diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzjmho" "b/data_all_eng_slimpj/shuffled/split2/finalzzjmho" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzjmho" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{sec.a}\nLet $k=k(x,y)$ be a complex valued function of two variables $x>0$ and $y>0$, which satisfies two conditions:\n\\begin{itemize}\n\\item\n$k$ is Hermitian: $k(x,y)=\\overline{k(y,x)}$; \n\\item\n$k$ is homogeneous of degree $-1$: $k(a x,a y)=k(x,y)\/a$.\n\\end{itemize}\nWe will call $k(x,y)$ a \\emph{Hardy kernel}; see remarks below about the terminology. \n\nThe purpose of this paper is to consider some spectral properties of the infinite matrices \n\\[\nK=\\{k(n,m)\\}_{n,m=1}^\\infty \\quad \\text{ in $\\ell^2({\\mathbb N})$}\n\\label{eq:3}\n\\]\nas well as their finite truncations\n$$\nK_N=\\{k(n,m)\\}_{n,m=1}^N \\quad \\text{ in ${\\mathbb C}^N$}\n$$\nas $N\\to\\infty$. \nOur point of view is the comparison between the Hardy kernel matrices $K$ and $K_N$ and their ``continuous analogues'', i.e. integral operators $T$ in $L^2(1,\\infty)$, \n$$\nTf(x)=\\int_1^\\infty k(x,y)f(y)dy, \\quad f\\in L^2(1,\\infty)\n$$\nas well as their truncations $T_N$ in $L^2(1,N)$, \n$$\nT_Nf(x)=\\int_1^N k(x,y)f(y)dy, \\quad f\\in L^2(1,N).\n$$\nIn fact, $T$ and $T_N$ are Wiener-Hopf operators in disguise. Indeed, an exponential change of variable $x=e^u$, $y=e^v$, $u,v\\in (0,\\infty)$, effects a unitary transformation which transforms $T$ (resp. $T_N$) into the integral operator on $L^2(0,\\infty)$ (resp. $L^2(0,\\log N)$) with the Wiener-Hopf kernel $k(e^{u-v},1)e^{(u-v)\/2}$. \n\n\n\nObviously, such exponential change of variable is not available on integers, \nand therefore the spectral analysis of the Hardy kernel matrices $K$ and $K_N$ presents considerable challenges. \nHowever, we will show that under suitable restrictions on the kernels:\n\\begin{itemize}\n\\item\nthe essential spectra of $K$ and $T$ coincide;\n\\item\nthe absolutely continuous (a.c.) parts of $K$ and $T$ are unitarily equivalent;\n\\item\nthe asymptotic spectral density of $K_N$ coincides with that of $T_N$ and is therefore given by the First Szeg\\H{o} Limit Theorem;\n\\item\n$K_N$ may have some eigenvalues (in contrast to $T_N$). \n\\end{itemize}\n\n\nTo enable meaningful analysis, we need to impose some constraints on the kernels $k$. First observe that the homogeneity of $k$ is equivalent to the representation \n$$\nk(x,y)=\\frac1{\\sqrt{xy}}\\widetilde k(x\/y)\n$$\nwith some function $\\widetilde k$ on $(0,\\infty)$. In what follows, we shall assume that \n$$\nk(x,y)=\\frac1{\\sqrt{xy}}\\frac1{2\\pi} \\int_{-\\infty}^\\infty \\varphi(t)(x\/y)^{-it}dt=\\frac1{\\sqrt{xy}}\\widehat \\varphi(\\log \\tfrac{x}{y}), \n$$\nwhere $\\varphi\\in L^1({\\mathbb R})$ is real-valued and \n$$\n\\widehat\\varphi(u)=\\frac1{2\\pi}\\int_{-\\infty}^\\infty \\varphi(t)e^{-itu}dt\n$$\nis the Fourier transform of $\\varphi$.\nWe shall call $\\varphi$ the \\emph{symbol} in this context. \nWith this notation, after the exponential change of variable $T$ becomes the integral operator with the kernel $\\widehat\\varphi(u-v)$, in agreement with the standard notion of a symbol of a Wiener-Hopf integral operator. \nWe shall henceforth indicate explicitly the dependence on the symbol and denote the above operators by $K(\\varphi)$, $K_N(\\varphi)$, $T(\\varphi)$ and $T_N(\\varphi)$. \n\n\nWe finish this section with some remarks on the terminology and the history of the problem; see Section~\\ref{sec.exa} for further discussion of related literature. \n\n\nHardy kernels are often associated with integral operators on $L^2(0,\\infty)$ rather than on $L^2(1,\\infty)$. By the same exponential change of variables, such integral operators are unitarily equivalent to the operators of convolution with $\\widehat\\varphi$ on $L^2({\\mathbb R})$. Thus, by the Fourier transform, they are unitarily equivalent to the operators of multiplication by the symbol $\\varphi$ in $L^2({\\mathbb R})$, and so the spectral theory of this class of operators is extremely simple. \n\n\nThe most famous Hardy kernel is \n$$\nk(x,y)=\\frac1{x+y}\\ ,\n$$\nwhose study, both in discrete and continuous versions, goes back to Hilbert and Schur. More generally, sufficient conditions for boundedness of Hardy kernel matrices, as well as (sometimes sharp) operator norm bounds for them on $\\ell^p$ spaces are discussed in detail in Chapter 9 of the classical monograph \\cite{HLP} by Hardy, Littlewood and Polya. They mostly argue by comparing matrices $K$ with the corresponding integral operators on $L^2(0,\\infty)$ and impose the condition \n\\[\n\\int_0^\\infty \\frac{\\abs{k(x,1)}}{\\sqrt{x}}dx<\\infty\n\\label{eq:14}\n\\]\ntogether with some monotonicity conditions. \n\n\nThe term ``Hardy kernel'' is usually applied to real-valued kernels which are symmetric, homogeneous of degree $-1$ and satisfy \\eqref{eq:14}.\nWe will not need \\eqref{eq:14}, but instead we will impose some conditions on the symbol $\\varphi$. \n\n\n\n\n\\section{Main results}\\label{sec.b}\nWe start with some general remarks. \nLet us write the quadratic form of $K_N(\\varphi)$ on a vector $a=\\{a_n\\}_{n=1}^N\\in {\\mathbb C}^N$ as follows:\n\\begin{align}\n\\jap{K_N(\\varphi)a,a}_{{\\mathbb C}^N}\n&=\n\\sum_{n,m=1}^N k(n,m)a_n\\overline{a}_m\n=\n\\frac1{2\\pi}\\sum_{n,m=1}^N a_n\\overline{a}_m \\int_{-\\infty}^\\infty\\varphi(t) n^{-\\frac12-it}m^{-\\frac12+it}dt\n\\notag\n\\\\\n&=\\frac1{2\\pi}\\int_{-\\infty}^\\infty\\varphi(t) \\Abs{\\sum_{n=1}^N a_nn^{-\\frac12-it}}^2dt\\ .\n\\label{eq:6}\n\\end{align}\nThis formula suggests the following:\n\\begin{itemize}\n\\item\n$K_N(\\varphi)$ depends monotonically on $\\varphi$ in the quadratic form sense; in particular, if $\\varphi\\geq0$, then $K_N(\\varphi)$ is positive semi-definite. \n\\item\nThe study of $K_N(\\varphi)$ is related to the theory of Dirichlet series. \nWe will touch upon this aspect of the problem only briefly in Lemma~\\ref{lma.8}. \n\\item\nIt is easy to see that $\\varphi\\in L^1({\\mathbb R})$ is a necessary condition for the definition of $K_N(\\varphi)$ to make sense. \n\\end{itemize}\nAs a warm-up, let us compute the asymptotics of the trace of $K_N(\\varphi)$. \nWe have\n$$\nk(n,n)=\\frac1{n}\\widehat\\varphi(0)=\\frac1{n}\\frac1{2\\pi}\\int_{-\\infty}^\\infty\\varphi(t)dt,\n$$\nand therefore \n$$\n\\Tr K_N(\\varphi)\n=\n\\frac1{2\\pi}\\int_{-\\infty}^\\infty \\varphi(t)dt \\sum_{n=1}^N \\frac1n\n=\n\\frac1{2\\pi}\\int_{-\\infty}^\\infty \\varphi(t)dt \\ \\bigl(\\log N+O(1)\\bigr), \\quad N\\to\\infty.\n$$\nNow, considering the case $\\varphi\\geq0$, it follows that for any $\\varepsilon>0$ we have \n$$\n\\#\\{j: \\lambda_j(K_N(\\varphi))>\\varepsilon\\}\n\\leq \\Tr(K_N(\\varphi)\/\\varepsilon)=O(\\log N), \\quad N\\to\\infty,\n$$\nwhere $\\{\\lambda_j(K_N(\\varphi))\\}_{j=1}^N$ are the eigenvalues of $K_N(\\varphi)$ and $\\#$ is the number of elements in a given set. So we see that ``most'' of the $N$ eigenvalues of $K_N(\\varphi)$ are located near zero and only $O(\\log N)$ eigenvalues are located above $\\varepsilon>0$. Our first result concerns this logarithmically small proportion of the eigenvalues of $K_N(\\varphi)$ and gives their asymptotic density.\n\n\n\\begin{theorem}\\label{thm2}\nLet $\\varphi\\in L^1({\\mathbb R})$ be a real-valued symbol, and let $g$ be a Lipschitz continuous function on ${\\mathbb R}$ with $g(0)=0$. As above, set \n$$\nK_N(\\varphi)=\\{k(n,m)\\}_{n,m=1}^N, \\quad k(n,m)=\\frac1{\\sqrt{nm}}\\widehat\\varphi(\\log\\tfrac{n}{m}). \n$$\nThen \n\\[\n\\lim_{N\\to\\infty}(\\log N)^{-1}\\Tr g(K_N(\\varphi))=\\frac1{2\\pi}\\int_{-\\infty}^\\infty g(\\varphi(t))dt\\ .\n\\label{eq:1}\n\\]\n\\end{theorem}\nSince by our assumptions\n$$\n\\abs{g(\\varphi(t))}\\leq C\\abs{\\varphi(t)}, \n$$\nthe integral in the right hand side of \\eqref{eq:1} converges absolutely. \n\n\nFormula \\eqref{eq:1} with $T_N(\\varphi)$ in place of $K_N(\\varphi)$ is well known (after the exponential change of variable reducing $T_N(\\varphi)$ to a Wiener-Hopf operator), see e.g. \\cite[Section~8.6]{GS}. It is more commonly used for Toeplitz matrices and in that context it is known as the First Szeg\\H{o} Limit Theorem, see e.g. \\cite[Section 5.4]{BS}. \n\n\n\nAs it is standard in this circle of questions, one can replace a Lipschitz function $g$ in \\eqref{eq:1} by the characteristic function of an interval $(\\lambda,\\infty)$, $\\lambda>0$, as long as the set $\\{t\\in{\\mathbb R}: \\varphi(t)=\\lambda\\}$ has zero Lebesgue measure. This leads to a more expressive formula\n$$\n\\lim_{N\\to\\infty}(\\log N)^{-1}\\#\\{j: \\lambda_j(K_N(\\varphi))>\\lambda)\\}\n=\n\\frac1{2\\pi}\\meas\\{t: \\varphi(t)>\\lambda\\}\\ ,\n$$\nwhere $\\meas$ is the Lebesgue measure on ${\\mathbb R}$. \n\n\n\nOur second result characterises the essential and the absolutely continuous spectra of $K(\\varphi)$. \nHere for simplicity we restrict ourselves to bounded positive semi-definite operators. \n\n\\begin{theorem}\\label{thm1}\nLet $\\varphi\\in L^\\infty({\\mathbb R})$ be a non-negative function satisfying \n\\[\n\\int_{-\\infty}^\\infty \\varphi(t)(\\log(2+\\abs{t}))^\\delta dt<\\infty\n\\label{eq:2}\n\\]\nfor some $\\delta>2$; assume that $\\varphi$ is not identically equal to zero. \nThen $K(\\varphi)$, defined by the matrix \\eqref{eq:3}, is a bounded positive semi-definite operator on $\\ell^2({\\mathbb N})$.\nThe kernel of $K(\\varphi)$ is trivial. The essential spectrum of $K(\\varphi)$ coincides with the essential spectrum of $T(\\varphi)$, and the a.c. part of $K(\\varphi)$ is unitarily equivalent to $T(\\varphi)$. \n\\end{theorem}\n\n\\begin{remark*}\n\\begin{enumerate}[1.]\n\\item\nBy a theorem of M.~Rosenblum \\cite{R2}, the spectrum of a self-adjoint Toeplitz operator is purely a.c. (unless the symbol is constant). The same applies to $T(\\varphi)$, since it is unitarily equivalent to a Toeplitz operator. \n\\item\nThe location of the a.c. spectrum of a Toeplitz operator and its multiplicity function can be explicitly described in terms of the symbol; see \\cite{I,R3} and \\cite{SY1,SY2}. Therefore, the same applies to $T(\\varphi)$. In particular, if the symbol $\\varphi\\in L^1({\\mathbb R})$ is continuous, then the multiplicity function of $T(\\varphi)$ coincides with $1\/2$ times the multiplicity function of the operator of multiplication by $\\varphi$ in $L^2({\\mathbb R})$. For example, if $\\varphi(t)$ is positive, strictly increasing on $(-\\infty,0)$ and strictly decreasing on $(0,\\infty)$, then the spectrum of $T(\\varphi)$ is $[0,\\varphi(0)]$ with multiplicity one. \n\\item\nIt seems to be an interesting open question to determine the class of symbols $\\varphi$ that corresponds to bounded operators $K(\\varphi)$; it is unlikely that \\eqref{eq:2} is optimal and it is not clear what the optimal condition should be. \n\\end{enumerate}\n\\end{remark*}\n\n\nTheorem~\\ref{thm1} may suggest that the spectrum of $K(\\varphi)$ coincides with that of $T(\\varphi)$. In fact, this is false; we demonstrate this below by showing that in some natural asymptotic regime, eigenvalues of $K(\\varphi)$ always appear (in contrast to $T(\\varphi)$). For a self-adjoint operator $A$ let us denote by $n(\\lambda;A)$ the rank of the spectral projection $E_A(\\lambda,\\infty)$. \nIn other words, $n(\\lambda;A)$ is the number of eigenvalues (counting multiplicities) of $A$ \nin $(\\lambda,\\infty)$ and $n(\\lambda;A)=\\infty$, if $A$ has some essential spectrum in $(\\lambda,\\infty)$. \n\\begin{theorem}\\label{thm3}\nLet $\\varphi$ be as in Theorem~\\ref{thm1} and for $\\alpha>0$, let \n$$\n\\varphi_\\alpha(t)=\\frac1\\alpha\\varphi(t\/\\alpha).\n$$\nThen for all sufficiently large $\\alpha$ we have \n$$\nn(\\lambda; K(\\varphi_\\alpha))\\geq\\#\\{j\\in{\\mathbb N}:\\frac1j\\widehat\\varphi(0)>\\lambda\\}.\n$$\nIn particular, the number of eigenvalues of $K(\\varphi_\\alpha)$ above the essential spectrum tends to infinity as $\\alpha\\to\\infty$.\n\\end{theorem}\n\\begin{proof}\nBy min-max, for all $\\lambda>0$ we have \n$n(\\lambda;K(\\varphi_\\alpha))\\geq n(\\lambda; K_N(\\varphi_\\alpha))$.\nBy the Riemann-Lebesgue lemma, $\\widehat \\varphi$ tends to zero at infinity and therefore, for all off-diagonal elements of $K_N(\\varphi_\\alpha)$ we have\n$$\nk_\\alpha(n,m)=\\frac1{\\sqrt{nm}}\\widehat\\varphi(\\alpha\\log\\tfrac{n}{m})\\to0, \\quad \\alpha\\to\\infty. \n$$\nIt follows that $\\norm{K_N(\\varphi_\\alpha)-K_{N,\\infty}}\\to0$ as $\\alpha\\to\\infty$, where $K_{N,\\infty}$ is the diagonal matrix with elements $\\{\\widehat\\varphi(0), \\frac12\\widehat\\varphi(0),\\dots,\\frac1N\\widehat\\varphi(0)\\}$ on the diagional. Thus, for any $\\varepsilon>0$,\n$$\nn(\\lambda;K_N(\\varphi_\\alpha))\\geq n(\\lambda+\\varepsilon; K_{N,\\infty})\n$$\nfor all sufficiently large $\\alpha$. Putting this together and sending $\\varepsilon\\to0$ and $N\\to\\infty$,\nwe get the desired inequality. \nBy Theorem~\\ref{thm1}, the essential spectrum of $K(\\varphi_\\alpha)$ is $[0,\\norm{\\varphi_\\alpha}_{L^\\infty}]=[0,\\frac1\\alpha\\norm{\\varphi}_{L^\\infty}]$; as it shrinks to zero, the number of eigenvalues above it tends to infinity. \n\\end{proof}\n\n\\begin{remark*}\n\\begin{enumerate}[1.]\n\\item\nObserve that $T(\\varphi)$ satisfies the estimate $\\norm{T(\\varphi)}\\leq\\norm{\\varphi}_{L^\\infty({\\mathbb R})}$. \nIn contrast to this, the last theorem shows that the norm of $K(\\varphi)$ may be strictly greater than $\\norm{\\varphi}_{L^\\infty({\\mathbb R})}$. Moreover, considering the asymptotics $\\alpha\\to\\infty$, we see that the estimate \n$$\n\\norm{K(\\varphi)}\\leq C\\norm{\\varphi}_{L^\\infty({\\mathbb R})}\n$$\nis false. \n\\item\nIf $\\widehat\\varphi$ tends to zero sufficiently fast at infinity, one can upgrade the above reasoning to conclude that if the eigenvalues of $K(\\varphi_\\alpha)$ are ordered non-increasingly, then the $j$'th eigenvalue satisfies \n\\[\n\\lambda_j(K(\\varphi_\\alpha))=\\frac1j\\widehat\\varphi(0)+O(1\/\\alpha), \\quad \\alpha\\to\\infty.\n\\label{eq:13}\n\\]\nThis is exactly what was proven in \\cite{BPP}, see Example~\\ref{exa.2} below. \n\\end{enumerate}\n\\end{remark*}\n\n\\section{Examples}\\label{sec.exa}\nHere we consider several examples of Hardy kernels with references to existing literature. \n\\begin{example}\nLet \n$$\nk(x,y)=\\frac1{x+y}, \\quad \\varphi(t)=\\frac{\\pi}{\\cosh \\pi t}. \n$$\nThis example is very special because in this case $k(x,y)$ is a Hankel kernel, i.e. it depends on the sum $x+y$. \nThe corresponding matrix $K(\\varphi)$ is the classical Hilbert's matrix, which was explicitly diagonalised by M.~Rosenblum in \\cite{R1,R2} in terms of special functions (see also \\cite{KS}). \nRosenblum proved that $K(\\varphi)$ has a purely a.c. spectrum $[0,\\pi]$ of multiplicity one.\n\nThe asymptotic spectral density of the truncated Hilbert matrix was determined by Widom in \\cite[Theorem 4.3]{Widom} (note that there is a factor of $2\\pi$ missing in \\cite{Widom}). See also \\cite{Fedele} for an alternative proof and for a more general class of Hankel matrices. \n\nOur Theorems~\\ref{thm2} and \\ref{thm1} specialised to this case are in agreement with all of the above but do not add anything new. \n\nHowever, the scaled version of this example\n$$\nk_\\alpha(x,y)=\\frac1{\\sqrt{xy}}\\frac1{(x\/y)^{\\alpha\/2}+(y\/x)^{\\alpha\/2}}, \n\\quad\n\\varphi_\\alpha(t)=\\frac{\\pi}{\\alpha\\cosh(\\frac{\\pi t}{\\alpha})}, \\quad \\alpha>0,\n$$\nseems to be new. By Theorem~\\ref{thm1}, the a.c. spectrum of $K(\\varphi_\\alpha)$ is $[0,\\pi\/\\alpha]$ and has multiplicity one. \n\\end{example}\n\n\\begin{example}\\label{exa.2}\nFor $\\alpha>0$, let \n$$\nk_\\alpha(x,y)\n=\\frac1{\\sqrt{xy}}\\min\\{(x\/y)^\\alpha,(y\/x)^\\alpha\\}, \n\\quad \n\\varphi_\\alpha(t)=\\dfrac{2}{\\alpha(1+(t\/\\alpha)^2)}.\n$$\nThe operator $K(\\varphi_\\alpha)$ was introduced in \\cite{Brevig} in connection with a question about composition operators on the Hardy space of Dirichlet series. Some estimates for the norm of $K(\\varphi_\\alpha)$ were given in \\cite{Brevig}, and a detailed spectral analysis of this operator was accomplished in \\cite{BPP}. \nIt was established that $K(\\varphi_\\alpha)$ has a.c. spectrum $[0,2\/\\alpha]$ of multiplicity one, no singular continuous spectrum and finitely many eigenvalues above $2\/\\alpha$, satisfying \\eqref{eq:13}. \nThese facts are in full agreement with Theorems~\\ref{thm1} and \\ref{thm3}, as the Wiener-Hopf operator $T(\\varphi_\\alpha)$ has a purely a.c. spectrum $[0,2\/\\alpha]$ of multiplicity one. \n\n\n\nThis example is very special because, as shown in \\cite{BPP}, the operator $K(\\varphi_\\alpha)$ is an inverse of a Jacobi matrix. In particular, since the spectrum of any Jacobi matrix is simple, all eigenvalues of $K(\\varphi_\\alpha)$ are simple. \nIt was also shown that if a Hardy kernel matrix with a continuous kernel is an inverse of a Jacobi matrix, then it coincides, up to a factor, with $K(\\varphi_\\alpha)$ for some $\\alpha>0$. \n\nThe spectral density of $K(\\varphi_\\alpha)$ was not computed in \\cite{BPP}, and Theorem~\\ref{thm2} in this case seems to be new. The author is grateful to Uzy Smilansky for asking the question about spectral density in this context. \n\\end{example}\n\n\\begin{example}\nFor $\\alpha>0$, let \n$$\nk_\\alpha(x,y)=\\frac1{\\sqrt{xy}}\\biggl(\\sqrt{\\frac{x}{y}}+\\sqrt{\\frac{y}{x}}\\biggr)^{-\\alpha}\n=\\frac1{\\sqrt{xy}}\\frac{(xy)^{\\alpha\/2}}{(x+y)^{\\alpha}}, \n\\quad\n\\varphi_\\alpha(t)=\\frac1{\\Gamma(\\alpha)}\\abs{\\Gamma(\\tfrac{\\alpha}{2}+it)}^2, \n$$\nwhere $\\Gamma$ is the Gamma-function. \nWe note that the scaling in $\\alpha$ here is different from the one in Theorem~\\ref{thm3}. \nAccording to Theorem~\\ref{thm1}, the essential spectrum and the a.c. spectrum of $K(\\varphi_\\alpha)$ is $[0,\\varphi_\\alpha(0)]$, with multiplicity one. \n\nFor $\\alpha=1$ this is the Hilbert matrix, and for $\\alpha=2$ this is a variant of the so-called Bergman-Hilbert matrix, considered in \\cite{G,DG,KS}. \nMore generally, for all $\\alpha\\in{\\mathbb N}$, the following matrix was considered in \\cite{KS} (as part of a larger three-parameter family of infinite matrices): \n$$\nB_\\alpha=\\{b_{n,m}\\}_{n,m=1}^\\infty, \\quad\nb_{n,m}=\\frac{\\sqrt{(n)_{\\alpha-1}(m)_{\\alpha-1}}}{(n+m-1)_{\\alpha}}, \n$$\nwhere $(x)_\\alpha=x(x+1)\\cdots(x+\\alpha-1)$ is the Pochhammer symbol. \nThis is not a Hardy kernel matrix, but for large $n,m$ it has the same asymptotics as $K(\\varphi_\\alpha)$. \nThis matrix was explicitly diagonalised in \\cite{KS} in terms of orthogonal polynomials; it was found that the a.c. spectrum of $B_\\alpha$ is $[0,\\varphi_\\alpha(0)]$ with multiplicity one, that it has no singular continuous spectrum and that for large $\\alpha$ there are some eigenvalues above the continuous spectrum. \n\nIt is easy to see (cf. the argument of \\cite[Proposition 9]{KS}) that $K(\\varphi_{2})-B_2$ is a trace class operator, and therefore, by the Kato-Rosenblum theorem, the a.c. parts of $B_2$ and $K(\\varphi_{2})$ are unitary equivalent and by Weyl's theorem (invariance of essential spectrum under compact perturbations) the essential spectra of $B_2$ and $K(\\varphi_2)$ coincide. This is in agreement with Theorem~\\ref{thm1}. \nIt is not clear whether the trace class argument works for $\\alpha>2$, but in any case comparing Theorem~\\ref{thm1} with the results of \\cite{KS} we see that the essential spectra and the a.c. spectra of $B_\\alpha$ and $K(\\varphi_\\alpha)$ coincide. \n\nWe are not aware of the spectral density of $K(\\varphi_\\alpha)$ having been discussed in the literature. \n\\end{example}\n\n\n\n\\begin{example}\nLet \n$$\nk(x,y)=\\frac{\\log(x\/y)}{x-y}, \\quad\n\\varphi(t)=\\frac{\\pi^2}{(\\cosh \\pi t)^2},\n$$\nand $k(x,x)=1\/x$ by continuity.\nAs far as we are aware, the spectral properties of the corresponding Hardy kernel matrix $K(\\varphi)$ have not been considered in the literature, except for the norm bound in \\cite[Inequality 342]{HLP} as a ``miscellaneous example''. (The norm bound there is $\\pi^2$.)\nTheorem~\\ref{thm1} in this case ensures that the a.c. spectrum of $K(\\varphi)$ is $[0,\\pi^2]$ and has multiplicity one.\n\\end{example}\n\n\\begin{example}\nA more general version of the previous example is \n$$\nk(x,y)=\\frac{\\omega(\\log(x\/y))}{x-y}, \\quad\n\\widehat \\varphi(u)=\\frac{\\omega(u)}{2\\sinh(u\/2)}, \n$$\nwhere $\\omega$ is a smooth odd function which satisfies the growth condition \n$$\n\\omega(u)=O(e^{\\alpha\\abs{u}}), \\quad \\alpha<1\/2,\n$$\nas $\\abs{u}\\to\\infty$. For example, $\\omega(u)=2\\sinh(\\alpha u)$, $0<\\alpha<1\/2$ gives\n$$\nk(x,y)=\\frac{(x\/y)^\\alpha-(y\/x)^\\alpha}{x-y}, \\quad \\varphi(t)=\\frac{2\\pi\\sin(2\\pi \\alpha)}{\\cos(2\\pi\\alpha)+\\cosh(2\\pi t)} \n$$\nwith $k(x,x)=2\\alpha\/x$ by continuity.\n\\end{example}\n\n\\begin{example}\nLet \n$$\nk(x,y)=\\frac1{\\sqrt{xy}}\\frac{\\sin (\\log(x\/y))}{\\log(x\/y)}, \n\\quad \n\\varphi(t)=\\pi\\chi_{(-1,1)}(t),\n$$\nand $k(x,x)=1\/x$ by continuity. \nThis example may be of interest because the symbol here is discontinuous. \nBy Theorem~\\ref{thm1}, the a.c. spectrum of $K(\\varphi)$ is $[0,\\pi]$ and has multiplicity one. \nBy Theorem~\\ref{thm2}, the asymptotic spectral density of $K(\\varphi)$ is concentrated at the points $\\pi$ and $0$. \n\\end{example}\n\n\n\\section{Key notation and the outline of the proof}\\label{sec.c}\nFirst we discuss the proof of Theorem~\\ref{thm2}. \nIn order to motivate what comes next, we factorise $\\varphi=\\abs{\\varphi}^{1\/2}\\varphi^{1\/2}$, where $\\varphi^{1\/2}=\\abs{\\varphi}^{1\/2}\\sign(\\varphi)$, and rewrite formula \\eqref{eq:6} for the \nthe quadratic form of $K_N(\\varphi)$ as follows:\n$$\n\\jap{K_N(\\varphi)a,a}_{{\\mathbb C}^N}\n=\\frac1{2\\pi}\\int_{-\\infty}^\\infty\n\\biggl(\\abs{\\varphi(t)}^{1\/2}\\sum_{n=1}^N a_nn^{-\\frac12-it}\\biggr)\n\\overline{\\biggl(\\varphi(t)^{1\/2}\\sum_{n=1}^N a_nn^{-\\frac12-it}\\biggr)}\ndt \\ .\n$$\nThis suggests the following factorisation of $K_N(\\varphi)$. \nFor $\\psi\\in L^2({\\mathbb R})$, let us define an operator $A_N(\\psi):L^2({\\mathbb R})\\to{\\mathbb C}^N$, \n$$\n(A_N(\\psi)f)_n=\\frac1{\\sqrt{2\\pi}}\\int_{-\\infty}^\\infty f(t)\\psi(t)n^{-\\frac12+it}dt\\ , \\quad n=1,\\dots,N.\n$$\nThen \n$$\nK_N(\\varphi)=A_N(\\varphi^{1\/2})A_N(\\abs{\\varphi}^{1\/2})^*.\n$$\nIn a similar way, we factorise the operator $T_N(\\varphi)$ \n$$\nT_N(\\varphi)=B_N(\\varphi^{1\/2})B_N(\\abs{\\varphi}^{1\/2})^*, \n$$\nwhere $B_N(\\psi):L^2({\\mathbb R})\\to L^2(1,N)$, \n$$\n(B_N(\\psi)f)(x)=\\frac1{\\sqrt{2\\pi}}\\int_{-\\infty}^\\infty f(t)\\psi(t)x^{-\\frac12+it}dt\\ ,\\quad x\\in(1,N).\n$$\nThe main step of the proof of Theorem~\\ref{thm2} is the proof of \\eqref{eq:1} for $g(\\lambda)=\\lambda^m$, $m\\in{\\mathbb N}$ and for a suitable dense class of symbols $\\varphi$. \nBy the cyclicity of trace,\n\\begin{align*}\n\\Tr \\bigl(K_N(\\varphi)^m\\bigr)\n&=\\Tr\\bigl((A_N(\\varphi^{1\/2})A_N(\\abs{\\varphi}^{1\/2})^*)^m\\bigr)\n=\\Tr\\bigl((A_N(\\abs{\\varphi}^{1\/2})^*A_N(\\varphi^{1\/2}))^m\\bigr)\\ ,\n\\\\\n\\Tr \\bigl(T_N(\\varphi)^m\\bigr)\n&=\\Tr\\bigl((B_N(\\varphi^{1\/2})B_N(\\abs{\\varphi}^{1\/2})^*)^m\\bigr)\n=\\Tr\\bigl((B_N(\\abs{\\varphi}^{1\/2})^*B_N(\\varphi^{1\/2}))^m\\bigr)\\ .\n\\end{align*}\nNow let us compare $A_N^*A_N$ and $B_N^*B_N$. Observe that the integral kernel of $A_N(\\abs{\\varphi}^{1\/2})^*A_N(\\varphi^{1\/2})$ is\n$$\n\\frac1{2\\pi}\\abs{\\varphi}^{1\/2}(t_1)\\varphi^{1\/2}(t_2)\\zeta_N(1+i(t_1-t_2)), \n\\quad\n\\zeta_N(s):=\\sum_{n=1}^N n^{-s}\n$$\nand similarly the integral kernel of $B_N(\\abs{\\varphi}^{1\/2})^*B_N(\\varphi^{1\/2})$ is\n$$\n\\frac1{2\\pi}\\abs{\\varphi}^{1\/2}(t_1)\\varphi^{1\/2}(t_2)\\eta_N(1+i(t_1-t_2)), \n\\quad\n\\eta_N(s):=\\int_{1}^N x^{-s}dx.\n$$\nUsing elementary estimates of $\\zeta_N(s)-\\eta_N(s)$, we shall prove\nthe trace norm estimate\n$$\n\\norm{A_N(\\abs{\\varphi}^{1\/2})^*A_N(\\varphi^{1\/2})-B_N(\\abs{\\varphi}^{1\/2})^*B_N(\\varphi^{1\/2})}_{\\mathbf{S}_1}=O(1), \\quad N\\to\\infty,\n$$\nfor a suitable dense subclass of symbols $\\varphi$. Here and in what follows $\\norm{\\cdot}_{\\mathbf{S}_1}$ is the trace norm.\nUsing this estimate, it is easy to derive the relation \n$$\n\\Tr \\bigl((K_N(\\varphi))^m\\bigr)=\\Tr\\bigl((T_N(\\varphi))^m\\bigr)+O(1), \\quad N\\to\\infty.\n$$\nThe asymptotics of the trace in the right hand side is well known, see e.g. \\cite[Section~8.6]{GS}: \n\\[\n\\lim_{N\\to\\infty}(\\log N)^{-1}\\Tr\\bigl((T_N(\\varphi))^m\\bigr)\n=\n\\frac1{2\\pi}\\int_{-\\infty}^\\infty \\varphi(t)^mdt \\ .\n\\label{eq:5}\n\\]\nBy linearity, we obtain formula \\eqref{eq:1} for polynomials $g$. \nStandard arguments allow us to extend this to all symbols $\\varphi\\in L^1({\\mathbb R})$ and all Lipschitz continuous functions $g$. \n\n\nThe outline of the proof of Theorem~\\ref{thm1} is similar. Recall that here $\\varphi\\geq0$ by hypothesis. We denote $\\psi=\\varphi^{1\/2}$ and write \n$$\nK(\\varphi)=A(\\psi)A(\\psi)^*, \\quad\nT(\\varphi)=B(\\psi)B(\\psi)^*, \n$$\nwhere $A(\\psi):L^2({\\mathbb R})\\to\\ell^2({\\mathbb N})$ and $B(\\psi):L^2({\\mathbb R})\\to L^2(0,\\infty)$ are defined by \n\\begin{align}\n(A(\\psi)f)_n&=\\frac1{\\sqrt{2\\pi}}\\int_{-\\infty}^\\infty \\psi(t) f(t)n^{-\\frac12+it}dt\\ ,\n\\label{eq:7}\n\\\\\n(B(\\psi)f)(x)&=\\frac1{\\sqrt{2\\pi}}\\int_{-\\infty}^\\infty \\psi(t) f(t)x^{-\\frac12+it}dt \\ .\n\\label{eq:8}\n\\end{align}\nOne slight technical complication is that $A(\\psi)$ and $B(\\psi)$\nare not automatically bounded for $\\varphi\\in L^1({\\mathbb R})$; but they are bounded under the additional assumptions on $\\varphi$, listed in the hypothesis of Theorem~\\ref{thm1}. \nWe prove that under these assumptions, the difference\n$$\nA(\\psi)^*A(\\psi)-B(\\psi)^*B(\\psi)\n$$\nis a trace class operator. After this, Theorem~\\ref{thm1} follows by an application of the Kato-Rosenblum theorem (invariance of a.c. spectrum under trace class perturbations) and Weyl's theorem (invariance of essential spectrum under compact perturbations). \n\n\n\n\\section{Proof of Theorem~\\ref{thm2}}\\label{sec.d}\n\nFor $\\psi\\in L^2({\\mathbb R})$, let $A_N(\\psi):L^2({\\mathbb R})\\to{\\mathbb C}^N$ and $B_N(\\psi):L^2({\\mathbb R})\\to L^2(1,N)$ be the operators defined in the previous section. \n\\begin{lemma}\\label{lma.1}\nLet $\\varphi\\in L^1({\\mathbb R})$ satisfy \n\\[\n\\int_{-\\infty}^\\infty \\abs{\\varphi(t)}(\\log(2+\\abs{t}))^\\delta dt<\\infty\n\\label{eq:4}\n\\]\nfor some $\\delta>2$. Let \n\\[\nD_N(\\varphi)=A_N(\\abs{\\varphi}^{1\/2})^*A_N(\\varphi^{1\/2})-B_N(\\abs{\\varphi}^{1\/2})^*B_N(\\varphi^{1\/2}). \n\\label{eq:9}\n\\]\nThen $D_N(\\varphi)$ converges in the trace norm as $N\\to\\infty$; in particular, $\\norm{D_N(\\varphi)}_{\\mathbf{S}_1}=O(1)$. \n\\end{lemma}\n\\begin{proof}\nFor $x\\geq1$, denote \n$$\nu_x(t)=\\varphi^{1\/2}(t)x^{-it}, \\quad\nv_x(t)=\\abs{\\varphi}^{1\/2}(t)x^{-it}. \n$$\nWe consider $u_x$ and $v_x$ as elements of $L^2({\\mathbb R})$; observe that \n$$\n\\norm{v_x}^2_{L^2({\\mathbb R})}=\\norm{u_x}^2_{L^2({\\mathbb R})}=\\norm{\\varphi}_{L^1({\\mathbb R})}.\n$$\nBelow $\\jap{\\cdot,\\cdot}$ is the inner product in $L^2({\\mathbb R})$. \nWith this notation we can write\n\\begin{align*}\nA_N(\\abs{\\varphi}^{1\/2})^*A_N(\\varphi^{1\/2})&=\n\\frac1{2\\pi}\\sum_{n=1}^N \\frac1n \\jap{\\cdot,u_n}v_n, \n\\\\\nB_N(\\abs{\\varphi}^{1\/2})^*B_N(\\varphi^{1\/2})&=\n\\frac1{2\\pi}\\int_1^N \\frac1x\\jap{\\cdot,u_x}v_x \\ dx\n\\\\\n&=\\frac1{2\\pi}\\sum_{n=2}^N\\int_{n-1}^n \\frac1x\\jap{\\cdot,u_x}v_x\\ dx\\ .\n\\end{align*}\nOur aim is to estimate the trace norm of the difference of $n$'th terms in the two sums in the right hand sides. For $n\\geq2$ we have \n\\begin{align*}\n\\frac1n\\jap{\\cdot,u_n}v_n-&\\int_{n-1}^n \\frac1x\\jap{\\cdot,u_x}v_x\\ dx\n=\\int_{n-1}^n \\biggl(\\frac1n-\\frac1x\\biggr)\\jap{\\cdot,u_n}v_n\\ dx\n\\\\\n&+\\int_{n-1}^n \\frac1x\\jap{\\cdot,u_n-u_x}v_n\\ dx\n+\\int_{n-1}^n\\frac1x\\jap{\\cdot,u_x}(v_n-v_x)\\ dx\\ .\n\\end{align*}\nThe first term is easy to estimate:\n$$\n\\Norm{\\int_{n-1}^n \\biggl(\\frac1n-\\frac1x\\biggr)\\jap{\\cdot,u_n}v_n\\ dx}_{\\mathbf{S}_1}\n\\leq \n\\frac1{n(n-1)}\\norm{\\jap{\\cdot,u_n}v_n}_{\\mathbf{S}_1}=\\frac1{n(n-1)}\\norm{\\varphi}_{L^1({\\mathbb R})}.\n$$\nIn order to estimate the second and third terms, we need to consider the differences $u_n-u_x$ and $v_n-v_x$. We have\n$$\nu_n(t)-u_x(t)=\\varphi^{1\/2}(t)(n^{-it}-x^{-it})=\\varphi^{1\/2}(t)(e^{-it\\log n}-e^{-it\\log x})\\ .\n$$\nWe use the elementary estimate $\\abs{e^{ia}-1}\\leq \\min\\{\\abs{a},2\\}$. \nThen, for $n-1\\leq x\\leq n$, we have \n$$\n\\abs{u_n(t)-u_x(t)}\n\\leq \n\\abs{\\varphi}^{1\/2}(t)\\min\\{\\abs{t}\\Abs{\\log \\tfrac{x}{n}}, 2\\}\n\\leq \n2\\abs{\\varphi}^{1\/2}(t)\\min\\{\\tfrac{\\abs{t}}{n}, 1\\},\n$$\nbecause $\\Abs{\\log \\tfrac{x}{n}}\\leq \\Abs{\\log (1-\\frac1n)}\\leq 2\/n$. \nIt follows that \n$$\n\\norm{u_n-u_x}_{L^2({\\mathbb R})}^2\n\\leq\n4\\int_{-\\infty}^\\infty \\abs{\\varphi(t)}\\min\\{\\tfrac{t^2}{n^2}, 1\\}dt\n=\n4\\int_{-\\infty}^\\infty \\abs{\\varphi(t)}(\\log(2+\\abs{t}))^\\delta F_n(t)dt,\n$$\nwhere \n$$\nF_n(t)=(\\log(2+\\abs{t}))^{-\\delta}\\min\\{\\tfrac{t^2}{n^2}, 1\\}. \n$$\nElementary considerations show that \n$$\nF_n(t)\\leq C(\\log n)^{-\\delta},\n$$\nand therefore \n$$\n\\norm{u_n-u_x}_{L^2({\\mathbb R})}^2\n\\leq\nC(\\log n)^{-\\delta}\\int_{-\\infty}^\\infty \\abs{\\varphi(t)}(\\log(2+\\abs{t}))^\\delta dt.\n$$\nOf course, exactly the same estimate holds for $v_n-v_x$. \nPutting this together, we find\n$$\n\\Norm{\\frac1n\\jap{\\cdot,u_n}v_n-\\int_{n-1}^n \\frac1x\\jap{\\cdot,u_x}v_x\\ dx}_{\\mathbf{S}_1}\n\\leq C(\\varphi)(n^{-2}+n^{-1}(\\log n)^{-\\delta\/2}). \n$$\nSince by assumption $\\delta>2$, it follows that the operator $D_N(\\varphi)$ is represented as a partial sum of a series that converges absolutely in trace norm. This yields the required statement. \n\\end{proof}\n\n\\begin{lemma}\\label{lma.2}\nLet $\\psi\\in L^\\infty({\\mathbb R})$; then the operator $B_N(\\psi)$ satisfies the operator norm estimate\n$$\n\\norm{B_N(\\psi)}\\leq \\norm{\\psi}_{L^\\infty}. \n$$\n\\end{lemma}\n\\begin{proof}\nWe have \n$$\n(B_N(\\psi)f)(x)=\\sqrt{2\\pi}\\frac1{\\sqrt{x}}\\widehat{\\psi f}(-\\log x), \n$$\nand therefore\n\\begin{align*}\n\\int_1^N\\abs{(B_N(\\psi)f)(x)}^2dx\n&=\n2\\pi\\int_1^N\\abs{\\widehat{\\psi f}(-\\log x)}^2\\frac{dx}{x}\n=\n2\\pi\\int_0^{\\log N}\\abs{\\widehat{\\psi f}(-t)}^2dt\n\\\\\n&\\leq\n2\\pi \\int_{-\\infty}^\\infty \\abs{\\widehat{\\psi f}(t)}^2dt\n=\n\\norm{\\psi f}_{L^2({\\mathbb R})}^2\n\\leq \n\\norm{\\psi}_{L^\\infty({\\mathbb R})}^2\\norm{f}_{L^2({\\mathbb R})}^2,\n\\end{align*}\nas required.\n\\end{proof}\n\n\\begin{lemma}\\label{lma.3}\nLet $\\varphi\\in L^\\infty({\\mathbb R})$ satisfy condition \\eqref{eq:4} for some $\\delta>2$. Then \nthe operator $A_N(\\abs{\\varphi}^{1\/2})$ satisfies the operator norm estimate \n$\\norm{A_N(\\abs{\\varphi}^{1\/2})}=O(1)$ as $N\\to\\infty$. \n\\end{lemma}\n\\begin{proof}\nOf course, it suffices to prove the statement for $\\varphi\\geq0$. In this case, \ncombining the results of two previous Lemmas, we find\n\\begin{align*}\n\\norm{A_N(\\abs{\\varphi}^{1\/2})}^2\n&=\n\\norm{A_N(\\abs{\\varphi}^{1\/2})^*A_N(\\abs{\\varphi}^{1\/2})}\n\\\\\n&\\leq\n\\norm{D_N(\\varphi)}+\\norm{B_N(\\abs{\\varphi}^{1\/2})^*B_N(\\abs{\\varphi}^{1\/2})}\n=O(1)\n\\end{align*}\nas $N\\to\\infty$. \n\\end{proof}\n\n\n\\begin{lemma}\\label{lma.4}\nLet $\\varphi\\in L^\\infty({\\mathbb R})$ satisfy \\eqref{eq:4} for some $\\delta>2$.\nThen for any $m\\in{\\mathbb N}$ we have \n$$\n\\lim_{N\\to\\infty}(\\log N)^{-1}\\Tr\\bigl((K_N(\\varphi))^m\\bigr)\n=\n\\frac1{2\\pi}\\int_{-\\infty}^\\infty \\varphi(t)^mdt \\ .