\\frac{2 \\gamma}{\\theta^2}$.\nIf $\\kappa>0$ then $s({\\mathbb R}) = (-\\infty, 1\/\\kappa)$, whereas if\n$\\kappa<0$ then $s({\\mathbb R}) = (1\/\\kappa, \\infty)$. Then, if\n$M=s(X)$ we\nhave $\\mathrm{d}M _t = \\theta(1 - \\kappa M_t) \\,\\mathrm{d}W _t$, and $M$ is a martingale.\n\nSuppose the aim is to embed $\\rho$.\nLet $F_\\rho$ be the distribution\nfunction of $\\rho$ and write $\\nu=\\rho\\circ s^{-1}$.\nSince $\\rho$ is a measure on ${\\mathbb\\R}$, $\\nu$ is a measure on\n$(l,r)$ and any embedding $\\tau$ is finite.\nNote that\\vspace*{-1pt}\n\\[\n\\nu^* = \\int s(x) \\rho(\\mathrm{d}x) = \\frac{1}{\\kappa} \\biggl( 1 - \\int\n_\\R \\mathrm{e}^{- \\kappa x} \\rho(\\mathrm{d}x) \\biggr).\n\\]\nThen, by Proposition~\\ref{prop:mneqbarnu} there is an embedding of\n$\\nu$ if and only if one of the following conditions is satisfied\n\\begin{enumerate}[3.]\n\\item$\\nu^* \\geq0$ and $\\kappa> 0$.\n\\item$\\nu^* \\leq0$ and $\\kappa< 0$.\n\\item$\\kappa= 0$.\n\\end{enumerate}\nCondition 1 and 2 simplify to\n\n0 \\leq\\nu^* \\kappa= 1 - \\int_\\R \\mathrm{e}^{- \\kappa x} \\rho(\\mathrm{d}x)\n$\nand hence $\\int_\\R \\mathrm{e}^{- \\kappa x} \\rho(\\mathrm{d}x) \\leq1$\nis necessary for the existence of an embedding if $\\kappa\\neq0$.\n\n\\subsubsection{The centred case}\n\nSuppose $\\int \\mathrm{e}^{- \\kappa x} \\rho(\\mathrm{d}x) = 1$. Then $\\nu$ has zero mean.\n\n\\begin{proposition}\nFor $\\kappa\\neq0$ ($\\kappa=0$) there exists an integrable stopping time\nembedding $\\rho$ into $X$ if and only if $x$ ($ x^2$) is integrable with\nrespect to $\\rho$. In this case, any minimal and integrable stopping time\n$\\tau$ satisfies\\vspace*{-1pt}\n\\[\n\\EE[\\tau] =\n\\cases{ \\displaystyle\\frac{1}{\\gamma} \\int x \\rho(\\mathrm{d}x) &\n\\quad$\\mbox{for } \\kappa\\neq0$,\\vspace*{2pt}\n\\cr\n\\displaystyle\\frac{1}{v^2} \\int x^2\\rho(\n\\mathrm{d}x) &\\quad$\\mbox{for } \\kappa= 0$. }\n\\]\n\\end{proposition}\n\n\\begin{pf}\nNote that $q$ is given by\\vspace*{-1pt}\n\\[\nq(x) =\n\\cases{ - \\displaystyle\\frac{2}{\\kappa\\theta^2} \\biggl( \\frac{1}{\\kappa} \\ln\n(1 - \\kappa x ) + x \\biggr) &\\quad$\\mbox{for } \\kappa\\neq0$,\\vspace*{2pt}\n\\cr\n\\displaystyle\\frac{x^2}{\\theta^2} &\\quad$\\mbox{for } \\kappa= 0$. }\n\\]\nMoreover,\\vspace*{-1pt}\n\\[\n\\int q(x) \\nu(\\mathrm{d}x) = \\int q\\bigl(s(x)\\bigr) \\rho(\\mathrm{d}x) =\n\\cases{ \\displaystyle\\frac{1}{\\gamma} \\int x \\rho(\\mathrm{d}x) &\\quad$\\mbox{for\n} \\kappa\\neq0$,\\vspace*{2pt}\n\\cr\n\\displaystyle\\frac{1}{v^2} \\int x^2\\rho(\\mathrm{d}x) &\n\\quad$\\mbox{for } \\kappa= 0$. }\n\\]\nThe result\nfollows now\nfrom Theorem~\\ref{thm:intcondidelta} and Proposition~\\ref{thm:intimplintofq}.\\vspace*{-1pt}\n\\end{pf}\n\nFinally, we consider sufficient conditions for there to exist a bounded\nembedding.