\n$$\n\\end{lemma}\n\\begin{proof}\nThe case $m=1$ is a direct calculation (see Section~\\ref{sec.b}); below we assume $m\\geq2$. \nUsing the cyclicity of trace, we find \n\\begin{align*}\n\\Tr \\bigl((K_N(\\varphi))^m\\bigr)&-\\Tr \\bigl((T_N(\\varphi))^m\\bigr)\n\\\\\n&=\n\\Tr\\bigl((A_N(\\varphi^{1\/2})A_N(\\abs{\\varphi}^{1\/2})^*)^m\\bigr)\n-\n\\Tr\\bigl((B_N(\\varphi^{1\/2})B_N(\\abs{\\varphi}^{1\/2})^*)^m\\bigr)\n\\\\\n&=\n\\Tr\\bigl((A_N(\\varphi^{1\/2})^*A_N(\\abs{\\varphi}^{1\/2}))^m\\bigr)\n-\n\\Tr\\bigl((B_N(\\varphi^{1\/2})^*B_N(\\abs{\\varphi}^{1\/2}))^m\\bigr)\\ .\n\\end{align*}\nDenoting \n$$\nX=A_N(\\varphi^{1\/2})^*A_N(\\abs{\\varphi}^{1\/2}), \n\\quad\nY=B_N(\\varphi^{1\/2})^*B_N(\\abs{\\varphi}^{1\/2}), \n$$\nwe have \n$$\nX^m-Y^m=(X-Y)X^{m-1}+\\cdots+Y^{m-1}(X-Y),\n$$\nand therefore\n$$\n\\abs{\\Tr(X^m-Y^m)}\\leq \\norm{X-Y}_{\\mathbf{S}_1}(\\norm{X}^{m-1}+\\cdots+\\norm{Y}^{m-1}). \n$$\nBy Lemma~\\ref{lma.1}, we have $\\norm{X-Y}_{\\mathbf{S}_1}=O(1)$ and by Lemmas~\\ref{lma.2} and \\ref{lma.3} the sum in the brackets above is also $O(1)$ as $N\\to\\infty$. We conclude that \n$$\n\\Tr (K_N(\\varphi)^m)-\\Tr (T_N(\\varphi)^m)=O(1), \\quad N\\to\\infty. \n$$\nUsing \\eqref{eq:5}, we conclude the proof. \n\\end{proof}\n\nThe rest of the proof of Theorem~\\ref{thm2} is a standard approximation argument. \n\n\\begin{lemma}\\label{lma.5}\nLet $g$ be a Lipschitz continuous function with $g(0)=0$, and let $\\varphi\\in L^\\infty({\\mathbb R})$ be a real-valued symbol with compact support. Then the asymptotic formula \\eqref{eq:1} holds true. \n\\end{lemma}\n\\begin{proof}\nFirst let us choose $\\Lambda>0$ such that $\\norm{\\varphi}_{L^\\infty({\\mathbb R})}\\leq \\Lambda$ and $\\norm{K_N(\\varphi)}\\leq \\Lambda$ for all $N$ (this can be done by Lemma~\\ref{lma.3}). Next, let $R>0$ be such that $\\supp\\varphi\\subset[-R,R]$. \n\nIt suffices to consider the case of real-valued $g$. \nLet $\\varepsilon>0$ be given. Using the Weierstrass approximation theorem, it is not difficult to construct two polynomials $p_+$ and $p_-$ with real coefficients such that $p_+(0)=p_-(0)=0$, \n$$\np_-(\\lambda)\\leq g(\\lambda)\\leq p_+(\\lambda), \\quad \\abs{\\lambda}\\leq \\Lambda\n$$\nand\n\\[\n0\\leq p_+(\\lambda)-p_-(\\lambda)\\leq \\varepsilon, \\quad \\abs{\\lambda}\\leq \\Lambda.\n\\label{eq:11}\n\\]\nThen \n$$\n\\Tr p_-(K_N(\\varphi))\\leq \\Tr g(K_N(\\varphi))\\leq \\Tr p_+(K_N(\\varphi)). \n$$\nBy taking linear combinations of monomials in Lemma~\\ref{lma.4}, we find that \\eqref{eq:1} holds for all polynomials vanishing at zero, and therefore\n\\begin{align*}\n\\limsup_{N\\to\\infty}(\\log N)^{-1}\\Tr g(K_N(\\varphi))\n\\leq \n\\limsup_{N\\to\\infty}(\\log N)^{-1}\\Tr p_+(K_N(\\varphi))\n\\\\\n=\n\\frac1{2\\pi}\\int_{-\\infty}^\\infty p_+(\\varphi(t))dt\n=\n\\frac1{2\\pi}\\int_{-R}^R p_+(\\varphi(t))dt\n\\end{align*}\nand similarly \n$$\n\\liminf_{N\\to\\infty}(\\log N)^{-1}\\Tr g(K_N(\\varphi))\n\\geq \n\\frac1{2\\pi}\\int_{-R}^R p_-(\\varphi(t))dt.\n$$\nOn the other hand, by \\eqref{eq:11} we have\n$$\n\\int_{-R}^R (p_+(\\varphi(t))-p_-(\\varphi(t)))dt\\leq 2R\\varepsilon. \n$$\nSince $\\varepsilon>0$ is arbitrary, we find that \n$$\n\\limsup_{N\\to\\infty}(\\log N)^{-1}\\Tr g(K_N(\\varphi))\n=\n\\liminf_{N\\to\\infty}(\\log N)^{-1}\\Tr g(K_N(\\varphi))\n=\n\\frac1{2\\pi}\\int_{-R}^R g(\\varphi(t))dt,\n$$\nas required. \n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm2}]\nThroughout the proof, we fix a Lipschitz function $g$ with $g(0)=0$ and we denote by $\\norm{g}_{\\text{Lip}}$ the norm of $g$ in the Lipschitz class. \nOur task is to extend \\eqref{eq:1} from compactly supported bounded symbols to all real-valued symbols in $L^1({\\mathbb R})$. For $\\varphi=\\overline{\\varphi}\\in L^1({\\mathbb R})$\n we denote \n$$\nM_N(\\varphi)=(\\log N)^{-1}\\Tr g(K_N(\\varphi)), \n\\quad\nM_\\infty(\\varphi)=\\frac1{2\\pi}\\int_{-\\infty}^\\infty g(\\varphi(t))dt.\n$$\nFurther, we set \n$$\n\\overline{M}(\\varphi)=\\limsup_{N\\to\\infty}M_N(\\varphi), \\quad\n\\underline{M}(\\varphi)=\\liminf_{N\\to\\infty}M_N(\\varphi).\n$$\nNow \\eqref{eq:1} is equivalent to \n$$\n\\overline{M}(\\varphi)=\\underline{M}(\\varphi)=M_\\infty(\\varphi). \n$$\nLet us discuss the continuity of the (nonlinear) functionals $\\overline{M}(\\varphi)$, $\\underline{M}(\\varphi)$, $M_\\infty(\\varphi)$ with respect to $\\varphi\\in L^1({\\mathbb R})$. \nBelow $\\varphi_1$, $\\varphi_2$ are two real-valued symbols. By the Lipschitz continuity of $g$, we have\n$$\n\\abs{M_\\infty(\\varphi_2)-M_\\infty(\\varphi_1)}\\leq \\frac1{2\\pi}\\norm{g}_{\\text{Lip}}\\norm{\\varphi_2-\\varphi_1}_{L^1({\\mathbb R})}, \n$$\nand so $M_\\infty(\\varphi)$ is Lipschitz continuous in $\\varphi\\in L^1({\\mathbb R})$. \n\nNext, suppose $\\varphi_1\\leq\\varphi_2$ for a.e. $t\\in{\\mathbb R}$; then by \\eqref{eq:6} we have\n$$\nK_N(\\varphi_1)\\leq K_N(\\varphi_2)\n$$\nin the quadratic form sense. By min-max, it follows that for all eigenvalues of $K_N(\\varphi_1)$ and $K_N(\\varphi_2)$ (labelled in non-decreasing order and counted with multiplicities) we have\n$$\n\\lambda_j(K_N(\\varphi_1))\\leq \\lambda_j(K_N(\\varphi_2)), \\quad j=1,\\dots,N\n$$\nand therefore for all $j$\n\\begin{multline*}\n\\abs{g(\\lambda_j(K_N(\\varphi_2)))-g(\\lambda_j(K_N(\\varphi_1)))}\n\\\\\n\\leq\n\\norm{g}_{\\text{Lip}}\\abs{\\lambda_j(K_N(\\varphi_2))-\\lambda_j(K_N(\\varphi_1))}\n=\n\\norm{g}_{\\text{Lip}}(\\lambda_j(K_N(\\varphi_2))-\\lambda_j(K_N(\\varphi_1))).\n\\end{multline*}\nSumming over $j$, we obtain\n\\begin{multline*}\n\\abs{\\Tr g(K_N(\\varphi_2))-\\Tr g(K_N(\\varphi_1))}\n\\leq \n\\norm{g}_{\\text{Lip}}(\\Tr K_N(\\varphi_2)-\\Tr K_N(\\varphi_1))\n\\\\\n=\n(\\log N)\\norm{g}_{\\text{Lip}}\\frac1{2\\pi}\\int_{-\\infty}^\\infty (\\varphi_2(t)-\\varphi_1(t))dt\n=\n(\\log N)\\norm{g}_{\\text{Lip}}\\frac1{2\\pi} \\norm{\\varphi_2-\\varphi_1}_{L^1({\\mathbb R})}\\ .\n\\end{multline*}\nIt follows that for $\\varphi_1\\leq\\varphi_2$ we have\n$$\n\\abs{M_N(\\varphi_2)-M_N(\\varphi_1)}\\leq \\frac1{2\\pi}\\norm{g}_{\\text{Lip}}\\norm{\\varphi_2-\\varphi_1}_{L^1({\\mathbb R})}\\ .\n$$\nTaking upper and lower limits, we finally conclude that \n\\begin{align*}\n\\abs{\\overline{M}(\\varphi_2)-\\overline{M}(\\varphi_1)}\n&\\leq \\frac1{2\\pi}\\norm{g}_{\\text{Lip}}\\norm{\\varphi_2-\\varphi_1}_{L^1({\\mathbb R})}, \n\\\\\n\\abs{\\underline{M}(\\varphi_2)-\\underline{M}(\\varphi_1)}\n&\\leq \\frac1{2\\pi}\\norm{g}_{\\text{Lip}}\\norm{\\varphi_2-\\varphi_1}_{L^1({\\mathbb R})}, \n\\end{align*}\nfor $\\varphi_1\\leq\\varphi_2$. Of course, the same is true for $\\varphi_1\\geq\\varphi_2$. \nThus, the functionals $\\overline{M}(\\varphi)$ and $\\underline{M}(\\varphi)$ are Lipschitz continuous with respect to monotone convergence of $\\varphi$ in $L^1({\\mathbb R})$. \n\nNow it remains to approximate a given $\\varphi\\in L^1({\\mathbb R})$ by compactly supported bounded symbols while using monotone convergence. This is an easy exercise, we leave out the details. \n\\end{proof}\n\n\n\\section{Proof of Theorem~\\ref{thm1}}\\label{sec.\u0435}\nLet $\\varphi$ be as in the hypothesis of the theorem and let $\\psi=\\varphi^{1\/2}$. \nOur first task is to prove that the operators $A(\\psi)$ and $B(\\psi)$, formally defined by \\eqref{eq:7} and \\eqref{eq:8}, are well-defined and bounded.\nFor $B(\\psi)$ this is an easy task, given by the same calculation as in the proof of Lemma~\\ref{lma.2}:\n\\begin{align*}\n\\int_1^\\infty\\abs{(B(\\psi)f)(x)}^2dx\n&=\n2\\pi\\int_1^\\infty\\abs{\\widehat{\\psi f}(-\\log x)}^2\\frac{dx}{x}\n=\n2\\pi\\int_0^{\\infty}\\abs{\\widehat{\\psi f}(-t)}^2dt\n\\\\\n&\\leq\n\\norm{\\psi f}_{L^2({\\mathbb R})}^2\n\\leq \n\\norm{\\psi}_{L^\\infty({\\mathbb R})}^2\\norm{f}_{L^2({\\mathbb R})}^2,\n\\end{align*}\nand so $B(\\psi)$ is bounded if $\\psi$ is bounded. \n\n\\begin{comment}\nBefore stating the next lemma, we need a short discussion on the convergence of Dirichlet series. \nLet $a=\\{a_n\\}_{n=1}^\\infty\\in\\ell^2({\\mathbb N})$; consider the corresponding Dirichlet series\n\\[\n\\sum_{n=1}^\\infty a_n n^{-\\frac12-it}, \\quad t\\in{\\mathbb R}.\n\\label{eq:10}\n\\]\nThe question of convergence of this series is not trivial. It has been proven in \\cite{HS} (see also an alternative proof in \\cite{KQ}) that for any $a\\in\\ell^2({\\mathbb N})$, the series \\eqref{eq:10} converges for a.e. $t\\in{\\mathbb R}$; this fact is in many ways parallel to the classical Carleson's theorem on a.e. convergence of Fourier series of a square integrable function. \n\\end{comment}\n\n\\begin{lemma}\\label{lma.6}\nThe operator $A(\\psi):L^2({\\mathbb R})\\to\\ell^2({\\mathbb N})$ is well defined by \\eqref{eq:7} and bounded. On the set of finitely supported elements $a\\in\\ell^2({\\mathbb N})$, \nthe adjoint is given by the formula\n$$\n(A(\\psi)^*a)(t)=\\frac1{\\sqrt{2\\pi}}\\psi(t)\\sum_{n=1}^\\infty a_n n^{-\\frac12-it}\n$$\nfor a.e. $t\\in{\\mathbb R}$. \nThe factorisation\n$$\nK(\\varphi)=A(\\psi)A(\\psi)^*\n$$\nholds true in the sense that on any finitely supported elements $a,b\\in\\ell^2({\\mathbb N})$, we have\n$$\n\\jap{K(\\varphi)a,b}_{\\ell^2({\\mathbb N})}=\\jap{A(\\psi)^*a,A(\\psi)^*b}_{L^2({\\mathbb R})}. \n$$\nIn particular, the operator $K(\\varphi)$, defined initially on finitely supported elements,\n extends to a bounded positive semi-definite operator on $\\ell^2({\\mathbb N})$. \n\\end{lemma}\n\\begin{proof}\nThe sequence $(A(\\psi)f)_n$, $n\\in{\\mathbb N}$, is clearly well defined. By Lemma~\\ref{lma.3}, there exists a constant $C>0$ independent of $N$, such that for all $f\\in L^2({\\mathbb R})$, \n$$\n\\sum_{n=1}^N \\abs{(A(\\psi)f)_n}^2\\leq C\\norm{f}^2.\n$$\nIt follows that $A(\\psi)f\\in\\ell^2({\\mathbb N})$ and the operator $A(\\psi)$ is bounded. \nComputing the adjoint and checking the factorisation of $K(\\varphi)$ is a direct calculation. \n\\end{proof}\n\nThe following lemma is the only point in the paper where we use some results from the theory of Hardy spaces of Dirichlet series. \n\\begin{lemma}\\label{lma.8}\nThe kernel of $K(\\varphi)$ is trivial. \n\\end{lemma}\n\\begin{proof}\nFirst let $a\\in\\ell^2({\\mathbb N})$ and let $f=f(s)$ be the corresponding Dirichlet series\n$$\nf(s)=\\sum_{n=1}^\\infty a_n n^{-s}, \\quad \\Re s>1\/2.\n$$\nThe space of all such functions $f$ is known as the Hardy space of Dirichlet series $\\mathscr{H}^2$. \nIt is known (see e.g. \\cite[Theorem 4.11]{HLS}) that $\\mathscr{H}^2$ is embedded into the locally uniform Hardy space in the half-plane $\\Re s>1\/2$, i.e. \n$$\n\\sup_{\\tau\\in{\\mathbb R}}\\sup_{\\sigma>1\/2}\\int_{\\tau}^{\\tau+1}\\abs{f(\\sigma+it)}^2dt\\leq C\\norm{a}_{\\ell^2({\\mathbb N})}^2.\n$$\nIn particular, the boundary values $f(\\frac12+it):=\\lim\\limits_{\\varepsilon\\to0_+}f(\\frac12+\\varepsilon+it)$ exist and are non-zero for a.e. $t\\in{\\mathbb R}$. Moreover, if \n$$\nf_N(s)=\\sum_{n=1}^N a_n n^{-s}, \n$$\nthen for every $\\tau\\in{\\mathbb R}$, \n\\[\n\\lim_{N\\to\\infty}\n\\int_{\\tau}^{\\tau+1}\\abs{f(\\tfrac12+it)-f_N(\\tfrac12+it)}^2dt=0.\n\\label{eq:12}\n\\]\nSuppose $A^*(\\psi)a=0$; let us prove that $a=0$. \nFor $a^{(N)}=(a_1,\\dots,a_N,0,\\dots)$ we have \n$\\norm{A^*(\\psi)a^{(N)}}_{L^2({\\mathbb R})}\\to\\norm{A^*(\\psi)a}_{L^2({\\mathbb R})}=0$ as $N\\to\\infty$. \nBy the formula for the adjoint of $A(\\psi)$ in the previous lemma, this means that \n$$\n\\lim_{N\\to\\infty}\\int_{-\\infty}^\\infty\\abs{\\psi(t)}^2\\Abs{\\sum_{n=1}^N a_nn^{-\\frac12-it}}^2dt\n=\\lim_{N\\to\\infty}\\int_{-\\infty}^\\infty \\abs{\\psi(t)}^2\\abs{f_N(\\tfrac12+it)}^2dt\n=0.\n$$\nCombining this with \\eqref{eq:12}, we find that $\\psi(t)f(\\frac12+it)=0$ for a.e. $t\\in{\\mathbb R}$. \nSince by our assumption $\\psi$ is not identically zero, we find that $f(\\frac12+it)=0$ on a set of a positive measure; hence $f$ must vanish identically and so $a=0$. \n\\end{proof}\n\n\n\\begin{lemma}\\label{lma.7}\nThe operator \n$$\nD(\\varphi)=A(\\psi)^*A(\\psi)-B(\\psi)^*B(\\psi)\n$$\nis trace class. \n\\end{lemma}\n\\begin{proof}\nLet us assume that ${\\mathbb C}^N$ is embedded in $\\ell^2({\\mathbb N})$ in a natural way; then the operator $A_N(\\psi)$ can be regarded as the operator from $L^2({\\mathbb R})$ to $\\ell^2({\\mathbb N})$. In the same way, $B_N(\\psi)$ can be regarded as an operator from $L^2({\\mathbb R})$ to $L^2(1,\\infty)$. \nFrom the boundedness of $A(\\psi)$ and $B(\\psi)$ it is clear that we have the convergence $A_N(\\psi)\\to A(\\psi)$ and $B_N(\\psi)\\to B(\\psi)$ in the strong operator topology as $N\\to\\infty$. \nIt follows that $D_N(\\varphi)\\to D(\\varphi)$ in the weak operator topology; here $D_N(\\varphi)$ is defined in \\eqref{eq:9}. From Lemma~\\ref{lma.1} we know that $D_N(\\varphi)$ converges to a trace class operator; therefore, by the uniqueness of the weak limit, $D(\\varphi)$ is trace class. \n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm1}]\nWe first note that by Weyl's theorem about the invariance of essential spectrum under compact perturbations, the essential spectra of $A(\\psi)^*A(\\psi)$ and $B(\\psi)^*B(\\psi)$ coincide. Similarly, by the Kato-Rosenblum theorem on trace class perturbations, the a.c. parts of $A(\\psi)^*A(\\psi)$ and $B(\\psi)^*B(\\psi)$ are unitarily equivalent. \nNext, we recall that we have already proved that \n$$\nK(\\varphi)=A(\\psi)A(\\psi)^*, \\quad\nT(\\varphi)=B(\\psi)B(\\psi)^*.\n$$\nIt is well known that for any bounded operator $X$ in a Hilbert space, the operators \n$$\nX^*X|_{(\\Ker X)^\\perp} \\text{ and } XX^*|_{(\\Ker X^*)^\\perp}\n$$\nare unitarily equivalent.\nWe conclude that the essential spectra of $K(\\varphi)|_{(\\Ker K(\\varphi))^\\perp}$ and $T(\\varphi)|_{(\\Ker T(\\varphi))^\\perp}$ coincide. Since the kernels of both $T(\\varphi)$ and $K(\\varphi)$ are trivial, we find that the essential spectra of $K(\\varphi)$ and $T(\\varphi)$ coincide. \nSimilarly, the a.c. parts of $K(\\varphi)$ and $T(\\varphi)$ are unitarily equivalent; since $T(\\varphi)$ is purely a.c., we find that the a.c. part of $K(\\varphi)$ is unitarily equivalent to $T(\\varphi)$. \n\\end{proof}\n\n\\begin{comment}\nBefore proving the part about the essential spectrum, we need a lemma. \nThe author is indebted to E.~Shargorodsky for the proof of this lemma. \n\n\\begin{lemma}\nLet $X$ and $Y$ be bounded operators in a Hilbert space. \nThen $I+XY$ is Fredholm if and only if $I+YX$ is Fredholm. \n\\end{lemma}\n\\begin{proof}\nWe recall that a bounded operator $V$ is Fredholm if and only if there exists an operator $W$ (called a \\emph{regulariser} for $V$) such that $I-VW$ and $I-WV$ are compact. \n\nSuppose $I+XY$ is Fredholm. Then there exists an operator $R$ \nsuch that $(I+XY)R=I+{\\mathcal K}_1$, $R(I+XY)=I+{\\mathcal K}_2$, where ${\\mathcal K}_1$ and ${\\mathcal K}_2$ are \ncompact operators. Then \n\\begin{multline*}\n(I+YX)(I-YRX)=(I+YX)-(I+YX)YRX\n\\\\\n=(I+YX)-Y(I+XY)RX=(I+YX)-YX-Y{\\mathcal K}_1 X=I-Y{\\mathcal K}_1 X,\n\\end{multline*}\nand \n\\begin{multline*}\n(I-YRX)(I+YX)=(I+YX)-YRX(I+YX)\n\\\\\n=(I+YX)-YR(I+XY)X=(I+YX)-YX-Y{\\mathcal K}_2 X=I-Y{\\mathcal K}_2 X.\n\\end{multline*}\nSince the operators $Y{\\mathcal K}_1 X$ and $Y{\\mathcal K}_2 X$ are compact, $I-YRX$ is a regulariser for $I+YX$. So the latter operator is Fredholm. \n\\end{proof}\n\\end{comment}\n\n\n\\section*{Acknowledgements}\nThe author is grateful to Uzy Smilansky for asking the question about the spectral density in Example~\\ref{exa.2} and to Ole Brevig, Nikolai Nikolski, Eugene Shargorodsky and Alexander Sobolev for useful discussions. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nLifetime distributions are often used in reliability theory and survival analysis for modelling real data. They play a fundamental role in reliability in diverse disciplines such as finance, manufacture, biological sciences, physics and engineering. The exponential distribution is a basic model in reliability theory and\nsurvival analysis. It is often used to model system reliability at a component level, assuming the failure rate is constant \\citep{Balakrishnan1995,Barlow1975,Sinha1980}. In recent years, a growing number of scholarly papers has been devoted to accommodate lifetime distributions with increasing or decreasing failure rate functions. The motivation is to give a parametric fit for real data sets where the underlying failure rates, arising on a latent competing risk problem base, present monotone shapes (nonconstant hazard rates).\nThe proposed distributions are introduced as extensions of the exponential distribution, following \\cite{Adamidis1998} and \\cite{Kus2007}, by compounding some useful lifetime and truncated discrete distributions (for review, see \\cite{Barreto-Souza2009,Chahkandi2009,Silva2010,Barreto-Souza2011,Cancho2011,Louzada-Neto2011,Morais2011,Hemmati2011,Nadarajah2013,Bakouch2014}, and others).\nThe genesis is stated on competing risk scenarios in presence of latent\nrisks, i.e. there is no information about the causes of the component's failure \\citep{Basu1982}. In fact, a system may experience multiple failure processes that compete against each other, and whichever occurs first can cause the system to fail \\citep{Rafiee2017,Kalbfleisch2002,Andersen2002,Tsiatis1998}. The term competing risks refers to duration data where two or more causes are competing to determine the observed time-to-failure. The potential multiple causes of failure are not mutually exclusive but the interest lies in the time to the first coming one \\citep{Putter2007,Bakoyannis2012}. For further details see \\citet{Basu1981}.\n\n\nIn the same way, the exponential-logarithmic (EL) distribution was proposed by \\cite{Tahmasbi2008} as a log-series mixture of exponential random variables. This two-parameter distribution with decreasing failure rate (DFR) is obtained by mixing the exponential and logarithmic distributions.\nIt is based on the idea of modelling the system's reliability where the time-to-failure occurs due to the presence of an unknown number of initial defects of some components is considered. Suppose the breakdown (failure) of a system of components occurs due to the presence of a non-observable number, $Z$, of initial defects of the same kind, that can be identifiable only after causing failure and are repaired perfectly \\citep{Adamidis1998,Kus2007}. Let $T_{i}$ be the failure time of the system due to the $i^{th}$ defect, for $i \\geq 1$. If we assume that $T=(T_{1}, T_{2}, ..., T_{Z})$ are iid exponential random variables independent of $Z$, that follows a truncated logarithmic distribution,\nthen the time to the first failure is adequately modelled by the\nEL distribution \\citep{Barreto-Souza2015,Bourguignon2014,Ross1976}.\nFor reliability studies, $X_{(1)}=min\\{T_i\\}_{i=1}^{Z}$ and $X_{(Z)}=max\\{T_i\\}_{i=1}^{Z}$ are used respectively in serial and parallel systems with identical components \\citep{Chahkandi2009,Ramos2015}.\nHowever, one may determine the distribution of the $k^{th}$ smallest value of the time-to-failure ($k^{th}$ order\nstatistic), instead of the minimum lifetime (first order statistic).\n\n\nThere is a huge literature on the order\nstatistics for reliability engineering (for review, see \\cite{Barlow1975,Sarhan1962,Barlow1965,pyke1965,Gnedenko1969,Pledger1971,Barlow1981,David1981,Bain1991}, and references contained therein).\nThe motivation arises in\nreliability theory, where the so-called $k$-out-of-$n$ systems\nare studied \\citep{Xie2008,Xie2005,Proschan1976,Kim1988}.\nAn engineering system consisting of $n$ components is working if at least $k$ out of the total $n$ components are operating and it breaks down if $(n-k+1)$ or more components fail. Hence, a $k$-out-of-$n$ system fails at the time of the $(n-k+1)^{th}$ component failure \\citep{Barlow1975,Kamps1995,Cramer2001}. This binary-state context is based on the assumption that a system or its components can be either fully working or completely failing.\nHowever, in reality, a system may provide its specified function at less than full capacity when\nsome of its components operate in a degraded state \\citep{Ramirez-Marquez2005}.\nThe binary $k$-out-of-$n$ system reliability models have been extended to multi-state $k$-out-of-$n$ system reliability models by allowing more than two performance levels for the system and its components \\citep{Eryilmaz2014}. Multi-state systems contain units presenting two or more failed states with multiple modes of failure and one working state \\citep{Anzanello2009}.\nReliability models provide, through multi-state, more realistic representations of engineering systems \\citep{Yingkui2012}.\nMany authors have made contributions about the reliability estimation approaches for multi-state systems \\citep{Jenney1986,Page1988,Rocco2005,Ramirez-Marquez2004,Ramirez-Marquez2008,Levitin2007,Levitin2008}.\n\nIn this paper, we generalize the EL distribution \\citep{Tahmasbi2008} modelling the time to the first failure, to a\ndistribution more appropriate for modelling any order statistic (second, third, or any $k^{th}$ lifetime). For instance, suppose a machine produces a random number, $Z$ units, of light bulbs or wire fuses which are put through a life test. Each item has a random lifetime $T_{i}$, $i=1, 2, ...,Z$.\nThe EL distribution \\citep{Tahmasbi2008} is focused only on the minimum time-to-failure of the first of the $Z$ functioning components.\nHowever, we may be interested in the $k^{th}$ duration and then determine the lifetime distribution for the order statistics, assuming the system will fail if $k$ of the units fail. We may let $X_{(1)} < X_{(2)} < ... 0$ and a pdf given by: $f(t)=\\theta e^{-\\theta t}$ , for $t>0$, where $Z$ is a discrete random variable following a logarithmic-series distribution with parameter $0 0\n\\end{equation}\n\nFrom equations (\\ref{eq:tr-logarithmic}) and (\\ref{eq:order}) the joint probability density is derived as\\footnote{The proofs of all steps and equations are presented in the appendix.}:\n\n\\begin{equation}\\label{eq:joint-dist}\n g_{k}(x,z\/p,\\theta)=\\frac{\\Gamma(z)}{\\Gamma(k)\\Gamma(z-k+1)}\\frac{\\theta p^{z} e^{-\\theta (z-k+1) x} (1-e^{-\\theta x})^{k-1}}{A(p,k)}\n\\end{equation}\n\nwhere, $x$ is the lifetime of a system and $z$ is the last order statistic. In equation (\\ref{eq:joint-dist}) we consider the ascending order $X_{(1)} < X_{(2)} < ... x) = 1 - G(x) = \\int_{x}^{\\infty}f(t)d t$ which is the probability that the event of interest has not occurred by duration $x$. So, the reliability $S(x)$ is the probability that a system will be successful in the interval from time $0$ to time $x$, where $X$ is a random variable denoting the time-to-failure or failure time. One may refer to the literature on reliability theory \\citep{Barlow1975,Barlow1981,Basu1988}.\nThe survival function, corresponding to the pdf in equation (\\ref{eq:marginal-x-dist}), is given by equation (\\ref{eq:survivior}). Table (\\ref{tab:survivior}) presents the reliability function for some special cases.\n\n\n\\begin{equation} \\label{eq:survivior}\n S_{k}(x\/p,\\theta, k) =\n 1 - \\frac{\\ln\\big(1-p\\frac{1-e^{-\\theta x}}{1-pe^{-\\theta x}}\\big)+\\psi(k)\\sum_{j=1}^{k-1}\\frac{1}{j}\n \\big(p\\frac{1-e^{-\\theta x}}{1-pe^{-\\theta x}}\\big)^{j}}{\\ln(1-p)+\\psi(k)\\sum_{j=1}^{k-1}\\frac{p^{j}}{j}}\n\\end{equation}\n\n\n\\begin{table}[htp]\n\\begin{center}\n\\small{\n\\caption{Reliability function for some special cases \\label{tab:survivior}}\n\\begin{tabular}{lcc}\n \\hline\n \\textbf{order statistic} & $\\textbf{k}$ & $\\textbf{S(x)}$\\\\\n \\hline\n &&\\\\\n first & $k=1$ & $\\frac{\\ln (1-pe^{-\\theta x})}{\\ln (1-p)}$ \\\\\n &&\\\\\n second & $k=2$ & $\\frac{\\ln (1-pe^{-\\theta x})-p\\big[\\frac{1-e^{-\\theta x}}{1-pe^{-\\theta x}}-1\\big]}{\\ln (1-p)+p}$ \\\\\n &&\\\\\n third & $k=3$ & $\\frac{\\ln (1-pe^{-\\theta x})+p+\\frac{p^{2}}{2}-p\\frac{1-e^{-\\theta x}}{1-pe^{-\\theta x}}-\n \\frac{p^{2}}{2}\\big(\\frac{1-e^{-\\theta x}}{1-pe^{-\\theta x}}\\big)^{2}}\n {\\ln (1-p)+p+\\frac{p^{2}}{2}}$ \\\\\n &&\\\\\n \\hline\n\\end{tabular}}\n\\end{center}\n\\end{table}\n\nThe failure rate, known as hazard rate function $h(x)$, is the instantaneous rate of occurrence of the event of interest at duration $x$ (i.e. the rate of event occurrence per unit of time). Mathematically, it is equal to the pdf of events at $x$, divided by the probability of surviving to that duration without experiencing the event. Thus, we define the failure rate function as in \\cite{Barlow1963} by $h(x)=g(x)\/S(x)$.\nThe hazard function for some special cases is given in table (\\ref{tab:hazard}).\n\n\n\\begin{table}[htp]\n\\begin{center}\n\\small{\n\\caption{Failure rate function for some special cases \\label{tab:hazard}}\n\\begin{tabular}{lcc}\n \\hline\n \\textbf{order statistic} & $\\textbf{k}$ & $\\textbf{h(x)}$\\\\\n \\hline\n &&\\\\\n first & $k=1$ & $\\frac{-p \\theta e^{-\\theta x}}{(1-pe^{-\\theta x}) \\ln(1-pe^{-\\theta x})}$ \\\\\n &&\\\\\n second & $k=2$ & $\\frac{-p^{2}\\theta e^{-\\theta x}(1-e^{-\\theta x})}{(1-pe^{-\\theta x})\\bigg[\\ln (1-pe^{-\\theta x})-p\\big[\\frac{1-e^{-\\theta x}}{1-pe^{-\\theta x}}-1\\big]\\bigg]}$ \\\\\n &&\\\\\n third & $k=3$ & $\\frac{-p^{3}\\theta e^{-\\theta x}(1-e^{-\\theta x})^{2}}{(1-pe^{-\\theta x})^{3}\\bigg[\\ln (1-pe^{-\\theta x})+p+\\frac{p^{2}}{2}-p\\frac{1-e^{-\\theta x}}{1-pe^{-\\theta x}}-\n \\frac{p^{2}}{2}\\big(\\frac{1-e^{-\\theta x}}{1-pe^{-\\theta x}}\\big)^{2}\\bigg]}$ \\\\\n &&\\\\\n \\hline\n\\end{tabular}}\n\\end{center}\n\\end{table}\n\nThe failure rate function is analytically related to the failure's probability distribution. It leads to the examination of the increasing (IFR) or decreasing failure rate (DFR) properties of life-length distributions. $G$ is an IFR distribution, if $h(x)$ increases for all $X$ such that $G(X)< 1$. The motivation of the EGTL lifetime distribution is the realistic features of the hazard rate in many real-life physical and non-physical systems, which is not monotonically increasing, decreasing or constant hazard rate.\nIf $k=1$, the hazard rate function is decreasing following \\cite{Tahmasbi2008}. In fact, if $x \\rightarrow 0$ then $h(x\/ p,\\theta,k)=\\frac{-p \\theta}{(1-p) \\ln (1-p)}$ and if $x \\rightarrow \\infty$ then $h(x\/p,\\theta ,k) \\rightarrow \\theta$.\nFor $k>1$, there is an increasing failure rate. Indeed, if $x \\rightarrow 0$ then $h(x\/p,\\theta ,k) \\rightarrow 0$. If $x \\rightarrow \\infty$ then $h(x\/p,\\theta ,k) > 0$ (see Figure \\ref{graph:hazard-rate}).\n\n\n\\begin{figure}[htp]\n\\begin{center}\n\\begin{tabular}[c]{cc}\n\\fbox{\\includegraphics[scale=0.23]{hazard-k=1.eps}} &\n\\fbox{\\includegraphics[scale=0.23]{hazard-k=2.eps}} \\\\\n\\fbox{\\includegraphics[scale=0.23]{hazard-k=3.eps}} &\n\\fbox{\\includegraphics[scale=0.23]{hazard-k=4.eps}} \\\\\n\\end{tabular}\n\\caption{Hazard functions of the EGTL distribution for $k=1,2,3,4$\n\\label{graph:hazard-rate}}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Random number generation}\\label{sec:random-generation}\n\nWe can generate a random variable from the cdf of $x$ in equation (\\ref{eq:cdf-reduced}) using the following steps:\n\n\\begin{itemize}\n \\item Generate a random variable $U$ from the standard uniform distribution.\n \\item Solve the non linear equation in $y$:\n\n \\begin{equation} \\label{eq:generate-y}\n U = \\frac{\\ln(1-py)+\\psi(k)\\sum_{j=1}^{k-1}\\frac{(py)^{j}}{j}}{\\ln(1-p)+\\psi(k)\\sum_{j=1}^{k-1}\\frac{p^{j}}{j}}\n \\end{equation}\n\n where,\n\n \\begin{equation*}\\label{eq:psi-k}\n \\psi(k) =\n \\begin{cases}\n 0 & \\mbox{if } k=1 \\\\\n 1 & \\mbox{if } k=2, 3, ...,z\n \\end{cases}\n \\end{equation*}\n\n \\item Calculate the values of $X$ such as:\n \\begin{equation} \\label{eq:generate-x}\n X=-\\frac{1}{\\theta}\\ln\\bigg(\\frac{1-y}{1-py}\\bigg)\n \\end{equation}\n\\end{itemize}\n\nwhere $X$ is EGTL random variable with parameters $\\theta$ and $p$. Note that for the special case $k=1$, we generate $X$ directly from the following equation:\n\n\\begin{equation}\\label{generate-x2}\n X=-\\frac{1}{\\theta}\\ln\\bigg(\\frac{1-(1-p)^{1-u}}{p}\\bigg)\n\\end{equation}\n\n\n\\section{Estimation of the parameters}\n\nIn this section, we will determine the estimates of the parameters $p$ and $\\theta$ for the EGTL new family of distributions. There are many methods available for estimating the parameters of interest. We present here the three most popular methods: Maximum likelihood, method of moments and Bayesian estimations.\n\n\\subsection{Maximum Likelihood estimation}\n\nLet ($X_{1},X_{2}, \\dots,X_{n}$) be a random sample from the EGTL distribution. The log-likelihood function given the observed values, $x_{obs}=(x_{1},x_{2}, \\dots,x_{n})$, is:\n\n\\begin{multline}\n \\ln L= n\\ln(\\theta)+nk\\ln(p)-\\theta \\sum_{i=1}^{n}x_{i}+(k-1)\\sum_{i=1}^{n}\\ln(1-e^{-\\theta x_{i}})\n \\\\ -n\\ln A(p,k)-k\\sum_{i=1}^{n}\\ln(1-pe^{-\\theta x_{i}})\n\\end{multline}\n\nWe subsequently derive the associated gradients:\n\n\\[\n\\frac{\\partial \\ln L}{\\partial p}\n = \\frac{np^{k-1}}{(1-p)\\big[\\ln(1-p)+\\psi(k)\\sum_{j=1}^{k-1}\\frac{p^{j}}{j}\\big]}+\\frac{nk}{p}+\\sum_{i=1}^{n}\\frac{1}{e^{\\theta x_{i}}-p}\n\\]\n\n\n\\[\n\\frac{\\partial \\ln L}{\\partial \\theta}\n=\n\\frac{n}{\\theta}-\\sum_{i=1}^{n}x_{i}+(k-1)\\sum_{i=1}^{n}\\frac{x_{i}}{e^{\\theta x_{i}}-1}-kp\\sum_{i=1}^{n}\\frac{x_{i}}{e^{\\theta x_{i}}-p}\n\\]\n\nWe need the Fisher information matrix for interval estimation and tests of hypotheses on the parameters.\nIt can be expressed in terms of the second derivatives of the log-likelihood function:\n\n\\[\n\\mathcal{I}(\\widehat{p},\\widehat{\\theta})\n = -\n\\left.\n\\left( \\begin{array}{cc}\n E\\Big(\\tfrac{\\partial^2 \\ln L}{\\partial p^2}\\Big)\n & E\\Big(\\tfrac{\\partial^2 \\ln L}{\\partial p \\partial \\theta}\\Big) \\\\\n E\\Big(\\tfrac{\\partial^2 \\ln L}{\\partial \\theta \\partial p}\\Big)\n & E\\Big(\\tfrac{\\partial^2 \\ln L}{\\partial \\theta^2}\\Big) \\\\\n\\end{array} \\right)\n\\right|_{\\theta = (\\widehat{p},\\widehat{\\theta})}\n\\]\n\n\nThe maximum likelihood estimates (MLEs) $\\widehat{p}$ and $\\widehat{\\theta}$ of the EGTL parameters $p$ and $\\theta$, respectively, can be found analytically using the iterative EM algorithm to handle the incomplete data problems \\citep{Dempster1977,Krishnan1997}. The iterative method consists in repeatedly updating the parameter estimates by replacing the \"missing data\" with the new estimated values. The standard method used to determine the MLEs is the Newton-Raphson algorithm that requires second derivatives of the log-likelihood function for all iterations. The main drawback of the EM algorithm is its rather slow convergence, compared to the Newton-Raphson method, when the \"missing data\" contain a relatively large amount of information \\citep{Little1983}. Recently, several researchers have used the EM method such as \\cite{Adamidis1998}, \\cite{Adamidis2005}, \\cite{Karlis2003}, \\cite{Ng2002} and others. Newton-Raphson is required for the M-step of the EM algorithm.\nTo start the algorithm, a hypothetical distribution of complete-data is defined with the pdf in equation (\\ref{eq:joint-dist}) and then, we drive the conditional mass function as:\n\n\\begin{equation}\\label{eq:prob-z}\n p(z\/x,p,\\theta)=\\frac{\\Gamma(z)}{\\Gamma(k)\\Gamma(z-k+1)}p^{z-k} e^{-\\theta (z-k)x}(1-pe^{-\\theta x})^{k}\n\\end{equation}\n\n\\textbf{E-step:}\n\n\\begin{equation}\\label{eq:E-step}\n E(z\/x,p,\\theta)=\\frac{k}{1-pe^{-\\theta x}}\n\\end{equation}\n\n\\textbf{M-step:}\n\n\\begin{equation}\\label{eq:M-step1}\n\\widehat{p}^{(r+1)}\n =\n \\frac{-\\big(1-p^{(r+1)}\\big)\\Big[\\ln\\big(1-p^{(r+1)}\\big)+\\psi(k)\\sum_{j=1}^{k-1}\\frac{\\big(p^{(r+1)}\\big)^{j}}{j}\\Big]}{n\\big(p^{(r+1)}\\big)^{k-1}}\\sum_{i=1}^{n}\\frac{k}{1-p^{(r)}e^{-\\theta^{(r)} x_{i}}}\n\\end{equation}\n\n\\begin{equation}\\label{eq:M-step2}\n\\widehat{\\theta}^{(r+1)} =\nn\\bigg[\\sum_{i=1}^{n}\\frac{kx_{i}}{1-p^{(r)}e^{-\\theta^{(r)} x_{i}}}-(k-1)\\sum_{i=1}^{n}\\frac{x_{i}}{1-e^{-\\theta^{(r+1)} x_{i}}}\\bigg]^{-1}\n\\end{equation}\n\n\\subsection{Method of moments estimation}\n\nThe method of moments involves equating theoretical with sample moments. The estimate of $r^{th}$ moment is\n$\\widehat{\\mu}_{r}=\\frac{1}{n}\\sum_{i=1}^{n}x_{i}^{r}$. For the EGTL distribution, the $r^{th}$ moment is given by equation (\\ref{eq:rth-moment}). The corresponding first and second moments are given by:\n\n\\begin{equation}\\label{eq:first-moment}\n \\frac{1}{n}\\sum_{i=1}^{n}x_{i}=E(x)= \\frac{1}{\\theta}\\frac{p^{k}}{A(p,k)}\\sum_{i=0}^{\\infty} \\sum_{j=0}^{k-1}{k-1+i\\choose i} {k-1\\choose j} p^{i} (-1)^{j} \\frac{1}{(i+j+1)^{2}}\n\\end{equation}\n\n\\begin{equation}\\label{eq:second-moment}\n \\frac{1}{n}\\sum_{i=1}^{n}x_{i}^{2}= E(x^{2})= \\frac{2}{\\theta^{2}}\\frac{p^{k}}{A(p,k)}\\sum_{i=0}^{\\infty} \\sum_{j=0}^{k-1}{k-1+i\\choose i} {k-1 \\choose j}p^{i} (-1)^{j} \\frac{1}{(i+j+1)^{3}}\n\\end{equation}\n\nFrom equation (\\ref{eq:first-moment}) we obtain:\n\n\\begin{equation}\\label{eq:p-moment}\n \\theta = \\frac{p^{k}}{\\overline{x}A(p,k)}\\sum_{i=0}^{\\infty} \\sum_{j=0}^{k-1}{k-1+i\\choose i} {k-1\\choose j}p^{i} (-1)^{j} \\frac{1}{(i+j+1)^{2}}\n\\end{equation}\n\nand then, we should solve the following equation in $p$:\n\n\\begin{equation}\\label{eq:theta-moment}\n \\frac{1}{n}\\sum_{i=1}^{n}x_{i}^{2}-\\frac{2\\overline{x}^{2}A(p,k)\\sum_{i=0}^{\\infty} \\sum_{j=0}^{k-1}{k-1+i\\choose i} {k-1 \\choose j}p^{i} (-1)^{j} \\frac{1}{(i+j+1)^{3}}} {p^{k}\\big[\\sum_{i=0}^{\\infty} \\sum_{j=0}^{k-1}{k-1+i\\choose i} {k-1 \\choose j}p^{i} (-1)^{j} \\frac{1}{(i+j+1)^{2}}\\big]^2}=0\n\\end{equation}\n\nThereafter, we determine $\\widehat{\\theta}$ by replacing $p$ with its estimated value, $\\widehat{p}$, in the equation (\\ref{eq:p-moment}).\n\n\\subsection{Bayesian estimation}\n\nIn the Bayesian approach inferences are expressed in a posterior distribution for the\nparameters which is, according to Bayes' theorem, given in terms of the likelihood and a prior density function by:\n\n\\begin{equation}\\label{eq:bayes-posterior}\n P_{k}(p,\\theta\/x_{1}, x_{2}, ..., x_{n})=\\frac{g_{k}(x\/p,\\theta).\\pi_{k}(p,\\theta)}{g_{k}(x)}\n\\end{equation}\n\nwhere, $\\pi_{k}(p,\\theta)$ is a prior probability distribution function and $g_{k}(x\/p,\\theta)$\nis the likelihood of observations $(x_{1}, x_{2}, ..., x_{n})$. Note that $g_{k}(x)$ is the normalizing constant for the function $\\pi_{k}(p,\\theta)g_{k}(x\/p,\\theta)$ given by:\n\n\\begin{equation}\\label{eq:bayes-normalizing}\n \\int\\int \\pi_{k}(p,\\theta)g_{k}(x\/p,\\theta) dp d\\theta\n\\end{equation}\n\nWe should first specify our initial beliefs or other sorts of knowledge on the prior distribution $\\pi_{k}(p,\\theta)$. Here, we suppose that the standard uniform distribution on the interval $[0, 1]$ is a prior distribution for the parameter $p$ and gamma, $G(a,b)$, is a prior distribution for the parameter $\\theta$, where $a$ is a shape parameter and $b$ is a scale parameter. The prior probability function is then equal to:\n\n\\begin{equation}\\label{eq:bayes-prior}\n \\pi_{k}(p,\\theta)=\\pi_{1,k}(p)\\pi_{2,k}(\\theta)\n\\end{equation}\n\nwhere, $\\pi_{1,k}(p)=1$ and $\\pi_{2,k}(\\theta)=\\frac{b^a}{\\Gamma(a)}\\theta^{a-1}e^{-b\\theta}$.\n\n\nUsing the mean square error as a risk function, we obtain the Bayes estimates as the means of the posterior distribution:\n\n\\begin{equation}\\label{eq:Bayes-p-estimate}\n \\widehat{\\theta} =E(\\theta)= \\int_{0}^{\\infty}\\int_{0}^{1}\\theta P_{k}(p,\\theta\/x)d\\theta dp\n\\end{equation}\n\n\\begin{equation}\\label{eq:Bayes-theta-estimate}\n \\widehat{p} =E(p)= \\int_{0}^{1}\\int_{0}^{\\infty}p P_{k}(p,\\theta\/x)dp d\\theta\n\\end{equation}\n\n\n\\section{Simulation study}\n\nAs an illustration of the three last methods of estimation, numerical computations have been performed using the steps presented in section \\ref{sec:random-generation} for the random number generation. The numerical study was\nbased on $1000$ random samples of the sizes $20$, $50$ and $100$ from the EGTL distribution for each of the $3$ values of $\\lambda=(p,\\theta)$ and the three cases $k=\\{1, 2, 3\\}$. We have considered the initial values $(0.5 ; 0.5)$, $(0.7 ; 1.5)$ and $(0.3 ; 2)$. For this purpose, we have used the program\nMathcad 14.0.\nAfter determining the parameter estimates $\\widehat{\\lambda}=(\\widehat{p},\\widehat{\\theta})$ we compute the biases, the variances and the mean square errors (MSEs), where\n$MSE(\\widehat{\\lambda})= E(\\widehat{\\lambda}-\\lambda)^{2}= Bias^{2}(\\widehat{\\lambda}) + var(\\widehat{\\lambda})$\nand $Bias(\\widehat{\\lambda}) = E(\\widehat{\\lambda})-\\lambda$.\nAn estimator $\\widehat{\\lambda}$ is said to be efficient if its mean square error (MSE) is minimum\namong all competitors. In fact, $\\widehat{\\lambda}_{1}$ is more efficient than $\\widehat{\\lambda}_{2}$ if $MSE(\\widehat{\\lambda}_{1}) < MSE(\\widehat{\\lambda}_{2})$.\n\nTable \\ref{tab:simulation} reports the results from the simulated data where the variances and the MSEs of the parameters are given. The results show that, for each case $k=\\{1, 2, 3\\}$, the variances and the MSEs decrease when the sample size increases.\nWe see that the values from the Bayesian method are generally lower than those obtained using the ML approach.\n\n\n\\begin{sidewaystable}[htp]\n \\centering\n \\caption{Results from the numerical computation}\n \\scriptsize{\n \\begin{tabular}{lllcccccccccccc}\n & & & \\multicolumn{4}{c}{\\textbf{Maximum likelihood }} & \\multicolumn{4}{c}{\\textbf{Method of Moments}} & \\multicolumn{4}{c}{\\textbf{Bayesian methods}}\\\\\n\\cline{4-15} \\textbf{n} & \\textbf{k} & \\textbf{$(p,\\theta)$} & \\textbf{$var(\\widehat{p})$} & \\textbf{$var(\\widehat{\\theta})$} & \\textbf{$MSE(\\widehat{p})$} & \\textbf{$MSE(\\widehat{\\theta})$} & \\textbf{$var(\\widehat{p})$} & \\textbf{$var(\\widehat{\\theta})$} & \\textbf{$MSE(\\widehat{p})$} & \\textbf{$MSE(\\widehat{\\theta})$} & \\textbf{$var(\\widehat{p})$} & \\textbf{$var(\\widehat{\\theta})$} & \\textbf{$MSE(\\widehat{p})$} & \\textbf{$MSE(\\widehat{\\theta})$} \\\\\n \\hline\n \\textbf{n=20} & & & & & & & & & & & & & & \\\\\n & \\textbf{1} & (0.5 ; 0.5) & 0.2091 & 0.2008 & 0.2136 & 0.2043 & 0.2775 & 0.2241 & 0.2849 & 0.2351 & 0.1737 & 0.1414 & 0.1835 & 0.1586 \\\\\n & & (0.7 ; 1.5) & 0.2779 & 0.6980 & 0.2861 & 0.7145 & 0.3066 & 1.0227 & 0.3078 & 1.0361 & 0.2208 & 0.5794 & 0.2287 & 0.6104 \\\\\n & & (0.3 ; 2) & 0.4250 & 1.0930 & 0.4680 & 1.1160 & 0.6890 & 1.7260 & 0.7270 & 1.8930 & 0.2110 & 0.5010 & 0.2420 & 0.5870 \\\\\n & & & & & & & & & & & & & & \\\\\n & \\textbf{2} & (0.5 ; 0.5) & 0.2725 & 0.2013 & 0.2773 & 0.2027 & 0.3087 & 0.2327 & 0.3347 & 0.2398 & 0.1934 & 0.1227 & 0.1957 & 0.1488 \\\\\n & & (0.7 ; 1.5) & 0.1105 & 1.1326 & 0.1114 & 1.1380 & 0.1318 & 1.2050 & 0.1409 & 1.2078 & 0.0742 & 0.9730 & 0.0796 & 1.0117 \\\\\n & & (0.3 ; 2) & 0.2700 & 0.6230 & 0.3660 & 0.6520 & 0.5750 & 1.0000 & 0.6810 & 1.1900 & 0.1000 & 0.3690 & 0.1230 & 0.4320 \\\\\n & & & & & & & & & & & & & & \\\\\n & \\textbf{3} & (0.5 ; 0.5) & 0.1407 & 0.1164 & 0.1193 & 0.1536 & 0.2186 & 0.1498 & 0.2245 & 0.1515 & 0.1054 & 0.0490 & 0.1065 & 0.0533 \\\\\n & & (0.7 ; 1.5) & 0.1782 & 0.6809 & 0.1938 & 0.7694 & 0.2247 & 0.9003 & 0.2269 & 0.9290 & 0.1653 & 0.5840 & 0.1682 & 0.5880 \\\\\n & & (0.3 ; 2) & 0.4090 & 0.8630 & 0.4410 & 0.9040 & 1.2390 & 0.6680 & 1.2930 & 0.7200 & 0.1590 & 0.3380 & 0.2090 & 0.3810 \\\\\n \\textbf{n=50} & & & & & & & & & & & & & & \\\\\n & \\textbf{1} & (0.5 ; 0.5) & 0.1099 & 0.1032 & 0.1119 & 0.1049 & 0.1534 & 0.1280 & 0.1848 & 0.1308 & 0.0681 & 0.0493 & 0.0723 & 0.0506 \\\\\n & & (0.7 ; 1.5) & 0.1221 & 0.5182 & 0.1257 & 0.5431 & 0.1426 & 0.6526 & 0.1532 & 0.6929 & 0.0567 & 0.3863 & 0.0625 & 0.3878 \\\\\n & & (0.3 ; 2) & 0.3630 & 0.8440 & 0.3820 & 0.9340 & 0.6330 & 1.6030 & 0.6580 & 1.6480 & 0.1380 & 0.3710 & 0.1730 & 0.4240 \\\\\n & & & & & & & & & & & & & & \\\\\n & \\textbf{2} & (0.5 ; 0.5) & 0.1687 & 0.1097 & 0.1915 & 0.1239 & 0.2252 & 0.1422 & 0.2257 & 0.1544 & 0.1014 & 0.0443 & 0.1119 & 0.0456 \\\\\n & & (0.7 ; 1.5) & 0.1051 & 1.0626 & 0.1072 & 1.0985 & 0.1287 & 1.1868 & 0.1381 & 1.1923 & 0.0691 & 0.9795 & 0.0735 & 0.9807 \\\\\n & & (0.3 ; 2 & 0.2350 & 0.5620 & 0.2510 & 0.5770 & 0.5330 & 0.8980 & 0.5650 & 0.9390 & 0.0340 & 0.2670 & 0.0740 & 0.3050 \\\\\n & & & & & & & & & & & & & & \\\\\n & \\textbf{3} & (0.5 ; 0.5) & 0.0941 & 0.0782 & 0.0947 & 0.0896 & 0.1782 & 0.1134 & 0.1810 & 0.1167 & 0.0624 & 0.0185 & 0.0686 & 0.0224 \\\\\n & & (0.7 ; 1.5) & 0.0606 & 0.5371 & 0.0608 & 0.5459 & 0.1005 & 0.6989 & 0.1067 & 0.7072 & 0.0433 & 0.4644 & 0.0438 & 0.4685 \\\\\n & & (0.3 ; 2 & 0.3360 & 0.7130 & 0.3790 & 0.8170 & 0.5770 & 1.1460 & 0.6340 & 1.1800 & 0.1310 & 0.2140 & 0.1430 & 0.2760 \\\\\n \\textbf{n=100} & & & & & & & & & & & & & & \\\\\n & \\textbf{1} & (0.5 ; 0.5) & 0.0652 & 0.0398 & 0.0663 & 0.0592 & 0.1049 & 0.0827 & 0.1098 & 0.0879 & 0.0185 & 0.0022 & 0.0327 & 0.0082 \\\\\n & & (0.7 ; 1.5) & 0.0531 & 0.4206 & 0.0605 & 0.4663 & 0.0870 & 0.5859 & 0.0901 & 0.5876 & 0.0013 & 0.1941 & 0.0052 & 0.2201 \\\\\n & & (0.3 ; 2) & 0.2588 & 0.6168 & 0.2828 & 0.7518 & 0.4918 & 1.2348 & 0.5318 & 1.3888 & 0.0118 & 0.0058 & 0.0618 & 0.1188 \\\\\n & & & & & & & & & & & & & & \\\\\n & \\textbf{2} & (0.5 ; 0.5) & 0.1059 & 0.0733 & 0.1140 & 0.0804 & 0.1897 & 0.1097 & 0.1971 & 0.1162 & 0.0479 & 0.0051 & 0.0574 & 0.0185 \\\\\n & & (0.7 ; 1.5) & 0.0522 & 0.9943 & 0.0614 & 1.0109 & 0.0849 & 1.1288 & 0.0889 & 1.1395 & 0.0002 & 0.8165 & 0.0039 & 0.8951 \\\\\n & & (0.3 ; 2) & 0.1739 & 0.4719 & 0.2009 & 0.5239 & 0.4229 & 0.7759 & 0.4979 & 0.8269 & 0.0049 & 0.0149 & 0.0219 & 0.0859 \\\\\n & & & & & & & & & & & & & & \\\\\n & \\textbf{3} & (0.5 ; 0.5) & 0.0749 & 0.0560 & 0.0814 & 0.0613 & 0.1255 & 0.0888 & 0.1471 & 0.0901 & 0.0247 & 0.0005 & 0.0505 & 0.0059 \\\\\n & & (0.7 ; 1.5) & 0.0174 & 0.4517 & 0.0199 & 0.4680 & 0.0543 & 0.6477 & 0.0586 & 0.6599 & 0.0006 & 0.3827 & 0.0011 & 0.4055 \\\\\n & & (0.3 ; 2) & 0.2183 & 0.4733 & 0.2873 & 0.5973 & 0.4423 & 1.0013 & 0.4923 & 1.0643 & 0.0043 & 0.0073 & 0.0663 & 0.0323 \\\\\n \\hline\n \\end{tabular}}\n \\label{tab:simulation}\n\\end{sidewaystable}\n\n\n\\section{Application examples}\n\nIn this section, we fit the EGTL distribution to two real data sets using the MLEs. The first set (table \\ref{tab:data-Barlow-Campo}) consists of \"$107$ failure times for right rear brakes on D9G-66A caterpillar tractors\", reproduced from \\cite{Barlow1975a} and used also by \\cite{Chang1993}. These data are used in many applications of reliability \\citep{Adamidis2005,Tsokos2012,Shahsanaei2012}.\nThe second set of data involves $100$ observations (table \\ref{tab:data-Quesenberry}) of the results from an experiment concerning \"the tensile fatigue characteristics of a polyester\/viscose yarn\". These data were presented by \\cite{Picciotto1970} to study the problem of warp breakage during weaving. The observations were obtained on the cycles to failure of a $100$ cm yarn sample put to test under $2.3\\%$ strain level. The sample is used in \\cite{Quesenberry1982} as an example to illustrate selection procedure among probability distributions used in reliability. The reliability function of these two data sets belongs to the increasing failure rate class \\citep{Doksum1984,Adamidis2005}. In addition to our class of distributions, the gamma and Weibull distributions were fitted these data sets. The respective densities of gamma and Weibull distributions are $f_{1}(x,\\lambda_{1},\\beta_{1})=\\lambda_{1}^{\\beta_{1}}x^{\\beta_{1}-1}exp(-\\lambda_{1}x)\\Gamma(\\beta_{1})^{-1}$ and $f_{2}(x,\\lambda_{2},\\beta_{2})=\\beta_{2}\\lambda_{2}^{\\beta_{2}}x^{\\beta_{2}-1}exp(-\\lambda_{2}x)^{\\beta_{2}}$.\n\n\n\n\\begin{table}[htp]\n\\begin{center}\n\\scriptsize{\n\\caption{\"Ordered Failure Times (in hours) of 107 Right Rear Brakes on D9G-66A Caterpillar Tractors\" \\cite{Barlow1975a,Chang1993} \\label{tab:data-Barlow-Campo}}\n\\begin{tabular}{lllllllllllllll}\n \\hline\n56&\t753&\t1153&\t1586&\t2150&\t2624&\t3826&\t83&\t763&\t1154&\t1599&\t2156&\t2675&\t3995&\t104\\\\\n806&\t1193&\t1608&\t2160&\t2701&\t4007&\t116&\t834&\t1201&\t1723&\t2190&\t2755&\t4159&\t244&\t838\\\\\n1253&\t1769&\t2210&\t2877&\t4300&\t305&\t862&\t1313&\t1795&\t2220&\t2879&\t4487&\t429&\t897&\t1329\\\\\n1927&\t2248&\t2922&\t5074&\t452\t&904&\t1347&\t1957&\t2285&\t2986&\t5579&\t453&\t981&\t1454&\t2005\\\\\n2325&\t3092&\t5623&\t503&\t1007&\t1464&\t2010&\t2337&\t3160&\t6869&\t552&\t1008&\t1490&\t2016&\t2351\\\\\n3185&\t7739&\t614&\t1049&\t1491&\t2022&\t2437&\t3191&\t661&\t1069&\t1532&\t2037&\t2454&\t3439&\t673\\\\\n1107&\t1549&\t2065&\t2546&\t3617&\t683&\t1125&\t1568&\t2096&\t2565&\t3685&\t685&\t1141&\t1574&\t2139\\\\\n2584&\t3756&&&&&&&&&&&&&\\\\\t\t\t\t\t\t\t\t\t\t\t\t\t\n \\hline\n\\end{tabular}}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}[htp]\n\\begin{center}\n\\scriptsize{\n\\caption{\"Results of Model Selection Program on Yarn Data\" \\citep{Quesenberry1982} \\label{tab:data-Quesenberry}}\n\\begin{tabular}{lllllllllllllll}\n \\hline\n86&\t146&\t251&\t653\t&98&\t249&\t400&\t292&\t131&\t169&175&\t176&\t76&\t264&\t15\\\\\t\n364&\t195&\t262&\t88&\t264& 157&\t220&\t42&\t321&\t180&\t198&\t38&\t20&\t61&\t121\\\\\n282&\t224&\t149\t&180&\t325&\t250&\t196&\t90&\t229&\t166&38&\t337&\t65&\t151&\t341\\\\\n40&\t40&\t135&\t597&\t246& 211&\t180&\t93&\t315&\t353&\t571&\t124&\t279&\t81&\t186\\\\\n497&\t182&\t423&\t185&\t229&\t400&\t338&\t290&\t398&\t71& 246&\t185&\t188&\t568& 55\\\\\t\n55&\t61&\t244&\t20&\t284& 393&\t396&\t203&\t829&\t239&\t286&\t194&\t277&\t143&\t198\\\\\n264&\t105\t&203&\t124&\t137&\t135&\t350&\t193\t&188&\t236&&&&&\\\\\n \\hline\n\\end{tabular}}\n\\end{center}\n\\end{table}\n\nTable \\ref{tab:fitted} shows the fitted parameters, the calculated values of Kolmogorov-Smirnov (K-S) and their respective \\emph{p-values} for the two sets of data. It should be noted that the K-S test compares\nan empirical distribution with a known (not estimated) one. It is used to decide if a sample comes from a population with a specific distribution ($H_{0}$: the data follow a specified distribution). We estimate some special cases ($k=1, 2, 3, 4$) of the EGTL family of distributions at $5\\%$ significant level. The \\emph{p-values} are only significant for the case $k=1$ for the \\cite{Barlow1975a} and \\cite{Quesenberry1982} data sets. In fact, the data exhibit increasing failure rates but, the EGTL distribution is a decreasing failure rate if $k=1$ (see figure \\ref{graph:hazard-rate}). The new lifetime distribution provides good fit to the data sets. The K-S test shows that the EGTL distribution is an attractive alternative to the popular gamma and Weibull distributions. It generalizes the reliability lifetime distributions to any $k^{th}$ order statistics.\nIndeed, as shown in section \\ref{sec:reliability}, If $k=1$, the hazard rate function is decreasing following \\cite{Tahmasbi2008} and there is an increasing hazard rate for $k>1$.\n\n\n\\begin{table}[htp]\n\\begin{center}\n\\small{\n\\caption{The Goodness of Fit for some Special Cases \\label{tab:fitted}}\n\\begin{tabular}{|l|c|c|c|c|}\n \\hline\n Distributions & $\\widehat{p}$ & $\\widehat{\\theta}$ & K-S value & p-value \\\\\n \\hline\n\\textbf{\\cite{Barlow1975a} data set ($n=107$):} &&&&\\\\\n\tFirst order (k=1) &$0.0500$ &$5.00$ $10^{-6}$ & $0.9611$ & $0.0000$ \\\\\n\tSecond order (k=2) &$0.0232$ &$7.32$ $10^{-4}$ &$0.0639$ &$0.7746$ \\\\\n\tThird order (k=3) &$0.8811$ &$4.38$ $10^{-4}$ &$0.1106$ &$0.1456$ \\\\\n\tFourth order (k=4) &$0.4209$ &$8.84$ $10^{-4}$ &$0.0723$ &$0.6305$ \\\\\n Gamma & \\multicolumn{2}{|c|}{$(0.943;1.908)$}& $0.0680$ & $0.7343$\\\\\n Weibull & \\multicolumn{2}{|c|}{$(0.447;1.486)$}& $0.0490$ & $0.9999$\\\\\n \\hline\n \\textbf{\\cite{Quesenberry1982} data set ($n=100$):} &&&&\\\\\n\tFirst order (k=1) &$0.1901$ &$4.22$ $10^{-3}$ & $0.1955$ & $0.0009$ \\\\\n\tSecond order (k=2) &$0.0248$ &$6.65$ $10^{-3}$ &$0.1078$ &$0.1952$ \\\\\n\tThird order (k=3) &$0.2127$ &$7.66$ $10^{-3}$ &$0.0879$ &$0.4218$ \\\\\n\tFourth order (k=4) &$0.1031$ &$9.10$ $10^{-3}$ &$0.0786$ &$0.5657$ \\\\\n Gamma & \\multicolumn{2}{|c|}{$(1.008;2.239)$}& $0.0950$ & $0.3118$\\\\\n Weibull & \\multicolumn{2}{|c|}{$(0.403;1.604)$}& $0.0760$ & $0.6080$\\\\\n \\hline\n\\end{tabular}}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\\section{Conclusion}\nWe define a new two-parameter lifetime distribution so-called EGTL distribution. Our procedure generalizes the EL distribution proposed by \\cite{Tahmasbi2008}. We derive some mathematical properties and we present the plots of the pdf and the failure rate functions for some special cases. The estimation of the parameters is attained by the maximum likelihood, EM algorithm, the method of moments and the Bayesian approach, with numerical computations performed as illustration of the different methods of estimation. The application study is illustrated based on two real data sets used in many applications of reliability.\nWe have shown that our proposed EGTL distribution is suitable for modelling the time to any failure and not only the time to the first or the last failure. It is very competitive compared with its standard counterpart's distributions.\n\nOrdered random variables are already known for their ascending order. The paper may be extended to the concept of dual generalized ordered statistics, introduced by \\cite{Burkschat2003}, that enables a common approach to the descending ordered spacings like the reverse ordered statistics and the lower record values.\n\n\n\n\n\n\\section*{Appendix}\n\\small\n\nLet $T=(T_{1}, T_{2}, ..., T_{Z})$ be iid exponential r.v. with pdf given by: $f(t)=\\theta e^{-\\theta t}$ , for $t>0$, where $Z$ is a log-series r.v. with pmf, $P(Z=z)$, given by:\n\n\\begin{equation}\\label{eq:logarithmic-appendix}\n P(Z=z) = \\frac{1}{-\\ln(1-p)} \\frac{p^z}{z} ; z \\in \\{1,2,3,\\dots\\}\n\\end{equation}\n\nFrom the Taylor series, for $|x|<1$ we have:\n\\begin{equation*}\n ln(1+x)=\\sum_{j=1}^{\\infty}\\frac{(-1)^{j+1}}{j}x^{j}\n\\end{equation*}\n\nthen,\n\\begin{equation*}\n \\ln(1-p)= - \\sum_{j=1}^{\\infty}\\frac{p^{j}}{j}\n\\end{equation*}\n\n\nThe truncated at $k-1$ logarithmic distribution with parameter $p$ is:\n\n\\begin{equation}\\label{eq:tr-logarithmic-appendix}\n P_{k-1}(Z=z) = \\frac{1}{A(p,k)} \\frac{p^{z}}{z} ; k = 1,2,3,\\dots\\ , z \\mbox{ and } z=k, k+1,\\dots\\\n\\end{equation}\n\nwhere,\n\\begin{equation}\\label{eq:tr-logarithmic-A-appendix}\n A(p,k) = \\sum_{j=k}^{\\infty}\\frac{p^{j}}{j}\n = - \\ln(1-p)-\\psi(k)\\sum_{j=1}^{k-1}\\frac{p^{j}}{j}\n\\end{equation}\n\nand,\n\\begin{equation}\\label{eq:psi-k-appendix}\n \\psi(k) =\n \\begin{cases}\n 0 & \\mbox{if } k=1 \\\\\n 1 & \\mbox{if } k=2, 3, ...,z\n \\end{cases}\n\\end{equation}\n\n\nThe pdf of the $k^{th}$ order statistic is:\n\n\n\\begin{equation}\\label{eq:order-appendix}\n f_{k}(x\/z,\\theta)=\\frac{\\theta \\Gamma(z+1)}{\\Gamma(k)\\Gamma(z-k+1)}e^{-\\theta(z-k+1)x}(1-e^{-\\theta x})^{k-1} \\mbox{ ; } \\theta, x > 0\n\\end{equation}\n\nThen, the joint distribution from eq. (\\ref{eq:tr-logarithmic-appendix}) and (\\ref{eq:order-appendix}) is:\n\n\\begin{equation}\\label{eq:joint-dist-appendix}\n\\begin{split}\ng_{k}(x,z\/p,\\theta) & =f_{k}(x\/z,\\theta)P_{k-1}(z\/p)\\\\\n & = \\frac{\\theta \\Gamma(z+1)}{\\Gamma(k)\\Gamma(z-k+1)}e^{-\\theta(z-k+1)x}(1-e^{-\\theta x})^{k-1}\\frac{1}{A(p,k)} \\frac{p^{z}}{z}\\\\\n & = \\frac{\\Gamma(z)}{\\Gamma(k)\\Gamma(z-k+1)}\\frac{\\theta p^{z} e^{-\\theta (z-k+1) x} (1-e^{-\\theta x})^{k-1}}{A(p,k)}\n\\end{split}\n\\end{equation}\n\nLet, $f=\\theta e^{-\\theta x}$, $F=1- e^{-\\theta x}$, and $a=pe^{-\\theta x}=p(1-F)$\n\n\\begin{equation}\n g_{k}(x,z\/p,\\theta) = \\frac{1}{A(p,k)}\\frac{(z-1)!p^k}{(z-k)!(k-1)!}fF^{k-1}a^{z-k}\n\\end{equation}\n\n\nthe marginal density of $x$ is:\n\\begin{equation}\ng_{k}(x\/p,\\theta) = \\frac{p^{k}fF^{k-1}}{A(p,k)}\\sum_{z=k}^{\\infty}\\frac{(z-1)!}{(z-k)!(k-1)!}a^{z-k}\n\\end{equation}\n\nLet, $z-k=s$ , $z=k+s$, $k-1=z-s-1$\n\n\\begin{equation}\n\\begin{split}\ng_{k}(x\/p,\\theta) &= \\frac{p^{k}fF^{k-1}}{A(p,k)}\\sum_{s=0}^{\\infty}\\frac{(s+k-1)!}{s!(k-1)!}a^{s}\\\\\n &= \\frac{p^{k}fF^{k-1}}{A(p,k)}\\sum_{s=0}^{\\infty}{s+k-1\\choose s}a^{s}\\\\\n &= \\frac{p^{k}fF^{k-1}}{A(p,k)}\\frac{1}{(1-a)^{k}}\\\\\n &= \\frac{\\theta p^{k} e^{-\\theta x} (1-e^{-\\theta x})^{k-1}}{A(p,k)(1-pe^{-\\theta x})^{k}}\n \\mbox{ ; } \\quad x \\in [0, \\infty)\n\\end{split}\n\\end{equation}\n\n\\begin{equation}\n\\begin{split}\nG_{k}(x\/p,\\theta) &= \\int_{0}^{U}\\frac{p^{k}}{A(p,k)}U^{k-1}\\frac{1}{1-p(1-U)^{k}}dU\\\\\n &= \\frac{p^{k}}{A(p,k)}\\int_{0}^{U}\\frac{U^{k-1}}{1-p(1-U)^{k}}dU\\\\\n\\end{split}\n\\end{equation}\n\nlet $y=1-p(1-U)$ ; $J=\\frac{1}{p}$ ; $q=1-p$\n\n\\begin{equation}\n\\begin{split}\nG_{k}(x\/p,\\theta) &= \\frac{p^{k}}{A(p,k)}\\int_{q}^{y}\\big(\\frac{y-q}{p}\\big)^{k-1}\\frac{1}{y^{k}}dy\\\\\n &= \\frac{1}{A(p,k)}\\int_{q}^{y}(y-q)^{k-1}y^{-k}dy\\\\\n &= \\frac{1}{A(p,k)}\\int_{q}^{y}\\sum_{r=0}^{k-1} {k-1\\choose r}(-q)^{r}y^{-(r+1)}dy\\\\\n &= \\frac{1}{A(p,k)}\\sum_{r=0}^{k-1} {k-1\\choose r}(-q)^{r}\\int_{q}^{y}y^{-(r+1)}dy\\\\\n &= \\frac{1}{A(p,k)}\\sum_{r=0}^{k-1} {k-1\\choose r}(-q)^{r}\\int_{q}^{y}I_{(x,r)}\\\\\n &= \\frac{1}{A(p,k)}\\sum_{r=0}^{k-1} {k-1\\choose r} (-1)^{r} (1-p)^{r}I_{(x,r)}\n\\end{split}\n\\end{equation}\n\nwhere,\n\\begin{equation*}\n I_{(x,r)}=\\int_{1-p}^{1-pe^{-\\theta x}}t^{-(r+1)}dt\n\\end{equation*}\n\nOtherwise,\n\nLet $y=\\frac{U}{1-p(1-U)}$ , then $U=\\frac{y(1-p)}{1-py}$ and $1-p(1-U)=\\frac{1-p}{1-py}$\n\n$dU = \\frac{1-p}{(1-py)^{2}}$\n\n\\begin{equation}\n G(x)=\\int_{0}^{y}\\frac{p^{k}y^{k-1}}{1-py} dy\n\\end{equation}\n\nif $v=py$ ; $dv=pdy$\n\n\\begin{equation}\n\\begin{split}\n G(x) &=\\frac{1}{A(p,k)}\\int_{0}^{v}\\frac{v^{k-1}}{1-v} dv\\\\\n &=\\frac{1}{A(p,k)}\\sum_{r=0}^{\\infty}\\int_{0}^{v}v^{k+r-1} dv\\\\\n &=\\frac{1}{A(p,k)}\\sum_{r=0}^{\\infty}\\frac{v^{k+r}}{k+r}\\\\\n &=\\frac{1}{A(p,k)}\\sum_{z=k}^{\\infty}\\frac{v^{z}}{z} \\mbox{ ; } z=k+r\\\\\n &=\\frac{-ln(1-v)-\\psi(k)\\sum_{j=1}^{k-1}\\frac{v^{j}}{j}}{A(p,k)}\\\\\n &=\\frac{A(py,k)}{A(p,k)}\n\\end{split}\n\\end{equation}\n\n\\[\ny=\\frac{1-e^{-\\theta x}}{1-pe^{-\\theta x}}\n\\]\n\n\nfor $k=1$\n\\begin{equation}\n G(x)=\\frac{ln\\big(\\frac{1-p}{1-pe^{-\\theta x}}\\big)}{ln(1-p)}\n\\end{equation}\n\nThe median is a solution of $G(x)=0.5$, then:\n\n\\begin{equation}\n x_\\text{median}=-\\frac{1}{\\theta}\\ln\\big(\\frac{1-\\sqrt{1-p}}{p}\\big)\n\\end{equation}\n\n\n\nThe mgf is $E(e^{t x})=E[(e^{-\\theta x})^{-t\/\\theta}]$ ; let $u=e^{-\\theta x}$\n\n\n\\begin{equation}\\label{eq:generating-f-appendix}\n\\begin{split}\n E(e^{t x}) &= \\frac{1}{A(p,k)}\\int_{u=0}^{1}\\frac{p^{k}u^{-t\/\\theta}(1-u)^{k-1}}{(1-pu)^{k}} du\\\\\n &= \\frac{p^{k}}{A(p,k)}\\int_{u=0}^{1}\\sum_{i=0}^{\\infty}{k-1+i\\choose i}(pu)^{i}u^{-t\/\\theta}(1-u)^{k-1} du\\\\\n &= \\frac{p^{k}}{A(p,k)}\\sum_{i=0}^{\\infty}{k-1+i\\choose i}p^{i}\\int_{u=0}^{1}u^{i-t\/\\theta}(1-u)^{k-1} du\\\\\n &=\\frac{p^{k}}{A(p,k)}\\sum_{i=0}^{\\infty} {k-1+i\\choose i} p^{i}\\beta (i-\\frac{t}{\\theta}+1,k)\n\\end{split}\n\\end{equation}\n\nwhere,\n\\begin{equation*}\n \\beta (a,b)=\\int_{0}^{1} t^{a-1} (1-t)^{b-1} dt\n\\end{equation*}\n\nfrom the binomial theorem, we have:\n\\begin{equation*}\n (1-u)^{k-1}= \\sum_{j=0}^{k-1}{k-1\\choose j}(-u)^{j}=\\sum_{j=0}^{k-1}{k-1\\choose j}(-1)^{j} u^{j}\n\\end{equation*}\n\nthen,\n\\begin{equation}\\label{eq:generating-f-appendix}\n\\begin{split}\n E(e^{t x}) &= \\frac{p^{k}}{A(p,k)}\\sum_{i=0}^{\\infty}{k-1+i\\choose i}p^{i}\\int_{u=0}^{1}u^{i-t\/\\theta}(1-u)^{k-1} du\\\\\n &= \\frac{p^{k}}{A(p,k)}\\sum_{i=0}^{\\infty}{k-1+i\\choose i}p^{i}\\sum_{j=0}^{k-1}{k-1\\choose j}(-1)^{j}\\int_{u=0}^{1}u^{i+j-t\/\\theta} du\\\\\n &= \\frac{p^{k}}{A(p,k)}\\sum_{i=0}^{\\infty} \\sum_{j=0}^{k-1}{k-1+i\\choose i} {k-1\\choose j}p^{i} (-1)^{j} \\frac{1}{i+j-\\frac{t}{\\theta}+1}\n\\end{split}\n\\end{equation}\n\n\n\\begin{eqnarray*}\n E(x) &=& \\frac{\\partial}{\\partial t} E(e^{t x})\/_{t=0}\\\\\n &=& \\frac{p^{k}}{\\theta A(p,k)}\\sum_{i=0}^{\\infty} \\sum_{j=0}^{k-1}{k-1+i\\choose i} {k-1\\choose j}p^{i} (-1)^{j} \\frac{1}{(i+j-\\frac{t}{\\theta}+1)^2}\/_{t=0} \\\\\n &=& \\frac{p^{k}}{\\theta A(p,k)}\\sum_{i=0}^{\\infty} \\sum_{j=0}^{k-1}{k-1+i\\choose i} {k-1\\choose j}p^{i} (-1)^{j} \\frac{1}{(i+j+1)^{2}}\n\\end{eqnarray*}\n\nThe $r^{th}$ moment is given by:\n\n\\begin{equation}\\label{eq:rth-moment-appendix}\n E(x^{r})= \\frac{\\Gamma(r+1)}{\\theta^{r}}\\frac{p^{k}}{A(p,k)}\\sum_{i=0}^{\\infty} \\sum_{j=0}^{k-1}{k-1+i\\choose i} {k-1\\choose j} p^{i} (-1)^{j} \\frac{1}{(i+j+1)^{r+1}}\n\\end{equation}\n\n\nReliability or survival function:\n\n\\begin{equation}\n\\begin{split}\n S(x) &= Pr(X \\geq x) = 1 - G(x) = \\int_{x}^{\\infty}f(t)d t\\\\\n &= 1-\\frac{A(py,k)}{A(p,k)}\\\\\n &= 1-\\frac{\\ln(1-py)+\\psi(k)\\sum_{j=1}^{k-1}\\frac{(py)^{j}}{j}}{\\ln(1-p)+\\psi(k)\\sum_{j=1}^{k-1}\\frac{p^{j}}{j}}\\\\\n &= 1-\\frac{\\ln\\big(1-p\\frac{1-e^{-\\theta x}}{1-pe^{-\\theta x}}\\big)+\\psi(k)\\sum_{j=1}^{k-1}\\frac{1}{j}\n \\big(p\\frac{1-e^{-\\theta x}}{1-pe^{-\\theta x}}\\big)^{j}}{\\ln(1-p)+\\psi(k)\\sum_{j=1}^{k-1}\\frac{p^{j}}{j}}\n\\end{split}\n\\end{equation}\n\n\nEM algorithm:\n\nUsing the joint distribution $g_{k}(z,x\/p,\\theta)$ we drive the conditional mass function as:\n\n\\begin{equation}\\label{eq:prob-z-appendix}\n p(z\/x,p,\\theta)=\\frac{\\Gamma(z)}{\\Gamma(k)\\Gamma(z-k+1)}p^{z-k} e^{-\\theta (z-k)x}(1-pe^{-\\theta x})^{k}\n\\end{equation}\n\nFor $|a|<1$, we have\n\n\\begin{equation*}\n \\sum_{i=0}^{\\infty}{n+i\\choose i}a^{i}=\\frac{1}{(1-a)^{n+1}}\n\\end{equation*}\n\n\n\nE-step:\n\n\n\\begin{equation}\\label{eq:E-step-appendix}\n\\begin{split}\n E(z\/x,p,\\theta) &= \\sum_{z=k}^{\\infty} z p(z\/x,p,\\theta)\\\\\n &= \\sum_{z=k}^{\\infty} k {z\\choose k} (pe^{-\\theta x})^{z-k}(1-pe^{-\\theta x})^{k}\\\\\n &= k (1-pe^{-\\theta x})^{k}\\sum_{z=k}^{\\infty} {z\\choose k} (pe^{-\\theta x})^{z-k}\\\\\n &= k (1-pe^{-\\theta x})^{k}\\sum_{z=k}^{\\infty} {z\\choose z-k} (pe^{-\\theta x})^{z-k}\\\\\n &= k (1-pe^{-\\theta x})^{k}\\sum_{t=0}^{\\infty} {k+t\\choose t} (pe^{-\\theta x})^{t}\\\\\n &= \\frac{k}{1-pe^{-\\theta x}}\n\\end{split}\n\\end{equation}\n\n\nM-step:\n\nLet $(x_{1}, x_{2},... ,x_{n})$ a random sample. The likelihood given the joint distribution $g_{k}(z,x\/p,\\theta)$ is:\n\n\\begin{equation}\\label{eq:Lik-M-step-appendix}\n\\begin{split}\n L(p,\\theta) &= \\prod_{i=1}^{n}g_{k}(z,x\/p,\\theta)\\\\\n &= \\prod_{i=1}^{n}\\frac{\\Gamma(z)}{\\Gamma(k)\\Gamma(z-k+1)}\\frac{\\theta p^{z} e^{-\\theta (z-k+1) x} (1-e^{-\\theta x})^{k-1}}{A(p,k)}\\\\\n &= \\propto\\prod_{i=1}^{n}\\frac{\\theta p^{z} e^{-\\theta (z-k+1) x} (1-e^{-\\theta x})^{k-1}}{A(p,k)}\\\\\n &= \\propto \\theta^{n} p^{\\sum_{i=1}^{n}z_{i}} e^{-\\theta\\sum_{i=1}^{n}(z_{i}-k+1)x_{i}}A(p,k)^{-n}\\prod_{i=1}^{n}(1-e^{-\\theta x_{i}})^{k-1}\\\\\n\\end{split}\n\\end{equation}\n\nThe log-likelihood is:\n\n\\begin{equation}\\label{eq:Lik-M-step-appendix}\n LL = n \\ln(\\theta)+\\ln(p)\\sum_{i=1}^{n}z_{i}-\\theta\\sum_{i=1}^{n}(z_{i}-k+1)x_{i}+\\sum_{i=1}^{n}(1-e^{-\\theta x_{i}})^{k-1}-n\\ln(A(p,k))\n\\end{equation}\n\n\n\\begin{equation*}\n \\frac{\\partial LL}{\\partial \\theta} = \\frac{n}{\\theta}-\\sum_{i=1}^{n}(z_{i}-k+1)x_{i}+(k-1)\\sum_{i=1}^{n}\\frac{x_{i}e^{-\\theta x_{i}}}{1-e^{-\\theta x_{i}}}=0\n\\end{equation*}\n\n\\begin{equation*}\n\\widehat{\\theta}^{(r+1)} =\nn\\bigg[\\sum_{i=1}^{n}z_{i}x_{i}-(k-1)\\sum_{i=1}^{n}\\frac{x_{i}}{1-e^{-\\theta^{(r+1)} x_{i}}}\\bigg]^{-1}\n\\end{equation*}\n\nwe replace $z_{i}$ with $E(z\/x,p^{(r)},\\theta^{(r)})=\\frac{k}{1-p^{(r)}e^{-\\theta^{(r)} x_{i}}}$\n\n\\begin{equation}\\label{eq:M-step1-appendix}\n\\widehat{\\theta}^{(r+1)} =\nn\\bigg[\\sum_{i=1}^{n}\\frac{kx_{i}}{1-p^{(r)}e^{-\\theta^{(r)} x_{i}}}-(k-1)\\sum_{i=1}^{n}\\frac{x_{i}}{1-e^{-\\theta^{(r+1)} x_{i}}}\\bigg]^{-1}\n\\end{equation}\n\n\n\n\\begin{equation*}\n \\frac{\\partial LL}{\\partial p} = \\frac{\\sum_{i=1}^{n}z_{i}}{p}-n\\frac{A(p,k)'}{A(p,k)}=0\n\\end{equation*}\n\n\nwhere,\n\\begin{equation*}\n A(p,k) = \\sum_{j=k}^{\\infty}\\frac{p^{j}}{j}\n = - \\ln(1-p)-\\psi(k)\\sum_{j=1}^{k-1}\\frac{p^{j}}{j}\n\\end{equation*}\n\n\\begin{equation*}\nA(p,k)' =\\frac{\\partial A(p,k)}{\\partial p} = \\sum_{j=k}^{\\infty}p^{j-1}=\\frac{p^{k-1}}{1-p}\n\\end{equation*}\n\nthen,\n\n\\begin{equation*}\n\\frac{\\sum_{i=1}^{n}z_{i}}{p}=n\\frac{A(p,k)'}{A(p,k)}\n\\end{equation*}\n\n\\begin{equation}\n\\begin{split}\n p^{(r+1)} &= \\frac{A(p,k)}{nA(p,k)'}\\sum_{i=1}^{n}z_{i}\\\\\n &= \\bigg[\\frac{1-p^{(r+1)}}{n\\Big(p^{(r+1)}\\Big)^{k-1}}\\sum_{j=k}^{\\infty}\\frac{\\Big(p^{(r+1)}\\Big)^{j}}{j}\\bigg]\\sum_{i=1}^{n}z_{i}\\\\\n &= \\bigg[\\frac{1-p^{(r+1)}}{n\\Big(p^{(r+1)}\\Big)^{k-1}}\\sum_{j=k}^{\\infty}\\frac{\\Big(p^{(r+1)}\\Big)^{j}}{j}\\bigg]\\sum_{i=1}^{n}\\frac{k}{1-p^{(r)}e^{-\\theta^{(r)} x_{i}}}\n\\end{split}\n\\end{equation}\n\n\n\\begin{equation}\\label{eq:M-step2-appendix}\n\\widehat{p}^{(r+1)}\n =\n \\frac{-\\big(1-p^{(r+1)}\\big)\\Big[\\ln\\big(1-p^{(r+1)}\\big)+\\psi(k)\\sum_{j=1}^{k-1}\\frac{\\big(p^{(r+1)}\\big)^{j}}{j}\\Big]}{n\\big(p^{(r+1)}\\big)^{k-1}}\\sum_{i=1}^{n}\\frac{k}{1-p^{(r)}e^{-\\theta^{(r)} x_{i}}}\n\\end{equation}\n\n\n\n\n\n\\newpage\n\\bibliographystyle{agsm}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nA compact Riemann surface $M_g$ of genus $g\\geq 0$ is determined by its space of holomorphic 1--forms. \nFor $g \\geq 1$, the Jacobian variety $J(M_g)$ is defined by using the vector space of holomorphic 1--forms. \nVery roughly speaking, Torelli's theorem says that we can recover $M_g$ from $J(M_g)$; \nsee \\cite[p.~359]{Griffith}. \nMoreover, for $g \\geq 2$, \na classical result of Hurwitz asserts that the automorphism group $Aut(M_g)$ is finite; \nsee \\cite[Ch. V]{farkas}. \nLooking at the moduli space of compact Riemann surfaces $\\mathcal{M}_{g,0}$, \ngenerically $Aut(M_g)$ is trivial. \nFor $g=0$, three special features appear. \nObviously there exists only one complex structure, set the Riemann sphere $\\ensuremath{{\\widehat{\\mathbb C}}}$. \nSecondly, \nany holomorphic 1--form over $\\ensuremath{{\\widehat{\\mathbb C}}}$ is identically zero. \nFinally, \nthe automorphism group $PSL(2,\\mathbb{C})$ of $\\ensuremath{{\\widehat{\\mathbb C}}}$ is the biggest between the automorphism groups of any Riemann surface. \\vspace{0cm}\n\\begin{flushleft}\n\\textit{A natural problem is the classification of rational 1--forms on $\\ensuremath{{\\widehat{\\mathbb C}}}$ with $-s \\leq -2$ poles up to the automorphism group $PSL(2,\\mathbb{C})$.}\n\\end{flushleft}\\vspace{0cm}\nThe family of rational 1--forms on $\\ensuremath{{\\widehat{\\mathbb C}}}$ is an infinite dimensional vector space. \nA natural idea is to consider the stratification by the multiplicities of zeros and poles. \nLet $\\racs{s}$ be the family of rational 1--forms having exactly $-s \\leq -2$ simple poles.\nOur techniques naturally allows us to study rational 1--forms with zeros of any multiplicity and simple poles. \nWe recognized three equivalent $(2s-1)$--dimensional complex atlases on $\\racs{s}$, \nlooking at different expressions of the rational 1--forms coefficients $\\Omega^1_{coef}(-1^s)$, zeros--poles $\\Omega^1_{zp}(-1^s)$ and residues--poles $\\Omega^1_{rp}(-1^s)$. \nOur main result is the following theorem:\n\\begin{rtheorem}[\\ref{teorema-equivalencia-parametros}]\n The complex manifolds $\\Omega_{coef}^{1}(-1^s)$, $\\Omega_{zp}^{1}(-1^s)$ and $\\Omega_{rp}^{1}(-1^s)$ are biholomorphic\n\\end{rtheorem}\nFor the proof, \nwe construct explicitly the biholomorphisms $\\Omega_{rp}^1(1^s) \\longrightarrow \\Omega_{coef}^1(1^s)$ and $\\Omega_{zp}^1(1^s) \\longrightarrow \\Omega_{coef}^1(1^s)$ using the \\textit{Vi\\`ete} map defined in \\eqref{viete-map}. \n\nThe group $PSL(2,\\mathbb{C})$ acts naturally on $\\racs{s}$ by coordinate changes. \nAs a result, \nthis $PSL(2,\\mathbb{C})$--action is proper for $s \\geq 3$; \nsee Lemma \\ref{proper-action}.\nBy using the classical theory of proper Lie group actions as in \\cite[Ch.