\nIt turns out that the embedding stopping time\n$\\delta(1)$ is bounded if $h=F^{-1}\\circ\\Phi=s \\circ F_\\rho^{-1}\n\\circ\\Phi$ is Lipschitz\ncontinuous\nwith parameter $L$. We can thus recover the sufficient condition from\nSection~3.2 in \\cite{as11}.\n\n\n\\begin{proposition}\nSuppose that $F_\\rho^{-1}\\circ\\Phi$ is Lipschitz\ncontinuous\nwith Lipschitz constant $L \\in\\Rp$. Then there exists an embedding\n$\\tau$ of $\\rho$ in $X$ such that $\\tau\\le\n\\frac{L^2}{\\theta^2}$.\n\\end{proposition}\n\n\\begin{pf}\nFor this example, $x \\mapsto- 2 \\beta\/\\alpha+ \\alpha'$ is the constant\nmap. Hence, the result follows from Theorem~\\ref{thm:gendiff}.\n\\end{pf}\n\n\\subsubsection{The non-centred case}\n\nSuppose $\\int \\mathrm{e}^{-\\kappa x} \\rho(\\mathrm{d}x) < 1$. It is clear that $\\delta^*$\nis finite almost surely, but the arguments of Section~\\ref{ss:Bnc}\nshow that\nthere can\nbe no embedding of $\\rho$ which is bounded. Further, if $\\int\n\\mathrm{e}^{-\\kappa\nx} \\rho(\\mathrm{d}x) < 1$ then it follows that $\\nu\\in L^1$ and that\n$\\nu^* \\in(0, 1\/\\kappa)$ (or $-1\/\\kappa,0)$.\n\nNow consider integrable embeddings. By Theorem~\\ref{thm:integrableNC}, there\nexists an\nintegrable embedding if and only if $\\EE[H^M_{\\nu^*}] < \\infty$\nand $\\int q(x) \\nu(\\mathrm{d}x)<\\infty$. But\n\\( \\EE[H^M_{\\nu^*}] = \\EE[H^X_{s^{-1}(\\nu^*)}] \\)\nand, since $X$ is drifting Brownian motion, provided $\\sgn(z)=\n\\sgn(\\kappa) = \\sgn(\\gamma)$, $X$ hits $z$ in finite mean time.\nHence, $\\EE[H^M_{\\nu^*}]<\\infty$. Further\n\\begin{eqnarray*}\n\\int q(x)\\nu(\\mathrm{d}x) & = & \\int q\\bigl(s(x)\\bigr) \\rho(\\mathrm{d}x)\n\\\\\n& = & - \\frac{2}{\\kappa\\theta^2} \\int \\biggl[ \\biggl( \\frac\n{1}{\\kappa} \\ln \\bigl(1 -\n\\kappa s(x) \\bigr) + s(x) \\biggr) \\biggr] \\rho(\\mathrm{d}x)\n\\\\\n& = & - \\frac{2}{\\kappa\\theta^2} \\int \\biggl[ -x + \\frac{1}{\\kappa} \\bigl(1 -\n\\mathrm{e}^{-\\kappa x}\\bigr) \\biggr] \\rho(\\mathrm{d}x)\n\\\\\n& = & \\int\\frac{x}{\\gamma} \\rho(\\mathrm{d}x) - \\frac{1}{\\gamma} \\int\n\\frac{1}{\\kappa} \\bigl(1 - \\mathrm{e}^{-\\kappa x}\\bigr) \\rho(\\mathrm{d}x)\n\\\\\n& = & \\frac{1}{\\gamma} \\biggl( \\int x \\rho(\\mathrm{d}x) - \\nu^* \\biggr).\n\\end{eqnarray*}\nHence, there is an integrable embedding if $\\int x \\rho(\\mathrm{d}x) < \\infty$ and\n$\\delta^*$ is integrable.\n\n\n\\subsection{Bessel process}\n\nLet $R$ be the radial part of 3-dimensional Brownian motion so that $R$ solves\n$\\mathrm{d}R_t = \\mathrm{d}B_t + R_t^{-1}\\,\\mathrm{d}t $\nand suppose that $R_0=1$. Then the scale function is given by $s(r)= 1\n- r^{-1}$, and we can embed any\ndistribution $\\rho$ on $\\Rp$ in $R$ provided $\\int r^{-1} \\rho(\\mathrm{d}r)\n\\le1$ (see Proposition~\\ref{prop:mneqbarnu}).