~2]{diustermaat}, \nwe recognize a principal $PSL(2,\\mathbb{C})$--bundle $\\pi_s: \\mathcal{G}(-1^s) \\to \\mathcal{G}(-1^s)\/PSL(2,\\mathbb{C})$, \nwhere $\\mathcal{G}(-1^s) \\subset \\racs{s}$ is the open and dense subset of generic rational 1--forms with trivial isotropy group in $PSL(2,\\mathbb{C})$, \nand $\\pi_s$ denotes the natural projection to the orbit space. \nFor $s=3 \\ (\\text{resp. } s=4)$, we prove that the principal $PSL(2,\\mathbb{C})$--bundle is trivial (resp. nontrivial). \nOn the other hand, \nfor all $\\omega \\in \\racs{s}\\setminus \\mathcal{G}(-1^s)$, \nthe isotropy group $PSL(2,\\mathbb{C})_{\\omega}$ is a nontrivial finite subgroup of $PSL(2,\\mathbb{C})$. \nA classical result of F. Klein classifies the finite subgroups of $PSL(2,\\mathbb{C})$; see \\cite[p. 126]{klein20}. \nIn our framework, \nthe realization problem is which finite subgroups of $PSL(2,\\mathbb{C})$ are realizable as the isotropy groups of suitable $\\omega \\in \\racs{s}$? \nA positive answer is as follows.\n\\begin{rproposition}[\\ref{realizacion-grupos-finitos}]\n Every finite subgroup $G < PSL(2,\\mathbb{C})$ appears as the isotropy group of suitable $\\omega \\in \\racs{s}$.\n\\end{rproposition}\n\\noindent Obviously, \nthe degree of the divisor of poles $s$ depends on the order of $G$. \nRecalling that the rotation groups of a pyramid, bypyramid and platonic solids, \nthe proof is done. \n\nAs a second goal, \nrecall that $\\omega \\in \\racs{s}$ is isochronous when all their residues are purely imaginary. \nWe study the subfamily $\\RI{s} \\subset \\racs{s}$ of isochronous 1--forms. \nOur result is below.\n\\begin{rcorollary}[\\ref{teorema-RI-variedad}]\n \\textit{The subfamily $\\RI{s}$ is a $(3s-1)$--dimensional real analytic submanifold of $\\racs{s}$.}\n\\end{rcorollary}\nFor the proof, \nwe use the complex atlas by residues--poles. \nProposition \\ref{realizacion-grupos-finitos} is fulfilled for suitable $\\omega \\in \\RI{s}$. \nThe residues are naturally a set of $PSL(2,\\mathbb{C})$--invariant functions. \nThey can be used in order to describe a realization of 1--forms; \nsee \\cite{isocrono}. \nThe $PSL(2,\\mathbb{C})$--action on $\\RI{s}$ is well--defined. \nHence, \nthe quotients $\\racs{s}\/PSL(2,\\mathbb{C})$ and $\\RI{s}\/PSL(2,\\mathbb{C})$ admit a complex and a real stratification by orbit types, repectively. \nFor $s \\geq 4$, in order to get a complete set of $PSL(2,\\mathbb{C})$--invariant functions we enlarge the set of residues by adding the cross--ratio of poles and explicitly recognize realizations for the quotients $\\racs{s}\/PSL(2,\\mathbb{C})$ and $\\RI{s}\/PSL(2,\\mathbb{C})$; \nsee respectively Proposition \\ref{quotient} and Corollary \\ref{quotient2}.\\\\\nIn Section \\ref{sec-surfaces}, for $\\omega \\in \\racs{s}$, resp. $\\omega \\in \\RI{s}$, we obtain a realization for the quotient of the associated flat surfaces $S_{\\omega}$ up to isometries $\\mathfrak{M}(-s)$, resp. $\\mathcal{RI}\\mathfrak{M}(-s)$, extending naturally the $PSL(2,\\mathbb{C})$--action.\\\\\n \nOur results have applications in dynamical systems and the geometry of flat surfaces since there is a one--to--one \\textit{correspondence} between rational 1--forms $\\omega = (Q(z)\/P(z)) dz$, \noriented rational quadratic differentials $\\omega \\otimes \\omega = (Q^2(z)\/ P^2(z))dz^2$, \nrational complex vector fields $X_{\\omega} = P(z)\/Q(z) \\ \\partial\/\\partial z$, \npairs of singular real analytic vector fields $(\\Re{X_{\\omega}},$ $\\Im{X_{\\omega}})$ on $\\ensuremath{{\\widehat{\\mathbb C}}} \\setminus \\{ Q = 0\\}$, \nand singular flat surfaces $S_{\\omega}=(\\ensuremath{{\\widehat{\\mathbb C}}} , g_{\\omega})$ provided with two real singular geodesic foliations. \nThe metrics $g_{\\omega}$ are the associated to the quadratic differentials $\\omega \\otimes \\omega$, \nand the foliations come from the horizontal and vertical trajectories. \nThis correspondence is used in many works, \\textit{e.g.} \\cite{kerckhoff,jesus1, alvaro1}.\nFor $\\omega \\in \\racs{s}$, \nits associated quadratic differential $\\omega \\otimes \\omega$ has poles of multiplicitie 2 and they were studied by K. Strebel in \\cite{strebelB, strebel}.\n\\\\\n\nHistorically, C. Huygens \\cite[p.~72]{whittaker} gave formulas for the period of isochronous centers in the model of a simple pendulum as differential form $(i\\lambda\/z)dz$.\nTrying to reach a contemporary point of view, we recall the following statements.\\\\\nQuadratic differentials with closed trajectories were first considered by O. Teich-m\\\"uller in his \\textit{``Habilitationsschrift\"} \\cite{teichmuller}. \nOn $M_g$, \nK. Strebel \\cite{strebelB, strebel} proved that certain quadratic differentials with closed regular horizontal trajectories realize the extremal metric problem introduced by J. A. Jenkins \\cite{jenkins}. \\\\\nAnother interesting facet of isochronous 1--forms $\\omega \\in \\RI{s}$, comes from dynamical systems. \nThe phase portrait of the associated vector field $\\Re{X_{\\omega}}$ is a union of isochronous centers or annulus. \nIn other words, any pair of trajectories of $\\Re{X_{\\omega}}$ that share a center basin have the same period. \nLooking at isochronous centers on $\\mathbb{R}^2$; \nP. Marde{\\v{s}}i\\'c, C. Rousseau and B. Toni \\cite{mardesic} studied the linearization problem, \nand L. Gavrilov \\cite{gavrilov} considered the appearance of isochronous centers in polynomial of Hamiltonian systems on $\\mathbb{C}^2$ and its relation to the famous Jacobian conjecture. \nA constructive result for isochronous vector fields on $\\mathbb{C}$, by using the residues, is provided by J. Muci\\~no-Raymundo in \\cite[\\S~8]{jesus1}. \nA topological and analytic classification of complex polynomials vector fields on $\\mathbb{C}$ with only isochronous centers was performed by M. E. Fr\\'ias-Armenta and J. Muci\\~no--Raymundo in \\cite{isocrono}.\n\n\\section{Rational 1--forms with simple poles}\n\\label{S-families}\n\n\\subsection{Stratification}\n\nWe define a stratification on the set $\\racs{s}$ of rational 1--forms on the Riemann sphere \\ensuremath{{\\widehat{\\mathbb C}}}, \nhaving exactly $-s \\geq -2$ simple poles. \nFirst, recall that the infinite dimensional vector space of rational 1--forms admits a stratification fixing the multiplicities of the zeros $\\{k_1, \\ldots, k_m\\}$ and poles $\\{-s_1, \\ldots, -s_n\\}$, \nwhere $k_j, s_{\\iota} \\in \\mathbb{N}$. \nThe stratum of rational 1--forms with these multiplicities is denoted by $\\Omega^1\\{k_1, \\ldots, k_m ; -s_1, \\ldots, -s_n \\}$, \nthey are connected $(m+n+1)$--dimensional complex manifolds in the vector space of rational 1--forms; \nour notation is similar to \\cite{jesus1, moguel2}.\nOn the other hand, \nfor $g \\geq 2$ the stratum are not necessarily connected. \nM. Kontsevich and A. Zorich describe the connected components of holomorphic 1--forms in each stratum on $M_g$; \nsee \\cite{kontsevich}.\\\\\nOur framework allows us to study \n\\begin{equation}\\label{omega-variedad}\n\\racs{s} = \\bigsqcup \\Omega^1\\{ k_1, \\ldots, k_m ; \\underbrace{-1,\\ldots, -1}_{\\text{$s$}} \\},\n\\end{equation}\nwhere the union takes all the multiplicities $\\{k_1, \\ldots, k_m; -1, \\ldots, -1 \\}$ such that $\\{k_1, \\ldots , k_m \\}$ is an integer partition of $s-2$, \n\\textit{i.e.} the sum $k_1 + \\ldots + k_m = s - 2$. \nThe expression $(-1^s)$ in \\eqref{omega-variedad} is motivated by the ``exponential\" notation for multiple poles of the same degree in the stratification by multiplicities of rational 1--forms; see \\cite{kontsevich, corentin}. \n\n\\subsection{Polynomials}\\label{polynomials}\n\nLet us recall the existence of two natural complex atlases for polynomials $\\mathbb{C}[z]_{=s}$ with degree $s$. For complex manifolds, we use notation as in \\cite[Ch.~IV]{fritzsche}. \nFirst, given the natural homeomorphism\n$$\n\\begin{array}{rcl}\nf_1: \\mathbb{C}[z]_{=s} & \\longrightarrow & \\mathbb{C}^{s+1} \\setminus \\{ b_s = 0\\} \\\\\nb_sz^s + \\ldots + b_0 & \\longmapsto & (b_s, \\ldots, b_0),\n\\end{array}\n$$\nwe obtain that $(\\mathbb{C}[z]_{=s}, f_1)$ is an $(s+1)$--dimensional complex coordinate system in $\\mathbb{C}[z]_{=s}$. \nClearly, the subset of polynomials with degree $s$ and simple roots $\\mathbb{C}[z]_{=s} \\setminus \\mathcal{D}(P, P')$ is open, \nwhere $\\mathcal{D}(P, P')$ denotes the discriminant of $P$. \\\\\nSecond, the action of the symmetric group $\\Sim{s}$ of $s$ elements on $\\ensuremath{{\\widehat{\\mathbb C}}}^{s} \\setminus \\Delta := \\{(p_1, \\ldots, p_s) \\in \\ensuremath{{\\widehat{\\mathbb C}}}^{s} \\ | \\ p_{\\iota} \\not= p_{\\kappa}, \\ \\text{for all }\\iota \\not= \\kappa \\}$ is properly discontinous. \nIn fact, the quotient $(\\ensuremath{{\\widehat{\\mathbb C}}}^s \\setminus \\Delta) \/ \\Sim{s}$ is an $s$--dimensional complex manifold. \nMoreover, \n$\\left(\\mathbb{C}[z]_{=s}\\setminus \\mathcal{D}(P,P'), \\nu_s \\circ f_2 \\right)$ is an $(s+1)$--dimensional complex coordinate system in $\\mathbb{C}[z]_{=s} \\setminus \\mathcal{D}(P,P')$, \nwhere the map\n$$\n\\begin{array}{rcl}\nf_2: \\mathbb{C}[z]_{=s} \\setminus \\mathcal{D}(P,P') & \\longrightarrow & \\mathbb{C}^* \\times \\left(\\frac{\\ensuremath{{\\widehat{\\mathbb C}}}^{s} \\setminus \\Delta }{\\Sim{s}} \\right) \\\\ \nb_s\\prod_{\\iota=1}^s (z - p_{\\iota}) & \\longmapsto & (b_s, \\{p_1, \\ldots, p_s \\})\n\\end{array}\n$$\nis a natural bijection and\n\\begin{equation}\\label{viete-map}\n\\begin{array}{rcl}\n\\nu_s : \\mathbb{C}^* \\times \\left( \\frac{\\ensuremath{{\\widehat{\\mathbb C}}}^s \\setminus \\Delta}{\\Sim{s}} \\right) & \\longrightarrow & \\mathbb{C}^{s+1} \\setminus \\{ b_s = 0 \\} \\\\\n(b_s, \\{p_1, \\ldots, p_s\\}) & \\longmapsto & \\left(b_s, -b_s\\left(\\sum_{\\iota=1}^s p_{\\iota}\\right), \\ldots, (-1)^sb_s\\prod_{\\iota=1}^s p_{\\iota} \\right),\n\\end{array}\n\\end{equation} \nis called the \\textit{Vi\\`ete} map. \\\\\nFinally, \nit is easy to prove that the coordinate systems $(\\mathbb{C}[z]_{=s}\\setminus \\mathcal{D}(P,P'), f_1)$ and $(\\mathbb{C}[z]_{=s} \\setminus \\mathcal{D}(P,P'), \\nu_s \\circ f_2)$ are hollomophically compatible; \nsee \\cite{katz}. \nA remarkable fact is that some properties of polynomials are easy to see in one coordinate system but others are kept hidden.\\\\\n\n\\subsection{Complex atlases on $\\racs{s}$}\n\nSimilarly as in Section \\ref{polynomials}, \nthree equivalents $(2s-1)$--dimensional complex atlases on $\\racs{s}$ will be constructed.\n\n\\begin{description}[leftmargin=0cm]\n\\item[1) \\textbf{Coefficients.}] We consider the Zariski open subset of $\\ensuremath{{\\mathbb{CP}}}^{2s-1}$, \n$$\n\\Omega_{coef}^{1}(-1^s) := \\left\\{ [a_{s-2} : \\ldots : a_0 : b_s : \\ldots : b_0 ] \\in \\ensuremath{{\\mathbb{CP}}}^{2s-1} \\ \\ \\left| \\ \\ \\begin{array}{rcl}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\mathcal{D}(P,Q) &\\not=& 0, \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\mathcal{D}(P,P') &\\not=& 0\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\end{array} \\right. \\right\\},\n$$\nwhere $Q(z) = a_{s-2}z^{s-2} + \\ldots + a_0$, $P(z) = b_sz^{s} + \\ldots + b_0$ and $\\mathcal{D}(P,Q)$ denotes the resultant of the polynomials $P$ and $Q$. \nIf $\\{ (U^{coef}_j, \\varphi^{coef}_j)\\}$ denotes the complex atlas on $\\Omega^1_{coef}(-1^s)$, \nthen $\\mathfrak{A}_{coef} := \\{ (f_{coef}^{-1}(U^{coef}_j), \\varphi^{coef}_j \\circ f_{coef} )\\}$ is a complex atlas on $\\racs{s}$, \nwhere\n$$\n\\begin{array}{rcl}\nf_{coef}: \\racs{s} & \\longrightarrow & \\Omega^1_{coef}(-1^s) \\\\\n\\displaystyle \\omega = \\frac{a_{s-2}z^{s-2} + \\ldots + a_0}{b_sz^s + \\ldots + b_0}dz & \\longmapsto & [a_{s-2} : \\ldots : a_0 : b_s : \\ldots : b_0],\n\\end{array}\n$$\nis a natural bijection map. \n\\item[2) \\textbf{Zeros--poles.}] We consider\n$$\nM_s := \\left.\\left\\{\\left\\{c_1, \\ldots, c_{s-2}, p_1, \\ldots, p_s\\right\\} \\in \\frac{\\ensuremath{{\\widehat{\\mathbb C}}}^{s-2}}{\\Sim{s-2}} \\times \\frac{\\ensuremath{{\\widehat{\\mathbb C}}}^s \\setminus \\Delta}{\\Sim{s}} \\ \\right| c_j \\not= p_{\\iota} \\ \\right\\}.\n$$\nRecalling that there exists a biholomorphism between $\\ensuremath{{\\widehat{\\mathbb C}}}^s\/\\Sim{s}$ and $\\ensuremath{{\\mathbb{CP}}}^s$, $M_s$ is a $(2s-2)$--dimensional open and dense complex submanifold of $\\ensuremath{{\\mathbb{CP}}}^{s-2}\\times \\ensuremath{{\\mathbb{CP}}}^s$. \nConsider the transition functions for a nontrivial principal $\\mathbb{C}^{*}$--bundle over $M_s$ as follows. \nLet $u= \\left\\{ c_1, \\ldots, c_{s-2}, p_1, \\ldots, p_s \\right\\}\t\\in M_s$ and set\n$$\n\tQ_u(z) := \\left\\{ \\begin{array}{ll}\n\t\t\t(z-c_1)\\cdots (z-c_{s-2}) & \\text{where $c_j\\not= \\infty$ for all $j$,} \\\\\n\t\t\t(z-c_1)\\cdots(z - c_{j-1})(z- c_{j+1})\\cdots(z-c_{s-2}) & \\text{if $c_j=\\infty$,}\n\t\t \\end{array}\\right.\n$$\n$$\n\tP_u(z) := \\left\\{ \\begin{array}{ll}\n\t\t\t(z-p_1)\\cdots (z-p_{s}) & \\text{where $p_{\\iota}\\not= \\infty$ for all $\\iota$,} \\\\\n\t\t\t(z-p_1)\\cdots(z - p_{\\iota-1})(z- p_{\\iota+1})\\cdots(z-p_{s}) & \\text{if $p_{\\iota}=\\infty$.}\n\t\t \\end{array}\\right.\n$$\nFor $\\alpha \\in I := \\{1, 2, \\ldots, 2s-1 \\}$, \nwe define $\\Omega_{zp}^{1}(-1^s)$ as the total space of the principal $\\mathbb{C}^*$--bundle over $M_s$ with the transition functions \n\\begin{equation}{\\label{funcion-transicion}}\n\\begin{array}{rcl}\n\t\t\\phi_{\\alpha \\beta} : \\mathcal{U}_{\\alpha} \\cap \\mathcal{U}_{\\beta} &\\longrightarrow & \\mathbb{C}^{*} \\\\\n\t\tu & \\longmapsto &\\displaystyle \\frac{Q_u(\\alpha)}{P_u(\\alpha)}\\left(\\frac{P_u(\\beta)}{Q_u(\\beta)}\\right),\n\\end{array}\n\\end{equation} \nwhere $\\mathcal{U}_\\alpha := \\left\\{ u \\in M_s \\ | \\ Q_u(\\alpha) \\not= 0 \\text{ and } P_u(\\alpha) \\not= 0 \\right\\}$. \nIf $\\{(U^{zp}_j, \\varphi^{zp}_j)\\}$ denotes the complex atlas on $\\Omega^1_{zp}(-1^s)$, \nthen $\\mathfrak{A}_{zp} = \\{ (f_{zp}^{-1}(U^{zp}_j), \\varphi^{zp}_j \\circ f_{zp} ) \\}$ is a complex atlas on $\\racs{s}$ where \n$$\n\\begin{array}{rcl}\nf_{zp}: \\racs{s} & \\longrightarrow & \\Omega^1_{zp}(-1^s) \\\\\n\\displaystyle \\omega = \\lambda\\frac{(z-c_1)\\cdots(z-c_{s-2})}{(z-p_1)\\cdots(z-p_s)} dz & \\longmapsto & [\\{c_1, \\ldots c_{s-2}, p_1, \\ldots, p_s\\}, \\lambda],\n\\end{array}\n$$\nis a bijection map.\n\\item[3) \\textbf{Residues--poles.}] Let $H_s := \\{(r_1, \\ldots, r_s) \\in (\\mathbb{C}^*)^s \\ | \\ r_1 + \\ldots + r_s = 0 \\}$ and recall that $\\ensuremath{{\\widehat{\\mathbb C}}}^{s} \\setminus \\Delta = \\{(p_1, \\ldots, p_s) \\in \\ensuremath{{\\widehat{\\mathbb C}}}^{s} \\ | \\ p_{\\iota} \\not= p_{\\kappa}, \\ \\text{for all }\\iota \\not= \\kappa \\}$. \nWe consider a diagonal action of the symmetric group $\\Sim{s}$ of $s$ elements\n\t\\begin{equation}\\label{accion-residuos-polos}\n\t \\begin{array}{lll}\n\t\\Sim{s} \\times (H_s \\times (\\ensuremath{{\\widehat{\\mathbb C}}}^s\\setminus \\Delta)) &\\longrightarrow &H_s \\times (\\ensuremath{{\\widehat{\\mathbb C}}}^s\\setminus \\Delta) \\\\\n\t(\\sigma, (r_1, \\ldots, r_s, p_1, \\ldots, p_s)) &\\longmapsto & (r_{\\sigma(1)}, \\ldots, r_{\\sigma(s)}, p_{\\sigma(1)}, \\ldots, p_{\\sigma(s)}).\n\t\\end{array}\n\t\\end{equation}\nClearly, \nthe action above is properly discontinuous and the quotient\n$$\n\\Omega_{rp}^{1}(-1^s) := \\frac{H_s \\times (\\ensuremath{{\\widehat{\\mathbb C}}}^{s} \\setminus \\Delta)}{\\Sim{s}}.\n$$\nis a $(2s-1)$--dimensional open complex manifold. \nWe denote the equivalence class under the action \\eqref{accion-residuos-polos} as $\\rp{r_1, \\ldots, r_s; p_1, \\ldots, p_s}$. \nGeometrically, \nan element in $\\Omega_{rp}^{1}(-1^s)$ is a configuration\\footnote{We convene that a configuration is an unordered set of points different between them.} \nof $s$ points $\\{p_{\\iota}\\}$ in the Riemann sphere with weights $\\{r_{\\iota}\\} \\subset \\mathbb{C}^*$ which satisfy the residue theorem.\nIf $\\{(U^{rp}_j, \\varphi^{rp}_j)\\}$ denotes the complex atlas on $\\Omega^1_{rp}(-1^s)$, \nthen $\\mathfrak{A}_{rp} := \\{ (f_{rp}^{-1}(U^{rp}_j), \\varphi^{zp}_j \\circ f_{zp} ) \\}$ is a complex atlas on $\\racs{s}$ where \n$$\n\\begin{array}{rcl}\nf_{rp}: \\racs{s} & \\longrightarrow & \\Omega^1_{rp}(-1^s) \\\\\n\\displaystyle \\omega = \\sum_{\\iota = 1}^s \\frac{r_{\\iota}}{z-p_{\\iota}} dz & \\longmapsto & \\rp{r_1,\\ldots,r_s;p_1, \\ldots ,p_s}, \n\\end{array}\n$$\nis a bijection map. \nFor a 1--form $\\omega$ with the pole $p_{\\kappa} = \\infty$, \nthe term $r_{\\kappa}\/(z - p_{\\kappa})$ is omitted in the sum above. \nObviously, \nthe residue theorem is the unique obstruction to realize $\\omega \\in \\racs{s}$. \nMoreover, \n$$\n2s-1 = dim_{\\mathbb{C}} (\\racs{s}) \\geq dim_{\\mathbb{C}} (\\Omega^1 \\{ k_1, \\ldots, k_m ; \\underbrace{-1,\\ldots, -1}_{\\text{$s$}} \\}) = m+s+1.\n$$\n\\end{description}\nRecall that the complex atlases above are valid only for rational 1--forms with simple poles. \nThe study of rational 1--forms with poles of multiplicitie grater or equal than 2, \nwill be consider in a future work. \nOur main result is as follows. \n\n\\begin{theorem}\\label{teorema-equivalencia-parametros}\n\tThe complex manifolds $\\Omega_{coef}^{1}(-1^s)$, $\\Omega_{zp}^{1}(-1^s)$ and $\\Omega_{rp}^{1}(-1^s)$ are biholomorphic\n\\end{theorem}\n\n\\begin{proof}\nWe construct explicitly two biholomorphisms from $\\Omega_{coef}^1(-1^s)$ to $\\Omega_{rp}^1(-1^s)$ and $\\Omega_{zp}^1(-1^s)$ to $\\Omega_{coef}^1(-1^s)$.\\\\ \nFirst, \nwe show that $\\Omega^{1}_{rp}(-1^s)$ and $\\Omega^{1}_{coef}(-1^s)$ are biholomorphic. \nFor $\\rp{P} = \\left\\langle r_1, \\ldots,\\right.$ $\\left. r_s; p_1, \\ldots, p_s \\right\\rangle \\in \\Omega_{rp}^{1}(-1^s)$, \nconsider $C_s:\\Omega_{rp}^{1}(-1^s)$ $\\longrightarrow \\Omega_{coef}^{1}(-1^s)$ such that\n$$\nC_s\\rp{P} := \\left\\{\\begin{array}{lll}\n\t\t\t\t\t\t\t\t\t\t\\displaystyle \\left[-\\sum_{j= 1}^s r_j \\sum_{\\iota \\not= j} p_{\\iota} :\\sum_{j=1}^{s} r_j \\sum_{\\iota, \\kappa \\not= j} p_{\\iota}p_{\\kappa}:\\ldots: \\right.&& \\\\\n\\displaystyle \\left. (-1)^{s-1}\\sum_{j=1}^{s}r_j \\prod_{\\iota \\not= j}p_{\\iota}: \\nu_{s}(1, \\left\\{p_1,\\ldots,p_s\\right\\} )\\right] &\\text{for }p_{\\iota} \\in \\mathbb{C}, & \\\\\n\t\t\t\t\t\t\t\t\t\t&& \\\\\n\t\t\t\t\t\t\t\t\t\t\\displaystyle \\left[\\sum_{j \\not= \\kappa }^s r_j :-\\sum_{j \\not= \\kappa }^{s} r_j \\sum_{\\iota \\not= j} p_{\\iota}:\\ldots:\\right.& &\\\\\n\\displaystyle \\left. (-1)^{s-2} \\sum_{j \\not= \\kappa }^{s} r_j \\prod_{\\iota \\not= j} p_{\\iota}:0: \\nu_{s-1}(1,\\left\\{p_1,\\ldots,\\widehat{p}_{\\kappa}, \\ldots, p_s\\right\\}) \\right] &\\text{for }p_{\\kappa} = \\infty, & \n\t\t\t\t\t\t\t\t\t\t\\end{array}\\right.\t\n$$\nwhere $\\nu_{s}$ is the \\textit{Vi\\`ete} map in \\eqref{viete-map}. \nThe hat over the pole $p_{\\kappa}$ indicates that it is omitted. \nA direct computation prove that the map $C_s$ is a bijective map. \nIf $p_j \\not= \\infty$ for all $j=1,\\ldots s$, \nthen the Jacobian matrix is\n$$\nD{C}_s(r_1,\\ldots,r_s,p_1,\\ldots,p_s) = \\left(\\begin{array}{c|c} \n\t\t\t\t\t\t\t\tA & * \\\\ \\hline\n\t\t\t\t\t\t\t\t0 & D{\\nu}^{o}_{s}\\{p_1,\\ldots,p_s\\} \n\t\t\t\t\t\t\t \\end{array}\\right),\n$$\nwhere\n\\begin{equation}\\label{matriz-vieta}\n\t\t\t\t\t\t\tA = \\left(\\begin{array}{cccc}\n\t\t\t\t\t\t\t\\displaystyle \\sum_{j\\not=1} p_j &\\displaystyle \\sum_{j\\not=2} p_j & \\ldots &\\displaystyle \\sum_{j\\not=s} p_j \\\\\n\t\t\t\t\t\t\t\\displaystyle -\\sum_{j, \\iota \\not=1} p_jp_{\\iota} &\\displaystyle -\\sum_{j, \\iota \\not=2} p_jp_{\\iota} & \\ldots &\\displaystyle -\\sum_{j, \\iota \\not=s} p_jp_{\\iota} \\\\\n\t\t\t\t\t\t\t\\vdots & \\vdots & & \\vdots \\\\\n\t\t\t\t\t\t\t\\displaystyle (-1)^{s}\\prod_{j\\not=1} p_j &\\displaystyle (-1)^{s}\\prod_{j\\not=2} p_j & \\ldots &\\displaystyle (-1)^{s}\\prod_{j\\not=s} p_j \\\\\n\t\t\t\t\t \\end{array}\\right),\n\\end{equation}\nand $\\nu^o_s$ denotes the \\textit{Vi\\`ete} map $\\nu_s$ by removing the first coordinate. \nThe rows of $D{C}_s$ are linear independent, \nand the map $C_s$ is a biholomorphism. \nThe case $p_j = \\infty$ is analogous. \nWe are done, $\\Omega^{1}_{coef}(-1^s)$ is biholomorphic to $\\Omega^{1}_{rp}(-1^s)$.\\\\\n\nSecondly, we prove that $\\Omega_{zp}^{1}(-1^s)$ and $\\Omega_{coef}^{1}(-1^s)$ are biholomorphic. \nFor $u = \\left\\{ c_1,\\ldots,\\right.$ $\\left.c_{s-2},p_1,\\ldots,p_s \\right\\} \\in \\mathcal{U}_{\\alpha} \\subset M_s$, \nwe consider the map\n\\begin{eqnarray*}\n\\mathcal{F}_s: \\Omega_{zp}^{1}(-1^s) & \\longrightarrow & \\Omega_{coef}^{1}(-1^s) \\\\\n \\left[ u, \\lambda \\right] & \\longmapsto & \\left[\\lambda \\frac{P_u(\\alpha)}{Q_{u}(\\alpha)}\\nu_{s-2}(1, \\left\\{c_1, \\ldots, c_{s-2}\\right\\}) : \\nu_s(1, \\left\\{p_1, \\ldots, p_s \\right\\}) \\right].\n\\end{eqnarray*} \nFor all $\\alpha, \\beta \\in I$, the transition functions $\\{ \\phi_{\\beta \\alpha} \\}$ make that the diagram below conmutes.\n$$\n\t\t\\xymatrix{ (\\mathcal{U}_{\\alpha} \\cap \\mathcal{U}_{\\beta}) \\times \\mathbb{C}^{*} \\ar@{->}^{\\phi_{\\beta \\alpha}}[rr] \\ar@{->}_{\\mathcal{F}_s}[rd] & & (\\mathcal{U}_{\\alpha} \\cap \\mathcal{U}_{\\beta}) \\times \\mathbb{C}^{*} \\ar@{->}^{\\mathcal{F}_s}[ld] \\\\\n &\\Omega_{coef}^{1}(-s), &\n\t\t\t\t}\n$$ \nIn fact, the map $\\mathcal{F}_s$ is a biholomorphism. \n\\end{proof}\n\nFrom now, \nwe only use the complex atlas by residues--poles $\\mathfrak{A}_{rp}$ on $\\racs{s}$. \nHowever, \nby Theorem \\ref{teorema-equivalencia-parametros} the results in this paper are valid independently of the complex atlas.\n\n\\begin{definition}\nA rational 1--form $\\omega \\in \\racs{s}$ is \\emph{isochronous} when all their residues are purely imaginary. \nThe family of rational isochronous 1--forms is denoted by\n$$\n \\RI{s} :=\\left\\{ \\omega \\in \\racs{s} \\ \\left| \\ \\omega \\text{ is isochronous} \\right. \\right\\}.\n$$\n\\end{definition}\n\n\\begin{corollary}\\label{teorema-RI-variedad}\nThe subfamily $\\RI{s}$ is a $(3s-1)$--dimensional real analytic submanifold of $\\racs{s}$.\n\\end{corollary}\n\n\\begin{proof}\nThe result follows using the complex atlas by residues--poles $\\mathfrak{A}_{rp}$ and the $(s-1)$--dimensional real analytic submanifold $\\Im{H_s} := \\{(ir_1, \\ldots , ir_s) \\in H_s \\ | \\ r_{\\iota} \\in \\mathbb{R}^*\\}$ of $H_s$. \n\\end{proof}\n\n\\section{Classification of isotropy groups}\n\\label{Sec:action}\n\n\\subsection{The $PSL(2,\\mathbb{C})$-action}\n\nIn this section, \nwe prove that the natural holomorphic $PSL(2,\\mathbb{C})$--action on $\\racs{s}$, \ndefined as\n\\begin{equation}\\label{PSL-action}\n\t\\begin{array}{rcl}\n\t\t\\mathcal{A}_s: PSL(2,\\mathbb{C}) \\times \\racs{s} &\\longrightarrow& \\racs{s} \\\\\n\t\t(T, \\omega) &\\longmapsto& T_*\\omega,\n\t\\end{array}\n\\end{equation}\nis proper for $s\\geq 3$. \n\\begin{remark}\n\\begin{upshape}\nUsing the complex atlas by residues--poles $\\mathfrak{A}_{rp}$, \nthe expression for the action is \n$$\n\\mathcal{A}_s(T, \\rp{r_1, \\ldots r_s; p_1, \\ldots, p_s}) = \\rp{r_1, \\ldots, r_s ; T(p_1), \\ldots, T(p_s)}.\n$$\nThe residues are a set of $PSL(2,\\mathbb{C})$--invariant functions under the above action.\n\\end{upshape}\n\\end{remark}\nThe class of an $\\omega$ is denoted by $\\rp{\\rp{\\omega}} \\in \\racs{s}\/PSL(2,\\mathbb{C})$. \nRecall the definition of proper action as in \\cite[p.~53]{diustermaat}, \nwe have the next result. \n\\begin{lemma}\\label{proper-action}\n For $s \\geq 3$, the holomorphic (resp. real analytic) $PSL(2,\\mathbb{C})$--action $\\mathcal{A}_s$ on $\\racs{s}$ (resp. on $\\RI{s}$) is proper.\n\\end{lemma}\n\n\\begin{proof}\nWe will show that the map $\\tilde{\\mathcal{A}}_s : PSL(2,\\mathbb{C}) \\times \\racs{s} \\longrightarrow \\racs{s} \\times \\racs{s}$, \ndefined as $\\tilde{\\mathcal{A}}_s(T, \\omega) := (T_*\\omega, \\omega)$, \nis closed and the preimage for all points is a compact set. \nApplying Thm. 1 in \\cite[Sec.~\\textsection~10.2~p.~101]{bourbaki}, \nthe action $\\mathcal{A}_s$ is proper.\n\nFirst, we want to prove that the map $\\tilde{\\mathcal{A}}_s$ is closed. \nConsider a closed subset $C \\subset PSL(2,\\mathbb{C}) \\times \\racs{s}$ and a convergent sequence $\\{(\\eta_m, \\omega_m )\\} \\subset \\tilde{\\mathcal{A}}_s(C)$ with a limit point $(\\eta, \\omega) \\in \\racs{s} \\times \\racs{s}$. \nSince $(\\eta_m, \\omega_m) \\in \\tilde{\\mathcal{A}}_s(C)$, the sets of resdiues for $\\omega_m$ and $\\eta_m$ coincide. \nExplicity, for each $m$ we choose $\\eta_m =\\rp{r'_{m1}, \\ldots, r'_{ms}; q_{m1} ,\\ldots, q_{ms}}$ and $\\omega_m = \\rp{r_{m1}, \\ldots, r_{ms}; p_{m1} ,\\ldots, p_{ms}}$, \nwithout loss of generality $r'_{m\\iota}=r_{m\\iota}$; \nhere our assertions will work for all $\\iota =1, \\ldots, s$. \nIf $s \\geq 3$, then there exists a unique $T_m \\in PSL(2,\\mathbb{C})$ such that $T(p_{m\\iota})=q_{m\\iota}$. \nSince $C$ is closed and $(\\eta, \\omega)$ is the limit point of the sequence $\\{(\\eta_m, \\omega_m)\\}$, \nsay $\\eta = \\rp{r'_1, \\ldots, r'_s; q_1 ,\\ldots, q_s}$ and $\\omega = \\rp{r_1, \\ldots, r_s; p_1 ,\\ldots, p_s}$; \nit follows that there is a unique limit transformation $T \\in PSL(2,\\mathbb{C})$ with $T(p_{\\iota})=q_{\\iota}$ and $r'_{\\iota} = r_{\\iota}$, \nthus, the sequence $\\{T_{m}\\}$ converges to $T$. \nTherefore $(\\eta,\\omega) \\in \\tilde{\\mathcal{A}}_s(C)$, and the map $\\tilde{\\mathcal{A}}_s$ is closed.\n\nSecondly, \nwe prove that $\\mathcal{A}^{-1}_s(\\eta,\\omega)$ is a compact set. \nSince there are at most $s!$ permutations of the configuration of poles $\\{p_{\\iota}\\}$ to $\\{q_{\\iota}\\}$, \na configuration of poles with residues $\\rp{r_1, \\ldots, r_s; p_1, \\ldots, p_s}$ has at least two residues satisfying $r_{\\iota} \\not= r_j$; \nhence there are at most $(s-1)!$ admissible permutations of $\\rp{r_1, \\ldots, r_s; p_1, \\ldots, p_s}$ to $\\rp{r'_1,\\ldots, r'_s; q_1, \\ldots, q_s}$. \nIn fact, $\\tilde{\\mathcal{A}}_s^{-1}(\\eta,\\omega)$ is a finite set, \nhence compact in $PSL(2,\\mathbb{C}) \\times \\racs{s}$. \n\\end{proof}\n\n\\subsection{Nontrivial isotropy groups}\n\nFor $\\omega \\in \\racs{s}$, \nwe denote by\n\n\\centerline{$PSL(2,\\mathbb{C})_{\\omega} := \\{ T \\in PSL(2,\\mathbb{C}) \\ | \\ T_*\\omega = \\omega \\}$, } \n\n\\noindent its isotropy group. \n\n\\noindent A direct computation prove that $\\tilde{\\mathcal{A}}_s^{-1}(\\omega,\\omega) = PSL(2,\\mathbb{C})_{\\omega} \\times \\{\\omega \\}$ is a finite set when $s \\geq 3$; \nsee prove of Lemma \\ref{proper-action}. \nIn fact, \nevery $\\omega \\in \\racs{s}$ has finite isotropy group. \nA well--known result of F. Klein \\cite[p.~126]{klein20} classifies the finite subgroups of $PSL(2,\\mathbb{C})$; \nfor a modern reference see \\cite[Sec.~2.13]{singerman}. \nThese finite subgroups are cyclic $\\mathbb{Z}_n$, \ndihedral $D_n$ and the rotation groups $G(S)$ of platonic solids $S$; \n$A_4$ for tetrahedron, \n$\\Sim{4}$ for octahedron and cube, \nand $A_5$ for dodecahedron and icosahedron. \nA natural question is which finite subgroups of $PSL(2,\\mathbb{C})$ are realizable as isotropy groups of $\\omega \\in \\RI{s}$? \nThe answer is as follows.\n\\begin{proposition}\\label{realizacion-grupos-finitos}\n Every finite subgroup $G < PSL(2,\\mathbb{C})$ appears as the isotropy group of suitable $\\omega \\in \\RI{s}$.\n\\end{proposition}\n\n\\begin{proof}\nFixing $G < PSL(2,\\mathbb{C})$ finite subgroup, \nwe construct explicitly a rational 1--form $\\omega \\in \\RI{s}$ such that $PSL(2,\\mathbb{C})_{\\omega} \\cong G$. \nConsider $\\zeta_1, \\ldots, \\zeta_n$ the $nth$ roots of unity; $n \\geq 2$.\\\\\n\\textit{Case $G=\\mathbb{Z}_n$.} Recall that the rotation group of a pyramid with polygonal base and triangular faces is $\\mathbb{Z}_n$, \nwhere $n$ is the number of sides on the base. \nIn particular, \nthe set $\\{\\zeta_1, \\ldots, \\zeta_n, 0 \\}$ in the Riemann sphere are the vertices of a pyramid as above. \nIn fact, \nif \n$$\n\\omega = \\rp{\\underbrace{i, \\ldots, i}_{n}, -ni; \\zeta_1, \\ldots, \\zeta_n, 0} \\in \\RI{(n+1)},\n$$ \nthen its isotropy group is $PSL(2,\\mathbb{C})_{\\omega} \\cong \\mathbb{Z}_n$. \\\\ \n\\textit{Case $G=D_n$.} A bypyramid is a polihedron defined by two pyramids glued together by their basis. \nIf all their faces are isosceles triangles, \nthen its rotation group is $D_n$ where $n$ are the number of sides in the base for both pyramids. \nIn particular, \nthe set $\\{\\zeta_1, \\ldots, \\zeta_n, 0, \\infty \\}$ in the Riemann sphere are the vertices of a bypyramid as above. \nIn fact, \nif\n\\vspace{-0.2cm}\n$$\n\\omega=\\rp{\\underbrace{i,\\ldots, i}_{\\text{$n$}},-\\frac{n}{2}i, -\\frac{n}{2}i; \\zeta_1,\\ldots,\\zeta_n, 0, \\infty} \\in \\RI{(n+2)},\n$$\nthen its isotropy group is $PSL(2,\\mathbb{C})_{\\omega} \\cong D_n$.\\\\\n\\textit{Case $G=G(S)$.} We consider the union of the vertices of a platonic solid $S$ and its dual $S^{*}$ in the Riemann sphere. The suitable 1--form $\\omega$ has poles in both sets of vertices. \nThe choice of the residues is as follows, \nresidue $i$ at the vertices of $S$ and $-ki$ at the vertices of $S^{*}$, where\n$$\nk := \t\\left\\{\\begin{array}{ll}\n\t\t1 & \\text{for $S$ the tetrahedron}, \\\\\n \t\\vspace{-0.4cm} & \\\\\n\t\t{4}\/{3} \\hspace{1cm}& \\text{for $S$ the cube}, \\\\\n \t\\vspace{-0.4cm} & \\\\\n {3}\/{4} & \\text{for $S$ the octahedron}, \\\\\n\t\t\\vspace{-0.4cm} & \\\\\n\t\t{3}\/{5} & \\text{for $S$ the dodecahedron}, \\\\\n \t\\vspace{-0.4cm} & \\\\\n {5}\/{3} & \\text{for $S$ the icosahedron}, \\\\ \n\t\\end{array}\\right. \n$$\nwhence the isotropy group $PSL(2,\\mathbb{C})_{\\omega} \\cong G(S)$. \nConcrete examples are provided below. \n\\end{proof}\n\n\\begin{example}\n\\begin{upshape}\n1. For $\\epsilon_1, \\epsilon_2, \\epsilon_3$ roots of $z^{3} + 1 = 0$, the isotropy group of\n$$\n\\begin{array}{ll}\n\\displaystyle \\omega=\\rp{\\underbrace{i,\\ldots, i}_{\\text{4}},\\underbrace{-i,\\ldots, -i}_{\\text{4}}; \\frac{\\sqrt{2}}{2}\\zeta_1, \\frac{\\sqrt{2}}{2}\\zeta_2, \\frac{\\sqrt{2}}{2}\\zeta_3, \\infty, {\\sqrt{2}}{\\epsilon_1}, {\\sqrt{2}}{\\epsilon_2}, {\\sqrt{2}}{\\epsilon_3}, 0 } &\\\\\n&\\hspace{-1.2cm} \\in \\RI{8}\n\\end{array}\n$$\nis isomorphic to the rotation $A_4$ group of a tetrahedron.\n\n\\noindent 2. For $\\epsilon_{1}, \\epsilon_2, \\epsilon_3, \\epsilon_4$ roots of $z^4+1=0$ and $\\lambda={(\\sqrt{6} - \\sqrt{2})}\/{2}$, the isotropy group of \n$$\n\\begin{array}{ll}\n \\omega= \\left\\langle \\underbrace{i,\\ldots, i}_{\\text{8}}, \\underbrace{-\\frac{4}{3}i, \\ldots, -\\frac{4}{3}i}_{\\text{6}}; \\right. & \\\\\n&\\displaystyle \\hspace{-2.8cm} \\left. \\lambda, -\\lambda, i\\lambda, -i\\lambda, \\frac{1}{\\lambda}, -\\frac{1}{\\lambda}, \\frac{i}{\\lambda}, -\\frac{i}{\\lambda}, \\epsilon_1, \\epsilon_2, \\epsilon_3, \\epsilon_4, 0, \\infty \\right\\rangle \\in \\RI{14}\n\\end{array}\n$$\nis isomorphic to the rotation group $\\Sim{4}$ of a cube (octahedron).\n\\end{upshape}\n\\end{example}\n\n Obviously, the degree $s$ depends on the order of $G$ and the $1$--forms in the above proposition are isochronous. \nA. Solynin \\cite{solynin} constructs quadratic differentials on compact Riemann surface $\\mathcal{R}$ associated with a weight graph embedded in $\\mathcal{R}$. \nHe explicitly gives quadratic differentials, with zeros in the vertices of a platonic solid and poles in the center of each face. \nThey are different from our 1--forms. \nA. Alvarez--Parrila, M. E. Fr\\'ias--Armenta and C. Yee--Romero \\cite{alvaro3} classify the isotropy groups of rational 1--forms on the Riemann sphere using the complex atlas by zeros--poles. \n\nFor $2 \\leq s \\leq 11$, we classify the isotropy groups $PSL(2,\\mathbb{C})_{\\omega}$. These results will help in Section \\ref{S-examplesquotient}.\n\n\\begin{example}\\label{ejemplo-S2}\n\\begin{upshape}\n1. For all $\\omega= \\rp{r, -r; p_1, p_2} \\in \\racs{2}$, \nthe isotropy group is $PSL(2,\\mathbb{C})_{\\omega} \\cong \\mathbb{C}^* \\cong \\{T(z) = az \\}$.\n\n\\noindent 2. For $\\omega \\in \\racs{3}$, the isotropy group is $PSL(2,\\mathbb{C})_{\\omega}\\cong \\mathbb{Z}_2$ if and only if $\\omega= \\rp{r_1, r_1, r_2; p_1, p_2, p_3}$, \\textit{i.e.} two residues are equal.\n\\end{upshape}\n\\end{example}\n\n\\begin{lemma}\\label{isotropia-residuos-4}\n\tConsider $\\omega=\\rp{r_1,r_2 ,r_3, r_4; p_1, p_2, p_3, p_4} \\in \\racs{4}$.\\\\\n\\noindent 1. If $\\omega$ has exactly two equal residues and the cross--ratio\\footnote{The cross--ratio is defined as $(p_1, p_2, p_3, p_4) := \\frac{(p_4-p_1)(p_3-p_2)}{(p_4-p_2)(p_3-p_1)}.$} \n\\begin{itemize}\n\\item[] $(p_1, p_2, p_3, p_4) \\in \\{-1, \\frac{1}{2}, 2\\}$, then $PSL(2,\\mathbb{C})_{\\omega} \\cong \\mathbb{Z}_2$.\n\\end{itemize}\n\\noindent 2. If $\\omega$ has two pairs of equal residues and\n\t\t\t\\begin{itemize}\n\t\t\t\t\\item[] $(p_1, p_2, p_3, p_4) \\not\\in \\{-1, \\frac{1}{2}, 2\\}$, then $PSL(2,\\mathbb{C})_{\\omega} \\cong \\mathbb{Z}_2$,\n\t\t\t\t\\item[] $(p_1, p_2, p_3, p_4) \\in \\{-1, \\frac{1}{2}, 2\\}$, then $PSL(2,\\mathbb{C})_{\\omega} \\cong \\mathbb{Z}_2 \\times \\mathbb{Z}_2 \\cong D_2$.\n\t\t\t\\end{itemize}\n\\noindent 3. If $\\omega$ has three equal residues and \n\\begin{itemize}\n\\item[] $(p_1, p_2, p_3, p_4) \\in \\left\\{ (1 \\pm i \\sqrt{3})\/ 2 \\right\\}$, then $PSL(2,\\mathbb{C})_{\\omega} \\cong \\mathbb{Z}_3$.\n\\end{itemize}\n\\noindent 4. For any other case the isotropy group $PSL(2,\\mathbb{C})_{\\omega}$ is trivial.\n\\end{lemma}\n\n\\begin{proof}\n\\noindent Case 1. \nConsider $\\omega = \\rp{r_1, r_2, r_3, r_4; p_1, p_2, p_3, p_4} \\in \\racs{4}$ with $r_1=r_2$, $r_3\\not=r_4$ and $(p_1, p_2, p_3, p_4)=-1$. \nWe can verify that $(p_1, p_2, p_3, p_4)=(p_2, p_1, p_3, p_4)$, \nand there is a nontrivial $T \\in PSL(2,\\mathbb{C})$ such that $T_{*}\\omega = \\left\\langle r_1, r_2, r_3, \\right.$ $\\left. r_4; p_2, p_1, p_3, p_4 \\right\\rangle = \\omega$. \nIn fact, \n$T \\in PSL(2,\\mathbb{C})_{\\omega}$. \nSince $r_3\\not= r_4$, \nthere are no more elements in the isotropy; \ntherefore $PSL(2,\\mathbb{C})_{\\omega} \\cong \\mathbb{Z}_2$. \nThe result is analogous for\n$$\n\\begin{array}{cccccc}\nr_1 \\not=r_2, & r_3=r_4, & \\lambda=-1, & r_1 = r_4, & r_2\\not=r_3, & \\lambda = 2, \\\\\nr_1 = r_3, & r_2\\not=r_4, & \\lambda = {1}\/{2}, & r_1 \\not= r_4, & r_2=r_3, & \\lambda = 2,\\\\\nr_1 \\not= r_3, & r_2=r_4, & \\lambda = {1}\/{2}.& & & \\\\\n\\end{array}\n$$\nWe leave the reader to perform the other cases. \n\\end{proof}\n\nObviously, for $s \\geq 5$ the specific conditions to determine the isotropy groups are more complicated.\n\n\\begin{example}{\\label{isotropia-residuos-5}}\n\\begin{upshape}\n\tFor $\\omega \\in \\racs{5}$, the nontrivial isotropy groups $PSL(2,\\mathbb{C})_{\\omega}$ are isomorphic to $\\mathbb{Z}_{2}, \\mathbb{Z}_{3}, \\mathbb{Z}_{4}$ or $D_3$.\n\nLet us explictly describe it. \nIf $\\omega=\\rp{r_1,\\ldots,r_5; p_1,\\ldots,p_5}$ has nontrivial isotropy group, \nthen there are at least two pairs of equal residues or three equal residues. \\\\\nCase \\textit{$r_1 = r_2$ and $r_3=r_4$}. \nSince $r_5$ is different from the other residues, \nthe pole $p_5$ is a fixed point in the action of $PSL(2,\\mathbb{C})_{\\omega}$ on $\\ensuremath{{\\widehat{\\mathbb C}}}$. \nIn fact, \nthe isotropy group is cyclic.\nIf $r_1 \\not= r_3$, \nthen $PSL(2,\\mathbb{C})_{\\omega} \\cong \\mathbb{Z}_2$. \nIf $r_1 = r_3$, \nthen $PSL(2,\\mathbb{C})_{\\omega} \\cong \\mathbb{Z}_2, \\mathbb{Z}_3, \\text{ or } \\mathbb{Z}_4$. \\\\\nCase \\textit{$r_1=r_2=r_3$}. \nWe suppose that $PSL(2,\\mathbb{C})_{\\omega}$ is not isomorphic to a cyclic group. \nSince $PSL(2,\\mathbb{C})_{\\omega}$ is nontrivial, \n$r_4=r_5$, and $\\{p_4, p_5 \\}$ is a orbit of order 2 in the action of $PSL(2,\\mathbb{C})_{\\omega}$ on $\\ensuremath{{\\widehat{\\mathbb C}}}$. \nIn fact, the isotropy group is dihedral.\nSince $r_1=r_2=r_3$, \nthe isotropy group $PSL(2,\\mathbb{C})_{\\omega} \\cong D_3$.\n\\end{upshape}\n\\end{example}\n\n\nNumerical conditions on $s \\geq 3$, \nto realize $G < PSL(2,\\mathbb{C})$ as an isotropy group for some $\\omega \\in \\racs{s}$, \nare as follow. \n \n\\begin{proposition}\n\\label{prop-numerica}\nConsider $n \\geq 2$ and $n_1, n_2 \\in \\mathbb{N}\\cup\\{ 0 \\}$ such that $n_1 + n_2 \\geq 2$. \nThere exists $\\omega \\in \\racs{s}$ such that\n\\begin{enumerate}\n\\item $PSL(2,\\mathbb{C})_{\\omega} \\cong \\mathbb{Z}_n$ if and only if $s \\equiv 0, 1 \\text{ or } 2 \\ (\\text{mod } n)$, where $s > n$. \n\n\\item $PSL(2,\\mathbb{C})_{\\omega} \\cong D_n$ if and only if $s \\equiv 0 \\text{ or } 2 \\ (\\text{mod } n)$, where $s > n$. \n\n\\item $PSL(2,\\mathbb{C})_{\\omega} \\cong A_4$ if and only if $s = 12n_1 + n_2$, where $n_2 \\in \\{ 0, 8, 10, 14, 16, 18 \\}$.\n\n\\item $PSL(2,\\mathbb{C})_{\\omega} \\cong \\Sim{4}$ if and only if $s = 24n_1 + n_2$, where $n_2 \\in \\left\\{ 0, 14, 18, 20, 26, \\right.$ $\\left. 30,32, 36 \\right\\}$.\n\n\\item $PSL(2,\\mathbb{C})_{\\omega} \\cong A_5$ if and only if $s = 60n_1 + n_2$, where $n_2 \\in \\left\\{ 0, 32, 42, 50, 62, \\right.$ $\\left. 72, 80, 90 \\right\\}$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nCase 1. \nConsider $\\eta = \\rp{-ni, i, \\ldots, i; \\infty, \\zeta_1, \\ldots, \\zeta_n}$, \nwhere $\\zeta_{\\iota}$ are the $nth$ roots of unity. \nObviously, its isotropy group $PSL(2,\\mathbb{C})_{\\eta} = \\{e^{2k\\pi i \/ n}z\\}\\cong \\mathbb{Z}_n$. \nFor $\\omega \\in \\racs{s}$, \nif $PSL(2,\\mathbb{C})_{\\omega} \\cong \\mathbb{Z}_n$ then there is $T \\in PSL(2,\\mathbb{C})$ such that $PSL(2,\\mathbb{C})_{T_*\\omega} = T \\cdot PSL(2,\\mathbb{C})_{\\omega} \\cdot T^{-1} = PSL(2,\\mathbb{C})_{\\eta}$; \nsee \\cite[p.~107]{diustermaat} and \\cite[p.~44]{singerman}. \nIt is easy to see that for $p_{\\iota} \\in \\ensuremath{{\\widehat{\\mathbb C}}}$ pole of $T_{*}\\omega$, \nits orbit $PSL(2,\\mathbb{C})_{T_*\\omega} \\cdot p_{\\iota}$, \nunder the action of $PSL(2,\\mathbb{C})_{T_*\\omega}$ on $\\ensuremath{{\\widehat{\\mathbb C}}}$, \nis a set of poles for $T_*\\omega$. \nIn other words, \nif $\\ell$ is the number of poles $p_{\\iota}$ with different orbits, \nthen \n$$\ns = \\# \\{\\text{poles of }\\omega\\} = \\# \\{\\text{poles of }T_*\\omega\\} = \\sum_{\\iota =1}^{\\ell} \\#(PSL(2,\\mathbb{C})_{T_*\\omega} \\cdot p_{\\iota}).\n$$\nSince $\\#(PSL(2,\\mathbb{C})_{T_*\\omega} \\cdot p_{\\iota}) = n$, for $p_{\\iota} \\not= 0$ or $\\infty$ and $\\#(PSL(2,\\mathbb{C})_{T_*\\omega} \\cdot 0) = \\#(PSL(2,\\mathbb{C})_{T_*\\omega} \\cdot \\infty ) = 1$, \nthe result is proved. \nThe other cases are analogous. \n\\end{proof}\nUsing Proposition \\ref{prop-numerica}, \nwe complete the Table \\ref{tab:1}.\n\n\\begin{table}[h]\n\\caption{Finite subgroups of $PSL(2,\\mathbb{C})$ that appear as isotropy for $\\omega \\in \\racs{s}$.}\n\\label{tab:1} \n\\begin{tabular}{ll}\n\\hline\\noalign{\\smallskip}\n$s$ & Nontrivial isotropy groups for $\\omega \\in \\racs{s}$ \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n3 & $\\mathbb{Z}_2$ \\\\ \n4 & $\\mathbb{Z}_2, \\mathbb{Z}_3, \\mathbb{Z}_2\\times \\mathbb{Z}_2$ \\\\\n5 & $\\mathbb{Z}_2, \\mathbb{Z}_3, \\mathbb{Z}_4, D_3$ \\\\ \n6 & $\\mathbb{Z}_2, \\mathbb{Z}_3, \\mathbb{Z}_4, \\mathbb{Z}_5, \\mathbb{Z}_2 \\times \\mathbb{Z}_2, D_3, D_4$ \\\\ \n7 & $\\mathbb{Z}_2, \\mathbb{Z}_3, \\mathbb{Z}_5, \\mathbb{Z}_6, D_5$ \\\\\n8 & $\\mathbb{Z}_2, \\mathbb{Z}_3, \\mathbb{Z}_4, \\mathbb{Z}_6, \\mathbb{Z}_7, \\mathbb{Z}_2 \\times \\mathbb{Z}_2, D_3, D_4, D_6, A_4$ \\\\ \n9 & $\\mathbb{Z}_2, \\mathbb{Z}_3, \\mathbb{Z}_4, \\mathbb{Z}_7, \\mathbb{Z}_8, D_3, D_7$ \\\\ \n10 & $\\mathbb{Z}_2, \\mathbb{Z}_3, \\mathbb{Z}_4, \\mathbb{Z}_5, \\mathbb{Z}_8, \\mathbb{Z}_9, \\mathbb{Z}_2 \\times \\mathbb{Z}_2, D_4, D_5, D_8$ \\\\ \n11 & $\\mathbb{Z}_2, \\mathbb{Z}_3, \\mathbb{Z}_5, \\mathbb{Z}_9, \\mathbb{Z}_{10}, D_3, D_9$ \\\\ \n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table}\n\n\\section{Quotients}\n\\label{S-examplesquotient}\n\n\\subsection{Stratification by orbit types}\n\nFor $s\\geq 3$, \nthe $PSL(2,\\mathbb{C})$--action is proper, \nand the classical theory of Lie groups can be applied. \nMainly, we follow the theory and notation of J. J. Duistermaat and J. A. Kolk in \\cite{diustermaat}. \nIn order to describe the quotients $\\racs{s}\/PSL(2,\\mathbb{C})$ and $\\RI{s}\/PSL(2,\\mathbb{C})$, \nrecall that if a Lie group $G$ acts properly on a manifold $M$, \nthen every closed subgroup $H$ of $G$ acts in a proper and free way on $G$; \nsee \\cite[p.~93]{diustermaat}. \nMoreover, \nthe right coset $G\/G_x$ is a manifold of dimension $dim(G) - dim(G_x)$ diffeomorphic to the orbit $G\\cdot x$. \\\\\nIn our case, \nsince the $PSL(2,\\mathbb{C})$--action is proper and all isotropy groups are finites, \nthe orbits $PSL(2,\\mathbb{C}) \\cdot \\omega$ under $\\mathcal{A}_s$ are 3--dimensional complex submanifolds of $\\racs{s}$, \nbiholomorphic to the right coset ${PSL(2,\\mathbb{C})}\/{\\ PSL(2,\\mathbb{C})_{\\omega}}$. \nSimilarly, \nfor $\\omega \\in \\RI{s}$ its orbit $PSL(2,\\mathbb{C}) \\cdot \\omega$ is a 6--dimensional real analytic submanifold of $\\RI{s}$, \nand $PSL(2,\\mathbb{C}) \\cdot \\omega$ is diffeomorphic to the right coset ${PSL(2,\\mathbb{C})}\/{\\ PSL(2,\\mathbb{C})_{\\omega}}$. \\\\\nFurthermore, \nby applying Theorem 2.7.4 in \\cite{diustermaat} the quotients $\\racs{s}\/PSL(2,\\mathbb{C})$ and ${\\RI{s}}\/{PSL(2,\\mathbb{C})}$ admit a stratification by orbit types. \nThe action $\\mathcal{A}_s$ is proper and free in the generic open and dense subset\n\\begin{equation}\\label{generic-forms}\n\\mathcal{G}(-1^s) := \\{\\omega \\in \\racs{s} \\ | \\ PSL(2,\\mathbb{C})_{\\omega} \\cong \\{Id \\} \\}.\n\\end{equation}\n\\noindent The quotient $\\mathcal{E}(-1^s) := {\\mathcal{G}(-1^s)}\/{PSL(2,\\mathbb{C})}$ is a $(2s-4)$--dimensional complex manifold. \nBy following \\cite[p.~107]{diustermaat}, \nfor each $\\omega \\in \\racs{s}$ its orbit type is\n$$\n\\racs{s}^{\\sim}_{\\omega} := \\left\\{ \\eta \\in \\racs{s} \\ | PSL(2,\\mathbb{C})_{\\eta} \\cong PSL(2,\\mathbb{C})_{\\omega} \\right\\}.\n$$\nNote that in our case the isotropy groups are isomorphic in $\\racs{s}^{\\sim}_{\\omega}$ instead of conjugates since for finites subgroups of $PSL(2,\\mathbb{C})$, they are equivalents; see \\cite[p.~50]{singerman}.\nSimilarly, its orbit type on the quotient is $\\racs{s}^{\\sim}_{\\omega}\/PSL(2,\\mathbb{C})$. \nLooking at $\\racs{s}^{\\sim}_{\\omega}$, its connected components $\\{E_j\\}$ are the stratum and they are complex submanifolds of $\\racs{s}$ with dimension $dim(E_j) \\leq 2s-1$. \nThe higher dimensional stratum is $\\mathcal{G}(-1^s)$. \nFor the quotient, \nthe connected components of $\\racs{s}^{\\sim}_{\\omega} \/ PSL(2,\\mathbb{C}) $ are the stratum and they are complex manifolds with dimension less or equal to $2s-4$. \nThe higher dimensional stratum is $\\mathcal{E}(-1^s)$. \n\n\\begin{remark}\nThere exists a holomorphic principal $PSL(2,\\mathbb{C})$--bundle\n$$\n\\xymatrix{PSL(2,\\mathbb{C}) \\ar@{->}[r] & \\mathcal{G}(-1^s) \\ar@{->}^-{\\pi_s}[d] \\\\\n & \\mathcal{E}(-1^s) \\ ,}\n$$\nwhere $\\pi_s$ denotes the natural projection to the $PSL(2,\\mathbb{C})$--orbits. \n\\end{remark}\n\nFor $\\RI{s}$, the generic open and dense real analytic submanifold is\n$$\n\\mathcal{RIG}(-1^s) := \\{\\omega \\in \\RI{s} \\ | \\ PSL(2,\\mathbb{C})_{\\omega} \\cong \\{Id \\} \\}.\n$$\nThe quotient $\\mathcal{RIE}(-1^s) := \\mathcal{RIG}(-1^s)\/PSL(2,\\mathbb{C})$ is a $(3s-7)$--dimensional real analytic manifold. \nFor $\\RI{s}$ and $\\RI{s}\/PSL(2,\\mathbb{C})$, their stratification by orbit types are analogous as for $\\racs{s}$ and $\\racs{s}\/PSL(2,\\mathbb{C})$, respectively. \\\\\n\n\\subsection{Realizations}\n\nLet us define a realization\\footnote{We use definition of realization as in \\cite[p.~6]{yoshida}.} for the quotient $\\racs{s}\/PSL(2,\\mathbb{C})$ by using a complete set of $PSL(2,\\mathbb{C})$--invariant functions. \nFirst, we consider the ordered set of residues as the complement of an arrangement of $s$ hyperplanes\n$$\n\\mathbb{A}_s = \\mathbb{C}_{(r_1, \\ldots, r_{s-1})}^{s-1} \\setminus \\left\\{r_1+\\ldots + r_{s-1}=0, \\ r_{\\iota}=0 \\ \\iota=1,\\ldots, s-1 \\right\\}.\n$$\nFor $s=2,3$, \nthe residues are a complete set of $PSL(2,\\mathbb{C})$--invariant functions.\n\n\\begin{example}\n\\begin{upshape}\nCase $s=2$, \nthe natural projection \n\n\\centerline{$\\pi_2 : \\racs{2} \\longrightarrow {\\racs{2}}\/{PSL(2,\\mathbb{C})}: \\ \\rp{r_1, r_2; p_1, p_2} \\longmapsto r_1$}\n\n\\noindent determines a fiber bundle. \nObviously, the base space is biholomorphic to $\\mathbb{C}^*\/\\mathbb{Z}_2$. \nSimilarly, the quotient $\\RI{2}\/PSL(2,\\mathbb{C})$ is diffeomorphic to $\\mathbb{R}^{+} = \\{r_1 \\ | \\ r_1 > 0 \\}$. \nFor both cases, the fibers are $\\ensuremath{{\\widehat{\\mathbb C}}}^2 \\setminus \\Delta = \\{(p_1, p_2) \\ | \\ p_1 \\not= p_2 \\}$. \n\\end{upshape}\n\\end{example}\n\n\\begin{example}\\label{cociente-3}\n\\begin{upshape}\n\tCase $s=3$, using Example \\ref{ejemplo-S2}.2 the quotient $\\racs{3}\/PSL(2,\\mathbb{C})$ admits a stratification with two orbit types. \nIt is homemorphic to $\\mathbb{A}_3\/\\Sim{3}$, where the symetric group $\\Sim{3}$ acts linearly on $\\mathbb{A}_3$ using the isomorphism \n$$\n\\Sim{3} \\cong \\rp{\\left(\\begin{smallmatrix}\n 0 & 1 \\\\\n 1 & 0\n \\end{smallmatrix}\\right), \\left(\\begin{smallmatrix}\n \\ \\ 1 & \\ \\ 0 \\\\\n -1 & -1 \n \\end{smallmatrix}\\right)}.\n$$\nSimilarly, the quotient ${\\RI{3}}\/{PSL(2,\\mathbb{C})}$ has two connected components. \nA fundamental domain is\n$$\n\\{ (r_1, r_2) \\ | \\ r_1r_2 >0 \\text{ and } r_1 \\leq r_2 \\} \\subset \\mathbb{A}_3.\n$$\nHere $(r_1,r_2)$ determines the 1--form $\\rp{ir_1, ir_2, -i(r_1+r_2); 0, \\infty, 1}$. \nTheir connected components come from $\\{r_1 > 0\\}$ and $\\{r_1 < 0\\}$. \nThe orbit types are $\\{r_1=r_2\\}$ and $\\{r_1 < r_2\\}$. \nSince the connected components are contractibles, \nthe corresponding principal $PSL(2,\\mathbb{C})$--bundle $\\pi_3: \\mathcal{RIG}(-3) \\longrightarrow \\mathcal{RIE}(-3)$ is trivial.\n\\end{upshape}\n\\end{example}\n\nFor $s \\geq 4$, the residues are not a complete set of $PSL(2,\\mathbb{C})$--invariant functions. \nIn order to enlarge our set, \nwe fix three poles in $\\{0, \\infty, 1\\}$ and consider the ordered set of poles with \n$$\n\\left[ \\mathbb{C}^* \\setminus \\{ 1 \\}\\right]^{s-3}\\setminus \\Delta := \\left. \\left\\{(p_4, \\ldots, p_s) \\in \\left[ \\mathbb{C}^* \\setminus \\{ 1 \\}\\right]^{s-3} \\ \\right| \\ p_{\\iota} \\not= p_{\\kappa} \\text{ for } \\iota \\not= \\kappa \\right\\} . \n$$\n\nGiven a configuration $\\{ q_1, \\ldots, q_s \\} \\subset \\ensuremath{{\\widehat{\\mathbb C}}}$, \nthere exist $\\left(\\begin{smallmatrix} s \\\\ s-3 \\end{smallmatrix}\\right)3!$ M\\\"obius transformations $T \\in PSL(2,\\mathbb{C})$ such that $\\{T(q_1), \\ldots, T(q_s) \\} = \\{0, \\infty, 1, p_4, \\ldots, p_s \\}$.\n\n\\begin{remark}\nFor an ordered collection\n\n\\centerline{$(r_1, \\ldots, r_{s-1},p_4, \\ldots, p_s) \\in \\mathbb{A}_s \\times \\left[ \\mathbb{C}^* \\setminus \\{ 1 \\}\\right]^{s-3}\\setminus \\Delta : = \\mathcal{M}(-s)$,}\n\n\\noindent and each permutation $\\sigma \\in \\Sim{s}$, there exists a unique $T_{\\sigma} \\in PSL(2,\\mathbb{C})$ that \n$$\n\\begin{array}{rl}\n(r_1, \\ldots, r_s, p_4, \\ldots, p_s) & \\longmapsto \\\\\n&\\vspace{-0.2cm}\\\\\n&\\hspace{-2cm} \\left\\{(r_{\\sigma(1)}, 0), (r_{\\sigma(2)}, \\infty), (r_{\\sigma(3)}, 1), (r_{\\sigma(4)}, T_{\\sigma}(p_4)), \\ldots, (r_{\\sigma(s)}, T_{\\sigma}(p_s))\\right\\} \\\\\n&\\vspace{-0.2cm}\\\\\n&\\hspace{-2cm} := \\rp{\\rp{ r_1, \\ldots, r_s; 0, \\infty, 1, p_4, \\ldots, p_s }} \\in \\racs{s}\/PSL(2,\\mathbb{C}).\n\\end{array}\n$$ \n\\end{remark}\n\nNote the appearance of $r_s= -(r_1+ \\ldots + r_{s-1})$ and $0, \\infty, 1$ on the right side. \nThere is a natural $\\Sim{s}$--action on $\\mathcal{M}(-s)$. \nIn order to recognize it, \nwe define a group representation in the Coexeter generators of $\\Sim{s}$, \nsee \\cite[Sec.~1.2]{coxeter}, \nas\n$$\n\\begin{array}{rcl}\n\\rho_s : \\Sim{s} & \\longrightarrow & GL_{s-1}( \\mathbb{Z}) \\times Bir(\\ensuremath{{\\widehat{\\mathbb C}}}^{s-3}) \\\\\n&&\\vspace{-0.2cm}\\\\\n\\sigma_j = (j \\ j+1) & \\longmapsto & {(A_j, f_j) = \\left\\{ \\begin{array}{ll}\n \\left(A_1, \\left( \\frac{1}{z_4}, \\ldots, \\frac{1}{z_s} \\right) \\right) &\\\\\n&\\vspace{-0.2cm}\\\\\n \\left(A_2, \\left( \\frac{z_4}{z_4-1}, \\ldots, \\frac{z_s}{z_s-1} \\right) \\right) &\\\\\n& \\vspace{-0.2cm}\\\\\n \\left(A_3, \\left( \\frac{1}{z_4}, \\frac{z_5}{z_4}, \\ldots, \\frac{z_s}{z_4} \\right) \\right) &\\\\\n& \\vspace{-0.2cm}\\\\\n \\left(A_{j}, \\left( z_{\\sigma_{\\iota}(4)}, \\ldots, z_{\\sigma_{j}(s)}\\right) \\right),&\\\\\n&\\hspace{-1.2cm} \\text{where } j = 4, \\ldots, s-1.\n \\end{array} \\right. }\n\\end{array}\n$$\nHere $Bir(\\ensuremath{{\\widehat{\\mathbb C}}}^{s-3})$ denotes the group of complex birational maps on $\\ensuremath{{\\widehat{\\mathbb C}}}^{s-3}$; \nthe birational map $f_1$ from $\\sigma_1$ must be understood as $f_{1}: (z_4, \\ldots, z_s) \\longmapsto ( {1}\/{z_4}, \\ldots, {1}\/{z_s} )$. \nSince there is a biholomorphism between the Torelli space of the $s$--punctured sphere and $[\\mathbb{C}^* \\setminus\\{1 \\}]^{s-3} \\setminus \\Delta$, \nthe subgroup of birational maps $\\{ f_{\\sigma} \\}$ is the corresponding Torelli modular group; \nsee \\cite{patterson}. \\\\\nFor $j=1, \\ldots, s-2$, the matrices $A_j$ come from the identity matrix by exchanging the $j\\text{th}$--row with the $(j+1)\\text{th}$--row; \nfor $j=s-1$, $A_{s-1}$ results from replacing, \nin the identity matrix, \nthe $(s-1){\\text{th}}$--row with $(-1, \\ldots, -1)$. \\\\\nIt is a straighforward computation that $\\{\\rho_s(\\sigma_{j})\\}$ satisfy the relations in Coxeter's presentation. \nBy using $\\rho_s$, we define a $\\Sim{s}$--action on $\\mathcal{M}(-s)$ as\n\\begin{equation}\\label{S-action}\n\\begin{array}{rcl}\n\\Sim{s} \\times \\mathcal{M}(-s) & \\longrightarrow & \\mathcal{M}(-s)\\\\\n(\\sigma, (r_1, \\ldots, r_{s-1}, p_4, \\ldots, p_s)) & \\longmapsto & \\left({A_{\\sigma}}\\left(\\begin{smallmatrix}\n r_1 \\\\\n \\vdots \\\\\n r_{s-1}\n \\end{smallmatrix}\\right),\\ f_{\\sigma}(p_4, \\ldots, p_s)\\right).\n\\end{array}\n\\end{equation}\nIn order to recognize the quotient $\\racs{s}\/PSL(2,\\mathbb{C})$, the map\n$$\n\\begin{array}{rcl}\n\\mu_s: \\mathcal{M}(-s) & \\longrightarrow & \\racs{s}\\\\\n(r_1, \\ldots, r_{s-1}, p_4, \\ldots, p_s) & \\longmapsto & \\rp{r_1, \\ldots, r_{s}; 0, \\infty, 1, p_4, \\ldots, p_s} = \\omega,\n\\end{array}\n$$\nwill be useful. \nThe number of preimages $ \\mu_s^{-1}(\\omega)$ is $(s-3)!$ \nFurthermore, \nthe number of preimages $(\\pi_s \\circ \\mu_s)^{-1}\\rp{\\rp{\\omega}}$ is less than or equal to $s!$ and the equality is fulfilled when $\\omega \\in \\mathcal{G}(-1^s)$; \nrecall \\eqref{generic-forms}.\n\n\\begin{proposition}\\label{quotient}\n\tFor $s\\geq 4$, the realization of the quotient $\\racs{s}\/PSL(2,\\mathbb{C})$ is $\\mathcal{M}(-s)\/\\Sim{s}$.\n\\end{proposition}\n\n\\begin{proof}\nLet us prove that $\\pi_s \\circ \\mu_s$ is a $\\Sim{s}$--equivariant map, \\textit{i. e.} \n\n\\centerline{$(\\pi_s \\circ \\mu_s)(\\sigma \\cdot (r_1, \\ldots, r_{s-1}, p_4, \\ldots, p_s)) = (\\pi_s \\circ \\mu_s)(r_1, \\ldots, r_{s-1}, p_4, \\ldots, p_s)$,}\n\n\\noindent for all $\\sigma \\in \\Sim{s}$. \nFor example, consider $\\sigma_{1} = (1 \\ 2) \\in \\Sim{s}$, the explicit calculation is\n$$\n\\begin{array}{rl}\n(\\pi_s \\circ \\mu_s)(\\sigma_1 \\cdot (r_1, \\ldots, r_{s-1}, p_4, \\ldots, p_s)) &\\\\\n&\\hspace{-1.7cm}= (\\pi_s \\circ \\mu_s)(r_2, r_1, r_3, \\ldots, r_{s-1}, 1\/p_4, \\ldots, 1\/p_s)\\\\\n&\\hspace{-1.7cm}= \\rp{\\rp{ r_2, r_1, r_3, \\ldots, r_s;0, \\infty, 1, 1\/p_4, \\ldots, 1\/p_s }} \\\\\n&\\hspace{-1.7cm}= \\rp{\\rp{(1\/z)_*\\rp{r_2, r_1, r_3, \\ldots, r_s; \\infty, 0, 1, p_4, \\ldots, p_s} }}\\\\\n&\\hspace{-1.7cm}= \\rp{\\rp{r_1, \\ldots, r_s; 0, \\infty, 1, p_4, \\ldots, p_s }}\\\\\n&\\hspace{-1.7cm}= (\\pi_s \\circ \\mu_s)(r_1, \\ldots, r_{s-1}, p_4, \\ldots, p_s).\n\\end{array}\n$$\nOn the other hand, \n$(\\pi_s \\circ \\mu_s)$ is surjective. \nTherefore, \nthere exists a homeomorphism $\\tilde{\\mu}_s: \\mathcal{M}(-s)\/\\Sim{s} \\longrightarrow \\racs{s}\/PSL(2,\\mathbb{C})$ such that the diagram below conmutes. \n$$\n\\xymatrix{& \\mathcal{M}(-s) \\ar@{->}[d] \\ar@{->}^-{\\pi_s \\circ \\mu_s}[d] \\ar@{->}[dl]\\\\\n \\frac{\\mathcal{M}(-s)}{\\Sim{s}} \\ar@{-->}_-{\\tilde{\\mu}_s}[r]& \\frac{\\racs{s}}{PSL(2,\\mathbb{C})}.}\n$$\n\n\\end{proof}\n\nSimilarly, we define $\\Im{\\mathbb{A}_s} := \\{(ir_1, \\ldots, ir_{s-1}) \\in \\mathbb{A}_s \\ | \\ r_{\\iota} \\in \\mathbb{R}^*, \\ \\iota=1,\\ldots, s-1 \\}$. \nSince the $\\Sim{s}$--action \\eqref{S-action} is well--defined on $\\Im{\\mathcal{M}(-s)} := \\Im{\\mathbb{A}_s} \\times [\\mathbb{C}^*\\setminus \\{1\\}]^{s-3} \\setminus \\Delta$, \nThe result below was proved.\n\\begin{corollary}\\label{quotient2}\n\tFor $s\\geq 4$, the realization of the quotient $\\RI{s}\/PSL(2,\\mathbb{C})$ is $\\Im{\\mathcal{M}(-s)}\/\\Sim{s}$.\n\\end{corollary}\nFor $s=4$, \nthe number of connected components of $\\Im{\\mathcal{M}(-4)}$ depends only on the number of connected components of $\\Im{\\mathbb{A}_4}$. \nIn fact, $\\Im{\\mathcal{M}(-4)}$ has 14 connected components,\n$$\n\t\\begin{array}{lcl}\n\t\tX_{j}^{+} &:=& \\left\\{(ir_1,ir_2,ir_3, p_4) \\in \\Im{\\mathcal{M}(-4)} \\ \\left| \\ r_4 > 0, \\ r_{j} > 0, \\ r_{\\iota} < 0 \\ \\iota \\not= j\\right\\}\\right. ,\\\\\n\t\tX_{j}^{-} &:=& \\left\\{(ir_1,ir_2,ir_3, p_4) \\in \\Im{\\mathcal{M}(-4)} \\ \\left| \\ r_4 < 0, \\ r_{j} > 0, \\ r_{\\iota} < 0 \\ \\iota \\not= j\n\\right\\}\\right. ,\\\\\n\t\tX_{j_1j_2}^{+} &:=& \\left\\{(ir_1,ir_2,ir_3, p_4) \\in \\Im{\\mathcal{M}(-4)} \\ \\left| \\ r_4 > 0, \\ r_{j_1} > 0, r_{j_2} > 0 \\ r_{j_3} < 0 \\right\\}\\right. ,\\\\\n\t\tX_{j_1j_2}^{-} &:=& \\left\\{(ir_1,ir_2,ir_3, p_4) \\in \\Im{\\mathcal{M}(-4)} \\ \\left| \\ r_4 < 0, \\ r_{j_1} > 0, r_{j_2} > 0 \\ r_{j_3} < 0 \\right\\}\\right. ,\\\\\n\t\tX_{+} &:=& \\left\\{(ir_1, ir_2,ir_3, p_4) \\in \\Im{\\mathcal{M}(-4)} \\ | \\ r_j > 0, \\ \\ j=1,2,3\\right\\} ,\\\\\n\t\tX_{-} &:=& \\left\\{(ir_1, ir_2,ir_3, p_4) \\in \\Im{\\mathcal{M}(-4)} \\ | \\ r_j < 0, \\ \\ j=1,2,3\\right\\} .\\\\\n\t\\end{array}\n$$\nBy applying the $\\Sim{4}$--action \\eqref{S-action}, \nthese components are identified as\n$$\n\\begin{array}{c}\nX_+ \\sim X^{+}_{23} \\sim X^{+}_{13} \\sim X^{+}_{12}, \\\\\nX_- \\sim X^{-}_{1} \\sim X^{-}_{2} \\sim X^{-}_{3}, \\\\\nX^{+}_{1} \\sim X^{+}_{2} \\sim X^{+}_{3} \\sim X^{-}_{23} \\sim X^{-}_{13} \\sim X^{-}_{12}.\n\\end{array}\n$$\nUsing Proposition \\ref{quotient}, Corollary \\ref{quotient2} and Table \\ref{tab:1}, \nresult below was proved.\n\\begin{lemma}\n1. The quotient $\\racs{4}\/PSL(2,\\mathbb{C})$ is connected and it admits a stratification with 4 orbit types and 5 stratum.\\\\\n2. The quotient ${\\RI{4}}\/{PSL(2,\\mathbb{C})}$ admits a stratification with 4 orbit types, 3 connected components and 10 stratum. The corresponding principal $PSL(2,\\mathbb{C})$--bundle $\\pi_4: \\mathcal{RIG}(-4) \\longrightarrow \\mathcal{RIE}(-4)$ is nontrivial.\n\\end{lemma}\n\n\\begin{remark}\n1. The complex manifold $\\racs{s}$ and $\\racs{s}\/PSL(2,\\mathbb{C})$ are connected.\\\\\n2. The real analityc manifold $\\RI{s}$ and the quotient $\\RI{s}\/PSL(2,\\mathbb{C})$ have $s-1$ connected components.\n\\end{remark}\nFor odd numbers $3 \\leq s \\leq 11$, the Table \\ref{tab:2} shows the number of orbit types and stratum on the quotient $\\RI{s}\/PSL(2,\\mathbb{C})$.\n\\begin{table}[h]\n\\caption{\n\\label{tab:2} \n\\begin{tabular}{lll}\n\\hline\\noalign{\\smallskip}\n s & \\text{Orbit types} & \\text{Stratum} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n3 & 2 & 4 \\\\ \n5 & 5 & 16 \\\\ \n7 & 6 & 24 \\\\ \n9 & 8 & 32 \\\\ \n11 & 8 & 48 \\\\ \n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table}\n\n\\section{The associated singular flat surfaces $S_{\\omega}$}\n\\label{sec-surfaces}\n\n\\subsection{Isometries}\n\nRecall that for $\\omega \\in \\racs{s}$, \nthere is a complex atlas $\\{(V_j, \\Psi_j )\\}$ on $X_{\\omega} = \\ensuremath{{\\widehat{\\mathbb C}}} \\setminus \\{\\text{zeros and poles of } \\omega \\}$, \nwhere $\\{ V_j \\}$ is an open cover by simply connected sets and the functions \n$$\n\\Psi_{j}(z) = \\int_{z_0}^{z}{\\omega}: V_j \\longrightarrow \\mathbb{C}\n$$\nare well--defined for all $j$. \nMoreover, \n$\\Psi_{jk}(z) = z + a_{jk}$, for $a_{jk} \\in \\mathbb{C}$. \nIf $\\omega \\in \\Omega^1\\{k_1, \\ldots, k_m; -1, \\ldots, -1\\} \\subset \\racs{s}$, \nthen the zero of multiplicity $k_j$ is a singularity of cone angle $(2k_j+2)\\pi$ and the pole $p_{\\iota}$ is a cylindrical end of diameter $T_{\\iota} = 2\\pi |r_{\\iota}|$, \nwhere $r_{\\iota} = Res(\\omega, p_{\\iota})$, \n$j = 1, \\ldots, m$ and $\\iota =1, \\ldots , s$; \nsee \\cite{jesus1, valero}. \n\nFor $\\omega=({Q(z)}\/{P(z)})dz \\in \\racs{s}$, \nits associated singular flat surface $S_{\\omega} = (\\ensuremath{{\\widehat{\\mathbb C}}}, g_{\\omega})$ \nhas the riemannian metric\n$$\ng_{\\omega}(z) := \\left(\\begin{smallmatrix}\n \\left| \\frac{Q(z)}{P(z)} \\right|^2 & 0 \\\\\n 0 & \\left| \\frac{Q(z)}{P(z)} \\right|^2\n \\end{smallmatrix}\\right).\n$$\nWe denote by $S^1$ the unit circle on $\\mathbb{C}$. \nThe result below is well--known; \nsee \\cite{valero}. \n\n\\begin{proposition}\n\\label{isometric-surfaces}\n For $\\omega, \\eta \\in \\racs{s}$, their associated singular flat surfaces $S_{\\omega}$ and $S_{\\eta}$ are isometric if and only if there exist $\\lambda \\in S^1$ and $T \\in PSL(2,\\mathbb{C})$ such that $\\eta = \\lambda T_*\\omega$.\n\\end{proposition}\n\n\\subsection{The $(S^1 \\times PSL(2,\\mathbb{C}))$--action}\n\nBy applying Proposition \\ref{isometric-surfaces}, \nwe can extend naturally the $PSL(2,\\mathbb{C})$--action \\eqref{PSL-action} as follows.\n\\begin{equation}\\label{PSL-action2}\n\\begin{array}{rcl}\n\\widehat{\\mathcal{A}}_s: (S^1 \\times PSL(2,\\mathbb{C})) \\times \\racs{s} & \\longrightarrow & \\racs{s} \\\\\n((\\lambda, T), \\omega) & \\longmapsto & \\lambda T_*\\omega.\n\\end{array}\n\\end{equation}\nSimilarly, \nwe can extend the $\\Sim{s}$--action \\eqref{S-action} as\n\\begin{equation}\\label{S-action2}\n\\begin{array}{rcl}\n(S^1 \\times \\Sim{s}) \\times \\mathcal{M}(-s) & \\longrightarrow & \\mathcal{M}(-s) \\\\\n((\\lambda, \\sigma), (r_1, \\ldots, r_{s-1}, p_4, \\ldots, p_s)) & \\longmapsto & \\left(\\lambda{A_{\\sigma}}\\left(\\begin{smallmatrix}\n r_1 \\\\\n \\vdots \\\\\n r_{s-1}\n \\end{smallmatrix}\\right),\\ f_{\\sigma}(p_4, \\ldots, p_s)\\right).\n\\end{array}\n\\end{equation}\nThe expression for the $(S^1 \\times PSL(2,\\mathbb{C}))$--action, \nusing the complex atlas by residues--poles on $\\racs{s}$, \nis \n$$\n\\widehat{\\mathcal{A}}_s(\\lambda, T, \\rp{r_1, \\ldots, r_s; p_1, \\ldots, p_s }) = \\rp{\\lambda r_1,\\ldots, \\lambda r_s; T(p_1), \\ldots, T(p_s)}.\n$$\nWe use the techniques developed in Section \\ref{Sec:action} and \\ref{S-examplesquotient} to prove the results below. \n\n\\begin{remark} \nSince the $(S^1 \\times PSL(2,\\mathbb{C}))$--action is proper, \nthe quotient\n$$\n\\frac{\\racs{s}}{S^1 \\times PSL(2,\\mathbb{C})} = \\frac{\\{S_{\\omega} \\ | \\ \\omega \\in \\racs{s} \\}}{\\{\\text{Isometries}\\}} := \\mathfrak{M}(-s)\n$$\nadmits a stratification by orbit types.\nFurthermore, \nthe realization for the quotient $\\mathfrak{M}(-s)$ is $\\mathcal{M}(-s)\/ S^1 \\times \\Sim{s}$.\n\\end{remark}\nFor the subgroups $\\mathbb{Z}_2 \\times PSL(2,\\mathbb{C}) < S^1 \\times PSL(2,\\mathbb{C})$ and $\\mathbb{Z}_2 \\times \\Sim{s} < S^1 \\times \\Sim{s}$, \nthe actions \\eqref{PSL-action2} and \\eqref{S-action2} are well--defined on $\\RI{s}$ and $\\Im{\\mathcal{M}(-s)}$, \nrespectively. \n\\begin{remark} \nThe $(\\mathbb{Z}_2 \\times PSL(2,\\mathbb{C}))$--action on $\\RI{s}$ is proper, \ntherefore the quotient\n$$\n\\frac{\\RI{s}}{\\mathbb{Z}_2 \\times PSL(2,\\mathbb{C})} = \\frac{\\{S_{\\omega} \\ | \\ \\omega \\in \\RI{s} \\}}{\\{\\text{Isometries}\\}} := \\mathcal{RI}\\mathfrak{M}(-s)\n$$\nadmits a stratification by orbit types.\nFurthermore, \nthe realization for the quotient $\\mathcal{RI}\\mathfrak{M}(-s)$ is homeomorphic to $\\Im{\\mathcal{M}(-s)}\/ \\mathbb{Z}_2 \\times \\Sim{s}$.\n\\end{remark}\n\n\\begin{example}\n\\begin{upshape}\nThe quotient $\\mathfrak{M}(-3)$ and $\\mathcal{RI}\\mathfrak{M}(-3)$ are connected and their admit a stratification with two orbit types. \nFor $\\mathcal{RI}\\mathfrak{M}(-3)$, \na fundamental domain is \n$$\n\\{(r_1, r_2) \\ | \\ 0 < r_1 \\leq r_2 \\},\n$$\nand the orbit types are $\\{r_1 = r_2 \\}$ and $\\{ r_1 < r_2 \\}$.\n\\end{upshape}\n\\end{example}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection*{\\small{Acknowledgements}}\n\\small{The author would like to thank his advisor Jes\\'us Muci\\~no--Raymundo for all fruitful discussions with him during the preparation of this paper and the PhD thesis. \nThis work was supported by a PhD scholarship provided by CONACyT at the Centro de Ciencias Matem\\'aticas, UNAM and Instituto de F\\'isica y Matem\\'aticas de la Universidad Michoacana de San Nicol\\'as de Hidalgo. }\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{plain} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nDetermining the main source of ionizing radiation and the star-formation rate (SFR) of galaxies are essential quests in the study of galaxy evolution. \nWhile optical diagnostic diagrams \\citep[e.g.][]{BPT1981} allows a rather clear distinction between star-formation (SF) and accretion disk processes, \nthey are limited to - by definition - visible components and are at this point extremely difficult to apply at high redshifts. \nThe same caveats apply to the \nmeasurement of SFRs from optical lines. Mid-infrared (MIR) spectroscopy offers a potent alternative, much less sensitive to \ninterstellar exinction. MIR galaxy spectra exhibit an array of features \narising essentially from \n(1) a continuous distribution of dust grains, the smallest of which (VSGs for Very Small Grains) produce the continuum longward \nof $\\sim 10\\mu m$ \\citep{Desert_etal1990} while larger ones containing silicates produce absorption features at 9.7 and 18$\\mu$m\\ \\citep{LebofskyRieke1979};\n(2) ionized interstellar gas producing fine-structure lines; \nand (3) molecular gas producing most notably a series of broad emission features, most prominent in the $6-17$$\\mu$m\\ range, which were\npreviously referred to as ``Unidentified Infrared Bands'' but are now commonly attributed to vibrational emission of large polycyclic \naromatic hydocarbon (PAH) molecules \\citep{LegerPuget1984, Allamandola_etal1985, PugetLeger1989}. Rotational lines of molecular hydrogen\nare also detected \\citep[and references therein]{Roussel_etal2007}.