\n\n\\subsubsection{The centred case}\n\nSuppose that $\\int r^{-1} \\rho(\\mathrm{d}r) = 1$. Then $\\nu$ has zero mean.\n\n\\begin{proposition}\nThere exists an integrable stopping time that embeds $\\rho$ into $R$\nif and only if $\\int r^2 \\rho(\\mathrm{d}r) < \\infty$. In this case, any\nminimal and integrable stopping time $\\tau$ satisfies\n$\\EE[\\tau] = - \\frac{1}3 + \\int\\frac{1}3 r^2 \\rho(\\mathrm{d}r)$.\n\\end{proposition}\n\n\\begin{pf}\nNote that $\\eta(x) = (s' \\alpha) \\circ s^{-1}(x) = (1-x)^2$. Moreover,\n\\[\nq(x) = - \\frac{2}3 x + \\frac{1}3 \\frac{1}{(1-x)^2} -\n\\frac{1}3.\n\\]\nNotice that $\\int q(x) (\\rho\\circ s^{-1})(\\mathrm{d}x) = \\int ( \\frac{1}3\nr^2 + \\frac{2}3 r^{-1} - 1 ) \\rho(\\mathrm{d}r)$, and hence the result\nfollows from Theorem~\\ref{thm:intcondidelta} and Proposition~\\ref{thm:intimplintofq}.\n\\end{pf}\n\nBy Remark~\\ref{rem:felkot},\nwe have that $M_t = 1\n-R_t^{-1}$ is not a martingale (this is the Johnson--Helms example of a\nstrict local\nmartingale). Further,\nthe map $r \\mapsto- 2 \\beta(r)\/\\alpha(r) + \\alpha'(r) = - 2\/ r$ is\nincreasing.\n\nHowever, suppose we want to embed a target law $\\rho$ in $R$ in\nbounded time, where the support of $\\rho$\nis bounded away from both $0$ and $\\infty$ by $\\hat{l}$ and $\\hat\n{r}$, respectively. Let $\\bar{l} =\ns(\\hat{l})$ and $\\bar{r}=s(\\hat{r})$. Let $\\hat{R}$ be the stopped\nBessel process $\\hat{R}_t = R_{t \\wedge\nH_{\\hat{l}} \\wedge H_{\\hat{r}}}$ and let $\\bar{M} = s(\\hat{R})$.\nThen $\\bar{M}$ is a martingale, which is\nabsorbed at both $\\bar{l}$ and $\\bar{r}$. Then a necessary condition\nfor there to exist an embedding of\n$\\nu$ in $\\bar{M}$ in bounded time is that $\\nu$ has support in\n$[\\bar{l},\\bar{r}]$ and\n$\\int_{\\bar{l}}^{\\bar{r}}\nx \\nu(\\mathrm{d}x) = 0$. Hence, a necessary condition for it to be possible to\nembed $\\rho$ in $R$ in bounded time\nis\nthat $\\int_{\\hat{l}}^{\\hat{r}} r^{-1} \\rho(\\mathrm{d}r) = 1$. By Remark~\\ref\n{rem:notconcave}, a sufficient\ncondition is\nthat $\\int_{\\hat{l}}^{\\hat{r}} r^{-1} \\rho(\\mathrm{d}r) = 1$ and $\\log\nF_\\rho^{-1} \\circ\\Phi$ is Lipschitz\ncontinuous.\n\n\\subsubsection{The non-centred case}\n\nSuppose $\\int r^{-1} \\rho(\\mathrm{d}r) < 1$. It is clear that $\\delta^*$\nis finite almost surely, but the arguments of Section~\\ref{ss:Bnc}\nshow that there can be no embedding of $\\rho$ which is bounded.\n\nConsider integrable embeddings. By Theorem~\\ref{thm:integrableNC}, there\nexists an\nintegrable embedding if and only if $\\lim_{n \\to\\infty} \\frac\n{q(n)}{n} < \\infty$ and $\\int q(x) \\nu(\\mathrm{d}x)<\\infty$.\nFor the first part, we have that\n\\[\n\\lim_{n \\to\\infty} \\frac{q(n)}{n}= \\lim_{n \\to\\infty}-\n\\frac{2}3 + \\frac{1}3 \\frac{1}{(1-n)^2 n} - \\frac{1}{3 n} =\n-\\frac{2}{3} < \\infty.