\nMIR diagnostics have been devised to unveil the ionizing source heating these components\n\\citep[e.g.][]{Voit1992_diagnostics,Genzel_etal1998, Laurent_etal2000, Spoon_etal2007}\nand calibrations have been proposed to derive SFRs from their luminosities \\citep[e.g.][]{HoKeto2007, Zhu_etal2008, Rieke_etal2009, HernanCaballero_etal2009}. \nAs these calibrations and the resolving power of the various diagnostic diagrams vary with galaxy types, it is important to review the MIR spectral properties of \nwell defined classes of objects. The Infrared Spectrograph (IRS) on board the {\\it Spitzer} satellite has allowed many such investigations, building on earlier fundamental \nresults from the Infrared Space Observatory \\citep{CesarskySauvage1999, GenzelCesarsky2000}. Much attention has been devoted to extreme sources such as ULIRGs \n\\citep{Armus_etal2007, Farrah_etal2007, Desai_etal2007}, \nstarburst galaxies \\citep{Brandl_etal2006}, \nAGNs \\citep{Weedman_etal2005, Deo_etal2009, Thompson_etal2009} or \nQSOs \\citep{Cao_etal2008}. \nIRS observations of the SINGS sample \\citep{SINGS2003} have also provided many new results about the central region of nearby galaxies \nspanning a broad range of physical properties \\citep{Dale_etal2006, Smith_etal2007, Dale_etal2009}. \nHowever few studies have yet focused on `normal' galaxies. Still, questions remain open on this seemingly unexciting class of objects. \n\nWhether VSG or PAH emission can be used to trace SF in normal galaxies has been often debated in recent years \n\\citep{Roussel_etal2001, ForsterSchreiber_etal2004, Peeters_etal2004, Calzetti_etal2007, Kennicutt_etal2009}. \nResolved observations of star-forming regions have shown that the VSG continuum strongly peaks inside \\newion{H}{ii}\\ regions\nwhile PAH features dominate in photodissociation regions (PDRs) and get weaker nearer the core of \\newion{H}{ii}\\ regions, where the\nmolecules are thought to be destroyed by the intense radiation fields \n\\citep[e.g.][]{Boulanger_etal1988, Giard_etal1994, Cesarsky_etal1996, Verstraete_etal1996, Povich_etal2007,Gordon_etal2008}.\nHowever neutral PAH emission has recently been reported inside \\newion{H}{ii}\\ region \\citep{Compiegne_etal2007}.\nThey are also found in the interstellar medium (ISM), indicating that they must also be excited by softer \nnear-UV or optical photons \\citep[e.g.][]{LiDraine2002, Calzetti_etal2007}, making them perhaps better tracer of B stars than of SF \\citep{Peeters_etal2004}.\nVSG emission is also observed in the ISM but with higher PAH\/VSG surface brightness ratios than in SF regions \\citep{Bendo_etal2008}.\nDespite much complexity on small scales however, integrated MIR luminosities at 24$\\mu$m\\ and 8$\\mu$m\\, tracing the VSG and PAH emissions\nrespectively, are found to correlate with H$\\alpha$\\ luminosities \\citep[e.g.][]{Zhu_etal2008}, though not linearly and with scatter \n\\citep{Kennicutt_etal2009} leading to uncertain SFR estimates. \n \nAn additional source of uncertainty is the common occurence of AGN in normal galaxies. \nPAH molecules are also thought to get destroyed near the hard radiation fields of AGNs \\citep{DesertDennefeld1988, Voit1992_pahdestruction} \nhowever not totally and as was shown recently from IRS spectroscopy, preferentially at short wavelengths \\citep{Smith_etal2007, ODowd_etal2009}. \nThere is in fact no a priori reason why PAH emission could not be excited by UV photons from an AGN \\citep{Farrah_etal2007}.\nThis further compromises the use of PAH bands as SFR indicators, unless AGNs can be reliably detected in the MIR spectra of normal galaxies. \n\nWe have obtained IRS spectra for a sample of 101 normal galaxies at $z\\sim 0.1$ with the goal to tackle the above issues, making use\nof additional multi-wavelength (ultraviolet to far-infrared) photometric data and optical spectroscopic data.\nThe first results of this survey have been reported by \\cite{ODowd_etal2009} who analyzed the dependence of the\nrelative strength of PAH emission features with optical measures of SF and AGN activity. \nWe are pursuing this study by comparing optical and MIR diagnostic diagrams to detect AGN presence in normal galaxies and by investigating \nthe use of PAH, MIR continuum and emission line luminosities as tracer of the total IR luminosity and H$\\alpha$\\ derived SFRs. \nThe sample, IRS data and spectral decomposition method are described in section 2. Section 3 presents the continuum, PAH and emission line properties\nof the galaxies as a function of SF and AGN activity. In particular we analyze the dependency of PAH equivalent widths with age, metallicity and \nradiation field hardness, as well as the efficiency of MIR diagnostics to detect optically classified AGNs in these galaxies.\nWe present correlations between the luminosities of MIR components and the total IR luminosity in section 4 and between these components\nand SFR estimates in section 5. Our conclusion are summarized in section 6.\nThroughout the paper we assume a flat $\\Lambda$CDM cosmology with $H_0=70~{\\rm km~ s^{-1} Mpc^{-1}}$, \n$\\Omega_M=0.3$ and $\\Omega_{\\Lambda}=0.7$, and a Kroupa IMF \\citep{KroupaIMF} for SFR calibrations.\n\n\\begin{figure*}[tb]\n\\plottwo{cmd.eps}{ircol_col.eps}\n\\caption{{\\it Left:} $NUV-r$ color vs $r$-band magnitude diagram showing the location of SSGSS galaxies with respect to the underlying \nlocal population shown as volume density contours \\citep{Wyder_etal2007}. Galaxies separate into two well-defined blue and red sequences. \nIn this and all following figures, star-forming galaxies are represented as black dots, composite galaxies as pink stars and AGNs \nas open red triangles. The SSGSS sample is dominated by blue sequence galaxies with a small fraction (mostly AGNs) on the red sequence.\n{\\it Right:} Infrared color-color diagram: $f_8\/f_{24}$ versus $f_{70}\/f_{160}$ flux ratios. `Normal' quiescent SF galaxies are found \ntowards the bottom right corner (stronger PAH emission and cooler dust in the FIR) while starburst galaxies populate the top left corner \n(strong hot dust continuum in the MIR and warm dust emission in the FIR) .\nThe {\\it Spitzer} data for NGC 35321, NGC 337 and Mrk 33 are taken from \\cite{Dale_etal2007}. \n\\label{fig:cmd}\n}\n\\end{figure*}\n\n\\section{The SSGSS Sample}\n\nThe {\\it Spitzer}-SDSS-GALEX Spectroscopic Survey (SSGSS) is a MIR spectroscopic survey of 101 local star-forming\ngalaxies using the Infrared Spectrograph (IRS) \\citep{Houck_etal2004} aboard the {\\it Spitzer} Satellite. The IRS\nand corollary data are available at: http:\/\/www.astro.columbia.edu\/ssgss\/.\n\n\\subsection{The parent sample}\n\nThe sample is drawn from the Lockman Hole region which has been extensively surveyed at multiple wavelengths. \nIn particular UV photometry from GALEX (1500 and 2300\\AA), optical imaging and spectroscopic observations from SDSS and \ninfrared photometry (IRAC and MIPS channels) from {\\it Spitzer} (SWIRE) are available for all SSGSS galaxies. \nThe redshifts span $0.030.05$ and the 8.6$\\mu$m\\ PAH feature for galaxies at $z<0.05$. Both the observed and restframe fluxes in this band\nare largely dominated by PAH emission for most SF and composite galaxies. The continuum dominates only for 1 AGN.\nThe 16$\\mu$m\\ IRS Peak-Up band collects photons from the 17$\\mu$m\\ PAH complex \nplus other smaller PAH features around 14$\\mu$m\\ and the large 12.7$\\mu$m\\ complex for galaxies at $z>0.06$. The observed \n16$\\mu$m\\ flux includes more PAH emission than the restframe flux, which is continuum dominated ($\\sim 70\\%$) for all galaxies.\nThe observed 24$\\mu$m\\ MIPS channel is vastly dominated by the continuum for all sources. The highest PAH contribution (18\\%)\ncomes from the redshifted 18.92$\\mu$m\\ PAH feature and the red wing of the 17$\\mu$m\\ PAH feature for the highest redshift object \n($z=0.217$). PAH emission starts to dominate the 24$\\mu$m\\ channel for galaxies at $z>1$.\n\n \n\\begin{figure}\n\\plotone{apcorr_rad2.eps}\n\\caption{Aperture corrections in magnitude at 6, 8, 16 and 24$\\mu$m\\ as a function of $r$-band Petrosian diameter in arcseconds. \nThe dotted lines mark the average aperture corrections. The vertical lines show the slit widths of the SL and LL \nmodules (upper and lower panels respectively). \n\\label{fig:apcorr-rad}\n}\n\\end{figure}\n\n\\subsection{Aperture corrections}\n\nThe SL and LL IRS modules have slit widths of 3.6'' and 10.5'' respectively, while the mean angular size of the\nsample is 10''.\nThe corrections applied to stitch the two modules together in the overlap region ($14.0-14.5\\mu m$) are explained \nin detail by O'Dowd et al. (in preparation).\nWavelength dependent aperture effects also arise from the wavelength dependent PSF \n(increased sampling of the central regions of extended galaxies with increasing wavelength).\nTo remedy these effects, we compute spectral magnitudes in the MIR {\\it Spitzer} bands\nboth from the data and from the PAHFIT SEDs using the {\\it Spitzer Synthetic Photometry} cookbook. \nWe find excellent agreement between data and fits except in the 6$\\mu$m\\ IRAC bands, the data being noisy and\nthe fits unreliable below 5.8$\\mu$m. For this reason we do not make use of fluxes in this part of the spectrum. \nThe difference between the PAHFIT spectral magnitudes and the photometric magnitudes are used as \naperture corrections at the effective wavelengths. These corrections are shown in Figure \\ref{fig:apcorr-rad}\nas a function of $r$-band petrosian diameter and listed in Table \\ref{table:aper}. The vertical lines in the upper\nand lower panels show the slit widths of the SL and LL modules respectively. It is clear that flux is lost at 8$\\mu$m,\nhowever at longer wavelengths we do not find that much flux is lost even when the optical petrosian diameter is larger \nthan the slit width, which we attribute to the larger PSF.\nCorrections at intermediate wavelengths are obtained by interpolation. The mean corrections are $\\sim 1.2$ mag at 6$\\mu$m, \n$\\sim 0.5$ mag at 8$\\mu$m\\ and $< 0.1$ at 16 and 24$\\mu$m.\nIn the following, all MIR luminosities computed from the PAHFIT decomposition (PAH, continuum and emission line luminosities \nas well as total restframe luminosities in the {\\it Spitzer} bands) are corrected for aperture as described in this section. \n\n\\begin{figure}\n\\plotone{3meanspec.eps}\n\\caption{The mean spectra of SF galaxies (solid line), composite galaxies (dotted line) and AGNs (dashed line) \nnormalized at 10$\\mu$m. The dot-dashed spectrum is the average starburst spectrum of \\cite{Brandl_etal2006}.\nThe transition from starburst to SF galaxy to AGN is marked by a declining continuum slope, decreased [\\newion{Ne}{ii}]12.8$\\mu$m\\ and [\\newion{S}{iii}]18.7$\\mu$m\\ \nand enhanced [\\newion{O}{iv}]25.9$\\mu$m. The AGN and starburst spectra also show depleted PAH emission at low wavelength compared to \nthe SF spectrum.\n\\label{fig:meanspec}\n}\n\\end{figure}\n\n\n\\section{MIR Spectral properties}\n\nFigure \\ref{fig:meanspec} shows the mean spectra of our SF galaxies (solid line), composite galaxies \n(dotted line) and AGNs (dashed line) as well as the average starburst spectrum of \\cite{Brandl_etal2006} (dot-dash), \nnormalized at 10$\\mu$m.\nThe transition from starburst to SF galaxy to AGN is associated with a declining continuum slope,\nmost dramatic between the starburst spectrum and the normal SF spectrum. Indeed \\newion{H}{ii}\\ regions and starburst galaxies are found to \nexhibit a steep rising VSG continuum component longward of $\\sim 9\\mu m$ \\citep[e.g.][]{Cesarsky_etal1996, Laurent_etal2000, \nDale_etal2001, Peeters_etal2004}. The transition is also marked by\ndecreased [\\newion{Ne}{ii}]12.8$\\mu$m\\ and [\\newion{S}{iii}]18.7$\\mu$m\\ line emission and enhanced [\\newion{O}{iv}]25.9$\\mu$m\\ line emission \\citep[e.g.][]{Genzel_etal1998}.\nThe AGN spectrum, and to a lesser extent the starburst spectrum, show weaker PAH emission at low wavelength than \nthe SF spectrum, an effect attributed to the destruction of PAHs in intense far-UV radiation fields. \n\n\\subsection{PAH features, continuum and emission lines}\n\n\n\\begin{figure}\n\\plotone{specdiff_pah.eps}\n\\caption{{\\it Top:} The mean PAH component of `young' SF galaxies with $1.1=1.2$) and that of `old' SF galaxies with $1.3=1.4$).\nThe 4000\\AA\\ break D$_n$(4000) \\citep{Balogh98} is a measure of the average age of the stellar populations.\nThe separating value is simply the median of the distribution. \nThe bottom panel shows the average PAH components of SF galaxies and AGNs in the $1.31.6$) do tend to populate the low end of the distribution (i.e. have shallower slopes) in both cases, \nas do AGNs in the red part of the spectrum however a flatter continuum could not be used as a criterion to separate \nAGNs from SF galaxies, as previously reported by \\cite{Weedman_etal2005}. \nThe correlation with FIR color for SF galaxies is more striking, especially at longer MIR wavelengths. This may be expected if the peak \nof the dust SED (a blackbody modified by the emissivity) is located shortward of $\\sim$ 100$\\mu$m. In this case as the peak wavelength \ndecreases, the MIR continuum slope gets closer to the peak and therefore steepens while $f_{70}\/f_{160}$ increases. \n\n\\begin{figure}\n\\plotone{lr_hr.eps}\n\\caption{Comparison between the low and high resolution line fluxes of [\\newion{Ne}{ii}]12.8$\\mu$m\\ (in black) and [\\newion{Ne}{iii}]15.5$\\mu$m\\ (in blue)\nfor a subsample of 31 galaxies. Excluding the objects marked with a red cross, the rms of the correlation is 0.22. \n\\label{fig:lr-hr}\n}\n\\end{figure}\n\n\nFinally we look at variations in the emission line components.\nThe lines modeled by PAHFIT in the low resolution spectra are meant to provide a realistic decomposition of \nthe blended PAH features and the continuum (S07) but the spectral resolution is of the same order as the FWHM of the lines. \nFigure \\ref{fig:lr-hr} shows the comparison between the high and low resolution fluxes of the [\\newion{Ne}{ii}]12.8$\\mu$m\\ and [\\newion{Ne}{iii}]15.5$\\mu$m\\ lines (black and blue error\nbars respectively) for the subsample observed with the SH module. The high resolution lines were also measured using PAHFIT with the\ndefault settings. We make no attempt at aperture correction on this plot. \nExcluding the 3 extreme error bars among the [\\newion{Ne}{iii}]15.5$\\mu$m\\ fluxes at high resolution and the outlier among the [\\newion{Ne}{ii}]12.8$\\mu$m\\ fluxes\n(marked as red crosses in Figure \\ref{fig:lr-hr}), the fitting procedure at low resolution recovers \nthe high resolution fluxes with an rms of 0.22 dex, a reasonable estimate considering the factor of 10 difference in spectral resolution.\nIn particular the PAH contamination for the [\\newion{Ne}{ii}]12.8$\\mu$m\\ line does not seem to be a significant problem in the SL data using PAHFIT.\nFor the purpose of the present statistical analysis we use the low resolution line \nmeasurements which are available for the full sample and over the full range of wavelengths.\nWe refer to O'Dowd et al. (in preparation) for a detailed comparison between the high and low resolution data.\n \n\\begin{figure}\n\\plotone{specdiff_el.eps}\n\\caption{{\\it Top:} The difference - $\\Delta I_{\\nu}$ - between the mean emission line component of `young' SF galaxies \nwith $1.1=1.2$) and that of `old' SF galaxies ($=1.4$) as defined earlier,\nwhile the bottom panel shows the difference between the mean emission line component of SF galaxies and that of AGNs in their overlapping \nrange of D$_n$(4000)\\ ($1.31 \\mu m$, EW(7.7$\\mu$m) $>4\\mu m$\nand EW(8.6$\\mu$m) $>1\\mu m$). The dotted lines in the right panel are approximate lower limits for the SF population\nEW(11.3$\\mu$m) $>1.8 \\mu m$, EW(12.7$\\mu$m) $>0.9\\mu m$ and EW(17$\\mu$m) $>0.8\\mu m$). AGN EWs become increasingly undistinguishable \nfrom those of SF galaxies towards longer wavelengths. \n\\label{fig:ew-o3hb}\n}\n\\end{figure*}\n\n\\subsection{PAH Equivalent Widths}\n\nWe compute equivalent widths (EWs) as the integrated intensity of the Drude profile(s) fitting a particular PAH feature,\ndivided by the continuum intensity below the peak of that feature. Using Eq. 3 from S07 for the integrated\nintensity of a Drude profile, the EW of a PAH feature with central wavelength $\\lambda_r$, full width at half-maximum $FWHM_r$ \n(as listed in S07, their Table 3), and central intensity $b_r$ (PAHFIT output), can be written as:\n\\begin{equation}\nEW(\\lambda_r)={\\pi \\over 2} { b_r \\over I_{\\nu}^{cont}(\\lambda_r)} ~{FWHM}_r \n\\label{eq:ew}\n\\end{equation}\nwhere $I_{\\nu}^{cont}(\\lambda_r)$ is the continuum component of Eq. \\ref{eq:pahfit}. \nThis definition is different from that of S07 in PAHFIT which computes the integral $ \\int (I_{\\nu}^{PAH}\/I_{\\nu}^{cont}) d\\lambda$\nin the range $\\lambda_r \\pm 6\\times FWHM_r$. In the case of the 7.7 micron feature whose FWHM is large and extends \nthe limit of the integral to regions beyond the IRS range where the continuum vanishes arbitrarily, the profile weighted \naverage continuum is used. Despite this caveat, both methods agree within 10\\% \nand the discrespancies \nvirtually disappear when increasing the limits of the integral for all other PAHs\\footnote{In the process of making these comparisons, \nwe discovered two bugs in PAHFIT: \n1\/ the code was mistakenly calling gaussian profiles instead of Drude profiles to compute the EW integral, thus underestimating\nEWs by $\\sim 1.4$, and 2\/ it was applying silicate extinction to the continuum while using the extinction corrected PAH features\n(according to Eq. 1, both components are equally affected by the extinction term).\nThese bugs are being corrected (J.D. Smith, private communication).}. \nHowever the EWs measured as above differ significantly from those estimated with the spline method, which consists in fitting a\nspline function to the continuum from anchor points around the PAH feature, and a Gaussian profile to the continuum-subtracted feature. \nThis method yields considerably smaller EW values as it assigns a non negligible fraction of the PAH flux extracted by PAHFIT to the continuum.\nFigure \\ref{fig:ew-methods} shows our EWs (Eq. \\ref{eq:ew}) against the 6.2$\\mu$m\\ PAH EWs computed by \\cite{SargsyanWeedman2009} \nfor the SSGSS sample assuming a single gaussian on a linear continuum between 5.5$\\mu$m\\ and 6.9$\\mu$m. \nTheir published sample is restricted to SF galaxies defined as having EW(6.2$\\mu$m) $>0.4$$\\mu$m\\ \\citep{WeedmanHouck2009}. The measurements\nfor the remaining galaxies were kindly provided by L. Sargsyan. Their formal uncertainty is estimated to be $\\sim 10\\%$. \nThe two methods are obviously strongly divergent. The spline EWs strongly peak around a value of $\\sim 0.6$$\\mu$m\\ with no apparent \ncorrelation with the PAHFIT estimates, which reach $\\sim 15\\mu m$ and can be up to 25 times larger than the \\cite{SargsyanWeedman2009} values.\nOur EWs for the main PAH features are listed in Table \\ref{table:eqw}.\n\nThe strength of a PAH feature depends on several interwined properties of the ISM: metallicity, radiation field\nhardness, dust column density, size and ionization state distributions of the dust grains (Dale et al. 2006 \\nocite{Dale_etal2006} and \nreferences therein). In particular it is shown to be reduced in extreme far-UV radiation fields, such as AGN-dominated environment\n\\citep{Genzel_etal1998, Sturm_etal2000, Weedman_etal2005}, near the sites of SF \\citep{Geballe_etal1989, Cesarsky_etal1996, \nTacconiGarman_etal2005, Beirao_etal2006, Povich_etal2007,Gordon_etal2008} or in very low metallicity environments \n\\citep{Dwek2005, Wu_etal2005, OHalloran_etal2006, Madden_etal2006}, where the PAH molecules are thought to get destroyed \n\\citep[e.g.][]{Voit1992_pahdestruction}.\n\n\\begin{figure*}[tb]\n\\plottwo{shortEW_d4.eps}{longEW_d4.eps}\n\\caption{The equivalent widths of the main PAH features at short and long wavelengths (left and right panels respectively)\nas a function of D$_n$(4000). Symbols and lines are as described in Figure \\ref{fig:ew-o3hb}. \n\\label{fig:ew-d4}\n}\n\\end{figure*}\n\\begin{figure*}[tb]\n\\plottwo{shortEW_n2ha.eps}{longEW_n2ha.eps}\n\\caption{The equivalent widths of the main PAH features at short and long wavelengths (left and right panels respectively)\nas a function of [\\newion{N}{ii}]$\\lambda 6583$\/H$\\alpha$. Symbols and lines are as described in Figure \\ref{fig:ew-o3hb}. The short wavelength PAH EWs \nsignificantly decrease with [\\newion{N}{ii}]$\\lambda 6583$\/H$\\alpha$.\n\\label{fig:ew-n2ha}\n}\n\\end{figure*}\n\nFigure \\ref{fig:ew-o3hb} shows the EWs of the main PAH features as a function of [\\newion{O}{iii}]$\\lambda 5007$\/H$\\beta$.\nThe Pearson correlation coefficients $r$ are indicated at the top right of each panel. \nAGNs do exhibit noticeably smaller EWs than SF galaxies at short wavelengths (6.2, 7.7 and 8.6$\\mu$m, left panel), \nhowever seemingly uncorrelated with radiation field hardness. \nThe range of EWs spanned by AGNs becomes increasingly similar to that of SF galaxies towards longer wavelengths (11.3, 12.7 and 17$\\mu$m, right panel)\nwhile at the same time a correlation seems to appear with radiation field hardness. The Pearson coefficients for the AGN population alone \nat long wavelengths are -0.97, -0.89 and -0.81 respectively from top to bottom, though admittedly they are boosted by the rightmost data point. \nA larger sample of AGNs is needed to confirm this correlation.\nPAH strength remains largely independent of radiation field hardness for SF and composite galaxies. \nThese results complement the analysis of \\cite{ODowd_etal2009} who found a correlation between the long-to-short wavelength PAH ratios \nand [\\newion{O}{iii}]$\\lambda 5007$\/H$\\beta$ in AGNs.\nThese trends are consistent with the selective destruction of PAH molecules in the hard radiation fields of these sources \n([\\newion{O}{iii}]$\\lambda 5007$\/H$\\beta$ $>1.5$). The EW trends or lack thereof in Figure \\ref{fig:ew-o3hb} \nsuggest that the smallest PAH molecules effective at producing the short-wavelength PAH features get destroyed first, \nnear an AGN, while the larger molecules producing the larger wavelength PAHs require increasingly harder radiation fields for their \nPAH strength to drop below that of SF galaxies. \n\\cite{DesertDennefeld1988} first suggested that the absence of PAHs could be taken as evidence for the presence of an AGN.\nWeak PAH emission has since often been used to discriminate between photoionization and accretion disk processes. However the common boundaries \nfor a `pure starburt', e.g. EW(7.7$\\mu$m)$>1$ \\citep{Lutz_etal1998} or EW(6.2$\\mu$m) $>0.4$$\\mu$m\\ \\citep{WeedmanHouck2009} \nare significantly too weak here, due to the different method we use to compute the equivalent widths as shown above.\nBased on the PAHFIT decomposition, SF galaxies would be best isolated by EW(6.2$\\mu$m) $>1$$\\mu$m, EW(7.7$\\mu$m) $>4$$\\mu$m\\\nor EW(8.6$\\mu$m)$>$1$\\mu$m, the latter two criteria being more accurately determined in our sample. Those limits are shown as\ndotted lines in the left panel of Figure \\ref{fig:ew-o3hb}. The two \nSF exceptions below the 7.7$\\mu$m\\ and 8.6$\\mu$m\\ EW limits \n(\\#32 and \\#74) happen to have very strong silicate absorption parameters ($\\tau_{9.7}=$1.8 and 2.33) and still very distorted \nabsorption-corrected continua compared to the rest of the sample. The dotted lines in the right panel are approximate lower limits\nfor the SF population (EW(11.3$\\mu$m) $>1.8 \\mu m$, EW(12.7$\\mu$m) $>0.9\\mu m$ and EW(17$\\mu$m) $>0.8\\mu m$).\nIt is clear that the AGN population becomes increasingly difficult to isolate based on EW alone in\nthe red part of the spectrum. \n\n\nFigures \\ref{fig:ew-d4} and \\ref{fig:ew-n2ha} show the EWs of the main PAH features as a function D$_n$(4000)\\ and \n[\\newion{N}{ii}]$\\lambda 6583$\/H$\\alpha$\\ respectively. The EWs at short wavelengths show a mild downward trend \nwith increasing age (or decreasing SF activities) while they become independent of it at long wavelengths.\nThis again is consistent with the correlations between \nthe long-to-short wavelength PAH ratios and D$_n$(4000)\\ or H$\\alpha$\\ equivalent width found by \\cite{ODowd_etal2009}. \nThe short wavelength EWs decrease more notably with increasing [\\newion{N}{ii}]$\\lambda 6583$\/H$\\alpha$\\ ratios, which of course \nare related to D$_n$(4000)\\ but appear to be the property that most uniformally and significantly\naffects the sample as a whole. \nMetallicity and SF activity are known to affect PAH strength, however, as mentionned earlier, previous studies have demonstrated \nthe opposite effect, namely a decrease in PAH strength at very low metallicity and in intense SF environment.\nThese trends thus make normal blue sequence galaxies the sites of maximum PAH strength. \n\n\n\\begin{figure}\n\\plotone{laurent_diagnostic.eps}\n\\caption{The equivalent width of the 7.7$\\mu$m\\ feature as a function of continuum slope (a diagnostic diagram proposed by \\cite{Laurent_etal2000}).\nSymbols are as described in Figure \\ref{fig:cmd}. \nA clear trend is seen for SF and composite galaxies while AGNs appear more randomly distributed.\nAGNs are expected to populate the lower left side of the plot, PDRs the lower right corner and \\newion{H}{ii}\\ regions \nthe upper left corner of the diagram \\citep{Laurent_etal2000}.\n\\label{fig:laurent}}\n\\end{figure}\n\n\nOther than PAH destruction, another cause of decreasing PAH strength at low wavelengths may be a stronger continuum\nwhose strength may depend on the above parameters.\nA short wavelength continuum (3--10$\\mu$m) has been observed in AGNs, which is attributed to very hot dust heated by \ntheir intense radiation fields (Laurent et al. 2000 \\nocite{Laurent_etal2000} and references therein), however the continuum slopes of AGNs\nin our sample largely overlap those of the SF population (Figure \\ref{fig:slope-x}). Figure \\ref{fig:laurent} shows the\nEW of the 7.7$\\mu$m\\ feature as a function of continuum slope, a diagnostic diagram proposed by \\cite{Laurent_etal2000} to distinguish\nAGNs from PDRs and \\newion{H}{ii}\\ regions. \nA clear trend is seen for SF and composite galaxies, suggesting that decreased PAH strength in normal SF galaxies may be \nat least partly due to an increased continuum at low wavelength, which is itself loosely correlated with D$_n$(4000)\\ (Figure \\ref{fig:slope-x}).\nHowever the EWs of AGNs appear quite independent of their continuum slope, supporting the PAH destruction scenario.\nIn this diagram, AGNs are expected to populate the lower left side of the plot (shallow slopes and weak PAH features), PDRs the lower \nright corner (shallow slopes and strong PAH features) and \\newion{H}{ii}\\ regions the upper left corner of the diagram (steep slopes and weak PAH features). \nOur SF sequence is qualitatively similar to the location of quiescent SF regions on the Laurent et al. diagram (their Figure 6), which \nare modeled by a mix of PDR and \\newion{H}{ii}\\ region spectra, plus an AGN component towards the lower left corner where composite galaxies \nare indeed most concentrated. \n\n\n\\begin{figure*}[tb]\n\\plottwo{dale_diagnostic.eps}{mir_diagnostic.eps}\n\\caption{{\\it Left:} The spline derived 6.2$\\mu$m\\ PAH equivalent widths \\citep{SargsyanWeedman2009} against the [\\newion{O}{iv}]25.9$\\mu$m\/[\\newion{Ne}{ii}]12.8$\\mu$m\\\ncemission line ratios (a diagnostic diagram originally proposed by \\cite{Genzel_etal1998}). \nThe dotted line represents a variable mix of AGN and SF region; the short solid lines perpendicular\nto it delineate the AGN region on the left, the SF region at the bottom right, and in between a region of mixed classifications\nby \\cite{Dale_etal2006}. \nWe applied a cut in error bars for clarity. \nThis diagram is of limited resolving power for normal galaxies. \n{\\it Right:} The PAHFIT derived 8.6$\\mu$m\\ PAH equivalent widths against [\\newion{Ne}{ii}]12.8$\\mu$m\/[\\newion{O}{iv}]25.9$\\mu$m\\ (note the reversed\n$y$-axis). This version which resembles a flipped version of the optical \\cite{BPT1981} diagram better recovers the optical classification.\nThe short-dashed lower line and the dot-dashed upper line are optical boundaries translated into the MIR plane as explained in Section 3.3.\nThe long-dashed middle line is an empiral boundary marking the region below which we do not find any optically defined SF galaxy.\nThese boundaries are reported in Table \\ref{table:diagnostics}.\nThe circled galaxy and the lower limit in the SF corner are the two SF galaxies with EW lower than the\nSF limit in Figures \\ref{fig:ew-o3hb}, \\ref{fig:ew-d4} and \\ref{fig:ew-n2ha} (EW(8.6$\\mu$m)$<$1$\\mu$m). \n\\label{fig:mir-diagnostic}}\n\\end{figure*}\n\n\n\\subsection{Diagnostic Diagram} \n\nThe presence of an AGN is thought to be best verified by the detection of strong high-ionization lines such as [NeV]14.21$\\mu$m\\ or [\\newion{O}{iv}]25.9$\\mu$m. \n\\cite{Genzel_etal1998} were the first to show that the ratio of high to low excitation MIR emission lines combined with\nPAH strength could be used to distinguish AGN activity from star-formation in ULIRGs. This diagnostic was recently revisited\nby \\cite{Dale_etal2006} for the nuclear and extra-nuclear regions of normal star-forming galaxies in the SINGS sample \nobserved with the IRS. \\cite{Dale_etal2006}\nmade use of the [\\newion{O}{iv}]25.9$\\mu$m\/[\\newion{Ne}{ii}]12.8$\\mu$m\\ emission line ratios with spline derived EWs of the 6.2$\\mu$m\\ feature \n(they also proposed an alternative diagnostic using the [\\newion{Si}{ii}]34.8$\\mu$m\/[\\newion{Ne}{ii}]12.8$\\mu$m\\ emission line ratio but [\\newion{Si}{ii}]34.8$\\mu$m\\\nis beyond the usable range of our data). \nThe left panel of Figure \\ref{fig:mir-diagnostic} shows the Dale et al. diagram using\nthe spline derived EWs of the 6.2$\\mu$m\\ feature measured by \\cite{SargsyanWeedman2009} in the SSGSS sample. \nThe AGN with no detected [\\newion{O}{iv}]25.9$\\mu$m\\ line is plotted as an upper limit assuming an [\\newion{O}{iv}]25.9$\\mu$m\/[\\newion{Ne}{ii}]12.8$\\mu$m\\\nline ratio based on the correlation between [\\newion{Ne}{ii}]12.8$\\mu$m\/[\\newion{O}{iv}]25.9$\\mu$m\\ and [\\newion{O}{iii}]$\\lambda 5007$\/H$\\beta$\\ for other AGNs in \nFigure \\ref{fig:lineratios}. We applied a cut in error bars to this plot for clarity \n($\\Delta$log([\\newion{Ne}{ii}]12.8$\\mu$m\/[\\newion{O}{iv}]25.9$\\mu$m) $<1.5$, roughly the scale of the $y$-axis), which excludes 1 AGN, 1 composite galaxy\nand 1 SF galaxy. One other AGN is found with no measurable EW. \nThe dotted line represents a variable mix of AGN and SF region; the short solid lines perpendicular\nto it delineate the AGN region on the left, the SF region at the bottom right, and in between a region of mixed classifications\nwhose physical meaning remains unclear \\citep{Dale_etal2006}. Given the relative homogeneity of our sample (lacking in extreme types), \nthe very narrow range of spline EWs for ordinary galaxies and the rather large uncertainties in our emission line ratios derived \nfrom low resolution spectra, this diagnostic proves of limited use for normal galaxies. Most optically classified SF galaxies \ndo fall into the SF corner, but so do a few composite galaxies. The rest of the sample shows little spread within the mixed region.\n\nBased on the results of this and the previous sections, we revise this diagnostic using the PAHFIT based EWs (Eq. \\ref{eq:ew}) and the \ncorrelations between these EWs at low wavelength and [\\newion{N}{ii}]$\\lambda 6583$\/H$\\alpha$\\ (Figure \\ref{fig:ew-n2ha}) on the one hand, \nand between [\\newion{Ne}{ii}]12.8$\\mu$m\/[\\newion{O}{iv}]25.9$\\mu$m\\ and [\\newion{O}{iii}]$\\lambda 5007$\/H$\\beta$\\ (Figure \\ref{fig:lineratios}) on the other hand. \nThe right panel of Figure \\ref{fig:mir-diagnostic} shows the [\\newion{Ne}{ii}]12.8$\\mu$m\/[\\newion{O}{iv}]25.9$\\mu$m\\ emission line ratios against the \nPAHFIT EWs of the 8.6$\\mu$m\\ feature.\nNote that we inverted the $y$-axis with respect to the left panel (and the traditional Genzel et al. diagram), so that \nthe figure becomes a flipped version of the optical BPT diagram.\nThe short-dashed lower line is the theoretical optical boundary of \\cite{Kewley_etal2001} translated into this plane using the correlations between \nEW(8.6$\\mu$m) and [\\newion{N}{ii}]$\\lambda 6583$\/H$\\alpha$\\ in Figure \\ref{fig:ew-n2ha} and between [\\newion{Ne}{ii}]12.8$\\mu$m\/[\\newion{O}{iv}]25.9$\\mu$m\\ and [\\newion{O}{iii}]$\\lambda 5007$\/H$\\beta$\\\nin Figure \\ref{fig:lineratios} for the AGNs and composite galaxies. Its analytical form is:\n\\begin{equation}\ny = { 1.84 \\over x +1.51 } -0.88\n\\label{eq:mykewley}\n\\end{equation}\nwhere $x=$ log(EW(8.6$\\mu$m)) and $y=$ log([\\newion{Ne}{ii}]12.8$\\mu$m\/[\\newion{O}{iv}]25.9$\\mu$m). \nThe dotted upper line is the empirical boundary of \\cite{Kauffmann_etal2003} translated using these same correlations for the\ncomposite and SF galaxies:\n\\begin{equation}\ny = { 1.10 \\over x +0.32 } -1.27\n\\label{eq:mykauffmann}\n\\end{equation}\nAs expected from the poorer correlation between [\\newion{Ne}{ii}]12.8$\\mu$m\/[\\newion{O}{iv}]25.9$\\mu$m\\ and [\\newion{O}{iii}]$\\lambda 5007$\/H$\\beta$\\\nfor non AGNs, this boundary is less meaningful even though it does isolate the bulk of the SF galaxies.\nThe long-dashed line is an empiral boundary marking the region below which we do not find any SF galaxy:\n\\begin{equation}\ny = { 1.2 \\over x+0.8 } -0.7\n\\label{eq:myboundary}\n\\end{equation}\n\n\nDespite a mixed region of composite and SF galaxies, there is a clear sequence from the bottom left to the top right\nof the plots and 3 regions where each optical class is uniquely represented. In particular weak AGNs separate remarkably \nwell in this diagram. The mixed region may in fact be revealing an obscured AGN component in a large fraction ($\\ge 50\\%$) \nof the optically defined `pure' SF galaxies. Other dust insensitive AGN diagnostics such as X-ray or radio data are necessary to confirm this. \nDeep XMM observations are available only over a small region of the Lockman Hole and the FIRST radio limits are\ntoo bright to reliably test the presence of faint AGNs. Indeed we do not expect this hidden AGN contribution to be large since\nnone of the SF galaxies falls into the AGN corner of the diagram. These objects warrant a detailed study beyond the scope of \nthe present paper.\n\nAlternatively truly `pure' SF galaxies may be defined as lacking the [\\newion{O}{iv}]25.9$\\mu$m\\ emission line ($\\sim 25\\%$ of our SF\ncategory). These are not represented except for one, which is one of the two SF galaxies with EWs lower than the\nSF limit in Figs \\ref{fig:ew-o3hb}, \\ref{fig:ew-d4} and \\ref{fig:ew-n2ha} (EW(8.6$\\mu$m)$<$1$\\mu$m). The lower limit was calculated by \narbitrarily assigning it the lowest value of the [OIV] fluxes detected in the sample. The other one, which has a detected \n[OIV] line, is circled. These 2 galaxies which would have been misclassified as AGNs based on their EW alone sit well into \nthe SF category on this diagram.\nThe AGN with no detected [OIV] (plotted as a lower limit) happens to have the lowest 8.6$\\mu$m\\ EW in the sample. A significantly larger [\\newion{Ne}{ii}]12.8$\\mu$m\/[\\newion{O}{iv}]25.9$\\mu$m\\ \nflux ratio would move it into the LINER region of this flipped BPT diagram (although this particular AGN is not optically classified as a LINER). \nEquations \\ref{eq:mykewley}, \\ref{eq:mykauffmann} and \\ref{eq:myboundary} are reported in Table \\ref{table:diagnostics} as well as their equivalents \nfor the 6.2$\\mu$m\\ and 7.7$\\mu$m\\ PAH features.\nWe note that much larger samples, of AGNs in particular, are needed to confirm and\/or adjust these relations. \n\n\n\\section{MIR dust components and the total infrared luminosity}\n\nIn this section we investigate how individual dust components emitting in the narrow MIR region \ntrace the total dust emission in galaxies, which includes a very large FIR component. \nThe definition of the total infrared (TIR) luminosity and the methods used to estimate it \nvaries in the literature \\citep{Takeuchi_etal2005}. In this paper $L_{TIR}$\\ refers to $L(3-1100\\mu m)$ and has been \nderived by fitting the {\\it Spitzer} photometric points (IRAC+IRS Blue Peak Up+MIPS) \nwith \\cite{DraineLi2007} model SEDs\\footnote{http:\/\/www.astro.princeton.edu\/$\\sim$draine\/dust\/irem.html} and integrating \nthe best fit SED from 3 to 1100$\\mu$m. \nThis $L_{TIR}$\\ is in excellent agreement with the $3-1100$$\\mu$m\\ luminosity \nderived from the prescription of \\cite{DaleHelou2002} for MIPS data (their Eq. 4), with a standard deviation of 0.05 dex. This\nshows that the total IR luminosity really is driven by the MIPS points \\citep[e.g.][]{Dale_etal2007}.\nWe note also that integrating the SEDs between 8 and 1000$\\mu$m\\ (sometimes called the FIR luminosity) would decrease the luminosity \nby $\\sim 0.04$ dex in the present sample.\n\n\n\\begin{figure}\n\\plotone{ltir_all2.eps}\n\\caption{$L_{MIR}\/L_{TIR}$ ratios as a function of $L_{TIR}$\\ where $L_{MIR}$ equals - from top to bottom - \nthe luminosity of the PAH complexes at 7.7 and 17$\\mu$m, the luminosity of the continuum at 8 and 16$\\mu$m, and \nthe total restframe luminosities in the 8$\\mu$m\\ IRAC band, 16$\\mu$m\\ IRS band and 24$\\mu$m\\ MIPS bands.\nSymbols are as described in Figure \\ref{fig:cmd}. \nThe continuum and broadband luminosities are defined as $\\nu L_{\\nu}$.\nThe logarithmic scaling factors $\\kappa$ indicated in each panel are defined as the median \nof log($L_{TIR}\/L_{MIR}$) for the SF population alone (green dashed lines).\nThe rms and Pearson coefficient $r$ in each panel are also for the SF population alone.\n\\label{fig:ltir-all}\n}\n\\end{figure}\n\nFigure \\ref{fig:ltir-all} shows the correlations between $L_{TIR}$\\ and $L_{MIR}\/L_{TIR}$ ratios where $L_{MIR}$ equals - from top to bottom -\nthe luminosity of the PAH complexes at 7.7 and 17$\\mu$m, the luminosity of the continuum at 8 and 16$\\mu$m, \nand the total restframe luminosities in the 8$\\mu$m\\ IRAC band, 16$\\mu$m\\ IRS band and 24$\\mu$m\\ MIPS bands. \nThe continuum and broadband luminosities are defined as $\\nu L_{\\nu}$.\nAs in all previous figures, SF galaxies are shown as black dots, composite galaxies as pink stars and AGNs as red triangles.\nThe logarithmic scaling factors $\\kappa$ indicated in each panel are defined as the median of log($L_{TIR}\/L_{MIR}$) for the SF population\nalone and is represented by the green dashed lines (${\\rm log}(L_{MIR}\/L_{TIR}) +\\kappa =0$). \nThe rms and Pearson coefficients of the correlations are also quoted for the SF population alone. \n\nIt is striking that galaxies of all types follow the same tight, nearly linear correlations between $L_{TIR}$\\ and\nthe broadband luminosities in all 3 {\\it Spitzer} bands over 2 dex in luminosity. This implies that all the galaxies in our \nsample are assigned nearly the same SED from a few $\\mu$m\\ to a thousand $\\mu$m\\ and that the FIR component can be well predicted from any \none broadband luminosity in the MIR. This in turn suggests a common heating source for the small and large dust grains responsible for the\nMIR and FIR emissions respectively \\citep{Roussel_etal2001}. The same correlations apply whether this heating source is stellar or an AGN. \nAlthough this may result from the implicit stellar origin of the dust heating in the models, \nthe source of ionizing radiation may not significantly affect the broad SED, at least for weak AGNs. \nMany attempts have been made to derive calibrations between $L_{TIR}$\\ and single MIR broadband luminosities \\citep{CharyElbaz2001, \nElbaz_etal2002, Takeuchi_etal2005, Sajina_etal2006, Brandl_etal2006, Bavouzet_etal2008, Zhu_etal2008}.\nOur best fit slope at 16$\\mu$m\\ ($L_{TIR}$\\ $\\propto L_{16 \\mu m}^{0.98\\pm 0.02}$) is in good agreement \nwith that of \\cite{CharyElbaz2001} for the 15$\\mu$m\\ ISO fluxes. At 24$\\mu$m\\ our correlation for SF galaxies ($L_{TIR}$\\ $\\propto L_{24 \\mu m}^{0.94\\pm 0.025}$) \nis more linear than found in other studies \\citep{Takeuchi_etal2005, Sajina_etal2006, Zhu_etal2008, Bavouzet_etal2008} but the \ndiscrepancy with the first three calibrations \\citep{Takeuchi_etal2005, Sajina_etal2006, Zhu_etal2008} disappears\nwhen including composite galaxies into the fit ($L_{TIR}$\\ $\\propto L_{24 \\mu m}^{0.89 \\pm 0.03}$). \nOn the other hand our correlation is in excellent agreement with the \\cite{MoustakasKennicutt2006} sample (hereafter MK06).\n\n\n\n\\begin{figure}\n\\plotone{residuals_ltir2.eps}\n\\caption{$L_{MIR}\/L_{TIR}$ ratios as a function of PAH equivalent width at 7.7$\\mu$m\\ (left panels) and 17$\\mu$m\\ (right panels)\nwhere $L_{MIR}$ is defined at the top left of each panel. The symbols and $\\kappa$ are defined as in Figure \\ref{fig:ltir-all}. \nThe dotted lines show the expected relations when the broadband fluxes at 8 and 16$\\mu$m\\ are substituted for $L_{TIR}$\\ \nin the left and right panels respectively. \n\\label{fig:residuals-ltir}\n}\n\\end{figure}\n\n\nThe PAH and continuum luminosities also correlate remarkably tightly and nearly linearly with $L_{TIR}$, however with some distinctions \nbetween AGNs and SF galaxies and between the hot and cool parts of the spectrum. The scatter between $L_{TIR}$\\ and PAH luminosity for\nSF galaxies is larger for the 17$\\mu$m\\ PAH feature than for the 7.7$\\mu$m\\ PAH feature. AGNs and composite galaxies blend with the SF \npopulation in the 17$\\mu$m\\ feature correlation whereas they tend to have lower PAH luminosities at 7.7$\\mu$m\\ and stronger 8$\\mu$m\\ continua\nfor the same $L_{TIR}$. The residuals are shown in Figure \\ref{fig:residuals-ltir} \nas a function of the corresponding equivalent widths. The relations between these residuals and EWs are of course expected since \nthe total flux at 8 and 16$\\mu$m\\ can be nearly perfectly substituted for $L_{TIR}$\\ for SF galaxies and AGNs alike in the left and \nright panels respectively (the dotted lines show the predicted relations assuming these substitutions).\nThe most scattered correlation is found with the continuum luminosity at 8$\\mu$m. This may be due to larger measurement errors since\nthis continuum is faint and\/or a stellar contribution unrelated to $L_{TIR}$. A more speculative reason may be that this continuum \noriginates from high intensity radiation fields only and is thus uncorrelated with the cold component of $L_{TIR}$, unlike the PAH\nemission. \n\n\nThe scaling factors $\\kappa$ \nare listed in Table \\ref{table:coeffs} for the main PAH features and the {\\it Spitzer} band luminosities.\nWe also add to our list of MIR components the peak luminosity of the 7.7$\\mu$m\\ PAH feature, defined as $\\nu L_{\\nu}(7.7\\mu m)$, \nas it is a more easily measurable quantity at high redshift than the integrated flux of the PAH feature \\citep{WeedmanHouck2009, SargsyanWeedman2009}.\nFor galaxies with EW$>4$$\\mu$m\\ (most SF galaxies), the median ratio of this peak luminosity to \nthe total luminosity of the PAH complex, $\\nu L_{\\nu}(7.7\\mu m)\/L_{PAH}(7.7\\mu m)$, is $9.3 \\pm 0.9$ and the peak luminosity \nestimates the total PAH luminosity to within $\\sim$ 20\\%. However the overestimate can be as large as 50\\% for other galaxy types in this\nsample, in particular galaxies containing an AGN which may not be easily isolated in high redshift samples and may also have much smaller \nEWs leading to yet larger errors.\n\n\nFor the calibration that shows the strongest deviation from linearity in Figure \\ref{fig:ltir-all}, \nwhich is found for the 7.7$\\mu$m\\ PAH luminosity ($L_{TIR}\\propto L_{MIR}^{0.93\\pm 0.02}$),\nthe linear approximation ${\\rm log}(L_{TIR})= {\\rm log}(L_{PAH}(7.7\\mu m))+\\kappa$ (where $\\kappa=1.204$) recovers $L_{TIR}$\\ within a factor \nof 1.5 {\\it in this sample}. \nFor starbursts and ULIRG starbursts, \\cite{Rigopoulou_etal1999} found a mean log$(L_{TIR}\/L_{PAH}(7.7\\mu m))$ of 2.09 and 2.26 respectively,\nconsiderably larger than for normal galaxies. More recently \\cite{Lutz_etal2003} find a mean logarythmic ratio of 1.52 for \na sample of starburst nuclei, closer to our value.\nOur mean logarythmic ratio for the 6.2$\\mu$m\\ feature is 1.5 and 2.0 with and without aperture correction respectively \nwhile \\cite{Spoon_etal2004} find a value of 2.4 for a sample of normal and starburst nuclei. This ratio is yet higher (3.2) in Galactic \\newion{H}{ii}\\ regions \nwhile highly embedded star-forming regions can lack PAH emission altogether \\citep{Peeters_etal2004}. \nThese increased ratios for starburst regions compared to normal SF galaxies are generally attributed to PAH destruction near the site of on-going SF\ndue to intense radiation fields, making PAHs poor tracers of SF (Peeters et al. 2004 and references therein). \nThe EW dependence of the log$(L_{TIR}\/L_{PAH}(7.7\\mu m))$ ratio is clearly seen within our sample in the upper \nleft panel of Figure \\ref{fig:residuals-ltir}. This cautions against the use of a single linear relation between PAH luminosity and $L_{TIR}$\\ \nfor galaxies of unknown physical properties. \n\nHowever independently of galaxy type we expect to find lower values than these studies which all made use of interpolation methods to extract\nthe PAH features. Using a Lorentzian profile fitting method comparable to PAHFIT for a sample of starburst-dominated LIRGS at $z\\sim 0.5-3$, \n\\cite{HernanCaballero_etal2009}\nfind mean log$(L_{TIR}\/L_{PAH})$ ratios of $1.92\\pm 0.25$, $1.42\\pm 0.18$ and $1.96\\pm 0.27$ for the 6.2, 7.7 and 11.3$\\mu$m\\ features respectively.\nThese ratios are \n2.6, 1.8, and 1.4 times larger than ours respectively,\ncloser than previous studies despite the quite different galaxy type considered. The wavelength gradient \ncan be explained in the context of selective PAH destruction.\n\nFinally we note that in our sample the total $6.2-33$$\\mu$m\\ PAH luminosity amounts \nto $\\sim 15\\%$ of the total IR luminosity for SF galaxies, $\\sim 11\\%$ for composite galaxies and $\\sim 8\\%$ for AGNs.\nThe 7.7$\\mu$m\\ feature alone accounts for $\\sim 40\\%$ of the total PAH emission. These fractional contributions\nare in good agreement with those found in the SINGS sample (S07). \n\n\n\\begin{figure}\n\\plotone{lha_ltir.eps} \n\\caption{{\\it Left:} the extinction and $r$-band aperture corrected H$\\alpha$\\ luminosity against the TIR and 24$\\mu$m\\ continuum\nluminosities. $\\kappa$ is defined as the median of log($L_{{\\rm H}\\alpha}^{corr}\/L_{IR}$) for the SF population alone. \nThe rms and slope of the linear regressions (solid lines) are also shown for the SF population. The green dashed lines indicate \nequality. The open blue squares and crosses are SINGS data (integrated values and galaxies centers respectively); The open green squares \nrepresent the \\cite{MoustakasKennicutt2006} (MK06) sample. The dotted lines are fits to the MK06 sample. \n{\\it Right:} same as in the left panels but using the smaller H$\\alpha$\\ aperture corrections computed by \\cite{B04} (see text for details). \nThe new correlations (solid lines) are steeper, in better agreement with data that do not require aperture corrections.\n\\label{fig:lha-ltir}}\n\\end{figure}\n\n\n\\section{MIR components and the Star-Formation Rate}\n\nThe TIR luminosity is a robust tracer of the SFR for very dusty starbursts, whose stellar emission is dominated by \nyoung massive stars and almost entirely absorbed by dust (typically galaxies with depleted PAH emission),\nbut for more quiescent and\/or less dusty galaxies such as those in the present sample, it can include a non negligible \ncontribution from dust heated by evolved stars (`cirrus emission') as well as miss a non negligible fraction of the young stars' \nemission that is not absorbed by dust \\citep{LonsdaleHelou1987}.\nFor normal spiral galaxies the contribution of non ionizing photons may actually dominate the dust heating over \\newion{H}{ii}\\ regions \n\\citep{Dwek_etal2000, Dwek2005} while low dust opacity makes these galaxies H$\\alpha$\\ and UV bright.\nThe tight correlations between MIR luminosities and $L_{TIR}$\\ indicate that the same caveats apply from the MIR to the FIR\n\\citep{Boselli_etal2004}.\n\n\\subsection{MIR dust and H$\\alpha$}\n\nH$\\alpha$\\ emission is a more direct quantifier of young massive stars - in the absence of AGN -\nbut inversely it must be corrected for the fraction that gets absorbed by dust. \nThe SDSS line fluxes are corrected for foreground (galactic) reddening using \\cite{ODonnell1994}.\nThe correction for intrinsic extinction is usually done using the Balmer decrement and an extinction curve to first order, \nor more accurately with higher order hydrogen lines \\citep{B04}. \nHere we correct the SDSS H$\\alpha$\\ fluxes in the usual simple way using the stellar-absorption\ncorrected H$\\alpha$\/H$\\beta$\\ ratio and a Galactic extinction curve. We assumed an intrinsic H$\\alpha$\/H$\\beta$\\ ratio of 2.86\n(case B recombination at electron temperature $T_e=10 000K$ and density $N_e=100~{\\rm cm}^{-3}$) and\n$R_V=A(V)\/E(B-V)=3.1$ (the mean value for the diffuse ISM). \nThe H$\\alpha$\\ attenuations range from 0.4 to 2.3 mag in the SF galaxy subsample with a median value of 1.1 mag, \nmeaning that between 10 and 70\\% of the H$\\alpha$\\ photons do {\\it not} get reemitted in the IR.\n\n\nThe SDSS H$\\alpha$\\ measurements also require fiber aperture corrections. \nHere again we apply the usual method which consists in scaling the fiber-measured H$\\alpha$\\ fluxes using the $r$-band Petrosian-to-fiber \nflux ratios \\citep{Hopkins_etal2003}. The mean value for these ratios is 3.5. \nThe left panels of Figure \\ref{fig:lha-ltir} shows the extinction and aperture corrected H$\\alpha$\\ luminosity, $L_{{\\rm H}\\alpha}^{corr}$, \nagainst the TIR and 24$\\mu$m\\ continuum luminosities (top and bottom panel respectively). \nThe rms and slope $a$ of the linear regressions (solid lines) are indicated for the SF population alone. \nThe logarithmic scaling factors $\\kappa$ indicated in each panel are defined as the median of log($L_{{\\rm H}\\alpha}^{corr}\/L_{IR}$) also\nfor the SF population alone. The green dashed lines indicate equality (${\\rm log}(L_{{\\rm H}\\alpha}^{corr})={\\rm log}(L_{IR}) +\\kappa$). \nOverlaid are the MK06 data (open green squares) and SINGS data (open blue squares for the integrated measurements, \ncrosses for 20''x20'' galaxy center measurements), taken from \\cite{Kennicutt_etal2009} (hereafter K09). \n\nOur median $L_{TIR}$\\ to $L_{{\\rm H}\\alpha}^{corr}$ logarithmic ratio of $2.27 \\pm 0.2$ \nis in good agreement with the ratio of SFR calibration coefficients derived by \\cite{Ken98} for H$\\alpha$\\ and $L_{TIR}$\\ respectively,\nimplying that $L_{TIR}$\\ (and the MIR components that correlate with it) may be reasonable SFR tracers in normal SF galaxies \nafter all. This may actually be a coincidence due to the fact that the cirrus emission and the unattenuated ionizing flux\nroughly cancel each other in massive spiral galaxies (K09 and references therein).\nOur $L_{TIR}\/L_{{\\rm H}\\alpha}^{corr}$ ratio is also in good agreement with the MK06 sample ($2.32\\pm 0.19$). \nIn recent years several groups have exploited the capabilities of {\\it Spitzer} to re-investigate the relationship \nbetween MIR components and H$\\alpha$\\ emission. \nOur mean $\\nu L_\\nu(24\\mu m)$ to $L_{{\\rm H}\\alpha}^{corr}$ logarithmic ratio of $1.31 \\pm 0.14$ is in good agreememt with \nthese \\citep[e.g.][]{Wu_etal2005,Zhu_etal2008,Kennicutt_etal2009}, \nas is the higher mean $\\nu L_\\nu(24\\mu m)\/L_{{\\rm H}\\alpha}^{corr}$ \nratio for composite galaxies \\citep{Zhu_etal2008}. However the slope of our correlations for SF galaxies tend to be more linear \nthan found in these studies (the dotted lines in Figure \\ref{fig:lha-ltir} show fits to the MK06 sample).\nYet non linearity is expected from the positive correlation between attenuation and SFR. \nGiven the good agreement between our and the MK06 samples in the IR (cf. the $L_{TIR}$\\ -- $\\nu L_\\nu(24\\mu m)$ correlation\nin the previous section), differences in H$\\alpha$\\ measurements must be responsible for the discrepancy in slopes. In particular it is possible \nthat aperture corrections, which are not needed for the MK06 sample, are overestimated for all or a fraction of our galaxies. This would\nbe the case if SF is more intense at the center of the galaxies and\/or more attenuated, a common occurence in spiral galaxies\n\\citep[e.g.][]{Calzetti_etal2005}. \n \nAs a test we consider the smaller aperture corrections derived by \\cite{B04} (hereafter B04)\nthat rely on the likelihood distribution of the specific SFR as a function of colors. These corrections\ndepend on the galaxy colors outside the fiber which are not necessarily the same as inside, and are on average\n$\\sim 1.6$ smaller than the $r$-band corrections for SF galaxies.\nThe right panels of Figure \\ref{fig:lha-ltir} shows the same relations as in the left panels using these smaller aperture \ncorrections. The new correlations (solid lines) are indeed steeper while the higher mean $L_{TIR}\/L_{{\\rm H}\\alpha}^{corr}$ and \n$\\nu L_\\nu(24\\mu m)\/L_{{\\rm H}\\alpha}^{corr}$ logarithmic ratios of $2.47\\pm 0.14 $ and $1.50\\pm 0.15$ respectively \nremain within the range of the MK06 sample. \n\n\\begin{figure}\n\\plotone{tir_Aha.eps}\n\\caption{The ratio of $L_{TIR}$\\ to observed H$\\alpha$\\ luminosity as a function of H$\\alpha$\\ attenuation measured from the Balmer decrement\n(SF galaxies only).\nThe top panel assumes conventional $r$-band aperture corrections for H$\\alpha$, while the bottom panel assumes the B04\naperture corrections (see text for details). The solid lines are best fits to Eq. \\ref{eq:Aha} by K09 \nfor the SINGS+MK06 samples ($a_{TIR}=0.0024 \\pm 0006$).\nThe dashed lines are best fits to the SSGSS sample ($a_{TIR}=0.0033 \\pm 0.0014$ in the top panel, $0.0020 \\pm 0.0006$ in the bottom panel). \nThe smaller aperture corrections used in the bottom panel significantly improves the fit and the agreement between the three samples.\n\\label{fig:tir-Aha}\n}\n\\end{figure}\n\nMore dramatic is the effect on the relation between H$\\alpha$\\ attenuation and the ratio of $L_{TIR}$, or other IR luminosity,\nto $L_{{\\rm H}\\alpha}^{obs}$, the `observed' (aperture-corrected but attenuation-uncorrected) H$\\alpha$\\ luminosity.\nThis relation is shown in Figure \\ref{fig:tir-Aha} for both types of aperture correction. \nK09 modelled the H$\\alpha$\\ attenuation as: \n\\begin{equation}\nA_{{\\rm H}\\alpha}=2.5~{\\rm log}\\left[1+ a_{IR} {L_{IR}\\over L_{{\\rm H}\\alpha}^{obs}}\\right],\n\\label{eq:Aha}\n\\end{equation}\nequivalent to $L_{{\\rm H}\\alpha}^{corr}=L_{{\\rm H}\\alpha}^{obs}+ a_{IR} L_{IR}$. This energy balance approach \nwas introduced by \\cite{Calzetti_etal2007}, \\cite{Prescott_etal2007} and \\cite{Kennicutt_etal2007} to correct H$\\alpha$\\ fluxes\nbut has long been used to estimate UV attenuations from the $L_{TIR}\/L_{FUV}$ ratios \\citep[e.g.][]{Meurer99}.\nThe solid lines in both panels of Figure \\ref{fig:tir-Aha} show the best fits by K09 for the SINGS and MK06 samples\n($a_{TIR}=0.0024 \\pm 0.006 $). The dashed lines are best fits for the SSGSS sample ($a_{TIR}=0.0033 \\pm 0.0014$ in the top panel\nand $0.0020 \\pm 0.0006$ in the bottom panel). The smaller aperture corrections used in the bottom panel significantly improves \nthe fit and the agreement between the three samples. Unless otherwise stated we now assume these corrections in the rest of the\npaper. \n\n\\begin{figure}\n\\plotone{lha_sample_ken2.eps}\n\\caption{$L_{{\\rm H}\\alpha}^{obs}+a_{IR} L_{IR}$ to $L_{{\\rm H}\\alpha}^{corr}$ ratios as a function of $L_{{\\rm H}\\alpha}^{corr}$\nfor the TIR, 24$\\mu$m\\ continuum, 7.7$\\mu$m\\ and 17$\\mu$m\\ PAH luminosities, assuming the B04 aperture corrections for H$\\alpha$. \nThe $a_{IR}$ coefficients are indicated at the top left of each panel. For the TIR and 24$\\mu$m\\ luminosities, \n$a_{TIR}=0.0024$ and $a_{24}=0.020$ are best fits to Eq. \\ref{eq:Aha} for the SINGS+MK07 samples by K09.\nFor the PAH lumosities, $a_{7.7\\mu m}=0.034 \\pm 0.012$ and $a_{17\\mu m}=0.320 \\pm 0.159$ are best fits to Eq. \\ref{eq:Aha} for \nthe SSGSS sample. \n\\label{fig:lha-sample-ken}\n}\n\\end{figure}\n\nFigure \\ref{fig:lha-sample-ken} shows the $L_{{\\rm H}\\alpha}^{obs}+ a_{IR}L_{IR}$ to $L_{{\\rm H}\\alpha}^{corr}$ ratios as a function\nof $L_{{\\rm H}\\alpha}^{corr}$ for the TIR, 24$\\mu$m\\ continuum, 7.7$\\mu$m\\ and 17$\\mu$m\\ PAH luminosities.\nThe $a_{IR}$ coefficients are indicated at the top left of each panel for the SF population. The rms and Pearson coefficients are also \nindicated for the SF population.\nFor the TIR and 24$\\mu$m\\ luminosities, $a_{TIR}=0.0024\\pm 0.0006$ and $a_{24}=0.020\\pm 0.005$ are best fits to Eq. \\ref{eq:Aha} \nfor the SINGS+MK06 samples by K09. The combinations of $L_{{\\rm H}\\alpha}^{obs}$\\ and $L_{TIR}$\\ or $\\nu L_\\nu(24\\mu m)$ provide a very\ntight (rms=0.08) and perfectly linear fit to the total H$\\alpha$\\ luminosity\nfor all samples combined over 5 dex in luminosity, as was shown by K09 for\nthe SINGS and MK06 samples. Composite galaxies follow nearly the same\nrelation save for 2 over-corrected outliers. Although more\nscattered AGNs also follow the SF population. For the PAH \nluminosities, $a_{7.7\\mu m}=0.034 \\pm 0.012$ and $a_{17\\mu m}=0.320 \\pm 0.159$ are best fits to Eq. \\ref{eq:Aha} for the present sample. \nHere also the combinations of $L_{{\\rm H}\\alpha}^{obs}$ and PAH luminosities provide a much improved fit to the total H$\\alpha$\\ luminosity\ncompared to the raw $L_{PAH}\/L_{{\\rm H}\\alpha}^{corr}$ relations (not shown), including for composite galaxies and AGNs\nwith the exception of a few outliers, most notably a composite galaxy with no H$\\alpha$\\ attenuation and a large IR\/H$\\alpha$\\ ratio (\\#84).\n\n\nThe same exercise can be performed with similarly good results with any other MIR dust components. The $a_{TIR}$ and $a_{MIR}$ coefficients \nfor the SSGSS sample are listed in Table \\ref{table:coeffs}. Note that $a_{MIR} \\sim 10^{\\kappa}a_{TIR}$, using the scaling factors \n$\\kappa=<{\\rm log} (L_{TIR}\/L_{MIR})>$ listed in the first column \nof Table \\ref{table:coeffs}. Although the $\\kappa$ factors and $a_{TIR}$ depend on the specific definition of $L_{TIR}$ and on the models \nused to compute it, the $a_{MIR}$ coefficients for specific dust components or MIR broadband luminosities, which are easier to measure \nthan the total IR, are independent of these choices. \n\nAs stated above the smaller B04 corrections seem to be more appropriate than the usual $r$-band corrections given the agreement with data \nthat do not require aperture corrections. However they are not trivially calculated (see B04 for details of the modeling).\nMore importantly H$\\alpha$\\ is often not easily measurable at all. It is therefore useful to provide SFR recipes based on a single IR \nquantity, or on a combination of IR and UV measurements (see next section) as UV is more easily obtained at high redshifts. \nTable \\ref{table:coeffs} lists the mean $L_{{\\rm H}\\alpha}^{corr}\/L_{MIR}$ ratios of the SF population for the various MIR components.\nKeeping in mind the non linearities and scatter in the true relations,\nSFRs can be estimated from these approximate H$\\alpha$\\ luminosities using K09's calibration\n(derived from the latest Starburst99 models and assuming a Kroupa IMF and solar metallicity):\n\\begin{equation}\nSFR_{{\\rm H}\\alpha}~({\\rm M_\\odot~yr^{-1}})=5.5\\times 10^{-42} ~L_{{\\rm H}\\alpha}^{corr}~({\\rm ergs~s^{-1}}).\n\\label{eq:sfrha}\n\\end{equation}\nAs an example, the SFR derived from the MIPS 24$\\mu$m\\ luminosity would be \nSFR$({\\rm M_\\odot~yr^{-1}})= 6.5\\times 10^{-10} \\nu L_{\\nu}(24\\mu m)\/L_\\odot$ consistent with Rieke et al. (2009) for\ngalaxies in the range of TIR luminosities of the present sample.\n\n\\begin{figure}\n\\plotone{sfre_sample2.eps}\n\\caption{$L_{IR}\/SFR_e$ ratios as a function of $SFR_{e}$\\ (B04, see text for detail) where $L_{IR}$ equals the TIR, \n24$\\mu$m\\ continuum, 7.7$\\mu$m\\ and 17$\\mu$m\\ PAH luminosities. $\\kappa$ is defined as the median of log($SFR_e\/L_{MIR}$). \nOnly SF galaxies for which $SFR_{e}$\\ is computed from the Balmer lines are shown. \n\\label{fig:sfre-sample}\n}\n\\end{figure}\n\nAs Eq. \\ref{eq:sfrha} was shown by B04 to underestimate the SFR of massive galaxies and thus may not be appropriate \nfor this sample or at high $z$, \nwe also add to Table \\ref{table:coeffs} $SFR_e\/L_{MIR}$ calibrations where $SFR_{e}$\\ is the SFR derived by these \nauthors as follows: they computed SFR likelihood distributions of SF galaxies in the SDSS spectroscopic sample \nby fitting all strong emission lines simultaneously using the \\cite{CharlotLonghetti01} \nmodels and assuming a Kroupa IMF. Dust was accounted for using the \\cite{CharlotFall00} \nmulticomponent model which provides a consistent treatment of the attenuation of both continuum and emission \nline photons. $SFR_{e}$\\ refers to the medians of these SFR distributions.\nIn this model, the H$\\alpha$\\ attenuation increases with mass while the ratio of $L_{{\\rm H}\\alpha}^{corr}$ \nto SFR decreases with mass so that the same observed H$\\alpha$\\ luminosity signals a noticeably higher SFR \nin higher mass galaxies than predicted from Kennicutt's relation. We refer to B04 for full details.\n$SFR_{e}$\\ is found to be in good agreement with Eq. \\ref{eq:sfrha} for average local galaxies but \ndiverges from it for higher mass, higher metallicity galaxies such as found in the present sample where\n$SFR_{e}$\\ is on average twice larger {\\it within the SDSS fiber} than derived from the Kennicutt relation.\nHowever the aperture corrections in this study being $\\sim 1.6$ smaller than derived from the $r$-band \nmagnitudes for SF galaxies, the total $SFR_{e}$\\ are only $\\sim 1.3$ times larger than derived conventionally\nusing the Balmer decrements, $r$-band aperture corrrections and Eq. \\ref{eq:sfrha}. \nFor composite galaxies and AGNs, $SFR_{e}$\\ is not estimated from the emission lines which are contaminated by AGN \nemission, but in a statistical way based on the correlation between D$_n$(4000)\\ and the specific SFR. We exclude\nthose for clarity. \n\n\n\n\nFigure \\ref{fig:sfre-sample} shows the relations between $SFR_{e}$\\ and the $L_{IR}\/SFR_e$ ratios\nfor the TIR, 24$\\mu$m\\ continuum, 7.7$\\mu$m\\ and 17$\\mu$m\\ PAH luminosities. \nAs in previous figures the correlation parameters are quoted at the bottom right of each panel. \nThese correlations are more scattered and less linear (higher rms and Pearson coefficient) than with $L_{H\\alpha}^{corr}$.\nThe attenuations underlying $SFR_{e}$\\ being larger than those derived from the Balmer decrement for \nmassive galaxies, the $SFR_{e}$\\ to $L_{TIR}$\\ ratio: \n$SFR_e=3.98\\times 10^{-44}L_{TIR}$ is very close to that of Kennicutt et al. (1998) for opaque starburst galaxies\n(taking into account the difference in IMFs). The $SFR_e\/L_{MIR}$ calibrations are listed in Table \\ref{table:coeffs}.\n\n\n\n\\subsection{MIR dust and UV}\n\nTurning now to UV data where dust attenuation is an inevitable issue, we once again follow an energy balance\napproach \\citep{Meurer99, Gordon_etal2000, Kong04, Buat05, Cortese_etal2008, Zhu_etal2008, Kennicutt_etal2009}.\nSFRs can be estimated from dust corrected FUV luminosities using the following calibration by K09 assuming a \nKroupa IMF, solar metallicity, and adjusted to the GALEX FUV filter ($\\lambda_{eff} = 1538$\\AA).\n\\begin{equation}\nSFR_{FUV}~({\\rm M_\\odot~yr^{-1}})=4.5\\times 10^{-44} ~ L_{FUV}^{corr}~({\\rm ergs~s^{-1}}).\n\\label{fig:sfruv}\n\\end{equation}\nwhere $L_{FUV}^{corr}= \\nu L_{\\nu}^{corr} (1538{\\rm \\AA})$ is the dust-corrected GALEX FUV luminosity.\n\n\\begin{figure}\n\\plotone{Afuv_irx.eps}\n\\caption{The FUV attenuations of the SF population in the GALEX FUV band derived from Eq. \\ref{eq:Afuv-sfr} as a function \nof $L_{TIR}\/ \\nu L_{\\nu}^{obs}(1530\\AA)$ (the IRX) assuming SFR $=SFR_{{\\rm H}\\alpha}$ (Eq. \\ref{eq:sfrha}) and SFR=$SFR_{e}$\\ \n(top and bottom panels respectively). \nThe dotted line is a theorical relation by \\cite{Buat05}; the dashed lines shows a model derived \nby \\cite{Cortese_etal2008} for galaxies with $FUV-g=2.9$ corresponding to the mean color of our sample; \nthe solid lines are best fits to Eq. \\ref{eq:Afuv-irx}.