\n\\]\nFurthermore,\n\\begin{eqnarray*}\n\\int q(x)\\nu(\\mathrm{d}x) & = & \\int q\\bigl(s(x)\\bigr) \\rho(\\mathrm{d}x) = \\int\n_{\\Rp} \\biggl( \\frac{1}3 r^2 +\n\\frac{2}3 r^{-1} - 1 \\biggr) \\rho(\\mathrm{d}x)\n\\\\\n&=& \\int_{\\Rp} \\biggl( \\frac{1}3 r^2 -\n\\frac{1}3 - \\frac{2}3 s(x) \\biggr) \\rho(\\mathrm{d}x)\n\\\\\n&=& \\frac{1}{3} \\biggl(\\int_{\\Rp} r^2\n\\rho(\\mathrm{d}x) - 1 - 2 \\nu^* \\biggr).\n\\end{eqnarray*}\nHence, there exists an integrable stopping time if $r^2$ is integrable\nwith respect to $\\rho$.\n\n\\subsection{Ornstein--Uhlenbeck process}\n\nLet $X$ be an Ornstein--Uhlenbeck solving the SDE\n\\[\n\\mathrm{d}X_t = \\xi X_t\\,\\mathrm{d}t + \\sigma\\,\n\\mathrm{d}W _t,\n\\]\nwhere $\\xi\\in\\R$, $\\sigma> 0$ and $X_0=0$. The scale function is\ngiven by $s(x) = \\int_0^x \\mathrm{e}^{(-\\fraca{\\xi}{\\sigma^2} y^2)}\\,\\mathrm{d}y $.\n\n\\subsubsection*{The centred case}\n\nLet $\\rho$ be a distribution with $\\int s(x) \\rho(\\mathrm{d}x) = 0$. Then $\\nu\n$ has zero mean.\n\nWe next give sufficient conditions for $\\rho$ to be embeddable in\nbounded time. We need to distinguish between a positive and negative\nmean reversion speed $\\xi$.\n\nSuppose first that $\\xi> 0$, and the process is mean repelling. Then\nthe scale function\nis bounded. In this case,\n$- \\frac{2\n\\beta(x)}{\\alpha(x)} + \\alpha'(x) = -\n\\frac{2\\xi}{\\sigma} x$ is decreasing. Moreover,\\vspace*{2pt} for $g=F^{-1}_\\rho\n\\circ\\Phi$,\n$\\frac{g'}{\\alpha\\circ g} =\n\\frac{1}{\\sigma} g'$. Therefore, by Theorem~\\ref{thm:gendiff}, if\n$g$ is Lipschitz\ncontinuous with Lipschitz constant $L$, then there exists an embedding\nthat is bounded\nby $\\frac{L^2}{\\sigma^2}$.\n\nSuppose next that $\\xi< 0$. Then the derivative of the scale function\nsatisfies $s'(x)\n\\ge1$, $x \\in\\R$. Moreover, $\\eta(x) = (s' \\alpha) \\circ s^{-1}(x)\n\\ge\\sigma$ and the\nintensity of the time change satisfies $r^2(t,x) \\le\\frac{1}{\\sigma\n^2} b^2_x(t,x)$.\nTherefore, if $h = s \\circ F^{-1}_\\rho\\circ\\Phi$ is Lipschitz\ncontinuous with Lipschitz constant $L$, then there exists an embedding\nthat is bounded\nby $\\frac{L^2}{\\sigma^2}$. (Note that $h' = (s' \\circ g) g' \\geq g'$\nso that the\nrequirement that $h$ is Lipschitz is stronger than the requirement that\n$g$ is\nLipschitz.)\n\nFinally, suppose that $\\xi=0$. Then the scale function is the identity\nfunction, and\nthe arguments from each of the last two paragraphs apply and yield the\nsame sufficient\ncondition.\n\n\n\\section*{Acknowledgements}\n\nWe thank an anonymous referee for many helpful comments.\nStefan Ankirchner and Philipp Strack were supported by\nthe German Research Foundation (DFG) through the \\textit{Hausdorff\nCenter for Mathematics} and SFB TR 15.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}