\n\\label{fig:Afuv-irx}\n}\n\\end{figure}\n\\begin{figure}\n\\plotone{fuv_sample_ken2.eps}\n\\caption{The ratios of FUV to H$\\alpha$\\ SFRs against the H$\\alpha$\\ SFR: H$\\alpha$\\ is corrected using the Balmer decrement and the B04 aperture \ncorrections while the FUV is dust corrected using Eqs. \\ref{eq:Afuv-sfr} and \\ref{eq:Afuv-irx} for the TIR, 24$\\mu$m\\ continuum, 7.7$\\mu$m\\ \nand 17$\\mu$m\\ PAH luminosities. \n\\label{fig:fuv-sample-ken}\n}\n\\end{figure}\n\n\nAssuming equality with a known SFR estimate (e.g. $SFR_{{\\rm H}\\alpha}$ or $SFR_{e}$), we derive FUV attenuations as follows:\n\\begin{equation}\nA_{FUV}\t= 2.5~{\\rm log}\\left[ {SFR \\over SFR_{FUV}(L_{FUV}^{obs})} \\right].\n\\label{eq:Afuv-sfr}\n\\end{equation}\nwhere $L_{FUV}^{obs}=\\nu L_{\\nu}^{obs}(1538{\\rm \\AA})$ is the observed FUV luminosity in ${\\rm ergs~s^{-1}}$.\nFigure \\ref{fig:Afuv-irx} shows the FUV attenuations of the SF subsample derived from Eq. \\ref{eq:Afuv-sfr}\nas a function of $L_{TIR}\/ L_{FUV}^{obs}$ (known as the infrared excess or IRX) assuming assuming SFR=$SFR_{{\\rm H}\\alpha}$\\ (Eq. \\ref{eq:sfrha}, top panel) \nand $SFR_{e}$\\ (bottom panel).\nThe median FUV attenuations are 1.9 and 2.8 magnitudes respectively, corresponding to $\\sim$ 83 and 92\\% of the FUV light being \nabsorbed by dust (note that assuming conventional $r$-band aperture corrections for H$\\alpha$\\ yields exactly intermediate values). \nThe dotted line is a theoretical relation by \\cite{Buat05}; the dashed lines shows a model derived \nby \\cite{Cortese_etal2008} for galaxies with $FUV-g=2.9$ corresponding to the mean color of our sample \n(these authors modeled the dependence of the IRX\/$A_{FUV}$ relation with the age of the underlying stellar populations, \nor specific SFR, or color). The solid lines are best fits of the form: \n\\begin{equation}\nA_{FUV}\t= 2.5~{\\rm log}\\left[ 1+ b_{IR} {L_{IR} \\over L_{FUV}^{obs}} \\right] \n\\label{eq:Afuv-irx}\n\\end{equation}\nequivalent to $L_{FUV}^{corr}= L_{FUV}^{obs}+ b_{IR} L_{IR}$, i.e.\n$SFR= 4.5\\times10^{-44} [L_{FUV}^{obs}+ b_{IR} L_{IR}]$, following K09's method.\nOur best fit parameters are $b_{TIR}=0.317$ and 0.729 in the top and bottom panels respectively. However\nall three models are poor in the bottom panel. FUV attenuations assuming $SFR_{e}$\\ are best modeled by a linear\nfunction of log(IRX) or FUV-optical colors \\citep{Treyer_etal2007}. \nAssuming $SFR_{{\\rm H}\\alpha}$\\ (top panel) the FUV attenuations are well fit both by \\cite{Cortese_etal2008} and by Eq. \\ref{eq:Afuv-irx}.\nIn this case a linear combination of $L_{FUV}^{obs}$ and $L_{TIR}$\\ or $L_{MIR}$ recovers $SFR_{{\\rm H}\\alpha}$\\ with low scatter as\nshown in Figure \\ref{fig:fuv-sample-ken} using the TIR, 24$\\mu$m\\ continuum, 7.7$\\mu$m\\ and 17$\\mu$m\\ luminosities. \nAs with H$\\alpha$\\ in the previous section, similarly good corrections can be achieved using other MIR \ncomponents. The $b_{TIR}$ and $b_{MIR}$ coefficients are listed in Table \\ref{table:coeffs}. \n\n\\subsection{Neon emission lines}\n\n\\begin{figure} \n\\plotone{neon_sample2.eps}\n\\caption{The ratio of Ne luminosity (defined in Eq. \\ref{eq:cloudy}) to $L_{TIR}$\\ (a), $L_{{\\rm H}\\alpha}^{corr}$\\ (b) and $SFR_{e}$\\ (c) \nas a function $L_{TIR}$, $L_{{\\rm H}\\alpha}^{corr}$\\ and $SFR_{e}$\\ respectively. Only SF galaxies with measured metallicity are shown. \nThe lower right panel (d) shows the ratio of the linear combination of $L_{{\\rm H}\\alpha}^{obs}$\\ and $L_{Ne}$ that best fits $L_{{\\rm H}\\alpha}^{corr}$\\ \n(see text for details) to $L_{{\\rm H}\\alpha}^{corr}$\\ against $L_{{\\rm H}\\alpha}^{corr}$. \n\\label{fig:neon}\n}\n\\end{figure}\n\nAs put forward by \\cite{HoKeto2007}, [\\newion{Ne}{ii}]12.8$\\mu$m\\ is an excellent tracer of ionizing stars, being an abundant and dominant species \nin \\newion{H}{ii}\\ regions, quite insensitive to density, as well as to dust given its long wavelength. [\\newion{Ne}{iii}]15.5$\\mu$m\\ has similar properties\nbut can be the dominant species in e.g. low-mass, low-metallicity galaxies \\citep{OHalloran_etal2006, Wu_etal2006}. \nThus Ne emission is expected to be directly comparable to the dust corrected H$\\alpha$\\ emission. \nUsing the CLOUDY code \\citep{CLOUDY1998}, we find that the ionizing flux from stars hotter than 10K is best \nrepresented by the following weighted linear combination of [\\newion{Ne}{ii}]12.8$\\mu$m\\ and [\\newion{Ne}{iii}]15.5$\\mu$m\\\nalso including a metallicity dependence:\n\\begin{equation}\n{\\rm H}\\alpha = (8.8 {\\rm [Ne~II] } + 3.5 {\\rm [Ne~III] } ) \\times (1\/Z)^{0.8},\n\\label{eq:cloudy}\n\\end{equation}\nwhere $Z$ is the metallicity in solar units. We use the right hand side of this equation to define the neon flux and luminosity, $L_{\\rm Ne}$.\n\nFor the sake of comparison with the study of \\cite{HoKeto2007} who used $L_{TIR}$\\ as SFR estimate, as well as a straight sum of [\\newion{Ne}{ii}]12.8$\\mu$m\\ and [\\newion{Ne}{iii}]15.5$\\mu$m,\nwe note that our $L$([\\newion{Ne}{ii}]12.8$\\mu$m+[\\newion{Ne}{iii}]15.5$\\mu$m) to $L_{TIR}$\\ ratio is consistent with the IRS dataset used by these authors\n\\citep{OHalloran_etal2006, Wu_etal2006}. Our $L$([\\newion{Ne}{ii}]12.8$\\mu$m) to $L_{TIR}$\\ ratio is larger but this may be \nexplained by the large number of low metallicity galaxies in the samples used, in particular the \\cite{Wu_etal2006}\ndataset which specifically targets low-metallicity blue compact dwarf galaxies for which [\\newion{Ne}{iii}]15.5$\\mu$m\\ is the dominant\nNe species (cf. Figure \\ref{fig:Ne-Z}). \n\nThe left panels of Figure \\ref{fig:neon} shows the $L_{\\rm Ne}$\/$L_{TIR}$\\ and $L_{\\rm Ne}$\/$L_{{\\rm H}\\alpha}^{corr}$\\ ratios as a function of $L_{TIR}$\\ (a) and $L_{{\\rm H}\\alpha}^{corr}$\\ (b)\nrespectively. \nOnly SF galaxies with measured metallicity are represented (85\\%). Surprisingly $L_{\\rm Ne}$\\ behaves much like the MIR dust components.\nIt traces fairly linearly and tightly the total IR luminosity while we can define $a_{Ne}=0.073\\pm0.030$ using Eq. \\ref{eq:Aha} such \nthat $L_{{\\rm H}\\alpha}^{obs}+ a_{Ne} L_{Ne}$ provides the tightest and most linear correlation with $L_{{\\rm H}\\alpha}^{corr}$, as shown in\nthe lower right panel (d) of Figure \\ref{fig:neon}. Likewise we can define $b_{Ne}=11.05\\pm 5.13$ such that \n$4.5\\times10^{-44} [L_{FUV}^{obs}+ b_{Ne} L_{Ne}]$ provides a good fit to $SFR_{{\\rm H}\\alpha}$.\nThe upper right panel (c) shows the $L_{\\rm Ne}$\\ to $SFR_{e}$\\ ratio against $SFR_{e}$, which is significantly more\nscattered than the previous relations as with the MIR dust components. This correlation implies the following calibration:\n\\begin{equation}\nSFR({\\rm M_\\odot~yr^{-1}})=1.26\\times 10^{-42} ~ L({\\rm Ne})~({\\rm ergs~s^{-1}}).\n\\end{equation}\nThe $a_{Ne}$ and $b_{Ne}$ coefficient as well as the median $L_{TIR}$\/$L_{\\rm Ne}$, $L_{{\\rm H}\\alpha}^{corr}$\/$L_{\\rm Ne}$ \\ and $SFR_{e}$\/$L_{\\rm Ne}$\\ ratios are reported in Table \\ref{table:coeffs}.\n\n\\subsection{Molecular Hydrogen lines}\n\n\\begin{figure}\n\\plotone{H2_sample2.eps}\n\\caption{The ratio of H$_2$ luminosity - defined as the sum of the $S(0)$ to $S(2)$ rotational lines of H$_2$ - \nto $L_{TIR}$\\ (a), $L_{{\\rm H}\\alpha}^{corr}$\\ (b) and $SFR_{e}$\\ (c) as a function $L_{TIR}$, $L_{{\\rm H}\\alpha}^{corr}$\\ and $SFR_{e}$\\ respectively.\nThe lower right panel (d) shows the ratio of the linear combination of $L_{{\\rm H}\\alpha}^{obs}$\\ and $L_{{\\rm H}_2}$ that best fits $L_{{\\rm H}\\alpha}^{corr}$\\ \n(see text for details) to $L_{{\\rm H}\\alpha}^{corr}$\\ against $L_{{\\rm H}\\alpha}^{corr}$. \n\\label{fig:H2}\n}\n\\end{figure}\n\nThe rotational H$_2$ lines are fainter than the [NeII]12.9$\\mu$m, [\\newion{Ne}{iii}]15.5$\\mu$m\\ and [\\newion{S}{iii}]18.7$\\mu$m\\ lines for most galaxies in our sample but \nmolecular hydrogen represents a significant mass fraction of the ISM in normal galaxies. The main excitation source of the rotational transitions \nis expected to be FUV radiation from massive stars in PDRs \\citep[and references therein]{HollenbachTielens1997}, therefore these lines\nalso trace SF. The first study of warm molecular hydrogen ($T\\sim 100 - 1000$ K) in the nuclei of normal, low luminosity galaxies \nwas presented by \\cite{Roussel_etal2007} (hereafter R07) using the SINGS sample. A major result of their work is the tight correlation \nbetween the sum of the $S(0)$ to $S(2)$ rotational lines (noted $F(S0-S2)$) and the PAH emission, \nwith a $F(S0-S2)$\/PAH ratio insensitive to the intensity of the radiation field. This correlation is interpreted \nas supporting the origin of H$_2$ excitation within PDRs (defined by \\cite{HollenbachTielens1997} as including the neutral ISM illuminated by \nFUV photons), with fluorescence as the dominant excitation mechanism.\n\nOur median logarithmic ratios of $L(S0-S2)$ to the TIR, 24$\\mu$m\\ MIPS band and 7.7$\\mu$m\\ PAH luminosities\nfor the SF population are $-3.17\\pm 0.19$, $-2.18\\pm 0.23$ and $-1.95\\pm 0.19$ respectively.\nThe first two ratios are 1.6 and 1.8 times larger respectively than those of R07 for \\newion{H}{ii}\\ nuclei \n(taking into account that R07 assumed a filter width of 3.1THz instead of $\\nu F_{\\nu}$ for the 24$\\mu$m\\ band). \nOn the other hand our $F(S0-S2)$\/PAH ratio \nusing the stellar component subtracted 8$\\mu$m\\ IRAC flux instead of the PAHFIT extracted feature following R07, is only 1.2 times higher than that \nof R07. These differences are within uncertainties but the gradients may also reflect real differences between \\newion{H}{ii}\\ nuclei and disks (warm\nH$_2$ more abundant in disks), as well as support the physical link between warm H$_2$ and PAH emissions suggested by R07. \n\nUnlike R07 we do not find significantly higher $L(S0-S2)\/L_{TIR}$ or $L(S0-S2)\/L_{24}$ ratios for AGNs (see also the top panel of Figure \n\\ref{fig:lineratios}).\nThis may also be due to the much lower AGN contribution when disks are included. \nR07 interpret the higher AGN ratios as an excess of H$_2$ emission, attributed to additional mechanisms exciting H$_2$ molecules in AGNs. \nThe $L(S0-S2)\/L_{PAH}(7.7\\mu m)$ ratio does show a significant excess for AGNs, however this excess correlates\nwith PAH EWs, suggesting that depleted PAHs in AGNs contribute in part to the effect. \nOur dispersion for the $L(S0-S2)\/L_{PAH}(7.7\\mu m)$ ratio is also comparable to the other two while R07 find it to be significantly tighter\nin \\newion{H}{ii}\\ nuclei, especially than the $L(S0-S2)\/L_{24}$ ratio. Our results suggest that local complexities are largely washed out on galactic scale \nand that warm molecular hydrogen traces dust in all its forms when considering integrated measurements. \n\nThe left panels of Figure \\ref{fig:H2} show the $L(S0-S2)$ to $L_{TIR}$\\ and $L_{{\\rm H}\\alpha}^{corr}$\\ ratios as a function of $L_{TIR}$\\ (a) and $L_{{\\rm H}\\alpha}^{corr}$\\ (b).\nLike $L_{\\rm Ne}$, $L(S0-S2)$ traces reasonably linearly and tightly the total IR luminosity while the correlation with $L_{{\\rm H}\\alpha}^{corr}$\\ is improved with\na linear combination of $L_{{\\rm H}\\alpha}^{obs}$ and $L_{H_2}$ using Eq. \\ref{eq:Aha} ($a_{H_2}=3.16\\pm 2.74$) (c).\nLikewise the upper right panel (d) shows the $L(S0-S2)$ to $SFR_{e}$\\ ratio against $SFR_{e}$, which implies the following calibration:\n\\begin{equation}\nSFR({\\rm M_\\odot~yr^{-1}})= 6.31\\times 10^{-41} ~ L(S0-S2)~({\\rm ergs~s^{-1}}).\n\\end{equation}\nAll the coefficients are reported in Table \\ref{table:coeffs}.\n\n\n\\section{Summary and Conclusions}\n\nWe present a MIR spectroscopic survey of 100 `normal' galaxies at $z\\sim 0.1$ with the goal\nof investigating the use of mid-infrared PAH features, continuum and emission lines as probes of their \nstar-formation and AGN activity. Available data include GALEX UV photometry, \nSDSS optical photometry and spectroscopy, and {\\it Spitzer} near to far-infrared photometry. \nThe optical spectroscopic data in particular allow us to classify these galaxies into star-forming, \ncomposite and AGNs, according to the standard optical ``BPT'' diagnotic diagram.\nThe MIR spectra were obtained with the low resolution modules of the {\\it Spitzer} IRS and decomposed \ninto unattenuated features and continuum using the PAHFIT code of \\cite{Smith_etal2007}. \nA notable feature of this decomposition method is to extract a much larger PAH contribution\n(and proportionally smaller continuum contribution) from the total flux compared to standard spline fitting methods \nwhich anchor the continuum in the wings of the features where non negligible PAH power remain. \nAs a consequence, the PAH equivalenth widths are not only larger but extend over a considerably larger dynamic range\n(e.g. the equivalent widths of the 6.2 and 7.7$\\mu$m\\ PAH features in our sample extend to 15 and 32$\\mu$m\\ \nrespectively).\n\nWe study the variations of the various MIR spectral components as a function of the optically derived \nage (as measured by the 4000\\AA\\ break index), radiation field hardness (as measured by the\n[\\newion{O}{iii}]$\\lambda 5007$\/H$\\beta$\\ ratio) and metallicity (as measured by [\\newion{N}{ii}]$\\lambda 6583$\/H$\\alpha$\\ ratio) of the galaxies. \nSystematic trends are found despite the lack of extreme objects in the sample, in particular\nbetween PAH strength at low wavelength and gas phase metallicity, and between the ratio of high \nto low excitation lines (e.g. [\\newion{O}{iv}]25.9$\\mu$m\/[\\newion{Ne}{ii}]12.8$\\mu$m) and radiation field hardness. \nThese trends confirm earlier results detected in sources with higher surface brightnesses such as ULIRGS, \nstrong AGNs and \\newion{H}{ii}\\ nuclei. Our results are consistent with the selective destruction in AGN radiation \nfields of the smallest PAH molecules efficient at producing the low wavelength PAH features (6.2 to 8.6$\\mu$m). \nThey also suggest that radiation fields harder than those in the present sample would also destroy larger PAH \nmolecules responsible for the longer wavelength features (11.3 to 17$\\mu$m). Aging galaxies also tend to\nshow weaker low wavelength PAH features, consistent with their main origin in star-forming regions.\n\nWe revisit the MIR diagnostic diagram of Genzel et al. (1998) relating PAH equivalent widths and \n[\\newion{Ne}{ii}]12.8$\\mu$m\/[\\newion{O}{iv}]25.9$\\mu$m\\ emission line ratios. Based on the strongest trends we observed between these\nmeasurements and optical emission line ratios and thanks to the extended range of equivalent widths \nprovided by PAHFIT, we find this diagnostic to closely resemble the optical ``BPT'' diagram,\nwith a much improved resolving power for normal galaxies than previously found based on spline derived equivalent widths.\nA mixed region of star-forming and composite galaxies remains, which may be revealing obscured AGNs \nin a large fraction of the optically defined `pure' star-forming galaxies.\n\n\nWe find tight and nearly linear correlations between the total infrared luminosity of star-forming galaxies and the luminosity \nof individual MIR components, including PAH features, continuum, neon emission lines and molecular hydrogen lines.\nThis implies that these individual MIR components are good gauges of the total dust emission on galactic scale\ndespite different spatial and physical origins on the scale of star-forming regions. Following the approach of \\cite{Kennicutt_etal2009}\nbased on energy balance arguments, we show that like the total infrared luminosity, these individual components can \nbe used to estimate dust attenuation in the UV and in the H$\\alpha$\\ lines. Given the non negligible attenuation in\nthese IR selected galaxies, the correlations between the MIR and dust corrected H$\\alpha$\\ luminosities can also\nprovide first order estimates of the SFR. We thus propose average scaling relations between the various\nMIR components and H$\\alpha$\\ derived star-formation rates. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\n\n\n\\section{Introduction}\n\nGround-based gamma-ray telescopes rely on the imaging atmospheric Cherenkov technique (IACT), which uses the Cherenkov photons emitted by the charged particles of an extensive air shower (EAS) in order to reconstruct the arrival directions and energies of the initial photon or charged particle. The Cherenkov photons are detected on the ground with telescopes equipped with large mirrors. The image obtained by the telescope camera is then used to determine the type of the primary particle inducing the EAS, its energy and incoming direction. \\\\\n\nNowadays Cherenkov telescope cameras~\\cite{HESS-camera}~\\cite{MAGIC-camera} use most often Photo Multiplier Tubes (PMTs). The PMTs offer great sensitivity in the UV band and enable the detection of the faint and short lived flashes of Cherenkov light produced in the atmosphere. However, the PMTs have a shortened life time when exposed to background light. For this reason most current gamma-ray observations are performed in the darkest conditions while avoiding nights of high-Moon intensity. This reduces the duty cycle of the telescopes. \\\\\n\nTo increase the duty cycle of Cherenkov telescopes, Silicon Photo Multiplier (SiPM) based Cherenkov camera have been built and operated by the FACT collaboration~\\cite{FACT}. The FACT collaboration showed that SiPM-based cameras can be operated without damage in high night-sky background conditions. The recent progress in SiPM technology, such as the increased photodetection efficiency in the UV band, the improved photoelectron resolution and the reduction in optical crosstalk, have allowed SiPMs to be considered as a replacement for PMTs in next-generation Cherenkov cameras~\\cite{Otte:2016aaw}. \\\\\n\nThe performances of the instrument presented here are compared to the highly demanding requirements set by the next-generation ground-based gamma-ray observatory: the Cherenkov Telescope Array (CTA)~\\cite{CTA-concept}.\n \n\\section{The SST-1M camera and its Calibration Test Setup (CTS)}\n\nThe SST-1M camera~\\cite{CameraPaperHeller2017} is a SiPM-based Cherenkov camera for the SST-1M telescope~\\cite{SST1M-project-ICRC19}. It consists of 1296 hexagonal SiPMs from Hammamatsu (S10943-3739(X)) connected to a front-end electronics for signal shaping and amplification designed at the University of Geneva~\\cite{Frontend2016}. The front-end signals are recorded by the trigger and readout system DigiCam~\\cite{Rajdak2015}. Continuous digitization with 12-bit Flash Analog to Digital Converters (FADCs) of the signal is performed at a sampling frequency of 250~MHz. A ring buffer keeps the data within the system while continuing the observations, allowing dead time free operations. \\\\\n\nCalibration of the SiPMs is performed with a dedicated Camera Test Setup (CTS). A description of this calibration setup can be found in~\\cite{Alispach-ICRC17} which has since then been upgraded to an array of 1296 pairs of LED that covers the entire camera field of view. Each pair is placed in front of a camera pixel and consists of a pulsed and a continuously operated LED. The pulsed LED (AC LED) emulates the prompt Cherenkov signal while the continuous LED (DC LED) emulates the night-sky background (NSB). The CTS is now fully controllable by the telescope control software~\\cite{Sliusar2017-ACS} and does not require an external pulse generator. The pulse generation is carried out by one of the DigiCam microcrates which makes the CTS fully autonomous and usable on any observation site, only requiring a 230~V power plug.\\\\\n\nThis calibration system was developed in view of a mass production of the cameras (70 pieces at a rate of two cameras per month). The whole procedure described in the following sections would require a couple of days for the data acquisition and another day for the data analysis. The data analysis, presented in the following, was performed with a dedicated Python software: \\textit{digicampipe}~\\cite{digicampipe} based on the core functionaries of the proposed reconstruction pipeline for CTA: \\textit{ctapipe}~\\cite{ctapipe-ICRC19} \\\\\n\n\\section{Camera performances in the presence of night-sky background}\\label{sec:key_perf}\n\nIn order to fully characterize each pixel, the camera pixels are illuminated with the CTS mounted in front of it. The data sample used for calibration consists of a scan of pulsed light from 0 to $\\sim 10$~k photons and continuous light from 0 to $\\sim 1$~GHz photoelectron rate per pixel. For each light level a set of 10~k waveforms at 500~Hz are acquired. The subset without background continuous light is used to calibrate the pulsed LEDs using the photo counting capabilities of SiPM as performed in~\\cite{CameraPaperHeller2017}. It is also used to extract key SiPM parameters such as gain, dark count rate, gain smearing and optical crosstalk for each pixel. The subset without pulsed light is used to calibrate the continuous LEDs from the pedestal shift. \\\\\n\nOnce the LEDs are calibrated and the SiPM characterized the rest of the dataset is used to measure the time and charge resolution of the camera. The reconstructed time and charge of an EAS play a crucial role in the reconstruction of the impact parameter and the primary particle energy. Therefore time and charge resolution are both instrument response functions directly linked to the telescope performance on gamma\/hadron separation, angular resolution and energy resolution. The Monte Carlo evaluation of the telescope performances can be found in~\\cite{SST1M-monte-carlo-ICRC19}. In the following we will present the time and charge resolution of the SST-1M camera. \n\n\\subsection{Time resolution}\n\n\nThe flashes of the CTS were triggered by the digital readout itself which ensures the synchronicity between the readout and the light pulse. Doing so, the light arrives every time at the same position in the readout window. To measure the time offset $\\delta t$, we use the charge ratio $a$ between the pulse template function $f$ and the $N$ waveform samples $x_i$. This ratio $a$ is calculated using the sum over 8 consecutive samples (3 before the maximum and 4 after) individually for the samples and for the template:\n\\begin{equation}\n a(\\delta t) = \\frac{\\sum\\limits_{i=-3}^{4} x_{i - i_{max}}} {\\sum\\limits_{j=-3}^{4} f(t_j+\\delta t-t_{max})}\n\\end{equation}\n\nThe $\\chi^2$ giving the agreement between the pulse and the scaled template with an offset $\\delta t$ is calculated as:\n\\begin{equation}\n \\chi^2(\\delta t) = \\frac{1}{N} \\sum_{i=1}^{N} \\frac{(x_i - a(\\delta t) f(t_i+\\delta t))^2}{\\sigma_{f(t_i+\\delta t)}^2 + \\sigma_e^2}\n\\end{equation}\nwhere:\n\\begin{itemize}\n \n \n \\item $f(t_i+\\delta t)$ and $\\sigma_{f(t_i+\\delta t)}$ are respectively the amplitude and the uncertainty of the template for the sample $i$ and an offset $\\delta t$\n \\item $\\sigma_e$ is the electronic noise per time slice\n\\end{itemize}\n\nThe $\\chi^2$ is calculated for a range of offsets with 0.1~ns steps (as smaller steps did not improve the results), for each pixel and each flash. The time offset $\\delta s$ corresponding to the lowest $\\chi^2$ is chosen as the measurement of the reconstructed time offset for that flash. The time offset (respectively the time resolution) for each pixel is obtained as the mean value (respectively the standard deviation) over the sample. Only flashes where the measured amplitude is above 3.5~p.e. are used.\n\n\nFig. \\ref{fig:time_resol} shows for the whole camera the evolution of the timing resolution with the charge for two NSB levels. We see that without any NSB, the resolution is below 1~ns and reaches 0.1~ns at 400 p.e. With 125 MHz NSB, the resolution is mainly affected below 50 p.e. and goes above 1~ns only for pulse amplitude below 7 p.e. (~30 photons).\n\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{images\/time_resolution_all_pixels_dark.png}\n\\includegraphics[width=0.49\\textwidth]{images\/time_resolution_all_pixels_nsb.png}\n\\end{center}\n\\caption{Time resolution as a function of the reconstructed charge in dark conditions (left) and with 125~MHz NSB equivalent (right). The time resolution is compared to the current CTA requirements for a 125~MHz NSB level.}\\label{fig:time_resol}\n\\end{figure}\n\n\n\n\n\n\n\n\\subsection{Charge resolution}\n\n\nThe data analysis performed here is similar to the one described in~\\cite{CameraPaperHeller2017}. Compared to the previous results obtained in~\\cite{CameraPaperHeller2017}, the measurements were performed for each of the 1296 camera pixels and are expressed in photon units rather than photoelectrons. They are compared to the current CTA requirements.\n\nThe charge resolution is defined as the ratio between the variance and mean of the reconstructed charge in units of photons.\n\\begin{equation}\n CR = \\frac{Var(N_{\\gamma})}{\\mathbb{E}(N_{\\gamma})}\n \\label{eq:charge_reso}\n\\end{equation}\nThe variance and mean are computed with the sample standard deviation and mean. The charge resolution presented here takes into account: Poisson fluctuations of the LED, electronic noise from the photodetection plane, SiPM gain smearing, optical crosstalk, precision of the computed baseline and non linearity of the amplification chain. \\\\\n\nThe readout chain starts to saturate at high illumination due to the pre-amplifier saturation. To cope with saturation effects, a look-up table of reconstructed charge (waveform integral) as a function of the true number of photoelectrons is used (see Fig. \\ref{fig:charge_resolution}) for all camera pixels. The number of photons is then assessed by correcting for the optical efficiency of the photodectection plane. The correction of the number of photoelectrons induced by a voltage drop from night-sky background are considered as described in~\\cite{VdropHeller}. \\\\\n\nThe charge resolution is presented in Fig.~\\ref{fig:charge_resolution}. The solid lines denote the average resolution over the camera pixels and the contoured area represents its $1\\sigma$ deviation. The theoretical Poisson limit (fluctuation of the light source only) is given in black. The charge resolution is given for two distinct night-sky background rates per pixel (in photoelectrons) corresponding to clear sky conditions (40~MHz) and to a half-Moon night (670~MHz) in Paranal, Chile. The corresponding CTA requirements for the small-sized telescopes are also drawn which correspond to two data processing levels (dotted and dashed lines). As shown the SST-1M camera is compliant with the CTA requirements. \\\\\n\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{images\/charge_linearity_integral_dynamic_bis.png}\n\\includegraphics[width=0.49\\textwidth]{images\/charge_resolution_vs_requirements_bis.png}\n\\end{center}\n\\caption{Left: Reconstructed charge as a function of the true number of photoelectrons for all camera pixels. Right: Charge resolution as a function of the true number of photons for different night-sky background levels. The solid lines represents the average charge resolution of the camera and the contoured band its 1-sigma deviation among camera pixels.}\\label{fig:charge_resolution}\n\\end{figure}\n\n\n\\section{On-site monitoring}\\label{sec:on-site}\n\nThe SST-1M camera has been subject to extensive integration and commissioning to the telescope prototype structure in IFJ Krakow during autumn 2018. A few nights were dedicated to observations of the Crab Nebula and its first analysis is described in~\\cite{SST1M-monte-carlo-ICRC19}. Prior to and during observations, calibration runs are performed to ensure the stability and operability of the camera. The safety and operability of the various camera components (such as temperatures of the SiPMs, readout rates, trigger rates, etc.) are monitored via a slow control link at 2~Hz. In this section we present the dark runs and trigger monitoring performed on site.\n\n\\subsection{Dark run}\\label{sec:dark_runs}\n\nBefore each night of observation, while the lid is closed, a dark run is acquired. Ten thousand waveforms of 50 samples for each pixel are registered. Their first use is to measure the baseline without any night-sky background to determine later the baseline shift per pixel (later used to compute the night-sky background level). Their second use is to monitor the sensor parameters evolution, e.g. gain, optical crosstalk and dark count rate. Fig.~\\ref{fig:dark_spe} shows the projection of the raw waveform in Least Significant Bit (LSB) for each camera pixel. Given that, the first five photoelectron peaks are easily identifiable within a dark run, the distribution shown on Fig.~\\ref{fig:dark_spe} is enough to extract with good precision the sensor parameters on a per night basis. Additionally, merging all the dark runs and\/or merging all the pixels allows the necessary statistics to be accumulated to correct the optical crosstalk modelization as shown in~\\cite{FACT}.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=0.49\\textwidth]{images\/AllRawADCDark.png} \\hfill\n \\includegraphics[width=0.49\\textwidth]{images\/SumRawADCDark.png}\n \\caption{Left: Normalized distribution of the projection of the raw waveform in LSB for all the pixels and for one run. Right: Merged distribution of all pixels and all nights.}\n \\label{fig:dark_spe}\n\\end{figure}\n\n\\subsection{Trigger rate}\n\n\nThe trigger logic is based on the clustering of neighboring signals in the camera pixels. The internal trigger of DigiCam sums the signal of each of the three neighboring pixels of the camera into so-called trigger patches. Each of the 432 patches are then summed with its neighboring six patches to form a trigger cluster. An event is triggered when at least one of the 432-cluster waveform signals passes the threshold. This trigger topology allows the rate of false triggers induced by night-sky background fluctuations to be reduced while capturing EAS events which illuminate simultaneously groups of nearby pixels. \\\\\n\nDuring the 2018 operation of the telescope, high night-sky background conditions of 600~MHz on average per pixel were observed~\\cite{SST1M-project-ICRC19}. On top of the high night-sky background level, external parasitic light from the surrounding inhabited area perturbed the observations. To limit the influence of such background during science runs, the fully digital trigger logic allows trigger clusters to be disabled such that they would not trigger the readout of the full camera. This is needed when a bright continuous light source falls into the field of view of a pixel or a group of pixels. Alternatively, the signal can be clipped at the pixel. Even though both SiPM and FADC gains have been equalized providing a 3\\% rms spread over the camera gain, the variation in optical efficiencies were not corrected. For instance, the transmissivity of the entrance window and of the light guides are so far not compensated. Once determined using an external light source (e.g. Flasher), the correction factors can be compensated as the trigger threshold can be set at the patch level. This has the advantage to maintain all SiPMs at the same working point and avoids the need to compensate with the bias voltage which would lead to sensors working with different characteristics (noise, gain, optical crosstalk, dark count rate, etc.). Fig.~\\ref{fig:trigger_uniformity} shows the fractional contribution of each trigger cluster to the camera internal trigger for one night of observation during the 2018 observation campaign. The non-uniformities observed here are mostly related to different NSB levels across the camera FoV. \n\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=0.49\\textwidth]{images\/trigger_uniformity_2018_10_12_run028.png}\n \\caption{Trigger uniformity of the EAS shower events during the observation night of the 12$^{th}$ october 2018}\n \\label{fig:trigger_uniformity}\n\\end{figure}\n\n\\section{Conclusion}\n\n\\input{conclusion.tex}\n\n\\acknowledgments\n\n\\input{acknowledgements.tex}\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}